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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> |
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The '''Scatchard equation''' is an equation for calculating the [[Dissociation_constant#Protein-Ligand_binding|affinity]] constant of a [[ligand]] with a [[protein]]. The Scatchard equation is given by |
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==Equation== |
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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 |
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where <em>r</em> is the ratio of the concentration of bound ligand to total available binding sites, <em>c</em> is the concentration of free ligand, and <em>n</em> is the number of binding sites per protein molecule. |
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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''. |
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<em>K<sub>a</sub></em> is the association (affinity) constant from the equation |
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==Derivation== |
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:<math>K_a = \frac{[Ab-Ag]}{[Ab][Ag]} </math> |
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===''n''=1 Ligand=== |
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where Ab is the binding site on the antibody, Ag is a monovalent antigen, and Ag-Ab is antigen-bound antibody. |
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When each receptor has a single ligand binding site, the system is described by |
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The Scatchard equation is sometimes referred to as the Rosenthal-Scatchard equation. |
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:<math> |
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[R] + [L] \underset{k_{\text{off}}}{\overset{k_{\text{on}}}{\rightleftharpoons}} [RL] |
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</math> |
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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, |
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:<math> |
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k_{\text{on}} [R] [L] = k_{\text{off}} [RL] |
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</math> |
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so that the average number of ligands bound to each receptor is given by |
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:<math> |
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\bar{n} = \frac{[RL]}{[R] + [RL]} = \frac{[L]}{K_d + [L]} = (1 - \bar{n}) \frac{[L]}{K_d} |
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</math> |
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which is the Scatchard equation for ''n''=1. |
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===''n''=2 Ligands=== |
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Plotting this data, <em>r/c</em> vs <em>r</em>, yields the Scatchard plot with a slope <em>-K<sub>a</sub></em> and an X-intercept of <em>nK<sub>a</sub></em>. Relative binding affinities between two sites can be distinguished with a line showing identical affinity and a curve showing different affinities. |
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When each receptor has two ligand binding sites, the system is governed by |
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The Scatchard equation is named after the former [[MIT Chemistry Department]] member George Scatchard, an American chemist, 1892–1973. |
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:<math> |
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[R] + [L] \underset{k_{\text{off}}}{\overset{2k_{\text{on}}}{\rightleftharpoons}} [RL] |
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</math> |
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:<math> |
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[RL] + [L] \underset{2k_{\text{off}}}{\overset{k_{\text{on}}}{\rightleftharpoons}} [RL_2]. |
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</math> |
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At equilibrium, the average number of ligands bound to each receptor is given by |
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:<math> |
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\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} |
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</math> |
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which is equivalent to the Scatchard equation. |
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===General Case of ''n'' Ligands=== |
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A real-life example: |
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A binding experiment using a regular spectrophotometer is a real-life application of this equation. Here we define he change in absorbance at differing concentrations of ligand as delta A (at a constant wavelength, typically whichever one gives you the largest number) The concentration of the protein is held constant, and the absorbance is measured at different concentrations of total ligand until everything is saturated. The fraction saturation (represented by the greek letter Ɵ) is equal to the absorbance minus the absorbance when the ligand concentration is zero, divided by the absorbance at saturation minus the absorbance when the ligand concentration is zero. Or to put it another way, |
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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 |
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<math>Ɵ=ΔA/ΔAmax</math> |
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:<math> |
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Now we have to take into account the number of binding sites per molecule of protein. This requires |
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\bar{n} = n \frac{[L]}{K_d + [L]} = (n - \bar{n}) \frac{[L]}{K_d}. |
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a second equation <math>nƟ=v</math> where n is the number of binding sites. If you don't know the |
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</math> |
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number of binding sites, take v to equal the amount of ligand bound (assume all ligand binds until |
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ligands bound to each receptor on average, from which the Scatchard equation follows. |
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saturated, and throw out the point where it is saturated) divided by the total protein. |
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If you plot v/L versus 1/v you will get a nice linear plot with a slope equal to -Ka, a y-intercept |
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==Problems with the method== |
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of nKa, and an x-intercept of n. CAUTION!!! If you wind up with a data set that is NOT linear by |
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the Scatchard plot, you have COOPERATIVITY. In case of cooperativity, please see the Hill equation. |
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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}} |
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== See == |
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* [http://www.graphpad.com/curvefit/scatchard_plots.htm some explanation] |
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==References== |
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* [http://www.unmc.edu/Pharmacology/receptortutorial/saturation/rosenthal.htm explanation] |
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{{reflist}} |
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* [http://www.unmc.edu/Pharmacology/receptortutorial/derivations/derivation_rosenthal.htm derivation] |
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== Further reading == |
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* [https://web.archive.org/web/20091007171045/http://www.biochem.oulu.fi/Biocomputing/juffer/Teaching/PhysicalBiochemistry/PhysBiochem-protein-ligand.pdf lecture with derivation] ([https://web.archive.org/web/20091007171045/http://www.biochem.oulu.fi/Biocomputing/juffer/Teaching/PhysicalBiochemistry/PhysBiochem-protein-ligand.pdf Archived version at web.archive.org]) |
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[[Category:Biochemistry methods]] |
[[Category:Biochemistry methods]] |
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[[Category:Proteins]] |
[[Category:Proteins]] |
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{{biochem-stub}} |
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[[de:Scatchard-Diagramm]] |
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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
[edit]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 Kd denote the dissociation constant between the ligand and receptor. The Scatchard equation is given by
![{\displaystyle {\frac {\bar {n}}{[L]}}={\frac {n}{K_{d}}}-{\frac {\bar {n}}{K_{d}}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a4f44dbadee55328a7503113c8517057355ffe5d)
By plotting n/[L] versus n, the Scatchard plot shows that the slope equals to -1/Kd while the x-intercept equals the number of ligand binding sites n.
Derivation
[edit]n=1 Ligand
[edit]When each receptor has a single ligand binding site, the system is described by
![{\displaystyle [R]+[L]{\underset {k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9125b7abb98ebb3371740699753f79d45558d2bf)
with an on-rate (kon) and off-rate (koff) related to the dissociation constant through Kd=koff/kon. When the system equilibrates,
![{\displaystyle k_{\text{on}}[R][L]=k_{\text{off}}[RL]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b3389e39bfe13d0e3f95b6ee07d3c4d6f7684ca4)
so that the average number of ligands bound to each receptor is given by
![{\displaystyle {\bar {n}}={\frac {[RL]}{[R]+[RL]}}={\frac {[L]}{K_{d}+[L]}}=(1-{\bar {n}}){\frac {[L]}{K_{d}}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/0749e8b8d666bfa1a6625feb26628f5f7da37089)
which is the Scatchard equation for n=1.
n=2 Ligands
[edit]When each receptor has two ligand binding sites, the system is governed by
![{\displaystyle [R]+[L]{\underset {k_{\text{off}}}{\overset {2k_{\text{on}}}{\rightleftharpoons }}}[RL]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/96a14fc3ea506b69bcd316cbc6f75975936c74d6)
![{\displaystyle [RL]+[L]{\underset {2k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL_{2}].}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b7bbf31f522c79d2e31ed1fbf5f5e07208d30f8f)
At equilibrium, the average number of ligands bound to each receptor is given by
![{\displaystyle {\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}}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b1c4c67528cdbd4b6f66ac91685b4c3e11fa90bb)
which is equivalent to the Scatchard equation.
General Case of n Ligands
[edit]For a receptor with n binding sites that independently bind to the ligand, each binding site will have an average occupancy of [L]/(Kd + [L]). Hence, by considering all n binding sites, there will
![{\displaystyle {\bar {n}}=n{\frac {[L]}{K_{d}+[L]}}=(n-{\bar {n}}){\frac {[L]}{K_{d}}}.}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/0b687cf3a777212ba0df3f51d3d6ec1eede8fc07)
ligands bound to each receptor on average, from which the Scatchard equation follows.
Problems with the method
[edit]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
[edit]- ^ 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.