Receptor–ligand kinetics

Inbiochemistry,receptor–ligand kineticsis a branch ofchemical kineticsin which the kinetic species are defined by different non-covalent bindings and/or conformations of the molecules involved, which are denoted asreceptor(s)andligand(s).Receptor–ligand bindingkinetics 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 thesteady-state approximation.A less ambitious goal is to determine the finalequilibriumconcentrations 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/ordissociation constantsof 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

Thebinding constantis a special case of theequilibrium 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 thedissociation constantK_d

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

wherek₁andk₋₁are the forward and backwardrate 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 fromR_tot,L_totand 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 byk₁and introducing the constant2E = 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 thequadratic formulaandDis 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 variablesand apartial-fraction expansionyield the integrableordinary 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 φ₀is 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.

See also

• Binding potential
• Patlak plot
• Scatchard plot

Further reading

• D.A. LauffenburgerandJ.J. Linderman(1993)Receptors: Models for Binding, Trafficking, and Signaling,Oxford University Press.ISBN0-19-506466-6(hardcover) and 0-19-510663-6 (paperback)