# 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 $K$. It is associated with the binding and unbinding reaction of receptor (R) and ligand (L) molecules, which is formalized as:
    ${\ce {{R}+ {L}<=> {RL}}}$.
The reaction is characterized by the on-rate constant $k_{\rm {on}}$ and the off-rate constant $k_{\rm {off}}$, which have units of 1/(concentration time) and 1/time, respectively. In equilibrium, the forward binding transition ${\ce {{R}+ {L}-> {RL}}}$ should be balanced by the backward unbinding transition ${\ce {{RL}-> {R}+ {L}}}$. That is,
    $k_{{\ce {on}}}\,[{\ce {R}}]\,[{\ce {L}}]=k_{{\ce {off}}}\,[{\ce {RL}}]$,
where ${\ce {[{R}]}}$, ${\ce {[{L}]}}$ and ${\ce {[{RL}]}}$ 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 $K_{\rm {a}}$ is defined by
    $K_{\rm {a}}={k_{\ce {on}} \over k_{\ce {off}}}={\ce {[{RL}] \over {[{R}]\,[{L}]}}}$.

## 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
    ${\ce {{R}+ {L}<-> {C}}}$
The equilibrium concentrations are related by the dissociation constant K$_d$
    $K_{d}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {k_{-1}}{k_{1}}}={\frac {[{\ce {R}}]_{eq}[{\ce {L}}]_{eq}}{[{\ce {C}}]_{eq}}}$
where k$_1$ and k$_{−1}$ are the forward and backward rate constants, respectively. The total concentrations of receptor and ligand in the system are constant
    $R_{tot}\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {R}}]+[{\ce {C}}]$
    $L_{tot}\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {L}}]+[{\ce {C}}]$
Thus, only one concentration of the three ([R], [L] and [C]) is independent; the other two concentrations may be determined from 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., $R\ {\stackrel {\mathrm {def} }{=}}\ [{\ce {R}}]$), the kinetic rate equation can be written
    ${\frac {dR}{dt}}=-k_{1}RL+k_{-1}C=-k_{1}R(L_{tot}-R_{tot}+R)+k_{-1}(R_{tot}-R)$
Dividing both sides by k$_1$ and introducing the constant 2E = R$_{tot}$ - L$_{tot}$ - K$_d$, the rate equation becomes
    ${\frac {1}{k_{1}}}{\frac {dR}{dt}}=-R^{2}+2ER+K_{d}R_{tot}=-\left(R-R_{+}\right)\left(R-R_{-}\right)$
where the two equilibrium concentrations $R_{\pm }\ {\stackrel {\mathrm {def} }{=}}\ E\pm D$ are given by the quadratic formula and D is defined
    $D\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {E^{2}+R_{tot}K_{d}}}$
However, only the $R_{+}$ 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
    $\left\{{\frac {1}{R-R_{+}}}-{\frac {1}{R-R_{-}}}\right\}dR=-2Dk_{1}dt$
whose solution is
    $\log \left|R-R_{+}\right|-\log \left|R-R_{-}\right|=-2Dk_{1}t+\phi _{0}$
or, equivalently,
    $g=exp(-2Dk_{1}t+\phi _{0})$
$R(t)={\frac {R_{+}-gR_{-}}{1-g}}$
for association, and
$R(t)={\frac {R_{+}+gR_{-}}{1+g}}$
for dissociation, respectively; where the integration constant φ$_0$ 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.