# Receptor-ligand kinetics

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In biochemistry, **receptor-ligand kinetics** is a branch of chemical kinetics in which the kinetic species are defined by different non-covalent bindings and/or conformations of the molecules involved, which are denoted as *receptor(s)* and *ligand(s)*.

A main goal of receptor-ligand kinetics is to determine the concentrations of the various kinetic species (i.e., the states of the receptor and ligand) at all times, from a given set of initial concentrations and a given set of rate constants. In a few cases, an analytical solution of the rate equations may be determined, but this is relatively rare. However, most rate equations can be integrated numerically, or approximately, using the steady-state approximation. A less ambitious goal is to determine the final *equilibrium* concentrations of the kinetic species, which is adequate for the interpretation of equilibrium binding data.

A converse goal of receptor-ligand kinetics is to estimate the rate constants and/or dissociation constants of the receptors and ligands from experimental kinetic or equilibrium data. The total concentrations of receptor and ligands are sometimes varied systematically to estimate these constants.

## Kinetics of single receptor/single ligand/single complex bindingEdit

The simplest example of receptor-ligand kinetics is that of a single ligand L binding to a single receptor R to form a single complex C

The equilibrium concentrations are related by the dissociation constant *K _{d}*

where *k _{1}* and

*k*are the forward and backward rate constants, respectively. The total concentrations of receptor and ligand in the system are constant

_{-1}Thus, only one concentration of the three ([R], [L] and [C]) is independent; the other two concentrations may be determined from *R _{tot}*,

*L*and the independent concentration.

_{tot}This system is one of the few systems whose kinetics can be determined analytically. Choosing [R] as the independent concentration and representing the concentrations by italic variables for brevity (e.g., ), the kinetic rate equation can be written

Dividing both sides by *k*_{1} and introducing the constant *2E = R _{tot} - L_{tot} - K_{d}*, the rate equation becomes

where the two equilibrium concentrations are given by the quadratic formula and the discriminant *D* is defined

However, only the equilibrium is stable, corresponding to the equilibrium observed experimentally.

Separation of variables and a partial-fraction expansion yield the integrable ordinary differential equation

whose solution is

or, equivalently,

where the integration constant φ_{0} is defined

From this solution, the corresponding solutions for the other concentrations and can be obtained.

## See also Edit

## Further readingEdit

- D.A. Lauffenburger and J.J. Linderman (1993)
*Receptors: Models for Binding, Trafficking, and Signaling*, Oxford University Press. ISBN 0-19-506466-6 (hardcover) and 0-19-510663-6 (paperback)

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