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| 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 (biochemistry)|receptor(s)]]'' and ''[[ligand (biochemistry)|ligand(s)]]''.
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| 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 (chemistry)|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.
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| A converse goal of receptor-ligand kinetics is to estimate the rate constants and/or [[dissociation constant]]s 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.
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| ==Kinetics of single receptor/single ligand/single complex binding==
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| 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
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| :<math>
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| \mathrm{R} + \mathrm{L} \leftrightarrow \mathrm{C}
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| </math> | |
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| The equilibrium concentrations are related by the [[dissociation constant]] ''K<sub>d</sub>''
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| :<math>
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| K_{d} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{-1}}{k_{1}} = \frac{[\mathrm{R}]_{eq} [\mathrm{L}]_{eq}}{[\mathrm{C}]_{eq}}
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| </math>
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| where ''k<sub>1</sub>'' and ''k<sub>-1</sub>'' are the forward and backward [[rate constant]]s, respectively. The total concentrations of receptor and ligand in the system are constant
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| :<math> | |
| R_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}] + [\mathrm{C}]
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| </math>
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| :<math> | |
| L_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{L}] + [\mathrm{C}]
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| </math>
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| Thus, only one concentration of the three ([R], [L] and [C]) is independent; the other two concentrations may be determined from ''R<sub>tot</sub>'', ''L<sub>tot</sub>'' and the independent concentration.
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| 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., <math>R \ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}]</math>), the kinetic rate equation can be written
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| :<math>
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| \frac{dR}{dt} = -k_{1} R L + k_{-1} C = -k_{1} R (L_{tot} - R_{tot} + R) + k_{-1} (R_{tot} - R)
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| </math>
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| Dividing both sides by ''k''<sub>1</sub> and introducing the constant ''2E = R<sub>tot</sub> - L<sub>tot</sub> - K<sub>d</sub>'', the rate equation becomes
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| :<math>
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| \frac{1}{k_{1}} \frac{dR}{dt} = -R^{2} + 2ER + K_{d}R_{tot} =
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| -\left( R - R_{+}\right) \left( R - R_{-}\right)
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| </math>
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| where the two equilibrium concentrations <math>R_{\pm} \ \stackrel{\mathrm{def}}{=}\ E \pm D</math> are given by the [[quadratic formula]] and the discriminant ''D'' is defined
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| :<math>
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| D \ \stackrel{\mathrm{def}}{=}\ \sqrt{E^{2} + R_{tot} K_{d}}
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| </math>
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| However, only the <math>R_{-}</math> equilibrium is stable, corresponding to the equilibrium observed experimentally.
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| [[Separation of variables]] and a [[partial fraction|partial-fraction expansion]] yield the integrable [[ordinary differential equation]]
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| :<math>
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| \left\{ \frac{1}{R - R_{+}} - \frac{1}{R - R_{-}} \right\} dR = -2 D k_{1} dt
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| </math>
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| whose solution is
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| :<math>
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| \log \left| R - R_{+} \right| - \log \left| R - R_{-} \right| = -2Dk_{1}t + \phi_{0}
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| </math>
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| or, equivalently,
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| :<math>
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| g = exp(-2Dk_{1}t+\phi_{0})
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| </math>
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| <math>
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| R(t) = \frac{R_{+} - gR_{-}}{1 - g}
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| </math>
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| where the integration constant φ<sub>0</sub> is defined
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| :<math>
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| \phi_{0} \ \stackrel{\mathrm{def}}{=}\ \log \left| R(t=0) - R_{+} \right| - \log \left| R(t=0) - R_{-} \right|
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| </math>
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| From this solution, the corresponding solutions for the other concentrations <math>C(t)</math> and <math>L(t)</math> can be obtained.
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| == See also ==
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| * [[Binding potential]]
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| * [[Patlak plot]]
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| * [[Scatchard plot]]
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| ==Further reading==
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| * [[D.A. Lauffenburger]] and [[J.J. Linderman]] (1993) ''Receptors: Models for Binding, Trafficking, and Signaling'', [[Oxford University Press]]. ISBN 0-19-506466-6 (hardcover) and 0-19-510663-6 (paperback)
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| [[Category:Receptors]]
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| [[Category:Chemical kinetics]]
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Friends contact her Claude Gulledge. My home is now in Kansas. My occupation is a messenger. The factor she adores most is to perform handball but she can't make it her profession.
Feel free to surf to my web blog: indianapolisfaith.org