Cooperative Binding - The Hill Coefficient

The Hill Coefficient

The Hill coefficient provides a quantitative method for characterizing binding cooperativity. The macromolecule is assumed to bind to ligands simultaneously (where is to be determined)

to form the complex C. Hence the dissociation constant equals

Let the variable represent the fraction of binding sites that are occupied on the macromolecule. Then, represents the fraction of binding sites that are not occupied, giving the ratio

Taking the logarithm yields an equation linear in

Hence, the slope of this line yields, whereas its intercept is determined by .

More generally, plotting versus and taking the slope gives the effective number of ligands that are binding cooperatively at a particular ligand concentration . In a non-cooperative system such as myoglobin, the plot is a straight line with slope at all ligand concentrations. By contrast, in a system with positive cooperativity such as hemoglobin, the plot begins as a line with slope, then ramps up to a new line (also with slope ) that is offset upwards. The degree of cooperativity is characterized by the maximum slope in the "ramping up" region, which is ~2.8 for hemoglobin; thus, at its most cooperative, hemoglobin effectively binds three ligands in concert. The "ramping up" corresponds to an increase in the affinity (decrease in ) that occurs as the amount of bound ligand increases. Such plots are sometimes characterized as "sigmoid" due to their subtle "S"-shape.