# 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

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

The equilibrium concentrations are related by the dissociation constant Kd

$K_{d} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{-1}}{k_{1}} = \frac{[\mathrm{R}]_{eq} [\mathrm{L}]_{eq}}{[\mathrm{C}]_{eq}}$

where k1 and k-1 are the forward and backward rate constants, 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 Rtot, Ltot 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 k1 and introducing the constant 2E = Rtot - Ltot - Kd, the rate equation becomes

$\frac{1}{k_{1}} \frac{dR}{dt} = -R^{2} + 2ER + K_{d}R_{tot} = -\left( R - R_{+}\right) \left( R - R_{-}\right)$

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 integrable ordinary differential equation

$\left\{ \frac{1}{R - R_{+}} - \frac{1}{R - R_{-}} \right\} dR = -2 D k_{1} dt$

whose solution is

$\log \left| R - R_{+} \right| - \log \left| R - R_{-} \right| = -2Dk_{1}t + \phi_{0}$

or, equivalently,

$R(t) = E + D \tanh(-Dk_{t} t + \phi_{0})$

where the integration constant φ0 is defined

$\phi_{0} \ \stackrel{\mathrm{def}}{=}\ \log \left| R(t=0) - R_{+} \right| - \log \left| R(t=0) - R_{-} \right|$

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