Editing Cooperative binding (section)

=== The Adair equation ===
[[Gilbert Smithson Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that binding affinity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>{{cite journal | vauthors = Adair GS | year = 1925 | title = 'The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin | journal = J Biol Chem | volume = 63 | issue = 2 | pages = 529–545 | doi = 10.1016/S0021-9258(18)85018-9 | doi-access = free }}</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:

:<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>

Or, for any protein with ''n'' ligand binding sites:

:<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>

where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
By combining the Adair treatment with the Hill plot, one arrives at the modern experimental definition of cooperativity (Hill, 1985, Abeliovich, 2005). The resultant Hill coefficient, or more correctly the slope of the Hill plot as calculated from the Adair Equation, can be shown to be the ratio between the variance of the binding number to the variance of the binding number in an equivalent system of non-interacting binding sites.<ref name=Abeliovich2005>{{cite journal | vauthors = Abeliovich H | title = An empirical extremum principle for the hill coefficient in ligand-protein interactions showing negative cooperativity | journal = Biophysical Journal | volume = 89 | issue = 1 | pages = 76–9 | date = July 2005 | pmid = 15834004 | pmc = 1366580 | doi = 10.1529/biophysj.105.060194 | bibcode = 2005BpJ....89...76A }}</ref> Thus, the Hill coefficient defines cooperativity as a statistical dependence of one binding site on the state of other site(s).