Receptor-ligand kinetics

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 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 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.