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class="vector-toc-link" href="#Response_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Response coefficient</span> </div> </a> <ul id="toc-Response_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Link_between_Hill_coefficient_and_response_coefficient" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Link_between_Hill_coefficient_and_response_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Link between Hill coefficient and response coefficient</span> </div> </a> <ul id="toc-Link_between_Hill_coefficient_and_response_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ultrasensitivity_in_function_composition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ultrasensitivity_in_function_composition"> <div class="vector-toc-text"> <span 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Enzyme kinetics and chemical bonding</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output 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print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Cooperativity" title="Special:EditPage/Cooperativity">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Cooperativity%22">"Cooperativity"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Cooperativity%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Cooperativity%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Cooperativity%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Cooperativity%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Cooperativity%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Cooperativity</b> is a phenomenon displayed by systems involving identical or near-identical elements, which act dependently of each other, relative to a hypothetical standard non-interacting system in which the individual elements are acting independently. One manifestation of this is <a href="/wiki/Enzyme" title="Enzyme">enzymes</a> or <a href="/wiki/Receptor_(biochemistry)" title="Receptor (biochemistry)">receptors</a> that have multiple <a href="/wiki/Binding_site" title="Binding site">binding sites</a> where the affinity of the binding sites for a ligand is <i>apparently</i> increased, <b>positive cooperativity</b>, or decreased, <b>negative cooperativity</b>, upon the binding of a ligand to a binding site. For example, when an oxygen atom binds to one of hemoglobin's four binding sites, the affinity to oxygen of the three remaining available binding sites increases; i.e. oxygen is more likely to bind to a hemoglobin bound to one oxygen than to an unbound hemoglobin. This is referred to as <a href="/wiki/Cooperative_binding" title="Cooperative binding">cooperative binding</a>.<sup id="cite_ref-Whitford_2005_1-0" class="reference"><a href="#cite_note-Whitford_2005-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>We also see cooperativity in large chain molecules made of many identical (or nearly identical) subunits (such as <a href="/wiki/DNA" title="DNA">DNA</a>, <a href="/wiki/Protein" title="Protein">proteins</a>, and <a href="/wiki/Phospholipid" title="Phospholipid">phospholipids</a>), when such molecules undergo <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a> such as melting, unfolding or unwinding. This is referred to as subunit cooperativity. However, the definition of cooperativity based on apparent increase or decrease in affinity to successive ligand binding steps is problematic, as the concept of "energy" must always be defined relative to a standard state. When we say that the affinity is increased upon binding of one ligand, it is empirically unclear what we mean since a non-cooperative binding curve is required to rigorously define binding energy and hence also affinity. A much more general and useful definition of positive cooperativity is: A process involving multiple identical incremental steps, in which intermediate states are statistically <i>underrepresented</i> relative to a hypothetical standard system (null hypothesis) where the steps occur independently of each other. </p><p>Likewise, a definition of negative cooperativity would be a process involving multiple identical incremental steps, in which the intermediate states are <i>overrepresented</i> relative to a hypothetical standard state in which individual steps occur independently.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> These latter definitions for positive and negative cooperativity easily encompass all processes which we call "cooperative", including conformational transitions in large molecules (such as proteins) and even psychological phenomena of large numbers of people (which can act independently of each other, or in a co-operative fashion). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Cooperative_binding">Cooperative binding</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=1" title="Edit section: Cooperative binding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When a substrate binds to one enzymatic subunit, the rest of the subunits are stimulated and become active. <a href="/wiki/Ligands" class="mw-redirect" title="Ligands">Ligands</a> can either have positive cooperativity, negative cooperativity, or non-cooperativity. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hemoglobin_saturation_curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Hemoglobin_saturation_curve.svg/220px-Hemoglobin_saturation_curve.svg.png" decoding="async" width="220" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Hemoglobin_saturation_curve.svg/330px-Hemoglobin_saturation_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Hemoglobin_saturation_curve.svg/440px-Hemoglobin_saturation_curve.svg.png 2x" data-file-width="639" data-file-height="541" /></a><figcaption>The sigmoidal shape of hemoglobin's oxygen-dissociation curve results from cooperative binding of oxygen to hemoglobin.</figcaption></figure> <p>An example of positive cooperativity is the binding of <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> to <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a>. One <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> molecule can bind to the <a href="/wiki/Ferrous_iron" class="mw-redirect" title="Ferrous iron">ferrous iron</a> of a heme molecule in each of the four chains of a <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a> molecule. Deoxy-hemoglobin has a relatively low affinity for <a href="/wiki/Oxygen" title="Oxygen">oxygen</a>, but when one molecule binds to a single heme, the <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> affinity increases, allowing the second molecule to bind more easily, and the third and fourth even more easily. The <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> affinity of 3-oxy-hemoglobin is ~300 times greater than that of deoxy-hemoglobin. This behavior leads the affinity curve of <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a> to be <a href="/wiki/Sigmoid_function" title="Sigmoid function">sigmoidal</a>, rather than <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic</a> as with the monomeric <a href="/wiki/Myoglobin" title="Myoglobin">myoglobin</a>. By the same process, the ability for <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a> to lose <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> increases as fewer oxygen molecules are bound.<sup id="cite_ref-Whitford_2005_1-1" class="reference"><a href="#cite_note-Whitford_2005-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i>See also <a href="/wiki/Oxygen-hemoglobin_dissociation_curve" class="mw-redirect" title="Oxygen-hemoglobin dissociation curve">Oxygen-hemoglobin dissociation curve</a>.</i> </p><p>Negative cooperativity means that the opposite will be true; as <a href="/wiki/Ligand" title="Ligand">ligands</a> bind to the <a href="/wiki/Protein" title="Protein">protein</a>, the <a href="/wiki/Protein" title="Protein">protein</a>'s affinity for the ligand will decrease, i.e. it becomes less likely for the ligand to bind to the protein. An example of this occurring is the relationship between <a href="/wiki/Glyceraldehyde-3-phosphate" class="mw-redirect" title="Glyceraldehyde-3-phosphate">glyceraldehyde-3-phosphate</a> and the <a href="/wiki/Enzyme" title="Enzyme">enzyme</a> glyceraldehyde-3-phosphate dehydrogenase. </p><p><a href="/wiki/Allosteric_regulation#Types_of_allosteric_regulation" title="Allosteric regulation">Homotropic</a> cooperativity refers to the fact that the molecule causing the cooperativity is the one that will be affected by it. <a href="/wiki/Allosteric_regulation#Types_of_allosteric_regulation" title="Allosteric regulation">Heterotropic</a> cooperativity is where a third party substance causes the change in affinity. Homotropic or heterotropic cooperativity could be of both positives as well as negative types depend upon whether it support or oppose further binding of the ligand molecules to the enzymes.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Subunit_cooperativity">Subunit cooperativity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=2" title="Edit section: Subunit cooperativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cooperativity is not only a phenomenon of ligand binding, but also applies anytime energetic interactions make it easier or more difficult for something to happen involving multiple units as opposed to with single units. (That is, easier or more difficult compared with what is expected when only accounting for the addition of multiple units). For example, unwinding of <a href="/wiki/DNA" title="DNA">DNA</a> involves cooperativity: Portions of DNA must unwind in order for DNA to carry out <a href="/wiki/DNA_replication" title="DNA replication">replication</a>, <a href="/wiki/Transcription_(genetics)" class="mw-redirect" title="Transcription (genetics)">transcription</a> and <a href="/wiki/Genetic_recombination" title="Genetic recombination">recombination</a>. Positive cooperativity among adjacent DNA <a href="/wiki/Nucleotide" title="Nucleotide">nucleotides</a> makes it easier to unwind a whole group of adjacent nucleotides than it is to unwind the same number of nucleotides spread out along the DNA chain. The <a href="/w/index.php?title=Cooperative_unit_size&action=edit&redlink=1" class="new" title="Cooperative unit size (page does not exist)">cooperative unit size</a> is the number of adjacent bases that tend to unwind as a single unit due to the effects of positive cooperativity. This phenomenon applies to other types of chain molecules as well, such as the folding and unfolding of proteins and in the "melting" of <a href="/wiki/Phospholipid" title="Phospholipid">phospholipid</a> chains that make up the <a href="/wiki/Cell_membrane" title="Cell membrane">membranes of cells</a>. Subunit cooperativity is measured on the relative scale known as Hill's Constant. </p> <div class="mw-heading mw-heading2"><h2 id="Hill_equation">Hill equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=3" title="Edit section: Hill equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple and widely used model for molecular interactions is the <a href="/wiki/Hill_equation_(biochemistry)" title="Hill equation (biochemistry)">Hill equation</a>, which provides a way to quantify cooperative binding by describing the fraction of saturated ligand binding sites as a function of the ligand concentration. </p> <div class="mw-heading mw-heading2"><h2 id="Hill_coefficient">Hill coefficient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=4" title="Edit section: Hill coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hill coefficient is a measure of <a href="/wiki/Ultrasensitivity" title="Ultrasensitivity">ultrasensitivity</a> (i.e. how steep is the response curve). </p><p>From an operational point of view the Hill coefficient can be estimated as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{H}={\frac {\log(81)}{\ce {\log(EC90/EC10)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>81</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>90</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{H}={\frac {\log(81)}{\ce {\log(EC90/EC10)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115b287443bcaebca535736ce1bb1727be475314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.626ex; height:6.676ex;" alt="{\displaystyle n_{H}={\frac {\log(81)}{\ce {\log(EC90/EC10)}}}}"></span>.</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {EC90}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>90</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {EC90}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac72ae0769eec387497bb8e3259467d5a5862b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.137ex; height:2.843ex;" alt="{\displaystyle {\ce {EC90}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {EC10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {EC10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e810a14e19e77bc3315bcc51481a9e5f00bdacc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.137ex; height:2.843ex;" alt="{\displaystyle {\ce {EC10}}}"></span> are the input values needed to produce the 10% and 90% of the maximal response, respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Response_coefficient">Response coefficient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=5" title="Edit section: Response coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Global sensitivity measures such as the Hill coefficient do not characterise the local behaviours of the s-shaped curves. Instead, these features are well captured by the response coefficient measure <sup id="cite_ref-Kholodenko_1997_4-0" class="reference"><a href="#cite_note-Kholodenko_1997-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)={\frac {x}{y}}{\frac {dy}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)={\frac {x}{y}}{\frac {dy}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf85c13f5d0e15263138958dda32135e0834d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.549ex; height:5.843ex;" alt="{\displaystyle R(x)={\frac {x}{y}}{\frac {dy}{dx}}}"></span></dd></dl> <p>In <a href="/wiki/Systems_biology" title="Systems biology">systems biology</a>, such responses are referred to as <a href="/wiki/Elasticity_coefficient" title="Elasticity coefficient">elasticities</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Link_between_Hill_coefficient_and_response_coefficient">Link between Hill coefficient and response coefficient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=6" title="Edit section: Link between Hill coefficient and response coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Altszyler et al. (2017) have shown that these ultrasensitivity measures can be linked by the following equation:<sup id="cite_ref-Altszyler_2017_5-0" class="reference"><a href="#cite_note-Altszyler_2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{H}=2{\frac {\int _{\ce {\log(EC10)}}^{\ce {\log(EC90)}}R_{f}(I)d(\log I)}{\ce {\log(EC90)-\log(EC10)}}}=2\langle R_{f}\rangle _{\ce {EC10,EC90}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>90</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>90</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mo>,</mo> <msubsup> <mtext>EC</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>90</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{H}=2{\frac {\int _{\ce {\log(EC10)}}^{\ce {\log(EC90)}}R_{f}(I)d(\log I)}{\ce {\log(EC90)-\log(EC10)}}}=2\langle R_{f}\rangle _{\ce {EC10,EC90}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d343a0fa73530dcc27a74791288337f7c82d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:48.596ex; height:8.509ex;" alt="{\displaystyle n_{H}=2{\frac {\int _{\ce {\log(EC10)}}^{\ce {\log(EC90)}}R_{f}(I)d(\log I)}{\ce {\log(EC90)-\log(EC10)}}}=2\langle R_{f}\rangle _{\ce {EC10,EC90}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle X\rangle _{a,b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X\rangle _{a,b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564a7ca47f5288cf30c3cb82631b81c91d0173d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.054ex; height:3.009ex;" alt="{\displaystyle \langle X\rangle _{a,b}}"></span> denoted the mean value of the variable x over the range [a,b]. </p> <div class="mw-heading mw-heading2"><h2 id="Ultrasensitivity_in_function_composition">Ultrasensitivity in function composition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=7" title="Edit section: Ultrasensitivity in function composition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider two coupled ultrasensitive modules, disregarding effects of sequestration of molecular components between layers. In this case, the expression for the system's dose-response curve, <span class="texhtml mvar" style="font-style:italic;">F</span>, results from the mathematical composition of the functions, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span>, which describe the input/output relationship of isolated modules <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b3c5e448c465889913a88a105e38e7316fba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.26ex; height:2.509ex;" alt="{\displaystyle i=1,2}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(I_{1})=f_{2}{\big (}f_{1}(I_{1}){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(I_{1})=f_{2}{\big (}f_{1}(I_{1}){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668d6460ba24e4300b63c430501a997cc071898f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.129ex; height:3.176ex;" alt="{\displaystyle F(I_{1})=f_{2}{\big (}f_{1}(I_{1}){\big )}}"></span></dd></dl> <p>Brown et al. (1997) <sup id="cite_ref-brown1997_6-0" class="reference"><a href="#cite_note-brown1997-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Altszyler_2017_5-1" class="reference"><a href="#cite_note-Altszyler_2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> have shown that the local ultrasensitivity of the different layers combines multiplicatively: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=R_{2}(f_{1}(x)).R_{1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=R_{2}(f_{1}(x)).R_{1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abaf0a109af08abcbb3ebdcfe5f6f5b074148f27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.953ex; height:2.843ex;" alt="{\displaystyle R(x)=R_{2}(f_{1}(x)).R_{1}(x)}"></span>.</dd></dl> <p>In connection with this result, Ferrell et al. (1997) <sup id="cite_ref-ferrell1997_7-0" class="reference"><a href="#cite_note-ferrell1997-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> showed, for Hill-type modules, that the overall cascade global ultrasensitivity had to be less than or equal to the product of the global ultrasensitivity estimations of each cascade's layer,<sup id="cite_ref-Altszyler_2017_5-2" class="reference"><a href="#cite_note-Altszyler_2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq n_{1}.n_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq n_{1}.n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45963dc64b8d96c692d13eeff6ae3052c00b256e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.425ex; height:2.343ex;" alt="{\displaystyle n\leq n_{1}.n_{2}}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"></span> are the Hill coefficient of modules 1 and 2 respectively. </p><p>Altszyler et al. (2017) <sup id="cite_ref-Altszyler_2017_5-3" class="reference"><a href="#cite_note-Altszyler_2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> have shown that the cascade's global ultrasensitivity can be analytically calculated: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2\overbrace {\langle R_{2}\rangle _{X10_{2},X90_{2}}} ^{\nu _{2}}\overbrace {\langle R_{1}\rangle _{X10_{1},X90_{1}}} ^{\nu _{1}}=2\,\nu _{2}\,\nu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <msub> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>X</mi> <msub> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <msub> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>X</mi> <msub> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mover> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2\overbrace {\langle R_{2}\rangle _{X10_{2},X90_{2}}} ^{\nu _{2}}\overbrace {\langle R_{1}\rangle _{X10_{1},X90_{1}}} ^{\nu _{1}}=2\,\nu _{2}\,\nu _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ac3e6b65dd256fcbd8fddb2c620d0471cdf81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.233ex; height:6.009ex;" alt="{\displaystyle n=2\overbrace {\langle R_{2}\rangle _{X10_{2},X90_{2}}} ^{\nu _{2}}\overbrace {\langle R_{1}\rangle _{X10_{1},X90_{1}}} ^{\nu _{1}}=2\,\nu _{2}\,\nu _{1}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X10_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msub> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X10_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d2d0c5add320a75dd1a6f0cc4898703c2f6f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.105ex; height:2.509ex;" alt="{\displaystyle X10_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X90_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msub> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X90_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71973a99bfe6c935eda80a7298eeb0858def657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.105ex; height:2.509ex;" alt="{\displaystyle X90_{i}}"></span> delimited the Hill input's working range of the composite system, i.e. the input values for the i-layer so that the last layer (corresponding to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4e89f84cbdca577f73238abfb2fa18119c6eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=2}"></span> in this case) reached the 10% and 90% of it maximal output level. It followed this equation that the system's Hill coefficient <span class="texhtml mvar" style="font-style:italic;">n</span> could be written as the product of two factors, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ba9b02669b6a4569a4b4bd2159ffa91f459fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle \nu _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20157e640b69bcfc79a73194f1e80dbb456ab254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle \nu _{2}}"></span>, which characterized local average sensitivities over the relevant input region for each layer: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X10_{i},X90_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <msub> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <msub> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X10_{i},X90_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6a9f701cc2d5bd5754f6206af5ce8e2b644e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.537ex; height:2.843ex;" alt="{\displaystyle [X10_{i},X90_{i}]}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b3c5e448c465889913a88a105e38e7316fba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.26ex; height:2.509ex;" alt="{\displaystyle i=1,2}"></span> in this case. </p><p>For the more general case of a cascade of <span class="texhtml mvar" style="font-style:italic;">N</span> modules, the Hill coefficient can be expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\nu _{N}\,\nu _{N-1}...\nu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\nu _{N}\,\nu _{N-1}...\nu _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa148c07ef46bc40040cb5b0a989a195374ca95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.965ex; height:2.009ex;" alt="{\displaystyle n=\nu _{N}\,\nu _{N-1}...\nu _{1}}"></span>,</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Supramultiplicativity">Supramultiplicativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=8" title="Edit section: Supramultiplicativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several authors have reported the existence of supramultiplicative behavior in signaling cascades <sup id="cite_ref-altszylerUltrasens2014_8-0" class="reference"><a href="#cite_note-altszylerUltrasens2014-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-racz2008_9-0" class="reference"><a href="#cite_note-racz2008-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>(i.e. the ultrasensitivity of the combination of layers is higher than the product of individual ultrasensitivities), but in many cases the ultimate origin of supramultiplicativity remained elusive. Altszyler et al. (2017)<sup id="cite_ref-Altszyler_2017_5-4" class="reference"><a href="#cite_note-Altszyler_2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> framework naturally suggested a general scenario where supramultiplicative behavior could take place. This could occur when, for a given module, the corresponding Hill's input working range was located in an input region with local ultrasensitivities higher than the global ultrasensitivity of the respective dose-response curve. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cooperativity&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Whitford_2005-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Whitford_2005_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Whitford_2005_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWhitford2005" class="citation book cs1">Whitford D (2005). <i>Proteins: structure and function</i>. 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href="https://api.semanticscholar.org/CorpusID:19466336">19466336</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=FEBS+Letters&rft.atitle=Quantification+of+information+transfer+via+cellular+signal+transduction+pathways&rft.volume=414&rft.issue=2&rft.pages=430-4&rft.date=1997-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A19466336%23id-name%3DS2CID&rft_id=info%3Apmid%2F9315734&rft_id=info%3Adoi%2F10.1016%2FS0014-5793%2897%2901018-1&rft.aulast=Kholodenko&rft.aufirst=BN&rft.au=Hoek%2C+JB&rft.au=Westerhoff%2C+HV&rft.au=Brown%2C+GC&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0014-5793%252897%252901018-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACooperativity" class="Z3988"></span></span> </li> <li id="cite_note-Altszyler_2017-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Altszyler_2017_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Altszyler_2017_5-1"><sup><i><b>b</b></i></sup></a> <a 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"How responses get more switch-like as you move down a protein kinase cascade". <i>Trends in Biochemical Sciences</i>. <b>22</b> (8): 288–9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0968-0004%2897%2982217-7">10.1016/s0968-0004(97)82217-7</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9270299">9270299</a>.</cite><span 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href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4233326">"Impact of upstream and downstream constraints on a signaling module's ultrasensitivity"</a>. <i>Physical Biology</i>. <b>11</b> (6): 066003. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhBio..11f6003A">2014PhBio..11f6003A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1478-3975%2F11%2F6%2F066003">10.1088/1478-3975/11/6/066003</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4233326">4233326</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25313165">25313165</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Biology&rft.atitle=Impact+of+upstream+and+downstream+constraints+on+a+signaling+module%27s+ultrasensitivity&rft.volume=11&rft.issue=6&rft.pages=066003&rft.date=2014-10&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4233326%23id-name%3DPMC&rft_id=info%3Apmid%2F25313165&rft_id=info%3Adoi%2F10.1088%2F1478-3975%2F11%2F6%2F066003&rft_id=info%3Abibcode%2F2014PhBio..11f6003A&rft.aulast=Altszyler&rft.aufirst=E&rft.au=Ventura%2C+A&rft.au=Colman-Lerner%2C+A&rft.au=Chernomoretz%2C+A&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4233326&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACooperativity" class="Z3988"></span></span> </li> <li id="cite_note-racz2008-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-racz2008_9-0">^</a></b></span> 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title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2675913">2675913</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18663279">18663279</a>.</cite><span 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activator">Enzyme activator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Enzyme_Commission_number" title="Enzyme Commission number">EC number</a></li> <li><a href="/wiki/Protein_superfamily" title="Protein superfamily">Enzyme superfamily</a></li> <li><a href="/wiki/Protein_family" title="Protein family">Enzyme family</a></li> <li><a href="/wiki/List_of_enzymes" title="List of enzymes">List of enzymes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinetics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Enzyme_kinetics" title="Enzyme kinetics">Enzyme kinetics</a></li> <li><a href="/wiki/Eadie%E2%80%93Hofstee_diagram" title="Eadie–Hofstee diagram">Eadie–Hofstee diagram</a></li> <li><a href="/wiki/Hanes%E2%80%93Woolf_plot" title="Hanes–Woolf plot">Hanes–Woolf plot</a></li> <li><a href="/wiki/Lineweaver%E2%80%93Burk_plot" title="Lineweaver–Burk plot">Lineweaver–Burk plot</a></li> <li><a href="/wiki/Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><b>EC1 <a href="/wiki/Oxidoreductase" title="Oxidoreductase">Oxidoreductases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_1)" title="List of EC numbers (EC 1)">list</a>)</li> <li><b>EC2 <a href="/wiki/Transferase" title="Transferase">Transferases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_2)" title="List of EC numbers (EC 2)">list</a>)</li> <li><b>EC3 <a href="/wiki/Hydrolase" title="Hydrolase">Hydrolases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_3)" title="List of EC numbers (EC 3)">list</a>)</li> <li><b>EC4 <a href="/wiki/Lyase" title="Lyase">Lyases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_4)" title="List of EC numbers (EC 4)">list</a>)</li> <li><b>EC5 <a href="/wiki/Isomerase" title="Isomerase">Isomerases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_5)" title="List of EC numbers (EC 5)">list</a>)</li> <li><b>EC6 <a href="/wiki/Ligase" title="Ligase">Ligases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_6)" title="List of EC numbers (EC 6)">list</a>)</li> <li><b>EC7 <a href="/wiki/Translocase" title="Translocase">Translocases</a></b> (<a href="/wiki/List_of_EC_numbers_(EC_7)" title="List of EC numbers (EC 7)">list</a>)</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: 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