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{{short description|Diagram showing the proportion of a receptor bound to a ligand}}
{{about|the '''Hill equation''' as an equation used in biochemical characterization||Hill differential equation}}
[[File:Hill-Langmuir equation.svg|thumb|upright=1.4|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.]]
In [[biochemistry]] and [[pharmacology]], the '''Hill equation''' refers to two closely related equations that reflect the binding of [[Ligand (biochemistry)|ligands]] to macromolecules, as a function of the ligand [[concentration]]. A [[Ligand (biochemistry)|ligand]] is "a substance that forms a complex with a biomolecule to serve a biological purpose", and a [[macromolecule]] 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.

The distinction between the two Hill equations is whether they measure ''occupancy'' or ''response''. The '''Hill equation''' reflects the occupancy of macromolecules: the fraction that is saturated or bound by the [[Ligand (biochemistry)|ligand]].<ref name="Terms">{{cite journal|title=International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification. XXXVIII. Update on Terms and Symbols in Quantitative Pharmacology|url=https://www.guidetopharmacology.org/pdfs/termsAndSymbols.pdf|last1=Neubig|first1=Richard R.|journal=Pharmacological Reviews|year=2003|volume=55|issue=4|pages=597–606|doi=10.1124/pr.55.4.4|pmid=14657418|s2cid=1729572}}</ref><ref name="Gesztelyi2012">{{cite journal|last1=Gesztelyi|first1=Rudolf|last2=Zsuga|first2=Judit|last3=Kemeny-Beke|first3=Adam|last4=Varga|first4=Balazs|last5=Juhasz|first5=Bela|last6=Tosaki|first6=Arpad|date=31 March 2012|title=The Hill equation and the origin of quantitative pharmacology|journal=Archive for History of Exact Sciences|language=en|volume=66|issue=4|pages=427–438|doi=10.1007/s00407-012-0098-5|s2cid=122929930|issn=0003-9519}}</ref><ref group = nb>For clarity, this article will use the [[International Union of Basic and Clinical Pharmacology]] convention of distinguishing between the Hill-Langmuir equation (for receptor saturation) and Hill equation (for tissue response)</ref> This equation is formally equivalent to the [[Langmuir adsorption model|Langmuir isotherm]].<ref name="Langmuir1918">{{cite journal|last1=Langmuir|first1=Irving|author-link=Irving Langmuir|date=1918|title=The adsorption of gases on plane surfaces of glass, mica and platinum.|journal=Journal of the American Chemical Society|volume=40|issue=9|pages=1361–1403|doi=10.1021/ja02242a004|url=https://zenodo.org/record/1429050}}</ref> Conversely, the '''Hill equation''' proper reflects the cellular or tissue response to the ligand: the physiological output of the system, such as muscle contraction.

The Hill equation was originally formulated by [[Archibald Hill]] in 1910 to describe the [[Sigmoid function|sigmoidal]] O<sub>2</sub> binding curve of [[hemoglobin]].<ref name="Hill1910">{{cite journal |last=Hill |first=A. V. |author-link=Archibald Hill |date=1910-01-22 |title=The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves |url=http://ci.nii.ac.jp/naid/10008305829/ |journal=[[J. Physiol.]] |volume=40 |issue=Suppl |pages=iv–vii |doi=10.1113/jphysiol.1910.sp001386 |s2cid=222195613}}</ref>

The binding of a [[ligand (biochemistry)|ligand]] to a [[macromolecule]] is often enhanced if there are already other ligands present on the same macromolecule (this is known as [[cooperative binding]]). The Hill equation is useful for determining the degree of [[cooperativity]] of the ligand(s) binding to the enzyme or receptor. The '''Hill coefficient''' provides a way to quantify the degree of interaction between ligand binding sites.<ref name="UsesMisuses">{{cite journal|date=1 September 1997|title=The Hill equation revisited: uses and misuses.|url=http://www.fasebj.org/content/11/11/835.short|journal=The FASEB Journal|language=en|volume=11|issue=11|pages=835–841|doi=10.1096/fasebj.11.11.9285481|issn=0892-6638|last1=Weiss|first1=J. N.|doi-access=free |pmid=9285481|s2cid=827335}}</ref>

The Hill equation (for response) is important in the construction of [[dose-response curves]].

== Proportion of ligand-bound receptors ==
[[File:Plot of % saturation of O2 binding to haemoglobin as a function of O2 pressure - from 1910 Hill paper.png|alt=|thumb|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.<ref>{{Cite journal|date=1910|title=Proceedings of the Physiological Society: January 22, 1910|journal=The Journal of Physiology|language=en|volume=40|issue=suppl|pages=i–vii|doi=10.1113/jphysiol.1910.sp001386|s2cid=222195613|issn=1469-7793}}</ref>]]
The Hill equation is commonly expressed in the following ways:<ref name="Gesztelyi2012" /><ref name="CoopBind2013">{{cite journal|last1=Stefan|first1=Melanie I.|last2=Novère|first2=Nicolas Le| title=Cooperative Binding|journal=PLOS Computational Biology|date=27 June 2013| volume=9|issue=6|pages=e1003106| doi=10.1371/journal.pcbi.1003106| issn=1553-7358|pmid=23843752| pmc=3699289|bibcode=2013PLSCB...9E3106S |doi-access=free }}</ref><ref name=":0" />

:<math chem=""> \begin{align}\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{align} </math>,
where
*<math> \theta </math> is the fraction of the [[receptor (biochemistry)|receptor protein]] concentration that is bound by the [[Ligand (biochemistry)|ligand]],
*<chem>[L]</chem>is the total [[Ligand (biochemistry)|ligand]] concentration,
*<math>K_d</math> is the apparent [[dissociation constant]] derived from the [[law of mass action]],
*<math>K_A</math> is the ligand concentration producing half occupation,
*<math>n</math> is the Hill coefficient.

The special case where <math>n=1 </math> is a [[Monod equation]].

=== Constants ===
In pharmacology, <math>\theta</math> is often written as <math chem>p_\ce{AR}</math>, where <chem>A</chem> is the ligand, equivalent to L, and <chem>R</chem> is the receptor. <math>\theta</math> can be expressed in terms of the total amount of receptor and ligand-bound receptor concentrations: <math chem>\theta = \frac\ce{[LR]}\ce{[R_{\rm total}]}</math>. <math>K_d</math> is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate (<math display="inline">K_{\rm d} = {k_{\rm d} \over k_{\rm a}}</math>).<ref name=":0" /> Kd is the equilibrium constant for dissociation. <math display="inline">K_A</math> is defined so that <math display="inline">(K_A)^n = K_{\rm d} = {k_{\rm d} \over k_{\rm a}}</math>, this is also known as the microscopic [[dissociation constant]] and is the ligand concentration occupying half of the binding sites. In recent literature, this constant is sometimes referred to as <math display="inline">K_D</math>.<ref name=":0">{{cite book | last1=Nelson|first1=David L.| last2=Cox|first2=Michael M.| title=Lehninger principles of biochemistry | date=2013 | publisher=W.H. Freeman | location=New York | isbn=978-1429234146 | pages=158–162 | edition=6th}}</ref>

=== Gaddum equation ===
The [[John Gaddum|Gaddum]] equation is a further generalisation of the Hill-equation, incorporating the presence of a reversible competitive antagonist.<ref name = Terms/> 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

=== Hill plot ===
[[File:Hill_Plot.png|thumb|upright=1.2|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.]]

The Hill plot is the rearrangement of the Hill equation into a straight line.

Taking the reciprocal of both sides of the Hill equation, rearranging, and inverting again yields: <math chem="">  {\theta\over 1-\theta} = {[\ce L]^n \over K_d } = {[\ce L]^n \over (K_A)^n } </math>. Taking the logarithm of both sides of the equation leads to an alternative formulation of the Hill-Langmuir equation:
:<math chem=""> \begin{align}\log\left( {\theta\over 1-\theta} \right)
&= n \log{[\ce L]} - \log{K_d}\\
&= n \log{[\ce L]} - n \log{K_A} 
\end{align}</math>.

This last form of the Hill equation is advantageous because a plot of '''<math chem="" display="inline"> \log\left( {\theta\over 1-\theta} \right)  </math>''' versus <math chem=""> \log{[\ce L]}  </math> yields a [[Linear equation|linear plot]], which is called a '''Hill plot.'''<ref name="CoopBind2013" /><ref name=":0" /> Because the slope of a Hill plot is equal to the Hill coefficient for the biochemical interaction, the slope is denoted by <math>n_H</math>. 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.

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.{{refn|group=nb|See [[Propagation of uncertainty]]. The function <math>f(\theta)=\log_{10}\left(\frac{\theta}{1-\theta}\right)</math> propagates errors in <math>\theta</math> as <math>\delta_f=\delta_\theta\frac{\mathrm{d}f}{\mathrm{d}\theta}=\frac{\delta_\theta}{(\ln10)\,\theta(1-\theta)}</math>. Hence errors in values of <math>\theta</math> near <math>0</math> or <math>1</math> are given far more weight than those for <math>\theta\approx0.5</math>}} This impacts the parameters of linear regression lines fitted to the data. Furthermore, the use of computers enables more robust analysis involving [[nonlinear regression]].

== Tissue response ==
[[File:Dose_response_antagonist.jpg|thumb|upright=2|A trio of dose response curves]]

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 [[IUPHAR]] defines the Hill equation in terms of the tissue response <math>(E)</math>, as<ref name = Terms/>
<math display="block">\begin{align}
\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{align}</math>
where <chem>[A]</chem> is the drug concentration, <math>n</math> is the Hill coefficient, and [[EC50|<math>\text{EC}_{50}</math>]] is the drug concentration that produces a 50% maximal response. Dissociation constants (in the previous section) relate to ligand binding, while <math>\text{EC}_{50}</math> reflects tissue response.

This form of the equation can reflect tissue/cell/population responses to drugs and can be used to generate [[dose response curves]]. The relationship between <math>K_d</math> 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.

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 ''activated'' form of the ligand-bound receptor.

Statistical analysis of response as a function of stimulus may be performed by regression methods such as the [[probit model]] or [[logit model]], or other methods such as the [[Spearman–Kärber method]].<ref name="HamiltonRusso1977">{{cite journal |last1= Hamilton |first1= MA |last2= Russo |first2= RC |last3= Thurston |first3= RV |title= Trimmed Spearman–Karber method for estimating median lethal concentrations in toxicity bioassays |journal= [[Environmental Science & Technology]] |volume= 11 |issue= 7 |year= 1977 |pages= 714–9 |doi= 10.1021/es60130a004|bibcode= 1977EnST...11..714H }}</ref> Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the dose-response relationship.<ref name="BatesWatts1988">{{cite book |last1= Bates |first1= Douglas M. |last2= Watts |first2= Donald G. |title= Nonlinear Regression Analysis and its Applications |url= https://archive.org/details/nonlinearregress00bate_737 |url-access= limited |year= 1988 |publisher= [[John Wiley & Sons|Wiley]] |isbn= 9780471816430 |page= [https://archive.org/details/nonlinearregress00bate_737/page/n370 365]}}</ref>

== Hill coefficient ==
The Hill coefficient is a measure of [[ultrasensitivity]] (i.e. how steep is the response curve).

The Hill coefficient, <math>n</math> or <math>n_H</math>, may describe cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used). When appropriate,{{clarify|reason=when is it appropriate?|date=April 2019}} the value of the Hill coefficient describes the cooperativity of ligand binding in the following way:

*<math> n>1 </math>. '''Positively cooperative binding''': 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 [[haemoglobin]] (an example of positive cooperativity) falls within the range of 1.7–3.2.<ref name="UsesMisuses" />
*<math> n<1 </math>. '''Negatively cooperative binding''': Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
*<math> n=1 </math>. '''Noncooperative (completely independent) binding''': 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 [[Michaelis–Menten kinetics]],<ref name="Alon2007">{{cite book|title=An Introduction to Systems Biology: Design Principles of Biological Circuits|last1=Alon|first1=Uri|date=2007|publisher=Chapman & Hall|isbn=978-1-58488-642-6|edition=[Nachdr.]|location=Boca Raton, FL}}</ref> in which <math display="inline">K_D = K_A = K_M</math>, the [[Michaelis–Menten constant]].

The Hill coefficient can be calculated approximately in terms of the ''cooperativity index'' of Taketa and Pogell<ref>{{cite journal | journal= J. Biol. Chem.|last1 = Taketa|first1= K.|last2 = Pogell|first2 = B. M.|volume = 240|number=2| pages=651–662|year = 1965|title=Allosteric inhibition of rat-liver fructose 1,6-diphosphatase by adenosine 5'-monophosphate| doi=10.1016/S0021-9258(17)45224-0 | doi-access=free | pmid=14275118 }}</ref>  as follows:<ref name="altszylerUltrasens2017">{{cite journal|author1=Altszyler, E|author2=Ventura, A. C.|author3=Colman-Lerner, A.|author4=Chernomoretz, A.|year=2017|title=Ultrasensitivity in signaling cascades revisited: Linking local and global ultrasensitivity estimations.|journal=PLOS ONE|volume=12|issue=6|page=e0180083|doi=10.1371/journal.pone.0180083|bibcode=2017PLoSO..1280083A|arxiv=1608.08007|pmid=28662096|pmc=5491127|doi-access=free}}</ref>

:<math chem="">  n = \frac{ \log_{10}(81)}{\log_{10}(\ce{EC90}/\ce{EC10})} </math>.

where <chem>EC90</chem> and <chem>EC10</chem> are the input values needed to produce the 10% and 90% of the maximal response, respectively.

<!--== Derivation from mass action kinetics ==
The Hill-Langmuir equation is derived similarly to the [[Michaelis Menten equation]] but incorporates the Hill coefficient. Consider a protein (<chem>P</chem>), such as [[haemoglobin]] or a protein receptor, with <math>n</math> binding sites for ligands (<chem>L</chem>). The binding of the ligands to the protein can be represented by the chemical equilibrium expression:

:<chem>
 {P} + \mathit{n}{L} <=>[k_a][k_d] {P}{L}_\mathit{n},
</chem>

where <math>k_a</math> (forward rate, or the rate of association of the protein-ligand complex) and <math>k_d</math> (reverse rate, or the complex's rate of dissociation) are the reaction rate constants for the association of the ligands to the protein and their dissociation from the protein, respectively.<ref name=":0"/> From the [[law of mass action]], which in turn may be derived from the principles of [[collision theory]], the apparent dissociation constant <math>K_d</math>, an equilibrium constant, is given by:

:<math>K_{\rm d} = {k_{\rm d} \over k_{\rm a}} = { {[{\rm P}][{\rm L}]^\mathit{n}} \over [{\rm PL_\mathit{n}}] }.</math>
At the same time, <math> \theta </math>, the ratio of the concentration of occupied receptor to total receptor concentration, is given by:
:<math>\theta = {\text{Occupied Receptor} \over \text{Total Receptor}}= {[{\rm PL_\mathit{n}}] \over {[{\rm P}]\ +\ [{\rm PL_\mathit{n}}]}} . </math>

By using the expression obtained earlier for the dissociation constant, we can replace <math display="inline">[{\rm PL_\mathit{n}}] </math> with <math display="inline">{[{\rm P}][{\rm L}]^\mathit{n} \over K_{\rm d}}</math> to yield a simplified expression for <math display="inline">\theta </math>:

:<math>\theta = { \left({[{\rm P}][{\rm L}]^\mathit{n} \over K_{\rm d}}\right) \over {[{\rm P}]\ +\ \left({[{\rm P}][{\rm L}]^\mathit{n} \over K_{\rm d}}\right)} } = { {[{\rm P}][{\rm L}]^\mathit{n} } \over {K_{\rm d}[{\rm P}]\ +\ {[{\rm P}][{\rm L}]^\mathit{n} }} } = { {[{\rm L}]^\mathit{n} } \over {K_{\rm d}\ +\ {[{\rm L}]^\mathit{n}} } } , </math>
which is a common formulation of the Hill equation.<ref name="CoopBind2013">{{cite journal|last1=Stefan|first1=Melanie I.|last2=Novère|first2=Nicolas Le| title=Cooperative Binding|journal=PLOS Computational Biology|date=27 June 2013| volume=9|issue=6|pages=e1003106| doi=10.1371/journal.pcbi.1003106| issn=1553-7358|pmid=23843752| pmc=3699289|bibcode=2013PLSCB...9E3106S}}</ref><ref name="TextbookReceptorPharm">{{cite book |last=Foreman |first=John |date=2003 |title=Textbook of Receptor Pharmacology, Second Edition |url=https://archive.org/details/textbookreceptor00fore_185|url-access=limited |page=[https://archive.org/details/textbookreceptor00fore_185/page/n26 14] |isbn=9780849310294 }}</ref><ref name=":0" />

Assuming that the protein receptor was initially completely free (unbound) at a concentration <math display="inline">[{\rm P_0}] </math>, then at any time, <math display="inline">{[{\rm P}] + [{\rm PL_\mathit{n}}]} = [{\rm P_0}]</math> and <math display="inline">\theta = {[{\rm PL_\mathit{n}}] \over {[{\rm P_0}]\ }}  </math>. Consequently, the Hill equation is also commonly written as an expression for the concentration <math display="inline">[{\rm PL_\mathit{n}}]</math> of bound protein:

:<math>[{\rm PL_\mathit{n}}] = [{\rm P_0}] \cdot { {[{\rm L}]^\mathit{n} } \over {K_{\rm d}\ +\ {[{\rm L}]^\mathit{n}} } }.  </math><ref name="Gesztelyi2012" /> 
All of these formulations assume that the protein has <math>\mathit{n}
</math> sites to which ligands can bind. In practice, however, the Hill Coefficient <math>\mathit{n}</math> rarely provides an accurate approximation of the number of ligand binding sites on a protein.<ref name="UsesMisuses" /><ref name="CoopBind2013" /> Consequently, it has been observed that the Hill coefficient should instead be interpreted as an "interaction coefficient" describing the cooperativity among ligand binding sites.<ref name="UsesMisuses" /> -->

== Reversible form ==

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 [[Reversible Hill Equation|reversible Hill equation]].<ref>{{cite journal |last1=Hofmeyr |first1=Jan-Hendrik S. |last2=Cornish-Bowden |first2=Athel |title=The reversible Hill equation: how to incorporate cooperative enzymes into metabolic models |journal=Bioinformatics |date=1997 |volume=13 |issue=4 |pages=377–385 |doi=10.1093/bioinformatics/13.4.377|doi-access=free |pmid=9283752 }}</ref>

== Relationship to the elasticity coefficients == 

The Hill coefficient is also intimately connected to the [[elasticity coefficient]] where the Hill coefficient can be shown to equal:

<math> n = \varepsilon^v_s \frac{1}{1 - \theta} </math>

where <math>\theta</math> is the fractional saturation, <math>ES/E_t</math>, and <math> \varepsilon^v_s</math> the elasticity coefficient. 

This is derived by taking the slope of the Hill equation:

<math> n = \frac{d\log \frac{\theta}{1-\theta}}{d\log s}  </math>

and expanding the slope using the quotient rule. The result shows that the elasticity can never exceed <math> n </math> since the equation above can be rearranged to:

<math> \varepsilon^v_s = n (1 - \theta) </math>

== Applications ==

The Hill equation is used extensively in pharmacology to quantify the functional parameters of a drug{{citation needed|date=April 2019}} and are also used in other areas of biochemistry.

The Hill equation can be used to describe dose-response relationships, for example [[ion channel]] open-probability (P-open) vs. ligand concentration.<ref>{{cite journal|title=Single Channel Properties of P2X2 Purinoceptors |last1=Ding|first1=S|last2=Sachs|first2=F|publisher= The Rockefeller University Press |year=1999|journal=J. Gen. Physiol.|pmc=2222910|pmid=10228183|volume=113|issue=5|pages=695–720|doi=10.1085/jgp.113.5.695}}</ref>

=== Regulation of gene transcription ===
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 [[transcription factor]]s (e.g., [[Activator (genetics)|activators]] and/or [[repressor]]s).<ref name="Alon2007" /> 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.<ref>{{Cite journal|last1=Chu|first1=Dominique|last2=Zabet|first2=Nicolae Radu|last3=Mitavskiy|first3=Boris|date=2009-04-07|title=Models of transcription factor binding: Sensitivity of activation functions to model assumptions|journal=Journal of Theoretical Biology|volume=257|issue=3|pages=419–429|doi=10.1016/j.jtbi.2008.11.026|pmid=19121637|bibcode=2009JThBi.257..419C|url=http://kar.kent.ac.uk/24077/1/myzabetpaper.pdf}}</ref>

If the production of protein from gene {{mvar|X}} is up-regulated ('''activated''') by a transcription factor {{mvar|Y}}, then the rate of production of protein {{mvar|X}} can be modeled as a differential equation in terms of the concentration of activated {{mvar|Y}} protein:

:<math> {\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}} } }  </math>,

where {{mvar|k}} is the maximal transcription rate of gene {{mvar|X}}.

Likewise, if the production of protein from gene {{mvar|Y}} is down-regulated ('''repressed''') by a transcription factor {{mvar|Z}}, then the rate of production of protein {{mvar|Y}} can be modeled as a differential equation in terms of the concentration of activated {{mvar|Z}} protein:

:<math> {\mathrm{d} \over \mathrm{d}t} [{\rm Y_{produced}}]= k\ \cdot { {(K_A)^\mathit{n} } \over {(K_A)^n\ +\ {[{\rm Z_{active}}]^\mathit{n}} } }  </math>,
where {{mvar|k}} is the maximal transcription rate of gene {{mvar|Y}}.

==Limitations==
Because of its assumption that ligand molecules bind to a receptor simultaneously, the Hill equation has been criticized as a physically unrealistic model.<ref name="UsesMisuses" /> Moreover, the Hill coefficient should not be considered a reliable approximation of the number of cooperative ligand binding sites on a receptor<ref name="UsesMisuses" /><ref name="Monod 1965">{{cite journal|last1=Monod|first1=Jacques|last2=Wyman|first2=Jeffries|last3=Changeux|first3=Jean-Pierre|title=On the nature of allosteric transitions: A plausible model|journal=Journal of Molecular Biology|date=1 May 1965|volume=12|issue=1|pages=88–118|doi=10.1016/S0022-2836(65)80285-6|pmid=14343300}}</ref> except when the binding of the first and subsequent ligands results in extreme positive cooperativity.<ref name="UsesMisuses" />

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 ''a priori'' knowledge about the properties of either the protein or ligand being studied.<ref name="Gesztelyi2012" /> Nevertheless, other, more complex models of cooperative binding have been proposed.<ref name="CoopBind2013" /> For more information and examples of such models, see [[Cooperative binding]].

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.<ref name="Kholodenko1997">{{cite journal|author=Kholodenko, Boris N.|display-authors=etal|year=1997|title=Quantification of information transfer via cellular signal transduction pathways|journal=FEBS Letters|volume=414|issue=2|pages=430–434|doi=10.1016/S0014-5793(97)01018-1|pmid=9315734|s2cid=19466336|doi-access=free|bibcode=1997FEBSL.414..430K }}</ref>

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.<ref name="altszylerUltrasens2017" />

==See also==

* [[Binding coefficient]] 
* [[Bjerrum plot]]
* [[Cooperative binding]]
* [[Gompertz curve]]
* [[Langmuir adsorption model]]
* [[Logistic function]]
* [[Michaelis–Menten kinetics]]
* [[Monod equation]]

==Notes==
{{reflist|group=nb}}

==References==
{{reflist}}

==Further reading==
* ''[[Dorland's Illustrated Medical Dictionary]]''
*{{cite journal |last=Coval |first=ML |date=December 1970 |title=Analysis of Hill interaction coefficients and the invalidity of the Kwon and Brown equation |journal=[[J. Biol. Chem.]] |pmid=5484812 |volume=245 |issue=23 |pages=6335–6 |doi=10.1016/S0021-9258(18)62614-6 |url=http://www.jbc.org/content/245/23/6335|doi-access=free }}
*{{cite journal |last1=d'A Heck |first1=Henry |year=1971 |title=Statistical theory of cooperative binding to proteins. Hill equation and the binding potential |journal=J. Am. Chem. Soc. |volume=93 |issue=1 |pages=23–29 | doi= 10.1021/ja00730a004|pmid=5538860 }}
*{{cite journal |last1=Atkins |first1=Gordon L. |year=1973 |title=A simple digital-computer program for estimating the parameter of the Hill Equation | journal=Eur. J. Biochem. |volume=33 |issue=1 |pages=175–180 |doi=10.1111/j.1432-1033.1973.tb02667.x|pmid=4691349 |doi-access=free }}
*{{cite journal |title=Evaluation of Hill slopes and Hill coefficients when the saturation binding or velocity is not known |year=1975 | journal= Eur. J. Biochem. |doi=10.1111/j.1432-1033.1975.tb03931.x | volume=51 |issue=2 |pages=317–328 |first1=Laszlo |last1=Endrenyi |first2=F. H. F. |last2=Kwong |first3=Csaba |last3=Fajszi |pmid=1149734|doi-access=free }}
*{{cite book |last1=Voet |first1=Donald |last2=Voet |first2=Judith G. |title=Biochemistry|year=2004 |url=https://archive.org/details/biochemistry00voet_1 |url-access=registration }}
*{{cite journal |last1=Weiss |first1=J. N. |year=1997 |title=The Hill equation revisited: uses and misuses |journal=FASEB Journal |pmid=9285481|volume=11 |issue=11 |pages=835–841 |url=http://www.fasebj.org/content/11/11/835.short|doi=10.1096/fasebj.11.11.9285481 |doi-access=free |s2cid=827335 }}
*{{cite journal|title=Criterion for Hill equation validity for description of biosensor calibration curves 
| journal= Anal. Chim. Acta | year=2001 | doi=10.1016/S0003-2670(00)01167-3 |first1= B. I.
|last1= Kurganov |first2= A. V. | last2=Lobanov | volume=427 | issue =1|pages=11–19| bibcode= 2001AcAC..427...11K }}
*{{cite journal
|title =The Hill equation: a review of its capabilities in pharmacological modelling
|year=2008
|volume=22
|issue=6
|pages=633–648
|journal=Fundamental & Clinical Pharmacology
|first1=    Sylvain |last1= Goutelle
|first2=    Michel |last2=Maurin
|first3=    Florent |last3=Rougier
|first4=    Xavier |last4=Barbaut
|first5=    Laurent |last5=Bourguignon
|first6=    Michel |last6=Ducher
|first7=    Pascal|last7= Maire
|doi= 10.1111/j.1472-8206.2008.00633.x
|pmid=19049668
|s2cid=4979109
}}
* {{cite journal| author1=Gesztelyi R| author2=Zsuga J| author3=Kemeny-Beke A| author4=Varga B| author5=Juhasz B| author6=Tosaki A| title=The Hill equation and the origin of quantitative pharmacology| journal=[[Archive for History of Exact Sciences]]| year=2012| volume=66| issue=4| pages=427–38| doi=10.1007/s00407-012-0098-5| s2cid=122929930}}
* {{cite journal| author=Colquhoun D| title=The quantitative analysis of drug-receptor interactions: a short history| journal=Trends Pharmacol Sci| year=2006| volume=27| pages=149–57| doi=10.1016/j.tips.2006.01.008| pmid=16483674| issue=3}}
* {{cite journal| author=Rang HP| title=The receptor concept: pharmacology's big idea| journal=Br J Pharmacol| year=2006| volume=147| issue=Suppl 1| pages=S9–16|doi=10.1038/sj.bjp.0706457| pmid=16402126| pmc=1760743}}

==External links==
* [http://www.fxsolver.com/browse/formulas/Hill+equation Hill equation calculator]

{{DEFAULTSORT:Hill Equation}}
[[Category:Enzyme kinetics]]
[[Category:Pharmacology]]