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| {{about|The '''Hill equation''' is an equation used in biochemical characterization||Hill differential equation}}
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| {{more footnotes|date=March 2009}}
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| In [[biochemistry]], 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 coefficient''' provides a way to quantify this effect.
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| It describes the fraction of the macromolecule saturated by [[Ligand (biochemistry)|ligand]] as a function of the ligand [[concentration]]; it is used in determining the degree of cooperativeness of the ligand binding to the enzyme or receptor. It was originally formulated by [[Archibald Hill]] in 1910 to describe the [[Sigmoid function|sigmoidal]] O<sub>2</sub> binding curve of hemoglobin.<ref>{{cite journal|title=The possible effects of the aggregation of the molecules of hæmoglobin on its dissociation curves|journal=[[J. Physiol.]]|date=1910-01-22|first=A. V.|last=Hill|coauthors=|volume=40|issue=Suppl|pages=iv-vii|id= |url=http://jp.physoc.org/cgi/reprint/40/Suppl/i|format=PDF|accessdate=2009-03-18}}</ref>
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| A coefficient of 1 indicates completely independent binding, regardless of how many additional ligands are already bound. Numbers greater than one indicate positive cooperativity, while numbers less than one indicate negative cooperativity. The Hill coefficient of oxygen binding to [[hemoglobin]] is 2.3-3.0.
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| Hill equation:
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| <math> \theta = {[L]^n \over K_d + [L]^n} = {[L]^n \over (K_A)^n + [L]^n} = {1 \over ({K_A \over [L]})^n+1} </math>
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| <math> \theta </math> - fraction of occupied sites where the [[Ligand (biochemistry)|ligand]] can bind to the binding site of the [[receptor (biochemistry)|receptor protein]].
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| <math>[L]</math> - free (unbound) [[Ligand (biochemistry)|ligand]] concentration
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| <math>K_d</math> - Apparent [[dissociation constant]] derived from the [[law of mass action]] (equilibrium constant for dissociation)
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| <math>K_A</math> - ligand concentration producing half occupation (ligand concentration occupying half of the binding sites). This is also the microscopic [[dissociation constant]].
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| <math>n</math> - Hill coefficient, describing cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used)
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| Taking the reciprocal of both sides, rearranging, inverting again, and then taking the logarithm on both sides of the equation leads to an alternative formulation of the Hill equation:
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| <math> \log\left( {\theta\over 1-\theta} \right) = n\log{[L]} - \log{K_d}.</math>
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| When appropriate, the value of the Hill coefficient describes the cooperativity of ligand binding in the following way:
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| * <math> n>1 </math> - '''Positively cooperative binding''': Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
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| * <math> n<1 </math> - '''Negatively cooperative binding''': Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
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| * <math> n=1 </math> - '''Noncooperative binding''': The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound. In this case, the Hill equation (as a relationship between the concentration of a compound adsorbing to binding sites and the fractional occupancy of the binding sites) is equivalent to the [[Langmuir equation]].
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| ==See also==
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| The Hill equation is related to the [[logistic function]] and is in some ways a logarithmic transform of it, i.e. when you plot the Hill function on a log scale it looks identical to a logistic function. This is particularly important if the range of concentrations that results in saturation does not vary over several orders of magnitude. In such a case the logistic function would be a more appropriate equation to model the behavior.
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| * [[Logistic function]]
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| * [[Gompertz curve]]
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| * [[Sigmoid function]]
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| ==References==
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| <references/>
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| * ''[[Dorland's Illustrated Medical Dictionary]]''
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| * {{cite book | title=Lehninger Principles of Biochemistry | edition=4th |first1=David L. |last1= Nelson
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| |first2=Michael M. |last2=Cox}}
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| * {{cite journal |author=Coval ML |title=Analysis of Hill interaction coefficients and the invalidity of the Kwon and Brown equation |journal=[[J. Biol. Chem.]] |volume=245 |issue=23 |pages=6335–6 |date=December 1970 |pmid=5484812 |doi= |url=http://www.jbc.org/content/245/23/6335}}
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| * {{cite journal|title=Statistical theory of cooperative binding to proteins. Hill equation and the binding potential
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| |year=1971 | journal=J. Am. Chem. Soc. | volume=93|issue=1 | pages=23–29
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| |first1=Henry | last1=d'A Heck | doi= 10.1021/ja00730a004}}
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| * {{cite journal | title=A simple digital-computer program for estimating the parameter of the Hill Equation
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| |first1=Gordon L. | last1=Atkins | journal=Eur. J. Biochem.
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| |year=1973 | volume=33 |issue=1 | pages=175–180 | doi=10.1111/j.1432-1033.1973.tb02667.x}}
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| * {{cite journal| title=Evaluation of Hill slopes and Hill coefficients when the saturation binding or velocity is not known
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| |year=1975 | journal= Eur. J. Biochem. | doi= 10.1111/j.1432-1033.1975.tb03931.x | volume=51 |issue=2
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| |pages=317–328 | first1=Laszlo | last1=Endrenyi | first2=F. H. F. | last2=Kwong | first3=Csaba |last3=Fajszi}}
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| * {{cite book| title=Biochemistry|first1=Donald |last1=Voet |first2=Judith G. |last2=Voet}}
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| * {{cite journal| journal=FASEB | year=1997
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| |title=The Hill equation revisited: uses and misuses
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| |first1=J. N. | last1=Weiss | volume=11 | issue=11 | pages=835–841
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| |url=http://www.fasebj.org/content/11/11/835.short |pmid=9285481
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| }}
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| * {{cite journal|title=Criterion for Hill equation validity for description of biosensor calibration curves
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| | journal= Anal. Chimica Acta | year=2001 | doi=10.1016/S0003-2670(00)01167-3 |first1= B. I.
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| |last1= Kurganov |first2= A. V. | last2=Lobanov | volume=427 | issue =1|pages=11–19}}
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| * {{cite journal
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| |title =The Hill equation: a review of its capabilities in pharmacological modelling
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| |year=2008
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| |volume=22
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| |pages=633–648
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| |journal=Fund. Clinic. Pharmac.
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| |first1= Sylvain |last1= Goutelle1
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| |first2= Michel |last2=Maurin
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| |first3= Florent |last3=Rougier
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| |first4= Xavier |last4=Barbaut
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| |first5= Laurent |last5=Bourguignon
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| |first6= Michel |last6=Ducher
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| |first7= Pascal|last7= Maire
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| |doi= 10.1111/j.1472-8206.2008.00633.x
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| }}
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| * {{cite journal |title=Models of transcription factor binding: Sensitivity of activation functions to model assumptions. |first1=B.
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| |last1= Mitavskiy
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| |first2=D. |last2=Chu | first3=Radu |last3= Zabet. |journal=Journal of Theoretical Biology |volume=257|issue=3|pages=419–429|year= 2009|doi= 10.1016/j.jtbi.2008.11.026 |pmid=19121637}}
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| * {{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=Arch Hist Exact Sci| year=2012| volume=66| pages=427–38| url=http://link.springer.com/content/pdf/10.1007%2Fs00407-012-0098-5}}
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| * {{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}}
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| * {{cite journal| author=Rang HP| title=The receptor concept: pharmacology's big idea| journal=Br J Pharmacol| year=2006| volume=147| pages=S9-16}}
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| * {{cite journal| author1=Neubig RR| author2=Spedding M| author3=Kenakin T| author4=Christopoulos A| title=International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification. XXXVIII. update on terms and symbols in quantitative pharmacology| journal=Pharmacol Rev| year=2003| volume=55| pages=597–606}}
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| {{DEFAULTSORT:Hill Equation}}
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| [[Category:Enzyme kinetics]]
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The title of the author is Figures. Her family life in Minnesota. I am a meter reader but I plan on changing it. One of the very best things in the world for him is to gather badges but he is having difficulties to discover time for it.
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