Statistical potential - Revision history

2A02:8070:8681:31A0:D496:ABE4:D6E1:DEFD at 22:39, 19 July 2014

2014-07-19T22:39:07Z

← Older revision		Revision as of 00:39, 20 July 2014
Line 1:		Line 1:
	~~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)]]''.		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.<br><br>Feel free to surf to my web blog: [http://Indianapolisfaith.org/UserProfile/tabid/320/userId/242895/Default.aspx indianapolisfaith.org]

	~~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~~.

	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.

	~~==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~~

	:<~~math~~>
	~~\mathrm{R} + \mathrm{L} \leftrightarrow \mathrm{C}~~
	<~~/math~~>

	~~The equilibrium concentrations are related by the [[dissociation constant]] ''K<sub>d</sub>''~~

	~~:<math>~~
	~~K_{d} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{-1}}{k_{1}} = \frac{[\mathrm{R}]_{eq} [\mathrm{L}]_{eq}}{[\mathrm{C}]_{eq}}~~
	~~</math>~~

	~~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~~

	:~~<math>~~
	~~R_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}] +~~ [~~\mathrm{C}]~~
	~~</math>~~

	:~~<math>~~
	~~L_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{L}] + [\mathrm{C}]~~
	</~~math>~~

	~~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~~.

	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

	~~:<math>~~
	~~\frac{dR}{dt} = -k_{1} R L + k_{-1} C = -k_{1} R (L_{tot} - R_{tot} + R) + k_{-1} (R_{tot} - R)~~
	~~</math>~~

	~~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~~

	~~:<math>~~
	~~\frac{1}{k_{1}} \frac{dR}{dt} = -R^{2} + 2ER + K_{d}R_{tot} =~~
	~~-\left( R - R_{+}\right) \left( R - R_{-}\right)~~
	~~</math>~~

	~~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~~

	~~:<math>~~
	~~D \ \stackrel{\mathrm{def}}{=}\ \sqrt{E^{2} + R_{tot} K_{d}}~~
	~~</math>~~

	~~However, only the <math>R_{-}</math> equilibrium is stable, corresponding to the equilibrium observed experimentally.~~

	~~[[Separation of variables]] and a [[partial fraction\|partial-fraction expansion]] yield the integrable [[ordinary differential equation]]~~

	~~:<math>~~
	~~\left\{ \frac{1}{R - R_{+}} - \frac{1}{R - R_{-}} \right\} dR = -2 D k_{1} dt~~
	~~</math>~~

	~~whose solution is~~

	~~:<math>~~
	~~\log \left\| R - R_{+} \right\| - \log \left\| R - R_{-} \right\| = -2Dk_{1}t + \phi_{0}~~
	~~</math>~~

	~~or, equivalently,~~

	~~:<math>~~
	~~g = exp(-2Dk_{1}t+\phi_{0})~~
	</~~math>~~

	~~<math>~~
	~~R(t) = \frac{R_{+} - gR_{-}}{1 - g}~~
	</~~math>~~

	~~where the integration constant φ<sub>0<~~/~~sub> is defined~~

	~~:<math>~~
	~~\phi_{0} \ \stackrel{\mathrm{def}}{=}\ \log \left\| R(t=0) - R_{+} \right\| - \log \left\| R(t=0) - R_{-} \right\|~~
	</~~math>~~

	~~From this solution, the corresponding solutions for the other concentrations <math>C(t)<~~/~~math> and <math>L(t)<~~/~~math> can be obtained.~~

	~~== See also ==~~
	* [[Binding potential]]
	* [[Patlak plot]]
	* [[Scatchard plot]]

	~~==Further reading==~~
	* [[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)~~

	~~[[Category:Receptors]]~~
	~~[[Category:Chemical kinetics]~~]

en>Yobot: Reference before punctuation using AWB (9585)

2013-11-07T17:04:22Z

Reference before punctuation using AWB (9585)

← Older revision		Revision as of 19:04, 7 November 2013
Line 1:		Line 1:
	~~The writer~~ is ~~called Araceli Gulledge~~. ~~To perform croquet~~ is ~~something that I~~'~~ve carried out~~ for ~~many years~~. ~~I am a production~~ and ~~distribution officer~~. ~~Alabama has always been his house~~.<br><br>~~Also visit my website~~ [~~http~~://~~Valkyrjagc~~.~~org~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~14605 http~~://~~Valkyrjagc~~.~~org~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~14605~~]		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)]]''.

			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.

			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.

			==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

			:<math>
			\mathrm{R} + \mathrm{L} \leftrightarrow \mathrm{C}
			</math>

			The equilibrium concentrations are related by the [[dissociation constant]] ''K<sub>d</sub>''

			:<math>
			K_{d} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{-1}}{k_{1}} = \frac{[\mathrm{R}]_{eq} [\mathrm{L}]_{eq}}{[\mathrm{C}]_{eq}}
			</math>

			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

			:<math>
			R_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}] + [\mathrm{C}]
			</math>

			:<math>
			L_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mathrm{L}] + [\mathrm{C}]
			</math>

			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.

			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

			:<math>
			\frac{dR}{dt} = -k_{1} R L + k_{-1} C = -k_{1} R (L_{tot} - R_{tot} + R) + k_{-1} (R_{tot} - R)
			</math>

			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

			:<math>
			\frac{1}{k_{1}} \frac{dR}{dt} = -R^{2} + 2ER + K_{d}R_{tot} =
			-\left( R - R_{+}\right) \left( R - R_{-}\right)
			</math>

			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

			:<math>
			D \ \stackrel{\mathrm{def}}{=}\ \sqrt{E^{2} + R_{tot} K_{d}}
			</math>

			However, only the <math>R_{-}</math> equilibrium is stable, corresponding to the equilibrium observed experimentally.

			[[Separation of variables]] and a [[partial fraction\|partial-fraction expansion]] yield the integrable [[ordinary differential equation]]

			:<math>
			\left\{ \frac{1}{R - R_{+}} - \frac{1}{R - R_{-}} \right\} dR = -2 D k_{1} dt
			</math>

			whose solution is

			:<math>
			\log \left\| R - R_{+} \right\| - \log \left\| R - R_{-} \right\| = -2Dk_{1}t + \phi_{0}
			</math>

			or, equivalently,

			:<math>
			g = exp(-2Dk_{1}t+\phi_{0})
			</math>

			<math>
			R(t) = \frac{R_{+} - gR_{-}}{1 - g}
			</math>

			where the integration constant φ<sub>0</sub> is defined

			:<math>
			\phi_{0} \ \stackrel{\mathrm{def}}{=}\ \log \left\| R(t=0) - R_{+} \right\| - \log \left\| R(t=0) - R_{-} \right\|
			</math>

			From this solution, the corresponding solutions for the other concentrations <math>C(t)</math> and <math>L(t)</math> can be obtained.

			== See also ==
			* [[Binding potential]]
			* [[Patlak plot]]
			* [[Scatchard plot]]

			==Further reading==
			* [[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)

			[[Category:Receptors]]
			[[Category:Chemical kinetics]]

en>My very best wishes at 19:56, 19 March 2012

2012-03-19T19:56:04Z

New page

The writer is called Araceli Gulledge. To perform croquet is something that I've carried out for many years. I am a production and distribution officer. Alabama has always been his house.<br><br>Also visit my website [http://Valkyrjagc.org/index.php?mod=users&action=view&id=14605 http://Valkyrjagc.org/index.php?mod=users&action=view&id=14605]