Scatchard equation: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 16:26, 2 August 2022

The Scatchard equation is an equation used in molecular biology to calculate the affinity and number of binding sites of a receptor for a ligand.^[1] It is named after the American chemist George Scatchard.^[2]

Equation

Throughout this article, [RL] denotes the concentration of a receptor-ligand complex, [R] the concentration of free receptor, and [L] the concentration of free ligand (so that the total concentration of the receptor and ligand are [R]+[RL] and [L]+[RL], respectively). Let n be the number of binding sites for ligand on each receptor molecule, and let n represent the average number of ligands bound to a receptor. Let K_d denote the dissociation constant between the ligand and receptor. The Scatchard equation is given by

{\frac {\bar {n}}{[L]}}={\frac {n}{K_{d}}}-{\frac {\bar {n}}{K_{d}}}

By plotting n/[L] versus n, the Scatchard plot shows that the slope equals to -1/K_d while the x-intercept equals the number of ligand binding sites n.

Derivation

n=1 Ligand

When each receptor has a single ligand binding site, the system is described by

[R]+[L]{\underset {k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL]

with an on-rate (k_on) and off-rate (k_off) related to the dissociation constant through K_d=k_off/k_on. When the system equilibrates,

k_{\text{on}}[R][L]=k_{\text{off}}[RL]

so that the average number of ligands bound to each receptor is given by

{\bar {n}}={\frac {[RL]}{[R]+[RL]}}={\frac {[L]}{K_{d}+[L]}}=(1-{\bar {n}}){\frac {[L]}{K_{d}}}

which is the Scatchard equation for n=1.

n=2 Ligands

When each receptor has two ligand binding sites, the system is governed by

[R]+[L]{\underset {k_{\text{off}}}{\overset {2k_{\text{on}}}{\rightleftharpoons }}}[RL]

[RL]+[L]{\underset {2k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL_{2}].

At equilibrium, the average number of ligands bound to each receptor is given by

{\bar {n}}={\frac {[RL]+2[RL_{2}]}{[R]+[RL]+[RL_{2}]}}={\frac {2{\frac {[L]}{K_{d}}}+2\left({\frac {[L]}{K_{d}}}\right)^{2}}{\left(1+{\frac {[L]}{K_{d}}}\right)^{2}}}={\frac {2[L]}{K_{d}+[L]}}=(2-{\bar {n}}){\frac {[L]}{K_{d}}}

which is equivalent to the Scatchard equation.

General Case of n Ligands

For a receptor with n binding sites that independently bind to the ligand, each binding site will have an average occupancy of [L]/(K_d + [L]). Hence, by considering all n binding sites, there will

{\bar {n}}=n{\frac {[L]}{K_{d}+[L]}}=(n-{\bar {n}}){\frac {[L]}{K_{d}}}.

ligands bound to each receptor on average, from which the Scatchard equation follows.

Problems with the method

The Scatchard method is less used nowadays because of the availability of computer programs that directly fit parameters to binding data. Mathematically, the Scatchard equation is related to Eadie-Hofstee method, which is used to infer kinetic properties from enzyme reaction data. Many modern methods for measuring binding such as surface plasmon resonance and isothermal titration calorimetry provide additional binding parameters that are globally fit by computer-based iterative methods.^{[citation needed]}

References

^ Scatchard, George (1949). "The Attraction of Proteins for Small Molecules and Ions". Annals of the New York Academy of Sciences. 51 (4): 660–672. Bibcode:1949NYASA..51..660S. doi:10.1111/j.1749-6632.1949.tb27297.x. S2CID 83567741.
^ Voet, Donald (1995). Biochemistry, 3rd Ed. John Wiley & Sons, Inc. ISBN 978-0-471-39223-1.

@@ Line 1: / Line 1: @@
-The '''Scatchard equation''' is an equation used in [[molecular biology]] for calculating the [[Receptor affinity|affinity]] constant of a [[ligand]] with a [[protein]].<ref>{{Cite journal|last=Scatchard|first=George|year=1949|title=The Attraction of Proteins for Small Molecules and Ions|url=http://onlinelibrary.wiley.com/wol1/doi/10.1111/j.1749-6632.1949.tb27297.x/abstract|journal=Annals of the New York Academy of Sciences|volume=51|issue=4|pages=660–672|doi=10.1111/j.1749-6632.1949.tb27297.x}}</ref>
+The '''Scatchard equation''' is an equation used in [[molecular biology]] to calculate the [[Receptor affinity|affinity]] and number of binding sites of a receptor for a [[ligand]].<ref>{{Cite journal|last=Scatchard|first=George|year=1949|title=The Attraction of Proteins for Small Molecules and Ions|journal=Annals of the New York Academy of Sciences|volume=51|issue=4|pages=660–672|doi=10.1111/j.1749-6632.1949.tb27297.x|bibcode=1949NYASA..51..660S|s2cid=83567741}}</ref> It is named after the American chemist George Scatchard.<ref name="text">{{cite book |author1=Voet, Donald |title=Biochemistry, 3rd Ed. |year=1995 |publisher=John Wiley & Sons, Inc. |isbn=978-0-471-39223-1 |url-access=registration |url=https://archive.org/details/biochemistry00voet_1 }}</ref>
-It is named after the American chemist George Scatchard<ref name="text">{{cite book |author=Voet, Donald; |title=Biochemistry, 3rd Ed. |year=1995 |publisher= John Wiley & Sons, Inc. |isbn=0-471-39223-5}}</ref> and is sometimes referred to as the '''Rosenthal-Scatchard equation'''.{{fact|date=October 2016}}
 ==Equation==
-The Scatchard equation is given by
+Throughout this article, [''RL''] denotes the concentration of a receptor-ligand complex, [''R''] the concentration of free receptor, and [''L''] the concentration of free ligand (so that the total concentration of the receptor and ligand are [''R'']+[''RL''] and [''L'']+[''RL''], respectively). Let ''n'' be the number of binding sites for ligand on each receptor molecule, and let {{overline|''n''}} represent the average number of ligands bound to a receptor. Let ''K<sub>d</sub>'' denote the [[dissociation constant]] between the ligand and receptor. The Scatchard equation is given by
-:<math>\frac{r}{[L]} = nK_a - rK_a </math>
+:<math>\frac{\bar{n}}{[L]} = \frac{n}{K_d} - \frac{\bar{n}}{K_d} </math>
+By plotting {{overline|''n''}}/[''L''] versus {{overline|''n''}}, the Scatchard plot shows that the slope equals to -1/''K<sub>d</sub>'' while the x-intercept equals the number of ligand binding sites ''n''.
-where
+==Derivation==
-:<math>r = \frac{[L]_{bound}}{[P]_0} </math>
+===''n''=1 Ligand===
-is the ratio of the concentration of bound ligand to total available binding sites, and ''n'' is the number of binding sites per protein molecule.
+When each receptor has a single ligand binding site, the system is described by
-''K<sub>a</sub>'' is the association (affinity) constant from the equation
+:<math>
+[R] + [L] \underset{k_{\text{off}}}{\overset{k_{\text{on}}}{\rightleftharpoons}} [RL]
+</math>
+with an on-rate (''k''<sub>on</sub>) and off-rate (''k''<sub>off</sub>) related to the dissociation constant through ''K<sub>d</sub>''=''k''<sub>off</sub>/''k''<sub>on</sub>. When the system equilibrates,
+:<math>
+k_{\text{on}} [R] [L] = k_{\text{off}} [RL]
+</math>
+so that the average number of ligands bound to each receptor is given by
+:<math>
+\bar{n} = \frac{[RL]}{[R] + [RL]} = \frac{[L]}{K_d + [L]} = (1 - \bar{n}) \frac{[L]}{K_d}
+</math>
+which is the Scatchard equation for ''n''=1.
+===''n''=2 Ligands===
-:<math>K_a = \frac{[LP]}{[L][P]} </math>
+When each receptor has two ligand binding sites, the system is governed by
-Plotting these data, ''r/[L]'' vs ''r'', yields the Scatchard plot with a slope ''-K<sub>a</sub>'' and a Y-intercept of ''nK<sub>a</sub>''. Relative binding affinities between two sites can be distinguished with a line showing identical affinity and a curve showing different affinities.
+:<math>
+[R] + [L] \underset{k_{\text{off}}}{\overset{2k_{\text{on}}}{\rightleftharpoons}} [RL]
+</math>
+:<math>
+[RL] + [L] \underset{2k_{\text{off}}}{\overset{k_{\text{on}}}{\rightleftharpoons}} [RL_2].
+</math>
+At equilibrium, the average number of ligands bound to each receptor is given by
+:<math>
+\bar{n} = \frac{[RL] + 2[RL_2]}{[R] + [RL] + [RL_2]} = \frac{2\frac{[L]}{K_d} + 2 \left( \frac{[L]}{K_d} \right)^2}{\left( 1 + \frac{[L]}{K_d} \right)^2} = \frac{2[L]}{K_d + [L]} = (2 - \bar{n}) \frac{[L]}{K_d}
+</math>
+which is equivalent to the Scatchard equation.
+===General Case of ''n'' Ligands===
-==Scatchard plot==
+For a receptor with ''n'' binding sites that independently bind to the ligand, each binding site will have an average occupancy of [''L'']/(''K<sub>d</sub>'' + [''L'']). Hence, by considering all ''n'' binding sites, there will
+:<math>
+\bar{n} = n \frac{[L]}{K_d + [L]} = (n - \bar{n}) \frac{[L]}{K_d}.
+</math>
+ligands bound to each receptor on average, from which the Scatchard equation follows.
+==Problems with the method==
-[[File:Scatchard plot.svg|thumb|Scatchard Plot showing positive, negative, and no cooperativity.]]
+The Scatchard method is less used nowadays because of the availability of computer programs that directly fit parameters to binding data. Mathematically, the Scatchard equation is related to [[Eadie-Hofstee plot|Eadie-Hofstee method]], which is used to infer kinetic properties from enzyme reaction data. Many modern methods for measuring binding such as [[surface plasmon resonance]] and [[isothermal titration calorimetry]] provide additional binding parameters that are globally fit by computer-based iterative methods.{{CN|date=August 2022}}
-A '''Scatchard plot''' is a plot of the ratio of [[molar concentration|concentration]]s of bound [[ligand (biochemistry)|ligand]] to unbound ligand versus the bound ligand concentration. It is a method for analyzing data for freely reversible ligand/receptor binding interactions. The plot yields a straight line of slope -K, where K is the [[affinity constant]] for ligand binding. The affinity constant is the inverse of the dissociation constant. The intercept on the X axis is Bmax.<ref name="text" /> It is sometimes the case that binding data does not form a straight line when plotted in a Scatchard plot. Such is the case when ligand bound to [[Substrate (biochemistry)|substrate]] is not allowed to achieve equilibrium before the binding is measured or binding is cooperative.<ref name="website">{{cite book |author=Gross, David |title=Physical Chemistry: Applications in the Life Sciences |url=http://owl.cs.umass.edu/extapps/chemistry/ebook/ch7/Sec13.htm }}{{dead link|date=May 2018 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
-In a Scatchard plot, assumptions of independence in [[linear regression]] model is violated because B (bound ligand) is used in the X and Y axes.  Generally, Scatchard and [[Lineweaver-Burk plot]]s are outdated.  Their original intention was to transform the data into [[linear representation]]s of the original data such that linear regression methods could be applied.  These transformations frequently distort [[experimental error]] and can be misleading if results are not accurate.<ref name="website2">{{failed verification|date=February 2014}}
-{{cite web| url= http://www.graphpad.com/support/faqid/1748/| title = GraphPad FAQ: Saturation Binding Curves and Scatchard Plots}}</ref>
 ==References==
@@ Line 39: / Line 61: @@
 [[Category:Biochemistry methods]]
 [[Category:Proteins]]
-{{biochem-stub}}