# 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n**
provides a quantitative method for characterizing binding cooperativity. The macromolecule is assumed to bind to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n**
ligands simultaneously (where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n**
is to be determined)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{P} + n\mathrm{L} \leftrightarrow \mathrm{C}**

to form the complex C. Hence the dissociation constant equals

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K_{d} = \frac{\left[ \mathrm{P} \right]\left[ \mathrm{L} \right]^{n}}{\left[ \mathrm{C} \right]}**

The variable **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta**
represents the fraction of binding sites that are occupied on the macromolecule. Therefore, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1-\theta**
represents the fraction of binding sites that are *not* occupied, giving the ratio

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log \left[ \frac{\theta}{1 - \theta} \right] = n \log \left[ \mathrm{L} \right] - \log K_{d}**

Hence, the slope of this line yields **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log \ K_{d}**
.

More generally, plotting **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log \left[ \frac{\theta}{1 - \theta} \right]**
versus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log \left[ \mathrm{L} \right]**
and taking the slope gives the effective number of ligands **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \mathrm{L} \right]**
. In a non-cooperative system such as myoglobin, the plot is a straight line with slope **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=1**
, then ramps up to a new line (also with slope **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=1**
) that is offset upwards. The degree of cooperativity is characterized by the maximum slope **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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.

## Additional information

In non-cooperative binding, the way the affinity depends on the concentration of ligand in solution often is described as "hyperbolic," because a graph of this dependence traces a hyperbola.

## See also

## References

- Changeux J.-P. (1964). Allosteric interactions interpreted in terms of quaternary structure. Brookhaven Symposia in Biology, 17: 232-249.

- Monod J., Wyman J., and Changeux J.-P. (1965). On the nature of allosteric transitions: a plausible model. J. Mol. Biol. 12: 88-118.

## External links