Hill equation (biochemistry)

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</nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the <b>Hill equation</b> as an equation used in biochemical characterization. For other uses, see <a href="/wiki/Hill_differential_equation" title="Hill differential equation">Hill differential equation</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hill-Langmuir_equation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Hill-Langmuir_equation.svg/310px-Hill-Langmuir_equation.svg.png" decoding="async" width="310" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Hill-Langmuir_equation.svg/465px-Hill-Langmuir_equation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Hill-Langmuir_equation.svg/620px-Hill-Langmuir_equation.svg.png 2x" data-file-width="450" data-file-height="270" /></a><figcaption>Binding curves showing the characteristically sigmoidal curves generated by using the Hill equation to model cooperative binding. Each curve corresponds to a different Hill coefficient, labeled to the curve's right. The vertical axis displays the proportion of the total number of receptors that have been bound by a ligand. The horizontal axis is the concentration of the ligand. As the Hill coefficient is increased, the saturation curve becomes steeper.</figcaption></figure> <p>In <a href="/wiki/Biochemistry" title="Biochemistry">biochemistry</a> and <a href="/wiki/Pharmacology" title="Pharmacology">pharmacology</a>, the <b>Hill equation</b> refers to two closely related equations that reflect the binding of <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligands</a> to macromolecules, as a function of the ligand <a href="/wiki/Concentration" title="Concentration">concentration</a>. A <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligand</a> is "a substance that forms a complex with a biomolecule to serve a biological purpose", and a <a href="/wiki/Macromolecule" title="Macromolecule">macromolecule</a> is a very large molecule, such as a protein, with a complex structure of components. Protein-ligand binding typically changes the structure of the target protein, thereby changing its function in a cell. </p><p>The distinction between the two Hill equations is whether they measure <i>occupancy</i> or <i>response</i>. The <b>Hill equation</b> reflects the occupancy of macromolecules: the fraction that is saturated or bound by the <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligand</a>.<sup id="cite_ref-Terms_1-0" class="reference"><a href="#cite_note-Terms-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gesztelyi2012_2-0" class="reference"><a href="#cite_note-Gesztelyi2012-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> This equation is formally equivalent to the <a href="/wiki/Langmuir_adsorption_model" title="Langmuir adsorption model">Langmuir isotherm</a>.<sup id="cite_ref-Langmuir1918_4-0" class="reference"><a href="#cite_note-Langmuir1918-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Conversely, the <b>Hill equation</b> proper reflects the cellular or tissue response to the ligand: the physiological output of the system, such as muscle contraction. </p><p>The Hill equation was originally formulated by <a href="/wiki/Archibald_Hill" title="Archibald Hill">Archibald Hill</a> in 1910 to describe the <a href="/wiki/Sigmoid_function" title="Sigmoid function">sigmoidal</a> O<sub>2</sub> binding curve of <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a>.<sup id="cite_ref-Hill1910_5-0" class="reference"><a href="#cite_note-Hill1910-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The binding of a <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligand</a> to a <a href="/wiki/Macromolecule" title="Macromolecule">macromolecule</a> is often enhanced if there are already other ligands present on the same macromolecule (this is known as <a href="/wiki/Cooperative_binding" title="Cooperative binding">cooperative binding</a>). The Hill equation is useful for determining the degree of <a href="/wiki/Cooperativity" title="Cooperativity">cooperativity</a> of the ligand(s) binding to the enzyme or receptor. The <b>Hill coefficient</b> provides a way to quantify the degree of interaction between ligand binding sites.<sup id="cite_ref-UsesMisuses_6-0" class="reference"><a href="#cite_note-UsesMisuses-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The Hill equation (for response) is important in the construction of <a href="/wiki/Dose-response_curves" class="mw-redirect" title="Dose-response curves">dose-response curves</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proportion_of_ligand-bound_receptors">Proportion of ligand-bound receptors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=1" title="Edit section: Proportion of ligand-bound receptors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png/220px-Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png/330px-Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png/440px-Plot_of_%25_saturation_of_O2_binding_to_haemoglobin_as_a_function_of_O2_pressure_-_from_1910_Hill_paper.png 2x" data-file-width="4200" data-file-height="2700" /></a><figcaption>Plot of the % saturation of oxygen binding to haemoglobin, as a function of the amount of oxygen present (expressed as an oxygen pressure). Data (red circles) and Hill equation fit (black curve) from original 1910 paper of Hill.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The Hill equation is commonly expressed in the following ways:<sup id="cite_ref-Gesztelyi2012_2-1" class="reference"><a href="#cite_note-Gesztelyi2012-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CoopBind2013_8-0" class="reference"><a href="#cite_note-CoopBind2013-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_9-0" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>8<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 {\begin{aligned}\theta &={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}\\&={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}\\&={1 \over 1+\left({K_{A} \over [{\ce {L}}]}\right)^{n}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta &={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}\\&={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}\\&={1 \over 1+\left({K_{A} \over [{\ce {L}}]}\right)^{n}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/386ff885fdf558b83ca54868e1c90d7a27431451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:19.047ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}\theta &={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}\\&={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}\\&={1 \over 1+\left({K_{A} \over [{\ce {L}}]}\right)^{n}}\end{aligned}}}" /></span>,</dd></dl> <p>where </p> <ul><li><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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> is the fraction of the <a href="/wiki/Receptor_(biochemistry)" title="Receptor (biochemistry)">receptor protein</a> concentration that is bound by the <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligand</a>,</li> <li><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 {[L]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mtext>L</mtext> <mo stretchy="false">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {[L]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d168a8fcf5a74047be127a23620e6c9a5534c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.746ex; height:2.843ex;" alt="{\displaystyle {\ce {[L]}}}" /></span>is the total <a href="/wiki/Ligand_(biochemistry)" title="Ligand (biochemistry)">ligand</a> concentration,</li> <li><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 K_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af467d41e25cebc9640cba0240778eaae4057181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.065ex; height:2.509ex;" alt="{\displaystyle K_{d}}" /></span> is the apparent <a href="/wiki/Dissociation_constant" title="Dissociation constant">dissociation constant</a> derived from the <a href="/wiki/Law_of_mass_action" title="Law of mass action">law of mass action</a>,</li> <li><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 K_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfa73ba3893b9a169d62f5d8f6164832963ccd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.438ex; height:2.509ex;" alt="{\displaystyle K_{A}}" /></span> is the ligand concentration producing half occupation,</li> <li><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is the Hill coefficient.</li></ul> <p>The special case 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"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </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/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" /></span> is a <a href="/wiki/Monod_equation" title="Monod equation">Monod equation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Constants">Constants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=2" title="Edit section: Constants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In pharmacology, <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> is often written as <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 p_{{\ce {AR}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>AR</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{{\ce {AR}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77baa9fff375e16275f7ac17e837af210895b422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.933ex; height:2.009ex;" alt="{\displaystyle p_{{\ce {AR}}}}" /></span>, 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 {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7683201425b7130b4f6bc52be92b6de7ccbf648f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\ce {A}}}" /></span> is the ligand, equivalent to L, 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 {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cc62e6f7df24645658eb13f4f5cbad4afaf967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.711ex; height:2.176ex;" alt="{\displaystyle {\ce {R}}}" /></span> is the receptor. <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> can be expressed in terms of the total amount of receptor and ligand-bound receptor concentrations: <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 \theta ={\frac {\ce {[LR]}}{\ce {[R_{\rm {total}}]}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mtext>LR</mtext> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msubsup> <mtext>R</mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em"></mspace> </mrow> </msubsup> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {\ce {[LR]}}{\ce {[R_{\rm {total}}]}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24813a2bd5f5bc70a7568ffb6743a803bac97e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.642ex; height:6.676ex;" alt="{\displaystyle \theta ={\frac {\ce {[LR]}}{\ce {[R_{\rm {total}}]}}}}" /></span>. <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 K_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af467d41e25cebc9640cba0240778eaae4057181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.065ex; height:2.509ex;" alt="{\displaystyle K_{d}}" /></span> is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate (<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="{\textstyle K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd5de6ee1ffa16911030bf295d6735697065ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.817ex; height:4.343ex;" alt="{\textstyle K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}" /></span>).<sup id="cite_ref-:0_9-1" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Kd is the equilibrium constant for dissociation. <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="{\textstyle K_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d30e746c11eaadc8aeb24d4c61e73ccb06e7166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.438ex; height:2.509ex;" alt="{\textstyle K_{A}}" /></span> is defined so that <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="{\textstyle (K_{A})^{n}=K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (K_{A})^{n}=K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b3eed5846a4a1205e21da05422c63da616c3c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.381ex; height:4.343ex;" alt="{\textstyle (K_{A})^{n}=K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}" /></span>, this is also known as the microscopic <a href="/wiki/Dissociation_constant" title="Dissociation constant">dissociation constant</a> and is the ligand concentration occupying half of the binding sites. In recent literature, this constant is sometimes referred to as <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="{\textstyle K_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37a60f3f3ecd03cc777067d4dba74085afc2c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.566ex; height:2.509ex;" alt="{\textstyle K_{D}}" /></span>.<sup id="cite_ref-:0_9-2" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Gaddum_equation">Gaddum equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=3" title="Edit section: Gaddum equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/John_Gaddum" title="John Gaddum">Gaddum</a> equation is a further generalisation of the Hill-equation, incorporating the presence of a reversible competitive antagonist.<sup id="cite_ref-Terms_1-1" class="reference"><a href="#cite_note-Terms-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The Gaddum equation is derived similarly to the Hill-equation but with 2 equilibria: both the ligand with the receptor and the antagonist with the receptor. Hence, the Gaddum equation has 2 constants: the equilibrium constants of the ligand and that of the antagonist </p> <div class="mw-heading mw-heading3"><h3 id="Hill_plot">Hill plot</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=4" title="Edit section: Hill plot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hill_Plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Hill_Plot.png/330px-Hill_Plot.png" decoding="async" width="260" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Hill_Plot.png/500px-Hill_Plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Hill_Plot.png/520px-Hill_Plot.png 2x" data-file-width="613" data-file-height="538" /></a><figcaption>A Hill plot, where the x-axis is the logarithm of the ligand concentration and the y-axis is the transformed receptor occupancy. X represents L and Y represents theta.</figcaption></figure> <p>The Hill plot is the rearrangement of the Hill equation into a straight line. </p><p>Taking the reciprocal of both sides of the Hill equation, rearranging, and inverting again yields: <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 {\theta \over 1-\theta }={[{\ce {L}}]^{n} \over K_{d}}={[{\ce {L}}]^{n} \over (K_{A})^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\theta \over 1-\theta }={[{\ce {L}}]^{n} \over K_{d}}={[{\ce {L}}]^{n} \over (K_{A})^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62619f678cb95dde3c47e61e2aac4f144f10357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.229ex; height:6.509ex;" alt="{\displaystyle {\theta \over 1-\theta }={[{\ce {L}}]^{n} \over K_{d}}={[{\ce {L}}]^{n} \over (K_{A})^{n}}}" /></span>. Taking the logarithm of both sides of the equation leads to an alternative formulation of the Hill-Langmuir equation: </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 {\begin{aligned}\log \left({\theta \over 1-\theta }\right)&=n\log {[{\ce {L}}]}-\log {K_{d}}\\&=n\log {[{\ce {L}}]}-n\log {K_{A}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>−</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log \left({\theta \over 1-\theta }\right)&=n\log {[{\ce {L}}]}-\log {K_{d}}\\&=n\log {[{\ce {L}}]}-n\log {K_{A}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e17bac0e7828741f0ce8943abbf129033f3fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.478ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\log \left({\theta \over 1-\theta }\right)&=n\log {[{\ce {L}}]}-\log {K_{d}}\\&=n\log {[{\ce {L}}]}-n\log {K_{A}}\end{aligned}}}" /></span>.</dd></dl> <p>This last form of the Hill equation is advantageous because a plot of <b><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="{\textstyle \log \left({\theta \over 1-\theta }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log \left({\theta \over 1-\theta }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53cfdd7330683c601bf6a40ecd4046768e9660d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.455ex; height:4.843ex;" alt="{\textstyle \log \left({\theta \over 1-\theta }\right)}" /></span></b> versus <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 \log {[{\ce {L}}]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log {[{\ce {L}}]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44efcd14530f15374d04ede7e01db63da3ac6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.105ex; height:2.843ex;" alt="{\displaystyle \log {[{\ce {L}}]}}" /></span> yields a <a href="/wiki/Linear_equation" title="Linear equation">linear plot</a>, which is called a <b>Hill plot.</b><sup id="cite_ref-CoopBind2013_8-1" class="reference"><a href="#cite_note-CoopBind2013-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_9-3" class="reference"><a href="#cite_note-:0-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Because the slope of a Hill plot is equal to the Hill coefficient for the biochemical interaction, the slope is denoted by <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc0ce2be9bd5c14d06c294088b3b2e8a050d8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.086ex; height:2.009ex;" alt="{\displaystyle n_{H}}" /></span>. A slope greater than one thus indicates positively cooperative binding between the receptor and the ligand, while a slope less than one indicates negatively cooperative binding. </p><p>Transformations of equations into linear forms such as this were very useful before the widespread use of computers, as they allowed researchers to determine parameters by fitting lines to data. However, these transformations affect error propagation, and this may result in undue weight to error in data points near 0 or 1.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> This impacts the parameters of linear regression lines fitted to the data. Furthermore, the use of computers enables more robust analysis involving <a href="/wiki/Nonlinear_regression" title="Nonlinear regression">nonlinear regression</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Tissue_response">Tissue response</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=5" title="Edit section: Tissue response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dose_response_antagonist.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Dose_response_antagonist.jpg/440px-Dose_response_antagonist.jpg" decoding="async" width="440" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d2/Dose_response_antagonist.jpg 1.5x" data-file-width="587" data-file-height="279" /></a><figcaption>A trio of dose response curves</figcaption></figure> <p>A distinction should be made between quantification of drugs binding to receptors and drugs producing responses. There may not necessarily be a linear relationship between the two values. In contrast to this article's previous definition of the Hill equation, the <a href="/wiki/IUPHAR" class="mw-redirect" title="IUPHAR">IUPHAR</a> defines the Hill equation in terms of the tissue response <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 (E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ad56f168aab34124adbda79b9164eaacca8373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.585ex; height:2.843ex;" alt="{\displaystyle (E)}" /></span>, as<sup id="cite_ref-Terms_1-2" class="reference"><a href="#cite_note-Terms-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {E}{E_{\mathrm {max} }}}&={\frac {[A]^{n}}{{\text{EC}}_{50}^{n}+[A]^{n}}}\\&={\frac {1}{1+\left({\frac {{\text{EC}}_{50}}{[A]}}\right)^{n}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>A</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>EC</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">[</mo> <mi>A</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>EC</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> <mrow> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {E}{E_{\mathrm {max} }}}&={\frac {[A]^{n}}{{\text{EC}}_{50}^{n}+[A]^{n}}}\\&={\frac {1}{1+\left({\frac {{\text{EC}}_{50}}{[A]}}\right)^{n}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59f5002b0fc98cc03c0931e09ee8c937da865a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:23.166ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {E}{E_{\mathrm {max} }}}&={\frac {[A]^{n}}{{\text{EC}}_{50}^{n}+[A]^{n}}}\\&={\frac {1}{1+\left({\frac {{\text{EC}}_{50}}{[A]}}\right)^{n}}}\end{aligned}}}" /></span> 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 {[A]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mtext>A</mtext> <mo stretchy="false">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {[A]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881146b6653b24508d87e34a81c84832f1d5ffea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle {\ce {[A]}}}" /></span> is the drug concentration, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is the Hill coefficient, and <a href="/wiki/EC50" title="EC50"><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 {\text{EC}}_{50}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>EC</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{EC}}_{50}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c08ae0721895a790ec1a28f3cf456685bbc571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.137ex; height:2.509ex;" alt="{\displaystyle {\text{EC}}_{50}}" /></span></a> is the drug concentration that produces a 50% maximal response. Dissociation constants (in the previous section) relate to ligand binding, while <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 {\text{EC}}_{50}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>EC</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{EC}}_{50}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c08ae0721895a790ec1a28f3cf456685bbc571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.137ex; height:2.509ex;" alt="{\displaystyle {\text{EC}}_{50}}" /></span> reflects tissue response. </p><p>This form of the equation can reflect tissue/cell/population responses to drugs and can be used to generate <a href="/wiki/Dose_response_curves" class="mw-redirect" title="Dose response curves">dose response curves</a>. The relationship between <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 K_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af467d41e25cebc9640cba0240778eaae4057181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.065ex; height:2.509ex;" alt="{\displaystyle K_{d}}" /></span> and EC50 may be quite complex as a biological response will be the sum of myriad factors; a drug will have a different biological effect if more receptors are present, regardless of its affinity. </p><p>The Del-Castillo Katz model is used to relate the Hill equation to receptor activation by including a second equilibrium of the ligand-bound receptor to an <i>activated</i> form of the ligand-bound receptor. </p><p>Statistical analysis of response as a function of stimulus may be performed by regression methods such as the <a href="/wiki/Probit_model" title="Probit model">probit model</a> or <a href="/wiki/Logit_model" class="mw-redirect" title="Logit model">logit model</a>, or other methods such as the <a href="/w/index.php?title=Spearman%E2%80%93K%C3%A4rber_method&action=edit&redlink=1" class="new" title="Spearman–Kärber method (page does not exist)">Spearman–Kärber method</a>.<sup id="cite_ref-HamiltonRusso1977_11-0" class="reference"><a href="#cite_note-HamiltonRusso1977-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the dose-response relationship.<sup id="cite_ref-BatesWatts1988_12-0" class="reference"><a href="#cite_note-BatesWatts1988-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </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=Hill_equation_(biochemistry)&action=edit&section=6" 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>The Hill coefficient, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> or <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc0ce2be9bd5c14d06c294088b3b2e8a050d8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.086ex; height:2.009ex;" alt="{\displaystyle n_{H}}" /></span>, may describe cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used). When appropriate,<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="when is it appropriate? (April 2019)">clarification needed</span></a></i>]</sup> the value of the Hill coefficient describes the cooperativity of ligand binding in the following way: </p> <ul><li><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"> <mi>n</mi> <mo>></mo> <mn>1</mn> </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/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" /></span>. <b>Positively cooperative binding</b>: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases. For example, the Hill coefficient of oxygen binding to <a href="/wiki/Haemoglobin" class="mw-redirect" title="Haemoglobin">haemoglobin</a> (an example of positive cooperativity) falls within the range of 1.7–3.2.<sup id="cite_ref-UsesMisuses_6-1" class="reference"><a href="#cite_note-UsesMisuses-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><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"> <mi>n</mi> <mo><</mo> <mn>1</mn> </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/dffd3f7a2c3b20dc8e615dfd85f820287a8fc06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<1}" /></span>. <b>Negatively cooperative binding</b>: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.</li> <li><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"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </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/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" /></span>. <b>Noncooperative (completely independent) binding</b>: The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound. When n=1, we obtain a model that can be modeled by <a href="/wiki/Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a>,<sup id="cite_ref-Alon2007_13-0" class="reference"><a href="#cite_note-Alon2007-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> in which <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="{\textstyle K_{D}=K_{A}=K_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K_{D}=K_{A}=K_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2dc9d8e80a31b18bd4b863fdb7a6cf485bdbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.133ex; height:2.509ex;" alt="{\textstyle K_{D}=K_{A}=K_{M}}" /></span>, the <a href="/wiki/Michaelis%E2%80%93Menten_constant" class="mw-redirect" title="Michaelis–Menten constant">Michaelis–Menten constant</a>.</li></ul> <p>The Hill coefficient can be calculated approximately in terms of the <i>cooperativity index</i> of Taketa and Pogell<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> as follows:<sup id="cite_ref-altszylerUltrasens2017_15-0" class="reference"><a href="#cite_note-altszylerUltrasens2017-15"><span class="cite-bracket">[</span>13<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={\frac {\log _{10}(81)}{\log _{10}({\ce {EC90}}/{\ce {EC10}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>81</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <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"></mspace> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <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"></mspace> </mrow> </msubsup> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {\log _{10}(81)}{\log _{10}({\ce {EC90}}/{\ce {EC10}})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa698cfb48c145fd373445680b4042d8f6308a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.423ex; height:6.676ex;" alt="{\displaystyle n={\frac {\log _{10}(81)}{\log _{10}({\ce {EC90}}/{\ce {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"></mspace> </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"></mspace> </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><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Reversible_form">Reversible form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=7" title="Edit section: Reversible form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most common form of the Hill equation is its irreversible form. However, when building computational models a reversible form is often required in order to model product inhibition. For this reason, Hofmeyr and Cornish-Bowden devised the <a href="/wiki/Reversible_Hill_Equation" class="mw-redirect" title="Reversible Hill Equation">reversible Hill equation</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_the_elasticity_coefficients">Relationship to the elasticity coefficients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=8" title="Edit section: Relationship to the elasticity coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hill coefficient is also intimately connected to the <a href="/wiki/Elasticity_coefficient" title="Elasticity coefficient">elasticity coefficient</a> where the Hill coefficient can be shown to equal: </p><p><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=\varepsilon _{s}^{v}{\frac {1}{1-\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\varepsilon _{s}^{v}{\frac {1}{1-\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814e5e7cf5b44a56c309c3c18f11285145f6ecc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.536ex; height:5.509ex;" alt="{\displaystyle n=\varepsilon _{s}^{v}{\frac {1}{1-\theta }}}" /></span> </p><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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> is the fractional saturation, <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 ES/E_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ES/E_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8636a26b602b5d3f35656d0ae3b56dd977301093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.978ex; height:2.843ex;" alt="{\displaystyle ES/E_{t}}" /></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 \varepsilon _{s}^{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{s}^{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f114e9578dabbb7a7b367b5bda6401998856c8bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.113ex; height:2.509ex;" alt="{\displaystyle \varepsilon _{s}^{v}}" /></span> the elasticity coefficient. </p><p>This is derived by taking the slope of the Hill equation: </p><p><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={\frac {d\log {\frac {\theta }{1-\theta }}}{d\log s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mi>d</mi> <mi>log</mi> <mo>⁡</mo> <mi>s</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {d\log {\frac {\theta }{1-\theta }}}{d\log s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2684a47b321d0a5830a9b6b34eb2c74ec3878c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.999ex; height:7.176ex;" alt="{\displaystyle n={\frac {d\log {\frac {\theta }{1-\theta }}}{d\log s}}}" /></span> </p><p>and expanding the slope using the quotient rule. The result shows that the elasticity can never exceed <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> since the equation above can be rearranged to: </p><p><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 \varepsilon _{s}^{v}=n(1-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msubsup> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{s}^{v}=n(1-\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbb64e865adeec9063f1c79cec19612588ab75a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.509ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{s}^{v}=n(1-\theta )}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hill equation is used extensively in pharmacology to quantify the functional parameters of a drug<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2019)">citation needed</span></a></i>]</sup> and are also used in other areas of biochemistry. </p><p>The Hill equation can be used to describe dose-response relationships, for example <a href="/wiki/Ion_channel" title="Ion channel">ion channel</a> open-probability (P-open) vs. ligand concentration.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Regulation_of_gene_transcription">Regulation of gene transcription</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=10" title="Edit section: Regulation of gene transcription"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hill equation can be applied in modelling the rate at which a gene product is produced when its parent gene is being regulated by <a href="/wiki/Transcription_factor" title="Transcription factor">transcription factors</a> (e.g., <a href="/wiki/Activator_(genetics)" title="Activator (genetics)">activators</a> and/or <a href="/wiki/Repressor" title="Repressor">repressors</a>).<sup id="cite_ref-Alon2007_13-1" class="reference"><a href="#cite_note-Alon2007-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Doing so is appropriate when a gene is regulated by multiple binding sites for transcription factors, in which case the transcription factors may bind the DNA in a cooperative fashion.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>If the production of protein from gene <span class="texhtml mvar" style="font-style:italic;">X</span> is up-regulated (<b>activated</b>) by a transcription factor <span class="texhtml mvar" style="font-style:italic;">Y</span>, then the rate of production of protein <span class="texhtml mvar" style="font-style:italic;">X</span> can be modeled as a differential equation in terms of the concentration of activated <span class="texhtml mvar" style="font-style:italic;">Y</span> protein: </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 {\mathrm {d} \over \mathrm {d} t}[{\rm {X_{produced}}}]=k\ \cdot {{[{\rm {Y_{active}}}]^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Y_{active}}}]^{\mathit {n}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>k</mi> <mtext> </mtext> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} \over \mathrm {d} t}[{\rm {X_{produced}}}]=k\ \cdot {{[{\rm {Y_{active}}}]^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Y_{active}}}]^{\mathit {n}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8f1d4a7e5379a47f17d14efde6a9c497496bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.365ex; height:6.509ex;" alt="{\displaystyle {\mathrm {d} \over \mathrm {d} t}[{\rm {X_{produced}}}]=k\ \cdot {{[{\rm {Y_{active}}}]^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Y_{active}}}]^{\mathit {n}}}}}}" /></span>,</dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">k</span> is the maximal transcription rate of gene <span class="texhtml mvar" style="font-style:italic;">X</span>. </p><p>Likewise, if the production of protein from gene <span class="texhtml mvar" style="font-style:italic;">Y</span> is down-regulated (<b>repressed</b>) by a transcription factor <span class="texhtml mvar" style="font-style:italic;">Z</span>, then the rate of production of protein <span class="texhtml mvar" style="font-style:italic;">Y</span> can be modeled as a differential equation in terms of the concentration of activated <span class="texhtml mvar" style="font-style:italic;">Z</span> protein: </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 {\mathrm {d} \over \mathrm {d} t}[{\rm {Y_{produced}}}]=k\ \cdot {{(K_{A})^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Z_{active}}}]^{\mathit {n}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>k</mi> <mtext> </mtext> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} \over \mathrm {d} t}[{\rm {Y_{produced}}}]=k\ \cdot {{(K_{A})^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Z_{active}}}]^{\mathit {n}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fca0f3452f5c99014402ed5ed8f04e7286e4be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.042ex; height:6.509ex;" alt="{\displaystyle {\mathrm {d} \over \mathrm {d} t}[{\rm {Y_{produced}}}]=k\ \cdot {{(K_{A})^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Z_{active}}}]^{\mathit {n}}}}}}" /></span>,</dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">k</span> is the maximal transcription rate of gene <span class="texhtml mvar" style="font-style:italic;">Y</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Limitations">Limitations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=11" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because of its assumption that ligand molecules bind to a receptor simultaneously, the Hill equation has been criticized as a physically unrealistic model.<sup id="cite_ref-UsesMisuses_6-2" class="reference"><a href="#cite_note-UsesMisuses-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Moreover, the Hill coefficient should not be considered a reliable approximation of the number of cooperative ligand binding sites on a receptor<sup id="cite_ref-UsesMisuses_6-3" class="reference"><a href="#cite_note-UsesMisuses-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Monod_1965_19-0" class="reference"><a href="#cite_note-Monod_1965-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> except when the binding of the first and subsequent ligands results in extreme positive cooperativity.<sup id="cite_ref-UsesMisuses_6-4" class="reference"><a href="#cite_note-UsesMisuses-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Unlike more complex models, the relatively simple Hill equation provides little insight into underlying physiological mechanisms of protein-ligand interactions. This simplicity, however, is what makes the Hill equation a useful empirical model, since its use requires little <i>a priori</i> knowledge about the properties of either the protein or ligand being studied.<sup id="cite_ref-Gesztelyi2012_2-2" class="reference"><a href="#cite_note-Gesztelyi2012-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Nevertheless, other, more complex models of cooperative binding have been proposed.<sup id="cite_ref-CoopBind2013_8-2" class="reference"><a href="#cite_note-CoopBind2013-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For more information and examples of such models, see <a href="/wiki/Cooperative_binding" title="Cooperative binding">Cooperative binding</a>. </p><p>Global sensitivity measure such as 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-Kholodenko1997_20-0" class="reference"><a href="#cite_note-Kholodenko1997-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>There is a link between Hill Coefficient and Response coefficient, as follows. Altszyler et al. (2017) have shown that these ultrasensitivity measures can be linked.<sup id="cite_ref-altszylerUltrasens2017_15-1" class="reference"><a href="#cite_note-altszylerUltrasens2017-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Binding_coefficient" title="Binding coefficient">Binding coefficient</a></li> <li><a href="/wiki/Bjerrum_plot" title="Bjerrum plot">Bjerrum plot</a></li> <li><a href="/wiki/Cooperative_binding" title="Cooperative binding">Cooperative binding</a></li> <li><a href="/wiki/Gompertz_curve" class="mw-redirect" title="Gompertz curve">Gompertz curve</a></li> <li><a href="/wiki/Langmuir_adsorption_model" title="Langmuir adsorption model">Langmuir adsorption model</a></li> <li><a href="/wiki/Logistic_function" title="Logistic function">Logistic function</a></li> <li><a href="/wiki/Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a></li> <li><a href="/wiki/Monod_equation" title="Monod equation">Monod equation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=13" title="Edit section: Notes"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">For clarity, this article will use the <a href="/wiki/International_Union_of_Basic_and_Clinical_Pharmacology" title="International Union of Basic and Clinical Pharmacology">International Union of Basic and Clinical Pharmacology</a> convention of distinguishing between the Hill-Langmuir equation (for receptor saturation) and Hill equation (for tissue response)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">See <a href="/wiki/Propagation_of_uncertainty" title="Propagation of uncertainty">Propagation of uncertainty</a>. The function <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(\theta )=\log _{10}\left({\frac {\theta }{1-\theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=\log _{10}\left({\frac {\theta }{1-\theta }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb26ca9fd1810d739599bfe7e4a918d8ee12ab29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.475ex; height:6.176ex;" alt="{\displaystyle f(\theta )=\log _{10}\left({\frac {\theta }{1-\theta }}\right)}" /></span> propagates errors in <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> as <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 \delta _{f}=\delta _{\theta }{\frac {\mathrm {d} f}{\mathrm {d} \theta }}={\frac {\delta _{\theta }}{(\ln 10)\,\theta (1-\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{f}=\delta _{\theta }{\frac {\mathrm {d} f}{\mathrm {d} \theta }}={\frac {\delta _{\theta }}{(\ln 10)\,\theta (1-\theta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d03386b8499e0e1a6720f4f8ad1d1dc6866e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.486ex; height:6.176ex;" alt="{\displaystyle \delta _{f}=\delta _{\theta }{\frac {\mathrm {d} f}{\mathrm {d} \theta }}={\frac {\delta _{\theta }}{(\ln 10)\,\theta (1-\theta )}}}" /></span>. Hence errors in values of <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> near <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /></span> or <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /></span> are given far more weight than those for <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 \theta \approx 0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd2199ffbac37120309db096375fa5fb90403db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.161ex; height:2.176ex;" alt="{\displaystyle \theta \approx 0.5}" /></span></span> </li> </ol></div></div> <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=Hill_equation_(biochemistry)&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Terms-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Terms_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Terms_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Terms_1-2"><sup><i><b>c</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="CITEREFNeubig2003" class="citation journal cs1">Neubig, Richard R. 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Biol. Chem.">J. Biol. Chem.</a></i> <b>245</b> (23): <span class="nowrap">6335–</span>6. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9258%2818%2962614-6">10.1016/S0021-9258(18)62614-6</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/5484812">5484812</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Biol.+Chem.&rft.atitle=Analysis+of+Hill+interaction+coefficients+and+the+invalidity+of+the+Kwon+and+Brown+equation&rft.volume=245&rft.issue=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6335-%3C%2Fspan%3E6&rft.date=1970-12&rft_id=info%3Adoi%2F10.1016%2FS0021-9258%2818%2962614-6&rft_id=info%3Apmid%2F5484812&rft.aulast=Coval&rft.aufirst=ML&rft_id=http%3A%2F%2Fwww.jbc.org%2Fcontent%2F245%2F23%2F6335&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFd'A_Heck1971" class="citation journal cs1">d'A Heck, Henry (1971). "Statistical theory of cooperative binding to proteins. Hill equation and the binding potential". <i>J. Am. Chem. Soc</i>. <b>93</b> (1): <span class="nowrap">23–</span>29. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fja00730a004">10.1021/ja00730a004</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/5538860">5538860</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Am.+Chem.+Soc.&rft.atitle=Statistical+theory+of+cooperative+binding+to+proteins.+Hill+equation+and+the+binding+potential&rft.volume=93&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E29&rft.date=1971&rft_id=info%3Adoi%2F10.1021%2Fja00730a004&rft_id=info%3Apmid%2F5538860&rft.aulast=d%27A+Heck&rft.aufirst=Henry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAtkins1973" class="citation journal cs1">Atkins, Gordon L. (1973). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1432-1033.1973.tb02667.x">"A simple digital-computer program for estimating the parameter of the Hill Equation"</a>. <i>Eur. J. Biochem</i>. <b>33</b> (1): <span class="nowrap">175–</span>180. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1432-1033.1973.tb02667.x">10.1111/j.1432-1033.1973.tb02667.x</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/4691349">4691349</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Eur.+J.+Biochem.&rft.atitle=A+simple+digital-computer+program+for+estimating+the+parameter+of+the+Hill+Equation&rft.volume=33&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E175-%3C%2Fspan%3E180&rft.date=1973&rft_id=info%3Adoi%2F10.1111%2Fj.1432-1033.1973.tb02667.x&rft_id=info%3Apmid%2F4691349&rft.aulast=Atkins&rft.aufirst=Gordon+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1432-1033.1973.tb02667.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEndrenyiKwongFajszi1975" class="citation journal cs1">Endrenyi, Laszlo; Kwong, F. H. F.; Fajszi, Csaba (1975). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1432-1033.1975.tb03931.x">"Evaluation of Hill slopes and Hill coefficients when the saturation binding or velocity is not known"</a>. <i>Eur. J. Biochem</i>. <b>51</b> (2): <span class="nowrap">317–</span>328. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1432-1033.1975.tb03931.x">10.1111/j.1432-1033.1975.tb03931.x</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/1149734">1149734</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Eur.+J.+Biochem.&rft.atitle=Evaluation+of+Hill+slopes+and+Hill+coefficients+when+the+saturation+binding+or+velocity+is+not+known&rft.volume=51&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E317-%3C%2Fspan%3E328&rft.date=1975&rft_id=info%3Adoi%2F10.1111%2Fj.1432-1033.1975.tb03931.x&rft_id=info%3Apmid%2F1149734&rft.aulast=Endrenyi&rft.aufirst=Laszlo&rft.au=Kwong%2C+F.+H.+F.&rft.au=Fajszi%2C+Csaba&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1432-1033.1975.tb03931.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVoetVoet2004" class="citation book cs1">Voet, Donald; Voet, Judith G. (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/biochemistry00voet_1"><i>Biochemistry</i></a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Biochemistry&rft.date=2004&rft.aulast=Voet&rft.aufirst=Donald&rft.au=Voet%2C+Judith+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbiochemistry00voet_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeiss1997" class="citation journal cs1">Weiss, J. N. (1997). <a rel="nofollow" class="external text" href="http://www.fasebj.org/content/11/11/835.short">"The Hill equation revisited: uses and misuses"</a>. <i>FASEB Journal</i>. <b>11</b> (11): <span class="nowrap">835–</span>841. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1096%2Ffasebj.11.11.9285481">10.1096/fasebj.11.11.9285481</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/9285481">9285481</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:827335">827335</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=FASEB+Journal&rft.atitle=The+Hill+equation+revisited%3A+uses+and+misuses&rft.volume=11&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E835-%3C%2Fspan%3E841&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A827335%23id-name%3DS2CID&rft_id=info%3Apmid%2F9285481&rft_id=info%3Adoi%2F10.1096%2Ffasebj.11.11.9285481&rft.aulast=Weiss&rft.aufirst=J.+N.&rft_id=http%3A%2F%2Fwww.fasebj.org%2Fcontent%2F11%2F11%2F835.short&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKurganovLobanov2001" class="citation journal cs1">Kurganov, B. 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"The quantitative analysis of drug-receptor interactions: a short history". <i>Trends Pharmacol Sci</i>. <b>27</b> (3): <span class="nowrap">149–</span>57. <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%2Fj.tips.2006.01.008">10.1016/j.tips.2006.01.008</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/16483674">16483674</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trends+Pharmacol+Sci&rft.atitle=The+quantitative+analysis+of+drug-receptor+interactions%3A+a+short+history&rft.volume=27&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E149-%3C%2Fspan%3E57&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.tips.2006.01.008&rft_id=info%3Apmid%2F16483674&rft.au=Colquhoun+D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRang_HP2006" class="citation journal cs1">Rang HP (2006). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1760743">"The receptor concept: pharmacology's big idea"</a>. <i>Br J Pharmacol</i>. <b>147</b> (Suppl 1): S9–16. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsj.bjp.0706457">10.1038/sj.bjp.0706457</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/PMC1760743">1760743</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/16402126">16402126</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Br+J+Pharmacol&rft.atitle=The+receptor+concept%3A+pharmacology%27s+big+idea&rft.volume=147&rft.issue=Suppl+1&rft.pages=S9-16&rft.date=2006&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1760743%23id-name%3DPMC&rft_id=info%3Apmid%2F16402126&rft_id=info%3Adoi%2F10.1038%2Fsj.bjp.0706457&rft.au=Rang+HP&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1760743&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHill+equation+%28biochemistry%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hill_equation_(biochemistry)&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.fxsolver.com/browse/formulas/Hill+equation">Hill equation calculator</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Hill equation (biochemistry) - Wikipedia