Editing Cooperative binding (section)

=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[A V Hill|A.V. Hill]].<ref name=Hill1910>{{cite journal | vauthors = Hill AV | date = 1910 | title = The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves | journal = J Physiol | volume = 40 | pages = iv–vii }}</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[Hill equation (biochemistry)|named after him]]: [[File:Hill Plot.png|thumb|right|Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]

:<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} =  \frac{[X]^n}{K^* + [X]^n} = \frac{[X]^n}{K_d^n + [X]^n}
</math>

where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation), <math>K^*</math> is an empirical dissociation constant, and <math>K_d</math> a microscopic dissociation constant (used in modern forms of the equation, and equivalent to an <math>\mathrm{EC}_{50}</math>). If  <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:

:<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - n\cdot{}\log K_d
</math>

The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>n\cdot\log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.