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). Biochemistry is the study of the chemical processes in living Organisms It deals with the Structure and function of cellular components such as Chemical kinetics, also known as reaction kinetics is the study of rates of chemical processes In Biochemistry, a receptor is a Protein molecule embedded in either the Plasma membrane or Cytoplasm of a cell to which a mobile signaling In Biochemistry, a ligand ( latin ligare = to bind is a substance that is able to bind to and form a complex with a Biomolecule

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. For other uses of the term steady state see Steady state (disambiguation In Chemistry, a steady state is a situation in which all 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 K_d

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

where k₁ and k_-1 are the forward and backward rate constants, respectively. In Chemical kinetics a reaction rate constant k or \lambda quantifies the speed of a Chemical reaction. 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 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. , $R \ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}]$ ), 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 = 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 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. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree.

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,

g = e x p ( - 2 D k 1 t + φ 0)

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

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. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an In Algebra, the partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its

Kinetics of single receptor/single ligand/single complex binding

See also

Further reading