# Cooperative binding

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.

## The Hill coefficient

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

${\displaystyle \mathrm {P} +n\mathrm {L} \leftrightarrow \mathrm {C} }$

to form the complex C. Hence the dissociation constant equals

${\displaystyle K_{d}={\frac {\left[\mathrm {P} \right]\left[\mathrm {L} \right]^{n}}{\left[\mathrm {C} \right]}}}$

The variable ${\displaystyle \theta }$ represents the fraction of binding sites that are occupied on the macromolecule. Therefore, ${\displaystyle 1-\theta }$ represents the fraction of binding sites that are not occupied, giving the ratio

${\displaystyle {\frac {\theta }{1-\theta }}={\frac {\left[\mathrm {C} \right]}{\left[\mathrm {P} \right]}}={\frac {\left[\mathrm {L} \right]^{n}}{K_{d}}}}$

Taking the logarithm yields an equation linear in ${\displaystyle n}$

${\displaystyle \log \left[{\frac {\theta }{1-\theta }}\right]=n\log \left[\mathrm {L} \right]-\log K_{d}}$

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

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

## Mechanisms of cooperativity

Two models were hypothesized to account for the binding cooperativity observed in proteins, the MWC model and the KNF model.

The Monod-Wyman-Changeux (MWC) model was advanced by Jacques Monod, Jeffries Wyman and Jean-Pierre Changeux in 1965. It posits that the protein has only two states, a low-affinity state T and a high-affinity state R, where the T state is thermodynamically favored. Hence, at low amounts of bound ligand, the protein prefers the low-affinity T state; however, as the amount of bound ligand increases, the protein comes to prefer the high-affinity state. Structural studies have supported the MWC model and elucidated the R and T states; however, the model cannot explain negative cooperativity.

An alternative model is the sequential or "induced fit" model of Daniel Koshland, George Némethy and Filmer (KNF model), in which ligand binding at one site causes a local conformational change ("induced fit") that causes small conformational changes at nearby binding sites, affecting their affinity for the ligand. Thus, according to the KNF model, the protein has many slightly different conformational states, corresponding to all possible modes of ligand binding.