# Receptor–ligand kinetics

(Redirected from 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). Receptor–ligand binding kinetics also involves the on- and off-rates of binding.

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.

## Binding kinetics

The binding constant is a special case of the equilibrium constant ${\displaystyle K}$. It is associated with the binding and unbinding reaction of receptor (R) and ligand (L) molecules, which is formalized as:

${\displaystyle {\ce {{R}+{L}<=>{RL}}}}$.

The reaction is characterized by the on-rate constant ${\displaystyle k_{\rm {on}}}$ and the off-rate constant ${\displaystyle k_{\rm {off}}}$, which have units of 1/(concentration time) and 1/time, respectively. In equilibrium, the forward binding transition ${\displaystyle {\ce {{R}+{L}->{RL}}}}$ should be balanced by the backward unbinding transition ${\displaystyle {\ce {{RL}->{R}+{L}}}}$. That is,

${\displaystyle k_{\ce {on}}\,[{\ce {R}}]\,[{\ce {L}}]=k_{\ce {off}}\,[{\ce {RL}}]}$,

where ${\displaystyle {\ce {[{R}]}}}$, ${\displaystyle {\ce {[{L}]}}}$ and ${\displaystyle {\ce {[{RL}]}}}$ represent the concentration of unbound free receptors, the concentration of unbound free ligand and the concentration of receptor-ligand complexes. The binding constant, or the association constant ${\displaystyle K_{\rm {a}}}$ is defined by

${\displaystyle K_{\rm {a}}={k_{\ce {on}} \over k_{\ce {off}}}={\ce {[{RL}] \over {[{R}]\,[{L}]}}}}$.

## Simplest case: single receptor and single ligand bind to form a complex

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

${\displaystyle {\ce {{R}+{L}<->{C}}}}$

The equilibrium concentrations are related by the dissociation constant Kd

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

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

${\displaystyle R_{tot}\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {R}}]+[{\ce {C}}]}$
${\displaystyle L_{tot}\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {L}}]+[{\ce {C}}]}$

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., ${\displaystyle R\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {R}}]}$), the kinetic rate equation can be written

${\displaystyle {\frac {dR}{dt}}=-k_{1}RL+k_{-1}C=-k_{1}R(L_{tot}-R_{tot}+R)+k_{-1}(R_{tot}-R)}$

Dividing both sides by k1 and introducing the constant 2E = Rtot - Ltot - Kd, the rate equation becomes

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

where the two equilibrium concentrations ${\displaystyle R_{\pm }\ {\stackrel {\mathrm {def} }{=}}\ E\pm D}$ are given by the quadratic formula and D is defined

${\displaystyle D\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {E^{2}+R_{tot}K_{d}}}}$

However, only the ${\displaystyle R_{+}}$ equilibrium has a positive concentration, corresponding to the equilibrium observed experimentally.

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

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

whose solution is

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

or, equivalently,

${\displaystyle g=exp(-2Dk_{1}t+\phi _{0})}$

${\displaystyle R(t)={\frac {R_{+}-gR_{-}}{1-g}}}$

for association, and

${\displaystyle R(t)={\frac {R_{+}+gR_{-}}{1+g}}}$

for dissociation, respectively; where the integration constant φ0 is defined

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

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