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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Stem-cell differentiation underpins reproducible morphogenesis</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.19375v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S1" title="In Stem-cell differentiation underpins reproducible morphogenesis">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2" title="In Stem-cell differentiation underpins reproducible morphogenesis">2 Results</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS1" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.1 Multi-scale model</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS2" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.2 Reproducibility is not an intrinsic property of complex morphogenesis</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS3" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS4" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.4 Multiple SCCs in reproducible organisms resemble stem-cell systems</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS5" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS6" title="In 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S3" title="In Stem-cell differentiation underpins reproducible morphogenesis">3 Discussion</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4" title="In Stem-cell differentiation underpins reproducible morphogenesis">4 Methods</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS1" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.1 Cellular Potts Model</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS2" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.2 Development</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS3" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.3 Gene regulatory network and morphogens</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS4" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.4 Adhesion and contractile proteins</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS5" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.5 Evolution</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6 Cell state and state space</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS7" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.7 Reproducibility score</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS8" title="In 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.8 Momentum and anisotropy</a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Stem-cell differentiation underpins reproducible morphogenesis</h1> <div class="ltx_authors"> Dominic K Devlin Austen RD Ganley School of Biological Sciences, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Digital Life Institute, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Nobuto Takeuchi School of Biological Sciences, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Research Centre for Complex Systems Biology, Universal Biology Institute, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka, Japan </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Summary</h6> Morphogenesis of complex body shapes is reproducible despite the noise inherent in the underlying morphogenetic processes. However, how these morphogenetic processes work together to achieve this reproducibility remains unclear. Here, we ask how morphogenetic reproducibility is realised by developing a computational model that evolves complex morphologies. We find that evolved, complex morphologies are reproducible in a sizeable fraction of simulations, despite no direct selection for reproducibility. Strikingly, reproducible morphologies had also evolved stem-cell systems. We show that reproducibility is caused by a morphogenetic division of labour based on these stem-cell systems, where moving, dividing stem cells “shape” morphologies and irreversibly differentiate into stationary, non-dividing cells. These results suggest that stem-cell systems observed in natural development play fundamental roles in morphogenesis in addition to their known role of producing specialised cell types. This previously-unrecognised role of stem-cell systems has major implications for our understanding of how morphologies are generated and regenerated. </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> 1 Introduction</h2> <div class="ltx_para" id="S1.p1"> Morphogenesis is the multifaceted process that transforms a relatively homogeneous starting material — a fertilised zygote — into the complex morphological structures, such as organs, tissues and appendages, that constitute a mature organism <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>]</cite>. This transformation occurs through a combination of chemical-level pattern formation and cellular-level shape formation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib4" title="">4</a>]</cite>. At the chemical level, reacting and diffusing chemicals produce spatial patterns, such as stripes and segments <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib6" title="">6</a>]</cite>. At the cellular level, processes such as cell motion, division, contraction and differential adhesion interact with chemical-level pattern formation to produce morphological shapes, such as tails, tubes, branches and limbs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib9" title="">9</a>]</cite>. For extant animals, the morphological structures produced by these processes are not only complex, but also reproduced with astonishing precision across generations. Understanding how complex morphogenesis is made reproducible has intrigued the minds of thinkers since Pythagoras and Aristotle <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib10" title="">10</a>]</cite>, and is a focal point of developmental biology <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib13" title="">13</a>]</cite>. </div> <div class="ltx_para" id="S1.p2"> Reproducibility of complex morphologies requires both cell-level and chemical-level processes to be robust to noise <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib15" title="">15</a>]</cite>. Much attention has been devoted to understanding the robustness of chemical-level pattern formation to molecular sources of noise, such as fluctuations in chemical concentrations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib16" title="">16</a>]</cite>, resulting in the characterisation of chemical-level processes that enhance pattern reproducibility, such as genetic feedback loops and signal transduction pathways <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib24" title="">24</a>]</cite>. In contrast, much less is known about the cellular processes underlying the robustness of morphogenesis to cell-level sources of noise, such as stochasticity in the motion and geometry of cells <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib25" title="">25</a>]</cite>. To address this issue, previous studies have taken a targeted approach in which they selected a set of cell-level processes, such as cell-cell signalling and differential adhesion, and examined each process to determine whether it increases morphogenetic reproducibility <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib27" title="">27</a>]</cite>. </div> <div class="ltx_para" id="S1.p3"> Here, we instead asked whether morphogenetic reproducibility is an emergent by-product of complex morphogenesis that evolves even if reproducibility is not explicitly selected for, and, if so, how this reproducibility is realised—a non-prescriptive approach pioneered by Hogeweg <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib30" title="">30</a>]</cite>. To answer these questions, we computationally generated an ensemble of “morphogeneses” by repeatedly evolving a population of organisms selected for complex, multicellular shapes. We found that a sizeable fraction of evolved organisms had high reproducibility, even though reproducibility was not explicitly selected. Strikingly, these organisms shared one cell-level feature responsible for morphogenetic reproducibility: a “morphogenetic division of labour”, where moving and dividing stem cells “shape” morphologies, and non-moving and non-dividing differentiated cells spatially “anchor” this shaping process, thereby enhancing morphogenetic reproducibility. </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> 2 Results</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> 2.1 Multi-scale model</h3> <div class="ltx_para" id="S2.SS1.p1"> We set out to investigate whether complex, evolvable morphogenesis is intrinsically reproducible by developing a computational model consisting of cells equipped with only the basic properties necessary for morphogenesis that are not species-specific: cell motion, gene regulation, cell-cell signalling, differential adhesion, and unpolarised cell contractions. Although limiting the model to these basic properties constrains the types of morphologies we can evolve, it increases the likelihood that the morphologies we do evolve could be realised across a broad range of developmental contexts in nature. </div> <div class="ltx_para" id="S2.SS1.p2"> To capture the noisy dynamics of natural-world morphogenesis, we employed the Cellular Potts Model (CPM), which uses a Metropolis algorithm to simulate stochastic cell motion and cell geometry dynamics (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS1" title="4.1 Cellular Potts Model ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.1</a>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib32" title="">32</a>]</cite>. The CPM models the development of a single organism on a two-dimensional square grid (<math alttext="250\times 250" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mn id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">250</mn><mo id="S2.SS1.p2.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p2.1.m1.1.1.1.cmml">×</mo><mn id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml">250</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><times id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1"></times><cn id="S2.SS1.p2.1.m1.1.1.2.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.2">250</cn><cn id="S2.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.3">250</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">250\times 250</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">250 × 250</annotation></semantics></math> pixels) starting from a single cell. A collection of neighbouring pixels on the grid represents a cell (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>A). Pixels not occupied by cells represent the medium (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>A), which is akin to an extracellular matrix or fluid <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib33" title="">33</a>]</cite>. Cell motion occurs through stochastic extensions and retractions of cell boundaries, mimicking natural cell motion <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib34" title="">34</a>]</cite>. These extensions and retractions are generated by pixel copying at the cell boundaries (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>A; Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS1" title="4.1 Cellular Potts Model ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.1</a>). The probability that a pixel copy occurs is determined by free energy contributions from cell-cell adhesion, cell-medium adhesion, cell shape, and cell size, as described later. Pixels on the grid are chosen in a random order with replacement for copy attempts. The unit of time is the number of pixel copy attempts equal to the total number of pixels on the grid, hereon referred to as a developmental time step (DTS). </div> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="424" id="S2.F1.g1" src="x1.png" width="544"/> <figcaption class="ltx_caption ltx_centering">Figure 1: Multi-scale model of morphogenesis evolution. A Three neighboring cells on a CPM grid. A cell consists of one or more pixels. Each cell is colored by its “cell state” defined by the concentrations of all its proteins converted to boolean values (the medium is represented by white pixels). Pixels with alternating stripes indicate pixel copies at cell boundaries. Each cell contains a genome that encodes transcription factors (TFs; circles, squares, triangles), adhesion proteins (sticks with a lock or key) and contractile proteins (not shown). Arrows indicate regulation of gene expression by TFs (arrow head for activation, blunt head for inhibition). Double harpoon arrows indicate diffusion of morphogens (membrane-permeable TFs). B Adhesion proteins facilitate the binding of cells to each other via a lock and key mechanism, or to the surrounding medium (not shown). C. A population consists of 60 organisms (only three depicted). Organisms undergo a developmental phase on separate CPM grids for 12,000 DTS, and then a reproduction phase, where organisms with complex morphologies are selected. Reproduction can occur without mutation (black arrows) or with mutation (orange arrow), with mutation determined probabilistically. Mutations change the topology of the GRN (dashed orange arrow), with the example showing a change from inhibition of gene <math alttext="y" class="ltx_Math" display="inline" id="S2.F1.5.m1.1"><semantics id="S2.F1.5.m1.1b"><mi id="S2.F1.5.m1.1.1" xref="S2.F1.5.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.F1.5.m1.1c"><ci id="S2.F1.5.m1.1.1.cmml" xref="S2.F1.5.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.5.m1.1d">y</annotation><annotation encoding="application/x-llamapun" id="S2.F1.5.m1.1e">italic_y</annotation></semantics></math> by gene <math alttext="x" class="ltx_Math" display="inline" id="S2.F1.6.m2.1"><semantics id="S2.F1.6.m2.1b"><mi id="S2.F1.6.m2.1.1" xref="S2.F1.6.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.F1.6.m2.1c"><ci id="S2.F1.6.m2.1.1.cmml" xref="S2.F1.6.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.6.m2.1d">x</annotation><annotation encoding="application/x-llamapun" id="S2.F1.6.m2.1e">italic_x</annotation></semantics></math> to activation of gene <math alttext="y" class="ltx_Math" display="inline" id="S2.F1.7.m3.1"><semantics id="S2.F1.7.m3.1b"><mi id="S2.F1.7.m3.1.1" xref="S2.F1.7.m3.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.F1.7.m3.1c"><ci id="S2.F1.7.m3.1.1.cmml" xref="S2.F1.7.m3.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.7.m3.1d">y</annotation><annotation encoding="application/x-llamapun" id="S2.F1.7.m3.1e">italic_y</annotation></semantics></math> by gene <math alttext="x" class="ltx_Math" display="inline" id="S2.F1.8.m4.1"><semantics id="S2.F1.8.m4.1b"><mi id="S2.F1.8.m4.1.1" xref="S2.F1.8.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.F1.8.m4.1c"><ci id="S2.F1.8.m4.1.1.cmml" xref="S2.F1.8.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.8.m4.1d">x</annotation><annotation encoding="application/x-llamapun" id="S2.F1.8.m4.1e">italic_x</annotation></semantics></math>. DE Illustration of the two measurements used to select for morphological complexity (described in Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS5" title="4.5 Evolution ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.5</a>). The cells are all coloured grey to emphasise that only the shape, not cell states, determine the complexity score. (D) depicts the measurement for deviation of the morphology from a perfect circle; black arrows mark areas where the organism’s radius (measured from its centre of mass) is smaller than that of a circle, while yellow arrows indicate regions where the radius is larger. (E) depicts regions of inward folding using double sided arrows, with the number and length of arrows indicated the extent of inward folding.</figcaption> </figure> <div class="ltx_para" id="S2.SS1.p3"> To simulate the development of an organism, the CPM is run for 12,000 DTS, which is approximately the minimum time it takes for a fast-growing organism to reach the edge of the grid. Each organism starts as a “zygote” that rapidly divides six times to create a ball of 64 cells of approximately 75 pixels each (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>C, Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS2" title="4.2 Development ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.2</a>). This rapid cell division without growth mimics the earliest cell divisions that occur in animal embryos after fertilisation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>]</cite>. After the 64-cell stage, cells grow and divide if they are mechanically stretched, mimicking the mechanical induction of cell growth and division observed in natural development <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib37" title="">37</a>]</cite>. The size of each cell, measured in pixels, is energetically constrained to a target size: the more a cell’s size departs from the target size, the larger the free energy. The target size is initially set to each cell’s actual size following the six rapid divisions. When a cell is stretched to three pixels above its target size, its target size is increased to its current size (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS2" title="4.2 Development ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.2</a>). Cell stretching is induced by adhesion to neighbouring cells and the extracellular medium, although it can also occur stochastically. When a cell reaches a size of 100 pixels, it divides by splitting along its minor axis into two daughter cells of approximately equal size. Protein concentrations remain the same for both daughter cells upon cell division and the target sizes of the daughter cells are decreased to their current sizes. </div> <div class="ltx_para" id="S2.SS1.p4"> To model the dependency of morphogenesis on spatial patterns of protein concentrations, we coupled the CPM dynamics to gene expression dynamics, as previously done <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib41" title="">41</a>]</cite>. Specifically, we equip each organism with a set of protein-encoding genes, such as transcription factors (TFs), that form an gene regulatory network (GRN). A GRN is a graph consisting of nodes representing proteins and edges representing TF-mediated activation or inhibition of gene expression (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>A). Concentrations of proteins within a cell are determined by numerically integrating a set of ordinary differentiation equations given by the GRN (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS3" title="4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.3</a>). To model cell-cell signalling, three TFs (hereafter called morphogens) diffuse between cells and into the medium (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>A; Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS3" title="4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.3</a>). These morphogens model morphogens encountered in real biological development, such as Wnts and BMPs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib43" title="">43</a>]</cite>, and they allow different cells to express different proteins even though all cells within an organism have an identical GRN. The diffusivity is the same for all three morphogens (methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS3" title="4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.3</a>). </div> <div class="ltx_para" id="S2.SS1.p5"> The other way differential protein expression between cells can occur in real biological embryos is by the asymmetric distribution of “maternal factors” at the zygote stage <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib44" title="">44</a>]</cite>. To model this, two non-morphogen TFs are defined as maternal factors. One maternal factor is restricted to the left of the zygote’s vertical centre line; the other, to the bottom side of the zygote’s horizontal centre line. The first two of the six rapid cell divisions occur along these centre lines, so that each cell at the four-cell stage has a unique combination of maternal factor concentrations (illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>C). </div> <div class="ltx_para" id="S2.SS1.p6"> Organisms carry two other types of genes: those encoding adhesion proteins and contractile proteins (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS4" title="4.4 Adhesion and contractile proteins ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.4</a>). Adhesion proteins modulate the adhesiveness of cells to neighbouring cells and the extracellular medium (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>AB). These proteins model cell-surface proteins, such as cadherins and integrins, which are essential drivers of cell behaviours necessary for morphogenesis, such as cell migration and sorting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib45" title="">45</a>]</cite>. Adhesion proteins that determine the strength of cell-cell adhesion are either locks or keys, where the adhesion strength between two neighbouring cells scales with the number of compatible lock-key pairs that the two cells express. Similarly, adhesion of a cell to the medium scales with the number of medium-adhesion proteins expressed by the cell (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS4" title="4.4 Adhesion and contractile proteins ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.4</a>). Contractile proteins make the cell less deformable by energetically constraining the cell to an elliptic shape, where the length of the major axis of the elliptic shape scales with the number of expressed contractile proteins (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS4" title="4.4 Adhesion and contractile proteins ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.4</a>). These proteins model unpolarised contractions of the cell cortex, mirroring those observed in biological cells by the accumulation of actin filament stress <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib46" title="">46</a>]</cite>. </div> <div class="ltx_para" id="S2.SS1.p7"> We distinguish cells based on the proteins they express by assigning each cell a cell state defined as a vector of boolean values, where each boolean value indicates whether a contractile or adhesion protein is expressed or not expressed (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="4.6 Cell state and state space ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6</a>). All cells with the same state are shown with the same colour on the CPM grid (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>). Cell states are only used to visualise and analyse model outcomes and do not play any role in model dynamics. </div> <div class="ltx_para" id="S2.SS1.p8"> To simulate the evolution of morphogenesis, we established an initial population of 60 organisms (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>C; Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS5" title="4.5 Evolution ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.5</a>), with each assigned a different randomly generated GRN. Each organism develops on a separate CPM grid. We applied a genetic algorithm to select for morphological complexity by measuring inward folding and deviation of the morphological shape from a circle at the end of the 12,000 DTS (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>DE; Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS5" title="4.5 Evolution ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.5</a>). The 15 organisms with the highest morphological complexity reproduce four times to populate the next generation, and their GRNs undergo a single mutation with a probability of 50%. A mutation changes the regulatory effect of one TF on one gene, such as causing a TF to switch from inhibiting the expression of a gene to activating its expression (see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F1" title="Figure 1 ‣ 2.1 Multi-scale model ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>C). The GRNs used in our main set of simulations, described subsequently are composed of nine transcription factors (including the three morphogens), 15 adhesion proteins and two contractile proteins. However, we also run simulations with GRNs composed of different numbers of genes to confirm subsequent results (Fig. S7EFGH) </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> 2.2 Reproducibility is not an intrinsic property of complex morphogenesis</h3> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="388" id="S2.F2.g1" src="x2.png" width="342"/> <figcaption class="ltx_caption">Figure 2: Computational simulations showing the evolution of complex morphologies. AB The plots show a five-generation moving average of population fitness (morphological complexity) over evolution. The organism morphologies above the plots each depict a separate organism at the end of its development (12,000 DTS) from six different generations over the course of evolution. The generation number of each organism morphology is shown to its left. “Evolved organisms”, defined as the most morphologically complex organism from the final generation of the simulation, are shown on the far right. The evolved organisms in (A) and (B) are hereafter referred to as organism-1 and organism-2, respectively.</figcaption> </figure> <div class="ltx_para" id="S2.SS2.p1"> To investigate whether complex morphogenesis is intrinsically reproducible, we conducted 126 independent evolutionary simulations of our model. We ran these for at least <math alttext="2.5\times 10^{3}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mrow id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mn id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">2.5</mn><mo id="S2.SS2.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p1.1.m1.1.1.1.cmml">×</mo><msup id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3.cmml"><mn id="S2.SS2.p1.1.m1.1.1.3.2" xref="S2.SS2.p1.1.m1.1.1.3.2.cmml">10</mn><mn id="S2.SS2.p1.1.m1.1.1.3.3" xref="S2.SS2.p1.1.m1.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"><times id="S2.SS2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1.1"></times><cn id="S2.SS2.p1.1.m1.1.1.2.cmml" type="float" xref="S2.SS2.p1.1.m1.1.1.2">2.5</cn><apply id="S2.SS2.p1.1.m1.1.1.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.p1.1.m1.1.1.3">superscript</csymbol><cn id="S2.SS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1.3.2">10</cn><cn id="S2.SS2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">2.5\times 10^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">2.5 × 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> generations, which is usually sufficient to reach a plateau in fitness (Fig. S1). We then identified the fittest (i.e., most morphologically complex) organism from the final generation (hereafter referred to as an “evolved organism”) from each simulation. To ensure that we were analysing the reproducibility of complex morphologies, we removed 36 evolved organisms that did not reach an arbitrary threshold of complexity (our results are similar when a different threshold is used; Fig. S2A). The 90 organisms above this threshold each display a different morphology (Fig. S1), with the evolution of two representative organisms illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F2" title="Figure 2 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a>AB. </div> <figure class="ltx_figure" id="S2.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="381" id="S2.F3.g1" src="x3.png" width="692"/> <figcaption class="ltx_caption">Figure 3: Both highly and poorly reproducible organisms evolve in response to selection for complex morphologies. AB Two developmental replicates of organism-1 (A) and organism-2 (B) are depicted after 400, 2,600, 4,000, 8,000 and 12,000 DTS, showing a difference in their reproducibility. Dashed arrows in (B) indicate the presence (replicate-1) or absence (replicate-2) of a bifurcation in collective cell motion; asterisks indicate protrusions. Vector plots above the organisms show the displacement of the centre of mass of each cell during 2,000 DTS at each respective time point, with colours indicating magnitude (the lighter, the larger). C Reproducibility scores for the 90 organisms that evolved a complex morphology. Circles indicate organisms with a single strongly connected component (SCC, defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.SS5" title="2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">2.5</a>) (mean reproducibility<math alttext="=52.0\%" class="ltx_Math" display="inline" id="S2.F3.5.m1.1"><semantics id="S2.F3.5.m1.1b"><mrow id="S2.F3.5.m1.1.1" xref="S2.F3.5.m1.1.1.cmml"><mi id="S2.F3.5.m1.1.1.2" xref="S2.F3.5.m1.1.1.2.cmml"></mi><mo id="S2.F3.5.m1.1.1.1" xref="S2.F3.5.m1.1.1.1.cmml">=</mo><mrow id="S2.F3.5.m1.1.1.3" xref="S2.F3.5.m1.1.1.3.cmml"><mn id="S2.F3.5.m1.1.1.3.2" xref="S2.F3.5.m1.1.1.3.2.cmml">52.0</mn><mo id="S2.F3.5.m1.1.1.3.1" xref="S2.F3.5.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.5.m1.1c"><apply id="S2.F3.5.m1.1.1.cmml" xref="S2.F3.5.m1.1.1"><eq id="S2.F3.5.m1.1.1.1.cmml" xref="S2.F3.5.m1.1.1.1"></eq><csymbol cd="latexml" id="S2.F3.5.m1.1.1.2.cmml" xref="S2.F3.5.m1.1.1.2">absent</csymbol><apply id="S2.F3.5.m1.1.1.3.cmml" xref="S2.F3.5.m1.1.1.3"><csymbol cd="latexml" id="S2.F3.5.m1.1.1.3.1.cmml" xref="S2.F3.5.m1.1.1.3.1">percent</csymbol><cn id="S2.F3.5.m1.1.1.3.2.cmml" type="float" xref="S2.F3.5.m1.1.1.3.2">52.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.5.m1.1d">=52.0\%</annotation><annotation encoding="application/x-llamapun" id="S2.F3.5.m1.1e">= 52.0 %</annotation></semantics></math>, <math alttext="n=65" class="ltx_Math" display="inline" id="S2.F3.6.m2.1"><semantics id="S2.F3.6.m2.1b"><mrow id="S2.F3.6.m2.1.1" xref="S2.F3.6.m2.1.1.cmml"><mi id="S2.F3.6.m2.1.1.2" xref="S2.F3.6.m2.1.1.2.cmml">n</mi><mo id="S2.F3.6.m2.1.1.1" xref="S2.F3.6.m2.1.1.1.cmml">=</mo><mn id="S2.F3.6.m2.1.1.3" xref="S2.F3.6.m2.1.1.3.cmml">65</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.6.m2.1c"><apply id="S2.F3.6.m2.1.1.cmml" xref="S2.F3.6.m2.1.1"><eq id="S2.F3.6.m2.1.1.1.cmml" xref="S2.F3.6.m2.1.1.1"></eq><ci id="S2.F3.6.m2.1.1.2.cmml" xref="S2.F3.6.m2.1.1.2">𝑛</ci><cn id="S2.F3.6.m2.1.1.3.cmml" type="integer" xref="S2.F3.6.m2.1.1.3">65</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.6.m2.1d">n=65</annotation><annotation encoding="application/x-llamapun" id="S2.F3.6.m2.1e">italic_n = 65</annotation></semantics></math>); filled triangles indicate organisms with multiple SCCs with unidirectional transitions between them (mean reproducibility<math alttext="=72.1\%" class="ltx_Math" display="inline" id="S2.F3.7.m3.1"><semantics id="S2.F3.7.m3.1b"><mrow id="S2.F3.7.m3.1.1" xref="S2.F3.7.m3.1.1.cmml"><mi id="S2.F3.7.m3.1.1.2" xref="S2.F3.7.m3.1.1.2.cmml"></mi><mo id="S2.F3.7.m3.1.1.1" xref="S2.F3.7.m3.1.1.1.cmml">=</mo><mrow id="S2.F3.7.m3.1.1.3" xref="S2.F3.7.m3.1.1.3.cmml"><mn id="S2.F3.7.m3.1.1.3.2" xref="S2.F3.7.m3.1.1.3.2.cmml">72.1</mn><mo id="S2.F3.7.m3.1.1.3.1" xref="S2.F3.7.m3.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.7.m3.1c"><apply id="S2.F3.7.m3.1.1.cmml" xref="S2.F3.7.m3.1.1"><eq id="S2.F3.7.m3.1.1.1.cmml" xref="S2.F3.7.m3.1.1.1"></eq><csymbol cd="latexml" id="S2.F3.7.m3.1.1.2.cmml" xref="S2.F3.7.m3.1.1.2">absent</csymbol><apply id="S2.F3.7.m3.1.1.3.cmml" xref="S2.F3.7.m3.1.1.3"><csymbol cd="latexml" id="S2.F3.7.m3.1.1.3.1.cmml" xref="S2.F3.7.m3.1.1.3.1">percent</csymbol><cn id="S2.F3.7.m3.1.1.3.2.cmml" type="float" xref="S2.F3.7.m3.1.1.3.2">72.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.7.m3.1d">=72.1\%</annotation><annotation encoding="application/x-llamapun" id="S2.F3.7.m3.1e">= 72.1 %</annotation></semantics></math>, <math alttext="n=24" class="ltx_Math" display="inline" id="S2.F3.8.m4.1"><semantics id="S2.F3.8.m4.1b"><mrow id="S2.F3.8.m4.1.1" xref="S2.F3.8.m4.1.1.cmml"><mi id="S2.F3.8.m4.1.1.2" xref="S2.F3.8.m4.1.1.2.cmml">n</mi><mo id="S2.F3.8.m4.1.1.1" xref="S2.F3.8.m4.1.1.1.cmml">=</mo><mn id="S2.F3.8.m4.1.1.3" xref="S2.F3.8.m4.1.1.3.cmml">24</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.8.m4.1c"><apply id="S2.F3.8.m4.1.1.cmml" xref="S2.F3.8.m4.1.1"><eq id="S2.F3.8.m4.1.1.1.cmml" xref="S2.F3.8.m4.1.1.1"></eq><ci id="S2.F3.8.m4.1.1.2.cmml" xref="S2.F3.8.m4.1.1.2">𝑛</ci><cn id="S2.F3.8.m4.1.1.3.cmml" type="integer" xref="S2.F3.8.m4.1.1.3">24</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.8.m4.1d">n=24</annotation><annotation encoding="application/x-llamapun" id="S2.F3.8.m4.1e">italic_n = 24</annotation></semantics></math>). The blue diamond is an organism with multiple SCCs without unidirectional transitions (Fig. S3D). Numbered arrows refer to the organisms in panels A, B, D, and E. The data is overlaid on a violin plot that illustrates the bimodality in reproducibility scores. D Four organisms with highly reproducible morphologies. E Four organisms with poorly reproducible morphologies.</figcaption> </figure> <div class="ltx_para" id="S2.SS2.p2"> We examined whether the 90 evolved organisms with complex morphologies display reproducible morphogenesis by repeatedly simulating their development (60 replicates per organism, hereafter referred to as developmental replicates). To quantify reproducibility, we measured a “reproducibility score”, which indicates how geometrically similar the morphologies of replicates are to each other (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS7" title="4.7 Reproducibility score ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.7</a>). The reproducibility score depends on the geometry, size and time taken to generate the morphology but is invariant to reflection, rotation and translation of the morphology. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A and B illustrate two developmental replicates of an organism with high reproducibility and an organism with low reproducibility, respectively (hereafter referred to as organism-1 and organism-2). The distribution of reproducibility scores across all 90 evolved organisms is bimodal (bimodality coefficient <math alttext="=0.69" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml"></mi><mo id="S2.SS2.p2.1.m1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml">0.69</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><eq id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1"></eq><csymbol cd="latexml" id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">absent</csymbol><cn id="S2.SS2.p2.1.m1.1.1.3.cmml" type="float" xref="S2.SS2.p2.1.m1.1.1.3">0.69</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">=0.69</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">= 0.69</annotation></semantics></math>, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>C), with 19 organisms in the upper mode, which we term highly reproducible (including organism-1; Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>D shows four other examples), 65 organisms in the lower mode, which we term poorly reproducible (including organism-2; Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>E shows four other examples), and six that are between the two modes, which we term intermediately reproducible (see Fig. S3EF for information about intermediately reproducible organisms). The bimodal distribution implies that evolved organisms consist of a mixture of two populations, suggesting that variation in reproducibility is a consequence of the two groups having different properties. </div> <div class="ltx_para" id="S2.SS2.p3"> The observed variation in morphological reproducibility might trivially arise from variation in morphological complexity because morphogenesis can become more sensitive to noise as the resulting morphology becomes more complex <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib28" title="">28</a>]</cite>. To test this possibility, we compared the morphological complexity scores of evolved organisms. The result shows that highly reproducible organisms had 24% lower morphological complexity on average than poorly reproducible organisms (Fig. S2). To determine whether this difference in complexity scores was indeed responsible for differences in morphological reproducibility, we conducted a regression analysis of evolved organisms’ morphological reproducibility against their complexity scores (Text S1, which also includes an alternative test independent of our complexity and reproducibility measures). The result shows that, when matched for morphological complexity, the difference in reproducibility of organisms categorised as highly and poorly reproducible remains significant, although there is some correlation between complexity and reproducibility (Fig. S2E). These results indicate that variation in morphological reproducibility cannot be exclusively explained by variation in morphological complexity, suggesting the presence of reproducibility-conferring properties in a subset of the evolved organisms. </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics</h3> <div class="ltx_para" id="S2.SS3.p1"> We wondered if the property conferring high reproducibility onto organisms is related to cell motion, given that cell motion is a driver of morphogenesis. To investigate this, we visualised the velocity of every cell at multiple time points during development of evolved organisms. We found that in organism-1, which is highly reproducible, cells at the bottom of the organism collectively move consistently downwards and radiate slightly outwards, like a travelling wave (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A; Video S1), while cells at the top of the organism show little motion. These position-dependent collective cell motions generate a “cap” of moving cells and a “stalk” of stationary cells forming in the wake of the moving cap (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A). Cap and stalk cells are in distinct cell states as indicated by the colours in Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A. Importantly, these collective cell motions are consistent across developmental replicates (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A). By contrast, in organism-2, which is poorly reproducible, patterns of collective cell motion are inconsistent across developmental replicates (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>B; Video S2). These inconsistencies arise in two related ways, which we call “bifurcations” and “protrusions.” Bifurcations are the spontaneous splitting of a group of moving cells into two separate groups caused by some of the moving cells transitioning into stationary cells at the bifurcation point (Fig. S3AB). Bifurcations occur inconsistently across developmental replicates of organism-2 (dashed arrows, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>B). Protrusions are the spontaneous onset of collective cell motion caused by stationary cells transitioning into moving cells (Fig. S3AC). For instance, in replicate 1 of organism-2, two groups of initially stationary cells protrude to form bulges around 8,000 DTS (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>B, top), whereas in replicate 2 this occurred in three groups of cells at locations different from replicate 1 around 12,000 DTS (Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>B, bottom). Both bifurcations and protrusions are caused by transitions between moving and stationary cells, suggesting that poorly reproducible organisms may be more prone to stochastic transitions in cell motion. </div> <div class="ltx_para" id="S2.SS3.p2"> Transitions between moving and stationary cells appear to occur via cell-state transitions (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>AB; Fig. S4BF). Therefore, we next tested whether highly and poorly reproducible organisms have different cell-state transition dynamics. To this end, we recorded all cell-state transitions of all cells in each evolved organism to generate the “cell state space” of that organism, which is defined as the graph consisting of nodes representing cell states and edges representing all possible transitions between cell states (see Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="4.6 Cell state and state space ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6</a> for details). Since each cell state space contains numerous nodes and edges, we simplified the state spaces in two steps. First, we pruned infrequently observed cell states and cell-state transitions (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="4.6 Cell state and state space ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6</a>). Second, we split the cell state space into strongly connected components (SCCs), where an SCC is defined as a set of cell states for which a pathway of transitions exists from any cell state to any other cell state within that set. We found that almost all (64 out of 65) poorly reproducible organisms had just a single SCC, as illustrated for organism-2 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>B. In contrast, all 19 highly reproducible organisms and most (five out of six) intermediately reproducible organisms had cell state spaces containing multiple SCCs, as illustrated for organism-1 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>A. Moreover, most (24 out of the 25) organisms with multiple SCCs had at least one SCC that unidirectionally transitioned to another SCC, as illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>A (see Fig. S3D for details about the one organism without unidirectional SCC transitions). The difference in reproducibility between organisms with a single SCC and those with multiple SCCs is statistically significant (<math alttext="p<10^{-12}" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mrow id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml"><mi id="S2.SS3.p2.1.m1.1.1.2" xref="S2.SS3.p2.1.m1.1.1.2.cmml">p</mi><mo id="S2.SS3.p2.1.m1.1.1.1" xref="S2.SS3.p2.1.m1.1.1.1.cmml"><</mo><msup id="S2.SS3.p2.1.m1.1.1.3" xref="S2.SS3.p2.1.m1.1.1.3.cmml"><mn id="S2.SS3.p2.1.m1.1.1.3.2" xref="S2.SS3.p2.1.m1.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.p2.1.m1.1.1.3.3" xref="S2.SS3.p2.1.m1.1.1.3.3.cmml"><mo id="S2.SS3.p2.1.m1.1.1.3.3a" xref="S2.SS3.p2.1.m1.1.1.3.3.cmml">−</mo><mn id="S2.SS3.p2.1.m1.1.1.3.3.2" xref="S2.SS3.p2.1.m1.1.1.3.3.2.cmml">12</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><apply id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1"><lt id="S2.SS3.p2.1.m1.1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1.1"></lt><ci id="S2.SS3.p2.1.m1.1.1.2.cmml" xref="S2.SS3.p2.1.m1.1.1.2">𝑝</ci><apply id="S2.SS3.p2.1.m1.1.1.3.cmml" xref="S2.SS3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p2.1.m1.1.1.3.1.cmml" xref="S2.SS3.p2.1.m1.1.1.3">superscript</csymbol><cn id="S2.SS3.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS3.p2.1.m1.1.1.3.2">10</cn><apply id="S2.SS3.p2.1.m1.1.1.3.3.cmml" xref="S2.SS3.p2.1.m1.1.1.3.3"><minus id="S2.SS3.p2.1.m1.1.1.3.3.1.cmml" xref="S2.SS3.p2.1.m1.1.1.3.3"></minus><cn id="S2.SS3.p2.1.m1.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.p2.1.m1.1.1.3.3.2">12</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">p<10^{-12}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_p < 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT</annotation></semantics></math>, two-tailed t-test), suggesting that the presence of multiple SCCs is involved in morphogenetic reproducibility. </div> <figure class="ltx_figure" id="S2.F4"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="276" id="S2.F4.g1" src="x4.png" width="696"/> <figcaption class="ltx_caption">Figure 4: Highly reproducible organisms have moving and dividing cells that undergo unidirectional transitions to non-moving and non-dividing cells. AB Simplified cell state spaces, consisting of cell states (nodes) and cell-state transitions (arrows) after partitioning into strongly connected components (SCCs, coloured boxes) for (A) organism-1, which is highly reproducible, and (B) organism-2, which is poorly reproducible. Node sizes depict cell state frequency over all of development; node colours correspond to cell states from organism-1 and organism-2, respectively (shown to the left of each state space). See Fig. S4IJ for the state spaces without pruning of nodes and edges. C Average cell momentum magnitude for each SCC from the 25 organisms with multiple SCCs. Momentum is the distance travelled of a cell per DTS multiplied by its size in pixels (see Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS8" title="4.8 Momentum and anisotropy ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.8</a>). Black arrows indicate unidirectional transitions between SCCs. Grey lines connect SCCs from the same organism that do not have unidirectional SCC transitions between each other. Filled orange triangles are SCCs from organisms that have unidirectional SCC transitions. All transitory SCCs are excluded (see Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="4.6 Cell state and state space ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6</a> for information about transitory SCCs). Blue diamonds are SCCs from the highly reproducible organism that does not have unidirectional SCC transitions. The first five numbered organisms correspond to those from Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>D. D Stacked bar charts showing the proportion of developmental time spent in each cell state and the proportion of cell divisions undergone by each state across all cells during developmental replicate-1 of organism-1 from Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A and development replicate-1 of organism-2 from Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>B. Diagonal lattices are pruned states. E The rate at which cells divide per developmental time when their state belongs to an upstream SCC (left) or a downstream SCC (right). Each data point represents an SCC from (C). Black lines connect upstream SCCs to their counterpart downstream SCCs. Boxes show medians and interquartile ranges (IQR); the downstream SCC box is tiny because most division rates are either very low or 0. Numbers on top of the box plots are median cell division rates.</figcaption> </figure> <div class="ltx_para" id="S2.SS3.p3"> We next sought to determine whether the difference in the number of SCCs can explain why highly reproducible organisms have less stochastic (i.e., more consistent) transitions between stationary and moving cell states than poorly reproducible organisms. In organisms with one SCC, the single SCC includes both stationary and moving cell states. Thus, transitions in cell motion are reversible since a transition pathway exists between any two cell states within the same SCC (i.e., all cell-state transitions can be reversed). By contrast, in organisms with multiple SCCs, it is possible that stationary and moving cell states are partitioned into separate SCCs to limit cell-motion transitions. To test if this partitioning has evolved, we determined whether different SCCs have different cell-motion properties in organisms with multiple SCCs by measuring the momentum of cells in each SCC (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS8" title="4.8 Momentum and anisotropy ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.8</a>). The result shows a significant disparity (mean 6.7-fold difference) in the average magnitude of cell momentum between SCCs across all highly reproducible organisms (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>C), indicating that moving and stationary states are indeed partitioned into separate SCCs. Strikingly, the transitions are always from high momentum SCCs to low momentum SCCs (black arrows in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>C)). Thus, cells in upstream SCCs are motile, whereas those in downstream SCCs are stationary. This suggests that organisms with multiple SCCs establish a “morphogenetic division of labour”, whereby moving cells (in upstream SCCs) “shape” morphologies and unidirectionally transition into stationary cells (in downstream SCCs) that “maintain” the morphologies shaped by moving cells. Restricting cell-motion transitions to unidirectional SCC transitions increases reproducibility by preventing protrusions, which require transitions from stationary to moving cells. </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> 2.4 Multiple SCCs in reproducible organisms resemble stem-cell systems</h3> <div class="ltx_para" id="S2.SS4.p1"> Our results suggest that having a morphogenetic division of labour between SCCs is important for reproducibility. Intriguingly, these morphogenetic divisions of labour resemble stem-cell systems, as upstream SCCs (corresponding to stem or progenitor cells) unidirectionally transition to downstream SCCs (corresponding to differentiated cells). This unidirectional transitioning is equivalent to irreversible differentiation — a hallmark of stem-cell systems. Another hallmark of stem-cell systems is that stem or progenitor cells undergo more rapid cell division than differentiated cells <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib48" title="">48</a>]</cite>. To examine whether this other hallmark is also displayed by our organisms with a morphogenetic division of labour, we compared how often cells in upstream versus downstream SCCs divide (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F4" title="Figure 4 ‣ 2.3 Highly and poorly reproducible organisms have distinct cell-state transition dynamics ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a>DE). We found that cells in upstream SCCs divide a median of 49.6 times more frequently than those in downstream SCCs. Thus, the morphogenetic division of labour encompasses not only differential cell motion, but also differential cell divisions between SCCs. Hereafter, we refer to upstream SCCs and downstream SCCs as stem-cell types and differentiated-cell types, respectively. We define a stem-cell system as comprising one stem-cell type and its counterpart differentiated-cell types (some organisms have multiple stem-cell types, thus multiple stem-cell systems). </div> </section> <section class="ltx_subsection" id="S2.SS5"> <h3 class="ltx_title ltx_title_subsection"> 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries</h3> <div class="ltx_para" id="S2.SS5.p1"> For morphogenetic division of labour to elevate reproducibility, differentiation of stem cells (which shape morphology by moving and dividing) to differentiated cells (which maintain morphology by neither moving nor dividing) must be consistent during development and across developmental replicates. Thus, we next asked how this consistency is achieved. We noticed that the spatial layout of stem and differentiated cells tends to mirror the topology of the cell state space in organisms with stem-cell systems. Specifically, cells in states belonging to the same SCC are spatially clustered within an organism such that each cluster corresponds to a distinct stem or differentiated cell type (Figs. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A and S4K). For example, in organism-6, there are two clusters of cells at the top and bottom of the organism, each corresponding to a separate stem-cell type, denoted as type-1 and type-2 (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A). These stem-cell clusters are present from the start of organism-6 development (i.e., the 64-cell stage) because of the initial heterogeneous distribution of maternal factors. After the 64-cell stage, stem cells initially differentiate exclusively at the boundary between the type-1 and type-2 stem cells and, subsequently, exclusively at the boundary between stem cells and differentiated cells (Video S3). This boundary-localised differentiation results in a cluster of differentiated cells consistently forming between the stem-cell clusters (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A). </div> <figure class="ltx_figure" id="S2.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="613" id="S2.F5.g1" src="x5.png" width="622"/> <figcaption class="ltx_caption ltx_centering">Figure 5: Mechanisms underlying the consistency of stem-cell differentiation and motion. A The spatial structure of cell types in organism-6 reflects the topology of the cell state space. There are two stem-cell types, one differentiated-cell type, and a transitory SCC (see Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS6" title="4.6 Cell state and state space ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.6</a> for information about transitory SCCs). The morphology is shown after 2,000, 6,000 and 12,000 DTS. B Contours showing the concentrations of the three morphogens (<math alttext="x" class="ltx_Math" display="inline" id="S2.F5.7.m1.1"><semantics id="S2.F5.7.m1.1b"><mi id="S2.F5.7.m1.1.1" xref="S2.F5.7.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.F5.7.m1.1c"><ci id="S2.F5.7.m1.1.1.cmml" xref="S2.F5.7.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.7.m1.1d">x</annotation><annotation encoding="application/x-llamapun" id="S2.F5.7.m1.1e">italic_x</annotation></semantics></math>, <math alttext="y" class="ltx_Math" display="inline" id="S2.F5.8.m2.1"><semantics id="S2.F5.8.m2.1b"><mi id="S2.F5.8.m2.1.1" xref="S2.F5.8.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.F5.8.m2.1c"><ci id="S2.F5.8.m2.1.1.cmml" xref="S2.F5.8.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.8.m2.1d">y</annotation><annotation encoding="application/x-llamapun" id="S2.F5.8.m2.1e">italic_y</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="S2.F5.9.m3.1"><semantics id="S2.F5.9.m3.1b"><mi id="S2.F5.9.m3.1.1" xref="S2.F5.9.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.F5.9.m3.1c"><ci id="S2.F5.9.m3.1.1.cmml" xref="S2.F5.9.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.9.m3.1d">z</annotation><annotation encoding="application/x-llamapun" id="S2.F5.9.m3.1e">italic_z</annotation></semantics></math>) overlaid on organism-6 after 9,000 DTS. Each contour joins points of equal concentration of the same morphogen. C Schematic depicting type-1, type-2 and differentiated cell clusters from (B), with stem cell motion (grey arrows) and differentiation (black arrows) indicated. Vertical dashed lines indicate cell-type boundaries. Below, morphogen concentrations along the cross-section line are plotted, along with the sums of cell protein concentrations for cells along the cross-section (each cross is one cell). D Polar plots of momentum magnitude by angle of motion for each cell type summed over all cells over the 12,000 DTS of organism-6 development (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS8" title="4.8 Momentum and anisotropy ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.8</a>). E Development of type-1, type-2 and differentiated cells in isolation. Polar plots show distributions of cell momentum as in (D). F Motion anisotropy of stem cells within organisms (orange) and isolated (black) plotted against average cell momentum magnitude (<math alttext="n=30" class="ltx_Math" display="inline" id="S2.F5.10.m4.1"><semantics id="S2.F5.10.m4.1b"><mrow id="S2.F5.10.m4.1.1" xref="S2.F5.10.m4.1.1.cmml"><mi id="S2.F5.10.m4.1.1.2" xref="S2.F5.10.m4.1.1.2.cmml">n</mi><mo id="S2.F5.10.m4.1.1.1" xref="S2.F5.10.m4.1.1.1.cmml">=</mo><mn id="S2.F5.10.m4.1.1.3" xref="S2.F5.10.m4.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F5.10.m4.1c"><apply id="S2.F5.10.m4.1.1.cmml" xref="S2.F5.10.m4.1.1"><eq id="S2.F5.10.m4.1.1.1.cmml" xref="S2.F5.10.m4.1.1.1"></eq><ci id="S2.F5.10.m4.1.1.2.cmml" xref="S2.F5.10.m4.1.1.2">𝑛</ci><cn id="S2.F5.10.m4.1.1.3.cmml" type="integer" xref="S2.F5.10.m4.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.10.m4.1d">n=30</annotation><annotation encoding="application/x-llamapun" id="S2.F5.10.m4.1e">italic_n = 30</annotation></semantics></math> cell types). Grey lines connect a stem cell type within an organism to that stem cell type in isolation. G The same as (F), but for differentiated cells (<math alttext="n=26" class="ltx_Math" display="inline" id="S2.F5.11.m5.1"><semantics id="S2.F5.11.m5.1b"><mrow id="S2.F5.11.m5.1.1" xref="S2.F5.11.m5.1.1.cmml"><mi id="S2.F5.11.m5.1.1.2" xref="S2.F5.11.m5.1.1.2.cmml">n</mi><mo id="S2.F5.11.m5.1.1.1" xref="S2.F5.11.m5.1.1.1.cmml">=</mo><mn id="S2.F5.11.m5.1.1.3" xref="S2.F5.11.m5.1.1.3.cmml">26</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F5.11.m5.1c"><apply id="S2.F5.11.m5.1.1.cmml" xref="S2.F5.11.m5.1.1"><eq id="S2.F5.11.m5.1.1.1.cmml" xref="S2.F5.11.m5.1.1.1"></eq><ci id="S2.F5.11.m5.1.1.2.cmml" xref="S2.F5.11.m5.1.1.2">𝑛</ci><cn id="S2.F5.11.m5.1.1.3.cmml" type="integer" xref="S2.F5.11.m5.1.1.3">26</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.11.m5.1d">n=26</annotation><annotation encoding="application/x-llamapun" id="S2.F5.11.m5.1e">italic_n = 26</annotation></semantics></math> cell types). H The same as (F), but for cells from organisms with only one SCC (<math alttext="n=65" class="ltx_Math" display="inline" id="S2.F5.12.m6.1"><semantics id="S2.F5.12.m6.1b"><mrow id="S2.F5.12.m6.1.1" xref="S2.F5.12.m6.1.1.cmml"><mi id="S2.F5.12.m6.1.1.2" xref="S2.F5.12.m6.1.1.2.cmml">n</mi><mo id="S2.F5.12.m6.1.1.1" xref="S2.F5.12.m6.1.1.1.cmml">=</mo><mn id="S2.F5.12.m6.1.1.3" xref="S2.F5.12.m6.1.1.3.cmml">65</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F5.12.m6.1c"><apply id="S2.F5.12.m6.1.1.cmml" xref="S2.F5.12.m6.1.1"><eq id="S2.F5.12.m6.1.1.1.cmml" xref="S2.F5.12.m6.1.1.1"></eq><ci id="S2.F5.12.m6.1.1.2.cmml" xref="S2.F5.12.m6.1.1.2">𝑛</ci><cn id="S2.F5.12.m6.1.1.3.cmml" type="integer" xref="S2.F5.12.m6.1.1.3">65</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F5.12.m6.1d">n=65</annotation><annotation encoding="application/x-llamapun" id="S2.F5.12.m6.1e">italic_n = 65</annotation></semantics></math>). I Developmental trajectory for an arbitrary configuration of type-1 stem and differentiated cells from organism-6 for 15,000 DTS. The initial state of all cells was set to the differentiated blue state from (A), except for a subset of cells on the surface of the triangle that were set to the orange type-2 state from (A).</figcaption> </figure> <div class="ltx_para" id="S2.SS5.p2"> Boundary-localised differentiation suggests that stem-cell differentiation is spatially regulated. To determine how this regulation is achieved, we first focused on organism-6 as it has two separate examples of stem-cell differentiation due to it having two stem-cell types. We hypothesised that boundary-localised differentiation is caused by morphogen-mediated interactions between stem and differentiated cells as morphogens provide a means to spatially regulate gene expression. To test this, we made a contour plot of morphogen concentrations in organism-6 (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>B). The plot shows that the concentrations of different morphogens abruptly change at the boundaries between stem and differentiated cells (dashed lines in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>B). To determine whether gene expression responds to these changes in morphogen concentrations, we plotted the sum of protein concentrations (excluding morphogens) for cells along a cross-section of organism-6 (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>C). The plot shows that this sum changes abruptly wherever morphogen concentrations abruptly change, suggesting that morphogens regulate gene expression. These results suggest that a specific morphogen profile induces differentiation, with differentiated cells producing this profile and thus localising differentiation to the boundary between stem and differentiated cells. To directly test this, we isolated each stem-cell type from the other two cell types, thereby removing the effect of differentiated cells on morphogen profiles. To achieve this isolation, we created organisms in which all cells at the 64-cell stage were set to the state that is most frequently observed for each of the stem-cell types (e.g., the green cell state shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A for type-1 stem cells). We developed these organisms for 12,000 DTS to test whether isolated type-1 and type-2 stem cells differentiate. We found that neither stem-cell type differentiates (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>E). These results support the hypothesis that differentiated cells induce stem-cell differentiation, thus localising this differentiation to the boundaries between cell types. </div> <div class="ltx_para" id="S2.SS5.p3"> We next asked whether the above findings — that differentiated cells induce stem-cell differentiation — are generalisable to all organisms with stem-cell systems. To answer this, we isolated each of the 30 stem-cell types from the 24 organisms with stem-cell systems from all other cell types and tested whether the isolated stem cells differentiate through the same method as described in the previous paragraph. We found that the great majority (26 out of 30) do not differentiate when isolated from other cell types (Fig. S5 shows why the four exceptions do not counter our hypothesis). This result, along with the observation that every stem-cell type always differentiates at the boundary shared with differentiated cells (Figs. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A and S4K), indicates that differentiated cells induce stem-cell differentiation. </div> <div class="ltx_para" id="S2.SS5.p4"> The spatial consistency of stem-cell differentiation is crucial but not sufficient for morphogenetic reproducibility. The other critical factor is the motion of stem cells in a consistent direction (e.g., organism-1 Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A and organism-6 Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A), as this motion helps to confer consistent shape formation across developmental replicates. Given that stem-cell differentiation is induced by differentiated cells, we hypothesised that the directionality of stem-cell motion is also induced by differentiated cells. </div> <div class="ltx_para" id="S2.SS5.p5"> To test this, we determined how stem cells’ motion depends on the presence or absence of differentiated cells by measuring the magnitude of stem cells’ momentum as a function of the angle of momentum, either in normal development (differentiated cells present) or when each stem cell type is isolated (differentiated cells absent). The result shows that momentum is strongly directional (i.e., anisotropic) when a stem-cell type is in normal development, whereas it is radially symmetrically distributed (i.e., isotropic) when a stem-cell type is isolated (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>DE shows results for organism-6; see Fig. S5 for other organisms). We quantified this difference in anisotropy by calculating the ratio of the variance in momentum magnitude across angle to the mean momentum magnitude across angle (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS8" title="4.8 Momentum and anisotropy ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.8</a>). We found that anisotropy of stem-cell motion decreased an average of 65-fold across the 30 stem cell types when stem cells were isolated (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>F). These results support the hypothesis that the presence of differentiated cells induces directional stem-cell motion. </div> <div class="ltx_para" id="S2.SS5.p6"> We next asked how, mechanistically, the presence of differentiated cells induces directional stem-cell motion. We hypothesised that differential expression of adhesion proteins between stem and differentiated cells might drive this motion, as the adhesion energies determined by these proteins are a determinant of cell motion in our model. To test this, we measured the adhesion energy arising from stem-to-stem cell contact, stem-to-differentiated cell contact and differentiated-to-differentiated cell contact across all 30 stem-cell systems (Fig. S6AB). We found that stem-to-stem adhesion energies were lower than stem-to-differentiated adhesion energies in all 30 stem-cell systems (means of 4.1 and 8.1, respectively; Fig. S6AB; differentiated-to-differentiated adhesion energy is variable, with a mean of 10.6). The low stem-to-stem adhesion energies relative to stem-to-differentiated adhesion energies implies that stem cells preferentially adhere to each other rather than to differentiated cells, indicating that clusters of stem cells should move away from areas where they border differentiated cells (this effect is akin to a dewetting of fluid-like tissue from solid-like tissue, a phenomenon observed in both epithelial and mesenchymal tissues <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib50" title="">50</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib51" title="">51</a>]</cite>). However, ongoing stem-cell differentiation at this border has the opposite effect by creating more points of contact between the newly formed differentiated cells and yet-to-differentiate stem cells, thus increasing adhesion energy. To minimise this adhesion energy, stem cells move away from the points of contacts with the new differentiated cells. Thus, the location-specific differentiation induced by differentiated cells guides the direction of stem-cell motion. Together, these results suggest that interactions between stem and differentiated cells represent another aspect of the morphogenetic division of labour, whereby differentiated cells not only maintain the shape formed by stem cells but also make this shape formation consistent. </div> <div class="ltx_para" id="S2.SS5.p7"> A practical implication of our finding that reproducible morphogenesis is a consequence of interactions between stem and differentiated cells is that we should be able to program the same robust morphology by appropriately arranging stem and differentiated cells with different initial conditions. To determine if this is possible, we created arbitrary shapes of stem and differentiated cells and simulated their development. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>I shows an example that begins with a triangular shape of differentiated cells from organism-6 accompanied by a small cluster of type-2 stem cells from that organism on the diagonal surface. This is predicted to result in the stem cells moving diagonally away from the differentiated cells to generate a stalk of differentiated cells in their wake, which is indeed what is observed (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>I). Similar results were obtained for different initial shapes and different stem-cell systems (Fig. S6HI), demonstrating the robustness of stem-cell system morphogenesis to different initial conditions. These results suggest that repeating or even branching morphologies can be generated by the appropriate initial positioning of stem and differentiated cells, and that these morphologies should be highly reproducible due to the nature of stem-cell system morphogenesis. </div> </section> <section class="ltx_subsection" id="S2.SS6"> <h3 class="ltx_title ltx_title_subsection"> 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions</h3> <figure class="ltx_figure" id="S2.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="224" id="S2.F6.g1" src="x6.png" width="585"/> <figcaption class="ltx_caption ltx_centering">Figure 6: Reproducible morphogenesis with stem-cell systems is evolutionarily accessible. AB Development of two organism with stem-cell systems. Morphologies are shown after 2,000, 6,000 and 12,000 DTS. A Simplified state space of each organism is shown to the right of their respective developments, indicating multiple SCCs with unidirectional transitions. C The rate at which cells divide per developmental time when their state belongs to an upstream SCC (left) or a downstream SCC (right). Each data point represents an SCC from the 29 evolved organisms that had multiple SCCs with unidirectional transitions that were evolved under selection for directional motion and morphological complexity. Black lines connect upstream SCCs to their counterpart downstream SCCs. Boxes show medians and interquartile ranges (IQR). D Reproducibility scores for the 31 evolved organisms from simulations selecting for both directional motion and morphological complexity. Black circles indicate organisms with a single SCC. Orange triangles indicate organisms with stem-cell systems. Categories of “high”, “intermediate” and “poor” are copied from Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>C. Numbers (1) and (2) refer to the organisms in panels (B) and (C), respectively.</figcaption> </figure> <div class="ltx_para" id="S2.SS6.p1"> Our results show that stem-cell systems evolved only in a minority of simulations (24 out of the 90 organisms with complex morphologies). We asked whether this is because stem-cell systems are evolutionarily accessible from a restricted portion of genotype space, and thus determined by the initial conditions, or because they are selectively disfavoured by the particular fitness criterion we used. To address this, we tested an alternative selection criterion that favoured not only the morphological complexity but also the directional motion of organisms. This additional selection for directional motion is expected to favour stem-cell systems because directional motion is a property of morphogenesis with stem-cell systems (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>F), although it does not directly select for stem-cell systems or reproducibility. We quantified directional motion by determining how much an organism’s centre of mass (measured in pixels) shifts over the 12,000 DTS. We ran 35 simulations lasting at least <math alttext="2.5\times 10^{3}" class="ltx_Math" display="inline" id="S2.SS6.p1.1.m1.1"><semantics id="S2.SS6.p1.1.m1.1a"><mrow id="S2.SS6.p1.1.m1.1.1" xref="S2.SS6.p1.1.m1.1.1.cmml"><mn id="S2.SS6.p1.1.m1.1.1.2" xref="S2.SS6.p1.1.m1.1.1.2.cmml">2.5</mn><mo id="S2.SS6.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS6.p1.1.m1.1.1.1.cmml">×</mo><msup id="S2.SS6.p1.1.m1.1.1.3" xref="S2.SS6.p1.1.m1.1.1.3.cmml"><mn id="S2.SS6.p1.1.m1.1.1.3.2" xref="S2.SS6.p1.1.m1.1.1.3.2.cmml">10</mn><mn id="S2.SS6.p1.1.m1.1.1.3.3" xref="S2.SS6.p1.1.m1.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS6.p1.1.m1.1b"><apply id="S2.SS6.p1.1.m1.1.1.cmml" xref="S2.SS6.p1.1.m1.1.1"><times id="S2.SS6.p1.1.m1.1.1.1.cmml" xref="S2.SS6.p1.1.m1.1.1.1"></times><cn id="S2.SS6.p1.1.m1.1.1.2.cmml" type="float" xref="S2.SS6.p1.1.m1.1.1.2">2.5</cn><apply id="S2.SS6.p1.1.m1.1.1.3.cmml" xref="S2.SS6.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS6.p1.1.m1.1.1.3.1.cmml" xref="S2.SS6.p1.1.m1.1.1.3">superscript</csymbol><cn id="S2.SS6.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS6.p1.1.m1.1.1.3.2">10</cn><cn id="S2.SS6.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS6.p1.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS6.p1.1.m1.1c">2.5\times 10^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS6.p1.1.m1.1d">2.5 × 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> evolutionary generations, of which 31 evolved organisms surpassed our arbitrary fitness threshold (Methods <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.SS5" title="4.5 Evolution ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4.5</a>). We found the great majority of these organisms (29 out of 31) had evolved stem-cell systems (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F6" title="Figure 6 ‣ 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">6</a>ABC and S7ABCD). Of these 29 organisms, all but one displayed highly reproducible morphogenesis (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F6" title="Figure 6 ‣ 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">6</a>D). In contrast, the two organisms that did not evolve stem-cell systems displayed poorly reproducible morphogenesis (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F6" title="Figure 6 ‣ 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">6</a>D and S7E). In addition, we found that stem-cell systems are evolvable when the model assumes simpler gene regulatory networks that lack contractile proteins and have only half the number of the other genes (Fig S7FGH). Together, these results suggest that the evolution of stem-cell systems are robust to the structure of the model, and are likely “easy” to evolve across a wide range of starting genotypes and regulatory networks. </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> 3 Discussion</h2> <div class="ltx_para" id="S3.p1"> In this study, we found that reproducible morphogenesis evolves in a sizeable fraction of simulations with selection solely for complex morphology, indicating that morphogenetic reproducibility can be an emergent by-product of complex morphogenesis. We show that this reproducibility is realised by a morphogenetic division of labour based on stem-cell systems, where mobile, dividing stem cells “shape” morphologies and irreversibly differentiate into stationary, non-dividing cells that “maintain” these morphologies. This labour division allows differentiated cells to serve as “anchor points” that establish the locations of stem-cell differentiation and directions of stem-cell motion. These locations are made consistent by interactions between stem and differentiated cells mediated by morphogen gradients as well as differential cell adhesion. Thus, we propose that stem-cell systems have a previously unrecognised role in enabling reproducible complex morphogenesis, in addition to their well-known role of generating specialised cell types <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib54" title="">54</a>]</cite>. This proposal is supported by three key observations that we describe below. </div> <div class="ltx_para" id="S3.p2"> The first observation is that stem-cell-based morphogenesis appears to evolve from most regions of genotype space (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F6" title="Figure 6 ‣ 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">6</a>). Although all simulations evolved distinct gene regulatory networks (GRNs), each generating a different morphology (i.e., simulations took different evolutionary paths, a well-documented phenomenon in developmental evolution <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib56" title="">56</a>]</cite>), we found that stem-cell-based morphogenesis evolved repeatedly. This repeated evolution indicates that stem-cell-based morphogenesis is not constrained to just a few “special” GRN. Thus, we suggest that so long as gene regulation, cell adhesion, and cell-cell signalling are in place, such as in all animals and some multicellular microbes such as slime moulds <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib57" title="">57</a>]</cite>, stem-cell-based morphogenesis will be a frequent evolutionary outcome in nature when complex morphogenesis is selected for. </div> <div class="ltx_para" id="S3.p3"> The second observation supporting our proposal is that whenever stem-cell-based morphogenesis evolved in our model, it had three properties that have been shown to elevate morphogenetic reproducibility <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib27" title="">27</a>]</cite>: (1) boundary-localised differentiation mediated by morphogen signalling, (2) the immobility and non-division of differentiated cells, and (3) differential adhesion between stem and differentiated cells. These properties have been shown to enhance morphogenetic reproducibility by smoothing the boundaries between different cell types <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib58" title="">58</a>]</cite>, with smoother boundaries contributing to a more stable spatial organisation of cell types and, thus, more stable morphogenesis. Indeed, the boundaries between stem and differentiated cells in our simulated organisms appear to be smooth (Figs. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F3" title="Figure 3 ‣ 2.2 Reproducibility is not an intrinsic property of complex morphogenesis ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a>A, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>A, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F6" title="Figure 6 ‣ 2.6 Reproducible morphogenesis with stem-cell systems is evolutionarily accessible from most initial conditions ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">6</a>BC). We also found that these three properties enable differentiated cells to act as anchors that guide consistent stem-cell motion and differentiation, which again enhances morphogenetic reproducibility. These three properties are widespread at cell-type boundaries in extant developmental systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib59" title="">59</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib60" title="">60</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib62" title="">62</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib58" title="">58</a>]</cite>, implying they are generic properties of morphogenesis. Critically, we observed these three properties whenever stem-cell-based morphogenesis evolved, despite no direct selection for them. Therefore, stem-cell systems appear to be automatically equipped with these three properties, implying that stem-cell-based morphogenesis will be generally reproducible. </div> <div class="ltx_para" id="S3.p4"> The third observation supporting our proposal is that stem-cell systems enable modular morphogenesis. This modularity arises from the fact that stem-cell-based morphogenesis depends solely on the interactions between stem and differentiated cells (e.g., Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>I) and does not depend on external cues or a specific initial shape. Consequently, stem-cell-based morphogenesis is not limited to creating the elongated or bulge-like shapes that evolved in our model. Instead, these shapes could constitute sub-structures within a more complex morphology, such as organ building blocks, where sub-structures are repeated wherever the appropriate cell-type configuration is present. Intriguingly, the stem-cell-based morphogenesis we observed in our evolved organisms resembles the morphogenesis of sub-structures in a number of extant organisms, such as bilaterian body axis and limb elongation, intestinal crypts and villi, lung alveoli, mammary gland buds, salivary gland buds and kidney tubules <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib63" title="">63</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib65" title="">65</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib66" title="">66</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib67" title="">67</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib68" title="">68</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib69" title="">69</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib51" title="">51</a>]</cite>. For example, in the kidney, nephron tubule formation begins with the recruitment of mesenchymal progenitor (lineage-restricted stem) cells to a ureteric bud branch <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib70" title="">70</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib71" title="">71</a>]</cite>. These progenitor cells then undergo division and differentiation into epithelial cells to elongate the nephron, a process driven by signals that originate from already differentiated epithelial cells <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib72" title="">72</a>]</cite>, similar to the signalling between stem and differentiated cells that drives elongation in our evolved organisms. This process is repeated in a modular fashion at many locations in the kidney to generate a large number of nephrons. Although stem-cell system morphogenesis does not produce branching in our model, we hypothesise that branching occurs by combining stem-cell systems with species-specific mechanisms known to be involved in branching, such as polarised cell contractions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib73" title="">73</a>]</cite>. The abundance of examples in which modular sub-structures are reminiscent of stem-cell-based morphogenesis, suggests that stem-cell-based morphogenesis is frequently employed in extant developmental systems. </div> <div class="ltx_para" id="S3.p5"> In summary, our results suggest that stem-cell systems are a fundamental developmental phenomenon that, in addition to producing specialised cell types, can underpin multicellular morphogenesis and its reproducibility. Expanding our focus on stem-cell systems from solely cell-type specification to also encompassing morphogenesis has important implications for how we understand the generation and regeneration of tissue morphologies. The programming of tissue morphogenesis ex vivo, such as organoids, in an accurate and reproducible way is a critical challenge in synthetic biology <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib74" title="">74</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib75" title="">75</a>]</cite>. Our findings suggest that a better understanding of the roles the different cell types in stem-cell systems play in morphogenesis will enable us to more effectively manipulate the configuration of these cell types to elevate the accuracy and reproducibility of programmed morphogenesis. Finally, our findings open the possibility that some stem-cell systems originally evolved in animal development for a morphogenetic purpose, with their role in producing specialised cell types emerging as a later exaptation. </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> 4 Methods</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> 4.1 Cellular Potts Model</h3> <div class="ltx_para" id="S4.SS1.p1"> Our model extends a Cellular Potts Model (CPM) introduced by Hogeweg (2000) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib29" title="">29</a>]</cite>. The CPM dynamics are driven by pixel copying, where repeated random sampling of pixels on grid determines the location of these copies. For each chosen pixel, a random neighbouring pixel from its Moore neighbourhood is selected as the recipient of the pixel copy. Whether the pixel copy is accepted depends on its effect on the system’s energy, denoted as <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_H</annotation></semantics></math>, represented by <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H=\sum_{i,j}{J_{ij}}+\sum_{i,m}{J_{im}}+\lambda_{V}\sum_{\sigma}(\upsilon_{% 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xref="S4.E1.m1.5.5.1.1.2.2.1.1.1.1.1.3">subscript</csymbol><ci id="S4.E1.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.cmml" xref="S4.E1.m1.5.5.1.1.2.2.1.1.1.1.1.3.2">𝐿</ci><ci id="S4.E1.m1.5.5.1.1.2.2.1.1.1.1.1.3.3.cmml" xref="S4.E1.m1.5.5.1.1.2.2.1.1.1.1.1.3.3">𝜎</ci></apply></apply><cn id="S4.E1.m1.5.5.1.1.2.2.1.1.3.cmml" type="integer" xref="S4.E1.m1.5.5.1.1.2.2.1.1.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.5c">H=\sum_{i,j}{J_{ij}}+\sum_{i,m}{J_{im}}+\lambda_{V}\sum_{\sigma}(\upsilon_{% \sigma}-V_{\sigma})^{2}+\lambda^{\sigma}_{L}\sum_{\sigma}(l_{\sigma}-L_{\sigma% })^{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.5d">italic_H = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i , italic_m end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_l start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(1)</td> </tr></tbody> </table> Here, <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m1.1"><semantics id="S4.SS1.p1.2.m1.1a"><mi id="S4.SS1.p1.2.m1.1.1" xref="S4.SS1.p1.2.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m1.1b"><ci id="S4.SS1.p1.2.m1.1.1.cmml" xref="S4.SS1.p1.2.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m1.1d">italic_H</annotation></semantics></math> encompasses the total surface energy accumulated from cell-cell adhesion (<math alttext="J_{ij}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m2.1"><semantics id="S4.SS1.p1.3.m2.1a"><msub id="S4.SS1.p1.3.m2.1.1" xref="S4.SS1.p1.3.m2.1.1.cmml"><mi id="S4.SS1.p1.3.m2.1.1.2" xref="S4.SS1.p1.3.m2.1.1.2.cmml">J</mi><mrow id="S4.SS1.p1.3.m2.1.1.3" xref="S4.SS1.p1.3.m2.1.1.3.cmml"><mi id="S4.SS1.p1.3.m2.1.1.3.2" xref="S4.SS1.p1.3.m2.1.1.3.2.cmml">i</mi><mo id="S4.SS1.p1.3.m2.1.1.3.1" xref="S4.SS1.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.3.m2.1.1.3.3" xref="S4.SS1.p1.3.m2.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m2.1b"><apply id="S4.SS1.p1.3.m2.1.1.cmml" xref="S4.SS1.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m2.1.1.1.cmml" xref="S4.SS1.p1.3.m2.1.1">subscript</csymbol><ci id="S4.SS1.p1.3.m2.1.1.2.cmml" xref="S4.SS1.p1.3.m2.1.1.2">𝐽</ci><apply id="S4.SS1.p1.3.m2.1.1.3.cmml" xref="S4.SS1.p1.3.m2.1.1.3"><times id="S4.SS1.p1.3.m2.1.1.3.1.cmml" xref="S4.SS1.p1.3.m2.1.1.3.1"></times><ci id="S4.SS1.p1.3.m2.1.1.3.2.cmml" xref="S4.SS1.p1.3.m2.1.1.3.2">𝑖</ci><ci id="S4.SS1.p1.3.m2.1.1.3.3.cmml" xref="S4.SS1.p1.3.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m2.1c">J_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m2.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>) and cell-medium adhesion (<math alttext="J_{im}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m3.1"><semantics id="S4.SS1.p1.4.m3.1a"><msub id="S4.SS1.p1.4.m3.1.1" xref="S4.SS1.p1.4.m3.1.1.cmml"><mi id="S4.SS1.p1.4.m3.1.1.2" xref="S4.SS1.p1.4.m3.1.1.2.cmml">J</mi><mrow id="S4.SS1.p1.4.m3.1.1.3" xref="S4.SS1.p1.4.m3.1.1.3.cmml"><mi id="S4.SS1.p1.4.m3.1.1.3.2" xref="S4.SS1.p1.4.m3.1.1.3.2.cmml">i</mi><mo id="S4.SS1.p1.4.m3.1.1.3.1" xref="S4.SS1.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.4.m3.1.1.3.3" xref="S4.SS1.p1.4.m3.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m3.1b"><apply id="S4.SS1.p1.4.m3.1.1.cmml" xref="S4.SS1.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m3.1.1.1.cmml" xref="S4.SS1.p1.4.m3.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m3.1.1.2.cmml" xref="S4.SS1.p1.4.m3.1.1.2">𝐽</ci><apply id="S4.SS1.p1.4.m3.1.1.3.cmml" xref="S4.SS1.p1.4.m3.1.1.3"><times id="S4.SS1.p1.4.m3.1.1.3.1.cmml" xref="S4.SS1.p1.4.m3.1.1.3.1"></times><ci id="S4.SS1.p1.4.m3.1.1.3.2.cmml" xref="S4.SS1.p1.4.m3.1.1.3.2">𝑖</ci><ci id="S4.SS1.p1.4.m3.1.1.3.3.cmml" xref="S4.SS1.p1.4.m3.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m3.1c">J_{im}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m3.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT</annotation></semantics></math>), which are both determined dynamically as functions of protein concentrations (explained later). The index <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m4.1"><semantics id="S4.SS1.p1.5.m4.1a"><mi id="S4.SS1.p1.5.m4.1.1" xref="S4.SS1.p1.5.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m4.1b"><ci id="S4.SS1.p1.5.m4.1.1.cmml" xref="S4.SS1.p1.5.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m4.1d">italic_i</annotation></semantics></math> is of pixels at cell boundaries. The index <math alttext="j" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m5.1"><semantics id="S4.SS1.p1.6.m5.1a"><mi id="S4.SS1.p1.6.m5.1.1" xref="S4.SS1.p1.6.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m5.1b"><ci id="S4.SS1.p1.6.m5.1.1.cmml" xref="S4.SS1.p1.6.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m5.1d">italic_j</annotation></semantics></math> is of pixels neighbouring <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m6.1"><semantics id="S4.SS1.p1.7.m6.1a"><mi id="S4.SS1.p1.7.m6.1.1" xref="S4.SS1.p1.7.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m6.1b"><ci id="S4.SS1.p1.7.m6.1.1.cmml" xref="S4.SS1.p1.7.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m6.1d">italic_i</annotation></semantics></math> that are occupied by a different cell from <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m7.1"><semantics id="S4.SS1.p1.8.m7.1a"><mi id="S4.SS1.p1.8.m7.1.1" xref="S4.SS1.p1.8.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m7.1b"><ci id="S4.SS1.p1.8.m7.1.1.cmml" xref="S4.SS1.p1.8.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m7.1d">italic_i</annotation></semantics></math>. The index <math alttext="m" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m8.1"><semantics id="S4.SS1.p1.9.m8.1a"><mi id="S4.SS1.p1.9.m8.1.1" xref="S4.SS1.p1.9.m8.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m8.1b"><ci id="S4.SS1.p1.9.m8.1.1.cmml" xref="S4.SS1.p1.9.m8.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m8.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m8.1d">italic_m</annotation></semantics></math> is of pixels neighbouring <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m9.1"><semantics id="S4.SS1.p1.10.m9.1a"><mi id="S4.SS1.p1.10.m9.1.1" xref="S4.SS1.p1.10.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m9.1b"><ci id="S4.SS1.p1.10.m9.1.1.cmml" xref="S4.SS1.p1.10.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m9.1d">italic_i</annotation></semantics></math> that are occupied by the medium. Every pixel on the grid has an associated value, <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p1.11.m10.1"><semantics id="S4.SS1.p1.11.m10.1a"><mi id="S4.SS1.p1.11.m10.1.1" xref="S4.SS1.p1.11.m10.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.11.m10.1b"><ci id="S4.SS1.p1.11.m10.1.1.cmml" xref="S4.SS1.p1.11.m10.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.11.m10.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.11.m10.1d">italic_σ</annotation></semantics></math>, that represents either the cell that occupies that pixel (<math alttext="\sigma\geq 1" class="ltx_Math" display="inline" id="S4.SS1.p1.12.m11.1"><semantics id="S4.SS1.p1.12.m11.1a"><mrow id="S4.SS1.p1.12.m11.1.1" xref="S4.SS1.p1.12.m11.1.1.cmml"><mi id="S4.SS1.p1.12.m11.1.1.2" xref="S4.SS1.p1.12.m11.1.1.2.cmml">σ</mi><mo id="S4.SS1.p1.12.m11.1.1.1" xref="S4.SS1.p1.12.m11.1.1.1.cmml">≥</mo><mn id="S4.SS1.p1.12.m11.1.1.3" xref="S4.SS1.p1.12.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.12.m11.1b"><apply id="S4.SS1.p1.12.m11.1.1.cmml" xref="S4.SS1.p1.12.m11.1.1"><geq id="S4.SS1.p1.12.m11.1.1.1.cmml" xref="S4.SS1.p1.12.m11.1.1.1"></geq><ci id="S4.SS1.p1.12.m11.1.1.2.cmml" xref="S4.SS1.p1.12.m11.1.1.2">𝜎</ci><cn id="S4.SS1.p1.12.m11.1.1.3.cmml" type="integer" xref="S4.SS1.p1.12.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.12.m11.1c">\sigma\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.12.m11.1d">italic_σ ≥ 1</annotation></semantics></math>) or the medium (<math alttext="\sigma=0" class="ltx_Math" display="inline" id="S4.SS1.p1.13.m12.1"><semantics id="S4.SS1.p1.13.m12.1a"><mrow id="S4.SS1.p1.13.m12.1.1" xref="S4.SS1.p1.13.m12.1.1.cmml"><mi id="S4.SS1.p1.13.m12.1.1.2" xref="S4.SS1.p1.13.m12.1.1.2.cmml">σ</mi><mo id="S4.SS1.p1.13.m12.1.1.1" xref="S4.SS1.p1.13.m12.1.1.1.cmml">=</mo><mn id="S4.SS1.p1.13.m12.1.1.3" xref="S4.SS1.p1.13.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.13.m12.1b"><apply id="S4.SS1.p1.13.m12.1.1.cmml" xref="S4.SS1.p1.13.m12.1.1"><eq id="S4.SS1.p1.13.m12.1.1.1.cmml" xref="S4.SS1.p1.13.m12.1.1.1"></eq><ci id="S4.SS1.p1.13.m12.1.1.2.cmml" xref="S4.SS1.p1.13.m12.1.1.2">𝜎</ci><cn id="S4.SS1.p1.13.m12.1.1.3.cmml" type="integer" xref="S4.SS1.p1.13.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.13.m12.1c">\sigma=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.13.m12.1d">italic_σ = 0</annotation></semantics></math>). Each cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p1.14.m13.1"><semantics id="S4.SS1.p1.14.m13.1a"><mi id="S4.SS1.p1.14.m13.1.1" xref="S4.SS1.p1.14.m13.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.14.m13.1b"><ci id="S4.SS1.p1.14.m13.1.1.cmml" xref="S4.SS1.p1.14.m13.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.14.m13.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.14.m13.1d">italic_σ</annotation></semantics></math> (<math alttext="\sigma\geq 1" class="ltx_Math" display="inline" id="S4.SS1.p1.15.m14.1"><semantics id="S4.SS1.p1.15.m14.1a"><mrow id="S4.SS1.p1.15.m14.1.1" xref="S4.SS1.p1.15.m14.1.1.cmml"><mi id="S4.SS1.p1.15.m14.1.1.2" xref="S4.SS1.p1.15.m14.1.1.2.cmml">σ</mi><mo id="S4.SS1.p1.15.m14.1.1.1" xref="S4.SS1.p1.15.m14.1.1.1.cmml">≥</mo><mn id="S4.SS1.p1.15.m14.1.1.3" xref="S4.SS1.p1.15.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.15.m14.1b"><apply id="S4.SS1.p1.15.m14.1.1.cmml" xref="S4.SS1.p1.15.m14.1.1"><geq id="S4.SS1.p1.15.m14.1.1.1.cmml" xref="S4.SS1.p1.15.m14.1.1.1"></geq><ci id="S4.SS1.p1.15.m14.1.1.2.cmml" xref="S4.SS1.p1.15.m14.1.1.2">𝜎</ci><cn id="S4.SS1.p1.15.m14.1.1.3.cmml" type="integer" xref="S4.SS1.p1.15.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.15.m14.1c">\sigma\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.15.m14.1d">italic_σ ≥ 1</annotation></semantics></math>) with current size <math alttext="\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.16.m15.1"><semantics id="S4.SS1.p1.16.m15.1a"><msub id="S4.SS1.p1.16.m15.1.1" xref="S4.SS1.p1.16.m15.1.1.cmml"><mi id="S4.SS1.p1.16.m15.1.1.2" xref="S4.SS1.p1.16.m15.1.1.2.cmml">υ</mi><mi id="S4.SS1.p1.16.m15.1.1.3" xref="S4.SS1.p1.16.m15.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.16.m15.1b"><apply id="S4.SS1.p1.16.m15.1.1.cmml" xref="S4.SS1.p1.16.m15.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.16.m15.1.1.1.cmml" xref="S4.SS1.p1.16.m15.1.1">subscript</csymbol><ci id="S4.SS1.p1.16.m15.1.1.2.cmml" xref="S4.SS1.p1.16.m15.1.1.2">𝜐</ci><ci id="S4.SS1.p1.16.m15.1.1.3.cmml" xref="S4.SS1.p1.16.m15.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.16.m15.1c">\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.16.m15.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> (in pixels) is constrained to size <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.17.m16.1"><semantics id="S4.SS1.p1.17.m16.1a"><msub id="S4.SS1.p1.17.m16.1.1" xref="S4.SS1.p1.17.m16.1.1.cmml"><mi id="S4.SS1.p1.17.m16.1.1.2" xref="S4.SS1.p1.17.m16.1.1.2.cmml">V</mi><mi id="S4.SS1.p1.17.m16.1.1.3" xref="S4.SS1.p1.17.m16.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.17.m16.1b"><apply id="S4.SS1.p1.17.m16.1.1.cmml" xref="S4.SS1.p1.17.m16.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.17.m16.1.1.1.cmml" xref="S4.SS1.p1.17.m16.1.1">subscript</csymbol><ci id="S4.SS1.p1.17.m16.1.1.2.cmml" xref="S4.SS1.p1.17.m16.1.1.2">𝑉</ci><ci id="S4.SS1.p1.17.m16.1.1.3.cmml" xref="S4.SS1.p1.17.m16.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.17.m16.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.17.m16.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> with parameter <math alttext="\lambda_{V}" class="ltx_Math" display="inline" id="S4.SS1.p1.18.m17.1"><semantics id="S4.SS1.p1.18.m17.1a"><msub id="S4.SS1.p1.18.m17.1.1" xref="S4.SS1.p1.18.m17.1.1.cmml"><mi id="S4.SS1.p1.18.m17.1.1.2" xref="S4.SS1.p1.18.m17.1.1.2.cmml">λ</mi><mi id="S4.SS1.p1.18.m17.1.1.3" xref="S4.SS1.p1.18.m17.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.18.m17.1b"><apply id="S4.SS1.p1.18.m17.1.1.cmml" xref="S4.SS1.p1.18.m17.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.18.m17.1.1.1.cmml" xref="S4.SS1.p1.18.m17.1.1">subscript</csymbol><ci id="S4.SS1.p1.18.m17.1.1.2.cmml" xref="S4.SS1.p1.18.m17.1.1.2">𝜆</ci><ci id="S4.SS1.p1.18.m17.1.1.3.cmml" xref="S4.SS1.p1.18.m17.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.18.m17.1c">\lambda_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.18.m17.1d">italic_λ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="\lambda_{V}=0.5" class="ltx_Math" display="inline" id="S4.SS1.p1.19.m18.1"><semantics id="S4.SS1.p1.19.m18.1a"><mrow id="S4.SS1.p1.19.m18.1.1" xref="S4.SS1.p1.19.m18.1.1.cmml"><msub id="S4.SS1.p1.19.m18.1.1.2" xref="S4.SS1.p1.19.m18.1.1.2.cmml"><mi id="S4.SS1.p1.19.m18.1.1.2.2" xref="S4.SS1.p1.19.m18.1.1.2.2.cmml">λ</mi><mi id="S4.SS1.p1.19.m18.1.1.2.3" xref="S4.SS1.p1.19.m18.1.1.2.3.cmml">V</mi></msub><mo id="S4.SS1.p1.19.m18.1.1.1" xref="S4.SS1.p1.19.m18.1.1.1.cmml">=</mo><mn id="S4.SS1.p1.19.m18.1.1.3" xref="S4.SS1.p1.19.m18.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.19.m18.1b"><apply id="S4.SS1.p1.19.m18.1.1.cmml" xref="S4.SS1.p1.19.m18.1.1"><eq id="S4.SS1.p1.19.m18.1.1.1.cmml" xref="S4.SS1.p1.19.m18.1.1.1"></eq><apply id="S4.SS1.p1.19.m18.1.1.2.cmml" xref="S4.SS1.p1.19.m18.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.19.m18.1.1.2.1.cmml" xref="S4.SS1.p1.19.m18.1.1.2">subscript</csymbol><ci id="S4.SS1.p1.19.m18.1.1.2.2.cmml" xref="S4.SS1.p1.19.m18.1.1.2.2">𝜆</ci><ci id="S4.SS1.p1.19.m18.1.1.2.3.cmml" xref="S4.SS1.p1.19.m18.1.1.2.3">𝑉</ci></apply><cn id="S4.SS1.p1.19.m18.1.1.3.cmml" type="float" xref="S4.SS1.p1.19.m18.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.19.m18.1c">\lambda_{V}=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.19.m18.1d">italic_λ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 0.5</annotation></semantics></math> for all simulations). The longest axis <math alttext="l_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.20.m19.1"><semantics id="S4.SS1.p1.20.m19.1a"><msub id="S4.SS1.p1.20.m19.1.1" xref="S4.SS1.p1.20.m19.1.1.cmml"><mi id="S4.SS1.p1.20.m19.1.1.2" xref="S4.SS1.p1.20.m19.1.1.2.cmml">l</mi><mi id="S4.SS1.p1.20.m19.1.1.3" xref="S4.SS1.p1.20.m19.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.20.m19.1b"><apply id="S4.SS1.p1.20.m19.1.1.cmml" xref="S4.SS1.p1.20.m19.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.20.m19.1.1.1.cmml" xref="S4.SS1.p1.20.m19.1.1">subscript</csymbol><ci id="S4.SS1.p1.20.m19.1.1.2.cmml" xref="S4.SS1.p1.20.m19.1.1.2">𝑙</ci><ci id="S4.SS1.p1.20.m19.1.1.3.cmml" xref="S4.SS1.p1.20.m19.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.20.m19.1c">l_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.20.m19.1d">italic_l start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> (in pixels) of each cell is constrained to length <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.21.m20.1"><semantics id="S4.SS1.p1.21.m20.1a"><msub id="S4.SS1.p1.21.m20.1.1" xref="S4.SS1.p1.21.m20.1.1.cmml"><mi id="S4.SS1.p1.21.m20.1.1.2" xref="S4.SS1.p1.21.m20.1.1.2.cmml">L</mi><mi id="S4.SS1.p1.21.m20.1.1.3" xref="S4.SS1.p1.21.m20.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.21.m20.1b"><apply id="S4.SS1.p1.21.m20.1.1.cmml" xref="S4.SS1.p1.21.m20.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.21.m20.1.1.1.cmml" xref="S4.SS1.p1.21.m20.1.1">subscript</csymbol><ci id="S4.SS1.p1.21.m20.1.1.2.cmml" xref="S4.SS1.p1.21.m20.1.1.2">𝐿</ci><ci id="S4.SS1.p1.21.m20.1.1.3.cmml" xref="S4.SS1.p1.21.m20.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.21.m20.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.21.m20.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\lambda^{\sigma}_{L}" class="ltx_Math" display="inline" id="S4.SS1.p1.22.m21.1"><semantics id="S4.SS1.p1.22.m21.1a"><msubsup id="S4.SS1.p1.22.m21.1.1" xref="S4.SS1.p1.22.m21.1.1.cmml"><mi id="S4.SS1.p1.22.m21.1.1.2.2" xref="S4.SS1.p1.22.m21.1.1.2.2.cmml">λ</mi><mi id="S4.SS1.p1.22.m21.1.1.3" xref="S4.SS1.p1.22.m21.1.1.3.cmml">L</mi><mi id="S4.SS1.p1.22.m21.1.1.2.3" xref="S4.SS1.p1.22.m21.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.22.m21.1b"><apply id="S4.SS1.p1.22.m21.1.1.cmml" xref="S4.SS1.p1.22.m21.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.22.m21.1.1.1.cmml" xref="S4.SS1.p1.22.m21.1.1">subscript</csymbol><apply id="S4.SS1.p1.22.m21.1.1.2.cmml" xref="S4.SS1.p1.22.m21.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.22.m21.1.1.2.1.cmml" xref="S4.SS1.p1.22.m21.1.1">superscript</csymbol><ci id="S4.SS1.p1.22.m21.1.1.2.2.cmml" xref="S4.SS1.p1.22.m21.1.1.2.2">𝜆</ci><ci id="S4.SS1.p1.22.m21.1.1.2.3.cmml" xref="S4.SS1.p1.22.m21.1.1.2.3">𝜎</ci></apply><ci id="S4.SS1.p1.22.m21.1.1.3.cmml" xref="S4.SS1.p1.22.m21.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.22.m21.1c">\lambda^{\sigma}_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.22.m21.1d">italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.23.m22.1"><semantics id="S4.SS1.p1.23.m22.1a"><msub id="S4.SS1.p1.23.m22.1.1" xref="S4.SS1.p1.23.m22.1.1.cmml"><mi id="S4.SS1.p1.23.m22.1.1.2" xref="S4.SS1.p1.23.m22.1.1.2.cmml">V</mi><mi id="S4.SS1.p1.23.m22.1.1.3" xref="S4.SS1.p1.23.m22.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.23.m22.1b"><apply id="S4.SS1.p1.23.m22.1.1.cmml" xref="S4.SS1.p1.23.m22.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.23.m22.1.1.1.cmml" xref="S4.SS1.p1.23.m22.1.1">subscript</csymbol><ci id="S4.SS1.p1.23.m22.1.1.2.cmml" xref="S4.SS1.p1.23.m22.1.1.2">𝑉</ci><ci id="S4.SS1.p1.23.m22.1.1.3.cmml" xref="S4.SS1.p1.23.m22.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.23.m22.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.23.m22.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS1.p1.24.m23.1"><semantics id="S4.SS1.p1.24.m23.1a"><msub id="S4.SS1.p1.24.m23.1.1" xref="S4.SS1.p1.24.m23.1.1.cmml"><mi id="S4.SS1.p1.24.m23.1.1.2" xref="S4.SS1.p1.24.m23.1.1.2.cmml">L</mi><mi id="S4.SS1.p1.24.m23.1.1.3" xref="S4.SS1.p1.24.m23.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.24.m23.1b"><apply id="S4.SS1.p1.24.m23.1.1.cmml" xref="S4.SS1.p1.24.m23.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.24.m23.1.1.1.cmml" xref="S4.SS1.p1.24.m23.1.1">subscript</csymbol><ci id="S4.SS1.p1.24.m23.1.1.2.cmml" xref="S4.SS1.p1.24.m23.1.1.2">𝐿</ci><ci id="S4.SS1.p1.24.m23.1.1.3.cmml" xref="S4.SS1.p1.24.m23.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.24.m23.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.24.m23.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lambda^{\sigma}_{L}" class="ltx_Math" display="inline" id="S4.SS1.p1.25.m24.1"><semantics id="S4.SS1.p1.25.m24.1a"><msubsup id="S4.SS1.p1.25.m24.1.1" xref="S4.SS1.p1.25.m24.1.1.cmml"><mi id="S4.SS1.p1.25.m24.1.1.2.2" xref="S4.SS1.p1.25.m24.1.1.2.2.cmml">λ</mi><mi id="S4.SS1.p1.25.m24.1.1.3" xref="S4.SS1.p1.25.m24.1.1.3.cmml">L</mi><mi id="S4.SS1.p1.25.m24.1.1.2.3" xref="S4.SS1.p1.25.m24.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.25.m24.1b"><apply id="S4.SS1.p1.25.m24.1.1.cmml" xref="S4.SS1.p1.25.m24.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.25.m24.1.1.1.cmml" xref="S4.SS1.p1.25.m24.1.1">subscript</csymbol><apply id="S4.SS1.p1.25.m24.1.1.2.cmml" xref="S4.SS1.p1.25.m24.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.25.m24.1.1.2.1.cmml" xref="S4.SS1.p1.25.m24.1.1">superscript</csymbol><ci id="S4.SS1.p1.25.m24.1.1.2.2.cmml" xref="S4.SS1.p1.25.m24.1.1.2.2">𝜆</ci><ci id="S4.SS1.p1.25.m24.1.1.2.3.cmml" xref="S4.SS1.p1.25.m24.1.1.2.3">𝜎</ci></apply><ci id="S4.SS1.p1.25.m24.1.1.3.cmml" xref="S4.SS1.p1.25.m24.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.25.m24.1c">\lambda^{\sigma}_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.25.m24.1d">italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> are determined dynamically (explained later). </div> <div class="ltx_para" id="S4.SS1.p2"> If a pixel copy attempt increases <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_H</annotation></semantics></math>, it is accepted with probability <math alttext="e^{-\frac{\Delta H}{T}}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><msup id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">e</mi><mrow id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml"><mo id="S4.SS1.p2.2.m2.1.1.3a" xref="S4.SS1.p2.2.m2.1.1.3.cmml">−</mo><mfrac id="S4.SS1.p2.2.m2.1.1.3.2" xref="S4.SS1.p2.2.m2.1.1.3.2.cmml"><mrow id="S4.SS1.p2.2.m2.1.1.3.2.2" xref="S4.SS1.p2.2.m2.1.1.3.2.2.cmml"><mi id="S4.SS1.p2.2.m2.1.1.3.2.2.2" mathvariant="normal" xref="S4.SS1.p2.2.m2.1.1.3.2.2.2.cmml">Δ</mi><mo id="S4.SS1.p2.2.m2.1.1.3.2.2.1" xref="S4.SS1.p2.2.m2.1.1.3.2.2.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.3.2.2.3" xref="S4.SS1.p2.2.m2.1.1.3.2.2.3.cmml">H</mi></mrow><mi id="S4.SS1.p2.2.m2.1.1.3.2.3" xref="S4.SS1.p2.2.m2.1.1.3.2.3.cmml">T</mi></mfrac></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝑒</ci><apply id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3"><minus id="S4.SS1.p2.2.m2.1.1.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3"></minus><apply id="S4.SS1.p2.2.m2.1.1.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2"><divide id="S4.SS1.p2.2.m2.1.1.3.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2"></divide><apply id="S4.SS1.p2.2.m2.1.1.3.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.2"><times id="S4.SS1.p2.2.m2.1.1.3.2.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.2.1"></times><ci id="S4.SS1.p2.2.m2.1.1.3.2.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.2.2">Δ</ci><ci id="S4.SS1.p2.2.m2.1.1.3.2.2.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.2.3">𝐻</ci></apply><ci id="S4.SS1.p2.2.m2.1.1.3.2.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">e^{-\frac{\Delta H}{T}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_e start_POSTSUPERSCRIPT - divide start_ARG roman_Δ italic_H end_ARG start_ARG italic_T end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\Delta H" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mrow id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml"><mi id="S4.SS1.p2.3.m3.1.1.2" mathvariant="normal" xref="S4.SS1.p2.3.m3.1.1.2.cmml">Δ</mi><mo id="S4.SS1.p2.3.m3.1.1.1" xref="S4.SS1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.3.m3.1.1.3" xref="S4.SS1.p2.3.m3.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><apply id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1"><times id="S4.SS1.p2.3.m3.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1"></times><ci id="S4.SS1.p2.3.m3.1.1.2.cmml" xref="S4.SS1.p2.3.m3.1.1.2">Δ</ci><ci id="S4.SS1.p2.3.m3.1.1.3.cmml" xref="S4.SS1.p2.3.m3.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">\Delta H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">roman_Δ italic_H</annotation></semantics></math> is the change in <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_H</annotation></semantics></math> made by a pixel copy, and <math alttext="T" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_T</annotation></semantics></math> is a temperature parameter (<math alttext="T=3" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mrow id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml"><mi id="S4.SS1.p2.6.m6.1.1.2" xref="S4.SS1.p2.6.m6.1.1.2.cmml">T</mi><mo id="S4.SS1.p2.6.m6.1.1.1" xref="S4.SS1.p2.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.6.m6.1.1.3" xref="S4.SS1.p2.6.m6.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><apply id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1"><eq id="S4.SS1.p2.6.m6.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1"></eq><ci id="S4.SS1.p2.6.m6.1.1.2.cmml" xref="S4.SS1.p2.6.m6.1.1.2">𝑇</ci><cn id="S4.SS1.p2.6.m6.1.1.3.cmml" type="integer" xref="S4.SS1.p2.6.m6.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">T=3</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_T = 3</annotation></semantics></math> for all simulations). Otherwise (<math alttext="\Delta H\leq 0" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><mrow id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml"><mrow id="S4.SS1.p2.7.m7.1.1.2" xref="S4.SS1.p2.7.m7.1.1.2.cmml"><mi id="S4.SS1.p2.7.m7.1.1.2.2" mathvariant="normal" xref="S4.SS1.p2.7.m7.1.1.2.2.cmml">Δ</mi><mo id="S4.SS1.p2.7.m7.1.1.2.1" xref="S4.SS1.p2.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p2.7.m7.1.1.2.3" xref="S4.SS1.p2.7.m7.1.1.2.3.cmml">H</mi></mrow><mo id="S4.SS1.p2.7.m7.1.1.1" xref="S4.SS1.p2.7.m7.1.1.1.cmml">≤</mo><mn id="S4.SS1.p2.7.m7.1.1.3" xref="S4.SS1.p2.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1"><leq id="S4.SS1.p2.7.m7.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1.1"></leq><apply id="S4.SS1.p2.7.m7.1.1.2.cmml" xref="S4.SS1.p2.7.m7.1.1.2"><times id="S4.SS1.p2.7.m7.1.1.2.1.cmml" xref="S4.SS1.p2.7.m7.1.1.2.1"></times><ci id="S4.SS1.p2.7.m7.1.1.2.2.cmml" xref="S4.SS1.p2.7.m7.1.1.2.2">Δ</ci><ci id="S4.SS1.p2.7.m7.1.1.2.3.cmml" xref="S4.SS1.p2.7.m7.1.1.2.3">𝐻</ci></apply><cn id="S4.SS1.p2.7.m7.1.1.3.cmml" type="integer" xref="S4.SS1.p2.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">\Delta H\leq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">roman_Δ italic_H ≤ 0</annotation></semantics></math>), the pixel copy attempt is always accepted. One developmental time step (DTS) is complete when the number of pixel copy attempts equals the number of pixels on the grid. </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> 4.2 Development</h3> <div class="ltx_para" id="S4.SS2.p1"> An organism starts as a single cell that undergoes six divisions equally spaced in time over the first 300 DTS. During the first 1,500 DTS (referred to as an equilibration phase), <math alttext="J_{ij}=30" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><msub id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2.2" xref="S4.SS2.p1.1.m1.1.1.2.2.cmml">J</mi><mrow id="S4.SS2.p1.1.m1.1.1.2.3" xref="S4.SS2.p1.1.m1.1.1.2.3.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2.3.2" xref="S4.SS2.p1.1.m1.1.1.2.3.2.cmml">i</mi><mo id="S4.SS2.p1.1.m1.1.1.2.3.1" xref="S4.SS2.p1.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.1.m1.1.1.2.3.3" xref="S4.SS2.p1.1.m1.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><eq id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1"></eq><apply id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.2.1.cmml" xref="S4.SS2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2.2">𝐽</ci><apply id="S4.SS2.p1.1.m1.1.1.2.3.cmml" xref="S4.SS2.p1.1.m1.1.1.2.3"><times id="S4.SS2.p1.1.m1.1.1.2.3.1.cmml" xref="S4.SS2.p1.1.m1.1.1.2.3.1"></times><ci id="S4.SS2.p1.1.m1.1.1.2.3.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2.3.2">𝑖</ci><ci id="S4.SS2.p1.1.m1.1.1.2.3.3.cmml" xref="S4.SS2.p1.1.m1.1.1.2.3.3">𝑗</ci></apply></apply><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">J_{ij}=30</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 30</annotation></semantics></math> and <math alttext="J_{im}=40" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mrow id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><msub id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2.2" xref="S4.SS2.p1.2.m2.1.1.2.2.cmml">J</mi><mrow id="S4.SS2.p1.2.m2.1.1.2.3" xref="S4.SS2.p1.2.m2.1.1.2.3.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2.3.2" xref="S4.SS2.p1.2.m2.1.1.2.3.2.cmml">i</mi><mo id="S4.SS2.p1.2.m2.1.1.2.3.1" xref="S4.SS2.p1.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.2.m2.1.1.2.3.3" xref="S4.SS2.p1.2.m2.1.1.2.3.3.cmml">m</mi></mrow></msub><mo id="S4.SS2.p1.2.m2.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml">40</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><eq id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></eq><apply id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.2.1.cmml" xref="S4.SS2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2.2">𝐽</ci><apply id="S4.SS2.p1.2.m2.1.1.2.3.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3"><times id="S4.SS2.p1.2.m2.1.1.2.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3.1"></times><ci id="S4.SS2.p1.2.m2.1.1.2.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3.2">𝑖</ci><ci id="S4.SS2.p1.2.m2.1.1.2.3.3.cmml" xref="S4.SS2.p1.2.m2.1.1.2.3.3">𝑚</ci></apply></apply><cn id="S4.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3">40</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">J_{im}=40</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = 40</annotation></semantics></math> for all cells. After the equilibration phase, <math alttext="J_{ij}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><msub id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml">J</mi><mrow id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml"><mi id="S4.SS2.p1.3.m3.1.1.3.2" xref="S4.SS2.p1.3.m3.1.1.3.2.cmml">i</mi><mo id="S4.SS2.p1.3.m3.1.1.3.1" xref="S4.SS2.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.3.m3.1.1.3.3" xref="S4.SS2.p1.3.m3.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2">𝐽</ci><apply id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3"><times id="S4.SS2.p1.3.m3.1.1.3.1.cmml" xref="S4.SS2.p1.3.m3.1.1.3.1"></times><ci id="S4.SS2.p1.3.m3.1.1.3.2.cmml" xref="S4.SS2.p1.3.m3.1.1.3.2">𝑖</ci><ci id="S4.SS2.p1.3.m3.1.1.3.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">J_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="J_{im}" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><msub id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">J</mi><mrow id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.3.2" xref="S4.SS2.p1.4.m4.1.1.3.2.cmml">i</mi><mo id="S4.SS2.p1.4.m4.1.1.3.1" xref="S4.SS2.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.4.m4.1.1.3.3" xref="S4.SS2.p1.4.m4.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2">𝐽</ci><apply id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3"><times id="S4.SS2.p1.4.m4.1.1.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3.1"></times><ci id="S4.SS2.p1.4.m4.1.1.3.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2">𝑖</ci><ci id="S4.SS2.p1.4.m4.1.1.3.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">J_{im}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT</annotation></semantics></math> are determined by protein concentrations in the cells (described later). At the beginning of the equilibration phase, <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><msub id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.p1.5.m5.1.1.2" xref="S4.SS2.p1.5.m5.1.1.2.cmml">V</mi><mi id="S4.SS2.p1.5.m5.1.1.3" xref="S4.SS2.p1.5.m5.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><apply id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2">𝑉</ci><ci id="S4.SS2.p1.5.m5.1.1.3.cmml" xref="S4.SS2.p1.5.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is set to the initial size of each of the 64 cells and does not change throughout the equilibration phase. After the equilibration phase, cell growth and shrinkage can occur through modulation of the target cell size, <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.1"><semantics id="S4.SS2.p1.6.m6.1a"><msub id="S4.SS2.p1.6.m6.1.1" xref="S4.SS2.p1.6.m6.1.1.cmml"><mi id="S4.SS2.p1.6.m6.1.1.2" xref="S4.SS2.p1.6.m6.1.1.2.cmml">V</mi><mi id="S4.SS2.p1.6.m6.1.1.3" xref="S4.SS2.p1.6.m6.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.1b"><apply id="S4.SS2.p1.6.m6.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2">𝑉</ci><ci id="S4.SS2.p1.6.m6.1.1.3.cmml" xref="S4.SS2.p1.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><msub id="S4.SS2.p1.7.m7.1.1" xref="S4.SS2.p1.7.m7.1.1.cmml"><mi id="S4.SS2.p1.7.m7.1.1.2" xref="S4.SS2.p1.7.m7.1.1.2.cmml">V</mi><mi id="S4.SS2.p1.7.m7.1.1.3" xref="S4.SS2.p1.7.m7.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2">𝑉</ci><ci id="S4.SS2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.p1.7.m7.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> increases when a cell is stretched, and decreases when a cell is squeezed, as follows. When <math alttext="\upsilon_{\sigma}\geq V_{\sigma}+3" class="ltx_Math" display="inline" id="S4.SS2.p1.8.m8.1"><semantics id="S4.SS2.p1.8.m8.1a"><mrow id="S4.SS2.p1.8.m8.1.1" xref="S4.SS2.p1.8.m8.1.1.cmml"><msub id="S4.SS2.p1.8.m8.1.1.2" xref="S4.SS2.p1.8.m8.1.1.2.cmml"><mi id="S4.SS2.p1.8.m8.1.1.2.2" xref="S4.SS2.p1.8.m8.1.1.2.2.cmml">υ</mi><mi id="S4.SS2.p1.8.m8.1.1.2.3" xref="S4.SS2.p1.8.m8.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.8.m8.1.1.1" xref="S4.SS2.p1.8.m8.1.1.1.cmml">≥</mo><mrow id="S4.SS2.p1.8.m8.1.1.3" xref="S4.SS2.p1.8.m8.1.1.3.cmml"><msub id="S4.SS2.p1.8.m8.1.1.3.2" xref="S4.SS2.p1.8.m8.1.1.3.2.cmml"><mi id="S4.SS2.p1.8.m8.1.1.3.2.2" xref="S4.SS2.p1.8.m8.1.1.3.2.2.cmml">V</mi><mi id="S4.SS2.p1.8.m8.1.1.3.2.3" xref="S4.SS2.p1.8.m8.1.1.3.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.8.m8.1.1.3.1" xref="S4.SS2.p1.8.m8.1.1.3.1.cmml">+</mo><mn id="S4.SS2.p1.8.m8.1.1.3.3" xref="S4.SS2.p1.8.m8.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.8.m8.1b"><apply id="S4.SS2.p1.8.m8.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1"><geq id="S4.SS2.p1.8.m8.1.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1.1"></geq><apply id="S4.SS2.p1.8.m8.1.1.2.cmml" xref="S4.SS2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.8.m8.1.1.2.1.cmml" xref="S4.SS2.p1.8.m8.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.8.m8.1.1.2.2.cmml" xref="S4.SS2.p1.8.m8.1.1.2.2">𝜐</ci><ci id="S4.SS2.p1.8.m8.1.1.2.3.cmml" xref="S4.SS2.p1.8.m8.1.1.2.3">𝜎</ci></apply><apply id="S4.SS2.p1.8.m8.1.1.3.cmml" xref="S4.SS2.p1.8.m8.1.1.3"><plus id="S4.SS2.p1.8.m8.1.1.3.1.cmml" xref="S4.SS2.p1.8.m8.1.1.3.1"></plus><apply id="S4.SS2.p1.8.m8.1.1.3.2.cmml" xref="S4.SS2.p1.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.8.m8.1.1.3.2.1.cmml" xref="S4.SS2.p1.8.m8.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p1.8.m8.1.1.3.2.2.cmml" xref="S4.SS2.p1.8.m8.1.1.3.2.2">𝑉</ci><ci id="S4.SS2.p1.8.m8.1.1.3.2.3.cmml" xref="S4.SS2.p1.8.m8.1.1.3.2.3">𝜎</ci></apply><cn id="S4.SS2.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S4.SS2.p1.8.m8.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.8.m8.1c">\upsilon_{\sigma}\geq V_{\sigma}+3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.8.m8.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ≥ italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT + 3</annotation></semantics></math> , <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.9.m9.1"><semantics id="S4.SS2.p1.9.m9.1a"><msub id="S4.SS2.p1.9.m9.1.1" xref="S4.SS2.p1.9.m9.1.1.cmml"><mi id="S4.SS2.p1.9.m9.1.1.2" xref="S4.SS2.p1.9.m9.1.1.2.cmml">V</mi><mi id="S4.SS2.p1.9.m9.1.1.3" xref="S4.SS2.p1.9.m9.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.9.m9.1b"><apply id="S4.SS2.p1.9.m9.1.1.cmml" xref="S4.SS2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.9.m9.1.1.1.cmml" xref="S4.SS2.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.p1.9.m9.1.1.2.cmml" xref="S4.SS2.p1.9.m9.1.1.2">𝑉</ci><ci id="S4.SS2.p1.9.m9.1.1.3.cmml" xref="S4.SS2.p1.9.m9.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.9.m9.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.9.m9.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is updated such that <math alttext="V_{\sigma}=\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.10.m10.1"><semantics id="S4.SS2.p1.10.m10.1a"><mrow id="S4.SS2.p1.10.m10.1.1" xref="S4.SS2.p1.10.m10.1.1.cmml"><msub id="S4.SS2.p1.10.m10.1.1.2" xref="S4.SS2.p1.10.m10.1.1.2.cmml"><mi id="S4.SS2.p1.10.m10.1.1.2.2" xref="S4.SS2.p1.10.m10.1.1.2.2.cmml">V</mi><mi id="S4.SS2.p1.10.m10.1.1.2.3" xref="S4.SS2.p1.10.m10.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.10.m10.1.1.1" xref="S4.SS2.p1.10.m10.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.10.m10.1.1.3" xref="S4.SS2.p1.10.m10.1.1.3.cmml"><mi id="S4.SS2.p1.10.m10.1.1.3.2" xref="S4.SS2.p1.10.m10.1.1.3.2.cmml">υ</mi><mi id="S4.SS2.p1.10.m10.1.1.3.3" xref="S4.SS2.p1.10.m10.1.1.3.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.10.m10.1b"><apply id="S4.SS2.p1.10.m10.1.1.cmml" xref="S4.SS2.p1.10.m10.1.1"><eq id="S4.SS2.p1.10.m10.1.1.1.cmml" xref="S4.SS2.p1.10.m10.1.1.1"></eq><apply id="S4.SS2.p1.10.m10.1.1.2.cmml" xref="S4.SS2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.10.m10.1.1.2.1.cmml" xref="S4.SS2.p1.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.10.m10.1.1.2.2.cmml" xref="S4.SS2.p1.10.m10.1.1.2.2">𝑉</ci><ci id="S4.SS2.p1.10.m10.1.1.2.3.cmml" xref="S4.SS2.p1.10.m10.1.1.2.3">𝜎</ci></apply><apply id="S4.SS2.p1.10.m10.1.1.3.cmml" xref="S4.SS2.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.10.m10.1.1.3.1.cmml" xref="S4.SS2.p1.10.m10.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.10.m10.1.1.3.2.cmml" xref="S4.SS2.p1.10.m10.1.1.3.2">𝜐</ci><ci id="S4.SS2.p1.10.m10.1.1.3.3.cmml" xref="S4.SS2.p1.10.m10.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.10.m10.1c">V_{\sigma}=\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.10.m10.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. When <math alttext="\upsilon_{\sigma}\leq V_{\sigma}-16" class="ltx_Math" display="inline" id="S4.SS2.p1.11.m11.1"><semantics id="S4.SS2.p1.11.m11.1a"><mrow id="S4.SS2.p1.11.m11.1.1" xref="S4.SS2.p1.11.m11.1.1.cmml"><msub id="S4.SS2.p1.11.m11.1.1.2" xref="S4.SS2.p1.11.m11.1.1.2.cmml"><mi id="S4.SS2.p1.11.m11.1.1.2.2" xref="S4.SS2.p1.11.m11.1.1.2.2.cmml">υ</mi><mi id="S4.SS2.p1.11.m11.1.1.2.3" xref="S4.SS2.p1.11.m11.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.11.m11.1.1.1" xref="S4.SS2.p1.11.m11.1.1.1.cmml">≤</mo><mrow id="S4.SS2.p1.11.m11.1.1.3" xref="S4.SS2.p1.11.m11.1.1.3.cmml"><msub id="S4.SS2.p1.11.m11.1.1.3.2" xref="S4.SS2.p1.11.m11.1.1.3.2.cmml"><mi id="S4.SS2.p1.11.m11.1.1.3.2.2" xref="S4.SS2.p1.11.m11.1.1.3.2.2.cmml">V</mi><mi id="S4.SS2.p1.11.m11.1.1.3.2.3" xref="S4.SS2.p1.11.m11.1.1.3.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.11.m11.1.1.3.1" xref="S4.SS2.p1.11.m11.1.1.3.1.cmml">−</mo><mn id="S4.SS2.p1.11.m11.1.1.3.3" xref="S4.SS2.p1.11.m11.1.1.3.3.cmml">16</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.11.m11.1b"><apply id="S4.SS2.p1.11.m11.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1"><leq id="S4.SS2.p1.11.m11.1.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1.1"></leq><apply id="S4.SS2.p1.11.m11.1.1.2.cmml" xref="S4.SS2.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.11.m11.1.1.2.1.cmml" xref="S4.SS2.p1.11.m11.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.11.m11.1.1.2.2.cmml" xref="S4.SS2.p1.11.m11.1.1.2.2">𝜐</ci><ci id="S4.SS2.p1.11.m11.1.1.2.3.cmml" xref="S4.SS2.p1.11.m11.1.1.2.3">𝜎</ci></apply><apply id="S4.SS2.p1.11.m11.1.1.3.cmml" xref="S4.SS2.p1.11.m11.1.1.3"><minus id="S4.SS2.p1.11.m11.1.1.3.1.cmml" xref="S4.SS2.p1.11.m11.1.1.3.1"></minus><apply id="S4.SS2.p1.11.m11.1.1.3.2.cmml" xref="S4.SS2.p1.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.11.m11.1.1.3.2.1.cmml" xref="S4.SS2.p1.11.m11.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p1.11.m11.1.1.3.2.2.cmml" xref="S4.SS2.p1.11.m11.1.1.3.2.2">𝑉</ci><ci id="S4.SS2.p1.11.m11.1.1.3.2.3.cmml" xref="S4.SS2.p1.11.m11.1.1.3.2.3">𝜎</ci></apply><cn id="S4.SS2.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S4.SS2.p1.11.m11.1.1.3.3">16</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.11.m11.1c">\upsilon_{\sigma}\leq V_{\sigma}-16</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.11.m11.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ≤ italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - 16</annotation></semantics></math>, <math alttext="V_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.12.m12.1"><semantics id="S4.SS2.p1.12.m12.1a"><msub id="S4.SS2.p1.12.m12.1.1" xref="S4.SS2.p1.12.m12.1.1.cmml"><mi id="S4.SS2.p1.12.m12.1.1.2" xref="S4.SS2.p1.12.m12.1.1.2.cmml">V</mi><mi id="S4.SS2.p1.12.m12.1.1.3" xref="S4.SS2.p1.12.m12.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.12.m12.1b"><apply id="S4.SS2.p1.12.m12.1.1.cmml" xref="S4.SS2.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.12.m12.1.1.1.cmml" xref="S4.SS2.p1.12.m12.1.1">subscript</csymbol><ci id="S4.SS2.p1.12.m12.1.1.2.cmml" xref="S4.SS2.p1.12.m12.1.1.2">𝑉</ci><ci id="S4.SS2.p1.12.m12.1.1.3.cmml" xref="S4.SS2.p1.12.m12.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.12.m12.1c">V_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.12.m12.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is updated such that <math alttext="V_{\sigma}=\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.13.m13.1"><semantics id="S4.SS2.p1.13.m13.1a"><mrow id="S4.SS2.p1.13.m13.1.1" xref="S4.SS2.p1.13.m13.1.1.cmml"><msub id="S4.SS2.p1.13.m13.1.1.2" xref="S4.SS2.p1.13.m13.1.1.2.cmml"><mi id="S4.SS2.p1.13.m13.1.1.2.2" xref="S4.SS2.p1.13.m13.1.1.2.2.cmml">V</mi><mi id="S4.SS2.p1.13.m13.1.1.2.3" xref="S4.SS2.p1.13.m13.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.13.m13.1.1.1" xref="S4.SS2.p1.13.m13.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.13.m13.1.1.3" xref="S4.SS2.p1.13.m13.1.1.3.cmml"><mi id="S4.SS2.p1.13.m13.1.1.3.2" xref="S4.SS2.p1.13.m13.1.1.3.2.cmml">υ</mi><mi id="S4.SS2.p1.13.m13.1.1.3.3" xref="S4.SS2.p1.13.m13.1.1.3.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.13.m13.1b"><apply id="S4.SS2.p1.13.m13.1.1.cmml" xref="S4.SS2.p1.13.m13.1.1"><eq id="S4.SS2.p1.13.m13.1.1.1.cmml" xref="S4.SS2.p1.13.m13.1.1.1"></eq><apply id="S4.SS2.p1.13.m13.1.1.2.cmml" xref="S4.SS2.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.13.m13.1.1.2.1.cmml" xref="S4.SS2.p1.13.m13.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.13.m13.1.1.2.2.cmml" xref="S4.SS2.p1.13.m13.1.1.2.2">𝑉</ci><ci id="S4.SS2.p1.13.m13.1.1.2.3.cmml" xref="S4.SS2.p1.13.m13.1.1.2.3">𝜎</ci></apply><apply id="S4.SS2.p1.13.m13.1.1.3.cmml" xref="S4.SS2.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.13.m13.1.1.3.1.cmml" xref="S4.SS2.p1.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.13.m13.1.1.3.2.cmml" xref="S4.SS2.p1.13.m13.1.1.3.2">𝜐</ci><ci id="S4.SS2.p1.13.m13.1.1.3.3.cmml" xref="S4.SS2.p1.13.m13.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.13.m13.1c">V_{\sigma}=\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.13.m13.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. When the cell volume reaches or exceeds a threshold (i.e., <math alttext="\upsilon_{\sigma}\geq 100" class="ltx_Math" display="inline" id="S4.SS2.p1.14.m14.1"><semantics id="S4.SS2.p1.14.m14.1a"><mrow id="S4.SS2.p1.14.m14.1.1" xref="S4.SS2.p1.14.m14.1.1.cmml"><msub id="S4.SS2.p1.14.m14.1.1.2" xref="S4.SS2.p1.14.m14.1.1.2.cmml"><mi id="S4.SS2.p1.14.m14.1.1.2.2" xref="S4.SS2.p1.14.m14.1.1.2.2.cmml">υ</mi><mi id="S4.SS2.p1.14.m14.1.1.2.3" xref="S4.SS2.p1.14.m14.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.14.m14.1.1.1" xref="S4.SS2.p1.14.m14.1.1.1.cmml">≥</mo><mn id="S4.SS2.p1.14.m14.1.1.3" xref="S4.SS2.p1.14.m14.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.14.m14.1b"><apply id="S4.SS2.p1.14.m14.1.1.cmml" xref="S4.SS2.p1.14.m14.1.1"><geq id="S4.SS2.p1.14.m14.1.1.1.cmml" xref="S4.SS2.p1.14.m14.1.1.1"></geq><apply id="S4.SS2.p1.14.m14.1.1.2.cmml" xref="S4.SS2.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.14.m14.1.1.2.1.cmml" xref="S4.SS2.p1.14.m14.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.14.m14.1.1.2.2.cmml" xref="S4.SS2.p1.14.m14.1.1.2.2">𝜐</ci><ci id="S4.SS2.p1.14.m14.1.1.2.3.cmml" xref="S4.SS2.p1.14.m14.1.1.2.3">𝜎</ci></apply><cn id="S4.SS2.p1.14.m14.1.1.3.cmml" type="integer" xref="S4.SS2.p1.14.m14.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.14.m14.1c">\upsilon_{\sigma}\geq 100</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.14.m14.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ≥ 100</annotation></semantics></math>), the cell undergoes division perpendicular to its longest axis. One daughter cell retains the same index <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p1.15.m15.1"><semantics id="S4.SS2.p1.15.m15.1a"><mi id="S4.SS2.p1.15.m15.1.1" xref="S4.SS2.p1.15.m15.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.15.m15.1b"><ci id="S4.SS2.p1.15.m15.1.1.cmml" xref="S4.SS2.p1.15.m15.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.15.m15.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.15.m15.1d">italic_σ</annotation></semantics></math> as the mother cell, while the other daughter cell is assigned a new index <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p1.16.m16.1"><semantics id="S4.SS2.p1.16.m16.1a"><msup id="S4.SS2.p1.16.m16.1.1" xref="S4.SS2.p1.16.m16.1.1.cmml"><mi id="S4.SS2.p1.16.m16.1.1.2" xref="S4.SS2.p1.16.m16.1.1.2.cmml">σ</mi><mo id="S4.SS2.p1.16.m16.1.1.3" xref="S4.SS2.p1.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.16.m16.1b"><apply id="S4.SS2.p1.16.m16.1.1.cmml" xref="S4.SS2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.16.m16.1.1.1.cmml" xref="S4.SS2.p1.16.m16.1.1">superscript</csymbol><ci id="S4.SS2.p1.16.m16.1.1.2.cmml" xref="S4.SS2.p1.16.m16.1.1.2">𝜎</ci><ci id="S4.SS2.p1.16.m16.1.1.3.cmml" xref="S4.SS2.p1.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.16.m16.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.16.m16.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. After division, the target area of the daughter cells are <math alttext="V_{\sigma}=\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS2.p1.17.m17.1"><semantics id="S4.SS2.p1.17.m17.1a"><mrow id="S4.SS2.p1.17.m17.1.1" xref="S4.SS2.p1.17.m17.1.1.cmml"><msub id="S4.SS2.p1.17.m17.1.1.2" xref="S4.SS2.p1.17.m17.1.1.2.cmml"><mi id="S4.SS2.p1.17.m17.1.1.2.2" xref="S4.SS2.p1.17.m17.1.1.2.2.cmml">V</mi><mi id="S4.SS2.p1.17.m17.1.1.2.3" xref="S4.SS2.p1.17.m17.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.17.m17.1.1.1" xref="S4.SS2.p1.17.m17.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.17.m17.1.1.3" xref="S4.SS2.p1.17.m17.1.1.3.cmml"><mi id="S4.SS2.p1.17.m17.1.1.3.2" xref="S4.SS2.p1.17.m17.1.1.3.2.cmml">υ</mi><mi id="S4.SS2.p1.17.m17.1.1.3.3" xref="S4.SS2.p1.17.m17.1.1.3.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.17.m17.1b"><apply id="S4.SS2.p1.17.m17.1.1.cmml" xref="S4.SS2.p1.17.m17.1.1"><eq id="S4.SS2.p1.17.m17.1.1.1.cmml" xref="S4.SS2.p1.17.m17.1.1.1"></eq><apply id="S4.SS2.p1.17.m17.1.1.2.cmml" xref="S4.SS2.p1.17.m17.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.17.m17.1.1.2.1.cmml" xref="S4.SS2.p1.17.m17.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.17.m17.1.1.2.2.cmml" xref="S4.SS2.p1.17.m17.1.1.2.2">𝑉</ci><ci id="S4.SS2.p1.17.m17.1.1.2.3.cmml" xref="S4.SS2.p1.17.m17.1.1.2.3">𝜎</ci></apply><apply id="S4.SS2.p1.17.m17.1.1.3.cmml" xref="S4.SS2.p1.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.17.m17.1.1.3.1.cmml" xref="S4.SS2.p1.17.m17.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.17.m17.1.1.3.2.cmml" xref="S4.SS2.p1.17.m17.1.1.3.2">𝜐</ci><ci id="S4.SS2.p1.17.m17.1.1.3.3.cmml" xref="S4.SS2.p1.17.m17.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.17.m17.1c">V_{\sigma}=\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.17.m17.1d">italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="V_{\sigma^{\prime}}=\upsilon_{\sigma^{\prime}}" class="ltx_Math" display="inline" id="S4.SS2.p1.18.m18.1"><semantics id="S4.SS2.p1.18.m18.1a"><mrow id="S4.SS2.p1.18.m18.1.1" xref="S4.SS2.p1.18.m18.1.1.cmml"><msub id="S4.SS2.p1.18.m18.1.1.2" xref="S4.SS2.p1.18.m18.1.1.2.cmml"><mi id="S4.SS2.p1.18.m18.1.1.2.2" xref="S4.SS2.p1.18.m18.1.1.2.2.cmml">V</mi><msup id="S4.SS2.p1.18.m18.1.1.2.3" xref="S4.SS2.p1.18.m18.1.1.2.3.cmml"><mi id="S4.SS2.p1.18.m18.1.1.2.3.2" xref="S4.SS2.p1.18.m18.1.1.2.3.2.cmml">σ</mi><mo id="S4.SS2.p1.18.m18.1.1.2.3.3" xref="S4.SS2.p1.18.m18.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S4.SS2.p1.18.m18.1.1.1" xref="S4.SS2.p1.18.m18.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.18.m18.1.1.3" xref="S4.SS2.p1.18.m18.1.1.3.cmml"><mi id="S4.SS2.p1.18.m18.1.1.3.2" xref="S4.SS2.p1.18.m18.1.1.3.2.cmml">υ</mi><msup id="S4.SS2.p1.18.m18.1.1.3.3" xref="S4.SS2.p1.18.m18.1.1.3.3.cmml"><mi id="S4.SS2.p1.18.m18.1.1.3.3.2" xref="S4.SS2.p1.18.m18.1.1.3.3.2.cmml">σ</mi><mo id="S4.SS2.p1.18.m18.1.1.3.3.3" xref="S4.SS2.p1.18.m18.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.18.m18.1b"><apply id="S4.SS2.p1.18.m18.1.1.cmml" xref="S4.SS2.p1.18.m18.1.1"><eq id="S4.SS2.p1.18.m18.1.1.1.cmml" xref="S4.SS2.p1.18.m18.1.1.1"></eq><apply id="S4.SS2.p1.18.m18.1.1.2.cmml" xref="S4.SS2.p1.18.m18.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m18.1.1.2.1.cmml" xref="S4.SS2.p1.18.m18.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.18.m18.1.1.2.2.cmml" xref="S4.SS2.p1.18.m18.1.1.2.2">𝑉</ci><apply id="S4.SS2.p1.18.m18.1.1.2.3.cmml" xref="S4.SS2.p1.18.m18.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m18.1.1.2.3.1.cmml" xref="S4.SS2.p1.18.m18.1.1.2.3">superscript</csymbol><ci id="S4.SS2.p1.18.m18.1.1.2.3.2.cmml" xref="S4.SS2.p1.18.m18.1.1.2.3.2">𝜎</ci><ci id="S4.SS2.p1.18.m18.1.1.2.3.3.cmml" xref="S4.SS2.p1.18.m18.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS2.p1.18.m18.1.1.3.cmml" xref="S4.SS2.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m18.1.1.3.1.cmml" xref="S4.SS2.p1.18.m18.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.18.m18.1.1.3.2.cmml" xref="S4.SS2.p1.18.m18.1.1.3.2">𝜐</ci><apply id="S4.SS2.p1.18.m18.1.1.3.3.cmml" xref="S4.SS2.p1.18.m18.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m18.1.1.3.3.1.cmml" xref="S4.SS2.p1.18.m18.1.1.3.3">superscript</csymbol><ci id="S4.SS2.p1.18.m18.1.1.3.3.2.cmml" xref="S4.SS2.p1.18.m18.1.1.3.3.2">𝜎</ci><ci id="S4.SS2.p1.18.m18.1.1.3.3.3.cmml" xref="S4.SS2.p1.18.m18.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.18.m18.1c">V_{\sigma^{\prime}}=\upsilon_{\sigma^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.18.m18.1d">italic_V start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_υ start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Protein concentrations remain unchanged during upon cell division. If <math alttext="\upsilon_{\sigma}=0" class="ltx_Math" display="inline" id="S4.SS2.p1.19.m19.1"><semantics id="S4.SS2.p1.19.m19.1a"><mrow id="S4.SS2.p1.19.m19.1.1" xref="S4.SS2.p1.19.m19.1.1.cmml"><msub id="S4.SS2.p1.19.m19.1.1.2" xref="S4.SS2.p1.19.m19.1.1.2.cmml"><mi id="S4.SS2.p1.19.m19.1.1.2.2" xref="S4.SS2.p1.19.m19.1.1.2.2.cmml">υ</mi><mi id="S4.SS2.p1.19.m19.1.1.2.3" xref="S4.SS2.p1.19.m19.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS2.p1.19.m19.1.1.1" xref="S4.SS2.p1.19.m19.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.19.m19.1.1.3" xref="S4.SS2.p1.19.m19.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.19.m19.1b"><apply id="S4.SS2.p1.19.m19.1.1.cmml" xref="S4.SS2.p1.19.m19.1.1"><eq id="S4.SS2.p1.19.m19.1.1.1.cmml" xref="S4.SS2.p1.19.m19.1.1.1"></eq><apply id="S4.SS2.p1.19.m19.1.1.2.cmml" xref="S4.SS2.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.19.m19.1.1.2.1.cmml" xref="S4.SS2.p1.19.m19.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.19.m19.1.1.2.2.cmml" xref="S4.SS2.p1.19.m19.1.1.2.2">𝜐</ci><ci id="S4.SS2.p1.19.m19.1.1.2.3.cmml" xref="S4.SS2.p1.19.m19.1.1.2.3">𝜎</ci></apply><cn id="S4.SS2.p1.19.m19.1.1.3.cmml" type="integer" xref="S4.SS2.p1.19.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.19.m19.1c">\upsilon_{\sigma}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.19.m19.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = 0</annotation></semantics></math>, the cell dies. </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> 4.3 Gene regulatory network and morphogens</h3> <div class="ltx_para" id="S4.SS3.p1"> In each cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mi id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><ci id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_σ</annotation></semantics></math>, there are <math alttext="N" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mi id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">italic_N</annotation></semantics></math> genes, indexed by <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mi id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><ci id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">italic_p</annotation></semantics></math>, that each encode a protein whose intracellular concentration is denoted by <math alttext="x_{p}^{\sigma}" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><msubsup id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml"><mi id="S4.SS3.p1.4.m4.1.1.2.2" xref="S4.SS3.p1.4.m4.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p1.4.m4.1.1.2.3" xref="S4.SS3.p1.4.m4.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p1.4.m4.1.1.3" xref="S4.SS3.p1.4.m4.1.1.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1">superscript</csymbol><apply id="S4.SS3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.1.2.1.cmml" xref="S4.SS3.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p1.4.m4.1.1.2.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2.2">𝑥</ci><ci id="S4.SS3.p1.4.m4.1.1.2.3.cmml" xref="S4.SS3.p1.4.m4.1.1.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.4.m4.1.1.3.cmml" xref="S4.SS3.p1.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">x_{p}^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math>. With the exception of <math alttext="p=1,2,3" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.3"><semantics id="S4.SS3.p1.5.m5.3a"><mrow id="S4.SS3.p1.5.m5.3.4" xref="S4.SS3.p1.5.m5.3.4.cmml"><mi id="S4.SS3.p1.5.m5.3.4.2" xref="S4.SS3.p1.5.m5.3.4.2.cmml">p</mi><mo id="S4.SS3.p1.5.m5.3.4.1" xref="S4.SS3.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S4.SS3.p1.5.m5.3.4.3.2" xref="S4.SS3.p1.5.m5.3.4.3.1.cmml"><mn id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml">1</mn><mo id="S4.SS3.p1.5.m5.3.4.3.2.1" xref="S4.SS3.p1.5.m5.3.4.3.1.cmml">,</mo><mn id="S4.SS3.p1.5.m5.2.2" xref="S4.SS3.p1.5.m5.2.2.cmml">2</mn><mo id="S4.SS3.p1.5.m5.3.4.3.2.2" xref="S4.SS3.p1.5.m5.3.4.3.1.cmml">,</mo><mn id="S4.SS3.p1.5.m5.3.3" xref="S4.SS3.p1.5.m5.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.3b"><apply id="S4.SS3.p1.5.m5.3.4.cmml" xref="S4.SS3.p1.5.m5.3.4"><eq id="S4.SS3.p1.5.m5.3.4.1.cmml" xref="S4.SS3.p1.5.m5.3.4.1"></eq><ci id="S4.SS3.p1.5.m5.3.4.2.cmml" xref="S4.SS3.p1.5.m5.3.4.2">𝑝</ci><list id="S4.SS3.p1.5.m5.3.4.3.1.cmml" xref="S4.SS3.p1.5.m5.3.4.3.2"><cn id="S4.SS3.p1.5.m5.1.1.cmml" type="integer" xref="S4.SS3.p1.5.m5.1.1">1</cn><cn id="S4.SS3.p1.5.m5.2.2.cmml" type="integer" xref="S4.SS3.p1.5.m5.2.2">2</cn><cn id="S4.SS3.p1.5.m5.3.3.cmml" type="integer" xref="S4.SS3.p1.5.m5.3.3">3</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.3c">p=1,2,3</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.3d">italic_p = 1 , 2 , 3</annotation></semantics></math> (which are morphogens, described later), the following equation determines the change in <math alttext="x_{p}^{\sigma}" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.1"><semantics id="S4.SS3.p1.6.m6.1a"><msubsup id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml"><mi id="S4.SS3.p1.6.m6.1.1.2.2" xref="S4.SS3.p1.6.m6.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p1.6.m6.1.1.2.3" xref="S4.SS3.p1.6.m6.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p1.6.m6.1.1.3" xref="S4.SS3.p1.6.m6.1.1.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.1b"><apply id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.1.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1">superscript</csymbol><apply id="S4.SS3.p1.6.m6.1.1.2.cmml" xref="S4.SS3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.1.1.2.1.cmml" xref="S4.SS3.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS3.p1.6.m6.1.1.2.2.cmml" xref="S4.SS3.p1.6.m6.1.1.2.2">𝑥</ci><ci id="S4.SS3.p1.6.m6.1.1.2.3.cmml" xref="S4.SS3.p1.6.m6.1.1.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.6.m6.1.1.3.cmml" xref="S4.SS3.p1.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.1c">x_{p}^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> over time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><mi id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><ci id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">italic_t</annotation></semantics></math>: <table class="ltx_equation ltx_eqn_table" id="S4.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{dx_{p}^{\sigma}}{dt}=\frac{a}{1+e^{-\beta f_{p}(x)}}-bx_{p}^{\sigma}" class="ltx_Math" display="block" id="S4.E2.m1.1"><semantics id="S4.E2.m1.1a"><mrow id="S4.E2.m1.1.2" xref="S4.E2.m1.1.2.cmml"><mfrac id="S4.E2.m1.1.2.2" xref="S4.E2.m1.1.2.2.cmml"><mrow id="S4.E2.m1.1.2.2.2" xref="S4.E2.m1.1.2.2.2.cmml"><mi id="S4.E2.m1.1.2.2.2.2" xref="S4.E2.m1.1.2.2.2.2.cmml">d</mi><mo id="S4.E2.m1.1.2.2.2.1" xref="S4.E2.m1.1.2.2.2.1.cmml">⁢</mo><msubsup id="S4.E2.m1.1.2.2.2.3" xref="S4.E2.m1.1.2.2.2.3.cmml"><mi id="S4.E2.m1.1.2.2.2.3.2.2" xref="S4.E2.m1.1.2.2.2.3.2.2.cmml">x</mi><mi id="S4.E2.m1.1.2.2.2.3.2.3" xref="S4.E2.m1.1.2.2.2.3.2.3.cmml">p</mi><mi id="S4.E2.m1.1.2.2.2.3.3" xref="S4.E2.m1.1.2.2.2.3.3.cmml">σ</mi></msubsup></mrow><mrow id="S4.E2.m1.1.2.2.3" xref="S4.E2.m1.1.2.2.3.cmml"><mi id="S4.E2.m1.1.2.2.3.2" xref="S4.E2.m1.1.2.2.3.2.cmml">d</mi><mo id="S4.E2.m1.1.2.2.3.1" xref="S4.E2.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E2.m1.1.2.2.3.3" xref="S4.E2.m1.1.2.2.3.3.cmml">t</mi></mrow></mfrac><mo id="S4.E2.m1.1.2.1" xref="S4.E2.m1.1.2.1.cmml">=</mo><mrow id="S4.E2.m1.1.2.3" xref="S4.E2.m1.1.2.3.cmml"><mfrac id="S4.E2.m1.1.1" xref="S4.E2.m1.1.1.cmml"><mi id="S4.E2.m1.1.1.3" xref="S4.E2.m1.1.1.3.cmml">a</mi><mrow id="S4.E2.m1.1.1.1" xref="S4.E2.m1.1.1.1.cmml"><mn id="S4.E2.m1.1.1.1.3" xref="S4.E2.m1.1.1.1.3.cmml">1</mn><mo id="S4.E2.m1.1.1.1.2" xref="S4.E2.m1.1.1.1.2.cmml">+</mo><msup id="S4.E2.m1.1.1.1.4" xref="S4.E2.m1.1.1.1.4.cmml"><mi id="S4.E2.m1.1.1.1.4.2" xref="S4.E2.m1.1.1.1.4.2.cmml">e</mi><mrow id="S4.E2.m1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.cmml"><mo id="S4.E2.m1.1.1.1.1.1a" xref="S4.E2.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E2.m1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.3.cmml"><mi id="S4.E2.m1.1.1.1.1.1.3.2" xref="S4.E2.m1.1.1.1.1.1.3.2.cmml">β</mi><mo id="S4.E2.m1.1.1.1.1.1.3.1" xref="S4.E2.m1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.E2.m1.1.1.1.1.1.3.3" xref="S4.E2.m1.1.1.1.1.1.3.3.cmml"><mi id="S4.E2.m1.1.1.1.1.1.3.3.2" xref="S4.E2.m1.1.1.1.1.1.3.3.2.cmml">f</mi><mi id="S4.E2.m1.1.1.1.1.1.3.3.3" xref="S4.E2.m1.1.1.1.1.1.3.3.3.cmml">p</mi></msub><mo id="S4.E2.m1.1.1.1.1.1.3.1a" xref="S4.E2.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow 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italic_a end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT - italic_β italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT end_ARG - italic_b italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> </table> with one unit of <math alttext="t" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m1.1"><semantics id="S4.SS3.p1.8.m1.1a"><mi id="S4.SS3.p1.8.m1.1.1" xref="S4.SS3.p1.8.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m1.1b"><ci id="S4.SS3.p1.8.m1.1.1.cmml" xref="S4.SS3.p1.8.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m1.1d">italic_t</annotation></semantics></math> being one DTS. The first term on the right-hand side represents the increase in <math alttext="x_{p}^{\sigma}" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m2.1"><semantics id="S4.SS3.p1.9.m2.1a"><msubsup id="S4.SS3.p1.9.m2.1.1" xref="S4.SS3.p1.9.m2.1.1.cmml"><mi id="S4.SS3.p1.9.m2.1.1.2.2" xref="S4.SS3.p1.9.m2.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p1.9.m2.1.1.2.3" xref="S4.SS3.p1.9.m2.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p1.9.m2.1.1.3" xref="S4.SS3.p1.9.m2.1.1.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m2.1b"><apply id="S4.SS3.p1.9.m2.1.1.cmml" xref="S4.SS3.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.9.m2.1.1.1.cmml" xref="S4.SS3.p1.9.m2.1.1">superscript</csymbol><apply id="S4.SS3.p1.9.m2.1.1.2.cmml" xref="S4.SS3.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.9.m2.1.1.2.1.cmml" xref="S4.SS3.p1.9.m2.1.1">subscript</csymbol><ci id="S4.SS3.p1.9.m2.1.1.2.2.cmml" xref="S4.SS3.p1.9.m2.1.1.2.2">𝑥</ci><ci id="S4.SS3.p1.9.m2.1.1.2.3.cmml" xref="S4.SS3.p1.9.m2.1.1.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.9.m2.1.1.3.cmml" xref="S4.SS3.p1.9.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m2.1c">x_{p}^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m2.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> due to gene expression, and is a sigmoidal function that depends on the regulation by transcription factors (<math alttext="f_{p}(x)" class="ltx_Math" display="inline" id="S4.SS3.p1.10.m3.1"><semantics id="S4.SS3.p1.10.m3.1a"><mrow id="S4.SS3.p1.10.m3.1.2" xref="S4.SS3.p1.10.m3.1.2.cmml"><msub id="S4.SS3.p1.10.m3.1.2.2" xref="S4.SS3.p1.10.m3.1.2.2.cmml"><mi id="S4.SS3.p1.10.m3.1.2.2.2" xref="S4.SS3.p1.10.m3.1.2.2.2.cmml">f</mi><mi id="S4.SS3.p1.10.m3.1.2.2.3" xref="S4.SS3.p1.10.m3.1.2.2.3.cmml">p</mi></msub><mo id="S4.SS3.p1.10.m3.1.2.1" xref="S4.SS3.p1.10.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.10.m3.1.2.3.2" xref="S4.SS3.p1.10.m3.1.2.cmml"><mo id="S4.SS3.p1.10.m3.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.10.m3.1.2.cmml">(</mo><mi id="S4.SS3.p1.10.m3.1.1" xref="S4.SS3.p1.10.m3.1.1.cmml">x</mi><mo id="S4.SS3.p1.10.m3.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.10.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.10.m3.1b"><apply id="S4.SS3.p1.10.m3.1.2.cmml" xref="S4.SS3.p1.10.m3.1.2"><times id="S4.SS3.p1.10.m3.1.2.1.cmml" xref="S4.SS3.p1.10.m3.1.2.1"></times><apply id="S4.SS3.p1.10.m3.1.2.2.cmml" xref="S4.SS3.p1.10.m3.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.10.m3.1.2.2.1.cmml" xref="S4.SS3.p1.10.m3.1.2.2">subscript</csymbol><ci id="S4.SS3.p1.10.m3.1.2.2.2.cmml" xref="S4.SS3.p1.10.m3.1.2.2.2">𝑓</ci><ci id="S4.SS3.p1.10.m3.1.2.2.3.cmml" xref="S4.SS3.p1.10.m3.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.10.m3.1.1.cmml" xref="S4.SS3.p1.10.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.10.m3.1c">f_{p}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.10.m3.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> is explained subsequently) with a maximum production rate <math alttext="a" class="ltx_Math" display="inline" id="S4.SS3.p1.11.m4.1"><semantics id="S4.SS3.p1.11.m4.1a"><mi id="S4.SS3.p1.11.m4.1.1" xref="S4.SS3.p1.11.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.11.m4.1b"><ci id="S4.SS3.p1.11.m4.1.1.cmml" xref="S4.SS3.p1.11.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.11.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.11.m4.1d">italic_a</annotation></semantics></math> and large <math alttext="\beta" class="ltx_Math" display="inline" id="S4.SS3.p1.12.m5.1"><semantics id="S4.SS3.p1.12.m5.1a"><mi id="S4.SS3.p1.12.m5.1.1" xref="S4.SS3.p1.12.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.12.m5.1b"><ci id="S4.SS3.p1.12.m5.1.1.cmml" xref="S4.SS3.p1.12.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.12.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.12.m5.1d">italic_β</annotation></semantics></math> (<math alttext="=20" class="ltx_Math" display="inline" id="S4.SS3.p1.13.m6.1"><semantics id="S4.SS3.p1.13.m6.1a"><mrow id="S4.SS3.p1.13.m6.1.1" xref="S4.SS3.p1.13.m6.1.1.cmml"><mi id="S4.SS3.p1.13.m6.1.1.2" xref="S4.SS3.p1.13.m6.1.1.2.cmml"></mi><mo id="S4.SS3.p1.13.m6.1.1.1" xref="S4.SS3.p1.13.m6.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.13.m6.1.1.3" xref="S4.SS3.p1.13.m6.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.13.m6.1b"><apply id="S4.SS3.p1.13.m6.1.1.cmml" xref="S4.SS3.p1.13.m6.1.1"><eq id="S4.SS3.p1.13.m6.1.1.1.cmml" xref="S4.SS3.p1.13.m6.1.1.1"></eq><csymbol cd="latexml" id="S4.SS3.p1.13.m6.1.1.2.cmml" xref="S4.SS3.p1.13.m6.1.1.2">absent</csymbol><cn id="S4.SS3.p1.13.m6.1.1.3.cmml" type="integer" xref="S4.SS3.p1.13.m6.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.13.m6.1c">=20</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.13.m6.1d">= 20</annotation></semantics></math>). The second term represents protein decay with rate <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.p1.14.m7.1"><semantics id="S4.SS3.p1.14.m7.1a"><mi id="S4.SS3.p1.14.m7.1.1" xref="S4.SS3.p1.14.m7.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.14.m7.1b"><ci id="S4.SS3.p1.14.m7.1.1.cmml" xref="S4.SS3.p1.14.m7.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.14.m7.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.14.m7.1d">italic_b</annotation></semantics></math>. We set <math alttext="a=b" class="ltx_Math" display="inline" id="S4.SS3.p1.15.m8.1"><semantics id="S4.SS3.p1.15.m8.1a"><mrow id="S4.SS3.p1.15.m8.1.1" xref="S4.SS3.p1.15.m8.1.1.cmml"><mi id="S4.SS3.p1.15.m8.1.1.2" xref="S4.SS3.p1.15.m8.1.1.2.cmml">a</mi><mo id="S4.SS3.p1.15.m8.1.1.1" xref="S4.SS3.p1.15.m8.1.1.1.cmml">=</mo><mi id="S4.SS3.p1.15.m8.1.1.3" xref="S4.SS3.p1.15.m8.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.15.m8.1b"><apply id="S4.SS3.p1.15.m8.1.1.cmml" xref="S4.SS3.p1.15.m8.1.1"><eq id="S4.SS3.p1.15.m8.1.1.1.cmml" xref="S4.SS3.p1.15.m8.1.1.1"></eq><ci id="S4.SS3.p1.15.m8.1.1.2.cmml" xref="S4.SS3.p1.15.m8.1.1.2">𝑎</ci><ci id="S4.SS3.p1.15.m8.1.1.3.cmml" xref="S4.SS3.p1.15.m8.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.15.m8.1c">a=b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.15.m8.1d">italic_a = italic_b</annotation></semantics></math> for all <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p1.16.m9.1"><semantics id="S4.SS3.p1.16.m9.1a"><mi id="S4.SS3.p1.16.m9.1.1" xref="S4.SS3.p1.16.m9.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.16.m9.1b"><ci id="S4.SS3.p1.16.m9.1.1.cmml" xref="S4.SS3.p1.16.m9.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.16.m9.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.16.m9.1d">italic_p</annotation></semantics></math>, so <math alttext="x_{p}^{\sigma}" class="ltx_Math" display="inline" id="S4.SS3.p1.17.m10.1"><semantics id="S4.SS3.p1.17.m10.1a"><msubsup id="S4.SS3.p1.17.m10.1.1" xref="S4.SS3.p1.17.m10.1.1.cmml"><mi id="S4.SS3.p1.17.m10.1.1.2.2" xref="S4.SS3.p1.17.m10.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p1.17.m10.1.1.2.3" xref="S4.SS3.p1.17.m10.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p1.17.m10.1.1.3" xref="S4.SS3.p1.17.m10.1.1.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.17.m10.1b"><apply id="S4.SS3.p1.17.m10.1.1.cmml" xref="S4.SS3.p1.17.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.17.m10.1.1.1.cmml" xref="S4.SS3.p1.17.m10.1.1">superscript</csymbol><apply id="S4.SS3.p1.17.m10.1.1.2.cmml" xref="S4.SS3.p1.17.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.17.m10.1.1.2.1.cmml" xref="S4.SS3.p1.17.m10.1.1">subscript</csymbol><ci id="S4.SS3.p1.17.m10.1.1.2.2.cmml" xref="S4.SS3.p1.17.m10.1.1.2.2">𝑥</ci><ci id="S4.SS3.p1.17.m10.1.1.2.3.cmml" xref="S4.SS3.p1.17.m10.1.1.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.17.m10.1.1.3.cmml" xref="S4.SS3.p1.17.m10.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.17.m10.1c">x_{p}^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.17.m10.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> equilibriates at 1 if the gene is constantly expressed or 0 if it is constantly not expressed. The values of <math alttext="a" class="ltx_Math" display="inline" id="S4.SS3.p1.18.m11.1"><semantics id="S4.SS3.p1.18.m11.1a"><mi id="S4.SS3.p1.18.m11.1.1" xref="S4.SS3.p1.18.m11.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.18.m11.1b"><ci id="S4.SS3.p1.18.m11.1.1.cmml" xref="S4.SS3.p1.18.m11.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.18.m11.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.18.m11.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.p1.19.m12.1"><semantics id="S4.SS3.p1.19.m12.1a"><mi id="S4.SS3.p1.19.m12.1.1" xref="S4.SS3.p1.19.m12.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.19.m12.1b"><ci id="S4.SS3.p1.19.m12.1.1.cmml" xref="S4.SS3.p1.19.m12.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.19.m12.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.19.m12.1d">italic_b</annotation></semantics></math> are small (specifically, <math alttext="6.25\times 10^{-3}" class="ltx_Math" display="inline" id="S4.SS3.p1.20.m13.1"><semantics id="S4.SS3.p1.20.m13.1a"><mrow id="S4.SS3.p1.20.m13.1.1" xref="S4.SS3.p1.20.m13.1.1.cmml"><mn id="S4.SS3.p1.20.m13.1.1.2" xref="S4.SS3.p1.20.m13.1.1.2.cmml">6.25</mn><mo id="S4.SS3.p1.20.m13.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p1.20.m13.1.1.1.cmml">×</mo><msup id="S4.SS3.p1.20.m13.1.1.3" xref="S4.SS3.p1.20.m13.1.1.3.cmml"><mn id="S4.SS3.p1.20.m13.1.1.3.2" xref="S4.SS3.p1.20.m13.1.1.3.2.cmml">10</mn><mrow id="S4.SS3.p1.20.m13.1.1.3.3" xref="S4.SS3.p1.20.m13.1.1.3.3.cmml"><mo id="S4.SS3.p1.20.m13.1.1.3.3a" xref="S4.SS3.p1.20.m13.1.1.3.3.cmml">−</mo><mn id="S4.SS3.p1.20.m13.1.1.3.3.2" xref="S4.SS3.p1.20.m13.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.20.m13.1b"><apply id="S4.SS3.p1.20.m13.1.1.cmml" xref="S4.SS3.p1.20.m13.1.1"><times id="S4.SS3.p1.20.m13.1.1.1.cmml" xref="S4.SS3.p1.20.m13.1.1.1"></times><cn id="S4.SS3.p1.20.m13.1.1.2.cmml" type="float" xref="S4.SS3.p1.20.m13.1.1.2">6.25</cn><apply id="S4.SS3.p1.20.m13.1.1.3.cmml" xref="S4.SS3.p1.20.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.20.m13.1.1.3.1.cmml" xref="S4.SS3.p1.20.m13.1.1.3">superscript</csymbol><cn id="S4.SS3.p1.20.m13.1.1.3.2.cmml" type="integer" xref="S4.SS3.p1.20.m13.1.1.3.2">10</cn><apply id="S4.SS3.p1.20.m13.1.1.3.3.cmml" xref="S4.SS3.p1.20.m13.1.1.3.3"><minus id="S4.SS3.p1.20.m13.1.1.3.3.1.cmml" xref="S4.SS3.p1.20.m13.1.1.3.3"></minus><cn id="S4.SS3.p1.20.m13.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p1.20.m13.1.1.3.3.2">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.20.m13.1c">6.25\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.20.m13.1d">6.25 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math>) to assume that the timescale of gene expression is slower than the timescale of CPM dynamics. Numerical integration of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a> occurs with <math alttext="\Delta t=40" class="ltx_Math" display="inline" id="S4.SS3.p1.21.m14.1"><semantics id="S4.SS3.p1.21.m14.1a"><mrow id="S4.SS3.p1.21.m14.1.1" xref="S4.SS3.p1.21.m14.1.1.cmml"><mrow id="S4.SS3.p1.21.m14.1.1.2" xref="S4.SS3.p1.21.m14.1.1.2.cmml"><mi id="S4.SS3.p1.21.m14.1.1.2.2" mathvariant="normal" xref="S4.SS3.p1.21.m14.1.1.2.2.cmml">Δ</mi><mo id="S4.SS3.p1.21.m14.1.1.2.1" xref="S4.SS3.p1.21.m14.1.1.2.1.cmml">⁢</mo><mi id="S4.SS3.p1.21.m14.1.1.2.3" xref="S4.SS3.p1.21.m14.1.1.2.3.cmml">t</mi></mrow><mo id="S4.SS3.p1.21.m14.1.1.1" xref="S4.SS3.p1.21.m14.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.21.m14.1.1.3" xref="S4.SS3.p1.21.m14.1.1.3.cmml">40</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.21.m14.1b"><apply id="S4.SS3.p1.21.m14.1.1.cmml" xref="S4.SS3.p1.21.m14.1.1"><eq id="S4.SS3.p1.21.m14.1.1.1.cmml" xref="S4.SS3.p1.21.m14.1.1.1"></eq><apply id="S4.SS3.p1.21.m14.1.1.2.cmml" xref="S4.SS3.p1.21.m14.1.1.2"><times id="S4.SS3.p1.21.m14.1.1.2.1.cmml" xref="S4.SS3.p1.21.m14.1.1.2.1"></times><ci id="S4.SS3.p1.21.m14.1.1.2.2.cmml" xref="S4.SS3.p1.21.m14.1.1.2.2">Δ</ci><ci id="S4.SS3.p1.21.m14.1.1.2.3.cmml" xref="S4.SS3.p1.21.m14.1.1.2.3">𝑡</ci></apply><cn id="S4.SS3.p1.21.m14.1.1.3.cmml" type="integer" xref="S4.SS3.p1.21.m14.1.1.3">40</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.21.m14.1c">\Delta t=40</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.21.m14.1d">roman_Δ italic_t = 40</annotation></semantics></math> via the Euler method. We chose this value of <math alttext="\Delta t" class="ltx_Math" display="inline" id="S4.SS3.p1.22.m15.1"><semantics id="S4.SS3.p1.22.m15.1a"><mrow id="S4.SS3.p1.22.m15.1.1" xref="S4.SS3.p1.22.m15.1.1.cmml"><mi id="S4.SS3.p1.22.m15.1.1.2" mathvariant="normal" xref="S4.SS3.p1.22.m15.1.1.2.cmml">Δ</mi><mo id="S4.SS3.p1.22.m15.1.1.1" xref="S4.SS3.p1.22.m15.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.22.m15.1.1.3" xref="S4.SS3.p1.22.m15.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.22.m15.1b"><apply id="S4.SS3.p1.22.m15.1.1.cmml" xref="S4.SS3.p1.22.m15.1.1"><times id="S4.SS3.p1.22.m15.1.1.1.cmml" xref="S4.SS3.p1.22.m15.1.1.1"></times><ci id="S4.SS3.p1.22.m15.1.1.2.cmml" xref="S4.SS3.p1.22.m15.1.1.2">Δ</ci><ci id="S4.SS3.p1.22.m15.1.1.3.cmml" xref="S4.SS3.p1.22.m15.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.22.m15.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.22.m15.1d">roman_Δ italic_t</annotation></semantics></math> instead of a smaller value to improve computational speed. The value of <math alttext="\Delta t" class="ltx_Math" display="inline" id="S4.SS3.p1.23.m16.1"><semantics id="S4.SS3.p1.23.m16.1a"><mrow id="S4.SS3.p1.23.m16.1.1" xref="S4.SS3.p1.23.m16.1.1.cmml"><mi id="S4.SS3.p1.23.m16.1.1.2" mathvariant="normal" xref="S4.SS3.p1.23.m16.1.1.2.cmml">Δ</mi><mo id="S4.SS3.p1.23.m16.1.1.1" xref="S4.SS3.p1.23.m16.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.23.m16.1.1.3" xref="S4.SS3.p1.23.m16.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.23.m16.1b"><apply id="S4.SS3.p1.23.m16.1.1.cmml" xref="S4.SS3.p1.23.m16.1.1"><times id="S4.SS3.p1.23.m16.1.1.1.cmml" xref="S4.SS3.p1.23.m16.1.1.1"></times><ci id="S4.SS3.p1.23.m16.1.1.2.cmml" xref="S4.SS3.p1.23.m16.1.1.2">Δ</ci><ci id="S4.SS3.p1.23.m16.1.1.3.cmml" xref="S4.SS3.p1.23.m16.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.23.m16.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.23.m16.1d">roman_Δ italic_t</annotation></semantics></math> can be large because <math alttext="a" class="ltx_Math" display="inline" id="S4.SS3.p1.24.m17.1"><semantics id="S4.SS3.p1.24.m17.1a"><mi id="S4.SS3.p1.24.m17.1.1" xref="S4.SS3.p1.24.m17.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.24.m17.1b"><ci id="S4.SS3.p1.24.m17.1.1.cmml" xref="S4.SS3.p1.24.m17.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.24.m17.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.24.m17.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.p1.25.m18.1"><semantics id="S4.SS3.p1.25.m18.1a"><mi id="S4.SS3.p1.25.m18.1.1" xref="S4.SS3.p1.25.m18.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.25.m18.1b"><ci id="S4.SS3.p1.25.m18.1.1.cmml" xref="S4.SS3.p1.25.m18.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.25.m18.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.25.m18.1d">italic_b</annotation></semantics></math> are very small. The initial concentrations are <math alttext="x_{p}^{\sigma}=1" class="ltx_Math" display="inline" id="S4.SS3.p1.26.m19.1"><semantics id="S4.SS3.p1.26.m19.1a"><mrow id="S4.SS3.p1.26.m19.1.1" xref="S4.SS3.p1.26.m19.1.1.cmml"><msubsup id="S4.SS3.p1.26.m19.1.1.2" xref="S4.SS3.p1.26.m19.1.1.2.cmml"><mi id="S4.SS3.p1.26.m19.1.1.2.2.2" xref="S4.SS3.p1.26.m19.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p1.26.m19.1.1.2.2.3" xref="S4.SS3.p1.26.m19.1.1.2.2.3.cmml">p</mi><mi id="S4.SS3.p1.26.m19.1.1.2.3" xref="S4.SS3.p1.26.m19.1.1.2.3.cmml">σ</mi></msubsup><mo id="S4.SS3.p1.26.m19.1.1.1" xref="S4.SS3.p1.26.m19.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.26.m19.1.1.3" xref="S4.SS3.p1.26.m19.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.26.m19.1b"><apply id="S4.SS3.p1.26.m19.1.1.cmml" xref="S4.SS3.p1.26.m19.1.1"><eq id="S4.SS3.p1.26.m19.1.1.1.cmml" xref="S4.SS3.p1.26.m19.1.1.1"></eq><apply id="S4.SS3.p1.26.m19.1.1.2.cmml" xref="S4.SS3.p1.26.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.26.m19.1.1.2.1.cmml" xref="S4.SS3.p1.26.m19.1.1.2">superscript</csymbol><apply id="S4.SS3.p1.26.m19.1.1.2.2.cmml" xref="S4.SS3.p1.26.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.26.m19.1.1.2.2.1.cmml" xref="S4.SS3.p1.26.m19.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.26.m19.1.1.2.2.2.cmml" xref="S4.SS3.p1.26.m19.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p1.26.m19.1.1.2.2.3.cmml" xref="S4.SS3.p1.26.m19.1.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.26.m19.1.1.2.3.cmml" xref="S4.SS3.p1.26.m19.1.1.2.3">𝜎</ci></apply><cn id="S4.SS3.p1.26.m19.1.1.3.cmml" type="integer" xref="S4.SS3.p1.26.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.26.m19.1c">x_{p}^{\sigma}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.26.m19.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = 1</annotation></semantics></math> for transcription factors (except for maternal factors, which are either <math alttext="x_{p}^{\sigma}=1" class="ltx_Math" display="inline" id="S4.SS3.p1.27.m20.1"><semantics id="S4.SS3.p1.27.m20.1a"><mrow id="S4.SS3.p1.27.m20.1.1" xref="S4.SS3.p1.27.m20.1.1.cmml"><msubsup id="S4.SS3.p1.27.m20.1.1.2" xref="S4.SS3.p1.27.m20.1.1.2.cmml"><mi id="S4.SS3.p1.27.m20.1.1.2.2.2" xref="S4.SS3.p1.27.m20.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p1.27.m20.1.1.2.2.3" xref="S4.SS3.p1.27.m20.1.1.2.2.3.cmml">p</mi><mi id="S4.SS3.p1.27.m20.1.1.2.3" xref="S4.SS3.p1.27.m20.1.1.2.3.cmml">σ</mi></msubsup><mo id="S4.SS3.p1.27.m20.1.1.1" xref="S4.SS3.p1.27.m20.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.27.m20.1.1.3" xref="S4.SS3.p1.27.m20.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.27.m20.1b"><apply id="S4.SS3.p1.27.m20.1.1.cmml" xref="S4.SS3.p1.27.m20.1.1"><eq id="S4.SS3.p1.27.m20.1.1.1.cmml" xref="S4.SS3.p1.27.m20.1.1.1"></eq><apply id="S4.SS3.p1.27.m20.1.1.2.cmml" xref="S4.SS3.p1.27.m20.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.27.m20.1.1.2.1.cmml" xref="S4.SS3.p1.27.m20.1.1.2">superscript</csymbol><apply id="S4.SS3.p1.27.m20.1.1.2.2.cmml" xref="S4.SS3.p1.27.m20.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.27.m20.1.1.2.2.1.cmml" xref="S4.SS3.p1.27.m20.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.27.m20.1.1.2.2.2.cmml" xref="S4.SS3.p1.27.m20.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p1.27.m20.1.1.2.2.3.cmml" xref="S4.SS3.p1.27.m20.1.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.27.m20.1.1.2.3.cmml" xref="S4.SS3.p1.27.m20.1.1.2.3">𝜎</ci></apply><cn id="S4.SS3.p1.27.m20.1.1.3.cmml" type="integer" xref="S4.SS3.p1.27.m20.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.27.m20.1c">x_{p}^{\sigma}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.27.m20.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = 1</annotation></semantics></math> or <math alttext="x_{p}^{\sigma}=0" class="ltx_Math" display="inline" id="S4.SS3.p1.28.m21.1"><semantics id="S4.SS3.p1.28.m21.1a"><mrow id="S4.SS3.p1.28.m21.1.1" xref="S4.SS3.p1.28.m21.1.1.cmml"><msubsup id="S4.SS3.p1.28.m21.1.1.2" xref="S4.SS3.p1.28.m21.1.1.2.cmml"><mi id="S4.SS3.p1.28.m21.1.1.2.2.2" xref="S4.SS3.p1.28.m21.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p1.28.m21.1.1.2.2.3" xref="S4.SS3.p1.28.m21.1.1.2.2.3.cmml">p</mi><mi id="S4.SS3.p1.28.m21.1.1.2.3" xref="S4.SS3.p1.28.m21.1.1.2.3.cmml">σ</mi></msubsup><mo id="S4.SS3.p1.28.m21.1.1.1" xref="S4.SS3.p1.28.m21.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.28.m21.1.1.3" xref="S4.SS3.p1.28.m21.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.28.m21.1b"><apply id="S4.SS3.p1.28.m21.1.1.cmml" xref="S4.SS3.p1.28.m21.1.1"><eq id="S4.SS3.p1.28.m21.1.1.1.cmml" xref="S4.SS3.p1.28.m21.1.1.1"></eq><apply id="S4.SS3.p1.28.m21.1.1.2.cmml" xref="S4.SS3.p1.28.m21.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.28.m21.1.1.2.1.cmml" xref="S4.SS3.p1.28.m21.1.1.2">superscript</csymbol><apply id="S4.SS3.p1.28.m21.1.1.2.2.cmml" xref="S4.SS3.p1.28.m21.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.28.m21.1.1.2.2.1.cmml" xref="S4.SS3.p1.28.m21.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.28.m21.1.1.2.2.2.cmml" xref="S4.SS3.p1.28.m21.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p1.28.m21.1.1.2.2.3.cmml" xref="S4.SS3.p1.28.m21.1.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.28.m21.1.1.2.3.cmml" xref="S4.SS3.p1.28.m21.1.1.2.3">𝜎</ci></apply><cn id="S4.SS3.p1.28.m21.1.1.3.cmml" type="integer" xref="S4.SS3.p1.28.m21.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.28.m21.1c">x_{p}^{\sigma}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.28.m21.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = 0</annotation></semantics></math> depending on <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS3.p1.29.m22.1"><semantics id="S4.SS3.p1.29.m22.1a"><mi id="S4.SS3.p1.29.m22.1.1" xref="S4.SS3.p1.29.m22.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.29.m22.1b"><ci id="S4.SS3.p1.29.m22.1.1.cmml" xref="S4.SS3.p1.29.m22.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.29.m22.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.29.m22.1d">italic_σ</annotation></semantics></math> at the four-cell stage), and <math alttext="x_{p}^{\sigma}=0" class="ltx_Math" display="inline" id="S4.SS3.p1.30.m23.1"><semantics id="S4.SS3.p1.30.m23.1a"><mrow id="S4.SS3.p1.30.m23.1.1" xref="S4.SS3.p1.30.m23.1.1.cmml"><msubsup id="S4.SS3.p1.30.m23.1.1.2" xref="S4.SS3.p1.30.m23.1.1.2.cmml"><mi id="S4.SS3.p1.30.m23.1.1.2.2.2" xref="S4.SS3.p1.30.m23.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p1.30.m23.1.1.2.2.3" xref="S4.SS3.p1.30.m23.1.1.2.2.3.cmml">p</mi><mi id="S4.SS3.p1.30.m23.1.1.2.3" xref="S4.SS3.p1.30.m23.1.1.2.3.cmml">σ</mi></msubsup><mo id="S4.SS3.p1.30.m23.1.1.1" xref="S4.SS3.p1.30.m23.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.30.m23.1.1.3" xref="S4.SS3.p1.30.m23.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.30.m23.1b"><apply id="S4.SS3.p1.30.m23.1.1.cmml" xref="S4.SS3.p1.30.m23.1.1"><eq id="S4.SS3.p1.30.m23.1.1.1.cmml" xref="S4.SS3.p1.30.m23.1.1.1"></eq><apply id="S4.SS3.p1.30.m23.1.1.2.cmml" xref="S4.SS3.p1.30.m23.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.30.m23.1.1.2.1.cmml" xref="S4.SS3.p1.30.m23.1.1.2">superscript</csymbol><apply id="S4.SS3.p1.30.m23.1.1.2.2.cmml" xref="S4.SS3.p1.30.m23.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.30.m23.1.1.2.2.1.cmml" xref="S4.SS3.p1.30.m23.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.30.m23.1.1.2.2.2.cmml" xref="S4.SS3.p1.30.m23.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p1.30.m23.1.1.2.2.3.cmml" xref="S4.SS3.p1.30.m23.1.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p1.30.m23.1.1.2.3.cmml" xref="S4.SS3.p1.30.m23.1.1.2.3">𝜎</ci></apply><cn id="S4.SS3.p1.30.m23.1.1.3.cmml" type="integer" xref="S4.SS3.p1.30.m23.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.30.m23.1c">x_{p}^{\sigma}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.30.m23.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = 0</annotation></semantics></math> for all other proteins (described later). Integration of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a> begins after the four-cell stage is reached (100 DTS), which is when the asymmetric distribution of maternal factors occurs. </div> <div class="ltx_para" id="S4.SS3.p2"> Function <math alttext="f_{p}(x)" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mrow id="S4.SS3.p2.1.m1.1.2" xref="S4.SS3.p2.1.m1.1.2.cmml"><msub id="S4.SS3.p2.1.m1.1.2.2" xref="S4.SS3.p2.1.m1.1.2.2.cmml"><mi id="S4.SS3.p2.1.m1.1.2.2.2" xref="S4.SS3.p2.1.m1.1.2.2.2.cmml">f</mi><mi id="S4.SS3.p2.1.m1.1.2.2.3" xref="S4.SS3.p2.1.m1.1.2.2.3.cmml">p</mi></msub><mo id="S4.SS3.p2.1.m1.1.2.1" xref="S4.SS3.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.1.m1.1.2.3.2" xref="S4.SS3.p2.1.m1.1.2.cmml"><mo id="S4.SS3.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS3.p2.1.m1.1.2.cmml">(</mo><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">x</mi><mo id="S4.SS3.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS3.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><apply id="S4.SS3.p2.1.m1.1.2.cmml" xref="S4.SS3.p2.1.m1.1.2"><times id="S4.SS3.p2.1.m1.1.2.1.cmml" xref="S4.SS3.p2.1.m1.1.2.1"></times><apply id="S4.SS3.p2.1.m1.1.2.2.cmml" xref="S4.SS3.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.1.2.2.1.cmml" xref="S4.SS3.p2.1.m1.1.2.2">subscript</csymbol><ci id="S4.SS3.p2.1.m1.1.2.2.2.cmml" xref="S4.SS3.p2.1.m1.1.2.2.2">𝑓</ci><ci id="S4.SS3.p2.1.m1.1.2.2.3.cmml" xref="S4.SS3.p2.1.m1.1.2.2.3">𝑝</ci></apply><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">f_{p}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a> sums the regulatory effects of <math alttext="n=9" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mrow id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mi id="S4.SS3.p2.2.m2.1.1.2" xref="S4.SS3.p2.2.m2.1.1.2.cmml">n</mi><mo id="S4.SS3.p2.2.m2.1.1.1" xref="S4.SS3.p2.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS3.p2.2.m2.1.1.3" xref="S4.SS3.p2.2.m2.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><eq id="S4.SS3.p2.2.m2.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1"></eq><ci id="S4.SS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1.2">𝑛</ci><cn id="S4.SS3.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS3.p2.2.m2.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">n=9</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">italic_n = 9</annotation></semantics></math> transcription factors (TFs, which include two maternal factors and three morphogens) as follows: <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{p}(x)=\left[\sum^{n}_{p^{\prime}}Z_{pp^{\prime}}x_{p^{\prime}}^{\sigma}% \right]+\theta" class="ltx_Math" display="block" id="S4.E3.m1.2"><semantics id="S4.E3.m1.2a"><mrow id="S4.E3.m1.2.2" xref="S4.E3.m1.2.2.cmml"><mrow id="S4.E3.m1.2.2.3" xref="S4.E3.m1.2.2.3.cmml"><msub id="S4.E3.m1.2.2.3.2" xref="S4.E3.m1.2.2.3.2.cmml"><mi id="S4.E3.m1.2.2.3.2.2" 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id="S4.E3.m1.2.2.1.1.1.1.2.3.1.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3">superscript</csymbol><apply id="S4.E3.m1.2.2.1.1.1.1.2.3.2.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.1.1.1.1.2.3.2.1.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3">subscript</csymbol><ci id="S4.E3.m1.2.2.1.1.1.1.2.3.2.2.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.2.2">𝑥</ci><apply id="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.2.3"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.1.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.2.3">superscript</csymbol><ci id="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.2.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.2">𝑝</ci><ci id="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.3.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.2.3.3">′</ci></apply></apply><ci id="S4.E3.m1.2.2.1.1.1.1.2.3.3.cmml" xref="S4.E3.m1.2.2.1.1.1.1.2.3.3">𝜎</ci></apply></apply></apply></apply><ci id="S4.E3.m1.2.2.1.3.cmml" xref="S4.E3.m1.2.2.1.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.2c">f_{p}(x)=\left[\sum^{n}_{p^{\prime}}Z_{pp^{\prime}}x_{p^{\prime}}^{\sigma}% \right]+\theta</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.2d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_x ) = [ ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ] + italic_θ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(3)</td> </tr></tbody> </table> where <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m1.1"><semantics id="S4.SS3.p2.3.m1.1a"><msub id="S4.SS3.p2.3.m1.1.1" xref="S4.SS3.p2.3.m1.1.1.cmml"><mi id="S4.SS3.p2.3.m1.1.1.2" xref="S4.SS3.p2.3.m1.1.1.2.cmml">Z</mi><mrow id="S4.SS3.p2.3.m1.1.1.3" xref="S4.SS3.p2.3.m1.1.1.3.cmml"><mi id="S4.SS3.p2.3.m1.1.1.3.2" xref="S4.SS3.p2.3.m1.1.1.3.2.cmml">p</mi><mo id="S4.SS3.p2.3.m1.1.1.3.1" xref="S4.SS3.p2.3.m1.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.p2.3.m1.1.1.3.3" xref="S4.SS3.p2.3.m1.1.1.3.3.cmml"><mi id="S4.SS3.p2.3.m1.1.1.3.3.2" xref="S4.SS3.p2.3.m1.1.1.3.3.2.cmml">p</mi><mo id="S4.SS3.p2.3.m1.1.1.3.3.3" xref="S4.SS3.p2.3.m1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m1.1b"><apply id="S4.SS3.p2.3.m1.1.1.cmml" xref="S4.SS3.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m1.1.1.1.cmml" xref="S4.SS3.p2.3.m1.1.1">subscript</csymbol><ci id="S4.SS3.p2.3.m1.1.1.2.cmml" xref="S4.SS3.p2.3.m1.1.1.2">𝑍</ci><apply id="S4.SS3.p2.3.m1.1.1.3.cmml" xref="S4.SS3.p2.3.m1.1.1.3"><times id="S4.SS3.p2.3.m1.1.1.3.1.cmml" xref="S4.SS3.p2.3.m1.1.1.3.1"></times><ci id="S4.SS3.p2.3.m1.1.1.3.2.cmml" xref="S4.SS3.p2.3.m1.1.1.3.2">𝑝</ci><apply id="S4.SS3.p2.3.m1.1.1.3.3.cmml" xref="S4.SS3.p2.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m1.1.1.3.3.1.cmml" xref="S4.SS3.p2.3.m1.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p2.3.m1.1.1.3.3.2.cmml" xref="S4.SS3.p2.3.m1.1.1.3.3.2">𝑝</ci><ci id="S4.SS3.p2.3.m1.1.1.3.3.3.cmml" xref="S4.SS3.p2.3.m1.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m1.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m1.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the regulatory effect of TF encoded by gene <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m2.1"><semantics id="S4.SS3.p2.4.m2.1a"><msup id="S4.SS3.p2.4.m2.1.1" xref="S4.SS3.p2.4.m2.1.1.cmml"><mi id="S4.SS3.p2.4.m2.1.1.2" xref="S4.SS3.p2.4.m2.1.1.2.cmml">p</mi><mo id="S4.SS3.p2.4.m2.1.1.3" xref="S4.SS3.p2.4.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m2.1b"><apply id="S4.SS3.p2.4.m2.1.1.cmml" xref="S4.SS3.p2.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.4.m2.1.1.1.cmml" xref="S4.SS3.p2.4.m2.1.1">superscript</csymbol><ci id="S4.SS3.p2.4.m2.1.1.2.cmml" xref="S4.SS3.p2.4.m2.1.1.2">𝑝</ci><ci id="S4.SS3.p2.4.m2.1.1.3.cmml" xref="S4.SS3.p2.4.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m2.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m2.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on the expression of gene <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m3.1"><semantics id="S4.SS3.p2.5.m3.1a"><mi id="S4.SS3.p2.5.m3.1.1" xref="S4.SS3.p2.5.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m3.1b"><ci id="S4.SS3.p2.5.m3.1.1.cmml" xref="S4.SS3.p2.5.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m3.1d">italic_p</annotation></semantics></math> (<math alttext="Z_{pp^{\prime}}\in\{0,\pm 1,\pm 2\}" class="ltx_Math" display="inline" id="S4.SS3.p2.6.m4.3"><semantics id="S4.SS3.p2.6.m4.3a"><mrow id="S4.SS3.p2.6.m4.3.3" xref="S4.SS3.p2.6.m4.3.3.cmml"><msub id="S4.SS3.p2.6.m4.3.3.4" xref="S4.SS3.p2.6.m4.3.3.4.cmml"><mi id="S4.SS3.p2.6.m4.3.3.4.2" xref="S4.SS3.p2.6.m4.3.3.4.2.cmml">Z</mi><mrow id="S4.SS3.p2.6.m4.3.3.4.3" xref="S4.SS3.p2.6.m4.3.3.4.3.cmml"><mi id="S4.SS3.p2.6.m4.3.3.4.3.2" xref="S4.SS3.p2.6.m4.3.3.4.3.2.cmml">p</mi><mo id="S4.SS3.p2.6.m4.3.3.4.3.1" xref="S4.SS3.p2.6.m4.3.3.4.3.1.cmml">⁢</mo><msup id="S4.SS3.p2.6.m4.3.3.4.3.3" xref="S4.SS3.p2.6.m4.3.3.4.3.3.cmml"><mi id="S4.SS3.p2.6.m4.3.3.4.3.3.2" xref="S4.SS3.p2.6.m4.3.3.4.3.3.2.cmml">p</mi><mo id="S4.SS3.p2.6.m4.3.3.4.3.3.3" xref="S4.SS3.p2.6.m4.3.3.4.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S4.SS3.p2.6.m4.3.3.3" xref="S4.SS3.p2.6.m4.3.3.3.cmml">∈</mo><mrow id="S4.SS3.p2.6.m4.3.3.2.2" xref="S4.SS3.p2.6.m4.3.3.2.3.cmml"><mo id="S4.SS3.p2.6.m4.3.3.2.2.3" stretchy="false" xref="S4.SS3.p2.6.m4.3.3.2.3.cmml">{</mo><mn id="S4.SS3.p2.6.m4.1.1" xref="S4.SS3.p2.6.m4.1.1.cmml">0</mn><mo id="S4.SS3.p2.6.m4.3.3.2.2.4" xref="S4.SS3.p2.6.m4.3.3.2.3.cmml">,</mo><mrow id="S4.SS3.p2.6.m4.2.2.1.1.1" xref="S4.SS3.p2.6.m4.2.2.1.1.1.cmml"><mo id="S4.SS3.p2.6.m4.2.2.1.1.1a" xref="S4.SS3.p2.6.m4.2.2.1.1.1.cmml">±</mo><mn id="S4.SS3.p2.6.m4.2.2.1.1.1.2" xref="S4.SS3.p2.6.m4.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS3.p2.6.m4.3.3.2.2.5" xref="S4.SS3.p2.6.m4.3.3.2.3.cmml">,</mo><mrow id="S4.SS3.p2.6.m4.3.3.2.2.2" xref="S4.SS3.p2.6.m4.3.3.2.2.2.cmml"><mo id="S4.SS3.p2.6.m4.3.3.2.2.2a" xref="S4.SS3.p2.6.m4.3.3.2.2.2.cmml">±</mo><mn id="S4.SS3.p2.6.m4.3.3.2.2.2.2" xref="S4.SS3.p2.6.m4.3.3.2.2.2.2.cmml">2</mn></mrow><mo id="S4.SS3.p2.6.m4.3.3.2.2.6" stretchy="false" xref="S4.SS3.p2.6.m4.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.6.m4.3b"><apply id="S4.SS3.p2.6.m4.3.3.cmml" xref="S4.SS3.p2.6.m4.3.3"><in id="S4.SS3.p2.6.m4.3.3.3.cmml" xref="S4.SS3.p2.6.m4.3.3.3"></in><apply id="S4.SS3.p2.6.m4.3.3.4.cmml" xref="S4.SS3.p2.6.m4.3.3.4"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m4.3.3.4.1.cmml" xref="S4.SS3.p2.6.m4.3.3.4">subscript</csymbol><ci id="S4.SS3.p2.6.m4.3.3.4.2.cmml" xref="S4.SS3.p2.6.m4.3.3.4.2">𝑍</ci><apply id="S4.SS3.p2.6.m4.3.3.4.3.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3"><times id="S4.SS3.p2.6.m4.3.3.4.3.1.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.1"></times><ci id="S4.SS3.p2.6.m4.3.3.4.3.2.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.2">𝑝</ci><apply id="S4.SS3.p2.6.m4.3.3.4.3.3.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m4.3.3.4.3.3.1.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.3">superscript</csymbol><ci id="S4.SS3.p2.6.m4.3.3.4.3.3.2.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.3.2">𝑝</ci><ci id="S4.SS3.p2.6.m4.3.3.4.3.3.3.cmml" xref="S4.SS3.p2.6.m4.3.3.4.3.3.3">′</ci></apply></apply></apply><set id="S4.SS3.p2.6.m4.3.3.2.3.cmml" xref="S4.SS3.p2.6.m4.3.3.2.2"><cn id="S4.SS3.p2.6.m4.1.1.cmml" type="integer" xref="S4.SS3.p2.6.m4.1.1">0</cn><apply id="S4.SS3.p2.6.m4.2.2.1.1.1.cmml" xref="S4.SS3.p2.6.m4.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS3.p2.6.m4.2.2.1.1.1.1.cmml" xref="S4.SS3.p2.6.m4.2.2.1.1.1">plus-or-minus</csymbol><cn id="S4.SS3.p2.6.m4.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS3.p2.6.m4.2.2.1.1.1.2">1</cn></apply><apply id="S4.SS3.p2.6.m4.3.3.2.2.2.cmml" xref="S4.SS3.p2.6.m4.3.3.2.2.2"><csymbol cd="latexml" id="S4.SS3.p2.6.m4.3.3.2.2.2.1.cmml" xref="S4.SS3.p2.6.m4.3.3.2.2.2">plus-or-minus</csymbol><cn id="S4.SS3.p2.6.m4.3.3.2.2.2.2.cmml" type="integer" xref="S4.SS3.p2.6.m4.3.3.2.2.2.2">2</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m4.3c">Z_{pp^{\prime}}\in\{0,\pm 1,\pm 2\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m4.3d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ { 0 , ± 1 , ± 2 }</annotation></semantics></math>). The TF encoded by <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p2.7.m5.1"><semantics id="S4.SS3.p2.7.m5.1a"><msup id="S4.SS3.p2.7.m5.1.1" xref="S4.SS3.p2.7.m5.1.1.cmml"><mi id="S4.SS3.p2.7.m5.1.1.2" xref="S4.SS3.p2.7.m5.1.1.2.cmml">p</mi><mo id="S4.SS3.p2.7.m5.1.1.3" xref="S4.SS3.p2.7.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.7.m5.1b"><apply id="S4.SS3.p2.7.m5.1.1.cmml" xref="S4.SS3.p2.7.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.7.m5.1.1.1.cmml" xref="S4.SS3.p2.7.m5.1.1">superscript</csymbol><ci id="S4.SS3.p2.7.m5.1.1.2.cmml" xref="S4.SS3.p2.7.m5.1.1.2">𝑝</ci><ci id="S4.SS3.p2.7.m5.1.1.3.cmml" xref="S4.SS3.p2.7.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.7.m5.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.7.m5.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> activates the expression of <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p2.8.m6.1"><semantics id="S4.SS3.p2.8.m6.1a"><mi id="S4.SS3.p2.8.m6.1.1" xref="S4.SS3.p2.8.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.8.m6.1b"><ci id="S4.SS3.p2.8.m6.1.1.cmml" xref="S4.SS3.p2.8.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.8.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.8.m6.1d">italic_p</annotation></semantics></math> if <math alttext="Z_{pp^{\prime}}>0" class="ltx_Math" display="inline" id="S4.SS3.p2.9.m7.1"><semantics id="S4.SS3.p2.9.m7.1a"><mrow id="S4.SS3.p2.9.m7.1.1" xref="S4.SS3.p2.9.m7.1.1.cmml"><msub id="S4.SS3.p2.9.m7.1.1.2" xref="S4.SS3.p2.9.m7.1.1.2.cmml"><mi id="S4.SS3.p2.9.m7.1.1.2.2" xref="S4.SS3.p2.9.m7.1.1.2.2.cmml">Z</mi><mrow id="S4.SS3.p2.9.m7.1.1.2.3" xref="S4.SS3.p2.9.m7.1.1.2.3.cmml"><mi id="S4.SS3.p2.9.m7.1.1.2.3.2" xref="S4.SS3.p2.9.m7.1.1.2.3.2.cmml">p</mi><mo id="S4.SS3.p2.9.m7.1.1.2.3.1" xref="S4.SS3.p2.9.m7.1.1.2.3.1.cmml">⁢</mo><msup id="S4.SS3.p2.9.m7.1.1.2.3.3" xref="S4.SS3.p2.9.m7.1.1.2.3.3.cmml"><mi id="S4.SS3.p2.9.m7.1.1.2.3.3.2" xref="S4.SS3.p2.9.m7.1.1.2.3.3.2.cmml">p</mi><mo id="S4.SS3.p2.9.m7.1.1.2.3.3.3" xref="S4.SS3.p2.9.m7.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S4.SS3.p2.9.m7.1.1.1" xref="S4.SS3.p2.9.m7.1.1.1.cmml">></mo><mn id="S4.SS3.p2.9.m7.1.1.3" xref="S4.SS3.p2.9.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.9.m7.1b"><apply id="S4.SS3.p2.9.m7.1.1.cmml" xref="S4.SS3.p2.9.m7.1.1"><gt id="S4.SS3.p2.9.m7.1.1.1.cmml" xref="S4.SS3.p2.9.m7.1.1.1"></gt><apply id="S4.SS3.p2.9.m7.1.1.2.cmml" xref="S4.SS3.p2.9.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.9.m7.1.1.2.1.cmml" xref="S4.SS3.p2.9.m7.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.9.m7.1.1.2.2.cmml" xref="S4.SS3.p2.9.m7.1.1.2.2">𝑍</ci><apply id="S4.SS3.p2.9.m7.1.1.2.3.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3"><times id="S4.SS3.p2.9.m7.1.1.2.3.1.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.1"></times><ci id="S4.SS3.p2.9.m7.1.1.2.3.2.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.2">𝑝</ci><apply id="S4.SS3.p2.9.m7.1.1.2.3.3.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.9.m7.1.1.2.3.3.1.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.3">superscript</csymbol><ci id="S4.SS3.p2.9.m7.1.1.2.3.3.2.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.3.2">𝑝</ci><ci id="S4.SS3.p2.9.m7.1.1.2.3.3.3.cmml" xref="S4.SS3.p2.9.m7.1.1.2.3.3.3">′</ci></apply></apply></apply><cn id="S4.SS3.p2.9.m7.1.1.3.cmml" type="integer" xref="S4.SS3.p2.9.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.9.m7.1c">Z_{pp^{\prime}}>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.9.m7.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT > 0</annotation></semantics></math>, inhibits if <math alttext="Z_{pp^{\prime}}<0" class="ltx_Math" display="inline" id="S4.SS3.p2.10.m8.1"><semantics id="S4.SS3.p2.10.m8.1a"><mrow id="S4.SS3.p2.10.m8.1.1" xref="S4.SS3.p2.10.m8.1.1.cmml"><msub id="S4.SS3.p2.10.m8.1.1.2" xref="S4.SS3.p2.10.m8.1.1.2.cmml"><mi id="S4.SS3.p2.10.m8.1.1.2.2" xref="S4.SS3.p2.10.m8.1.1.2.2.cmml">Z</mi><mrow id="S4.SS3.p2.10.m8.1.1.2.3" xref="S4.SS3.p2.10.m8.1.1.2.3.cmml"><mi id="S4.SS3.p2.10.m8.1.1.2.3.2" xref="S4.SS3.p2.10.m8.1.1.2.3.2.cmml">p</mi><mo id="S4.SS3.p2.10.m8.1.1.2.3.1" xref="S4.SS3.p2.10.m8.1.1.2.3.1.cmml">⁢</mo><msup id="S4.SS3.p2.10.m8.1.1.2.3.3" xref="S4.SS3.p2.10.m8.1.1.2.3.3.cmml"><mi id="S4.SS3.p2.10.m8.1.1.2.3.3.2" xref="S4.SS3.p2.10.m8.1.1.2.3.3.2.cmml">p</mi><mo id="S4.SS3.p2.10.m8.1.1.2.3.3.3" xref="S4.SS3.p2.10.m8.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S4.SS3.p2.10.m8.1.1.1" xref="S4.SS3.p2.10.m8.1.1.1.cmml"><</mo><mn id="S4.SS3.p2.10.m8.1.1.3" xref="S4.SS3.p2.10.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.10.m8.1b"><apply id="S4.SS3.p2.10.m8.1.1.cmml" xref="S4.SS3.p2.10.m8.1.1"><lt id="S4.SS3.p2.10.m8.1.1.1.cmml" xref="S4.SS3.p2.10.m8.1.1.1"></lt><apply id="S4.SS3.p2.10.m8.1.1.2.cmml" xref="S4.SS3.p2.10.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.10.m8.1.1.2.1.cmml" xref="S4.SS3.p2.10.m8.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.10.m8.1.1.2.2.cmml" xref="S4.SS3.p2.10.m8.1.1.2.2">𝑍</ci><apply id="S4.SS3.p2.10.m8.1.1.2.3.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3"><times id="S4.SS3.p2.10.m8.1.1.2.3.1.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.1"></times><ci id="S4.SS3.p2.10.m8.1.1.2.3.2.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.2">𝑝</ci><apply id="S4.SS3.p2.10.m8.1.1.2.3.3.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.10.m8.1.1.2.3.3.1.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.3">superscript</csymbol><ci id="S4.SS3.p2.10.m8.1.1.2.3.3.2.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.3.2">𝑝</ci><ci id="S4.SS3.p2.10.m8.1.1.2.3.3.3.cmml" xref="S4.SS3.p2.10.m8.1.1.2.3.3.3">′</ci></apply></apply></apply><cn id="S4.SS3.p2.10.m8.1.1.3.cmml" type="integer" xref="S4.SS3.p2.10.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.10.m8.1c">Z_{pp^{\prime}}<0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.10.m8.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT < 0</annotation></semantics></math> and has no effect if <math alttext="Z_{pp^{\prime}}=0" class="ltx_Math" display="inline" id="S4.SS3.p2.11.m9.1"><semantics id="S4.SS3.p2.11.m9.1a"><mrow id="S4.SS3.p2.11.m9.1.1" xref="S4.SS3.p2.11.m9.1.1.cmml"><msub id="S4.SS3.p2.11.m9.1.1.2" xref="S4.SS3.p2.11.m9.1.1.2.cmml"><mi id="S4.SS3.p2.11.m9.1.1.2.2" xref="S4.SS3.p2.11.m9.1.1.2.2.cmml">Z</mi><mrow id="S4.SS3.p2.11.m9.1.1.2.3" xref="S4.SS3.p2.11.m9.1.1.2.3.cmml"><mi id="S4.SS3.p2.11.m9.1.1.2.3.2" xref="S4.SS3.p2.11.m9.1.1.2.3.2.cmml">p</mi><mo id="S4.SS3.p2.11.m9.1.1.2.3.1" xref="S4.SS3.p2.11.m9.1.1.2.3.1.cmml">⁢</mo><msup id="S4.SS3.p2.11.m9.1.1.2.3.3" xref="S4.SS3.p2.11.m9.1.1.2.3.3.cmml"><mi id="S4.SS3.p2.11.m9.1.1.2.3.3.2" xref="S4.SS3.p2.11.m9.1.1.2.3.3.2.cmml">p</mi><mo id="S4.SS3.p2.11.m9.1.1.2.3.3.3" xref="S4.SS3.p2.11.m9.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S4.SS3.p2.11.m9.1.1.1" xref="S4.SS3.p2.11.m9.1.1.1.cmml">=</mo><mn id="S4.SS3.p2.11.m9.1.1.3" xref="S4.SS3.p2.11.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.11.m9.1b"><apply id="S4.SS3.p2.11.m9.1.1.cmml" xref="S4.SS3.p2.11.m9.1.1"><eq id="S4.SS3.p2.11.m9.1.1.1.cmml" xref="S4.SS3.p2.11.m9.1.1.1"></eq><apply id="S4.SS3.p2.11.m9.1.1.2.cmml" xref="S4.SS3.p2.11.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.11.m9.1.1.2.1.cmml" xref="S4.SS3.p2.11.m9.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.11.m9.1.1.2.2.cmml" xref="S4.SS3.p2.11.m9.1.1.2.2">𝑍</ci><apply id="S4.SS3.p2.11.m9.1.1.2.3.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3"><times id="S4.SS3.p2.11.m9.1.1.2.3.1.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.1"></times><ci id="S4.SS3.p2.11.m9.1.1.2.3.2.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.2">𝑝</ci><apply id="S4.SS3.p2.11.m9.1.1.2.3.3.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.11.m9.1.1.2.3.3.1.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.3">superscript</csymbol><ci id="S4.SS3.p2.11.m9.1.1.2.3.3.2.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.3.2">𝑝</ci><ci id="S4.SS3.p2.11.m9.1.1.2.3.3.3.cmml" xref="S4.SS3.p2.11.m9.1.1.2.3.3.3">′</ci></apply></apply></apply><cn id="S4.SS3.p2.11.m9.1.1.3.cmml" type="integer" xref="S4.SS3.p2.11.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.11.m9.1c">Z_{pp^{\prime}}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.11.m9.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>. The parameter <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS3.p2.12.m10.1"><semantics id="S4.SS3.p2.12.m10.1a"><mi id="S4.SS3.p2.12.m10.1.1" xref="S4.SS3.p2.12.m10.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.12.m10.1b"><ci id="S4.SS3.p2.12.m10.1.1.cmml" xref="S4.SS3.p2.12.m10.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.12.m10.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.12.m10.1d">italic_θ</annotation></semantics></math> sets the base level of gene expression. <math alttext="\theta=-0.3" class="ltx_Math" display="inline" id="S4.SS3.p2.13.m11.1"><semantics id="S4.SS3.p2.13.m11.1a"><mrow id="S4.SS3.p2.13.m11.1.1" xref="S4.SS3.p2.13.m11.1.1.cmml"><mi id="S4.SS3.p2.13.m11.1.1.2" xref="S4.SS3.p2.13.m11.1.1.2.cmml">θ</mi><mo id="S4.SS3.p2.13.m11.1.1.1" xref="S4.SS3.p2.13.m11.1.1.1.cmml">=</mo><mrow id="S4.SS3.p2.13.m11.1.1.3" xref="S4.SS3.p2.13.m11.1.1.3.cmml"><mo id="S4.SS3.p2.13.m11.1.1.3a" xref="S4.SS3.p2.13.m11.1.1.3.cmml">−</mo><mn id="S4.SS3.p2.13.m11.1.1.3.2" xref="S4.SS3.p2.13.m11.1.1.3.2.cmml">0.3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.13.m11.1b"><apply id="S4.SS3.p2.13.m11.1.1.cmml" xref="S4.SS3.p2.13.m11.1.1"><eq id="S4.SS3.p2.13.m11.1.1.1.cmml" xref="S4.SS3.p2.13.m11.1.1.1"></eq><ci id="S4.SS3.p2.13.m11.1.1.2.cmml" xref="S4.SS3.p2.13.m11.1.1.2">𝜃</ci><apply id="S4.SS3.p2.13.m11.1.1.3.cmml" xref="S4.SS3.p2.13.m11.1.1.3"><minus id="S4.SS3.p2.13.m11.1.1.3.1.cmml" xref="S4.SS3.p2.13.m11.1.1.3"></minus><cn id="S4.SS3.p2.13.m11.1.1.3.2.cmml" type="float" xref="S4.SS3.p2.13.m11.1.1.3.2">0.3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.13.m11.1c">\theta=-0.3</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.13.m11.1d">italic_θ = - 0.3</annotation></semantics></math> for all simulations, so protein concentrations equilibrate at 0 when they are not regulated by any TFs. </div> <div class="ltx_para" id="S4.SS3.p3"> To model cell-cell signalling, the first three (<math alttext="p=1,2,3" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.3"><semantics id="S4.SS3.p3.1.m1.3a"><mrow id="S4.SS3.p3.1.m1.3.4" xref="S4.SS3.p3.1.m1.3.4.cmml"><mi id="S4.SS3.p3.1.m1.3.4.2" xref="S4.SS3.p3.1.m1.3.4.2.cmml">p</mi><mo id="S4.SS3.p3.1.m1.3.4.1" xref="S4.SS3.p3.1.m1.3.4.1.cmml">=</mo><mrow id="S4.SS3.p3.1.m1.3.4.3.2" xref="S4.SS3.p3.1.m1.3.4.3.1.cmml"><mn id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml">1</mn><mo id="S4.SS3.p3.1.m1.3.4.3.2.1" xref="S4.SS3.p3.1.m1.3.4.3.1.cmml">,</mo><mn id="S4.SS3.p3.1.m1.2.2" xref="S4.SS3.p3.1.m1.2.2.cmml">2</mn><mo id="S4.SS3.p3.1.m1.3.4.3.2.2" xref="S4.SS3.p3.1.m1.3.4.3.1.cmml">,</mo><mn id="S4.SS3.p3.1.m1.3.3" xref="S4.SS3.p3.1.m1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.3b"><apply id="S4.SS3.p3.1.m1.3.4.cmml" xref="S4.SS3.p3.1.m1.3.4"><eq id="S4.SS3.p3.1.m1.3.4.1.cmml" xref="S4.SS3.p3.1.m1.3.4.1"></eq><ci id="S4.SS3.p3.1.m1.3.4.2.cmml" xref="S4.SS3.p3.1.m1.3.4.2">𝑝</ci><list id="S4.SS3.p3.1.m1.3.4.3.1.cmml" xref="S4.SS3.p3.1.m1.3.4.3.2"><cn id="S4.SS3.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS3.p3.1.m1.1.1">1</cn><cn id="S4.SS3.p3.1.m1.2.2.cmml" type="integer" xref="S4.SS3.p3.1.m1.2.2">2</cn><cn id="S4.SS3.p3.1.m1.3.3.cmml" type="integer" xref="S4.SS3.p3.1.m1.3.3">3</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.3c">p=1,2,3</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.3d">italic_p = 1 , 2 , 3</annotation></semantics></math>) of the nine transcription factors (TFs) are morphogens. These morphogens diffuse between cells and into the surrounding medium. 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id="S4.SS3.p3.3.m3.1d">italic_i</annotation></semantics></math> on the grid, <math alttext="x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.4.m4.1"><semantics id="S4.SS3.p3.4.m4.1a"><msubsup id="S4.SS3.p3.4.m4.1.1" xref="S4.SS3.p3.4.m4.1.1.cmml"><mi id="S4.SS3.p3.4.m4.1.1.2.2" xref="S4.SS3.p3.4.m4.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.4.m4.1.1.3" xref="S4.SS3.p3.4.m4.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.4.m4.1.1.2.3" xref="S4.SS3.p3.4.m4.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.4.m4.1b"><apply id="S4.SS3.p3.4.m4.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.4.m4.1.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1">subscript</csymbol><apply id="S4.SS3.p3.4.m4.1.1.2.cmml" xref="S4.SS3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.4.m4.1.1.2.1.cmml" xref="S4.SS3.p3.4.m4.1.1">superscript</csymbol><ci id="S4.SS3.p3.4.m4.1.1.2.2.cmml" xref="S4.SS3.p3.4.m4.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.4.m4.1.1.2.3.cmml" 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id="S4.E4.m1.1.1.1.1.1.5.3.cmml" xref="S4.E4.m1.1.1.1.1.1.5.3"><minus id="S4.E4.m1.1.1.1.1.1.5.3.1.cmml" xref="S4.E4.m1.1.1.1.1.1.5.3.1"></minus><ci id="S4.E4.m1.1.1.1.1.1.5.3.2.cmml" xref="S4.E4.m1.1.1.1.1.1.5.3.2">𝑁</ci><ci id="S4.E4.m1.1.1.1.1.1.5.3.3.cmml" xref="S4.E4.m1.1.1.1.1.1.5.3.3">𝑝</ci></apply></apply></apply></apply><apply id="S4.E4.m1.1.1.1.3.cmml" xref="S4.E4.m1.1.1.1.3"><times id="S4.E4.m1.1.1.1.3.1.cmml" xref="S4.E4.m1.1.1.1.3.1"></times><ci id="S4.E4.m1.1.1.1.3.2.cmml" xref="S4.E4.m1.1.1.1.3.2">𝜂</ci><apply id="S4.E4.m1.1.1.1.3.3.cmml" xref="S4.E4.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.3.3.1.cmml" xref="S4.E4.m1.1.1.1.3.3">subscript</csymbol><apply id="S4.E4.m1.1.1.1.3.3.2.cmml" xref="S4.E4.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.3.3.2.1.cmml" xref="S4.E4.m1.1.1.1.3.3">superscript</csymbol><ci id="S4.E4.m1.1.1.1.3.3.2.2.cmml" xref="S4.E4.m1.1.1.1.3.3.2.2">𝑥</ci><ci id="S4.E4.m1.1.1.1.3.3.2.3.cmml" xref="S4.E4.m1.1.1.1.3.3.2.3">𝑖</ci></apply><ci id="S4.E4.m1.1.1.1.3.3.3.cmml" xref="S4.E4.m1.1.1.1.3.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.1c">\frac{dx_{p}^{i}}{dt}=D\nabla^{2}x^{i}_{p}+\omega H\left(\sigma-1\right)x^{% \sigma}_{N-p}-\eta x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.1d">divide start_ARG italic_d italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG = italic_D ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_ω italic_H ( italic_σ - 1 ) italic_x start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N - italic_p end_POSTSUBSCRIPT - italic_η italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(4)</td> </tr></tbody> </table> where <math alttext="D" class="ltx_Math" display="inline" id="S4.SS3.p3.5.m1.1"><semantics id="S4.SS3.p3.5.m1.1a"><mi id="S4.SS3.p3.5.m1.1.1" xref="S4.SS3.p3.5.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.5.m1.1b"><ci id="S4.SS3.p3.5.m1.1.1.cmml" xref="S4.SS3.p3.5.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.5.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.5.m1.1d">italic_D</annotation></semantics></math> is a diffusion constant, <math alttext="\omega" class="ltx_Math" display="inline" id="S4.SS3.p3.6.m2.1"><semantics id="S4.SS3.p3.6.m2.1a"><mi id="S4.SS3.p3.6.m2.1.1" xref="S4.SS3.p3.6.m2.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.6.m2.1b"><ci id="S4.SS3.p3.6.m2.1.1.cmml" xref="S4.SS3.p3.6.m2.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.6.m2.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.6.m2.1d">italic_ω</annotation></semantics></math> is a production rate, <math alttext="\eta" class="ltx_Math" display="inline" id="S4.SS3.p3.7.m3.1"><semantics id="S4.SS3.p3.7.m3.1a"><mi id="S4.SS3.p3.7.m3.1.1" xref="S4.SS3.p3.7.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.7.m3.1b"><ci id="S4.SS3.p3.7.m3.1.1.cmml" xref="S4.SS3.p3.7.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.7.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.7.m3.1d">italic_η</annotation></semantics></math> is a decay rate and <math alttext="H(\sigma-1)" class="ltx_Math" display="inline" id="S4.SS3.p3.8.m4.1"><semantics id="S4.SS3.p3.8.m4.1a"><mrow id="S4.SS3.p3.8.m4.1.1" xref="S4.SS3.p3.8.m4.1.1.cmml"><mi id="S4.SS3.p3.8.m4.1.1.3" xref="S4.SS3.p3.8.m4.1.1.3.cmml">H</mi><mo id="S4.SS3.p3.8.m4.1.1.2" xref="S4.SS3.p3.8.m4.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.p3.8.m4.1.1.1.1" xref="S4.SS3.p3.8.m4.1.1.1.1.1.cmml"><mo id="S4.SS3.p3.8.m4.1.1.1.1.2" stretchy="false" xref="S4.SS3.p3.8.m4.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p3.8.m4.1.1.1.1.1" xref="S4.SS3.p3.8.m4.1.1.1.1.1.cmml"><mi id="S4.SS3.p3.8.m4.1.1.1.1.1.2" xref="S4.SS3.p3.8.m4.1.1.1.1.1.2.cmml">σ</mi><mo id="S4.SS3.p3.8.m4.1.1.1.1.1.1" xref="S4.SS3.p3.8.m4.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS3.p3.8.m4.1.1.1.1.1.3" xref="S4.SS3.p3.8.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.p3.8.m4.1.1.1.1.3" stretchy="false" xref="S4.SS3.p3.8.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.8.m4.1b"><apply id="S4.SS3.p3.8.m4.1.1.cmml" xref="S4.SS3.p3.8.m4.1.1"><times id="S4.SS3.p3.8.m4.1.1.2.cmml" xref="S4.SS3.p3.8.m4.1.1.2"></times><ci id="S4.SS3.p3.8.m4.1.1.3.cmml" xref="S4.SS3.p3.8.m4.1.1.3">𝐻</ci><apply id="S4.SS3.p3.8.m4.1.1.1.1.1.cmml" xref="S4.SS3.p3.8.m4.1.1.1.1"><minus id="S4.SS3.p3.8.m4.1.1.1.1.1.1.cmml" xref="S4.SS3.p3.8.m4.1.1.1.1.1.1"></minus><ci id="S4.SS3.p3.8.m4.1.1.1.1.1.2.cmml" xref="S4.SS3.p3.8.m4.1.1.1.1.1.2">𝜎</ci><cn id="S4.SS3.p3.8.m4.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.p3.8.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.8.m4.1c">H(\sigma-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.8.m4.1d">italic_H ( italic_σ - 1 )</annotation></semantics></math> is the Heaviside step function that evaluates to one if <math alttext="\sigma\geq 1" class="ltx_Math" display="inline" id="S4.SS3.p3.9.m5.1"><semantics id="S4.SS3.p3.9.m5.1a"><mrow id="S4.SS3.p3.9.m5.1.1" xref="S4.SS3.p3.9.m5.1.1.cmml"><mi id="S4.SS3.p3.9.m5.1.1.2" xref="S4.SS3.p3.9.m5.1.1.2.cmml">σ</mi><mo id="S4.SS3.p3.9.m5.1.1.1" xref="S4.SS3.p3.9.m5.1.1.1.cmml">≥</mo><mn id="S4.SS3.p3.9.m5.1.1.3" xref="S4.SS3.p3.9.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.9.m5.1b"><apply id="S4.SS3.p3.9.m5.1.1.cmml" xref="S4.SS3.p3.9.m5.1.1"><geq id="S4.SS3.p3.9.m5.1.1.1.cmml" xref="S4.SS3.p3.9.m5.1.1.1"></geq><ci id="S4.SS3.p3.9.m5.1.1.2.cmml" xref="S4.SS3.p3.9.m5.1.1.2">𝜎</ci><cn id="S4.SS3.p3.9.m5.1.1.3.cmml" type="integer" xref="S4.SS3.p3.9.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.9.m5.1c">\sigma\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.9.m5.1d">italic_σ ≥ 1</annotation></semantics></math> at pixel <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p3.10.m6.1"><semantics id="S4.SS3.p3.10.m6.1a"><mi id="S4.SS3.p3.10.m6.1.1" xref="S4.SS3.p3.10.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.10.m6.1b"><ci id="S4.SS3.p3.10.m6.1.1.cmml" xref="S4.SS3.p3.10.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.10.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.10.m6.1d">italic_i</annotation></semantics></math> (the pixel is occupied by a cell) and zero if <math alttext="\sigma=0" class="ltx_Math" display="inline" id="S4.SS3.p3.11.m7.1"><semantics id="S4.SS3.p3.11.m7.1a"><mrow id="S4.SS3.p3.11.m7.1.1" xref="S4.SS3.p3.11.m7.1.1.cmml"><mi id="S4.SS3.p3.11.m7.1.1.2" xref="S4.SS3.p3.11.m7.1.1.2.cmml">σ</mi><mo id="S4.SS3.p3.11.m7.1.1.1" xref="S4.SS3.p3.11.m7.1.1.1.cmml">=</mo><mn id="S4.SS3.p3.11.m7.1.1.3" xref="S4.SS3.p3.11.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.11.m7.1b"><apply id="S4.SS3.p3.11.m7.1.1.cmml" xref="S4.SS3.p3.11.m7.1.1"><eq id="S4.SS3.p3.11.m7.1.1.1.cmml" xref="S4.SS3.p3.11.m7.1.1.1"></eq><ci id="S4.SS3.p3.11.m7.1.1.2.cmml" xref="S4.SS3.p3.11.m7.1.1.2">𝜎</ci><cn id="S4.SS3.p3.11.m7.1.1.3.cmml" type="integer" xref="S4.SS3.p3.11.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.11.m7.1c">\sigma=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.11.m7.1d">italic_σ = 0</annotation></semantics></math> (the pixel is medium). The concentration <math alttext="x^{\sigma}_{N-p}" class="ltx_Math" display="inline" id="S4.SS3.p3.12.m8.1"><semantics id="S4.SS3.p3.12.m8.1a"><msubsup id="S4.SS3.p3.12.m8.1.1" xref="S4.SS3.p3.12.m8.1.1.cmml"><mi id="S4.SS3.p3.12.m8.1.1.2.2" xref="S4.SS3.p3.12.m8.1.1.2.2.cmml">x</mi><mrow id="S4.SS3.p3.12.m8.1.1.3" xref="S4.SS3.p3.12.m8.1.1.3.cmml"><mi id="S4.SS3.p3.12.m8.1.1.3.2" xref="S4.SS3.p3.12.m8.1.1.3.2.cmml">N</mi><mo id="S4.SS3.p3.12.m8.1.1.3.1" xref="S4.SS3.p3.12.m8.1.1.3.1.cmml">−</mo><mi id="S4.SS3.p3.12.m8.1.1.3.3" xref="S4.SS3.p3.12.m8.1.1.3.3.cmml">p</mi></mrow><mi id="S4.SS3.p3.12.m8.1.1.2.3" xref="S4.SS3.p3.12.m8.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.12.m8.1b"><apply id="S4.SS3.p3.12.m8.1.1.cmml" xref="S4.SS3.p3.12.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m8.1.1.1.cmml" xref="S4.SS3.p3.12.m8.1.1">subscript</csymbol><apply id="S4.SS3.p3.12.m8.1.1.2.cmml" xref="S4.SS3.p3.12.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m8.1.1.2.1.cmml" xref="S4.SS3.p3.12.m8.1.1">superscript</csymbol><ci id="S4.SS3.p3.12.m8.1.1.2.2.cmml" xref="S4.SS3.p3.12.m8.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.12.m8.1.1.2.3.cmml" xref="S4.SS3.p3.12.m8.1.1.2.3">𝜎</ci></apply><apply id="S4.SS3.p3.12.m8.1.1.3.cmml" xref="S4.SS3.p3.12.m8.1.1.3"><minus id="S4.SS3.p3.12.m8.1.1.3.1.cmml" xref="S4.SS3.p3.12.m8.1.1.3.1"></minus><ci id="S4.SS3.p3.12.m8.1.1.3.2.cmml" xref="S4.SS3.p3.12.m8.1.1.3.2">𝑁</ci><ci id="S4.SS3.p3.12.m8.1.1.3.3.cmml" xref="S4.SS3.p3.12.m8.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.12.m8.1c">x^{\sigma}_{N-p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.12.m8.1d">italic_x start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N - italic_p end_POSTSUBSCRIPT</annotation></semantics></math> represents a signalling protein that activates the expression of one morphogen. The concentration of the three signalling proteins is determined by Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a>. The operator <math alttext="\nabla^{2}x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.13.m9.1"><semantics id="S4.SS3.p3.13.m9.1a"><mrow id="S4.SS3.p3.13.m9.1.1" xref="S4.SS3.p3.13.m9.1.1.cmml"><msup id="S4.SS3.p3.13.m9.1.1.1" xref="S4.SS3.p3.13.m9.1.1.1.cmml"><mo id="S4.SS3.p3.13.m9.1.1.1.2" xref="S4.SS3.p3.13.m9.1.1.1.2.cmml">∇</mo><mn id="S4.SS3.p3.13.m9.1.1.1.3" xref="S4.SS3.p3.13.m9.1.1.1.3.cmml">2</mn></msup><msubsup id="S4.SS3.p3.13.m9.1.1.2" xref="S4.SS3.p3.13.m9.1.1.2.cmml"><mi id="S4.SS3.p3.13.m9.1.1.2.2.2" xref="S4.SS3.p3.13.m9.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p3.13.m9.1.1.2.3" xref="S4.SS3.p3.13.m9.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p3.13.m9.1.1.2.2.3" xref="S4.SS3.p3.13.m9.1.1.2.2.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.13.m9.1b"><apply id="S4.SS3.p3.13.m9.1.1.cmml" xref="S4.SS3.p3.13.m9.1.1"><apply id="S4.SS3.p3.13.m9.1.1.1.cmml" xref="S4.SS3.p3.13.m9.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.13.m9.1.1.1.1.cmml" xref="S4.SS3.p3.13.m9.1.1.1">superscript</csymbol><ci id="S4.SS3.p3.13.m9.1.1.1.2.cmml" xref="S4.SS3.p3.13.m9.1.1.1.2">∇</ci><cn id="S4.SS3.p3.13.m9.1.1.1.3.cmml" type="integer" xref="S4.SS3.p3.13.m9.1.1.1.3">2</cn></apply><apply id="S4.SS3.p3.13.m9.1.1.2.cmml" xref="S4.SS3.p3.13.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.13.m9.1.1.2.1.cmml" xref="S4.SS3.p3.13.m9.1.1.2">subscript</csymbol><apply id="S4.SS3.p3.13.m9.1.1.2.2.cmml" xref="S4.SS3.p3.13.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.13.m9.1.1.2.2.1.cmml" xref="S4.SS3.p3.13.m9.1.1.2">superscript</csymbol><ci id="S4.SS3.p3.13.m9.1.1.2.2.2.cmml" xref="S4.SS3.p3.13.m9.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p3.13.m9.1.1.2.2.3.cmml" xref="S4.SS3.p3.13.m9.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.13.m9.1.1.2.3.cmml" xref="S4.SS3.p3.13.m9.1.1.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.13.m9.1c">\nabla^{2}x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.13.m9.1d">∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is the Laplacian acting on <math alttext="x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.14.m10.1"><semantics id="S4.SS3.p3.14.m10.1a"><msubsup id="S4.SS3.p3.14.m10.1.1" xref="S4.SS3.p3.14.m10.1.1.cmml"><mi id="S4.SS3.p3.14.m10.1.1.2.2" xref="S4.SS3.p3.14.m10.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.14.m10.1.1.3" xref="S4.SS3.p3.14.m10.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.14.m10.1.1.2.3" xref="S4.SS3.p3.14.m10.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.14.m10.1b"><apply id="S4.SS3.p3.14.m10.1.1.cmml" xref="S4.SS3.p3.14.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.14.m10.1.1.1.cmml" xref="S4.SS3.p3.14.m10.1.1">subscript</csymbol><apply id="S4.SS3.p3.14.m10.1.1.2.cmml" xref="S4.SS3.p3.14.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.14.m10.1.1.2.1.cmml" xref="S4.SS3.p3.14.m10.1.1">superscript</csymbol><ci id="S4.SS3.p3.14.m10.1.1.2.2.cmml" xref="S4.SS3.p3.14.m10.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.14.m10.1.1.2.3.cmml" xref="S4.SS3.p3.14.m10.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.14.m10.1.1.3.cmml" xref="S4.SS3.p3.14.m10.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.14.m10.1c">x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.14.m10.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, which, in this context, is the difference between <math alttext="x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.15.m11.1"><semantics id="S4.SS3.p3.15.m11.1a"><msubsup id="S4.SS3.p3.15.m11.1.1" xref="S4.SS3.p3.15.m11.1.1.cmml"><mi id="S4.SS3.p3.15.m11.1.1.2.2" xref="S4.SS3.p3.15.m11.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.15.m11.1.1.3" xref="S4.SS3.p3.15.m11.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.15.m11.1.1.2.3" xref="S4.SS3.p3.15.m11.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.15.m11.1b"><apply id="S4.SS3.p3.15.m11.1.1.cmml" xref="S4.SS3.p3.15.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.15.m11.1.1.1.cmml" xref="S4.SS3.p3.15.m11.1.1">subscript</csymbol><apply id="S4.SS3.p3.15.m11.1.1.2.cmml" xref="S4.SS3.p3.15.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.15.m11.1.1.2.1.cmml" xref="S4.SS3.p3.15.m11.1.1">superscript</csymbol><ci id="S4.SS3.p3.15.m11.1.1.2.2.cmml" xref="S4.SS3.p3.15.m11.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.15.m11.1.1.2.3.cmml" xref="S4.SS3.p3.15.m11.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.15.m11.1.1.3.cmml" xref="S4.SS3.p3.15.m11.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.15.m11.1c">x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.15.m11.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and the average <math alttext="x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.16.m12.1"><semantics id="S4.SS3.p3.16.m12.1a"><msubsup id="S4.SS3.p3.16.m12.1.1" xref="S4.SS3.p3.16.m12.1.1.cmml"><mi id="S4.SS3.p3.16.m12.1.1.2.2" xref="S4.SS3.p3.16.m12.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.16.m12.1.1.3" xref="S4.SS3.p3.16.m12.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.16.m12.1.1.2.3" xref="S4.SS3.p3.16.m12.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.16.m12.1b"><apply id="S4.SS3.p3.16.m12.1.1.cmml" xref="S4.SS3.p3.16.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.16.m12.1.1.1.cmml" xref="S4.SS3.p3.16.m12.1.1">subscript</csymbol><apply id="S4.SS3.p3.16.m12.1.1.2.cmml" xref="S4.SS3.p3.16.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.16.m12.1.1.2.1.cmml" xref="S4.SS3.p3.16.m12.1.1">superscript</csymbol><ci id="S4.SS3.p3.16.m12.1.1.2.2.cmml" xref="S4.SS3.p3.16.m12.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.16.m12.1.1.2.3.cmml" xref="S4.SS3.p3.16.m12.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.16.m12.1.1.3.cmml" xref="S4.SS3.p3.16.m12.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.16.m12.1c">x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.16.m12.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> in its von Neumann neighbourhood (i.e., four nearest-neighbouring pixels) divided by a space step <math alttext="dx" class="ltx_Math" display="inline" id="S4.SS3.p3.17.m13.1"><semantics id="S4.SS3.p3.17.m13.1a"><mrow id="S4.SS3.p3.17.m13.1.1" xref="S4.SS3.p3.17.m13.1.1.cmml"><mi id="S4.SS3.p3.17.m13.1.1.2" xref="S4.SS3.p3.17.m13.1.1.2.cmml">d</mi><mo id="S4.SS3.p3.17.m13.1.1.1" xref="S4.SS3.p3.17.m13.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p3.17.m13.1.1.3" xref="S4.SS3.p3.17.m13.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.17.m13.1b"><apply id="S4.SS3.p3.17.m13.1.1.cmml" xref="S4.SS3.p3.17.m13.1.1"><times id="S4.SS3.p3.17.m13.1.1.1.cmml" xref="S4.SS3.p3.17.m13.1.1.1"></times><ci id="S4.SS3.p3.17.m13.1.1.2.cmml" xref="S4.SS3.p3.17.m13.1.1.2">𝑑</ci><ci id="S4.SS3.p3.17.m13.1.1.3.cmml" xref="S4.SS3.p3.17.m13.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.17.m13.1c">dx</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.17.m13.1d">italic_d italic_x</annotation></semantics></math>, with <math alttext="dx=1/250" class="ltx_Math" display="inline" id="S4.SS3.p3.18.m14.1"><semantics id="S4.SS3.p3.18.m14.1a"><mrow id="S4.SS3.p3.18.m14.1.1" xref="S4.SS3.p3.18.m14.1.1.cmml"><mrow id="S4.SS3.p3.18.m14.1.1.2" xref="S4.SS3.p3.18.m14.1.1.2.cmml"><mi id="S4.SS3.p3.18.m14.1.1.2.2" xref="S4.SS3.p3.18.m14.1.1.2.2.cmml">d</mi><mo id="S4.SS3.p3.18.m14.1.1.2.1" xref="S4.SS3.p3.18.m14.1.1.2.1.cmml">⁢</mo><mi id="S4.SS3.p3.18.m14.1.1.2.3" xref="S4.SS3.p3.18.m14.1.1.2.3.cmml">x</mi></mrow><mo id="S4.SS3.p3.18.m14.1.1.1" xref="S4.SS3.p3.18.m14.1.1.1.cmml">=</mo><mrow id="S4.SS3.p3.18.m14.1.1.3" xref="S4.SS3.p3.18.m14.1.1.3.cmml"><mn id="S4.SS3.p3.18.m14.1.1.3.2" xref="S4.SS3.p3.18.m14.1.1.3.2.cmml">1</mn><mo id="S4.SS3.p3.18.m14.1.1.3.1" xref="S4.SS3.p3.18.m14.1.1.3.1.cmml">/</mo><mn id="S4.SS3.p3.18.m14.1.1.3.3" xref="S4.SS3.p3.18.m14.1.1.3.3.cmml">250</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.18.m14.1b"><apply id="S4.SS3.p3.18.m14.1.1.cmml" xref="S4.SS3.p3.18.m14.1.1"><eq id="S4.SS3.p3.18.m14.1.1.1.cmml" xref="S4.SS3.p3.18.m14.1.1.1"></eq><apply id="S4.SS3.p3.18.m14.1.1.2.cmml" xref="S4.SS3.p3.18.m14.1.1.2"><times id="S4.SS3.p3.18.m14.1.1.2.1.cmml" xref="S4.SS3.p3.18.m14.1.1.2.1"></times><ci id="S4.SS3.p3.18.m14.1.1.2.2.cmml" xref="S4.SS3.p3.18.m14.1.1.2.2">𝑑</ci><ci id="S4.SS3.p3.18.m14.1.1.2.3.cmml" xref="S4.SS3.p3.18.m14.1.1.2.3">𝑥</ci></apply><apply id="S4.SS3.p3.18.m14.1.1.3.cmml" xref="S4.SS3.p3.18.m14.1.1.3"><divide id="S4.SS3.p3.18.m14.1.1.3.1.cmml" xref="S4.SS3.p3.18.m14.1.1.3.1"></divide><cn id="S4.SS3.p3.18.m14.1.1.3.2.cmml" type="integer" xref="S4.SS3.p3.18.m14.1.1.3.2">1</cn><cn id="S4.SS3.p3.18.m14.1.1.3.3.cmml" type="integer" xref="S4.SS3.p3.18.m14.1.1.3.3">250</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.18.m14.1c">dx=1/250</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.18.m14.1d">italic_d italic_x = 1 / 250</annotation></semantics></math> (250 is the length of the square grid in pixels). Numerical integration for Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E4" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a> occurs with with <math alttext="\Delta t=1" class="ltx_Math" display="inline" id="S4.SS3.p3.19.m15.1"><semantics id="S4.SS3.p3.19.m15.1a"><mrow id="S4.SS3.p3.19.m15.1.1" xref="S4.SS3.p3.19.m15.1.1.cmml"><mrow id="S4.SS3.p3.19.m15.1.1.2" xref="S4.SS3.p3.19.m15.1.1.2.cmml"><mi id="S4.SS3.p3.19.m15.1.1.2.2" mathvariant="normal" xref="S4.SS3.p3.19.m15.1.1.2.2.cmml">Δ</mi><mo id="S4.SS3.p3.19.m15.1.1.2.1" xref="S4.SS3.p3.19.m15.1.1.2.1.cmml">⁢</mo><mi id="S4.SS3.p3.19.m15.1.1.2.3" xref="S4.SS3.p3.19.m15.1.1.2.3.cmml">t</mi></mrow><mo id="S4.SS3.p3.19.m15.1.1.1" xref="S4.SS3.p3.19.m15.1.1.1.cmml">=</mo><mn id="S4.SS3.p3.19.m15.1.1.3" xref="S4.SS3.p3.19.m15.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.19.m15.1b"><apply id="S4.SS3.p3.19.m15.1.1.cmml" xref="S4.SS3.p3.19.m15.1.1"><eq id="S4.SS3.p3.19.m15.1.1.1.cmml" xref="S4.SS3.p3.19.m15.1.1.1"></eq><apply id="S4.SS3.p3.19.m15.1.1.2.cmml" xref="S4.SS3.p3.19.m15.1.1.2"><times id="S4.SS3.p3.19.m15.1.1.2.1.cmml" xref="S4.SS3.p3.19.m15.1.1.2.1"></times><ci id="S4.SS3.p3.19.m15.1.1.2.2.cmml" xref="S4.SS3.p3.19.m15.1.1.2.2">Δ</ci><ci id="S4.SS3.p3.19.m15.1.1.2.3.cmml" xref="S4.SS3.p3.19.m15.1.1.2.3">𝑡</ci></apply><cn id="S4.SS3.p3.19.m15.1.1.3.cmml" type="integer" xref="S4.SS3.p3.19.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.19.m15.1c">\Delta t=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.19.m15.1d">roman_Δ italic_t = 1</annotation></semantics></math> and is performed separately from Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a>. Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E4" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">4</a> is subject to the initial condition <math alttext="x^{i}_{p}=0" class="ltx_Math" display="inline" id="S4.SS3.p3.20.m16.1"><semantics id="S4.SS3.p3.20.m16.1a"><mrow id="S4.SS3.p3.20.m16.1.1" xref="S4.SS3.p3.20.m16.1.1.cmml"><msubsup id="S4.SS3.p3.20.m16.1.1.2" xref="S4.SS3.p3.20.m16.1.1.2.cmml"><mi id="S4.SS3.p3.20.m16.1.1.2.2.2" xref="S4.SS3.p3.20.m16.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p3.20.m16.1.1.2.3" xref="S4.SS3.p3.20.m16.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p3.20.m16.1.1.2.2.3" xref="S4.SS3.p3.20.m16.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p3.20.m16.1.1.1" xref="S4.SS3.p3.20.m16.1.1.1.cmml">=</mo><mn id="S4.SS3.p3.20.m16.1.1.3" xref="S4.SS3.p3.20.m16.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.20.m16.1b"><apply id="S4.SS3.p3.20.m16.1.1.cmml" xref="S4.SS3.p3.20.m16.1.1"><eq id="S4.SS3.p3.20.m16.1.1.1.cmml" xref="S4.SS3.p3.20.m16.1.1.1"></eq><apply id="S4.SS3.p3.20.m16.1.1.2.cmml" xref="S4.SS3.p3.20.m16.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.20.m16.1.1.2.1.cmml" xref="S4.SS3.p3.20.m16.1.1.2">subscript</csymbol><apply id="S4.SS3.p3.20.m16.1.1.2.2.cmml" xref="S4.SS3.p3.20.m16.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.20.m16.1.1.2.2.1.cmml" xref="S4.SS3.p3.20.m16.1.1.2">superscript</csymbol><ci id="S4.SS3.p3.20.m16.1.1.2.2.2.cmml" xref="S4.SS3.p3.20.m16.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p3.20.m16.1.1.2.2.3.cmml" xref="S4.SS3.p3.20.m16.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.20.m16.1.1.2.3.cmml" xref="S4.SS3.p3.20.m16.1.1.2.3">𝑝</ci></apply><cn id="S4.SS3.p3.20.m16.1.1.3.cmml" type="integer" xref="S4.SS3.p3.20.m16.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.20.m16.1c">x^{i}_{p}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.20.m16.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 0</annotation></semantics></math> for all <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p3.21.m17.1"><semantics id="S4.SS3.p3.21.m17.1a"><mi id="S4.SS3.p3.21.m17.1.1" xref="S4.SS3.p3.21.m17.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.21.m17.1b"><ci id="S4.SS3.p3.21.m17.1.1.cmml" xref="S4.SS3.p3.21.m17.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.21.m17.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.21.m17.1d">italic_i</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p3.22.m18.1"><semantics id="S4.SS3.p3.22.m18.1a"><mi id="S4.SS3.p3.22.m18.1.1" xref="S4.SS3.p3.22.m18.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.22.m18.1b"><ci id="S4.SS3.p3.22.m18.1.1.cmml" xref="S4.SS3.p3.22.m18.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.22.m18.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.22.m18.1d">italic_p</annotation></semantics></math>. The pixels at the boundary of the grid are subject to <math alttext="x_{i}^{p}=0" class="ltx_Math" display="inline" id="S4.SS3.p3.23.m19.1"><semantics id="S4.SS3.p3.23.m19.1a"><mrow id="S4.SS3.p3.23.m19.1.1" xref="S4.SS3.p3.23.m19.1.1.cmml"><msubsup id="S4.SS3.p3.23.m19.1.1.2" xref="S4.SS3.p3.23.m19.1.1.2.cmml"><mi id="S4.SS3.p3.23.m19.1.1.2.2.2" xref="S4.SS3.p3.23.m19.1.1.2.2.2.cmml">x</mi><mi id="S4.SS3.p3.23.m19.1.1.2.2.3" xref="S4.SS3.p3.23.m19.1.1.2.2.3.cmml">i</mi><mi id="S4.SS3.p3.23.m19.1.1.2.3" xref="S4.SS3.p3.23.m19.1.1.2.3.cmml">p</mi></msubsup><mo id="S4.SS3.p3.23.m19.1.1.1" xref="S4.SS3.p3.23.m19.1.1.1.cmml">=</mo><mn id="S4.SS3.p3.23.m19.1.1.3" xref="S4.SS3.p3.23.m19.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.23.m19.1b"><apply id="S4.SS3.p3.23.m19.1.1.cmml" xref="S4.SS3.p3.23.m19.1.1"><eq id="S4.SS3.p3.23.m19.1.1.1.cmml" xref="S4.SS3.p3.23.m19.1.1.1"></eq><apply id="S4.SS3.p3.23.m19.1.1.2.cmml" xref="S4.SS3.p3.23.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.23.m19.1.1.2.1.cmml" xref="S4.SS3.p3.23.m19.1.1.2">superscript</csymbol><apply id="S4.SS3.p3.23.m19.1.1.2.2.cmml" xref="S4.SS3.p3.23.m19.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.23.m19.1.1.2.2.1.cmml" xref="S4.SS3.p3.23.m19.1.1.2">subscript</csymbol><ci id="S4.SS3.p3.23.m19.1.1.2.2.2.cmml" xref="S4.SS3.p3.23.m19.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.p3.23.m19.1.1.2.2.3.cmml" xref="S4.SS3.p3.23.m19.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.23.m19.1.1.2.3.cmml" xref="S4.SS3.p3.23.m19.1.1.2.3">𝑝</ci></apply><cn id="S4.SS3.p3.23.m19.1.1.3.cmml" type="integer" xref="S4.SS3.p3.23.m19.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.23.m19.1c">x_{i}^{p}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.23.m19.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT = 0</annotation></semantics></math> for all <math alttext="t" class="ltx_Math" display="inline" id="S4.SS3.p3.24.m20.1"><semantics id="S4.SS3.p3.24.m20.1a"><mi id="S4.SS3.p3.24.m20.1.1" xref="S4.SS3.p3.24.m20.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.24.m20.1b"><ci id="S4.SS3.p3.24.m20.1.1.cmml" xref="S4.SS3.p3.24.m20.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.24.m20.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.24.m20.1d">italic_t</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.p3.25.m21.1"><semantics id="S4.SS3.p3.25.m21.1a"><mi id="S4.SS3.p3.25.m21.1.1" xref="S4.SS3.p3.25.m21.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.25.m21.1b"><ci id="S4.SS3.p3.25.m21.1.1.cmml" xref="S4.SS3.p3.25.m21.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.25.m21.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.25.m21.1d">italic_p</annotation></semantics></math>. The constants <math alttext="D=8\times 10^{-7}" class="ltx_Math" display="inline" id="S4.SS3.p3.26.m22.1"><semantics id="S4.SS3.p3.26.m22.1a"><mrow id="S4.SS3.p3.26.m22.1.1" xref="S4.SS3.p3.26.m22.1.1.cmml"><mi id="S4.SS3.p3.26.m22.1.1.2" xref="S4.SS3.p3.26.m22.1.1.2.cmml">D</mi><mo id="S4.SS3.p3.26.m22.1.1.1" xref="S4.SS3.p3.26.m22.1.1.1.cmml">=</mo><mrow id="S4.SS3.p3.26.m22.1.1.3" xref="S4.SS3.p3.26.m22.1.1.3.cmml"><mn id="S4.SS3.p3.26.m22.1.1.3.2" xref="S4.SS3.p3.26.m22.1.1.3.2.cmml">8</mn><mo id="S4.SS3.p3.26.m22.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.26.m22.1.1.3.1.cmml">×</mo><msup id="S4.SS3.p3.26.m22.1.1.3.3" xref="S4.SS3.p3.26.m22.1.1.3.3.cmml"><mn id="S4.SS3.p3.26.m22.1.1.3.3.2" xref="S4.SS3.p3.26.m22.1.1.3.3.2.cmml">10</mn><mrow id="S4.SS3.p3.26.m22.1.1.3.3.3" xref="S4.SS3.p3.26.m22.1.1.3.3.3.cmml"><mo id="S4.SS3.p3.26.m22.1.1.3.3.3a" xref="S4.SS3.p3.26.m22.1.1.3.3.3.cmml">−</mo><mn id="S4.SS3.p3.26.m22.1.1.3.3.3.2" xref="S4.SS3.p3.26.m22.1.1.3.3.3.2.cmml">7</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.26.m22.1b"><apply id="S4.SS3.p3.26.m22.1.1.cmml" xref="S4.SS3.p3.26.m22.1.1"><eq id="S4.SS3.p3.26.m22.1.1.1.cmml" xref="S4.SS3.p3.26.m22.1.1.1"></eq><ci id="S4.SS3.p3.26.m22.1.1.2.cmml" xref="S4.SS3.p3.26.m22.1.1.2">𝐷</ci><apply id="S4.SS3.p3.26.m22.1.1.3.cmml" xref="S4.SS3.p3.26.m22.1.1.3"><times id="S4.SS3.p3.26.m22.1.1.3.1.cmml" xref="S4.SS3.p3.26.m22.1.1.3.1"></times><cn id="S4.SS3.p3.26.m22.1.1.3.2.cmml" type="integer" xref="S4.SS3.p3.26.m22.1.1.3.2">8</cn><apply id="S4.SS3.p3.26.m22.1.1.3.3.cmml" xref="S4.SS3.p3.26.m22.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.26.m22.1.1.3.3.1.cmml" xref="S4.SS3.p3.26.m22.1.1.3.3">superscript</csymbol><cn id="S4.SS3.p3.26.m22.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p3.26.m22.1.1.3.3.2">10</cn><apply id="S4.SS3.p3.26.m22.1.1.3.3.3.cmml" xref="S4.SS3.p3.26.m22.1.1.3.3.3"><minus id="S4.SS3.p3.26.m22.1.1.3.3.3.1.cmml" xref="S4.SS3.p3.26.m22.1.1.3.3.3"></minus><cn id="S4.SS3.p3.26.m22.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS3.p3.26.m22.1.1.3.3.3.2">7</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.26.m22.1c">D=8\times 10^{-7}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.26.m22.1d">italic_D = 8 × 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\omega=2.4\times 10^{-3}" class="ltx_Math" display="inline" id="S4.SS3.p3.27.m23.1"><semantics id="S4.SS3.p3.27.m23.1a"><mrow id="S4.SS3.p3.27.m23.1.1" xref="S4.SS3.p3.27.m23.1.1.cmml"><mi id="S4.SS3.p3.27.m23.1.1.2" xref="S4.SS3.p3.27.m23.1.1.2.cmml">ω</mi><mo id="S4.SS3.p3.27.m23.1.1.1" xref="S4.SS3.p3.27.m23.1.1.1.cmml">=</mo><mrow id="S4.SS3.p3.27.m23.1.1.3" xref="S4.SS3.p3.27.m23.1.1.3.cmml"><mn id="S4.SS3.p3.27.m23.1.1.3.2" xref="S4.SS3.p3.27.m23.1.1.3.2.cmml">2.4</mn><mo id="S4.SS3.p3.27.m23.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.27.m23.1.1.3.1.cmml">×</mo><msup id="S4.SS3.p3.27.m23.1.1.3.3" xref="S4.SS3.p3.27.m23.1.1.3.3.cmml"><mn id="S4.SS3.p3.27.m23.1.1.3.3.2" xref="S4.SS3.p3.27.m23.1.1.3.3.2.cmml">10</mn><mrow id="S4.SS3.p3.27.m23.1.1.3.3.3" xref="S4.SS3.p3.27.m23.1.1.3.3.3.cmml"><mo id="S4.SS3.p3.27.m23.1.1.3.3.3a" xref="S4.SS3.p3.27.m23.1.1.3.3.3.cmml">−</mo><mn id="S4.SS3.p3.27.m23.1.1.3.3.3.2" xref="S4.SS3.p3.27.m23.1.1.3.3.3.2.cmml">3</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.27.m23.1b"><apply id="S4.SS3.p3.27.m23.1.1.cmml" xref="S4.SS3.p3.27.m23.1.1"><eq id="S4.SS3.p3.27.m23.1.1.1.cmml" xref="S4.SS3.p3.27.m23.1.1.1"></eq><ci id="S4.SS3.p3.27.m23.1.1.2.cmml" xref="S4.SS3.p3.27.m23.1.1.2">𝜔</ci><apply id="S4.SS3.p3.27.m23.1.1.3.cmml" xref="S4.SS3.p3.27.m23.1.1.3"><times id="S4.SS3.p3.27.m23.1.1.3.1.cmml" xref="S4.SS3.p3.27.m23.1.1.3.1"></times><cn id="S4.SS3.p3.27.m23.1.1.3.2.cmml" type="float" xref="S4.SS3.p3.27.m23.1.1.3.2">2.4</cn><apply id="S4.SS3.p3.27.m23.1.1.3.3.cmml" xref="S4.SS3.p3.27.m23.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.27.m23.1.1.3.3.1.cmml" xref="S4.SS3.p3.27.m23.1.1.3.3">superscript</csymbol><cn id="S4.SS3.p3.27.m23.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p3.27.m23.1.1.3.3.2">10</cn><apply id="S4.SS3.p3.27.m23.1.1.3.3.3.cmml" xref="S4.SS3.p3.27.m23.1.1.3.3.3"><minus id="S4.SS3.p3.27.m23.1.1.3.3.3.1.cmml" xref="S4.SS3.p3.27.m23.1.1.3.3.3"></minus><cn id="S4.SS3.p3.27.m23.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS3.p3.27.m23.1.1.3.3.3.2">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.27.m23.1c">\omega=2.4\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.27.m23.1d">italic_ω = 2.4 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\eta=2\times 10^{-3}" class="ltx_Math" display="inline" id="S4.SS3.p3.28.m24.1"><semantics id="S4.SS3.p3.28.m24.1a"><mrow id="S4.SS3.p3.28.m24.1.1" xref="S4.SS3.p3.28.m24.1.1.cmml"><mi id="S4.SS3.p3.28.m24.1.1.2" xref="S4.SS3.p3.28.m24.1.1.2.cmml">η</mi><mo id="S4.SS3.p3.28.m24.1.1.1" xref="S4.SS3.p3.28.m24.1.1.1.cmml">=</mo><mrow id="S4.SS3.p3.28.m24.1.1.3" xref="S4.SS3.p3.28.m24.1.1.3.cmml"><mn id="S4.SS3.p3.28.m24.1.1.3.2" xref="S4.SS3.p3.28.m24.1.1.3.2.cmml">2</mn><mo id="S4.SS3.p3.28.m24.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.28.m24.1.1.3.1.cmml">×</mo><msup id="S4.SS3.p3.28.m24.1.1.3.3" xref="S4.SS3.p3.28.m24.1.1.3.3.cmml"><mn id="S4.SS3.p3.28.m24.1.1.3.3.2" xref="S4.SS3.p3.28.m24.1.1.3.3.2.cmml">10</mn><mrow id="S4.SS3.p3.28.m24.1.1.3.3.3" xref="S4.SS3.p3.28.m24.1.1.3.3.3.cmml"><mo id="S4.SS3.p3.28.m24.1.1.3.3.3a" xref="S4.SS3.p3.28.m24.1.1.3.3.3.cmml">−</mo><mn id="S4.SS3.p3.28.m24.1.1.3.3.3.2" xref="S4.SS3.p3.28.m24.1.1.3.3.3.2.cmml">3</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.28.m24.1b"><apply id="S4.SS3.p3.28.m24.1.1.cmml" xref="S4.SS3.p3.28.m24.1.1"><eq id="S4.SS3.p3.28.m24.1.1.1.cmml" xref="S4.SS3.p3.28.m24.1.1.1"></eq><ci id="S4.SS3.p3.28.m24.1.1.2.cmml" xref="S4.SS3.p3.28.m24.1.1.2">𝜂</ci><apply id="S4.SS3.p3.28.m24.1.1.3.cmml" xref="S4.SS3.p3.28.m24.1.1.3"><times id="S4.SS3.p3.28.m24.1.1.3.1.cmml" xref="S4.SS3.p3.28.m24.1.1.3.1"></times><cn id="S4.SS3.p3.28.m24.1.1.3.2.cmml" type="integer" xref="S4.SS3.p3.28.m24.1.1.3.2">2</cn><apply id="S4.SS3.p3.28.m24.1.1.3.3.cmml" xref="S4.SS3.p3.28.m24.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.28.m24.1.1.3.3.1.cmml" xref="S4.SS3.p3.28.m24.1.1.3.3">superscript</csymbol><cn id="S4.SS3.p3.28.m24.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p3.28.m24.1.1.3.3.2">10</cn><apply id="S4.SS3.p3.28.m24.1.1.3.3.3.cmml" xref="S4.SS3.p3.28.m24.1.1.3.3.3"><minus id="S4.SS3.p3.28.m24.1.1.3.3.3.1.cmml" xref="S4.SS3.p3.28.m24.1.1.3.3.3"></minus><cn id="S4.SS3.p3.28.m24.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS3.p3.28.m24.1.1.3.3.3.2">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.28.m24.1c">\eta=2\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.28.m24.1d">italic_η = 2 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> were used for all simulations and are the same for all three morphogens. We chose these values for two reasons. The first is so that the maximum concentration is approximately 1 and thus similar to the concentrations of TFs per Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a>. The second is so that the characteristic diffusion length, <math alttext="\sqrt{D/\eta dx}" class="ltx_Math" display="inline" id="S4.SS3.p3.29.m25.1"><semantics id="S4.SS3.p3.29.m25.1a"><msqrt id="S4.SS3.p3.29.m25.1.1" xref="S4.SS3.p3.29.m25.1.1.cmml"><mrow id="S4.SS3.p3.29.m25.1.1.2" xref="S4.SS3.p3.29.m25.1.1.2.cmml"><mrow id="S4.SS3.p3.29.m25.1.1.2.2" xref="S4.SS3.p3.29.m25.1.1.2.2.cmml"><mi id="S4.SS3.p3.29.m25.1.1.2.2.2" xref="S4.SS3.p3.29.m25.1.1.2.2.2.cmml">D</mi><mo id="S4.SS3.p3.29.m25.1.1.2.2.1" xref="S4.SS3.p3.29.m25.1.1.2.2.1.cmml">/</mo><mi id="S4.SS3.p3.29.m25.1.1.2.2.3" xref="S4.SS3.p3.29.m25.1.1.2.2.3.cmml">η</mi></mrow><mo id="S4.SS3.p3.29.m25.1.1.2.1" xref="S4.SS3.p3.29.m25.1.1.2.1.cmml">⁢</mo><mi id="S4.SS3.p3.29.m25.1.1.2.3" xref="S4.SS3.p3.29.m25.1.1.2.3.cmml">d</mi><mo id="S4.SS3.p3.29.m25.1.1.2.1a" xref="S4.SS3.p3.29.m25.1.1.2.1.cmml">⁢</mo><mi id="S4.SS3.p3.29.m25.1.1.2.4" xref="S4.SS3.p3.29.m25.1.1.2.4.cmml">x</mi></mrow></msqrt><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.29.m25.1b"><apply id="S4.SS3.p3.29.m25.1.1.cmml" xref="S4.SS3.p3.29.m25.1.1"><root id="S4.SS3.p3.29.m25.1.1a.cmml" xref="S4.SS3.p3.29.m25.1.1"></root><apply id="S4.SS3.p3.29.m25.1.1.2.cmml" xref="S4.SS3.p3.29.m25.1.1.2"><times id="S4.SS3.p3.29.m25.1.1.2.1.cmml" xref="S4.SS3.p3.29.m25.1.1.2.1"></times><apply id="S4.SS3.p3.29.m25.1.1.2.2.cmml" xref="S4.SS3.p3.29.m25.1.1.2.2"><divide id="S4.SS3.p3.29.m25.1.1.2.2.1.cmml" xref="S4.SS3.p3.29.m25.1.1.2.2.1"></divide><ci id="S4.SS3.p3.29.m25.1.1.2.2.2.cmml" xref="S4.SS3.p3.29.m25.1.1.2.2.2">𝐷</ci><ci id="S4.SS3.p3.29.m25.1.1.2.2.3.cmml" xref="S4.SS3.p3.29.m25.1.1.2.2.3">𝜂</ci></apply><ci id="S4.SS3.p3.29.m25.1.1.2.3.cmml" xref="S4.SS3.p3.29.m25.1.1.2.3">𝑑</ci><ci id="S4.SS3.p3.29.m25.1.1.2.4.cmml" xref="S4.SS3.p3.29.m25.1.1.2.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.29.m25.1c">\sqrt{D/\eta dx}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.29.m25.1d">square-root start_ARG italic_D / italic_η italic_d italic_x end_ARG</annotation></semantics></math>, is similar to that of a paracrine morphogen, such as Wnts, which signal only to nearby cells <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib76" title="">76</a>]</cite>. This characteristic diffusion length is five pixels, which is approximately the diameter of a cell. Since <math alttext="x_{p}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p3.30.m26.1"><semantics id="S4.SS3.p3.30.m26.1a"><msubsup id="S4.SS3.p3.30.m26.1.1" xref="S4.SS3.p3.30.m26.1.1.cmml"><mi id="S4.SS3.p3.30.m26.1.1.2.2" xref="S4.SS3.p3.30.m26.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.30.m26.1.1.2.3" xref="S4.SS3.p3.30.m26.1.1.2.3.cmml">p</mi><mi id="S4.SS3.p3.30.m26.1.1.3" xref="S4.SS3.p3.30.m26.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.30.m26.1b"><apply id="S4.SS3.p3.30.m26.1.1.cmml" xref="S4.SS3.p3.30.m26.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.30.m26.1.1.1.cmml" xref="S4.SS3.p3.30.m26.1.1">superscript</csymbol><apply id="S4.SS3.p3.30.m26.1.1.2.cmml" xref="S4.SS3.p3.30.m26.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.30.m26.1.1.2.1.cmml" xref="S4.SS3.p3.30.m26.1.1">subscript</csymbol><ci id="S4.SS3.p3.30.m26.1.1.2.2.cmml" xref="S4.SS3.p3.30.m26.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.30.m26.1.1.2.3.cmml" xref="S4.SS3.p3.30.m26.1.1.2.3">𝑝</ci></apply><ci id="S4.SS3.p3.30.m26.1.1.3.cmml" xref="S4.SS3.p3.30.m26.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.30.m26.1c">x_{p}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.30.m26.1d">italic_x start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> regulates genes in each cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS3.p3.31.m27.1"><semantics id="S4.SS3.p3.31.m27.1a"><mi id="S4.SS3.p3.31.m27.1.1" xref="S4.SS3.p3.31.m27.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.31.m27.1b"><ci id="S4.SS3.p3.31.m27.1.1.cmml" xref="S4.SS3.p3.31.m27.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.31.m27.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.31.m27.1d">italic_σ</annotation></semantics></math>, we average <math alttext="x^{i}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.32.m28.1"><semantics id="S4.SS3.p3.32.m28.1a"><msubsup id="S4.SS3.p3.32.m28.1.1" xref="S4.SS3.p3.32.m28.1.1.cmml"><mi id="S4.SS3.p3.32.m28.1.1.2.2" xref="S4.SS3.p3.32.m28.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.32.m28.1.1.3" xref="S4.SS3.p3.32.m28.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.32.m28.1.1.2.3" xref="S4.SS3.p3.32.m28.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.32.m28.1b"><apply id="S4.SS3.p3.32.m28.1.1.cmml" xref="S4.SS3.p3.32.m28.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.32.m28.1.1.1.cmml" xref="S4.SS3.p3.32.m28.1.1">subscript</csymbol><apply id="S4.SS3.p3.32.m28.1.1.2.cmml" xref="S4.SS3.p3.32.m28.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.32.m28.1.1.2.1.cmml" xref="S4.SS3.p3.32.m28.1.1">superscript</csymbol><ci id="S4.SS3.p3.32.m28.1.1.2.2.cmml" xref="S4.SS3.p3.32.m28.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.32.m28.1.1.2.3.cmml" xref="S4.SS3.p3.32.m28.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.p3.32.m28.1.1.3.cmml" xref="S4.SS3.p3.32.m28.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.32.m28.1c">x^{i}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.32.m28.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> over all <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p3.33.m29.1"><semantics id="S4.SS3.p3.33.m29.1a"><mi id="S4.SS3.p3.33.m29.1.1" xref="S4.SS3.p3.33.m29.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.33.m29.1b"><ci id="S4.SS3.p3.33.m29.1.1.cmml" xref="S4.SS3.p3.33.m29.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.33.m29.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.33.m29.1d">italic_i</annotation></semantics></math> with the value <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS3.p3.34.m30.1"><semantics id="S4.SS3.p3.34.m30.1a"><mi id="S4.SS3.p3.34.m30.1.1" xref="S4.SS3.p3.34.m30.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.34.m30.1b"><ci id="S4.SS3.p3.34.m30.1.1.cmml" xref="S4.SS3.p3.34.m30.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.34.m30.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.34.m30.1d">italic_σ</annotation></semantics></math> to obtain <math alttext="x^{\sigma}_{p}" class="ltx_Math" display="inline" id="S4.SS3.p3.35.m31.1"><semantics id="S4.SS3.p3.35.m31.1a"><msubsup id="S4.SS3.p3.35.m31.1.1" xref="S4.SS3.p3.35.m31.1.1.cmml"><mi id="S4.SS3.p3.35.m31.1.1.2.2" xref="S4.SS3.p3.35.m31.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p3.35.m31.1.1.3" xref="S4.SS3.p3.35.m31.1.1.3.cmml">p</mi><mi id="S4.SS3.p3.35.m31.1.1.2.3" xref="S4.SS3.p3.35.m31.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.35.m31.1b"><apply id="S4.SS3.p3.35.m31.1.1.cmml" xref="S4.SS3.p3.35.m31.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.35.m31.1.1.1.cmml" xref="S4.SS3.p3.35.m31.1.1">subscript</csymbol><apply id="S4.SS3.p3.35.m31.1.1.2.cmml" xref="S4.SS3.p3.35.m31.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.35.m31.1.1.2.1.cmml" xref="S4.SS3.p3.35.m31.1.1">superscript</csymbol><ci id="S4.SS3.p3.35.m31.1.1.2.2.cmml" xref="S4.SS3.p3.35.m31.1.1.2.2">𝑥</ci><ci id="S4.SS3.p3.35.m31.1.1.2.3.cmml" xref="S4.SS3.p3.35.m31.1.1.2.3">𝜎</ci></apply><ci id="S4.SS3.p3.35.m31.1.1.3.cmml" xref="S4.SS3.p3.35.m31.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.35.m31.1c">x^{\sigma}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.35.m31.1d">italic_x start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. During the equilibration period (DTS < 1500), the parameters <math alttext="D,\omega,\eta" class="ltx_Math" display="inline" id="S4.SS3.p3.36.m32.3"><semantics id="S4.SS3.p3.36.m32.3a"><mrow id="S4.SS3.p3.36.m32.3.4.2" xref="S4.SS3.p3.36.m32.3.4.1.cmml"><mi id="S4.SS3.p3.36.m32.1.1" xref="S4.SS3.p3.36.m32.1.1.cmml">D</mi><mo id="S4.SS3.p3.36.m32.3.4.2.1" xref="S4.SS3.p3.36.m32.3.4.1.cmml">,</mo><mi id="S4.SS3.p3.36.m32.2.2" xref="S4.SS3.p3.36.m32.2.2.cmml">ω</mi><mo id="S4.SS3.p3.36.m32.3.4.2.2" xref="S4.SS3.p3.36.m32.3.4.1.cmml">,</mo><mi id="S4.SS3.p3.36.m32.3.3" xref="S4.SS3.p3.36.m32.3.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.36.m32.3b"><list id="S4.SS3.p3.36.m32.3.4.1.cmml" xref="S4.SS3.p3.36.m32.3.4.2"><ci id="S4.SS3.p3.36.m32.1.1.cmml" xref="S4.SS3.p3.36.m32.1.1">𝐷</ci><ci id="S4.SS3.p3.36.m32.2.2.cmml" xref="S4.SS3.p3.36.m32.2.2">𝜔</ci><ci id="S4.SS3.p3.36.m32.3.3.cmml" xref="S4.SS3.p3.36.m32.3.3">𝜂</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.36.m32.3c">D,\omega,\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.36.m32.3d">italic_D , italic_ω , italic_η</annotation></semantics></math> and <math alttext="\Delta t" class="ltx_Math" display="inline" id="S4.SS3.p3.37.m33.1"><semantics id="S4.SS3.p3.37.m33.1a"><mrow id="S4.SS3.p3.37.m33.1.1" xref="S4.SS3.p3.37.m33.1.1.cmml"><mi id="S4.SS3.p3.37.m33.1.1.2" mathvariant="normal" xref="S4.SS3.p3.37.m33.1.1.2.cmml">Δ</mi><mo id="S4.SS3.p3.37.m33.1.1.1" xref="S4.SS3.p3.37.m33.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p3.37.m33.1.1.3" xref="S4.SS3.p3.37.m33.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.37.m33.1b"><apply id="S4.SS3.p3.37.m33.1.1.cmml" xref="S4.SS3.p3.37.m33.1.1"><times id="S4.SS3.p3.37.m33.1.1.1.cmml" xref="S4.SS3.p3.37.m33.1.1.1"></times><ci id="S4.SS3.p3.37.m33.1.1.2.cmml" xref="S4.SS3.p3.37.m33.1.1.2">Δ</ci><ci id="S4.SS3.p3.37.m33.1.1.3.cmml" xref="S4.SS3.p3.37.m33.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.37.m33.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.37.m33.1d">roman_Δ italic_t</annotation></semantics></math> are multiplied by eight in order to speed up the time taken for morphogens to reach a steady state. The faster timescale does not have an effect on morphogenesis because there is no morphogenesis during the equilibration phase. </div> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> 4.4 Adhesion and contractile proteins</h3> <div class="ltx_para" id="S4.SS4.p1"> Cells encode 15 adhesion proteins constituting five pairs of lock-and-key proteins that determine cell-cell adhesion and five medium proteins that determine cell adhesion to the extracellular medium. Each lock protein has a complementary key protein to which it can bind. When two cells are in contact, the adhesion energy between them (<math alttext="J_{ij}" class="ltx_Math" display="inline" id="S4.SS4.p1.1.m1.1"><semantics id="S4.SS4.p1.1.m1.1a"><msub id="S4.SS4.p1.1.m1.1.1" xref="S4.SS4.p1.1.m1.1.1.cmml"><mi id="S4.SS4.p1.1.m1.1.1.2" xref="S4.SS4.p1.1.m1.1.1.2.cmml">J</mi><mrow id="S4.SS4.p1.1.m1.1.1.3" xref="S4.SS4.p1.1.m1.1.1.3.cmml"><mi id="S4.SS4.p1.1.m1.1.1.3.2" xref="S4.SS4.p1.1.m1.1.1.3.2.cmml">i</mi><mo id="S4.SS4.p1.1.m1.1.1.3.1" xref="S4.SS4.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.1.m1.1.1.3.3" xref="S4.SS4.p1.1.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.1.m1.1b"><apply id="S4.SS4.p1.1.m1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.1.m1.1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS4.p1.1.m1.1.1.2.cmml" xref="S4.SS4.p1.1.m1.1.1.2">𝐽</ci><apply id="S4.SS4.p1.1.m1.1.1.3.cmml" xref="S4.SS4.p1.1.m1.1.1.3"><times id="S4.SS4.p1.1.m1.1.1.3.1.cmml" xref="S4.SS4.p1.1.m1.1.1.3.1"></times><ci id="S4.SS4.p1.1.m1.1.1.3.2.cmml" xref="S4.SS4.p1.1.m1.1.1.3.2">𝑖</ci><ci id="S4.SS4.p1.1.m1.1.1.3.3.cmml" xref="S4.SS4.p1.1.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.1.m1.1c">J_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.1.m1.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E1" title="In 4.1 Cellular Potts Model ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>) decreases with the number of expressed pairs of compatible locks and keys. Each pair of lock and key reduces adhesion energy by the same amount. The adhesion energy to the extracellular medium (<math alttext="J_{im}" class="ltx_Math" display="inline" id="S4.SS4.p1.2.m2.1"><semantics id="S4.SS4.p1.2.m2.1a"><msub id="S4.SS4.p1.2.m2.1.1" xref="S4.SS4.p1.2.m2.1.1.cmml"><mi id="S4.SS4.p1.2.m2.1.1.2" xref="S4.SS4.p1.2.m2.1.1.2.cmml">J</mi><mrow id="S4.SS4.p1.2.m2.1.1.3" xref="S4.SS4.p1.2.m2.1.1.3.cmml"><mi id="S4.SS4.p1.2.m2.1.1.3.2" xref="S4.SS4.p1.2.m2.1.1.3.2.cmml">i</mi><mo id="S4.SS4.p1.2.m2.1.1.3.1" xref="S4.SS4.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.2.m2.1.1.3.3" xref="S4.SS4.p1.2.m2.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.2.m2.1b"><apply id="S4.SS4.p1.2.m2.1.1.cmml" xref="S4.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.2.m2.1.1.1.cmml" xref="S4.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS4.p1.2.m2.1.1.2.cmml" xref="S4.SS4.p1.2.m2.1.1.2">𝐽</ci><apply id="S4.SS4.p1.2.m2.1.1.3.cmml" xref="S4.SS4.p1.2.m2.1.1.3"><times id="S4.SS4.p1.2.m2.1.1.3.1.cmml" xref="S4.SS4.p1.2.m2.1.1.3.1"></times><ci id="S4.SS4.p1.2.m2.1.1.3.2.cmml" xref="S4.SS4.p1.2.m2.1.1.3.2">𝑖</ci><ci id="S4.SS4.p1.2.m2.1.1.3.3.cmml" xref="S4.SS4.p1.2.m2.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.2.m2.1c">J_{im}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.2.m2.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT</annotation></semantics></math> in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E1" title="In 4.1 Cellular Potts Model ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>) decreases with the number of medium proteins expressed by a cell. Medium proteins have graded adhesion strengths (described subsequently). Adhesion protein concentrations are booleanised to an ON or OFF state for adhesion energy calculations (i.e., ON if <math alttext="x^{\sigma}_{p}>0.5" class="ltx_Math" display="inline" id="S4.SS4.p1.3.m3.1"><semantics id="S4.SS4.p1.3.m3.1a"><mrow id="S4.SS4.p1.3.m3.1.1" xref="S4.SS4.p1.3.m3.1.1.cmml"><msubsup id="S4.SS4.p1.3.m3.1.1.2" xref="S4.SS4.p1.3.m3.1.1.2.cmml"><mi id="S4.SS4.p1.3.m3.1.1.2.2.2" xref="S4.SS4.p1.3.m3.1.1.2.2.2.cmml">x</mi><mi id="S4.SS4.p1.3.m3.1.1.2.3" xref="S4.SS4.p1.3.m3.1.1.2.3.cmml">p</mi><mi id="S4.SS4.p1.3.m3.1.1.2.2.3" xref="S4.SS4.p1.3.m3.1.1.2.2.3.cmml">σ</mi></msubsup><mo id="S4.SS4.p1.3.m3.1.1.1" xref="S4.SS4.p1.3.m3.1.1.1.cmml">></mo><mn id="S4.SS4.p1.3.m3.1.1.3" xref="S4.SS4.p1.3.m3.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.3.m3.1b"><apply id="S4.SS4.p1.3.m3.1.1.cmml" xref="S4.SS4.p1.3.m3.1.1"><gt id="S4.SS4.p1.3.m3.1.1.1.cmml" xref="S4.SS4.p1.3.m3.1.1.1"></gt><apply id="S4.SS4.p1.3.m3.1.1.2.cmml" xref="S4.SS4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.3.m3.1.1.2.1.cmml" xref="S4.SS4.p1.3.m3.1.1.2">subscript</csymbol><apply id="S4.SS4.p1.3.m3.1.1.2.2.cmml" xref="S4.SS4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.3.m3.1.1.2.2.1.cmml" xref="S4.SS4.p1.3.m3.1.1.2">superscript</csymbol><ci id="S4.SS4.p1.3.m3.1.1.2.2.2.cmml" xref="S4.SS4.p1.3.m3.1.1.2.2.2">𝑥</ci><ci id="S4.SS4.p1.3.m3.1.1.2.2.3.cmml" xref="S4.SS4.p1.3.m3.1.1.2.2.3">𝜎</ci></apply><ci id="S4.SS4.p1.3.m3.1.1.2.3.cmml" xref="S4.SS4.p1.3.m3.1.1.2.3">𝑝</ci></apply><cn id="S4.SS4.p1.3.m3.1.1.3.cmml" type="float" xref="S4.SS4.p1.3.m3.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.3.m3.1c">x^{\sigma}_{p}>0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.3.m3.1d">italic_x start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > 0.5</annotation></semantics></math> else OFF). Specifically, <math alttext="J_{ij}" class="ltx_Math" display="inline" id="S4.SS4.p1.4.m4.1"><semantics id="S4.SS4.p1.4.m4.1a"><msub id="S4.SS4.p1.4.m4.1.1" xref="S4.SS4.p1.4.m4.1.1.cmml"><mi id="S4.SS4.p1.4.m4.1.1.2" xref="S4.SS4.p1.4.m4.1.1.2.cmml">J</mi><mrow id="S4.SS4.p1.4.m4.1.1.3" xref="S4.SS4.p1.4.m4.1.1.3.cmml"><mi id="S4.SS4.p1.4.m4.1.1.3.2" xref="S4.SS4.p1.4.m4.1.1.3.2.cmml">i</mi><mo id="S4.SS4.p1.4.m4.1.1.3.1" xref="S4.SS4.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.4.m4.1.1.3.3" xref="S4.SS4.p1.4.m4.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.4.m4.1b"><apply id="S4.SS4.p1.4.m4.1.1.cmml" xref="S4.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.4.m4.1.1.1.cmml" xref="S4.SS4.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS4.p1.4.m4.1.1.2.cmml" xref="S4.SS4.p1.4.m4.1.1.2">𝐽</ci><apply id="S4.SS4.p1.4.m4.1.1.3.cmml" xref="S4.SS4.p1.4.m4.1.1.3"><times id="S4.SS4.p1.4.m4.1.1.3.1.cmml" xref="S4.SS4.p1.4.m4.1.1.3.1"></times><ci 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display="inline" id="S4.SS4.p1.6.m6.1"><semantics id="S4.SS4.p1.6.m6.1a"><mi id="S4.SS4.p1.6.m6.1.1" xref="S4.SS4.p1.6.m6.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.6.m6.1b"><ci id="S4.SS4.p1.6.m6.1.1.cmml" xref="S4.SS4.p1.6.m6.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.6.m6.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.6.m6.1d">italic_j</annotation></semantics></math> that belong to different cells is: <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="J_{ij}=J_{ij}^{\text{max}}-2\sum_{k=1}^{5}\left[\phi^{ij}_{k}+\phi^{ji}_{k}\right]" class="ltx_Math" display="block" id="S4.E5.m1.1"><semantics id="S4.E5.m1.1a"><mrow id="S4.E5.m1.1.1" xref="S4.E5.m1.1.1.cmml"><msub id="S4.E5.m1.1.1.3" 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xref="S4.E5.m1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.1.1.1.cmml"><msubsup id="S4.E5.m1.1.1.1.1.1.1.1.1.2" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.2.cmml">ϕ</mi><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.2.3" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.3.cmml">k</mi><mrow id="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.2" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.2.cmml">i</mi><mo id="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.1" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.3" xref="S4.E5.m1.1.1.1.1.1.1.1.1.2.2.3.3.cmml">j</mi></mrow></msubsup><mo id="S4.E5.m1.1.1.1.1.1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><msubsup id="S4.E5.m1.1.1.1.1.1.1.1.1.3" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.2" 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id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.2">italic-ϕ</ci><apply id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3"><times id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.1"></times><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.2.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.2">𝑗</ci><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.2.3.3">𝑖</ci></apply></apply><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.1c">J_{ij}=J_{ij}^{\text{max}}-2\sum_{k=1}^{5}\left[\phi^{ij}_{k}+\phi^{ji}_{k}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT - 2 ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT [ italic_ϕ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ϕ start_POSTSUPERSCRIPT italic_j italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(5)</td> </tr></tbody> </table> where <math alttext="\phi^{ij}_{k}=1" class="ltx_Math" display="inline" id="S4.SS4.p1.7.m1.1"><semantics id="S4.SS4.p1.7.m1.1a"><mrow id="S4.SS4.p1.7.m1.1.1" xref="S4.SS4.p1.7.m1.1.1.cmml"><msubsup id="S4.SS4.p1.7.m1.1.1.2" xref="S4.SS4.p1.7.m1.1.1.2.cmml"><mi id="S4.SS4.p1.7.m1.1.1.2.2.2" xref="S4.SS4.p1.7.m1.1.1.2.2.2.cmml">ϕ</mi><mi id="S4.SS4.p1.7.m1.1.1.2.3" xref="S4.SS4.p1.7.m1.1.1.2.3.cmml">k</mi><mrow id="S4.SS4.p1.7.m1.1.1.2.2.3" xref="S4.SS4.p1.7.m1.1.1.2.2.3.cmml"><mi id="S4.SS4.p1.7.m1.1.1.2.2.3.2" xref="S4.SS4.p1.7.m1.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS4.p1.7.m1.1.1.2.2.3.1" xref="S4.SS4.p1.7.m1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.7.m1.1.1.2.2.3.3" xref="S4.SS4.p1.7.m1.1.1.2.2.3.3.cmml">j</mi></mrow></msubsup><mo id="S4.SS4.p1.7.m1.1.1.1" xref="S4.SS4.p1.7.m1.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.7.m1.1.1.3" xref="S4.SS4.p1.7.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.7.m1.1b"><apply id="S4.SS4.p1.7.m1.1.1.cmml" xref="S4.SS4.p1.7.m1.1.1"><eq id="S4.SS4.p1.7.m1.1.1.1.cmml" xref="S4.SS4.p1.7.m1.1.1.1"></eq><apply id="S4.SS4.p1.7.m1.1.1.2.cmml" xref="S4.SS4.p1.7.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.7.m1.1.1.2.1.cmml" xref="S4.SS4.p1.7.m1.1.1.2">subscript</csymbol><apply id="S4.SS4.p1.7.m1.1.1.2.2.cmml" xref="S4.SS4.p1.7.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.7.m1.1.1.2.2.1.cmml" xref="S4.SS4.p1.7.m1.1.1.2">superscript</csymbol><ci id="S4.SS4.p1.7.m1.1.1.2.2.2.cmml" xref="S4.SS4.p1.7.m1.1.1.2.2.2">italic-ϕ</ci><apply id="S4.SS4.p1.7.m1.1.1.2.2.3.cmml" xref="S4.SS4.p1.7.m1.1.1.2.2.3"><times id="S4.SS4.p1.7.m1.1.1.2.2.3.1.cmml" xref="S4.SS4.p1.7.m1.1.1.2.2.3.1"></times><ci id="S4.SS4.p1.7.m1.1.1.2.2.3.2.cmml" xref="S4.SS4.p1.7.m1.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS4.p1.7.m1.1.1.2.2.3.3.cmml" xref="S4.SS4.p1.7.m1.1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS4.p1.7.m1.1.1.2.3.cmml" xref="S4.SS4.p1.7.m1.1.1.2.3">𝑘</ci></apply><cn id="S4.SS4.p1.7.m1.1.1.3.cmml" type="integer" xref="S4.SS4.p1.7.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.7.m1.1c">\phi^{ij}_{k}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.7.m1.1d">italic_ϕ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1</annotation></semantics></math> if the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS4.p1.8.m2.1"><semantics id="S4.SS4.p1.8.m2.1a"><mi id="S4.SS4.p1.8.m2.1.1" xref="S4.SS4.p1.8.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.8.m2.1b"><ci id="S4.SS4.p1.8.m2.1.1.cmml" xref="S4.SS4.p1.8.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.8.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.8.m2.1d">italic_k</annotation></semantics></math>-th lock in the cell of pixel <math alttext="i" class="ltx_Math" display="inline" id="S4.SS4.p1.9.m3.1"><semantics id="S4.SS4.p1.9.m3.1a"><mi id="S4.SS4.p1.9.m3.1.1" xref="S4.SS4.p1.9.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.9.m3.1b"><ci id="S4.SS4.p1.9.m3.1.1.cmml" xref="S4.SS4.p1.9.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.9.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.9.m3.1d">italic_i</annotation></semantics></math> and the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS4.p1.10.m4.1"><semantics id="S4.SS4.p1.10.m4.1a"><mi id="S4.SS4.p1.10.m4.1.1" xref="S4.SS4.p1.10.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.10.m4.1b"><ci id="S4.SS4.p1.10.m4.1.1.cmml" xref="S4.SS4.p1.10.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.10.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.10.m4.1d">italic_k</annotation></semantics></math>-th key in the cell of pixel <math alttext="j" class="ltx_Math" display="inline" id="S4.SS4.p1.11.m5.1"><semantics id="S4.SS4.p1.11.m5.1a"><mi id="S4.SS4.p1.11.m5.1.1" xref="S4.SS4.p1.11.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.11.m5.1b"><ci id="S4.SS4.p1.11.m5.1.1.cmml" xref="S4.SS4.p1.11.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.11.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.11.m5.1d">italic_j</annotation></semantics></math> are both ON and otherwise <math alttext="\phi^{ij}_{k}=0" class="ltx_Math" display="inline" id="S4.SS4.p1.12.m6.1"><semantics id="S4.SS4.p1.12.m6.1a"><mrow id="S4.SS4.p1.12.m6.1.1" xref="S4.SS4.p1.12.m6.1.1.cmml"><msubsup id="S4.SS4.p1.12.m6.1.1.2" xref="S4.SS4.p1.12.m6.1.1.2.cmml"><mi id="S4.SS4.p1.12.m6.1.1.2.2.2" xref="S4.SS4.p1.12.m6.1.1.2.2.2.cmml">ϕ</mi><mi id="S4.SS4.p1.12.m6.1.1.2.3" xref="S4.SS4.p1.12.m6.1.1.2.3.cmml">k</mi><mrow id="S4.SS4.p1.12.m6.1.1.2.2.3" xref="S4.SS4.p1.12.m6.1.1.2.2.3.cmml"><mi id="S4.SS4.p1.12.m6.1.1.2.2.3.2" xref="S4.SS4.p1.12.m6.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS4.p1.12.m6.1.1.2.2.3.1" xref="S4.SS4.p1.12.m6.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.12.m6.1.1.2.2.3.3" xref="S4.SS4.p1.12.m6.1.1.2.2.3.3.cmml">j</mi></mrow></msubsup><mo id="S4.SS4.p1.12.m6.1.1.1" xref="S4.SS4.p1.12.m6.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.12.m6.1.1.3" xref="S4.SS4.p1.12.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.12.m6.1b"><apply id="S4.SS4.p1.12.m6.1.1.cmml" xref="S4.SS4.p1.12.m6.1.1"><eq id="S4.SS4.p1.12.m6.1.1.1.cmml" xref="S4.SS4.p1.12.m6.1.1.1"></eq><apply id="S4.SS4.p1.12.m6.1.1.2.cmml" xref="S4.SS4.p1.12.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.12.m6.1.1.2.1.cmml" xref="S4.SS4.p1.12.m6.1.1.2">subscript</csymbol><apply id="S4.SS4.p1.12.m6.1.1.2.2.cmml" xref="S4.SS4.p1.12.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.12.m6.1.1.2.2.1.cmml" xref="S4.SS4.p1.12.m6.1.1.2">superscript</csymbol><ci id="S4.SS4.p1.12.m6.1.1.2.2.2.cmml" xref="S4.SS4.p1.12.m6.1.1.2.2.2">italic-ϕ</ci><apply id="S4.SS4.p1.12.m6.1.1.2.2.3.cmml" xref="S4.SS4.p1.12.m6.1.1.2.2.3"><times id="S4.SS4.p1.12.m6.1.1.2.2.3.1.cmml" xref="S4.SS4.p1.12.m6.1.1.2.2.3.1"></times><ci id="S4.SS4.p1.12.m6.1.1.2.2.3.2.cmml" xref="S4.SS4.p1.12.m6.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS4.p1.12.m6.1.1.2.2.3.3.cmml" xref="S4.SS4.p1.12.m6.1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS4.p1.12.m6.1.1.2.3.cmml" xref="S4.SS4.p1.12.m6.1.1.2.3">𝑘</ci></apply><cn id="S4.SS4.p1.12.m6.1.1.3.cmml" type="integer" xref="S4.SS4.p1.12.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.12.m6.1c">\phi^{ij}_{k}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.12.m6.1d">italic_ϕ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Similarly, <math alttext="J_{im}" class="ltx_Math" display="inline" id="S4.SS4.p1.13.m7.1"><semantics id="S4.SS4.p1.13.m7.1a"><msub id="S4.SS4.p1.13.m7.1.1" xref="S4.SS4.p1.13.m7.1.1.cmml"><mi id="S4.SS4.p1.13.m7.1.1.2" xref="S4.SS4.p1.13.m7.1.1.2.cmml">J</mi><mrow id="S4.SS4.p1.13.m7.1.1.3" xref="S4.SS4.p1.13.m7.1.1.3.cmml"><mi id="S4.SS4.p1.13.m7.1.1.3.2" xref="S4.SS4.p1.13.m7.1.1.3.2.cmml">i</mi><mo id="S4.SS4.p1.13.m7.1.1.3.1" xref="S4.SS4.p1.13.m7.1.1.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.13.m7.1.1.3.3" xref="S4.SS4.p1.13.m7.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.13.m7.1b"><apply id="S4.SS4.p1.13.m7.1.1.cmml" xref="S4.SS4.p1.13.m7.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.13.m7.1.1.1.cmml" xref="S4.SS4.p1.13.m7.1.1">subscript</csymbol><ci id="S4.SS4.p1.13.m7.1.1.2.cmml" xref="S4.SS4.p1.13.m7.1.1.2">𝐽</ci><apply id="S4.SS4.p1.13.m7.1.1.3.cmml" xref="S4.SS4.p1.13.m7.1.1.3"><times id="S4.SS4.p1.13.m7.1.1.3.1.cmml" xref="S4.SS4.p1.13.m7.1.1.3.1"></times><ci id="S4.SS4.p1.13.m7.1.1.3.2.cmml" xref="S4.SS4.p1.13.m7.1.1.3.2">𝑖</ci><ci id="S4.SS4.p1.13.m7.1.1.3.3.cmml" xref="S4.SS4.p1.13.m7.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.13.m7.1c">J_{im}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.13.m7.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT</annotation></semantics></math> between pixels <math alttext="i" class="ltx_Math" display="inline" id="S4.SS4.p1.14.m8.1"><semantics id="S4.SS4.p1.14.m8.1a"><mi id="S4.SS4.p1.14.m8.1.1" xref="S4.SS4.p1.14.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.14.m8.1b"><ci id="S4.SS4.p1.14.m8.1.1.cmml" xref="S4.SS4.p1.14.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.14.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.14.m8.1d">italic_i</annotation></semantics></math> and <math alttext="m" class="ltx_Math" display="inline" id="S4.SS4.p1.15.m9.1"><semantics id="S4.SS4.p1.15.m9.1a"><mi id="S4.SS4.p1.15.m9.1.1" xref="S4.SS4.p1.15.m9.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.15.m9.1b"><ci id="S4.SS4.p1.15.m9.1.1.cmml" xref="S4.SS4.p1.15.m9.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.15.m9.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.15.m9.1d">italic_m</annotation></semantics></math> belonging to a cell and the medium, respectively, is: <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="J_{im}=J_{im}^{\text{max}}-\sum_{k=1}^{5}k\psi_{k}" class="ltx_Math" display="block" id="S4.E6.m1.1"><semantics id="S4.E6.m1.1a"><mrow id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml"><msub id="S4.E6.m1.1.1.2" xref="S4.E6.m1.1.1.2.cmml"><mi id="S4.E6.m1.1.1.2.2" xref="S4.E6.m1.1.1.2.2.cmml">J</mi><mrow id="S4.E6.m1.1.1.2.3" xref="S4.E6.m1.1.1.2.3.cmml"><mi id="S4.E6.m1.1.1.2.3.2" xref="S4.E6.m1.1.1.2.3.2.cmml">i</mi><mo id="S4.E6.m1.1.1.2.3.1" xref="S4.E6.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.E6.m1.1.1.2.3.3" xref="S4.E6.m1.1.1.2.3.3.cmml">m</mi></mrow></msub><mo id="S4.E6.m1.1.1.1" xref="S4.E6.m1.1.1.1.cmml">=</mo><mrow id="S4.E6.m1.1.1.3" xref="S4.E6.m1.1.1.3.cmml"><msubsup id="S4.E6.m1.1.1.3.2" xref="S4.E6.m1.1.1.3.2.cmml"><mi id="S4.E6.m1.1.1.3.2.2.2" xref="S4.E6.m1.1.1.3.2.2.2.cmml">J</mi><mrow id="S4.E6.m1.1.1.3.2.2.3" xref="S4.E6.m1.1.1.3.2.2.3.cmml"><mi id="S4.E6.m1.1.1.3.2.2.3.2" xref="S4.E6.m1.1.1.3.2.2.3.2.cmml">i</mi><mo id="S4.E6.m1.1.1.3.2.2.3.1" xref="S4.E6.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S4.E6.m1.1.1.3.2.2.3.3" xref="S4.E6.m1.1.1.3.2.2.3.3.cmml">m</mi></mrow><mtext id="S4.E6.m1.1.1.3.2.3" xref="S4.E6.m1.1.1.3.2.3a.cmml">max</mtext></msubsup><mo id="S4.E6.m1.1.1.3.1" rspace="0.055em" xref="S4.E6.m1.1.1.3.1.cmml">−</mo><mrow id="S4.E6.m1.1.1.3.3" xref="S4.E6.m1.1.1.3.3.cmml"><munderover id="S4.E6.m1.1.1.3.3.1" xref="S4.E6.m1.1.1.3.3.1.cmml"><mo id="S4.E6.m1.1.1.3.3.1.2.2" movablelimits="false" xref="S4.E6.m1.1.1.3.3.1.2.2.cmml">∑</mo><mrow id="S4.E6.m1.1.1.3.3.1.2.3" xref="S4.E6.m1.1.1.3.3.1.2.3.cmml"><mi id="S4.E6.m1.1.1.3.3.1.2.3.2" xref="S4.E6.m1.1.1.3.3.1.2.3.2.cmml">k</mi><mo id="S4.E6.m1.1.1.3.3.1.2.3.1" xref="S4.E6.m1.1.1.3.3.1.2.3.1.cmml">=</mo><mn id="S4.E6.m1.1.1.3.3.1.2.3.3" xref="S4.E6.m1.1.1.3.3.1.2.3.3.cmml">1</mn></mrow><mn id="S4.E6.m1.1.1.3.3.1.3" xref="S4.E6.m1.1.1.3.3.1.3.cmml">5</mn></munderover><mrow id="S4.E6.m1.1.1.3.3.2" xref="S4.E6.m1.1.1.3.3.2.cmml"><mi id="S4.E6.m1.1.1.3.3.2.2" xref="S4.E6.m1.1.1.3.3.2.2.cmml">k</mi><mo id="S4.E6.m1.1.1.3.3.2.1" xref="S4.E6.m1.1.1.3.3.2.1.cmml">⁢</mo><msub id="S4.E6.m1.1.1.3.3.2.3" xref="S4.E6.m1.1.1.3.3.2.3.cmml"><mi id="S4.E6.m1.1.1.3.3.2.3.2" xref="S4.E6.m1.1.1.3.3.2.3.2.cmml">ψ</mi><mi id="S4.E6.m1.1.1.3.3.2.3.3" xref="S4.E6.m1.1.1.3.3.2.3.3.cmml">k</mi></msub></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.1b"><apply id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1"><eq id="S4.E6.m1.1.1.1.cmml" xref="S4.E6.m1.1.1.1"></eq><apply id="S4.E6.m1.1.1.2.cmml" xref="S4.E6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.2.1.cmml" xref="S4.E6.m1.1.1.2">subscript</csymbol><ci id="S4.E6.m1.1.1.2.2.cmml" xref="S4.E6.m1.1.1.2.2">𝐽</ci><apply id="S4.E6.m1.1.1.2.3.cmml" xref="S4.E6.m1.1.1.2.3"><times id="S4.E6.m1.1.1.2.3.1.cmml" xref="S4.E6.m1.1.1.2.3.1"></times><ci id="S4.E6.m1.1.1.2.3.2.cmml" xref="S4.E6.m1.1.1.2.3.2">𝑖</ci><ci id="S4.E6.m1.1.1.2.3.3.cmml" xref="S4.E6.m1.1.1.2.3.3">𝑚</ci></apply></apply><apply id="S4.E6.m1.1.1.3.cmml" xref="S4.E6.m1.1.1.3"><minus id="S4.E6.m1.1.1.3.1.cmml" xref="S4.E6.m1.1.1.3.1"></minus><apply id="S4.E6.m1.1.1.3.2.cmml" xref="S4.E6.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.2.1.cmml" xref="S4.E6.m1.1.1.3.2">superscript</csymbol><apply id="S4.E6.m1.1.1.3.2.2.cmml" xref="S4.E6.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.2.2.1.cmml" xref="S4.E6.m1.1.1.3.2">subscript</csymbol><ci id="S4.E6.m1.1.1.3.2.2.2.cmml" xref="S4.E6.m1.1.1.3.2.2.2">𝐽</ci><apply id="S4.E6.m1.1.1.3.2.2.3.cmml" xref="S4.E6.m1.1.1.3.2.2.3"><times id="S4.E6.m1.1.1.3.2.2.3.1.cmml" xref="S4.E6.m1.1.1.3.2.2.3.1"></times><ci id="S4.E6.m1.1.1.3.2.2.3.2.cmml" xref="S4.E6.m1.1.1.3.2.2.3.2">𝑖</ci><ci id="S4.E6.m1.1.1.3.2.2.3.3.cmml" xref="S4.E6.m1.1.1.3.2.2.3.3">𝑚</ci></apply></apply><ci id="S4.E6.m1.1.1.3.2.3a.cmml" xref="S4.E6.m1.1.1.3.2.3"><mtext id="S4.E6.m1.1.1.3.2.3.cmml" mathsize="70%" xref="S4.E6.m1.1.1.3.2.3">max</mtext></ci></apply><apply id="S4.E6.m1.1.1.3.3.cmml" xref="S4.E6.m1.1.1.3.3"><apply id="S4.E6.m1.1.1.3.3.1.cmml" xref="S4.E6.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.3.1.1.cmml" xref="S4.E6.m1.1.1.3.3.1">superscript</csymbol><apply id="S4.E6.m1.1.1.3.3.1.2.cmml" xref="S4.E6.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.3.1.2.1.cmml" xref="S4.E6.m1.1.1.3.3.1">subscript</csymbol><sum id="S4.E6.m1.1.1.3.3.1.2.2.cmml" xref="S4.E6.m1.1.1.3.3.1.2.2"></sum><apply id="S4.E6.m1.1.1.3.3.1.2.3.cmml" xref="S4.E6.m1.1.1.3.3.1.2.3"><eq id="S4.E6.m1.1.1.3.3.1.2.3.1.cmml" xref="S4.E6.m1.1.1.3.3.1.2.3.1"></eq><ci id="S4.E6.m1.1.1.3.3.1.2.3.2.cmml" xref="S4.E6.m1.1.1.3.3.1.2.3.2">𝑘</ci><cn id="S4.E6.m1.1.1.3.3.1.2.3.3.cmml" type="integer" xref="S4.E6.m1.1.1.3.3.1.2.3.3">1</cn></apply></apply><cn id="S4.E6.m1.1.1.3.3.1.3.cmml" type="integer" xref="S4.E6.m1.1.1.3.3.1.3">5</cn></apply><apply id="S4.E6.m1.1.1.3.3.2.cmml" xref="S4.E6.m1.1.1.3.3.2"><times id="S4.E6.m1.1.1.3.3.2.1.cmml" xref="S4.E6.m1.1.1.3.3.2.1"></times><ci id="S4.E6.m1.1.1.3.3.2.2.cmml" xref="S4.E6.m1.1.1.3.3.2.2">𝑘</ci><apply id="S4.E6.m1.1.1.3.3.2.3.cmml" xref="S4.E6.m1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.3.2.3.1.cmml" xref="S4.E6.m1.1.1.3.3.2.3">subscript</csymbol><ci id="S4.E6.m1.1.1.3.3.2.3.2.cmml" xref="S4.E6.m1.1.1.3.3.2.3.2">𝜓</ci><ci id="S4.E6.m1.1.1.3.3.2.3.3.cmml" xref="S4.E6.m1.1.1.3.3.2.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.1c">J_{im}=J_{im}^{\text{max}}-\sum_{k=1}^{5}k\psi_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT italic_k italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(6)</td> </tr></tbody> </table> where <math alttext="\psi_{k}=1" class="ltx_Math" display="inline" id="S4.SS4.p1.16.m1.1"><semantics id="S4.SS4.p1.16.m1.1a"><mrow id="S4.SS4.p1.16.m1.1.1" xref="S4.SS4.p1.16.m1.1.1.cmml"><msub id="S4.SS4.p1.16.m1.1.1.2" xref="S4.SS4.p1.16.m1.1.1.2.cmml"><mi id="S4.SS4.p1.16.m1.1.1.2.2" xref="S4.SS4.p1.16.m1.1.1.2.2.cmml">ψ</mi><mi id="S4.SS4.p1.16.m1.1.1.2.3" xref="S4.SS4.p1.16.m1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS4.p1.16.m1.1.1.1" xref="S4.SS4.p1.16.m1.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.16.m1.1.1.3" xref="S4.SS4.p1.16.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.16.m1.1b"><apply id="S4.SS4.p1.16.m1.1.1.cmml" xref="S4.SS4.p1.16.m1.1.1"><eq id="S4.SS4.p1.16.m1.1.1.1.cmml" xref="S4.SS4.p1.16.m1.1.1.1"></eq><apply id="S4.SS4.p1.16.m1.1.1.2.cmml" xref="S4.SS4.p1.16.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.16.m1.1.1.2.1.cmml" xref="S4.SS4.p1.16.m1.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.16.m1.1.1.2.2.cmml" xref="S4.SS4.p1.16.m1.1.1.2.2">𝜓</ci><ci id="S4.SS4.p1.16.m1.1.1.2.3.cmml" xref="S4.SS4.p1.16.m1.1.1.2.3">𝑘</ci></apply><cn id="S4.SS4.p1.16.m1.1.1.3.cmml" type="integer" xref="S4.SS4.p1.16.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.16.m1.1c">\psi_{k}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.16.m1.1d">italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1</annotation></semantics></math> if the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS4.p1.17.m2.1"><semantics id="S4.SS4.p1.17.m2.1a"><mi id="S4.SS4.p1.17.m2.1.1" xref="S4.SS4.p1.17.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.17.m2.1b"><ci id="S4.SS4.p1.17.m2.1.1.cmml" xref="S4.SS4.p1.17.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.17.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.17.m2.1d">italic_k</annotation></semantics></math>-th medium-adhesion protein is ON, and otherwise <math alttext="\psi_{k}=0" class="ltx_Math" display="inline" id="S4.SS4.p1.18.m3.1"><semantics id="S4.SS4.p1.18.m3.1a"><mrow id="S4.SS4.p1.18.m3.1.1" xref="S4.SS4.p1.18.m3.1.1.cmml"><msub id="S4.SS4.p1.18.m3.1.1.2" xref="S4.SS4.p1.18.m3.1.1.2.cmml"><mi id="S4.SS4.p1.18.m3.1.1.2.2" xref="S4.SS4.p1.18.m3.1.1.2.2.cmml">ψ</mi><mi id="S4.SS4.p1.18.m3.1.1.2.3" xref="S4.SS4.p1.18.m3.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS4.p1.18.m3.1.1.1" xref="S4.SS4.p1.18.m3.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.18.m3.1.1.3" xref="S4.SS4.p1.18.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.18.m3.1b"><apply id="S4.SS4.p1.18.m3.1.1.cmml" xref="S4.SS4.p1.18.m3.1.1"><eq id="S4.SS4.p1.18.m3.1.1.1.cmml" xref="S4.SS4.p1.18.m3.1.1.1"></eq><apply id="S4.SS4.p1.18.m3.1.1.2.cmml" xref="S4.SS4.p1.18.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.18.m3.1.1.2.1.cmml" xref="S4.SS4.p1.18.m3.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.18.m3.1.1.2.2.cmml" xref="S4.SS4.p1.18.m3.1.1.2.2">𝜓</ci><ci id="S4.SS4.p1.18.m3.1.1.2.3.cmml" xref="S4.SS4.p1.18.m3.1.1.2.3">𝑘</ci></apply><cn id="S4.SS4.p1.18.m3.1.1.3.cmml" type="integer" xref="S4.SS4.p1.18.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.18.m3.1c">\psi_{k}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.18.m3.1d">italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 0</annotation></semantics></math>. <math alttext="J_{ij}^{\text{max}}=24" class="ltx_Math" display="inline" id="S4.SS4.p1.19.m4.1"><semantics id="S4.SS4.p1.19.m4.1a"><mrow id="S4.SS4.p1.19.m4.1.1" xref="S4.SS4.p1.19.m4.1.1.cmml"><msubsup id="S4.SS4.p1.19.m4.1.1.2" xref="S4.SS4.p1.19.m4.1.1.2.cmml"><mi id="S4.SS4.p1.19.m4.1.1.2.2.2" xref="S4.SS4.p1.19.m4.1.1.2.2.2.cmml">J</mi><mrow id="S4.SS4.p1.19.m4.1.1.2.2.3" xref="S4.SS4.p1.19.m4.1.1.2.2.3.cmml"><mi id="S4.SS4.p1.19.m4.1.1.2.2.3.2" xref="S4.SS4.p1.19.m4.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS4.p1.19.m4.1.1.2.2.3.1" xref="S4.SS4.p1.19.m4.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.19.m4.1.1.2.2.3.3" xref="S4.SS4.p1.19.m4.1.1.2.2.3.3.cmml">j</mi></mrow><mtext id="S4.SS4.p1.19.m4.1.1.2.3" xref="S4.SS4.p1.19.m4.1.1.2.3a.cmml">max</mtext></msubsup><mo id="S4.SS4.p1.19.m4.1.1.1" xref="S4.SS4.p1.19.m4.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.19.m4.1.1.3" xref="S4.SS4.p1.19.m4.1.1.3.cmml">24</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.19.m4.1b"><apply id="S4.SS4.p1.19.m4.1.1.cmml" xref="S4.SS4.p1.19.m4.1.1"><eq id="S4.SS4.p1.19.m4.1.1.1.cmml" xref="S4.SS4.p1.19.m4.1.1.1"></eq><apply id="S4.SS4.p1.19.m4.1.1.2.cmml" xref="S4.SS4.p1.19.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.19.m4.1.1.2.1.cmml" xref="S4.SS4.p1.19.m4.1.1.2">superscript</csymbol><apply id="S4.SS4.p1.19.m4.1.1.2.2.cmml" xref="S4.SS4.p1.19.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.19.m4.1.1.2.2.1.cmml" xref="S4.SS4.p1.19.m4.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.19.m4.1.1.2.2.2.cmml" xref="S4.SS4.p1.19.m4.1.1.2.2.2">𝐽</ci><apply id="S4.SS4.p1.19.m4.1.1.2.2.3.cmml" xref="S4.SS4.p1.19.m4.1.1.2.2.3"><times id="S4.SS4.p1.19.m4.1.1.2.2.3.1.cmml" xref="S4.SS4.p1.19.m4.1.1.2.2.3.1"></times><ci id="S4.SS4.p1.19.m4.1.1.2.2.3.2.cmml" xref="S4.SS4.p1.19.m4.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS4.p1.19.m4.1.1.2.2.3.3.cmml" xref="S4.SS4.p1.19.m4.1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS4.p1.19.m4.1.1.2.3a.cmml" xref="S4.SS4.p1.19.m4.1.1.2.3"><mtext id="S4.SS4.p1.19.m4.1.1.2.3.cmml" mathsize="70%" xref="S4.SS4.p1.19.m4.1.1.2.3">max</mtext></ci></apply><cn id="S4.SS4.p1.19.m4.1.1.3.cmml" type="integer" xref="S4.SS4.p1.19.m4.1.1.3">24</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.19.m4.1c">J_{ij}^{\text{max}}=24</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.19.m4.1d">italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT = 24</annotation></semantics></math> and <math alttext="J_{im}^{max}=21" class="ltx_Math" display="inline" id="S4.SS4.p1.20.m5.1"><semantics id="S4.SS4.p1.20.m5.1a"><mrow id="S4.SS4.p1.20.m5.1.1" xref="S4.SS4.p1.20.m5.1.1.cmml"><msubsup id="S4.SS4.p1.20.m5.1.1.2" xref="S4.SS4.p1.20.m5.1.1.2.cmml"><mi id="S4.SS4.p1.20.m5.1.1.2.2.2" xref="S4.SS4.p1.20.m5.1.1.2.2.2.cmml">J</mi><mrow id="S4.SS4.p1.20.m5.1.1.2.2.3" xref="S4.SS4.p1.20.m5.1.1.2.2.3.cmml"><mi id="S4.SS4.p1.20.m5.1.1.2.2.3.2" xref="S4.SS4.p1.20.m5.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS4.p1.20.m5.1.1.2.2.3.1" xref="S4.SS4.p1.20.m5.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.20.m5.1.1.2.2.3.3" xref="S4.SS4.p1.20.m5.1.1.2.2.3.3.cmml">m</mi></mrow><mrow id="S4.SS4.p1.20.m5.1.1.2.3" xref="S4.SS4.p1.20.m5.1.1.2.3.cmml"><mi id="S4.SS4.p1.20.m5.1.1.2.3.2" xref="S4.SS4.p1.20.m5.1.1.2.3.2.cmml">m</mi><mo id="S4.SS4.p1.20.m5.1.1.2.3.1" xref="S4.SS4.p1.20.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.20.m5.1.1.2.3.3" xref="S4.SS4.p1.20.m5.1.1.2.3.3.cmml">a</mi><mo id="S4.SS4.p1.20.m5.1.1.2.3.1a" xref="S4.SS4.p1.20.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS4.p1.20.m5.1.1.2.3.4" xref="S4.SS4.p1.20.m5.1.1.2.3.4.cmml">x</mi></mrow></msubsup><mo id="S4.SS4.p1.20.m5.1.1.1" xref="S4.SS4.p1.20.m5.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.20.m5.1.1.3" xref="S4.SS4.p1.20.m5.1.1.3.cmml">21</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.20.m5.1b"><apply id="S4.SS4.p1.20.m5.1.1.cmml" xref="S4.SS4.p1.20.m5.1.1"><eq id="S4.SS4.p1.20.m5.1.1.1.cmml" xref="S4.SS4.p1.20.m5.1.1.1"></eq><apply id="S4.SS4.p1.20.m5.1.1.2.cmml" xref="S4.SS4.p1.20.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.20.m5.1.1.2.1.cmml" xref="S4.SS4.p1.20.m5.1.1.2">superscript</csymbol><apply id="S4.SS4.p1.20.m5.1.1.2.2.cmml" xref="S4.SS4.p1.20.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.20.m5.1.1.2.2.1.cmml" xref="S4.SS4.p1.20.m5.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.20.m5.1.1.2.2.2.cmml" xref="S4.SS4.p1.20.m5.1.1.2.2.2">𝐽</ci><apply id="S4.SS4.p1.20.m5.1.1.2.2.3.cmml" xref="S4.SS4.p1.20.m5.1.1.2.2.3"><times id="S4.SS4.p1.20.m5.1.1.2.2.3.1.cmml" xref="S4.SS4.p1.20.m5.1.1.2.2.3.1"></times><ci id="S4.SS4.p1.20.m5.1.1.2.2.3.2.cmml" xref="S4.SS4.p1.20.m5.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS4.p1.20.m5.1.1.2.2.3.3.cmml" xref="S4.SS4.p1.20.m5.1.1.2.2.3.3">𝑚</ci></apply></apply><apply id="S4.SS4.p1.20.m5.1.1.2.3.cmml" xref="S4.SS4.p1.20.m5.1.1.2.3"><times id="S4.SS4.p1.20.m5.1.1.2.3.1.cmml" xref="S4.SS4.p1.20.m5.1.1.2.3.1"></times><ci id="S4.SS4.p1.20.m5.1.1.2.3.2.cmml" xref="S4.SS4.p1.20.m5.1.1.2.3.2">𝑚</ci><ci id="S4.SS4.p1.20.m5.1.1.2.3.3.cmml" xref="S4.SS4.p1.20.m5.1.1.2.3.3">𝑎</ci><ci id="S4.SS4.p1.20.m5.1.1.2.3.4.cmml" xref="S4.SS4.p1.20.m5.1.1.2.3.4">𝑥</ci></apply></apply><cn id="S4.SS4.p1.20.m5.1.1.3.cmml" type="integer" xref="S4.SS4.p1.20.m5.1.1.3">21</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.20.m5.1c">J_{im}^{max}=21</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.20.m5.1d">italic_J start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_a italic_x end_POSTSUPERSCRIPT = 21</annotation></semantics></math> for all simulations. In Figure S8ABCD, we show that the morphogenesis of evolved organisms is not disrupted by changes to these parameters. </div> <div class="ltx_para" id="S4.SS4.p2"> Cells encode two contractile proteins that make the cell shape less deformable by energetically constraining it to an elliptic shape. Cell shapes are defined as ellipses that have a major axis of length <math alttext="l_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.1.m1.1"><semantics id="S4.SS4.p2.1.m1.1a"><msub id="S4.SS4.p2.1.m1.1.1" xref="S4.SS4.p2.1.m1.1.1.cmml"><mi id="S4.SS4.p2.1.m1.1.1.2" xref="S4.SS4.p2.1.m1.1.1.2.cmml">l</mi><mi id="S4.SS4.p2.1.m1.1.1.3" xref="S4.SS4.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.1.m1.1b"><apply id="S4.SS4.p2.1.m1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.1.m1.1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS4.p2.1.m1.1.1.2.cmml" xref="S4.SS4.p2.1.m1.1.1.2">𝑙</ci><ci id="S4.SS4.p2.1.m1.1.1.3.cmml" xref="S4.SS4.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.1.m1.1c">l_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.1.m1.1d">italic_l start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> constrained to a target length of <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.2.m2.1"><semantics id="S4.SS4.p2.2.m2.1a"><msub id="S4.SS4.p2.2.m2.1.1" xref="S4.SS4.p2.2.m2.1.1.cmml"><mi id="S4.SS4.p2.2.m2.1.1.2" xref="S4.SS4.p2.2.m2.1.1.2.cmml">L</mi><mi id="S4.SS4.p2.2.m2.1.1.3" xref="S4.SS4.p2.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.2.m2.1b"><apply id="S4.SS4.p2.2.m2.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.2.m2.1.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS4.p2.2.m2.1.1.2.cmml" xref="S4.SS4.p2.2.m2.1.1.2">𝐿</ci><ci id="S4.SS4.p2.2.m2.1.1.3.cmml" xref="S4.SS4.p2.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.2.m2.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\lambda^{\sigma}_{L}" class="ltx_Math" display="inline" id="S4.SS4.p2.3.m3.1"><semantics id="S4.SS4.p2.3.m3.1a"><msubsup id="S4.SS4.p2.3.m3.1.1" xref="S4.SS4.p2.3.m3.1.1.cmml"><mi id="S4.SS4.p2.3.m3.1.1.2.2" xref="S4.SS4.p2.3.m3.1.1.2.2.cmml">λ</mi><mi id="S4.SS4.p2.3.m3.1.1.3" xref="S4.SS4.p2.3.m3.1.1.3.cmml">L</mi><mi id="S4.SS4.p2.3.m3.1.1.2.3" xref="S4.SS4.p2.3.m3.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.3.m3.1b"><apply id="S4.SS4.p2.3.m3.1.1.cmml" xref="S4.SS4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.3.m3.1.1.1.cmml" xref="S4.SS4.p2.3.m3.1.1">subscript</csymbol><apply id="S4.SS4.p2.3.m3.1.1.2.cmml" xref="S4.SS4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.3.m3.1.1.2.1.cmml" xref="S4.SS4.p2.3.m3.1.1">superscript</csymbol><ci id="S4.SS4.p2.3.m3.1.1.2.2.cmml" xref="S4.SS4.p2.3.m3.1.1.2.2">𝜆</ci><ci id="S4.SS4.p2.3.m3.1.1.2.3.cmml" xref="S4.SS4.p2.3.m3.1.1.2.3">𝜎</ci></apply><ci id="S4.SS4.p2.3.m3.1.1.3.cmml" xref="S4.SS4.p2.3.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.3.m3.1c">\lambda^{\sigma}_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.3.m3.1d">italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> (see Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E1" title="In 4.1 Cellular Potts Model ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">1</a>). <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.4.m4.1"><semantics id="S4.SS4.p2.4.m4.1a"><msub id="S4.SS4.p2.4.m4.1.1" xref="S4.SS4.p2.4.m4.1.1.cmml"><mi id="S4.SS4.p2.4.m4.1.1.2" xref="S4.SS4.p2.4.m4.1.1.2.cmml">L</mi><mi id="S4.SS4.p2.4.m4.1.1.3" xref="S4.SS4.p2.4.m4.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.4.m4.1b"><apply id="S4.SS4.p2.4.m4.1.1.cmml" xref="S4.SS4.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.4.m4.1.1.1.cmml" xref="S4.SS4.p2.4.m4.1.1">subscript</csymbol><ci id="S4.SS4.p2.4.m4.1.1.2.cmml" xref="S4.SS4.p2.4.m4.1.1.2">𝐿</ci><ci id="S4.SS4.p2.4.m4.1.1.3.cmml" xref="S4.SS4.p2.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.4.m4.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> increases with the number of expressed contractile proteins. Contractile proteins are first booleanised to an ON or OFF state by the same method used for adhesion proteins. When one length protein is ON in cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS4.p2.5.m5.1"><semantics id="S4.SS4.p2.5.m5.1a"><mi id="S4.SS4.p2.5.m5.1.1" xref="S4.SS4.p2.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.5.m5.1b"><ci id="S4.SS4.p2.5.m5.1.1.cmml" xref="S4.SS4.p2.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.5.m5.1d">italic_σ</annotation></semantics></math>, <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.6.m6.1"><semantics id="S4.SS4.p2.6.m6.1a"><msub id="S4.SS4.p2.6.m6.1.1" xref="S4.SS4.p2.6.m6.1.1.cmml"><mi id="S4.SS4.p2.6.m6.1.1.2" xref="S4.SS4.p2.6.m6.1.1.2.cmml">L</mi><mi id="S4.SS4.p2.6.m6.1.1.3" xref="S4.SS4.p2.6.m6.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.6.m6.1b"><apply id="S4.SS4.p2.6.m6.1.1.cmml" xref="S4.SS4.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.6.m6.1.1.1.cmml" xref="S4.SS4.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS4.p2.6.m6.1.1.2.cmml" xref="S4.SS4.p2.6.m6.1.1.2">𝐿</ci><ci id="S4.SS4.p2.6.m6.1.1.3.cmml" xref="S4.SS4.p2.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.6.m6.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.6.m6.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is set to one-sixth of <math alttext="\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.7.m7.1"><semantics id="S4.SS4.p2.7.m7.1a"><msub id="S4.SS4.p2.7.m7.1.1" xref="S4.SS4.p2.7.m7.1.1.cmml"><mi id="S4.SS4.p2.7.m7.1.1.2" xref="S4.SS4.p2.7.m7.1.1.2.cmml">υ</mi><mi id="S4.SS4.p2.7.m7.1.1.3" xref="S4.SS4.p2.7.m7.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.7.m7.1b"><apply id="S4.SS4.p2.7.m7.1.1.cmml" xref="S4.SS4.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.7.m7.1.1.1.cmml" xref="S4.SS4.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS4.p2.7.m7.1.1.2.cmml" xref="S4.SS4.p2.7.m7.1.1.2">𝜐</ci><ci id="S4.SS4.p2.7.m7.1.1.3.cmml" xref="S4.SS4.p2.7.m7.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.7.m7.1c">\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.7.m7.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>; When two are ON, <math alttext="L_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.8.m8.1"><semantics id="S4.SS4.p2.8.m8.1a"><msub id="S4.SS4.p2.8.m8.1.1" xref="S4.SS4.p2.8.m8.1.1.cmml"><mi id="S4.SS4.p2.8.m8.1.1.2" xref="S4.SS4.p2.8.m8.1.1.2.cmml">L</mi><mi id="S4.SS4.p2.8.m8.1.1.3" xref="S4.SS4.p2.8.m8.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.8.m8.1b"><apply id="S4.SS4.p2.8.m8.1.1.cmml" xref="S4.SS4.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.8.m8.1.1.1.cmml" xref="S4.SS4.p2.8.m8.1.1">subscript</csymbol><ci id="S4.SS4.p2.8.m8.1.1.2.cmml" xref="S4.SS4.p2.8.m8.1.1.2">𝐿</ci><ci id="S4.SS4.p2.8.m8.1.1.3.cmml" xref="S4.SS4.p2.8.m8.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.8.m8.1c">L_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.8.m8.1d">italic_L start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is set to one-third of <math alttext="\upsilon_{\sigma}" class="ltx_Math" display="inline" id="S4.SS4.p2.9.m9.1"><semantics id="S4.SS4.p2.9.m9.1a"><msub id="S4.SS4.p2.9.m9.1.1" xref="S4.SS4.p2.9.m9.1.1.cmml"><mi id="S4.SS4.p2.9.m9.1.1.2" xref="S4.SS4.p2.9.m9.1.1.2.cmml">υ</mi><mi id="S4.SS4.p2.9.m9.1.1.3" xref="S4.SS4.p2.9.m9.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.9.m9.1b"><apply id="S4.SS4.p2.9.m9.1.1.cmml" xref="S4.SS4.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.9.m9.1.1.1.cmml" xref="S4.SS4.p2.9.m9.1.1">subscript</csymbol><ci id="S4.SS4.p2.9.m9.1.1.2.cmml" xref="S4.SS4.p2.9.m9.1.1.2">𝜐</ci><ci id="S4.SS4.p2.9.m9.1.1.3.cmml" xref="S4.SS4.p2.9.m9.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.9.m9.1c">\upsilon_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.9.m9.1d">italic_υ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> (these numbers are arbitrarily chosen). To implement the presence or absence of a length constraint depending on whether contractile proteins are expressed, we set <math alttext="\lambda^{\sigma}_{L}=0.1" class="ltx_Math" display="inline" id="S4.SS4.p2.10.m10.1"><semantics id="S4.SS4.p2.10.m10.1a"><mrow id="S4.SS4.p2.10.m10.1.1" xref="S4.SS4.p2.10.m10.1.1.cmml"><msubsup id="S4.SS4.p2.10.m10.1.1.2" xref="S4.SS4.p2.10.m10.1.1.2.cmml"><mi id="S4.SS4.p2.10.m10.1.1.2.2.2" xref="S4.SS4.p2.10.m10.1.1.2.2.2.cmml">λ</mi><mi id="S4.SS4.p2.10.m10.1.1.2.3" xref="S4.SS4.p2.10.m10.1.1.2.3.cmml">L</mi><mi id="S4.SS4.p2.10.m10.1.1.2.2.3" xref="S4.SS4.p2.10.m10.1.1.2.2.3.cmml">σ</mi></msubsup><mo id="S4.SS4.p2.10.m10.1.1.1" xref="S4.SS4.p2.10.m10.1.1.1.cmml">=</mo><mn id="S4.SS4.p2.10.m10.1.1.3" xref="S4.SS4.p2.10.m10.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.10.m10.1b"><apply id="S4.SS4.p2.10.m10.1.1.cmml" xref="S4.SS4.p2.10.m10.1.1"><eq id="S4.SS4.p2.10.m10.1.1.1.cmml" xref="S4.SS4.p2.10.m10.1.1.1"></eq><apply id="S4.SS4.p2.10.m10.1.1.2.cmml" xref="S4.SS4.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p2.10.m10.1.1.2.1.cmml" xref="S4.SS4.p2.10.m10.1.1.2">subscript</csymbol><apply id="S4.SS4.p2.10.m10.1.1.2.2.cmml" xref="S4.SS4.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p2.10.m10.1.1.2.2.1.cmml" xref="S4.SS4.p2.10.m10.1.1.2">superscript</csymbol><ci id="S4.SS4.p2.10.m10.1.1.2.2.2.cmml" xref="S4.SS4.p2.10.m10.1.1.2.2.2">𝜆</ci><ci id="S4.SS4.p2.10.m10.1.1.2.2.3.cmml" xref="S4.SS4.p2.10.m10.1.1.2.2.3">𝜎</ci></apply><ci id="S4.SS4.p2.10.m10.1.1.2.3.cmml" xref="S4.SS4.p2.10.m10.1.1.2.3">𝐿</ci></apply><cn id="S4.SS4.p2.10.m10.1.1.3.cmml" type="float" xref="S4.SS4.p2.10.m10.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.10.m10.1c">\lambda^{\sigma}_{L}=0.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.10.m10.1d">italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = 0.1</annotation></semantics></math> if either contractile protein in cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS4.p2.11.m11.1"><semantics id="S4.SS4.p2.11.m11.1a"><mi id="S4.SS4.p2.11.m11.1.1" xref="S4.SS4.p2.11.m11.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.11.m11.1b"><ci id="S4.SS4.p2.11.m11.1.1.cmml" xref="S4.SS4.p2.11.m11.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.11.m11.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.11.m11.1d">italic_σ</annotation></semantics></math> is ON, otherwise <math alttext="\lambda^{\sigma}_{L}=0" class="ltx_Math" display="inline" id="S4.SS4.p2.12.m12.1"><semantics id="S4.SS4.p2.12.m12.1a"><mrow id="S4.SS4.p2.12.m12.1.1" xref="S4.SS4.p2.12.m12.1.1.cmml"><msubsup id="S4.SS4.p2.12.m12.1.1.2" xref="S4.SS4.p2.12.m12.1.1.2.cmml"><mi id="S4.SS4.p2.12.m12.1.1.2.2.2" xref="S4.SS4.p2.12.m12.1.1.2.2.2.cmml">λ</mi><mi id="S4.SS4.p2.12.m12.1.1.2.3" xref="S4.SS4.p2.12.m12.1.1.2.3.cmml">L</mi><mi id="S4.SS4.p2.12.m12.1.1.2.2.3" xref="S4.SS4.p2.12.m12.1.1.2.2.3.cmml">σ</mi></msubsup><mo id="S4.SS4.p2.12.m12.1.1.1" xref="S4.SS4.p2.12.m12.1.1.1.cmml">=</mo><mn id="S4.SS4.p2.12.m12.1.1.3" xref="S4.SS4.p2.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.12.m12.1b"><apply id="S4.SS4.p2.12.m12.1.1.cmml" xref="S4.SS4.p2.12.m12.1.1"><eq id="S4.SS4.p2.12.m12.1.1.1.cmml" xref="S4.SS4.p2.12.m12.1.1.1"></eq><apply id="S4.SS4.p2.12.m12.1.1.2.cmml" xref="S4.SS4.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p2.12.m12.1.1.2.1.cmml" xref="S4.SS4.p2.12.m12.1.1.2">subscript</csymbol><apply id="S4.SS4.p2.12.m12.1.1.2.2.cmml" xref="S4.SS4.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p2.12.m12.1.1.2.2.1.cmml" xref="S4.SS4.p2.12.m12.1.1.2">superscript</csymbol><ci id="S4.SS4.p2.12.m12.1.1.2.2.2.cmml" xref="S4.SS4.p2.12.m12.1.1.2.2.2">𝜆</ci><ci id="S4.SS4.p2.12.m12.1.1.2.2.3.cmml" xref="S4.SS4.p2.12.m12.1.1.2.2.3">𝜎</ci></apply><ci id="S4.SS4.p2.12.m12.1.1.2.3.cmml" xref="S4.SS4.p2.12.m12.1.1.2.3">𝐿</ci></apply><cn id="S4.SS4.p2.12.m12.1.1.3.cmml" type="integer" xref="S4.SS4.p2.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.12.m12.1c">\lambda^{\sigma}_{L}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.12.m12.1d">italic_λ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = 0</annotation></semantics></math>. </div> </section> <section class="ltx_subsection" id="S4.SS5"> <h3 class="ltx_title ltx_title_subsection"> 4.5 Evolution</h3> <div class="ltx_para" id="S4.SS5.p1"> To simulate the evolution of morphogenesis, we established an initial population of 60 organisms with each assigned a different GRN and developing on a separate CPM grid. Each GRN is specified by 234 <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.1.m1.1"><semantics id="S4.SS5.p1.1.m1.1a"><msub id="S4.SS5.p1.1.m1.1.1" xref="S4.SS5.p1.1.m1.1.1.cmml"><mi id="S4.SS5.p1.1.m1.1.1.2" xref="S4.SS5.p1.1.m1.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.1.m1.1.1.3" xref="S4.SS5.p1.1.m1.1.1.3.cmml"><mi id="S4.SS5.p1.1.m1.1.1.3.2" xref="S4.SS5.p1.1.m1.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.1.m1.1.1.3.1" xref="S4.SS5.p1.1.m1.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.1.m1.1.1.3.3" xref="S4.SS5.p1.1.m1.1.1.3.3.cmml"><mi id="S4.SS5.p1.1.m1.1.1.3.3.2" xref="S4.SS5.p1.1.m1.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.1.m1.1.1.3.3.3" xref="S4.SS5.p1.1.m1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.1.m1.1b"><apply id="S4.SS5.p1.1.m1.1.1.cmml" xref="S4.SS5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.1.m1.1.1.1.cmml" xref="S4.SS5.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS5.p1.1.m1.1.1.2.cmml" xref="S4.SS5.p1.1.m1.1.1.2">𝑍</ci><apply id="S4.SS5.p1.1.m1.1.1.3.cmml" xref="S4.SS5.p1.1.m1.1.1.3"><times id="S4.SS5.p1.1.m1.1.1.3.1.cmml" xref="S4.SS5.p1.1.m1.1.1.3.1"></times><ci id="S4.SS5.p1.1.m1.1.1.3.2.cmml" xref="S4.SS5.p1.1.m1.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.1.m1.1.1.3.3.cmml" xref="S4.SS5.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.1.m1.1.1.3.3.1.cmml" xref="S4.SS5.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.1.m1.1.1.3.3.2.cmml" xref="S4.SS5.p1.1.m1.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.1.m1.1.1.3.3.3.cmml" xref="S4.SS5.p1.1.m1.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.1.m1.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.1.m1.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> values representing the regulatory effects of all transcription factors, as described in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E3" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">3</a> (nine transcription factors regulate 26 genes including themselves). The <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.2.m2.1"><semantics id="S4.SS5.p1.2.m2.1a"><msub id="S4.SS5.p1.2.m2.1.1" xref="S4.SS5.p1.2.m2.1.1.cmml"><mi id="S4.SS5.p1.2.m2.1.1.2" xref="S4.SS5.p1.2.m2.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.2.m2.1.1.3" xref="S4.SS5.p1.2.m2.1.1.3.cmml"><mi id="S4.SS5.p1.2.m2.1.1.3.2" xref="S4.SS5.p1.2.m2.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.2.m2.1.1.3.1" xref="S4.SS5.p1.2.m2.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.2.m2.1.1.3.3" xref="S4.SS5.p1.2.m2.1.1.3.3.cmml"><mi id="S4.SS5.p1.2.m2.1.1.3.3.2" xref="S4.SS5.p1.2.m2.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.2.m2.1.1.3.3.3" xref="S4.SS5.p1.2.m2.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.2.m2.1b"><apply id="S4.SS5.p1.2.m2.1.1.cmml" xref="S4.SS5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.2.m2.1.1.1.cmml" xref="S4.SS5.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS5.p1.2.m2.1.1.2.cmml" xref="S4.SS5.p1.2.m2.1.1.2">𝑍</ci><apply id="S4.SS5.p1.2.m2.1.1.3.cmml" xref="S4.SS5.p1.2.m2.1.1.3"><times id="S4.SS5.p1.2.m2.1.1.3.1.cmml" xref="S4.SS5.p1.2.m2.1.1.3.1"></times><ci id="S4.SS5.p1.2.m2.1.1.3.2.cmml" xref="S4.SS5.p1.2.m2.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.2.m2.1.1.3.3.cmml" xref="S4.SS5.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.2.m2.1.1.3.3.1.cmml" xref="S4.SS5.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.2.m2.1.1.3.3.2.cmml" xref="S4.SS5.p1.2.m2.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.2.m2.1.1.3.3.3.cmml" xref="S4.SS5.p1.2.m2.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.2.m2.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.2.m2.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> values of the GRNs assigned to the initial population are randomly generated according to the following probabilities: <math alttext="P(Z_{pp^{\prime}}=0)=0.54" class="ltx_Math" display="inline" id="S4.SS5.p1.3.m3.1"><semantics id="S4.SS5.p1.3.m3.1a"><mrow id="S4.SS5.p1.3.m3.1.1" xref="S4.SS5.p1.3.m3.1.1.cmml"><mrow id="S4.SS5.p1.3.m3.1.1.1" xref="S4.SS5.p1.3.m3.1.1.1.cmml"><mi id="S4.SS5.p1.3.m3.1.1.1.3" xref="S4.SS5.p1.3.m3.1.1.1.3.cmml">P</mi><mo id="S4.SS5.p1.3.m3.1.1.1.2" xref="S4.SS5.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS5.p1.3.m3.1.1.1.1.1" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS5.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS5.p1.3.m3.1.1.1.1.1.1" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.cmml"><msub id="S4.SS5.p1.3.m3.1.1.1.1.1.1.2" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S4.SS5.p1.3.m3.1.1.1.1.1.1.2.2" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.2.2.cmml">Z</mi><mrow id="S4.SS5.p1.3.m3.1.1.1.1.1.1.2.3" xref="S4.SS5.p1.3.m3.1.1.1.1.1.1.2.3.cmml"><mi 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 ) = 0.54</annotation></semantics></math>, <math alttext="P(Z_{pp^{\prime}}=1)=0.18" class="ltx_Math" display="inline" id="S4.SS5.p1.4.m4.1"><semantics id="S4.SS5.p1.4.m4.1a"><mrow id="S4.SS5.p1.4.m4.1.1" xref="S4.SS5.p1.4.m4.1.1.cmml"><mrow id="S4.SS5.p1.4.m4.1.1.1" xref="S4.SS5.p1.4.m4.1.1.1.cmml"><mi id="S4.SS5.p1.4.m4.1.1.1.3" xref="S4.SS5.p1.4.m4.1.1.1.3.cmml">P</mi><mo id="S4.SS5.p1.4.m4.1.1.1.2" xref="S4.SS5.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS5.p1.4.m4.1.1.1.1.1" xref="S4.SS5.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.SS5.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.SS5.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS5.p1.4.m4.1.1.1.1.1.1" xref="S4.SS5.p1.4.m4.1.1.1.1.1.1.cmml"><msub id="S4.SS5.p1.4.m4.1.1.1.1.1.1.2" xref="S4.SS5.p1.4.m4.1.1.1.1.1.1.2.cmml"><mi id="S4.SS5.p1.4.m4.1.1.1.1.1.1.2.2" xref="S4.SS5.p1.4.m4.1.1.1.1.1.1.2.2.cmml">Z</mi><mrow id="S4.SS5.p1.4.m4.1.1.1.1.1.1.2.3" 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xref="S4.SS5.p1.6.m6.1.1.1.1.1.1.2.3.3.3">′</ci></apply></apply></apply><cn id="S4.SS5.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS5.p1.6.m6.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="S4.SS5.p1.6.m6.1.1.3.cmml" type="float" xref="S4.SS5.p1.6.m6.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.6.m6.1c">P(Z_{pp^{\prime}}=2)=0.05</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.6.m6.1d">italic_P ( italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 2 ) = 0.05</annotation></semantics></math> and <math alttext="P(Z_{pp^{\prime}}=-2)=0.05" class="ltx_Math" display="inline" id="S4.SS5.p1.7.m7.1"><semantics id="S4.SS5.p1.7.m7.1a"><mrow id="S4.SS5.p1.7.m7.1.1" xref="S4.SS5.p1.7.m7.1.1.cmml"><mrow id="S4.SS5.p1.7.m7.1.1.1" xref="S4.SS5.p1.7.m7.1.1.1.cmml"><mi id="S4.SS5.p1.7.m7.1.1.1.3" xref="S4.SS5.p1.7.m7.1.1.1.3.cmml">P</mi><mo 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xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S4.SS5.p1.7.m7.1.1.1.1.1.1.1" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.cmml"><mo id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3a" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.cmml">−</mo><mn id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.2" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.2.cmml">2</mn></mrow></mrow><mo id="S4.SS5.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS5.p1.7.m7.1.1.2" xref="S4.SS5.p1.7.m7.1.1.2.cmml">=</mo><mn id="S4.SS5.p1.7.m7.1.1.3" xref="S4.SS5.p1.7.m7.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.7.m7.1b"><apply id="S4.SS5.p1.7.m7.1.1.cmml" xref="S4.SS5.p1.7.m7.1.1"><eq id="S4.SS5.p1.7.m7.1.1.2.cmml" xref="S4.SS5.p1.7.m7.1.1.2"></eq><apply id="S4.SS5.p1.7.m7.1.1.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1"><times id="S4.SS5.p1.7.m7.1.1.1.2.cmml" xref="S4.SS5.p1.7.m7.1.1.1.2"></times><ci id="S4.SS5.p1.7.m7.1.1.1.3.cmml" xref="S4.SS5.p1.7.m7.1.1.1.3">𝑃</ci><apply id="S4.SS5.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1"><eq id="S4.SS5.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.1"></eq><apply id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.2.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.2">𝑍</ci><apply id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3"><times id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.1"></times><ci id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.2.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.2">𝑝</ci><apply id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3">superscript</csymbol><ci id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.2.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.2">𝑝</ci><ci id="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.3.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.2.3.3.3">′</ci></apply></apply></apply><apply id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3"><minus id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3"></minus><cn id="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.SS5.p1.7.m7.1.1.1.1.1.1.3.2">2</cn></apply></apply></apply><cn id="S4.SS5.p1.7.m7.1.1.3.cmml" type="float" xref="S4.SS5.p1.7.m7.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.7.m7.1c">P(Z_{pp^{\prime}}=-2)=0.05</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.7.m7.1d">italic_P ( italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = - 2 ) = 0.05</annotation></semantics></math>. The 15 organisms with the highest morphological complexity reproduce four times to populate the next generation (the definition of morphological complexity is described in the next paragraph). Upon reproduction, there is a 50% chance that one of the <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.8.m8.1"><semantics id="S4.SS5.p1.8.m8.1a"><msub id="S4.SS5.p1.8.m8.1.1" xref="S4.SS5.p1.8.m8.1.1.cmml"><mi id="S4.SS5.p1.8.m8.1.1.2" xref="S4.SS5.p1.8.m8.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.8.m8.1.1.3" xref="S4.SS5.p1.8.m8.1.1.3.cmml"><mi id="S4.SS5.p1.8.m8.1.1.3.2" xref="S4.SS5.p1.8.m8.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.8.m8.1.1.3.1" xref="S4.SS5.p1.8.m8.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.8.m8.1.1.3.3" xref="S4.SS5.p1.8.m8.1.1.3.3.cmml"><mi id="S4.SS5.p1.8.m8.1.1.3.3.2" xref="S4.SS5.p1.8.m8.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.8.m8.1.1.3.3.3" xref="S4.SS5.p1.8.m8.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.8.m8.1b"><apply id="S4.SS5.p1.8.m8.1.1.cmml" xref="S4.SS5.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.8.m8.1.1.1.cmml" xref="S4.SS5.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS5.p1.8.m8.1.1.2.cmml" xref="S4.SS5.p1.8.m8.1.1.2">𝑍</ci><apply id="S4.SS5.p1.8.m8.1.1.3.cmml" xref="S4.SS5.p1.8.m8.1.1.3"><times id="S4.SS5.p1.8.m8.1.1.3.1.cmml" xref="S4.SS5.p1.8.m8.1.1.3.1"></times><ci id="S4.SS5.p1.8.m8.1.1.3.2.cmml" xref="S4.SS5.p1.8.m8.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.8.m8.1.1.3.3.cmml" xref="S4.SS5.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.8.m8.1.1.3.3.1.cmml" xref="S4.SS5.p1.8.m8.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.8.m8.1.1.3.3.2.cmml" xref="S4.SS5.p1.8.m8.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.8.m8.1.1.3.3.3.cmml" xref="S4.SS5.p1.8.m8.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.8.m8.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.8.m8.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> values in the GRN mutates. The specific <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.9.m9.1"><semantics id="S4.SS5.p1.9.m9.1a"><msub id="S4.SS5.p1.9.m9.1.1" xref="S4.SS5.p1.9.m9.1.1.cmml"><mi id="S4.SS5.p1.9.m9.1.1.2" xref="S4.SS5.p1.9.m9.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.9.m9.1.1.3" xref="S4.SS5.p1.9.m9.1.1.3.cmml"><mi id="S4.SS5.p1.9.m9.1.1.3.2" xref="S4.SS5.p1.9.m9.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.9.m9.1.1.3.1" xref="S4.SS5.p1.9.m9.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.9.m9.1.1.3.3" xref="S4.SS5.p1.9.m9.1.1.3.3.cmml"><mi id="S4.SS5.p1.9.m9.1.1.3.3.2" xref="S4.SS5.p1.9.m9.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.9.m9.1.1.3.3.3" xref="S4.SS5.p1.9.m9.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.9.m9.1b"><apply id="S4.SS5.p1.9.m9.1.1.cmml" xref="S4.SS5.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.9.m9.1.1.1.cmml" xref="S4.SS5.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS5.p1.9.m9.1.1.2.cmml" xref="S4.SS5.p1.9.m9.1.1.2">𝑍</ci><apply id="S4.SS5.p1.9.m9.1.1.3.cmml" xref="S4.SS5.p1.9.m9.1.1.3"><times id="S4.SS5.p1.9.m9.1.1.3.1.cmml" xref="S4.SS5.p1.9.m9.1.1.3.1"></times><ci id="S4.SS5.p1.9.m9.1.1.3.2.cmml" xref="S4.SS5.p1.9.m9.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.9.m9.1.1.3.3.cmml" xref="S4.SS5.p1.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.9.m9.1.1.3.3.1.cmml" xref="S4.SS5.p1.9.m9.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.9.m9.1.1.3.3.2.cmml" xref="S4.SS5.p1.9.m9.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.9.m9.1.1.3.3.3.cmml" xref="S4.SS5.p1.9.m9.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.9.m9.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.9.m9.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> that mutates is chosen at random with equal probability. The mutation alters the value of <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.10.m10.1"><semantics id="S4.SS5.p1.10.m10.1a"><msub id="S4.SS5.p1.10.m10.1.1" xref="S4.SS5.p1.10.m10.1.1.cmml"><mi id="S4.SS5.p1.10.m10.1.1.2" xref="S4.SS5.p1.10.m10.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.10.m10.1.1.3" xref="S4.SS5.p1.10.m10.1.1.3.cmml"><mi id="S4.SS5.p1.10.m10.1.1.3.2" xref="S4.SS5.p1.10.m10.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.10.m10.1.1.3.1" xref="S4.SS5.p1.10.m10.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.10.m10.1.1.3.3" xref="S4.SS5.p1.10.m10.1.1.3.3.cmml"><mi id="S4.SS5.p1.10.m10.1.1.3.3.2" xref="S4.SS5.p1.10.m10.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.10.m10.1.1.3.3.3" xref="S4.SS5.p1.10.m10.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.10.m10.1b"><apply id="S4.SS5.p1.10.m10.1.1.cmml" xref="S4.SS5.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.10.m10.1.1.1.cmml" xref="S4.SS5.p1.10.m10.1.1">subscript</csymbol><ci id="S4.SS5.p1.10.m10.1.1.2.cmml" xref="S4.SS5.p1.10.m10.1.1.2">𝑍</ci><apply id="S4.SS5.p1.10.m10.1.1.3.cmml" xref="S4.SS5.p1.10.m10.1.1.3"><times id="S4.SS5.p1.10.m10.1.1.3.1.cmml" xref="S4.SS5.p1.10.m10.1.1.3.1"></times><ci id="S4.SS5.p1.10.m10.1.1.3.2.cmml" xref="S4.SS5.p1.10.m10.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.10.m10.1.1.3.3.cmml" xref="S4.SS5.p1.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.10.m10.1.1.3.3.1.cmml" xref="S4.SS5.p1.10.m10.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.10.m10.1.1.3.3.2.cmml" xref="S4.SS5.p1.10.m10.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.10.m10.1.1.3.3.3.cmml" xref="S4.SS5.p1.10.m10.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.10.m10.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.10.m10.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, independent of its current value, according to same probabilities used to generate the <math alttext="Z_{pp^{\prime}}" class="ltx_Math" display="inline" id="S4.SS5.p1.11.m11.1"><semantics id="S4.SS5.p1.11.m11.1a"><msub id="S4.SS5.p1.11.m11.1.1" xref="S4.SS5.p1.11.m11.1.1.cmml"><mi id="S4.SS5.p1.11.m11.1.1.2" xref="S4.SS5.p1.11.m11.1.1.2.cmml">Z</mi><mrow id="S4.SS5.p1.11.m11.1.1.3" xref="S4.SS5.p1.11.m11.1.1.3.cmml"><mi id="S4.SS5.p1.11.m11.1.1.3.2" xref="S4.SS5.p1.11.m11.1.1.3.2.cmml">p</mi><mo id="S4.SS5.p1.11.m11.1.1.3.1" xref="S4.SS5.p1.11.m11.1.1.3.1.cmml">⁢</mo><msup id="S4.SS5.p1.11.m11.1.1.3.3" xref="S4.SS5.p1.11.m11.1.1.3.3.cmml"><mi id="S4.SS5.p1.11.m11.1.1.3.3.2" xref="S4.SS5.p1.11.m11.1.1.3.3.2.cmml">p</mi><mo id="S4.SS5.p1.11.m11.1.1.3.3.3" xref="S4.SS5.p1.11.m11.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p1.11.m11.1b"><apply id="S4.SS5.p1.11.m11.1.1.cmml" xref="S4.SS5.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS5.p1.11.m11.1.1.1.cmml" xref="S4.SS5.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS5.p1.11.m11.1.1.2.cmml" xref="S4.SS5.p1.11.m11.1.1.2">𝑍</ci><apply id="S4.SS5.p1.11.m11.1.1.3.cmml" xref="S4.SS5.p1.11.m11.1.1.3"><times id="S4.SS5.p1.11.m11.1.1.3.1.cmml" xref="S4.SS5.p1.11.m11.1.1.3.1"></times><ci id="S4.SS5.p1.11.m11.1.1.3.2.cmml" xref="S4.SS5.p1.11.m11.1.1.3.2">𝑝</ci><apply id="S4.SS5.p1.11.m11.1.1.3.3.cmml" xref="S4.SS5.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p1.11.m11.1.1.3.3.1.cmml" xref="S4.SS5.p1.11.m11.1.1.3.3">superscript</csymbol><ci id="S4.SS5.p1.11.m11.1.1.3.3.2.cmml" xref="S4.SS5.p1.11.m11.1.1.3.3.2">𝑝</ci><ci id="S4.SS5.p1.11.m11.1.1.3.3.3.cmml" xref="S4.SS5.p1.11.m11.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p1.11.m11.1c">Z_{pp^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p1.11.m11.1d">italic_Z start_POSTSUBSCRIPT italic_p italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> assigned to the initial population. Gene duplication and deletion do not occur. </div> <div class="ltx_para" id="S4.SS5.p2"> Morphological complexity was defined based on an algorithm that quantifies the complexity of two-dimensional shapes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib77" title="">77</a>]</cite>. Morphological complexity is a summation of two measures: the deviation of morphology from a circle (denoted by <math alttext="z_{1}" class="ltx_Math" display="inline" id="S4.SS5.p2.1.m1.1"><semantics id="S4.SS5.p2.1.m1.1a"><msub id="S4.SS5.p2.1.m1.1.1" xref="S4.SS5.p2.1.m1.1.1.cmml"><mi id="S4.SS5.p2.1.m1.1.1.2" xref="S4.SS5.p2.1.m1.1.1.2.cmml">z</mi><mn id="S4.SS5.p2.1.m1.1.1.3" xref="S4.SS5.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p2.1.m1.1b"><apply id="S4.SS5.p2.1.m1.1.1.cmml" xref="S4.SS5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p2.1.m1.1.1.1.cmml" xref="S4.SS5.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS5.p2.1.m1.1.1.2.cmml" xref="S4.SS5.p2.1.m1.1.1.2">𝑧</ci><cn id="S4.SS5.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS5.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p2.1.m1.1c">z_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p2.1.m1.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>) and the degree of inward folds (denoted by <math alttext="z_{2}" class="ltx_Math" display="inline" id="S4.SS5.p2.2.m2.1"><semantics id="S4.SS5.p2.2.m2.1a"><msub id="S4.SS5.p2.2.m2.1.1" xref="S4.SS5.p2.2.m2.1.1.cmml"><mi id="S4.SS5.p2.2.m2.1.1.2" xref="S4.SS5.p2.2.m2.1.1.2.cmml">z</mi><mn id="S4.SS5.p2.2.m2.1.1.3" xref="S4.SS5.p2.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p2.2.m2.1b"><apply id="S4.SS5.p2.2.m2.1.1.cmml" xref="S4.SS5.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS5.p2.2.m2.1.1.1.cmml" xref="S4.SS5.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS5.p2.2.m2.1.1.2.cmml" xref="S4.SS5.p2.2.m2.1.1.2">𝑧</ci><cn id="S4.SS5.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS5.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p2.2.m2.1c">z_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p2.2.m2.1d">italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>), as described below. </div> <div class="ltx_para" id="S4.SS5.p3"> The deviation of morphology from a circle <math alttext="z_{1}" class="ltx_Math" display="inline" id="S4.SS5.p3.1.m1.1"><semantics id="S4.SS5.p3.1.m1.1a"><msub id="S4.SS5.p3.1.m1.1.1" xref="S4.SS5.p3.1.m1.1.1.cmml"><mi id="S4.SS5.p3.1.m1.1.1.2" xref="S4.SS5.p3.1.m1.1.1.2.cmml">z</mi><mn id="S4.SS5.p3.1.m1.1.1.3" xref="S4.SS5.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.1.m1.1b"><apply id="S4.SS5.p3.1.m1.1.1.cmml" xref="S4.SS5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p3.1.m1.1.1.1.cmml" xref="S4.SS5.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS5.p3.1.m1.1.1.2.cmml" xref="S4.SS5.p3.1.m1.1.1.2">𝑧</ci><cn id="S4.SS5.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS5.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.1.m1.1c">z_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.1.m1.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is defined by the following equation: <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z_{1}=\langle|r_{c}-r(\theta)|\rangle_{\theta}" class="ltx_Math" display="block" id="S4.E7.m1.2"><semantics id="S4.E7.m1.2a"><mrow id="S4.E7.m1.2.2" xref="S4.E7.m1.2.2.cmml"><msub id="S4.E7.m1.2.2.3" xref="S4.E7.m1.2.2.3.cmml"><mi id="S4.E7.m1.2.2.3.2" xref="S4.E7.m1.2.2.3.2.cmml">z</mi><mn id="S4.E7.m1.2.2.3.3" xref="S4.E7.m1.2.2.3.3.cmml">1</mn></msub><mo id="S4.E7.m1.2.2.2" xref="S4.E7.m1.2.2.2.cmml">=</mo><msub id="S4.E7.m1.2.2.1" xref="S4.E7.m1.2.2.1.cmml"><mrow id="S4.E7.m1.2.2.1.1.1" xref="S4.E7.m1.2.2.1.1.2.cmml"><mo id="S4.E7.m1.2.2.1.1.1.2" stretchy="false" xref="S4.E7.m1.2.2.1.1.2.1.cmml">⟨</mo><mrow id="S4.E7.m1.2.2.1.1.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.2.cmml"><mo id="S4.E7.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.2.1.cmml">|</mo><mrow id="S4.E7.m1.2.2.1.1.1.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.1.1.cmml"><msub id="S4.E7.m1.2.2.1.1.1.1.1.1.2" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="S4.E7.m1.2.2.1.1.1.1.1.1.2.2" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2.2.cmml">r</mi><mi id="S4.E7.m1.2.2.1.1.1.1.1.1.2.3" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="S4.E7.m1.2.2.1.1.1.1.1.1.1" xref="S4.E7.m1.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E7.m1.2.2.1.1.1.1.1.1.3" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="S4.E7.m1.2.2.1.1.1.1.1.1.3.2" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.2.cmml">r</mi><mo id="S4.E7.m1.2.2.1.1.1.1.1.1.3.1" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E7.m1.2.2.1.1.1.1.1.1.3.3.2" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.cmml"><mo id="S4.E7.m1.2.2.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.cmml">(</mo><mi id="S4.E7.m1.1.1" xref="S4.E7.m1.1.1.cmml">θ</mi><mo id="S4.E7.m1.2.2.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E7.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.E7.m1.2.2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.E7.m1.2.2.1.1.1.3" stretchy="false" xref="S4.E7.m1.2.2.1.1.2.1.cmml">⟩</mo></mrow><mi id="S4.E7.m1.2.2.1.3" xref="S4.E7.m1.2.2.1.3.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m1.2b"><apply id="S4.E7.m1.2.2.cmml" xref="S4.E7.m1.2.2"><eq id="S4.E7.m1.2.2.2.cmml" xref="S4.E7.m1.2.2.2"></eq><apply id="S4.E7.m1.2.2.3.cmml" xref="S4.E7.m1.2.2.3"><csymbol cd="ambiguous" id="S4.E7.m1.2.2.3.1.cmml" xref="S4.E7.m1.2.2.3">subscript</csymbol><ci id="S4.E7.m1.2.2.3.2.cmml" xref="S4.E7.m1.2.2.3.2">𝑧</ci><cn id="S4.E7.m1.2.2.3.3.cmml" type="integer" xref="S4.E7.m1.2.2.3.3">1</cn></apply><apply id="S4.E7.m1.2.2.1.cmml" xref="S4.E7.m1.2.2.1"><csymbol cd="ambiguous" id="S4.E7.m1.2.2.1.2.cmml" xref="S4.E7.m1.2.2.1">subscript</csymbol><apply id="S4.E7.m1.2.2.1.1.2.cmml" xref="S4.E7.m1.2.2.1.1.1"><csymbol cd="latexml" id="S4.E7.m1.2.2.1.1.2.1.cmml" xref="S4.E7.m1.2.2.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S4.E7.m1.2.2.1.1.1.1.2.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1"><abs id="S4.E7.m1.2.2.1.1.1.1.2.1.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.2"></abs><apply id="S4.E7.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1"><minus id="S4.E7.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.1"></minus><apply id="S4.E7.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E7.m1.2.2.1.1.1.1.1.1.2.1.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.E7.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2.2">𝑟</ci><ci id="S4.E7.m1.2.2.1.1.1.1.1.1.2.3.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.2.3">𝑐</ci></apply><apply id="S4.E7.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3"><times id="S4.E7.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.1"></times><ci id="S4.E7.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S4.E7.m1.2.2.1.1.1.1.1.1.3.2">𝑟</ci><ci id="S4.E7.m1.1.1.cmml" xref="S4.E7.m1.1.1">𝜃</ci></apply></apply></apply></apply><ci id="S4.E7.m1.2.2.1.3.cmml" xref="S4.E7.m1.2.2.1.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.2c">z_{1}=\langle|r_{c}-r(\theta)|\rangle_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.2d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ⟨ | italic_r start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - italic_r ( italic_θ ) | ⟩ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(7)</td> </tr></tbody> </table> where <math alttext="r_{c}" class="ltx_Math" display="inline" id="S4.SS5.p3.2.m1.1"><semantics id="S4.SS5.p3.2.m1.1a"><msub id="S4.SS5.p3.2.m1.1.1" xref="S4.SS5.p3.2.m1.1.1.cmml"><mi id="S4.SS5.p3.2.m1.1.1.2" xref="S4.SS5.p3.2.m1.1.1.2.cmml">r</mi><mi id="S4.SS5.p3.2.m1.1.1.3" xref="S4.SS5.p3.2.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.2.m1.1b"><apply id="S4.SS5.p3.2.m1.1.1.cmml" xref="S4.SS5.p3.2.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p3.2.m1.1.1.1.cmml" xref="S4.SS5.p3.2.m1.1.1">subscript</csymbol><ci id="S4.SS5.p3.2.m1.1.1.2.cmml" xref="S4.SS5.p3.2.m1.1.1.2">𝑟</ci><ci id="S4.SS5.p3.2.m1.1.1.3.cmml" xref="S4.SS5.p3.2.m1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.2.m1.1c">r_{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.2.m1.1d">italic_r start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> is the hypothetical radius of an organism if the organism’s morphology were perfectly circular, and <math alttext="r(\theta)" class="ltx_Math" display="inline" id="S4.SS5.p3.3.m2.1"><semantics id="S4.SS5.p3.3.m2.1a"><mrow id="S4.SS5.p3.3.m2.1.2" xref="S4.SS5.p3.3.m2.1.2.cmml"><mi id="S4.SS5.p3.3.m2.1.2.2" xref="S4.SS5.p3.3.m2.1.2.2.cmml">r</mi><mo id="S4.SS5.p3.3.m2.1.2.1" xref="S4.SS5.p3.3.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS5.p3.3.m2.1.2.3.2" xref="S4.SS5.p3.3.m2.1.2.cmml"><mo id="S4.SS5.p3.3.m2.1.2.3.2.1" stretchy="false" xref="S4.SS5.p3.3.m2.1.2.cmml">(</mo><mi id="S4.SS5.p3.3.m2.1.1" xref="S4.SS5.p3.3.m2.1.1.cmml">θ</mi><mo id="S4.SS5.p3.3.m2.1.2.3.2.2" stretchy="false" xref="S4.SS5.p3.3.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.3.m2.1b"><apply id="S4.SS5.p3.3.m2.1.2.cmml" xref="S4.SS5.p3.3.m2.1.2"><times id="S4.SS5.p3.3.m2.1.2.1.cmml" xref="S4.SS5.p3.3.m2.1.2.1"></times><ci id="S4.SS5.p3.3.m2.1.2.2.cmml" xref="S4.SS5.p3.3.m2.1.2.2">𝑟</ci><ci id="S4.SS5.p3.3.m2.1.1.cmml" xref="S4.SS5.p3.3.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.3.m2.1c">r(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.3.m2.1d">italic_r ( italic_θ )</annotation></semantics></math> is the maximum distance from the centre of mass of the organism to any pixel in the direction specified by angle <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS5.p3.4.m3.1"><semantics id="S4.SS5.p3.4.m3.1a"><mi id="S4.SS5.p3.4.m3.1.1" xref="S4.SS5.p3.4.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.4.m3.1b"><ci id="S4.SS5.p3.4.m3.1.1.cmml" xref="S4.SS5.p3.4.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.4.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.4.m3.1d">italic_θ</annotation></semantics></math> (one pixel corresponds to one unit mass). The notation <math alttext="\langle...\rangle_{\theta}" class="ltx_Math" display="inline" id="S4.SS5.p3.5.m4.1"><semantics id="S4.SS5.p3.5.m4.1a"><msub id="S4.SS5.p3.5.m4.1.2" xref="S4.SS5.p3.5.m4.1.2.cmml"><mrow id="S4.SS5.p3.5.m4.1.2.2.2" xref="S4.SS5.p3.5.m4.1.2.2.1.cmml"><mo id="S4.SS5.p3.5.m4.1.2.2.2.1" stretchy="false" xref="S4.SS5.p3.5.m4.1.2.2.1.1.cmml">⟨</mo><mi id="S4.SS5.p3.5.m4.1.1" mathvariant="normal" xref="S4.SS5.p3.5.m4.1.1.cmml">…</mi><mo id="S4.SS5.p3.5.m4.1.2.2.2.2" stretchy="false" xref="S4.SS5.p3.5.m4.1.2.2.1.1.cmml">⟩</mo></mrow><mi id="S4.SS5.p3.5.m4.1.2.3" xref="S4.SS5.p3.5.m4.1.2.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.5.m4.1b"><apply id="S4.SS5.p3.5.m4.1.2.cmml" xref="S4.SS5.p3.5.m4.1.2"><csymbol cd="ambiguous" id="S4.SS5.p3.5.m4.1.2.1.cmml" xref="S4.SS5.p3.5.m4.1.2">subscript</csymbol><apply id="S4.SS5.p3.5.m4.1.2.2.1.cmml" xref="S4.SS5.p3.5.m4.1.2.2.2"><csymbol cd="latexml" id="S4.SS5.p3.5.m4.1.2.2.1.1.cmml" xref="S4.SS5.p3.5.m4.1.2.2.2.1">delimited-⟨⟩</csymbol><ci id="S4.SS5.p3.5.m4.1.1.cmml" xref="S4.SS5.p3.5.m4.1.1">…</ci></apply><ci id="S4.SS5.p3.5.m4.1.2.3.cmml" xref="S4.SS5.p3.5.m4.1.2.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.5.m4.1c">\langle...\rangle_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.5.m4.1d">⟨ … ⟩ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> indicates an average taken over all angles <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS5.p3.6.m5.1"><semantics id="S4.SS5.p3.6.m5.1a"><mi id="S4.SS5.p3.6.m5.1.1" xref="S4.SS5.p3.6.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.6.m5.1b"><ci id="S4.SS5.p3.6.m5.1.1.cmml" xref="S4.SS5.p3.6.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.6.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.6.m5.1d">italic_θ</annotation></semantics></math>, where <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS5.p3.7.m6.1"><semantics id="S4.SS5.p3.7.m6.1a"><mi id="S4.SS5.p3.7.m6.1.1" xref="S4.SS5.p3.7.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS5.p3.7.m6.1b"><ci id="S4.SS5.p3.7.m6.1.1.cmml" xref="S4.SS5.p3.7.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p3.7.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p3.7.m6.1d">italic_θ</annotation></semantics></math> is discretised into 360 degrees for computation. </div> <div class="ltx_para" id="S4.SS5.p4"> The degree of inward folds (<math alttext="z_{2}" class="ltx_Math" display="inline" id="S4.SS5.p4.1.m1.1"><semantics id="S4.SS5.p4.1.m1.1a"><msub id="S4.SS5.p4.1.m1.1.1" xref="S4.SS5.p4.1.m1.1.1.cmml"><mi id="S4.SS5.p4.1.m1.1.1.2" xref="S4.SS5.p4.1.m1.1.1.2.cmml">z</mi><mn id="S4.SS5.p4.1.m1.1.1.3" xref="S4.SS5.p4.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p4.1.m1.1b"><apply id="S4.SS5.p4.1.m1.1.1.cmml" xref="S4.SS5.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p4.1.m1.1.1.1.cmml" xref="S4.SS5.p4.1.m1.1.1">subscript</csymbol><ci id="S4.SS5.p4.1.m1.1.1.2.cmml" xref="S4.SS5.p4.1.m1.1.1.2">𝑧</ci><cn id="S4.SS5.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.SS5.p4.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p4.1.m1.1c">z_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p4.1.m1.1d">italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>) measures the sum of the sizes of regions of the medium surrounded by concave parts of an organism. To identify these regions, we begin by drawing horizontal parallel lines across the CPM grid spaced one pixel apart, resulting in a total of 250 lines. Next, we locate segments along the lines that intersect the extracellular medium in between cells. The segments were discarded if they did not exceed a minimum length of 20 pixels to filter out inward folds due to stochastic cell boundary fluctuations. The number of the retained segments was recorded. The above procedure was repeated by tilting the 250 parallel lines at 12 evenly-spaced angles across the range <math alttext="[-\pi/2,\pi/2]" class="ltx_Math" display="inline" id="S4.SS5.p4.2.m2.2"><semantics id="S4.SS5.p4.2.m2.2a"><mrow id="S4.SS5.p4.2.m2.2.2.2" xref="S4.SS5.p4.2.m2.2.2.3.cmml"><mo id="S4.SS5.p4.2.m2.2.2.2.3" stretchy="false" xref="S4.SS5.p4.2.m2.2.2.3.cmml">[</mo><mrow id="S4.SS5.p4.2.m2.1.1.1.1" xref="S4.SS5.p4.2.m2.1.1.1.1.cmml"><mo id="S4.SS5.p4.2.m2.1.1.1.1a" xref="S4.SS5.p4.2.m2.1.1.1.1.cmml">−</mo><mrow id="S4.SS5.p4.2.m2.1.1.1.1.2" xref="S4.SS5.p4.2.m2.1.1.1.1.2.cmml"><mi id="S4.SS5.p4.2.m2.1.1.1.1.2.2" xref="S4.SS5.p4.2.m2.1.1.1.1.2.2.cmml">π</mi><mo id="S4.SS5.p4.2.m2.1.1.1.1.2.1" xref="S4.SS5.p4.2.m2.1.1.1.1.2.1.cmml">/</mo><mn id="S4.SS5.p4.2.m2.1.1.1.1.2.3" xref="S4.SS5.p4.2.m2.1.1.1.1.2.3.cmml">2</mn></mrow></mrow><mo id="S4.SS5.p4.2.m2.2.2.2.4" xref="S4.SS5.p4.2.m2.2.2.3.cmml">,</mo><mrow id="S4.SS5.p4.2.m2.2.2.2.2" 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encoding="MathML-Content" id="S4.SS5.p4.3.m3.1b"><apply id="S4.SS5.p4.3.m3.1.1.cmml" xref="S4.SS5.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS5.p4.3.m3.1.1.1.cmml" xref="S4.SS5.p4.3.m3.1.1">subscript</csymbol><ci id="S4.SS5.p4.3.m3.1.1.2.cmml" xref="S4.SS5.p4.3.m3.1.1.2">𝑧</ci><cn id="S4.SS5.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.SS5.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p4.3.m3.1c">z_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p4.3.m3.1d">italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is defined as the square root of the total number of the retained segments (<math alttext="z_{2}" class="ltx_Math" display="inline" id="S4.SS5.p4.4.m4.1"><semantics id="S4.SS5.p4.4.m4.1a"><msub id="S4.SS5.p4.4.m4.1.1" xref="S4.SS5.p4.4.m4.1.1.cmml"><mi id="S4.SS5.p4.4.m4.1.1.2" xref="S4.SS5.p4.4.m4.1.1.2.cmml">z</mi><mn id="S4.SS5.p4.4.m4.1.1.3" xref="S4.SS5.p4.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p4.4.m4.1b"><apply id="S4.SS5.p4.4.m4.1.1.cmml" xref="S4.SS5.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS5.p4.4.m4.1.1.1.cmml" xref="S4.SS5.p4.4.m4.1.1">subscript</csymbol><ci id="S4.SS5.p4.4.m4.1.1.2.cmml" xref="S4.SS5.p4.4.m4.1.1.2">𝑧</ci><cn id="S4.SS5.p4.4.m4.1.1.3.cmml" type="integer" xref="S4.SS5.p4.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p4.4.m4.1c">z_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p4.4.m4.1d">italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is independent of the lengths of the retained segments). </div> <div class="ltx_para" id="S4.SS5.p5"> Morphological complexity was defined as <math alttext="2z_{1}+2.5z_{2}" class="ltx_Math" display="inline" id="S4.SS5.p5.1.m1.1"><semantics id="S4.SS5.p5.1.m1.1a"><mrow id="S4.SS5.p5.1.m1.1.1" xref="S4.SS5.p5.1.m1.1.1.cmml"><mrow id="S4.SS5.p5.1.m1.1.1.2" xref="S4.SS5.p5.1.m1.1.1.2.cmml"><mn id="S4.SS5.p5.1.m1.1.1.2.2" xref="S4.SS5.p5.1.m1.1.1.2.2.cmml">2</mn><mo id="S4.SS5.p5.1.m1.1.1.2.1" xref="S4.SS5.p5.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S4.SS5.p5.1.m1.1.1.2.3" xref="S4.SS5.p5.1.m1.1.1.2.3.cmml"><mi id="S4.SS5.p5.1.m1.1.1.2.3.2" xref="S4.SS5.p5.1.m1.1.1.2.3.2.cmml">z</mi><mn id="S4.SS5.p5.1.m1.1.1.2.3.3" xref="S4.SS5.p5.1.m1.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.SS5.p5.1.m1.1.1.1" xref="S4.SS5.p5.1.m1.1.1.1.cmml">+</mo><mrow id="S4.SS5.p5.1.m1.1.1.3" xref="S4.SS5.p5.1.m1.1.1.3.cmml"><mn id="S4.SS5.p5.1.m1.1.1.3.2" xref="S4.SS5.p5.1.m1.1.1.3.2.cmml">2.5</mn><mo id="S4.SS5.p5.1.m1.1.1.3.1" xref="S4.SS5.p5.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S4.SS5.p5.1.m1.1.1.3.3" xref="S4.SS5.p5.1.m1.1.1.3.3.cmml"><mi id="S4.SS5.p5.1.m1.1.1.3.3.2" xref="S4.SS5.p5.1.m1.1.1.3.3.2.cmml">z</mi><mn id="S4.SS5.p5.1.m1.1.1.3.3.3" xref="S4.SS5.p5.1.m1.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.p5.1.m1.1b"><apply id="S4.SS5.p5.1.m1.1.1.cmml" xref="S4.SS5.p5.1.m1.1.1"><plus id="S4.SS5.p5.1.m1.1.1.1.cmml" xref="S4.SS5.p5.1.m1.1.1.1"></plus><apply id="S4.SS5.p5.1.m1.1.1.2.cmml" xref="S4.SS5.p5.1.m1.1.1.2"><times id="S4.SS5.p5.1.m1.1.1.2.1.cmml" xref="S4.SS5.p5.1.m1.1.1.2.1"></times><cn id="S4.SS5.p5.1.m1.1.1.2.2.cmml" type="integer" xref="S4.SS5.p5.1.m1.1.1.2.2">2</cn><apply id="S4.SS5.p5.1.m1.1.1.2.3.cmml" xref="S4.SS5.p5.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS5.p5.1.m1.1.1.2.3.1.cmml" xref="S4.SS5.p5.1.m1.1.1.2.3">subscript</csymbol><ci id="S4.SS5.p5.1.m1.1.1.2.3.2.cmml" xref="S4.SS5.p5.1.m1.1.1.2.3.2">𝑧</ci><cn id="S4.SS5.p5.1.m1.1.1.2.3.3.cmml" type="integer" xref="S4.SS5.p5.1.m1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS5.p5.1.m1.1.1.3.cmml" xref="S4.SS5.p5.1.m1.1.1.3"><times id="S4.SS5.p5.1.m1.1.1.3.1.cmml" xref="S4.SS5.p5.1.m1.1.1.3.1"></times><cn id="S4.SS5.p5.1.m1.1.1.3.2.cmml" type="float" xref="S4.SS5.p5.1.m1.1.1.3.2">2.5</cn><apply id="S4.SS5.p5.1.m1.1.1.3.3.cmml" xref="S4.SS5.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS5.p5.1.m1.1.1.3.3.1.cmml" xref="S4.SS5.p5.1.m1.1.1.3.3">subscript</csymbol><ci id="S4.SS5.p5.1.m1.1.1.3.3.2.cmml" xref="S4.SS5.p5.1.m1.1.1.3.3.2">𝑧</ci><cn id="S4.SS5.p5.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.SS5.p5.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p5.1.m1.1c">2z_{1}+2.5z_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p5.1.m1.1d">2 italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2.5 italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. The weights of 2 and 2.5 were selected to ensure that the maximum value of either term is approximately 100 for shapes that are realistically achievable within this model, ensuring that neither term dominates the fitness criterion. In order to filter out noise, morphological complexity is taken as an average of 10 evenly-spaced measurements over the last 1,000 DTS. If cells lose physical contact with other cells during development (see Fig. S8EF for examples), we assign the organism a fitness of 0 as our quantification of morphological complexity was not designed to handle multiple shapes. </div> <div class="ltx_para" id="S4.SS5.p6"> The alternative fitness criterion that was used to determine whether stem-cell-based morphogenesis is evolutionary accessible was implemented in two stages. In the first stage, the fitness criterion is the displacement of an organism’s centre of mass, measured in pixels, from the start to the end of the 12,000 DTS, denoted <math alttext="z_{3}" class="ltx_Math" display="inline" id="S4.SS5.p6.1.m1.1"><semantics id="S4.SS5.p6.1.m1.1a"><msub id="S4.SS5.p6.1.m1.1.1" xref="S4.SS5.p6.1.m1.1.1.cmml"><mi id="S4.SS5.p6.1.m1.1.1.2" xref="S4.SS5.p6.1.m1.1.1.2.cmml">z</mi><mn id="S4.SS5.p6.1.m1.1.1.3" xref="S4.SS5.p6.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS5.p6.1.m1.1b"><apply id="S4.SS5.p6.1.m1.1.1.cmml" xref="S4.SS5.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS5.p6.1.m1.1.1.1.cmml" xref="S4.SS5.p6.1.m1.1.1">subscript</csymbol><ci id="S4.SS5.p6.1.m1.1.1.2.cmml" xref="S4.SS5.p6.1.m1.1.1.2">𝑧</ci><cn id="S4.SS5.p6.1.m1.1.1.3.cmml" type="integer" xref="S4.SS5.p6.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.p6.1.m1.1c">z_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p6.1.m1.1d">italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. Once the average fitness across all organisms in a population exceeded 15 pixels, the fitness criterion transitioned to the second stage. In the second stage, the fitness criterion is (<math alttext="2z_{1}+2.5z_{2})/2+z_{3}^{1.5}" class="ltx_math_unparsed" display="inline" id="S4.SS5.p6.2.m2.1"><semantics id="S4.SS5.p6.2.m2.1a"><mrow id="S4.SS5.p6.2.m2.1b"><mn id="S4.SS5.p6.2.m2.1.1">2</mn><msub id="S4.SS5.p6.2.m2.1.2"><mi id="S4.SS5.p6.2.m2.1.2.2">z</mi><mn id="S4.SS5.p6.2.m2.1.2.3">1</mn></msub><mo id="S4.SS5.p6.2.m2.1.3">+</mo><mn id="S4.SS5.p6.2.m2.1.4">2.5</mn><msub id="S4.SS5.p6.2.m2.1.5"><mi id="S4.SS5.p6.2.m2.1.5.2">z</mi><mn id="S4.SS5.p6.2.m2.1.5.3">2</mn></msub><mo id="S4.SS5.p6.2.m2.1.6" stretchy="false">)</mo><mo id="S4.SS5.p6.2.m2.1.7">/</mo><mn id="S4.SS5.p6.2.m2.1.8">2</mn><mo id="S4.SS5.p6.2.m2.1.9">+</mo><mi id="S4.SS5.p6.2.m2.1.10">z</mi><msub id="S4.SS5.p6.2.m2.1.11"><mi id="S4.SS5.p6.2.m2.1.11a"></mi><mn id="S4.SS5.p6.2.m2.1.11.1">3</mn></msub><msup id="S4.SS5.p6.2.m2.1.12"><mi id="S4.SS5.p6.2.m2.1.12a"></mi><mn id="S4.SS5.p6.2.m2.1.12.1">1.5</mn></msup></mrow><annotation encoding="application/x-tex" id="S4.SS5.p6.2.m2.1c">2z_{1}+2.5z_{2})/2+z_{3}^{1.5}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.p6.2.m2.1d">2 italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2.5 italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 + italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1.5 end_POSTSUPERSCRIPT</annotation></semantics></math>. The weightings of each criterion were selected so that the maximum value of each is approximately 100 for shapes that are realistically achievable within this model, ensuring that neither term dominates the fitness criterion. </div> <div class="ltx_para" id="S4.SS5.p7"> To determine whether an evolutionary simulation succeeded or failed, we used fitness thresholds. The thresholds were applied to the organism that recorded the most complex morphology in the final generation of the simulation. In the simulations selecting only for morphological complexity, we applied a threshold of 70 after averaging the morphological complexity over 60 developmental replicates in order to account for variance in the complexity score. However, this threshold does not affect our key findings, as there are still highly reproducible organisms with stem-cell systems below the threshold (See Fig. S2AB). In the simulations selecting for both shifting centre of mass and morphological complexity, the fitness threshold we used to determine whether an evolutionary simulation succeeded was whether the second stage was reached. </div> </section> <section class="ltx_subsection" id="S4.SS6"> <h3 class="ltx_title ltx_title_subsection"> 4.6 Cell state and state space</h3> <div class="ltx_para" id="S4.SS6.p1"> The cell state is an <math alttext="n" class="ltx_Math" display="inline" id="S4.SS6.p1.1.m1.1"><semantics id="S4.SS6.p1.1.m1.1a"><mi id="S4.SS6.p1.1.m1.1.1" xref="S4.SS6.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p1.1.m1.1b"><ci id="S4.SS6.p1.1.m1.1.1.cmml" xref="S4.SS6.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p1.1.m1.1d">italic_n</annotation></semantics></math>-dimensional boolean vector, where <math alttext="n" class="ltx_Math" display="inline" id="S4.SS6.p1.2.m2.1"><semantics id="S4.SS6.p1.2.m2.1a"><mi id="S4.SS6.p1.2.m2.1.1" xref="S4.SS6.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p1.2.m2.1b"><ci id="S4.SS6.p1.2.m2.1.1.cmml" xref="S4.SS6.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p1.2.m2.1d">italic_n</annotation></semantics></math> is the total number of adhesion and contractile proteins (although the results do not change when TFs are included in the vector as well). Each element in this vector is the concentration of each protein booleanised to either ON or OFF (as described previously for lock, key and contractile proteins). Each boolean vector is assigned a single colour on the CPM grid. These colours are chosen arbitrarily, and each organism has its own distinct colour set. While this approach means that the same cell state might appear in different colours across organisms, it is uncommon to encounter identical cell states in different evolved organisms. The cell state of each cell is determined after every numerical integration step of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a>. A change in a cell’s boolean vector corresponds to a cell state transition. </div> <div class="ltx_para" id="S4.SS6.p2"> To generate the cell state space, we recorded all cell state transitions for all cells after 6,000 DTS from the beginning of development. Although it makes no qualitative difference when the recording of transitions begins, starting at 6,000 reduces the appearance of “transient” cell states <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib78" title="">78</a>]</cite>, such as the cell state corresponding to the initial conditions of cell proteins, which makes it easier to visualise the cell state space. Cell state transitions are recorded from 10 developmental replicates in order to reduce the effect of noise on cell state space creation. The state space is the directed graph, where the nodes are the cell states and the edges are the transitions. Figure S4IJ shows two examples of these directed graphs. To simplify the graph, we prune rare transitions (those that occur less than five times across all cells per developmental replicate) and rare cell states. To identify rare cell states, we first count the number of cells in each state after each integration step of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E2" title="In 4.3 Gene regulatory network and morphogens ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">2</a> to obtain a frequency distribution of cell states. Rare cell states are those that have a frequency of less than 1% from the 10 developmental replicates. However, even without any pruning of cell state transitions and cell states, the cell state space of 22 out of 24 organisms designated as having stem-cell systems still exhibit irreversible differentiation (Fig. S8GHIJ shows one that does not follow this rule). </div> <div class="ltx_para" id="S4.SS6.p3"> We used a depth-first search algorithm to identify a graph’s strongly connected components (SCCs). Irreversible differentiation occurs when there is weak connectivity between SCCs (i.e., a path exists from SCC <math alttext="u" class="ltx_Math" display="inline" id="S4.SS6.p3.1.m1.1"><semantics id="S4.SS6.p3.1.m1.1a"><mi id="S4.SS6.p3.1.m1.1.1" xref="S4.SS6.p3.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p3.1.m1.1b"><ci id="S4.SS6.p3.1.m1.1.1.cmml" xref="S4.SS6.p3.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p3.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p3.1.m1.1d">italic_u</annotation></semantics></math> to SCC <math alttext="v" class="ltx_Math" display="inline" id="S4.SS6.p3.2.m2.1"><semantics id="S4.SS6.p3.2.m2.1a"><mi id="S4.SS6.p3.2.m2.1.1" xref="S4.SS6.p3.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p3.2.m2.1b"><ci id="S4.SS6.p3.2.m2.1.1.cmml" xref="S4.SS6.p3.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p3.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p3.2.m2.1d">italic_v</annotation></semantics></math>, but not <math alttext="v" class="ltx_Math" display="inline" id="S4.SS6.p3.3.m3.1"><semantics id="S4.SS6.p3.3.m3.1a"><mi id="S4.SS6.p3.3.m3.1.1" xref="S4.SS6.p3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p3.3.m3.1b"><ci id="S4.SS6.p3.3.m3.1.1.cmml" xref="S4.SS6.p3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p3.3.m3.1d">italic_v</annotation></semantics></math> to <math alttext="u" class="ltx_Math" display="inline" id="S4.SS6.p3.4.m4.1"><semantics id="S4.SS6.p3.4.m4.1a"><mi id="S4.SS6.p3.4.m4.1.1" xref="S4.SS6.p3.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS6.p3.4.m4.1b"><ci id="S4.SS6.p3.4.m4.1.1.cmml" xref="S4.SS6.p3.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS6.p3.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS6.p3.4.m4.1d">italic_u</annotation></semantics></math>). SCCs that do not have incoming paths from any other SCCs (source components) are designated as stem-cell types. SCCs that have no outgoing paths to any other SCCs are designated as differentiated-cell types. </div> </section> <section class="ltx_subsection" id="S4.SS7"> <h3 class="ltx_title ltx_title_subsection"> 4.7 Reproducibility score</h3> <div class="ltx_para" id="S4.SS7.p1"> We measured the morphogenetic reproducibility of organisms in a rotation-, reflection- and translation-invariant manner, as follows. We first replayed the development of an organism 60 times with different random seeds. We then performed pairwise comparisons of all developed morphologies (<math alttext="60\times 59/2" class="ltx_Math" display="inline" id="S4.SS7.p1.1.m1.1"><semantics id="S4.SS7.p1.1.m1.1a"><mrow id="S4.SS7.p1.1.m1.1.1" xref="S4.SS7.p1.1.m1.1.1.cmml"><mrow id="S4.SS7.p1.1.m1.1.1.2" xref="S4.SS7.p1.1.m1.1.1.2.cmml"><mn id="S4.SS7.p1.1.m1.1.1.2.2" xref="S4.SS7.p1.1.m1.1.1.2.2.cmml">60</mn><mo id="S4.SS7.p1.1.m1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS7.p1.1.m1.1.1.2.1.cmml">×</mo><mn id="S4.SS7.p1.1.m1.1.1.2.3" xref="S4.SS7.p1.1.m1.1.1.2.3.cmml">59</mn></mrow><mo id="S4.SS7.p1.1.m1.1.1.1" xref="S4.SS7.p1.1.m1.1.1.1.cmml">/</mo><mn id="S4.SS7.p1.1.m1.1.1.3" xref="S4.SS7.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.1.m1.1b"><apply id="S4.SS7.p1.1.m1.1.1.cmml" xref="S4.SS7.p1.1.m1.1.1"><divide id="S4.SS7.p1.1.m1.1.1.1.cmml" xref="S4.SS7.p1.1.m1.1.1.1"></divide><apply id="S4.SS7.p1.1.m1.1.1.2.cmml" xref="S4.SS7.p1.1.m1.1.1.2"><times id="S4.SS7.p1.1.m1.1.1.2.1.cmml" xref="S4.SS7.p1.1.m1.1.1.2.1"></times><cn id="S4.SS7.p1.1.m1.1.1.2.2.cmml" type="integer" xref="S4.SS7.p1.1.m1.1.1.2.2">60</cn><cn id="S4.SS7.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S4.SS7.p1.1.m1.1.1.2.3">59</cn></apply><cn id="S4.SS7.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS7.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.1.m1.1c">60\times 59/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.1.m1.1d">60 × 59 / 2</annotation></semantics></math> comparisons). For each pair of morphologies, we computed morphological similarity scores between the two CPM grids on which morphologies developed (denoted by <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p1.2.m2.1"><semantics id="S4.SS7.p1.2.m2.1a"><mi id="S4.SS7.p1.2.m2.1.1" xref="S4.SS7.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.2.m2.1b"><ci id="S4.SS7.p1.2.m2.1.1.cmml" xref="S4.SS7.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.2.m2.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p1.3.m3.1"><semantics id="S4.SS7.p1.3.m3.1a"><mi id="S4.SS7.p1.3.m3.1.1" xref="S4.SS7.p1.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.3.m3.1b"><ci id="S4.SS7.p1.3.m3.1.1.cmml" xref="S4.SS7.p1.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.3.m3.1d">italic_B</annotation></semantics></math>). We computed the morphological similarity scores (denoted Jac) over many rotations, reflections and translations of grid <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p1.5.m5.1"><semantics id="S4.SS7.p1.5.m5.1a"><mi id="S4.SS7.p1.5.m5.1.1" xref="S4.SS7.p1.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.5.m5.1b"><ci id="S4.SS7.p1.5.m5.1.1.cmml" xref="S4.SS7.p1.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.5.m5.1d">italic_B</annotation></semantics></math> relative to a fixed grid <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p1.6.m6.1"><semantics id="S4.SS7.p1.6.m6.1a"><mi id="S4.SS7.p1.6.m6.1.1" xref="S4.SS7.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.6.m6.1b"><ci id="S4.SS7.p1.6.m6.1.1.cmml" xref="S4.SS7.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.6.m6.1d">italic_A</annotation></semantics></math> to find the maximum possible similarity between them (denoted <math alttext="\text{Jac}_{\text{max}}" class="ltx_Math" display="inline" id="S4.SS7.p1.7.m7.1"><semantics id="S4.SS7.p1.7.m7.1a"><msub id="S4.SS7.p1.7.m7.1.1" xref="S4.SS7.p1.7.m7.1.1.cmml"><mtext id="S4.SS7.p1.7.m7.1.1.2" xref="S4.SS7.p1.7.m7.1.1.2a.cmml">Jac</mtext><mtext id="S4.SS7.p1.7.m7.1.1.3" xref="S4.SS7.p1.7.m7.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS7.p1.7.m7.1b"><apply id="S4.SS7.p1.7.m7.1.1.cmml" xref="S4.SS7.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS7.p1.7.m7.1.1.1.cmml" xref="S4.SS7.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS7.p1.7.m7.1.1.2a.cmml" xref="S4.SS7.p1.7.m7.1.1.2"><mtext id="S4.SS7.p1.7.m7.1.1.2.cmml" xref="S4.SS7.p1.7.m7.1.1.2">Jac</mtext></ci><ci id="S4.SS7.p1.7.m7.1.1.3a.cmml" xref="S4.SS7.p1.7.m7.1.1.3"><mtext id="S4.SS7.p1.7.m7.1.1.3.cmml" mathsize="70%" xref="S4.SS7.p1.7.m7.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p1.7.m7.1c">\text{Jac}_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p1.7.m7.1d">Jac start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>). </div> <div class="ltx_para" id="S4.SS7.p2"> To calculate Jac, grids <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p2.2.m2.1"><semantics id="S4.SS7.p2.2.m2.1a"><mi id="S4.SS7.p2.2.m2.1.1" xref="S4.SS7.p2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.2.m2.1b"><ci id="S4.SS7.p2.2.m2.1.1.cmml" xref="S4.SS7.p2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.2.m2.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.3.m3.1"><semantics id="S4.SS7.p2.3.m3.1a"><mi id="S4.SS7.p2.3.m3.1.1" xref="S4.SS7.p2.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.3.m3.1b"><ci id="S4.SS7.p2.3.m3.1.1.cmml" xref="S4.SS7.p2.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.3.m3.1d">italic_B</annotation></semantics></math> are transformed from Cartesian to polar coordinates, as follows. The polar coordinate <math alttext="(r,\theta)" class="ltx_Math" display="inline" id="S4.SS7.p2.4.m4.2"><semantics id="S4.SS7.p2.4.m4.2a"><mrow id="S4.SS7.p2.4.m4.2.3.2" xref="S4.SS7.p2.4.m4.2.3.1.cmml"><mo id="S4.SS7.p2.4.m4.2.3.2.1" stretchy="false" xref="S4.SS7.p2.4.m4.2.3.1.cmml">(</mo><mi id="S4.SS7.p2.4.m4.1.1" xref="S4.SS7.p2.4.m4.1.1.cmml">r</mi><mo id="S4.SS7.p2.4.m4.2.3.2.2" xref="S4.SS7.p2.4.m4.2.3.1.cmml">,</mo><mi id="S4.SS7.p2.4.m4.2.2" xref="S4.SS7.p2.4.m4.2.2.cmml">θ</mi><mo id="S4.SS7.p2.4.m4.2.3.2.3" stretchy="false" xref="S4.SS7.p2.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.4.m4.2b"><interval closure="open" id="S4.SS7.p2.4.m4.2.3.1.cmml" xref="S4.SS7.p2.4.m4.2.3.2"><ci id="S4.SS7.p2.4.m4.1.1.cmml" xref="S4.SS7.p2.4.m4.1.1">𝑟</ci><ci id="S4.SS7.p2.4.m4.2.2.cmml" xref="S4.SS7.p2.4.m4.2.2">𝜃</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.4.m4.2c">(r,\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.4.m4.2d">( italic_r , italic_θ )</annotation></semantics></math> is discretised into <math alttext="250\times 360" class="ltx_Math" display="inline" id="S4.SS7.p2.5.m5.1"><semantics id="S4.SS7.p2.5.m5.1a"><mrow id="S4.SS7.p2.5.m5.1.1" xref="S4.SS7.p2.5.m5.1.1.cmml"><mn id="S4.SS7.p2.5.m5.1.1.2" xref="S4.SS7.p2.5.m5.1.1.2.cmml">250</mn><mo id="S4.SS7.p2.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS7.p2.5.m5.1.1.1.cmml">×</mo><mn id="S4.SS7.p2.5.m5.1.1.3" xref="S4.SS7.p2.5.m5.1.1.3.cmml">360</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.5.m5.1b"><apply id="S4.SS7.p2.5.m5.1.1.cmml" xref="S4.SS7.p2.5.m5.1.1"><times id="S4.SS7.p2.5.m5.1.1.1.cmml" xref="S4.SS7.p2.5.m5.1.1.1"></times><cn id="S4.SS7.p2.5.m5.1.1.2.cmml" type="integer" xref="S4.SS7.p2.5.m5.1.1.2">250</cn><cn id="S4.SS7.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.SS7.p2.5.m5.1.1.3">360</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.5.m5.1c">250\times 360</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.5.m5.1d">250 × 360</annotation></semantics></math> pixels. Each pixel in <math alttext="(r,\theta)" class="ltx_Math" display="inline" id="S4.SS7.p2.6.m6.2"><semantics id="S4.SS7.p2.6.m6.2a"><mrow id="S4.SS7.p2.6.m6.2.3.2" xref="S4.SS7.p2.6.m6.2.3.1.cmml"><mo id="S4.SS7.p2.6.m6.2.3.2.1" stretchy="false" xref="S4.SS7.p2.6.m6.2.3.1.cmml">(</mo><mi id="S4.SS7.p2.6.m6.1.1" xref="S4.SS7.p2.6.m6.1.1.cmml">r</mi><mo id="S4.SS7.p2.6.m6.2.3.2.2" xref="S4.SS7.p2.6.m6.2.3.1.cmml">,</mo><mi id="S4.SS7.p2.6.m6.2.2" xref="S4.SS7.p2.6.m6.2.2.cmml">θ</mi><mo id="S4.SS7.p2.6.m6.2.3.2.3" stretchy="false" xref="S4.SS7.p2.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.6.m6.2b"><interval closure="open" id="S4.SS7.p2.6.m6.2.3.1.cmml" xref="S4.SS7.p2.6.m6.2.3.2"><ci id="S4.SS7.p2.6.m6.1.1.cmml" xref="S4.SS7.p2.6.m6.1.1">𝑟</ci><ci id="S4.SS7.p2.6.m6.2.2.cmml" xref="S4.SS7.p2.6.m6.2.2">𝜃</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.6.m6.2c">(r,\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.6.m6.2d">( italic_r , italic_θ )</annotation></semantics></math> is mapped to the pixel closest to <math alttext="(r\cos\theta,r\sin\theta)" class="ltx_Math" display="inline" id="S4.SS7.p2.7.m7.2"><semantics id="S4.SS7.p2.7.m7.2a"><mrow id="S4.SS7.p2.7.m7.2.2.2" xref="S4.SS7.p2.7.m7.2.2.3.cmml"><mo id="S4.SS7.p2.7.m7.2.2.2.3" stretchy="false" xref="S4.SS7.p2.7.m7.2.2.3.cmml">(</mo><mrow id="S4.SS7.p2.7.m7.1.1.1.1" xref="S4.SS7.p2.7.m7.1.1.1.1.cmml"><mi id="S4.SS7.p2.7.m7.1.1.1.1.2" xref="S4.SS7.p2.7.m7.1.1.1.1.2.cmml">r</mi><mo id="S4.SS7.p2.7.m7.1.1.1.1.1" lspace="0.167em" xref="S4.SS7.p2.7.m7.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS7.p2.7.m7.1.1.1.1.3" xref="S4.SS7.p2.7.m7.1.1.1.1.3.cmml"><mi id="S4.SS7.p2.7.m7.1.1.1.1.3.1" xref="S4.SS7.p2.7.m7.1.1.1.1.3.1.cmml">cos</mi><mo id="S4.SS7.p2.7.m7.1.1.1.1.3a" lspace="0.167em" xref="S4.SS7.p2.7.m7.1.1.1.1.3.cmml">⁡</mo><mi id="S4.SS7.p2.7.m7.1.1.1.1.3.2" xref="S4.SS7.p2.7.m7.1.1.1.1.3.2.cmml">θ</mi></mrow></mrow><mo id="S4.SS7.p2.7.m7.2.2.2.4" xref="S4.SS7.p2.7.m7.2.2.3.cmml">,</mo><mrow id="S4.SS7.p2.7.m7.2.2.2.2" xref="S4.SS7.p2.7.m7.2.2.2.2.cmml"><mi id="S4.SS7.p2.7.m7.2.2.2.2.2" xref="S4.SS7.p2.7.m7.2.2.2.2.2.cmml">r</mi><mo id="S4.SS7.p2.7.m7.2.2.2.2.1" lspace="0.167em" xref="S4.SS7.p2.7.m7.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS7.p2.7.m7.2.2.2.2.3" xref="S4.SS7.p2.7.m7.2.2.2.2.3.cmml"><mi id="S4.SS7.p2.7.m7.2.2.2.2.3.1" xref="S4.SS7.p2.7.m7.2.2.2.2.3.1.cmml">sin</mi><mo id="S4.SS7.p2.7.m7.2.2.2.2.3a" lspace="0.167em" xref="S4.SS7.p2.7.m7.2.2.2.2.3.cmml">⁡</mo><mi id="S4.SS7.p2.7.m7.2.2.2.2.3.2" xref="S4.SS7.p2.7.m7.2.2.2.2.3.2.cmml">θ</mi></mrow></mrow><mo id="S4.SS7.p2.7.m7.2.2.2.5" stretchy="false" xref="S4.SS7.p2.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.7.m7.2b"><interval closure="open" id="S4.SS7.p2.7.m7.2.2.3.cmml" xref="S4.SS7.p2.7.m7.2.2.2"><apply id="S4.SS7.p2.7.m7.1.1.1.1.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1"><times id="S4.SS7.p2.7.m7.1.1.1.1.1.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1.1"></times><ci id="S4.SS7.p2.7.m7.1.1.1.1.2.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1.2">𝑟</ci><apply id="S4.SS7.p2.7.m7.1.1.1.1.3.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1.3"><cos id="S4.SS7.p2.7.m7.1.1.1.1.3.1.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1.3.1"></cos><ci id="S4.SS7.p2.7.m7.1.1.1.1.3.2.cmml" xref="S4.SS7.p2.7.m7.1.1.1.1.3.2">𝜃</ci></apply></apply><apply id="S4.SS7.p2.7.m7.2.2.2.2.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2"><times id="S4.SS7.p2.7.m7.2.2.2.2.1.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2.1"></times><ci id="S4.SS7.p2.7.m7.2.2.2.2.2.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2.2">𝑟</ci><apply id="S4.SS7.p2.7.m7.2.2.2.2.3.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2.3"><sin id="S4.SS7.p2.7.m7.2.2.2.2.3.1.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2.3.1"></sin><ci id="S4.SS7.p2.7.m7.2.2.2.2.3.2.cmml" xref="S4.SS7.p2.7.m7.2.2.2.2.3.2">𝜃</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.7.m7.2c">(r\cos\theta,r\sin\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.7.m7.2d">( italic_r roman_cos italic_θ , italic_r roman_sin italic_θ )</annotation></semantics></math> in <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p2.8.m8.1"><semantics id="S4.SS7.p2.8.m8.1a"><mi id="S4.SS7.p2.8.m8.1.1" xref="S4.SS7.p2.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.8.m8.1b"><ci id="S4.SS7.p2.8.m8.1.1.cmml" xref="S4.SS7.p2.8.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.8.m8.1d">italic_A</annotation></semantics></math> or <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.9.m9.1"><semantics id="S4.SS7.p2.9.m9.1a"><mi id="S4.SS7.p2.9.m9.1.1" xref="S4.SS7.p2.9.m9.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.9.m9.1b"><ci id="S4.SS7.p2.9.m9.1.1.cmml" xref="S4.SS7.p2.9.m9.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.9.m9.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.9.m9.1d">italic_B</annotation></semantics></math>, where <math alttext="r" class="ltx_Math" display="inline" id="S4.SS7.p2.10.m10.1"><semantics id="S4.SS7.p2.10.m10.1a"><mi id="S4.SS7.p2.10.m10.1.1" xref="S4.SS7.p2.10.m10.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.10.m10.1b"><ci id="S4.SS7.p2.10.m10.1.1.cmml" xref="S4.SS7.p2.10.m10.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.10.m10.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.10.m10.1d">italic_r</annotation></semantics></math> is the distance from the midpoint of grid <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p2.11.m11.1"><semantics id="S4.SS7.p2.11.m11.1a"><mi id="S4.SS7.p2.11.m11.1.1" xref="S4.SS7.p2.11.m11.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.11.m11.1b"><ci id="S4.SS7.p2.11.m11.1.1.cmml" xref="S4.SS7.p2.11.m11.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.11.m11.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.11.m11.1d">italic_A</annotation></semantics></math> or one of 25 equally-spaced locations in grid <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.12.m12.1"><semantics id="S4.SS7.p2.12.m12.1a"><mi id="S4.SS7.p2.12.m12.1.1" xref="S4.SS7.p2.12.m12.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.12.m12.1b"><ci id="S4.SS7.p2.12.m12.1.1.cmml" xref="S4.SS7.p2.12.m12.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.12.m12.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.12.m12.1d">italic_B</annotation></semantics></math> (thus, multiple pixels in the polar coordinate can be mapped to the same pixel in the Cartesian coordinate). Let <math alttext="A_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.13.m13.1"><semantics id="S4.SS7.p2.13.m13.1a"><msubsup id="S4.SS7.p2.13.m13.1.1" xref="S4.SS7.p2.13.m13.1.1.cmml"><mi id="S4.SS7.p2.13.m13.1.1.2.2" xref="S4.SS7.p2.13.m13.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.13.m13.1.1.2.3" xref="S4.SS7.p2.13.m13.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.13.m13.1.1.3" xref="S4.SS7.p2.13.m13.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.13.m13.1b"><apply id="S4.SS7.p2.13.m13.1.1.cmml" xref="S4.SS7.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.13.m13.1.1.1.cmml" xref="S4.SS7.p2.13.m13.1.1">superscript</csymbol><apply id="S4.SS7.p2.13.m13.1.1.2.cmml" xref="S4.SS7.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.13.m13.1.1.2.1.cmml" xref="S4.SS7.p2.13.m13.1.1">subscript</csymbol><ci id="S4.SS7.p2.13.m13.1.1.2.2.cmml" xref="S4.SS7.p2.13.m13.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.13.m13.1.1.2.3.cmml" xref="S4.SS7.p2.13.m13.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.13.m13.1.1.3.cmml" xref="S4.SS7.p2.13.m13.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.13.m13.1c">A_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.13.m13.1d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="B_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.14.m14.1"><semantics id="S4.SS7.p2.14.m14.1a"><msubsup id="S4.SS7.p2.14.m14.1.1" xref="S4.SS7.p2.14.m14.1.1.cmml"><mi id="S4.SS7.p2.14.m14.1.1.2.2" xref="S4.SS7.p2.14.m14.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.14.m14.1.1.2.3" xref="S4.SS7.p2.14.m14.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.14.m14.1.1.3" xref="S4.SS7.p2.14.m14.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.14.m14.1b"><apply id="S4.SS7.p2.14.m14.1.1.cmml" xref="S4.SS7.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.14.m14.1.1.1.cmml" xref="S4.SS7.p2.14.m14.1.1">superscript</csymbol><apply id="S4.SS7.p2.14.m14.1.1.2.cmml" xref="S4.SS7.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.14.m14.1.1.2.1.cmml" xref="S4.SS7.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS7.p2.14.m14.1.1.2.2.cmml" xref="S4.SS7.p2.14.m14.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.14.m14.1.1.2.3.cmml" xref="S4.SS7.p2.14.m14.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.14.m14.1.1.3.cmml" xref="S4.SS7.p2.14.m14.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.14.m14.1c">B_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.14.m14.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> be one if the mapped pixel in <math alttext="A" class="ltx_Math" display="inline" id="S4.SS7.p2.15.m15.1"><semantics id="S4.SS7.p2.15.m15.1a"><mi id="S4.SS7.p2.15.m15.1.1" xref="S4.SS7.p2.15.m15.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.15.m15.1b"><ci id="S4.SS7.p2.15.m15.1.1.cmml" xref="S4.SS7.p2.15.m15.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.15.m15.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.15.m15.1d">italic_A</annotation></semantics></math> or <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.16.m16.1"><semantics id="S4.SS7.p2.16.m16.1a"><mi id="S4.SS7.p2.16.m16.1.1" xref="S4.SS7.p2.16.m16.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.16.m16.1b"><ci id="S4.SS7.p2.16.m16.1.1.cmml" xref="S4.SS7.p2.16.m16.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.16.m16.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.16.m16.1d">italic_B</annotation></semantics></math> belongs to a biological cell and 0 otherwise. We then calculated the Jaccard index, <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})" class="ltx_Math" display="inline" id="S4.SS7.p2.17.m17.2"><semantics id="S4.SS7.p2.17.m17.2a"><mrow id="S4.SS7.p2.17.m17.2.2" xref="S4.SS7.p2.17.m17.2.2.cmml"><mtext id="S4.SS7.p2.17.m17.2.2.4" xref="S4.SS7.p2.17.m17.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.17.m17.2.2.3" xref="S4.SS7.p2.17.m17.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.17.m17.2.2.2.2" xref="S4.SS7.p2.17.m17.2.2.2.3.cmml"><mo id="S4.SS7.p2.17.m17.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.17.m17.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.17.m17.1.1.1.1.1" xref="S4.SS7.p2.17.m17.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.17.m17.1.1.1.1.1.2.2" xref="S4.SS7.p2.17.m17.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.17.m17.1.1.1.1.1.2.3" xref="S4.SS7.p2.17.m17.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.17.m17.1.1.1.1.1.3" xref="S4.SS7.p2.17.m17.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.17.m17.2.2.2.2.4" xref="S4.SS7.p2.17.m17.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.17.m17.2.2.2.2.2" xref="S4.SS7.p2.17.m17.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.17.m17.2.2.2.2.2.2.2" xref="S4.SS7.p2.17.m17.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.17.m17.2.2.2.2.2.2.3" xref="S4.SS7.p2.17.m17.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.17.m17.2.2.2.2.2.3" xref="S4.SS7.p2.17.m17.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.17.m17.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.17.m17.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.17.m17.2b"><apply id="S4.SS7.p2.17.m17.2.2.cmml" xref="S4.SS7.p2.17.m17.2.2"><times id="S4.SS7.p2.17.m17.2.2.3.cmml" xref="S4.SS7.p2.17.m17.2.2.3"></times><ci id="S4.SS7.p2.17.m17.2.2.4a.cmml" xref="S4.SS7.p2.17.m17.2.2.4"><mtext id="S4.SS7.p2.17.m17.2.2.4.cmml" xref="S4.SS7.p2.17.m17.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.17.m17.2.2.2.3.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2"><apply id="S4.SS7.p2.17.m17.1.1.1.1.1.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.17.m17.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.17.m17.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.17.m17.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.17.m17.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.17.m17.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.17.m17.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.17.m17.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.17.m17.2.2.2.2.2.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.17.m17.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.17.m17.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.17.m17.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.17.m17.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.17.m17.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.17.m17.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.17.m17.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.17.m17.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.17.m17.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math>, which measures the similarity between <math alttext="A_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.18.m18.1"><semantics id="S4.SS7.p2.18.m18.1a"><msubsup id="S4.SS7.p2.18.m18.1.1" xref="S4.SS7.p2.18.m18.1.1.cmml"><mi id="S4.SS7.p2.18.m18.1.1.2.2" xref="S4.SS7.p2.18.m18.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.18.m18.1.1.2.3" xref="S4.SS7.p2.18.m18.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.18.m18.1.1.3" xref="S4.SS7.p2.18.m18.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.18.m18.1b"><apply id="S4.SS7.p2.18.m18.1.1.cmml" xref="S4.SS7.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.18.m18.1.1.1.cmml" xref="S4.SS7.p2.18.m18.1.1">superscript</csymbol><apply id="S4.SS7.p2.18.m18.1.1.2.cmml" xref="S4.SS7.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.18.m18.1.1.2.1.cmml" xref="S4.SS7.p2.18.m18.1.1">subscript</csymbol><ci id="S4.SS7.p2.18.m18.1.1.2.2.cmml" xref="S4.SS7.p2.18.m18.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.18.m18.1.1.2.3.cmml" xref="S4.SS7.p2.18.m18.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.18.m18.1.1.3.cmml" xref="S4.SS7.p2.18.m18.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.18.m18.1c">A_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.18.m18.1d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="B_{\theta}^{r})" class="ltx_math_unparsed" display="inline" id="S4.SS7.p2.19.m19.1"><semantics id="S4.SS7.p2.19.m19.1a"><mrow id="S4.SS7.p2.19.m19.1b"><msubsup id="S4.SS7.p2.19.m19.1.1"><mi id="S4.SS7.p2.19.m19.1.1.2.2">B</mi><mi id="S4.SS7.p2.19.m19.1.1.2.3">θ</mi><mi id="S4.SS7.p2.19.m19.1.1.3">r</mi></msubsup><mo id="S4.SS7.p2.19.m19.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.SS7.p2.19.m19.1c">B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.19.m19.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math>, as follows: <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})=\frac{A_{\theta}^{r}\cap B_{\theta}^% {r}}{A_{\theta}^{r}\cup B_{\theta}^{r}}" class="ltx_Math" display="block" id="S4.E8.m1.2"><semantics id="S4.E8.m1.2a"><mrow id="S4.E8.m1.2.2" xref="S4.E8.m1.2.2.cmml"><mrow id="S4.E8.m1.2.2.2" xref="S4.E8.m1.2.2.2.cmml"><mtext id="S4.E8.m1.2.2.2.4" xref="S4.E8.m1.2.2.2.4a.cmml">Jac</mtext><mo id="S4.E8.m1.2.2.2.3" xref="S4.E8.m1.2.2.2.3.cmml">⁢</mo><mrow id="S4.E8.m1.2.2.2.2.2" xref="S4.E8.m1.2.2.2.2.3.cmml"><mo id="S4.E8.m1.2.2.2.2.2.3" stretchy="false" xref="S4.E8.m1.2.2.2.2.3.cmml">(</mo><msubsup id="S4.E8.m1.1.1.1.1.1.1" xref="S4.E8.m1.1.1.1.1.1.1.cmml"><mi id="S4.E8.m1.1.1.1.1.1.1.2.2" xref="S4.E8.m1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.E8.m1.1.1.1.1.1.1.2.3" xref="S4.E8.m1.1.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.E8.m1.1.1.1.1.1.1.3" xref="S4.E8.m1.1.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.E8.m1.2.2.2.2.2.4" xref="S4.E8.m1.2.2.2.2.3.cmml">,</mo><msubsup id="S4.E8.m1.2.2.2.2.2.2" xref="S4.E8.m1.2.2.2.2.2.2.cmml"><mi id="S4.E8.m1.2.2.2.2.2.2.2.2" xref="S4.E8.m1.2.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.E8.m1.2.2.2.2.2.2.2.3" xref="S4.E8.m1.2.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.E8.m1.2.2.2.2.2.2.3" xref="S4.E8.m1.2.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.E8.m1.2.2.2.2.2.5" stretchy="false" xref="S4.E8.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.2.2.3" xref="S4.E8.m1.2.2.3.cmml">=</mo><mfrac id="S4.E8.m1.2.2.4" xref="S4.E8.m1.2.2.4.cmml"><mrow id="S4.E8.m1.2.2.4.2" xref="S4.E8.m1.2.2.4.2.cmml"><msubsup id="S4.E8.m1.2.2.4.2.2" xref="S4.E8.m1.2.2.4.2.2.cmml"><mi id="S4.E8.m1.2.2.4.2.2.2.2" xref="S4.E8.m1.2.2.4.2.2.2.2.cmml">A</mi><mi id="S4.E8.m1.2.2.4.2.2.2.3" xref="S4.E8.m1.2.2.4.2.2.2.3.cmml">θ</mi><mi id="S4.E8.m1.2.2.4.2.2.3" xref="S4.E8.m1.2.2.4.2.2.3.cmml">r</mi></msubsup><mo id="S4.E8.m1.2.2.4.2.1" xref="S4.E8.m1.2.2.4.2.1.cmml">∩</mo><msubsup id="S4.E8.m1.2.2.4.2.3" xref="S4.E8.m1.2.2.4.2.3.cmml"><mi id="S4.E8.m1.2.2.4.2.3.2.2" xref="S4.E8.m1.2.2.4.2.3.2.2.cmml">B</mi><mi id="S4.E8.m1.2.2.4.2.3.2.3" xref="S4.E8.m1.2.2.4.2.3.2.3.cmml">θ</mi><mi id="S4.E8.m1.2.2.4.2.3.3" xref="S4.E8.m1.2.2.4.2.3.3.cmml">r</mi></msubsup></mrow><mrow id="S4.E8.m1.2.2.4.3" xref="S4.E8.m1.2.2.4.3.cmml"><msubsup id="S4.E8.m1.2.2.4.3.2" xref="S4.E8.m1.2.2.4.3.2.cmml"><mi id="S4.E8.m1.2.2.4.3.2.2.2" xref="S4.E8.m1.2.2.4.3.2.2.2.cmml">A</mi><mi id="S4.E8.m1.2.2.4.3.2.2.3" xref="S4.E8.m1.2.2.4.3.2.2.3.cmml">θ</mi><mi id="S4.E8.m1.2.2.4.3.2.3" xref="S4.E8.m1.2.2.4.3.2.3.cmml">r</mi></msubsup><mo id="S4.E8.m1.2.2.4.3.1" xref="S4.E8.m1.2.2.4.3.1.cmml">∪</mo><msubsup id="S4.E8.m1.2.2.4.3.3" xref="S4.E8.m1.2.2.4.3.3.cmml"><mi id="S4.E8.m1.2.2.4.3.3.2.2" xref="S4.E8.m1.2.2.4.3.3.2.2.cmml">B</mi><mi id="S4.E8.m1.2.2.4.3.3.2.3" xref="S4.E8.m1.2.2.4.3.3.2.3.cmml">θ</mi><mi id="S4.E8.m1.2.2.4.3.3.3" xref="S4.E8.m1.2.2.4.3.3.3.cmml">r</mi></msubsup></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.E8.m1.2b"><apply id="S4.E8.m1.2.2.cmml" xref="S4.E8.m1.2.2"><eq id="S4.E8.m1.2.2.3.cmml" xref="S4.E8.m1.2.2.3"></eq><apply id="S4.E8.m1.2.2.2.cmml" xref="S4.E8.m1.2.2.2"><times id="S4.E8.m1.2.2.2.3.cmml" xref="S4.E8.m1.2.2.2.3"></times><ci id="S4.E8.m1.2.2.2.4a.cmml" xref="S4.E8.m1.2.2.2.4"><mtext id="S4.E8.m1.2.2.2.4.cmml" xref="S4.E8.m1.2.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.E8.m1.2.2.2.2.3.cmml" xref="S4.E8.m1.2.2.2.2.2"><apply id="S4.E8.m1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1">superscript</csymbol><apply id="S4.E8.m1.1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.1.1.2.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E8.m1.1.1.1.1.1.1.2.2.cmml" xref="S4.E8.m1.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.E8.m1.1.1.1.1.1.1.2.3.cmml" xref="S4.E8.m1.1.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.E8.m1.1.1.1.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.E8.m1.2.2.2.2.2.2.cmml" xref="S4.E8.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E8.m1.2.2.2.2.2.2.1.cmml" xref="S4.E8.m1.2.2.2.2.2.2">superscript</csymbol><apply id="S4.E8.m1.2.2.2.2.2.2.2.cmml" xref="S4.E8.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E8.m1.2.2.2.2.2.2.2.1.cmml" xref="S4.E8.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S4.E8.m1.2.2.2.2.2.2.2.2.cmml" xref="S4.E8.m1.2.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.E8.m1.2.2.2.2.2.2.2.3.cmml" xref="S4.E8.m1.2.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.E8.m1.2.2.2.2.2.2.3.cmml" xref="S4.E8.m1.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S4.E8.m1.2.2.4.cmml" 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id="S4.E8.m1.2.2.4.3.2.3.cmml" xref="S4.E8.m1.2.2.4.3.2.3">𝑟</ci></apply><apply id="S4.E8.m1.2.2.4.3.3.cmml" xref="S4.E8.m1.2.2.4.3.3"><csymbol cd="ambiguous" id="S4.E8.m1.2.2.4.3.3.1.cmml" xref="S4.E8.m1.2.2.4.3.3">superscript</csymbol><apply id="S4.E8.m1.2.2.4.3.3.2.cmml" xref="S4.E8.m1.2.2.4.3.3"><csymbol cd="ambiguous" id="S4.E8.m1.2.2.4.3.3.2.1.cmml" xref="S4.E8.m1.2.2.4.3.3">subscript</csymbol><ci id="S4.E8.m1.2.2.4.3.3.2.2.cmml" xref="S4.E8.m1.2.2.4.3.3.2.2">𝐵</ci><ci id="S4.E8.m1.2.2.4.3.3.2.3.cmml" xref="S4.E8.m1.2.2.4.3.3.2.3">𝜃</ci></apply><ci id="S4.E8.m1.2.2.4.3.3.3.cmml" xref="S4.E8.m1.2.2.4.3.3.3">𝑟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})=\frac{A_{\theta}^{r}\cap B_{\theta}^% {r}}{A_{\theta}^{r}\cup B_{\theta}^{r}}</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) = divide start_ARG italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_ARG start_ARG italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ∪ italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> </table> where <math alttext="A_{\theta}^{r}\cap B_{\theta}^{r}=\sum^{359^{\circ}}_{\theta=0^{\circ}}\sum_{r% 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xref="S4.SS7.p2.20.m1.4.4.2.2.2.6">𝛿</ci><interval closure="open" id="S4.SS7.p2.20.m1.4.4.2.2.2.2.2.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1"><apply id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.1.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1">superscript</csymbol><apply id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.1.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1">subscript</csymbol><ci id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.2.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.3.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.3.cmml" xref="S4.SS7.p2.20.m1.4.4.2.2.2.2.1.1.3">𝑟</ci></apply><cn id="S4.SS7.p2.20.m1.2.2.cmml" type="integer" xref="S4.SS7.p2.20.m1.2.2">1</cn></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.20.m1.4c">A_{\theta}^{r}\cap B_{\theta}^{r}=\sum^{359^{\circ}}_{\theta=0^{\circ}}\sum_{r% =0}^{249}r\delta(A_{\theta}^{r},1)\delta(B_{\theta}^{r},1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.20.m1.4d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT = ∑ start_POSTSUPERSCRIPT 359 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_θ = 0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_r = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 249 end_POSTSUPERSCRIPT italic_r italic_δ ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , 1 ) italic_δ ( italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , 1 )</annotation></semantics></math> is the number of pixels in the polar coordinate with a value of one (i.e., the pixel is occupied by a cell) on both <math alttext="A_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.21.m2.1"><semantics id="S4.SS7.p2.21.m2.1a"><msubsup id="S4.SS7.p2.21.m2.1.1" xref="S4.SS7.p2.21.m2.1.1.cmml"><mi id="S4.SS7.p2.21.m2.1.1.2.2" xref="S4.SS7.p2.21.m2.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.21.m2.1.1.2.3" xref="S4.SS7.p2.21.m2.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.21.m2.1.1.3" xref="S4.SS7.p2.21.m2.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.21.m2.1b"><apply id="S4.SS7.p2.21.m2.1.1.cmml" xref="S4.SS7.p2.21.m2.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.21.m2.1.1.1.cmml" xref="S4.SS7.p2.21.m2.1.1">superscript</csymbol><apply id="S4.SS7.p2.21.m2.1.1.2.cmml" xref="S4.SS7.p2.21.m2.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.21.m2.1.1.2.1.cmml" xref="S4.SS7.p2.21.m2.1.1">subscript</csymbol><ci id="S4.SS7.p2.21.m2.1.1.2.2.cmml" xref="S4.SS7.p2.21.m2.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.21.m2.1.1.2.3.cmml" xref="S4.SS7.p2.21.m2.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.21.m2.1.1.3.cmml" xref="S4.SS7.p2.21.m2.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.21.m2.1c">A_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.21.m2.1d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="B_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.22.m3.1"><semantics id="S4.SS7.p2.22.m3.1a"><msubsup id="S4.SS7.p2.22.m3.1.1" xref="S4.SS7.p2.22.m3.1.1.cmml"><mi id="S4.SS7.p2.22.m3.1.1.2.2" xref="S4.SS7.p2.22.m3.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.22.m3.1.1.2.3" xref="S4.SS7.p2.22.m3.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.22.m3.1.1.3" xref="S4.SS7.p2.22.m3.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.22.m3.1b"><apply id="S4.SS7.p2.22.m3.1.1.cmml" xref="S4.SS7.p2.22.m3.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.22.m3.1.1.1.cmml" xref="S4.SS7.p2.22.m3.1.1">superscript</csymbol><apply id="S4.SS7.p2.22.m3.1.1.2.cmml" xref="S4.SS7.p2.22.m3.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.22.m3.1.1.2.1.cmml" xref="S4.SS7.p2.22.m3.1.1">subscript</csymbol><ci id="S4.SS7.p2.22.m3.1.1.2.2.cmml" xref="S4.SS7.p2.22.m3.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.22.m3.1.1.2.3.cmml" xref="S4.SS7.p2.22.m3.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.22.m3.1.1.3.cmml" xref="S4.SS7.p2.22.m3.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.22.m3.1c">B_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.22.m3.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\delta" class="ltx_Math" display="inline" id="S4.SS7.p2.23.m4.1"><semantics id="S4.SS7.p2.23.m4.1a"><mi id="S4.SS7.p2.23.m4.1.1" xref="S4.SS7.p2.23.m4.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.23.m4.1b"><ci id="S4.SS7.p2.23.m4.1.1.cmml" xref="S4.SS7.p2.23.m4.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.23.m4.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.23.m4.1d">italic_δ</annotation></semantics></math> is the Kronecker delta. The multiplication by <math alttext="r" class="ltx_Math" display="inline" id="S4.SS7.p2.24.m5.1"><semantics id="S4.SS7.p2.24.m5.1a"><mi id="S4.SS7.p2.24.m5.1.1" xref="S4.SS7.p2.24.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.24.m5.1b"><ci id="S4.SS7.p2.24.m5.1.1.cmml" xref="S4.SS7.p2.24.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.24.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.24.m5.1d">italic_r</annotation></semantics></math> accounts for the fact that the length of an arc drawn by an increment in <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS7.p2.25.m6.1"><semantics id="S4.SS7.p2.25.m6.1a"><mi id="S4.SS7.p2.25.m6.1.1" xref="S4.SS7.p2.25.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.25.m6.1b"><ci id="S4.SS7.p2.25.m6.1.1.cmml" xref="S4.SS7.p2.25.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.25.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.25.m6.1d">italic_θ</annotation></semantics></math> increases as <math alttext="r" class="ltx_Math" display="inline" id="S4.SS7.p2.26.m7.1"><semantics id="S4.SS7.p2.26.m7.1a"><mi id="S4.SS7.p2.26.m7.1.1" xref="S4.SS7.p2.26.m7.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.26.m7.1b"><ci id="S4.SS7.p2.26.m7.1.1.cmml" xref="S4.SS7.p2.26.m7.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.26.m7.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.26.m7.1d">italic_r</annotation></semantics></math> increases. Similarly, <math alttext="A_{\theta}^{r}\cup B_{\theta}^{r}=\sum^{359^{\circ}}_{\theta=0^{\circ}}\sum_{r% =0}^{249}r[\delta(A_{\theta}^{r},1)+\delta(B_{\theta}^{r},1)-\delta(A_{\theta}% ^{r},B_{\theta}^{r})]" class="ltx_Math" display="inline" id="S4.SS7.p2.27.m8.3"><semantics id="S4.SS7.p2.27.m8.3a"><mrow id="S4.SS7.p2.27.m8.3.3" xref="S4.SS7.p2.27.m8.3.3.cmml"><mrow id="S4.SS7.p2.27.m8.3.3.3" xref="S4.SS7.p2.27.m8.3.3.3.cmml"><msubsup id="S4.SS7.p2.27.m8.3.3.3.2" xref="S4.SS7.p2.27.m8.3.3.3.2.cmml"><mi id="S4.SS7.p2.27.m8.3.3.3.2.2.2" xref="S4.SS7.p2.27.m8.3.3.3.2.2.2.cmml">A</mi><mi id="S4.SS7.p2.27.m8.3.3.3.2.2.3" xref="S4.SS7.p2.27.m8.3.3.3.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.27.m8.3.3.3.2.3" xref="S4.SS7.p2.27.m8.3.3.3.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.27.m8.3.3.3.1" xref="S4.SS7.p2.27.m8.3.3.3.1.cmml">∪</mo><msubsup id="S4.SS7.p2.27.m8.3.3.3.3" xref="S4.SS7.p2.27.m8.3.3.3.3.cmml"><mi id="S4.SS7.p2.27.m8.3.3.3.3.2.2" xref="S4.SS7.p2.27.m8.3.3.3.3.2.2.cmml">B</mi><mi id="S4.SS7.p2.27.m8.3.3.3.3.2.3" xref="S4.SS7.p2.27.m8.3.3.3.3.2.3.cmml">θ</mi><mi id="S4.SS7.p2.27.m8.3.3.3.3.3" xref="S4.SS7.p2.27.m8.3.3.3.3.3.cmml">r</mi></msubsup></mrow><mo id="S4.SS7.p2.27.m8.3.3.2" rspace="0.111em" xref="S4.SS7.p2.27.m8.3.3.2.cmml">=</mo><mrow id="S4.SS7.p2.27.m8.3.3.1" xref="S4.SS7.p2.27.m8.3.3.1.cmml"><msubsup id="S4.SS7.p2.27.m8.3.3.1.2" xref="S4.SS7.p2.27.m8.3.3.1.2.cmml"><mo id="S4.SS7.p2.27.m8.3.3.1.2.2.2" rspace="0em" xref="S4.SS7.p2.27.m8.3.3.1.2.2.2.cmml">∑</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.2.3" xref="S4.SS7.p2.27.m8.3.3.1.2.3.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.2.3.2" xref="S4.SS7.p2.27.m8.3.3.1.2.3.2.cmml">θ</mi><mo id="S4.SS7.p2.27.m8.3.3.1.2.3.1" xref="S4.SS7.p2.27.m8.3.3.1.2.3.1.cmml">=</mo><msup id="S4.SS7.p2.27.m8.3.3.1.2.3.3" xref="S4.SS7.p2.27.m8.3.3.1.2.3.3.cmml"><mn id="S4.SS7.p2.27.m8.3.3.1.2.3.3.2" xref="S4.SS7.p2.27.m8.3.3.1.2.3.3.2.cmml">0</mn><mo id="S4.SS7.p2.27.m8.3.3.1.2.3.3.3" xref="S4.SS7.p2.27.m8.3.3.1.2.3.3.3.cmml">∘</mo></msup></mrow><msup id="S4.SS7.p2.27.m8.3.3.1.2.2.3" xref="S4.SS7.p2.27.m8.3.3.1.2.2.3.cmml"><mn id="S4.SS7.p2.27.m8.3.3.1.2.2.3.2" xref="S4.SS7.p2.27.m8.3.3.1.2.2.3.2.cmml">359</mn><mo id="S4.SS7.p2.27.m8.3.3.1.2.2.3.3" xref="S4.SS7.p2.27.m8.3.3.1.2.2.3.3.cmml">∘</mo></msup></msubsup><mrow id="S4.SS7.p2.27.m8.3.3.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.cmml"><msubsup id="S4.SS7.p2.27.m8.3.3.1.1.2" xref="S4.SS7.p2.27.m8.3.3.1.1.2.cmml"><mo id="S4.SS7.p2.27.m8.3.3.1.1.2.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.2.2.2.cmml">∑</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.2.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.2" xref="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.2.cmml">r</mi><mo id="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.1" xref="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.1.cmml">=</mo><mn id="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.3" xref="S4.SS7.p2.27.m8.3.3.1.1.2.2.3.3.cmml">0</mn></mrow><mn id="S4.SS7.p2.27.m8.3.3.1.1.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.2.3.cmml">249</mn></msubsup><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.3.cmml">r</mi><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.2.cmml"><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.cmml"><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.cmml"><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.3.cmml">δ</mi><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.2.cmml">(</mo><msubsup id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><mn id="S4.SS7.p2.27.m8.1.1" xref="S4.SS7.p2.27.m8.1.1.cmml">1</mn><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.1.4" stretchy="false" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.3.cmml">+</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.3.cmml">δ</mi><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.2.cmml">⁢</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.2.cmml"><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.2" stretchy="false" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.2.cmml">(</mo><msubsup id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.2.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.2.cmml">,</mo><mn id="S4.SS7.p2.27.m8.2.2" xref="S4.SS7.p2.27.m8.2.2.cmml">1</mn><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.1.4" stretchy="false" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.5" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.5.cmml">−</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.cmml"><mi id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.4" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.4.cmml">δ</mi><mo id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.3" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.3.cmml">⁢</mo><mrow id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.3.cmml"><mo 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id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.1.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.1.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.2.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.3.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.3.cmml" xref="S4.SS7.p2.27.m8.3.3.1.1.1.1.1.1.4.2.2.2.3">𝑟</ci></apply></interval></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.27.m8.3c">A_{\theta}^{r}\cup B_{\theta}^{r}=\sum^{359^{\circ}}_{\theta=0^{\circ}}\sum_{r% =0}^{249}r[\delta(A_{\theta}^{r},1)+\delta(B_{\theta}^{r},1)-\delta(A_{\theta}% ^{r},B_{\theta}^{r})]</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.27.m8.3d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ∪ italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT = ∑ start_POSTSUPERSCRIPT 359 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_θ = 0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_r = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 249 end_POSTSUPERSCRIPT italic_r [ italic_δ ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , 1 ) + italic_δ ( italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , 1 ) - italic_δ ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) ]</annotation></semantics></math> is the number of pixels in the polar coordinate in which <math alttext="A_{\theta}^{r}=1" class="ltx_Math" display="inline" id="S4.SS7.p2.28.m9.1"><semantics id="S4.SS7.p2.28.m9.1a"><mrow id="S4.SS7.p2.28.m9.1.1" xref="S4.SS7.p2.28.m9.1.1.cmml"><msubsup id="S4.SS7.p2.28.m9.1.1.2" xref="S4.SS7.p2.28.m9.1.1.2.cmml"><mi id="S4.SS7.p2.28.m9.1.1.2.2.2" xref="S4.SS7.p2.28.m9.1.1.2.2.2.cmml">A</mi><mi id="S4.SS7.p2.28.m9.1.1.2.2.3" xref="S4.SS7.p2.28.m9.1.1.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.28.m9.1.1.2.3" xref="S4.SS7.p2.28.m9.1.1.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.28.m9.1.1.1" xref="S4.SS7.p2.28.m9.1.1.1.cmml">=</mo><mn id="S4.SS7.p2.28.m9.1.1.3" xref="S4.SS7.p2.28.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.28.m9.1b"><apply id="S4.SS7.p2.28.m9.1.1.cmml" xref="S4.SS7.p2.28.m9.1.1"><eq id="S4.SS7.p2.28.m9.1.1.1.cmml" xref="S4.SS7.p2.28.m9.1.1.1"></eq><apply id="S4.SS7.p2.28.m9.1.1.2.cmml" xref="S4.SS7.p2.28.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS7.p2.28.m9.1.1.2.1.cmml" xref="S4.SS7.p2.28.m9.1.1.2">superscript</csymbol><apply id="S4.SS7.p2.28.m9.1.1.2.2.cmml" xref="S4.SS7.p2.28.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS7.p2.28.m9.1.1.2.2.1.cmml" xref="S4.SS7.p2.28.m9.1.1.2">subscript</csymbol><ci id="S4.SS7.p2.28.m9.1.1.2.2.2.cmml" xref="S4.SS7.p2.28.m9.1.1.2.2.2">𝐴</ci><ci id="S4.SS7.p2.28.m9.1.1.2.2.3.cmml" xref="S4.SS7.p2.28.m9.1.1.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.28.m9.1.1.2.3.cmml" xref="S4.SS7.p2.28.m9.1.1.2.3">𝑟</ci></apply><cn id="S4.SS7.p2.28.m9.1.1.3.cmml" type="integer" xref="S4.SS7.p2.28.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.28.m9.1c">A_{\theta}^{r}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.28.m9.1d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT = 1</annotation></semantics></math> or <math alttext="B_{\theta}^{r}=1" class="ltx_Math" display="inline" id="S4.SS7.p2.29.m10.1"><semantics id="S4.SS7.p2.29.m10.1a"><mrow id="S4.SS7.p2.29.m10.1.1" xref="S4.SS7.p2.29.m10.1.1.cmml"><msubsup id="S4.SS7.p2.29.m10.1.1.2" xref="S4.SS7.p2.29.m10.1.1.2.cmml"><mi id="S4.SS7.p2.29.m10.1.1.2.2.2" xref="S4.SS7.p2.29.m10.1.1.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.29.m10.1.1.2.2.3" xref="S4.SS7.p2.29.m10.1.1.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.29.m10.1.1.2.3" xref="S4.SS7.p2.29.m10.1.1.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.29.m10.1.1.1" xref="S4.SS7.p2.29.m10.1.1.1.cmml">=</mo><mn id="S4.SS7.p2.29.m10.1.1.3" xref="S4.SS7.p2.29.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.29.m10.1b"><apply id="S4.SS7.p2.29.m10.1.1.cmml" xref="S4.SS7.p2.29.m10.1.1"><eq id="S4.SS7.p2.29.m10.1.1.1.cmml" xref="S4.SS7.p2.29.m10.1.1.1"></eq><apply id="S4.SS7.p2.29.m10.1.1.2.cmml" xref="S4.SS7.p2.29.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS7.p2.29.m10.1.1.2.1.cmml" xref="S4.SS7.p2.29.m10.1.1.2">superscript</csymbol><apply id="S4.SS7.p2.29.m10.1.1.2.2.cmml" xref="S4.SS7.p2.29.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS7.p2.29.m10.1.1.2.2.1.cmml" xref="S4.SS7.p2.29.m10.1.1.2">subscript</csymbol><ci id="S4.SS7.p2.29.m10.1.1.2.2.2.cmml" xref="S4.SS7.p2.29.m10.1.1.2.2.2">𝐵</ci><ci id="S4.SS7.p2.29.m10.1.1.2.2.3.cmml" xref="S4.SS7.p2.29.m10.1.1.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.29.m10.1.1.2.3.cmml" xref="S4.SS7.p2.29.m10.1.1.2.3">𝑟</ci></apply><cn id="S4.SS7.p2.29.m10.1.1.3.cmml" type="integer" xref="S4.SS7.p2.29.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.29.m10.1c">B_{\theta}^{r}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.29.m10.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT = 1</annotation></semantics></math> (or both). Thus, when <math alttext="A_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.30.m11.1"><semantics id="S4.SS7.p2.30.m11.1a"><msubsup id="S4.SS7.p2.30.m11.1.1" xref="S4.SS7.p2.30.m11.1.1.cmml"><mi id="S4.SS7.p2.30.m11.1.1.2.2" xref="S4.SS7.p2.30.m11.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.30.m11.1.1.2.3" xref="S4.SS7.p2.30.m11.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.30.m11.1.1.3" xref="S4.SS7.p2.30.m11.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.30.m11.1b"><apply id="S4.SS7.p2.30.m11.1.1.cmml" xref="S4.SS7.p2.30.m11.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.30.m11.1.1.1.cmml" xref="S4.SS7.p2.30.m11.1.1">superscript</csymbol><apply id="S4.SS7.p2.30.m11.1.1.2.cmml" xref="S4.SS7.p2.30.m11.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.30.m11.1.1.2.1.cmml" xref="S4.SS7.p2.30.m11.1.1">subscript</csymbol><ci id="S4.SS7.p2.30.m11.1.1.2.2.cmml" xref="S4.SS7.p2.30.m11.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.30.m11.1.1.2.3.cmml" xref="S4.SS7.p2.30.m11.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.30.m11.1.1.3.cmml" xref="S4.SS7.p2.30.m11.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.30.m11.1c">A_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.30.m11.1d">italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="B_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.31.m12.1"><semantics id="S4.SS7.p2.31.m12.1a"><msubsup id="S4.SS7.p2.31.m12.1.1" xref="S4.SS7.p2.31.m12.1.1.cmml"><mi id="S4.SS7.p2.31.m12.1.1.2.2" xref="S4.SS7.p2.31.m12.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.31.m12.1.1.2.3" xref="S4.SS7.p2.31.m12.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.31.m12.1.1.3" xref="S4.SS7.p2.31.m12.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.31.m12.1b"><apply id="S4.SS7.p2.31.m12.1.1.cmml" xref="S4.SS7.p2.31.m12.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.31.m12.1.1.1.cmml" xref="S4.SS7.p2.31.m12.1.1">superscript</csymbol><apply id="S4.SS7.p2.31.m12.1.1.2.cmml" xref="S4.SS7.p2.31.m12.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.31.m12.1.1.2.1.cmml" xref="S4.SS7.p2.31.m12.1.1">subscript</csymbol><ci id="S4.SS7.p2.31.m12.1.1.2.2.cmml" xref="S4.SS7.p2.31.m12.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.31.m12.1.1.2.3.cmml" xref="S4.SS7.p2.31.m12.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.31.m12.1.1.3.cmml" xref="S4.SS7.p2.31.m12.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.31.m12.1c">B_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.31.m12.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> are exactly the same, <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})=1" class="ltx_Math" display="inline" id="S4.SS7.p2.32.m13.2"><semantics id="S4.SS7.p2.32.m13.2a"><mrow id="S4.SS7.p2.32.m13.2.2" xref="S4.SS7.p2.32.m13.2.2.cmml"><mrow id="S4.SS7.p2.32.m13.2.2.2" xref="S4.SS7.p2.32.m13.2.2.2.cmml"><mtext id="S4.SS7.p2.32.m13.2.2.2.4" xref="S4.SS7.p2.32.m13.2.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.32.m13.2.2.2.3" xref="S4.SS7.p2.32.m13.2.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.32.m13.2.2.2.2.2" xref="S4.SS7.p2.32.m13.2.2.2.2.3.cmml"><mo id="S4.SS7.p2.32.m13.2.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.32.m13.2.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.32.m13.1.1.1.1.1.1" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.2" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.3" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.32.m13.1.1.1.1.1.1.3" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.32.m13.2.2.2.2.2.4" xref="S4.SS7.p2.32.m13.2.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.32.m13.2.2.2.2.2.2" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.2" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.3" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.32.m13.2.2.2.2.2.2.3" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.32.m13.2.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.32.m13.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS7.p2.32.m13.2.2.3" xref="S4.SS7.p2.32.m13.2.2.3.cmml">=</mo><mn id="S4.SS7.p2.32.m13.2.2.4" xref="S4.SS7.p2.32.m13.2.2.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.32.m13.2b"><apply id="S4.SS7.p2.32.m13.2.2.cmml" xref="S4.SS7.p2.32.m13.2.2"><eq id="S4.SS7.p2.32.m13.2.2.3.cmml" xref="S4.SS7.p2.32.m13.2.2.3"></eq><apply id="S4.SS7.p2.32.m13.2.2.2.cmml" xref="S4.SS7.p2.32.m13.2.2.2"><times id="S4.SS7.p2.32.m13.2.2.2.3.cmml" xref="S4.SS7.p2.32.m13.2.2.2.3"></times><ci id="S4.SS7.p2.32.m13.2.2.2.4a.cmml" xref="S4.SS7.p2.32.m13.2.2.2.4"><mtext id="S4.SS7.p2.32.m13.2.2.2.4.cmml" xref="S4.SS7.p2.32.m13.2.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.32.m13.2.2.2.2.3.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2"><apply id="S4.SS7.p2.32.m13.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.32.m13.1.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.32.m13.1.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.32.m13.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.32.m13.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.32.m13.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.32.m13.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.32.m13.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><cn id="S4.SS7.p2.32.m13.2.2.4.cmml" type="integer" xref="S4.SS7.p2.32.m13.2.2.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.32.m13.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.32.m13.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>. Next, we shift all values of <math alttext="B_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.33.m14.1"><semantics id="S4.SS7.p2.33.m14.1a"><msubsup id="S4.SS7.p2.33.m14.1.1" xref="S4.SS7.p2.33.m14.1.1.cmml"><mi id="S4.SS7.p2.33.m14.1.1.2.2" xref="S4.SS7.p2.33.m14.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.33.m14.1.1.2.3" xref="S4.SS7.p2.33.m14.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.33.m14.1.1.3" xref="S4.SS7.p2.33.m14.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.33.m14.1b"><apply id="S4.SS7.p2.33.m14.1.1.cmml" xref="S4.SS7.p2.33.m14.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.33.m14.1.1.1.cmml" xref="S4.SS7.p2.33.m14.1.1">superscript</csymbol><apply id="S4.SS7.p2.33.m14.1.1.2.cmml" xref="S4.SS7.p2.33.m14.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.33.m14.1.1.2.1.cmml" xref="S4.SS7.p2.33.m14.1.1">subscript</csymbol><ci id="S4.SS7.p2.33.m14.1.1.2.2.cmml" xref="S4.SS7.p2.33.m14.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.33.m14.1.1.2.3.cmml" xref="S4.SS7.p2.33.m14.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.33.m14.1.1.3.cmml" xref="S4.SS7.p2.33.m14.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.33.m14.1c">B_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.33.m14.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="B_{\theta+1\pmod{360}}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.34.m15.2"><semantics id="S4.SS7.p2.34.m15.2a"><msubsup id="S4.SS7.p2.34.m15.2.3" xref="S4.SS7.p2.34.m15.2.3.cmml"><mi id="S4.SS7.p2.34.m15.2.3.2.2" xref="S4.SS7.p2.34.m15.2.3.2.2.cmml">B</mi><mrow id="S4.SS7.p2.34.m15.2.2.2" xref="S4.SS7.p2.34.m15.2.2.2.cmml"><mrow id="S4.SS7.p2.34.m15.2.2.2.4" xref="S4.SS7.p2.34.m15.2.2.2.4.cmml"><mi id="S4.SS7.p2.34.m15.2.2.2.4.2" xref="S4.SS7.p2.34.m15.2.2.2.4.2.cmml">θ</mi><mo id="S4.SS7.p2.34.m15.2.2.2.4.1" xref="S4.SS7.p2.34.m15.2.2.2.4.1.cmml">+</mo><mn id="S4.SS7.p2.34.m15.2.2.2.4.3" xref="S4.SS7.p2.34.m15.2.2.2.4.3.cmml">1</mn></mrow><mspace id="S4.SS7.p2.34.m15.2.2.2a" width="0.949em" xref="S4.SS7.p2.34.m15.2.2.2.cmml"></mspace><mrow id="S4.SS7.p2.34.m15.2.2.2.2.2.2" xref="S4.SS7.p2.34.m15.2.2.2.2.3.cmml"><mo id="S4.SS7.p2.34.m15.2.2.2.2.2.2.2" stretchy="false" xref="S4.SS7.p2.34.m15.2.2.2.2.3.1.cmml">(</mo><mrow id="S4.SS7.p2.34.m15.2.2.2.2.2.2.1" xref="S4.SS7.p2.34.m15.2.2.2.2.3.cmml"><mo id="S4.SS7.p2.34.m15.2.2.2.2.2.2.1.1" lspace="0em" rspace="0.167em" xref="S4.SS7.p2.34.m15.2.2.2.2.3.1.cmml">mod</mo><mn id="S4.SS7.p2.34.m15.1.1.1.1.1.1" xref="S4.SS7.p2.34.m15.1.1.1.1.1.1.cmml">360</mn></mrow><mo id="S4.SS7.p2.34.m15.2.2.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.34.m15.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mi id="S4.SS7.p2.34.m15.2.3.3" xref="S4.SS7.p2.34.m15.2.3.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.34.m15.2b"><apply id="S4.SS7.p2.34.m15.2.3.cmml" xref="S4.SS7.p2.34.m15.2.3"><csymbol cd="ambiguous" id="S4.SS7.p2.34.m15.2.3.1.cmml" xref="S4.SS7.p2.34.m15.2.3">superscript</csymbol><apply id="S4.SS7.p2.34.m15.2.3.2.cmml" xref="S4.SS7.p2.34.m15.2.3"><csymbol cd="ambiguous" id="S4.SS7.p2.34.m15.2.3.2.1.cmml" xref="S4.SS7.p2.34.m15.2.3">subscript</csymbol><ci id="S4.SS7.p2.34.m15.2.3.2.2.cmml" xref="S4.SS7.p2.34.m15.2.3.2.2">𝐵</ci><apply id="S4.SS7.p2.34.m15.2.2.2.cmml" xref="S4.SS7.p2.34.m15.2.2.2"><csymbol cd="latexml" id="S4.SS7.p2.34.m15.2.2.2.3.cmml" xref="S4.SS7.p2.34.m15.2.2.2">annotated</csymbol><apply id="S4.SS7.p2.34.m15.2.2.2.4.cmml" xref="S4.SS7.p2.34.m15.2.2.2.4"><plus id="S4.SS7.p2.34.m15.2.2.2.4.1.cmml" xref="S4.SS7.p2.34.m15.2.2.2.4.1"></plus><ci id="S4.SS7.p2.34.m15.2.2.2.4.2.cmml" xref="S4.SS7.p2.34.m15.2.2.2.4.2">𝜃</ci><cn id="S4.SS7.p2.34.m15.2.2.2.4.3.cmml" type="integer" xref="S4.SS7.p2.34.m15.2.2.2.4.3">1</cn></apply><apply id="S4.SS7.p2.34.m15.2.2.2.2.3.cmml" xref="S4.SS7.p2.34.m15.2.2.2.2.2.2"><ci id="S4.SS7.p2.34.m15.2.2.2.2.3.1.cmml" xref="S4.SS7.p2.34.m15.2.2.2.2.2.2.2">pmod</ci><cn id="S4.SS7.p2.34.m15.1.1.1.1.1.1.cmml" type="integer" xref="S4.SS7.p2.34.m15.1.1.1.1.1.1">360</cn></apply></apply></apply><ci id="S4.SS7.p2.34.m15.2.3.3.cmml" xref="S4.SS7.p2.34.m15.2.3.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.34.m15.2c">B_{\theta+1\pmod{360}}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.34.m15.2d">italic_B start_POSTSUBSCRIPT italic_θ + 1 start_MODIFIER ( roman_mod start_ARG 360 end_ARG ) end_MODIFIER end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math>, corresponding to a one-degree rotation in Cartesian coordinates, and compute <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})" class="ltx_Math" display="inline" id="S4.SS7.p2.35.m16.2"><semantics id="S4.SS7.p2.35.m16.2a"><mrow id="S4.SS7.p2.35.m16.2.2" xref="S4.SS7.p2.35.m16.2.2.cmml"><mtext id="S4.SS7.p2.35.m16.2.2.4" xref="S4.SS7.p2.35.m16.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.35.m16.2.2.3" xref="S4.SS7.p2.35.m16.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.35.m16.2.2.2.2" xref="S4.SS7.p2.35.m16.2.2.2.3.cmml"><mo id="S4.SS7.p2.35.m16.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.35.m16.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.35.m16.1.1.1.1.1" xref="S4.SS7.p2.35.m16.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.35.m16.1.1.1.1.1.2.2" xref="S4.SS7.p2.35.m16.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.35.m16.1.1.1.1.1.2.3" xref="S4.SS7.p2.35.m16.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.35.m16.1.1.1.1.1.3" xref="S4.SS7.p2.35.m16.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.35.m16.2.2.2.2.4" xref="S4.SS7.p2.35.m16.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.35.m16.2.2.2.2.2" xref="S4.SS7.p2.35.m16.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.35.m16.2.2.2.2.2.2.2" xref="S4.SS7.p2.35.m16.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.35.m16.2.2.2.2.2.2.3" xref="S4.SS7.p2.35.m16.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.35.m16.2.2.2.2.2.3" xref="S4.SS7.p2.35.m16.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.35.m16.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.35.m16.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.35.m16.2b"><apply id="S4.SS7.p2.35.m16.2.2.cmml" xref="S4.SS7.p2.35.m16.2.2"><times id="S4.SS7.p2.35.m16.2.2.3.cmml" xref="S4.SS7.p2.35.m16.2.2.3"></times><ci id="S4.SS7.p2.35.m16.2.2.4a.cmml" xref="S4.SS7.p2.35.m16.2.2.4"><mtext id="S4.SS7.p2.35.m16.2.2.4.cmml" xref="S4.SS7.p2.35.m16.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.35.m16.2.2.2.3.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2"><apply id="S4.SS7.p2.35.m16.1.1.1.1.1.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.35.m16.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.35.m16.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.35.m16.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.35.m16.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.35.m16.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.35.m16.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.35.m16.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.35.m16.2.2.2.2.2.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.35.m16.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.35.m16.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.35.m16.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.35.m16.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.35.m16.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.35.m16.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.35.m16.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.35.m16.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.35.m16.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math> again. We repeat these one-degree rotations 360 times, computing <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})" class="ltx_Math" display="inline" id="S4.SS7.p2.36.m17.2"><semantics id="S4.SS7.p2.36.m17.2a"><mrow id="S4.SS7.p2.36.m17.2.2" xref="S4.SS7.p2.36.m17.2.2.cmml"><mtext id="S4.SS7.p2.36.m17.2.2.4" xref="S4.SS7.p2.36.m17.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.36.m17.2.2.3" xref="S4.SS7.p2.36.m17.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.36.m17.2.2.2.2" xref="S4.SS7.p2.36.m17.2.2.2.3.cmml"><mo id="S4.SS7.p2.36.m17.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.36.m17.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.36.m17.1.1.1.1.1" xref="S4.SS7.p2.36.m17.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.36.m17.1.1.1.1.1.2.2" xref="S4.SS7.p2.36.m17.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.36.m17.1.1.1.1.1.2.3" xref="S4.SS7.p2.36.m17.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.36.m17.1.1.1.1.1.3" xref="S4.SS7.p2.36.m17.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.36.m17.2.2.2.2.4" xref="S4.SS7.p2.36.m17.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.36.m17.2.2.2.2.2" xref="S4.SS7.p2.36.m17.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.36.m17.2.2.2.2.2.2.2" xref="S4.SS7.p2.36.m17.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.36.m17.2.2.2.2.2.2.3" xref="S4.SS7.p2.36.m17.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.36.m17.2.2.2.2.2.3" xref="S4.SS7.p2.36.m17.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.36.m17.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.36.m17.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.36.m17.2b"><apply id="S4.SS7.p2.36.m17.2.2.cmml" xref="S4.SS7.p2.36.m17.2.2"><times id="S4.SS7.p2.36.m17.2.2.3.cmml" xref="S4.SS7.p2.36.m17.2.2.3"></times><ci id="S4.SS7.p2.36.m17.2.2.4a.cmml" xref="S4.SS7.p2.36.m17.2.2.4"><mtext id="S4.SS7.p2.36.m17.2.2.4.cmml" xref="S4.SS7.p2.36.m17.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.36.m17.2.2.2.3.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2"><apply id="S4.SS7.p2.36.m17.1.1.1.1.1.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.36.m17.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.36.m17.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.36.m17.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.36.m17.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.36.m17.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.36.m17.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.36.m17.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.36.m17.2.2.2.2.2.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.36.m17.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.36.m17.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.36.m17.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.36.m17.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.36.m17.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.36.m17.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.36.m17.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.36.m17.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.36.m17.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math> for each. Next, we invert all values of <math alttext="B_{\theta}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.37.m18.1"><semantics id="S4.SS7.p2.37.m18.1a"><msubsup id="S4.SS7.p2.37.m18.1.1" xref="S4.SS7.p2.37.m18.1.1.cmml"><mi id="S4.SS7.p2.37.m18.1.1.2.2" xref="S4.SS7.p2.37.m18.1.1.2.2.cmml">B</mi><mi id="S4.SS7.p2.37.m18.1.1.2.3" xref="S4.SS7.p2.37.m18.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.37.m18.1.1.3" xref="S4.SS7.p2.37.m18.1.1.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.37.m18.1b"><apply id="S4.SS7.p2.37.m18.1.1.cmml" xref="S4.SS7.p2.37.m18.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.37.m18.1.1.1.cmml" xref="S4.SS7.p2.37.m18.1.1">superscript</csymbol><apply id="S4.SS7.p2.37.m18.1.1.2.cmml" xref="S4.SS7.p2.37.m18.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.37.m18.1.1.2.1.cmml" xref="S4.SS7.p2.37.m18.1.1">subscript</csymbol><ci id="S4.SS7.p2.37.m18.1.1.2.2.cmml" xref="S4.SS7.p2.37.m18.1.1.2.2">𝐵</ci><ci id="S4.SS7.p2.37.m18.1.1.2.3.cmml" xref="S4.SS7.p2.37.m18.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.37.m18.1.1.3.cmml" xref="S4.SS7.p2.37.m18.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.37.m18.1c">B_{\theta}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.37.m18.1d">italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="B_{180-\theta\pmod{360}}^{r}" class="ltx_Math" display="inline" id="S4.SS7.p2.38.m19.2"><semantics id="S4.SS7.p2.38.m19.2a"><msubsup id="S4.SS7.p2.38.m19.2.3" xref="S4.SS7.p2.38.m19.2.3.cmml"><mi id="S4.SS7.p2.38.m19.2.3.2.2" xref="S4.SS7.p2.38.m19.2.3.2.2.cmml">B</mi><mrow id="S4.SS7.p2.38.m19.2.2.2" xref="S4.SS7.p2.38.m19.2.2.2.cmml"><mrow id="S4.SS7.p2.38.m19.2.2.2.4" xref="S4.SS7.p2.38.m19.2.2.2.4.cmml"><mn id="S4.SS7.p2.38.m19.2.2.2.4.2" xref="S4.SS7.p2.38.m19.2.2.2.4.2.cmml">180</mn><mo id="S4.SS7.p2.38.m19.2.2.2.4.1" xref="S4.SS7.p2.38.m19.2.2.2.4.1.cmml">−</mo><mi id="S4.SS7.p2.38.m19.2.2.2.4.3" xref="S4.SS7.p2.38.m19.2.2.2.4.3.cmml">θ</mi></mrow><mspace id="S4.SS7.p2.38.m19.2.2.2a" width="0.949em" xref="S4.SS7.p2.38.m19.2.2.2.cmml"></mspace><mrow id="S4.SS7.p2.38.m19.2.2.2.2.2.2" xref="S4.SS7.p2.38.m19.2.2.2.2.3.cmml"><mo id="S4.SS7.p2.38.m19.2.2.2.2.2.2.2" stretchy="false" xref="S4.SS7.p2.38.m19.2.2.2.2.3.1.cmml">(</mo><mrow id="S4.SS7.p2.38.m19.2.2.2.2.2.2.1" xref="S4.SS7.p2.38.m19.2.2.2.2.3.cmml"><mo id="S4.SS7.p2.38.m19.2.2.2.2.2.2.1.1" lspace="0em" rspace="0.167em" xref="S4.SS7.p2.38.m19.2.2.2.2.3.1.cmml">mod</mo><mn id="S4.SS7.p2.38.m19.1.1.1.1.1.1" xref="S4.SS7.p2.38.m19.1.1.1.1.1.1.cmml">360</mn></mrow><mo id="S4.SS7.p2.38.m19.2.2.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.38.m19.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mi id="S4.SS7.p2.38.m19.2.3.3" xref="S4.SS7.p2.38.m19.2.3.3.cmml">r</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.38.m19.2b"><apply id="S4.SS7.p2.38.m19.2.3.cmml" xref="S4.SS7.p2.38.m19.2.3"><csymbol cd="ambiguous" id="S4.SS7.p2.38.m19.2.3.1.cmml" xref="S4.SS7.p2.38.m19.2.3">superscript</csymbol><apply id="S4.SS7.p2.38.m19.2.3.2.cmml" xref="S4.SS7.p2.38.m19.2.3"><csymbol cd="ambiguous" id="S4.SS7.p2.38.m19.2.3.2.1.cmml" xref="S4.SS7.p2.38.m19.2.3">subscript</csymbol><ci id="S4.SS7.p2.38.m19.2.3.2.2.cmml" xref="S4.SS7.p2.38.m19.2.3.2.2">𝐵</ci><apply id="S4.SS7.p2.38.m19.2.2.2.cmml" xref="S4.SS7.p2.38.m19.2.2.2"><csymbol cd="latexml" id="S4.SS7.p2.38.m19.2.2.2.3.cmml" xref="S4.SS7.p2.38.m19.2.2.2">annotated</csymbol><apply id="S4.SS7.p2.38.m19.2.2.2.4.cmml" xref="S4.SS7.p2.38.m19.2.2.2.4"><minus id="S4.SS7.p2.38.m19.2.2.2.4.1.cmml" xref="S4.SS7.p2.38.m19.2.2.2.4.1"></minus><cn id="S4.SS7.p2.38.m19.2.2.2.4.2.cmml" type="integer" xref="S4.SS7.p2.38.m19.2.2.2.4.2">180</cn><ci id="S4.SS7.p2.38.m19.2.2.2.4.3.cmml" xref="S4.SS7.p2.38.m19.2.2.2.4.3">𝜃</ci></apply><apply id="S4.SS7.p2.38.m19.2.2.2.2.3.cmml" xref="S4.SS7.p2.38.m19.2.2.2.2.2.2"><ci id="S4.SS7.p2.38.m19.2.2.2.2.3.1.cmml" xref="S4.SS7.p2.38.m19.2.2.2.2.2.2.2">pmod</ci><cn id="S4.SS7.p2.38.m19.1.1.1.1.1.1.cmml" type="integer" xref="S4.SS7.p2.38.m19.1.1.1.1.1.1">360</cn></apply></apply></apply><ci id="S4.SS7.p2.38.m19.2.3.3.cmml" xref="S4.SS7.p2.38.m19.2.3.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.38.m19.2c">B_{180-\theta\pmod{360}}^{r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.38.m19.2d">italic_B start_POSTSUBSCRIPT 180 - italic_θ start_MODIFIER ( roman_mod start_ARG 360 end_ARG ) end_MODIFIER end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math>, which corresponds to a reflection of grid <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.39.m20.1"><semantics id="S4.SS7.p2.39.m20.1a"><mi id="S4.SS7.p2.39.m20.1.1" xref="S4.SS7.p2.39.m20.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.39.m20.1b"><ci id="S4.SS7.p2.39.m20.1.1.cmml" xref="S4.SS7.p2.39.m20.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.39.m20.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.39.m20.1d">italic_B</annotation></semantics></math>, and repeat the 360 one-degree rotations again, computing <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})" class="ltx_Math" display="inline" id="S4.SS7.p2.40.m21.2"><semantics id="S4.SS7.p2.40.m21.2a"><mrow id="S4.SS7.p2.40.m21.2.2" xref="S4.SS7.p2.40.m21.2.2.cmml"><mtext id="S4.SS7.p2.40.m21.2.2.4" xref="S4.SS7.p2.40.m21.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.40.m21.2.2.3" xref="S4.SS7.p2.40.m21.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.40.m21.2.2.2.2" xref="S4.SS7.p2.40.m21.2.2.2.3.cmml"><mo id="S4.SS7.p2.40.m21.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.40.m21.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.40.m21.1.1.1.1.1" xref="S4.SS7.p2.40.m21.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.40.m21.1.1.1.1.1.2.2" xref="S4.SS7.p2.40.m21.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.40.m21.1.1.1.1.1.2.3" xref="S4.SS7.p2.40.m21.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.40.m21.1.1.1.1.1.3" xref="S4.SS7.p2.40.m21.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.40.m21.2.2.2.2.4" xref="S4.SS7.p2.40.m21.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.40.m21.2.2.2.2.2" xref="S4.SS7.p2.40.m21.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.40.m21.2.2.2.2.2.2.2" xref="S4.SS7.p2.40.m21.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.40.m21.2.2.2.2.2.2.3" xref="S4.SS7.p2.40.m21.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.40.m21.2.2.2.2.2.3" xref="S4.SS7.p2.40.m21.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.40.m21.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.40.m21.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.40.m21.2b"><apply id="S4.SS7.p2.40.m21.2.2.cmml" xref="S4.SS7.p2.40.m21.2.2"><times id="S4.SS7.p2.40.m21.2.2.3.cmml" xref="S4.SS7.p2.40.m21.2.2.3"></times><ci id="S4.SS7.p2.40.m21.2.2.4a.cmml" xref="S4.SS7.p2.40.m21.2.2.4"><mtext id="S4.SS7.p2.40.m21.2.2.4.cmml" xref="S4.SS7.p2.40.m21.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.40.m21.2.2.2.3.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2"><apply id="S4.SS7.p2.40.m21.1.1.1.1.1.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.40.m21.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.40.m21.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.40.m21.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.40.m21.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.40.m21.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.40.m21.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.40.m21.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.40.m21.2.2.2.2.2.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.40.m21.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.40.m21.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.40.m21.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.40.m21.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.40.m21.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.40.m21.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.40.m21.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.40.m21.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.40.m21.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math> for each. Furthermore, we repeated these 720 computations for 25 equally-spaced translations of grid <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.41.m22.1"><semantics id="S4.SS7.p2.41.m22.1a"><mi id="S4.SS7.p2.41.m22.1.1" xref="S4.SS7.p2.41.m22.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.41.m22.1b"><ci id="S4.SS7.p2.41.m22.1.1.cmml" xref="S4.SS7.p2.41.m22.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.41.m22.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.41.m22.1d">italic_B</annotation></semantics></math> (as mentioned previously), achieved by shifting the location on grid <math alttext="B" class="ltx_Math" display="inline" id="S4.SS7.p2.42.m23.1"><semantics id="S4.SS7.p2.42.m23.1a"><mi id="S4.SS7.p2.42.m23.1.1" xref="S4.SS7.p2.42.m23.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.42.m23.1b"><ci id="S4.SS7.p2.42.m23.1.1.cmml" xref="S4.SS7.p2.42.m23.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.42.m23.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.42.m23.1d">italic_B</annotation></semantics></math> chosen to be <math alttext="r=0" class="ltx_Math" display="inline" id="S4.SS7.p2.43.m24.1"><semantics id="S4.SS7.p2.43.m24.1a"><mrow id="S4.SS7.p2.43.m24.1.1" xref="S4.SS7.p2.43.m24.1.1.cmml"><mi id="S4.SS7.p2.43.m24.1.1.2" xref="S4.SS7.p2.43.m24.1.1.2.cmml">r</mi><mo id="S4.SS7.p2.43.m24.1.1.1" xref="S4.SS7.p2.43.m24.1.1.1.cmml">=</mo><mn id="S4.SS7.p2.43.m24.1.1.3" xref="S4.SS7.p2.43.m24.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.43.m24.1b"><apply id="S4.SS7.p2.43.m24.1.1.cmml" xref="S4.SS7.p2.43.m24.1.1"><eq id="S4.SS7.p2.43.m24.1.1.1.cmml" xref="S4.SS7.p2.43.m24.1.1.1"></eq><ci id="S4.SS7.p2.43.m24.1.1.2.cmml" xref="S4.SS7.p2.43.m24.1.1.2">𝑟</ci><cn id="S4.SS7.p2.43.m24.1.1.3.cmml" type="integer" xref="S4.SS7.p2.43.m24.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.43.m24.1c">r=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.43.m24.1d">italic_r = 0</annotation></semantics></math> for the polar coordinate. Translations occur in steps of five pixels at a time to make up a five-by-five square. The maximum <math alttext="\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})" class="ltx_Math" display="inline" id="S4.SS7.p2.44.m25.2"><semantics id="S4.SS7.p2.44.m25.2a"><mrow id="S4.SS7.p2.44.m25.2.2" xref="S4.SS7.p2.44.m25.2.2.cmml"><mtext id="S4.SS7.p2.44.m25.2.2.4" xref="S4.SS7.p2.44.m25.2.2.4a.cmml">Jac</mtext><mo id="S4.SS7.p2.44.m25.2.2.3" xref="S4.SS7.p2.44.m25.2.2.3.cmml">⁢</mo><mrow id="S4.SS7.p2.44.m25.2.2.2.2" xref="S4.SS7.p2.44.m25.2.2.2.3.cmml"><mo id="S4.SS7.p2.44.m25.2.2.2.2.3" stretchy="false" xref="S4.SS7.p2.44.m25.2.2.2.3.cmml">(</mo><msubsup id="S4.SS7.p2.44.m25.1.1.1.1.1" xref="S4.SS7.p2.44.m25.1.1.1.1.1.cmml"><mi id="S4.SS7.p2.44.m25.1.1.1.1.1.2.2" xref="S4.SS7.p2.44.m25.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS7.p2.44.m25.1.1.1.1.1.2.3" xref="S4.SS7.p2.44.m25.1.1.1.1.1.2.3.cmml">θ</mi><mi id="S4.SS7.p2.44.m25.1.1.1.1.1.3" xref="S4.SS7.p2.44.m25.1.1.1.1.1.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.44.m25.2.2.2.2.4" xref="S4.SS7.p2.44.m25.2.2.2.3.cmml">,</mo><msubsup id="S4.SS7.p2.44.m25.2.2.2.2.2" xref="S4.SS7.p2.44.m25.2.2.2.2.2.cmml"><mi id="S4.SS7.p2.44.m25.2.2.2.2.2.2.2" xref="S4.SS7.p2.44.m25.2.2.2.2.2.2.2.cmml">B</mi><mi id="S4.SS7.p2.44.m25.2.2.2.2.2.2.3" xref="S4.SS7.p2.44.m25.2.2.2.2.2.2.3.cmml">θ</mi><mi id="S4.SS7.p2.44.m25.2.2.2.2.2.3" xref="S4.SS7.p2.44.m25.2.2.2.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS7.p2.44.m25.2.2.2.2.5" stretchy="false" xref="S4.SS7.p2.44.m25.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.44.m25.2b"><apply id="S4.SS7.p2.44.m25.2.2.cmml" xref="S4.SS7.p2.44.m25.2.2"><times id="S4.SS7.p2.44.m25.2.2.3.cmml" xref="S4.SS7.p2.44.m25.2.2.3"></times><ci id="S4.SS7.p2.44.m25.2.2.4a.cmml" xref="S4.SS7.p2.44.m25.2.2.4"><mtext id="S4.SS7.p2.44.m25.2.2.4.cmml" xref="S4.SS7.p2.44.m25.2.2.4">Jac</mtext></ci><interval closure="open" id="S4.SS7.p2.44.m25.2.2.2.3.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2"><apply id="S4.SS7.p2.44.m25.1.1.1.1.1.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.44.m25.1.1.1.1.1.1.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1">superscript</csymbol><apply id="S4.SS7.p2.44.m25.1.1.1.1.1.2.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.44.m25.1.1.1.1.1.2.1.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1">subscript</csymbol><ci id="S4.SS7.p2.44.m25.1.1.1.1.1.2.2.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS7.p2.44.m25.1.1.1.1.1.2.3.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.44.m25.1.1.1.1.1.3.cmml" xref="S4.SS7.p2.44.m25.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.SS7.p2.44.m25.2.2.2.2.2.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.44.m25.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2">superscript</csymbol><apply id="S4.SS7.p2.44.m25.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS7.p2.44.m25.2.2.2.2.2.2.1.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2">subscript</csymbol><ci id="S4.SS7.p2.44.m25.2.2.2.2.2.2.2.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2.2.2">𝐵</ci><ci id="S4.SS7.p2.44.m25.2.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2.2.3">𝜃</ci></apply><ci id="S4.SS7.p2.44.m25.2.2.2.2.2.3.cmml" xref="S4.SS7.p2.44.m25.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.44.m25.2c">\text{Jac}(A_{\theta}^{r},B_{\theta}^{r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.44.m25.2d">Jac ( italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math> recorded across all rotation, reflections and translations for each pairwise comparison is the maximum possible similarity, denoted <math alttext="\text{Jac}_{\text{max}}" class="ltx_Math" display="inline" id="S4.SS7.p2.45.m26.1"><semantics id="S4.SS7.p2.45.m26.1a"><msub id="S4.SS7.p2.45.m26.1.1" xref="S4.SS7.p2.45.m26.1.1.cmml"><mtext id="S4.SS7.p2.45.m26.1.1.2" xref="S4.SS7.p2.45.m26.1.1.2a.cmml">Jac</mtext><mtext id="S4.SS7.p2.45.m26.1.1.3" xref="S4.SS7.p2.45.m26.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.45.m26.1b"><apply id="S4.SS7.p2.45.m26.1.1.cmml" xref="S4.SS7.p2.45.m26.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.45.m26.1.1.1.cmml" xref="S4.SS7.p2.45.m26.1.1">subscript</csymbol><ci id="S4.SS7.p2.45.m26.1.1.2a.cmml" xref="S4.SS7.p2.45.m26.1.1.2"><mtext id="S4.SS7.p2.45.m26.1.1.2.cmml" xref="S4.SS7.p2.45.m26.1.1.2">Jac</mtext></ci><ci id="S4.SS7.p2.45.m26.1.1.3a.cmml" xref="S4.SS7.p2.45.m26.1.1.3"><mtext id="S4.SS7.p2.45.m26.1.1.3.cmml" mathsize="70%" xref="S4.SS7.p2.45.m26.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.45.m26.1c">\text{Jac}_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.45.m26.1d">Jac start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math>. The reproducibility score for an organism is the average <math alttext="\text{Jac}_{\text{max}}" class="ltx_Math" display="inline" id="S4.SS7.p2.46.m27.1"><semantics id="S4.SS7.p2.46.m27.1a"><msub id="S4.SS7.p2.46.m27.1.1" xref="S4.SS7.p2.46.m27.1.1.cmml"><mtext id="S4.SS7.p2.46.m27.1.1.2" xref="S4.SS7.p2.46.m27.1.1.2a.cmml">Jac</mtext><mtext id="S4.SS7.p2.46.m27.1.1.3" xref="S4.SS7.p2.46.m27.1.1.3a.cmml">max</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.46.m27.1b"><apply id="S4.SS7.p2.46.m27.1.1.cmml" xref="S4.SS7.p2.46.m27.1.1"><csymbol cd="ambiguous" id="S4.SS7.p2.46.m27.1.1.1.cmml" xref="S4.SS7.p2.46.m27.1.1">subscript</csymbol><ci id="S4.SS7.p2.46.m27.1.1.2a.cmml" xref="S4.SS7.p2.46.m27.1.1.2"><mtext id="S4.SS7.p2.46.m27.1.1.2.cmml" xref="S4.SS7.p2.46.m27.1.1.2">Jac</mtext></ci><ci id="S4.SS7.p2.46.m27.1.1.3a.cmml" xref="S4.SS7.p2.46.m27.1.1.3"><mtext id="S4.SS7.p2.46.m27.1.1.3.cmml" mathsize="70%" xref="S4.SS7.p2.46.m27.1.1.3">max</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.46.m27.1c">\text{Jac}_{\text{max}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.46.m27.1d">Jac start_POSTSUBSCRIPT max end_POSTSUBSCRIPT</annotation></semantics></math> across all <math alttext="60\times 59/2" class="ltx_Math" display="inline" id="S4.SS7.p2.47.m28.1"><semantics id="S4.SS7.p2.47.m28.1a"><mrow id="S4.SS7.p2.47.m28.1.1" xref="S4.SS7.p2.47.m28.1.1.cmml"><mrow id="S4.SS7.p2.47.m28.1.1.2" xref="S4.SS7.p2.47.m28.1.1.2.cmml"><mn id="S4.SS7.p2.47.m28.1.1.2.2" xref="S4.SS7.p2.47.m28.1.1.2.2.cmml">60</mn><mo id="S4.SS7.p2.47.m28.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS7.p2.47.m28.1.1.2.1.cmml">×</mo><mn id="S4.SS7.p2.47.m28.1.1.2.3" xref="S4.SS7.p2.47.m28.1.1.2.3.cmml">59</mn></mrow><mo id="S4.SS7.p2.47.m28.1.1.1" xref="S4.SS7.p2.47.m28.1.1.1.cmml">/</mo><mn id="S4.SS7.p2.47.m28.1.1.3" xref="S4.SS7.p2.47.m28.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS7.p2.47.m28.1b"><apply id="S4.SS7.p2.47.m28.1.1.cmml" xref="S4.SS7.p2.47.m28.1.1"><divide id="S4.SS7.p2.47.m28.1.1.1.cmml" xref="S4.SS7.p2.47.m28.1.1.1"></divide><apply id="S4.SS7.p2.47.m28.1.1.2.cmml" xref="S4.SS7.p2.47.m28.1.1.2"><times id="S4.SS7.p2.47.m28.1.1.2.1.cmml" xref="S4.SS7.p2.47.m28.1.1.2.1"></times><cn id="S4.SS7.p2.47.m28.1.1.2.2.cmml" type="integer" xref="S4.SS7.p2.47.m28.1.1.2.2">60</cn><cn id="S4.SS7.p2.47.m28.1.1.2.3.cmml" type="integer" xref="S4.SS7.p2.47.m28.1.1.2.3">59</cn></apply><cn id="S4.SS7.p2.47.m28.1.1.3.cmml" type="integer" xref="S4.SS7.p2.47.m28.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS7.p2.47.m28.1c">60\times 59/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS7.p2.47.m28.1d">60 × 59 / 2</annotation></semantics></math> pairwise comparisons. </div> </section> <section class="ltx_subsection" id="S4.SS8"> <h3 class="ltx_title ltx_title_subsection"> 4.8 Momentum and anisotropy</h3> <div class="ltx_para" id="S4.SS8.p1"> The momentum, <math alttext="p_{\sigma}(t)" class="ltx_Math" display="inline" id="S4.SS8.p1.1.m1.1"><semantics id="S4.SS8.p1.1.m1.1a"><mrow id="S4.SS8.p1.1.m1.1.2" xref="S4.SS8.p1.1.m1.1.2.cmml"><msub id="S4.SS8.p1.1.m1.1.2.2" xref="S4.SS8.p1.1.m1.1.2.2.cmml"><mi id="S4.SS8.p1.1.m1.1.2.2.2" xref="S4.SS8.p1.1.m1.1.2.2.2.cmml">p</mi><mi id="S4.SS8.p1.1.m1.1.2.2.3" xref="S4.SS8.p1.1.m1.1.2.2.3.cmml">σ</mi></msub><mo id="S4.SS8.p1.1.m1.1.2.1" xref="S4.SS8.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS8.p1.1.m1.1.2.3.2" xref="S4.SS8.p1.1.m1.1.2.cmml"><mo id="S4.SS8.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS8.p1.1.m1.1.2.cmml">(</mo><mi id="S4.SS8.p1.1.m1.1.1" xref="S4.SS8.p1.1.m1.1.1.cmml">t</mi><mo id="S4.SS8.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS8.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.1.m1.1b"><apply id="S4.SS8.p1.1.m1.1.2.cmml" xref="S4.SS8.p1.1.m1.1.2"><times id="S4.SS8.p1.1.m1.1.2.1.cmml" xref="S4.SS8.p1.1.m1.1.2.1"></times><apply id="S4.SS8.p1.1.m1.1.2.2.cmml" xref="S4.SS8.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS8.p1.1.m1.1.2.2.1.cmml" xref="S4.SS8.p1.1.m1.1.2.2">subscript</csymbol><ci id="S4.SS8.p1.1.m1.1.2.2.2.cmml" xref="S4.SS8.p1.1.m1.1.2.2.2">𝑝</ci><ci id="S4.SS8.p1.1.m1.1.2.2.3.cmml" xref="S4.SS8.p1.1.m1.1.2.2.3">𝜎</ci></apply><ci id="S4.SS8.p1.1.m1.1.1.cmml" xref="S4.SS8.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.1.m1.1c">p_{\sigma}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.1.m1.1d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>, of cell <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS8.p1.2.m2.1"><semantics id="S4.SS8.p1.2.m2.1a"><mi id="S4.SS8.p1.2.m2.1.1" xref="S4.SS8.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.2.m2.1b"><ci id="S4.SS8.p1.2.m2.1.1.cmml" xref="S4.SS8.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.2.m2.1d">italic_σ</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS8.p1.3.m3.1"><semantics id="S4.SS8.p1.3.m3.1a"><mi id="S4.SS8.p1.3.m3.1.1" xref="S4.SS8.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.3.m3.1b"><ci id="S4.SS8.p1.3.m3.1.1.cmml" xref="S4.SS8.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.3.m3.1d">italic_t</annotation></semantics></math> (where <math alttext="t" class="ltx_Math" display="inline" id="S4.SS8.p1.4.m4.1"><semantics id="S4.SS8.p1.4.m4.1a"><mi id="S4.SS8.p1.4.m4.1.1" xref="S4.SS8.p1.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.4.m4.1b"><ci id="S4.SS8.p1.4.m4.1.1.cmml" xref="S4.SS8.p1.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.4.m4.1d">italic_t</annotation></semantics></math> is in DTS) was defined as the distance travelled by the cell’s centre of mass between <math alttext="t-t_{w}" class="ltx_Math" display="inline" id="S4.SS8.p1.5.m5.1"><semantics id="S4.SS8.p1.5.m5.1a"><mrow id="S4.SS8.p1.5.m5.1.1" xref="S4.SS8.p1.5.m5.1.1.cmml"><mi id="S4.SS8.p1.5.m5.1.1.2" xref="S4.SS8.p1.5.m5.1.1.2.cmml">t</mi><mo id="S4.SS8.p1.5.m5.1.1.1" xref="S4.SS8.p1.5.m5.1.1.1.cmml">−</mo><msub id="S4.SS8.p1.5.m5.1.1.3" xref="S4.SS8.p1.5.m5.1.1.3.cmml"><mi id="S4.SS8.p1.5.m5.1.1.3.2" xref="S4.SS8.p1.5.m5.1.1.3.2.cmml">t</mi><mi id="S4.SS8.p1.5.m5.1.1.3.3" xref="S4.SS8.p1.5.m5.1.1.3.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.5.m5.1b"><apply id="S4.SS8.p1.5.m5.1.1.cmml" xref="S4.SS8.p1.5.m5.1.1"><minus id="S4.SS8.p1.5.m5.1.1.1.cmml" xref="S4.SS8.p1.5.m5.1.1.1"></minus><ci id="S4.SS8.p1.5.m5.1.1.2.cmml" xref="S4.SS8.p1.5.m5.1.1.2">𝑡</ci><apply id="S4.SS8.p1.5.m5.1.1.3.cmml" xref="S4.SS8.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS8.p1.5.m5.1.1.3.1.cmml" xref="S4.SS8.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS8.p1.5.m5.1.1.3.2.cmml" xref="S4.SS8.p1.5.m5.1.1.3.2">𝑡</ci><ci id="S4.SS8.p1.5.m5.1.1.3.3.cmml" xref="S4.SS8.p1.5.m5.1.1.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.5.m5.1c">t-t_{w}</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.5.m5.1d">italic_t - italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" 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xref="S4.SS8.p1.7.m7.1.1.3.2.2.cmml">t</mi><mi id="S4.SS8.p1.7.m7.1.1.3.2.3" xref="S4.SS8.p1.7.m7.1.1.3.2.3.cmml">w</mi></msub><mo id="S4.SS8.p1.7.m7.1.1.3.1" xref="S4.SS8.p1.7.m7.1.1.3.1.cmml">/</mo><mn id="S4.SS8.p1.7.m7.1.1.3.3" xref="S4.SS8.p1.7.m7.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.7.m7.1b"><apply id="S4.SS8.p1.7.m7.1.1.cmml" xref="S4.SS8.p1.7.m7.1.1"><minus id="S4.SS8.p1.7.m7.1.1.1.cmml" xref="S4.SS8.p1.7.m7.1.1.1"></minus><ci id="S4.SS8.p1.7.m7.1.1.2.cmml" xref="S4.SS8.p1.7.m7.1.1.2">𝑡</ci><apply id="S4.SS8.p1.7.m7.1.1.3.cmml" xref="S4.SS8.p1.7.m7.1.1.3"><divide id="S4.SS8.p1.7.m7.1.1.3.1.cmml" xref="S4.SS8.p1.7.m7.1.1.3.1"></divide><apply id="S4.SS8.p1.7.m7.1.1.3.2.cmml" xref="S4.SS8.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS8.p1.7.m7.1.1.3.2.1.cmml" xref="S4.SS8.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S4.SS8.p1.7.m7.1.1.3.2.2.cmml" xref="S4.SS8.p1.7.m7.1.1.3.2.2">𝑡</ci><ci id="S4.SS8.p1.7.m7.1.1.3.2.3.cmml" xref="S4.SS8.p1.7.m7.1.1.3.2.3">𝑤</ci></apply><cn id="S4.SS8.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S4.SS8.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.7.m7.1c">t-t_{w}/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.7.m7.1d">italic_t - italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT / 2</annotation></semantics></math>, with unit mass represented by one pixel: <table class="ltx_equation ltx_eqn_table" id="S4.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p_{\sigma}(t)=m_{\sigma}\left(t-\frac{t_{w}}{2}\right)\frac{s_{\sigma}(t)-s_{% \sigma}(t-t_{w})}{t_{w}}" class="ltx_Math" display="block" id="S4.E9.m1.4"><semantics id="S4.E9.m1.4a"><mrow id="S4.E9.m1.4.4" xref="S4.E9.m1.4.4.cmml"><mrow id="S4.E9.m1.4.4.3" xref="S4.E9.m1.4.4.3.cmml"><msub 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id="S4.E9.m1.4c">p_{\sigma}(t)=m_{\sigma}\left(t-\frac{t_{w}}{2}\right)\frac{s_{\sigma}(t)-s_{% \sigma}(t-t_{w})}{t_{w}}</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.4d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t ) = italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t - divide start_ARG italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) divide start_ARG italic_s start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t ) - italic_s start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) end_ARG start_ARG italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(9)</td> </tr></tbody> </table> where <math alttext="s_{\sigma}(t)" class="ltx_Math" display="inline" id="S4.SS8.p1.8.m1.1"><semantics id="S4.SS8.p1.8.m1.1a"><mrow id="S4.SS8.p1.8.m1.1.2" xref="S4.SS8.p1.8.m1.1.2.cmml"><msub id="S4.SS8.p1.8.m1.1.2.2" xref="S4.SS8.p1.8.m1.1.2.2.cmml"><mi id="S4.SS8.p1.8.m1.1.2.2.2" xref="S4.SS8.p1.8.m1.1.2.2.2.cmml">s</mi><mi id="S4.SS8.p1.8.m1.1.2.2.3" xref="S4.SS8.p1.8.m1.1.2.2.3.cmml">σ</mi></msub><mo id="S4.SS8.p1.8.m1.1.2.1" xref="S4.SS8.p1.8.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS8.p1.8.m1.1.2.3.2" xref="S4.SS8.p1.8.m1.1.2.cmml"><mo id="S4.SS8.p1.8.m1.1.2.3.2.1" stretchy="false" xref="S4.SS8.p1.8.m1.1.2.cmml">(</mo><mi id="S4.SS8.p1.8.m1.1.1" xref="S4.SS8.p1.8.m1.1.1.cmml">t</mi><mo id="S4.SS8.p1.8.m1.1.2.3.2.2" stretchy="false" xref="S4.SS8.p1.8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.8.m1.1b"><apply id="S4.SS8.p1.8.m1.1.2.cmml" xref="S4.SS8.p1.8.m1.1.2"><times id="S4.SS8.p1.8.m1.1.2.1.cmml" xref="S4.SS8.p1.8.m1.1.2.1"></times><apply id="S4.SS8.p1.8.m1.1.2.2.cmml" xref="S4.SS8.p1.8.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS8.p1.8.m1.1.2.2.1.cmml" xref="S4.SS8.p1.8.m1.1.2.2">subscript</csymbol><ci id="S4.SS8.p1.8.m1.1.2.2.2.cmml" xref="S4.SS8.p1.8.m1.1.2.2.2">𝑠</ci><ci id="S4.SS8.p1.8.m1.1.2.2.3.cmml" xref="S4.SS8.p1.8.m1.1.2.2.3">𝜎</ci></apply><ci id="S4.SS8.p1.8.m1.1.1.cmml" xref="S4.SS8.p1.8.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.8.m1.1c">s_{\sigma}(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.8.m1.1d">italic_s start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is the cell’s centre of mass, <math alttext="m_{\sigma}(t" class="ltx_math_unparsed" display="inline" id="S4.SS8.p1.9.m2.1"><semantics id="S4.SS8.p1.9.m2.1a"><mrow id="S4.SS8.p1.9.m2.1b"><msub id="S4.SS8.p1.9.m2.1.1"><mi id="S4.SS8.p1.9.m2.1.1.2">m</mi><mi id="S4.SS8.p1.9.m2.1.1.3">σ</mi></msub><mrow id="S4.SS8.p1.9.m2.1.2"><mo id="S4.SS8.p1.9.m2.1.2.1" stretchy="false">(</mo><mi id="S4.SS8.p1.9.m2.1.2.2">t</mi></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS8.p1.9.m2.1c">m_{\sigma}(t</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.9.m2.1d">italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_t</annotation></semantics></math>) is the cell’s mass at <math alttext="t" class="ltx_Math" display="inline" id="S4.SS8.p1.10.m3.1"><semantics id="S4.SS8.p1.10.m3.1a"><mi id="S4.SS8.p1.10.m3.1.1" xref="S4.SS8.p1.10.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.10.m3.1b"><ci id="S4.SS8.p1.10.m3.1.1.cmml" xref="S4.SS8.p1.10.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.10.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.10.m3.1d">italic_t</annotation></semantics></math>, and <math alttext="t_{w}" class="ltx_Math" display="inline" id="S4.SS8.p1.11.m4.1"><semantics id="S4.SS8.p1.11.m4.1a"><msub id="S4.SS8.p1.11.m4.1.1" xref="S4.SS8.p1.11.m4.1.1.cmml"><mi id="S4.SS8.p1.11.m4.1.1.2" xref="S4.SS8.p1.11.m4.1.1.2.cmml">t</mi><mi id="S4.SS8.p1.11.m4.1.1.3" xref="S4.SS8.p1.11.m4.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.11.m4.1b"><apply id="S4.SS8.p1.11.m4.1.1.cmml" xref="S4.SS8.p1.11.m4.1.1"><csymbol cd="ambiguous" id="S4.SS8.p1.11.m4.1.1.1.cmml" xref="S4.SS8.p1.11.m4.1.1">subscript</csymbol><ci id="S4.SS8.p1.11.m4.1.1.2.cmml" xref="S4.SS8.p1.11.m4.1.1.2">𝑡</ci><ci id="S4.SS8.p1.11.m4.1.1.3.cmml" xref="S4.SS8.p1.11.m4.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.11.m4.1c">t_{w}</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.11.m4.1d">italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> is the waiting time. The waiting time <math alttext="t_{w}" class="ltx_Math" display="inline" id="S4.SS8.p1.12.m5.1"><semantics id="S4.SS8.p1.12.m5.1a"><msub id="S4.SS8.p1.12.m5.1.1" xref="S4.SS8.p1.12.m5.1.1.cmml"><mi id="S4.SS8.p1.12.m5.1.1.2" xref="S4.SS8.p1.12.m5.1.1.2.cmml">t</mi><mi id="S4.SS8.p1.12.m5.1.1.3" xref="S4.SS8.p1.12.m5.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.12.m5.1b"><apply id="S4.SS8.p1.12.m5.1.1.cmml" xref="S4.SS8.p1.12.m5.1.1"><csymbol cd="ambiguous" id="S4.SS8.p1.12.m5.1.1.1.cmml" xref="S4.SS8.p1.12.m5.1.1">subscript</csymbol><ci id="S4.SS8.p1.12.m5.1.1.2.cmml" xref="S4.SS8.p1.12.m5.1.1.2">𝑡</ci><ci id="S4.SS8.p1.12.m5.1.1.3.cmml" xref="S4.SS8.p1.12.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.12.m5.1c">t_{w}</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.12.m5.1d">italic_t start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> was set to 500 to average out the influence of stochastic membrane fluctuations and cell divisions on the centre of mass of the cell, thereby capturing the true mobility of cells. If a cell divides during the waiting time and the <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS8.p1.13.m6.1"><semantics id="S4.SS8.p1.13.m6.1a"><mi id="S4.SS8.p1.13.m6.1.1" xref="S4.SS8.p1.13.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS8.p1.13.m6.1b"><ci id="S4.SS8.p1.13.m6.1.1.cmml" xref="S4.SS8.p1.13.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS8.p1.13.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS8.p1.13.m6.1d">italic_σ</annotation></semantics></math> values of the parent and daughter cells differ, Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S4.E9" title="In 4.8 Momentum and anisotropy ‣ 4 Methods ‣ Stem-cell differentiation underpins reproducible morphogenesis">9</a> is modified by subtracting the position of the parent cell’s <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS8.p1.14.m7.1"><semantics 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We then assigned each momentum recording into an SCC depending on that cell state. To create the polar plots of cell momentum magnitude (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>DE), we categorised each momentum measurement assigned to an SCC into one of 36 bins based on its direction. Each bin encompasses momentum measurements within an angular width of 10∘. To measure anisotropic motion (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#S2.F5" title="Figure 5 ‣ 2.5 Stem-cell systems elevate morphogenetic reproducibility by regulating cell-motion transitions at cell-type boundaries ‣ 2 Results ‣ Stem-cell differentiation underpins reproducible morphogenesis">5</a>FGH), we calculated the variance across these 36 bins. 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Vascular Morphogenesis: Methods and Protocols, pages 67–127, 2015. </li> </ul> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Contributions</h2> <div class="ltx_para" id="Sx1.p1"> D.K.D. conceived and designed the study, implemented the model, performed the analyses, and drafted the manuscript. N.T. contributed to the study design, provided feedback on the results and implications of the study, and commented on the manuscript at all stages. A.R.D.G. provided feedback on the results and implications of the study, and commented on the manuscript at all stages. </div> </section> <section class="ltx_section" id="Sx2"> <h2 class="ltx_title ltx_title_section">Competing interests</h2> <div class="ltx_para" id="Sx2.p1"> The authors declare no competing interests. </div> <section class="ltx_subsection" id="Sx2.SSx1"> <h3 class="ltx_title ltx_title_subsection">Materials & Correspondence</h3> <div class="ltx_para" id="Sx2.SSx1.p1"> <ul class="ltx_itemize" id="Sx2.I1"> <li class="ltx_item" id="Sx2.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="Sx2.I1.i1.p1"> All original code to generate the data in this study is publicly available, and can be found at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/DominicDevlin/Stem-cell-differentiation-underpins-reproducible-morphogenesis" title="">https://github.com/DominicDevlin/Stem-cell-differentiation-underpins-reproducible-morphogenesis</a>. Our code was adapted from the Tissue Simulation Toolkit <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.19375v1#bib.bib79" title="">79</a>]</cite>. </div> </li> <li class="ltx_item" id="Sx2.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="Sx2.I1.i2.p1"> The genomes and associated data of all evolved organisms are also found in the above repository. </div> </li> <li class="ltx_item" id="Sx2.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="Sx2.I1.i3.p1"> Any additional information required to re-analyse the data reported in this paper is available from the lead contact upon request (ddev825@aucklanduni.ac.nz) </div> </li> </ul> </div> </section> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Mar 25 06:07:38 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/">LaTeXML<img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>