- Receptor-ligand kinetics
In

biochemistry ,**receptor-ligand kinetics**is a branch ofchemical 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 constant s 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 binding**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

:$mathrm\{R\}\; +\; mathrm\{L\}\; leftrightarrow\; mathrm\{C\}$

The equilibrium concentrations are related by the

dissociation constant "K_{d}":$K\_\{d\}\; stackrel\{mathrm\{def\{=\}\; frac\{k\_\{-1\{k\_\{1\; =\; frac\{\; [mathrm\{R\}]\; \_\{eq\}\; [mathrm\{L\}]\; \_\{eq\{\; [mathrm\{C\}]\; \_\{eq$

where "k

_{1}" and "k_{-1}" are the forward and backwardrate constant s, respectively. The total concentrations of receptor and ligand in the system are constant:$R\_\{tot\}\; stackrel\{mathrm\{def\{=\}\; [mathrm\{R\}]\; +\; [mathrm\{C\}]$

:$L\_\{tot\}\; stackrel\{mathrm\{def\{=\}\; [mathrm\{L\}]\; +\; [mathrm\{C\}]$

Thus, only one concentration of the three ( [R] , [L] and [C] ) is independent; the other two concentrations may be determined from "R

_{tot}", "L_{tot}" and the independent concentration.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., $R\; stackrel\{mathrm\{def\{=\}\; [mathrm\{R\}]$), the kinetic rate equation can be written

:$frac\{dR\}\{dt\}\; =\; -k\_\{1\}\; R\; L\; +\; k\_\{-1\}\; C\; =\; -k\_\{1\}\; R\; (L\_\{tot\}\; -\; R\_\{tot\}\; +\; R)\; +\; k\_\{-1\}\; (R\_\{tot\}\; -\; R)$

Dividing both sides by "k"

_{1}and introducing the constant "2E = R_{tot}- L_{tot}- K_{d}", the rate equation becomes:$frac\{1\}\{k\_\{1\; frac\{dR\}\{dt\}\; =\; -R^\{2\}\; +\; 2ER\; +\; K\_\{d\}R\_\{tot\}\; =-left(\; R\; -\; R\_\{+\}\; ight)\; left(\; R\; -\; R\_\{-\}\; ight)$

where the two equilibrium concentrations $R\_\{pm\}\; stackrel\{mathrm\{def\{=\}\; E\; pm\; D$ are given by the

quadratic formula and the discriminant "D" is defined:$D\; stackrel\{mathrm\{def\{=\}\; sqrt\{E^\{2\}\; +\; R\_\{tot\}\; K\_\{d$

However, only the $R\_\{-\}$ equilibrium is stable, corresponding to the equilibrium observed experimentally.

Separation of variables and a partial-fraction expansion yield the integrableordinary differential equation :$left\{\; frac\{1\}\{R\; -\; R\_\{+\; -\; frac\{1\}\{R\; -\; R\_\{-\; ight\}\; dR\; =\; -2\; D\; k\_\{1\}\; dt$

whose solution is

:$log\; left|\; R\; -\; R\_\{+\}\; ight|\; -\; log\; left|\; R\; -\; R\_\{-\}\; ight|\; =\; -2Dk\_\{1\}t\; +\; phi\_\{0\}$

or, equivalently,

:$g\; =\; exp(-2Dk\_\{1\}t+phi\_\{0\})$

$R(t)\; =\; frac\{R\_\{+\}\; -\; gR\_\{-\{1\; -\; g\}$

where the integration constant φ

_{0}is defined:$phi\_\{0\}\; stackrel\{mathrm\{def\{=\}\; log\; left|\; R(t=0)\; -\; R\_\{+\}\; ight|\; -\; log\; left|\; R(t=0)\; -\; R\_\{-\}\; ight$

From this solution, the corresponding solutions for the other concentrations $C(t)$ and $L(t)$ can be obtained.

**See also***

Binding potential

*Patlak plot

*Scatchard plot **Further reading***

D.A. Lauffenburger andJ.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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