# Cooperative binding

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In biochemistry, a macromolecule exhibits cooperative binding if its affinity for its ligand changes with the amount of ligand already bound.

Cooperative binding is a special case of allostery. Cooperative binding requires that the macromolecule have more than one binding site, since cooperativity results from the interactions between binding sites. If the binding of ligand at one site increases the affinity for ligand at another site, the macromolecule exhibits positive cooperativity. Conversely, if the binding of ligand at one site lowers the affinity for ligand at another site, the protein exhibits negative cooperativity. If the ligand binds at each site independently, the binding is non-cooperative.

## Contents

### The Hill coefficient

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

to form the complex C. Hence the dissociation constant equals

The variable θ represents the fraction of binding sites that are occupied on the macromolecule. Therefore, 1 − θ represents the fraction of binding sites that are not occupied, giving the ratio

Taking the logarithm yields an equation linear in n

Hence, the slope of this line yields n, whereas its intercept is determined by $\log \ K_{d}$.

More generally, plotting $\log \left[ \frac{\theta}{1 - \theta} \right]$ versus $\log \left[ \mathrm{L} \right]$ and taking the slope gives the effective number of ligands n that are binding cooperatively at a particular ligand concentration $\left[ \mathrm{L} \right]$. In a non-cooperative system such as myoglobin, the plot is a straight line with slope n = 1 at all ligand concentrations. By contrast, in a system with positive cooperativity such as hemoglobin, the plot begins as a line with slope n = 1, then ramps up to a new line (also with slope n = 1) that is offset upwards. The degree of cooperativity is characterized by the maximum slope n 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 Kd) that occurs as the amount of bound ligand increases. Such plots are sometimes characterized as "sigmoid" due to their subtle "S"-shape.