# Scatchard equation

(Redirected from Scatchard plot)

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 Kd denote the dissociation constant between the ligand and receptor. The Scatchard equation is given by

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

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

### n=1 Ligand

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

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

with an on-rate (kon) and off-rate (koff) related to the dissociation constant through Kd=koff/kon. When the system equilibrates,

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

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

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

which is the Scatchard equation for n=1.

### n=2 Ligands

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

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

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

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

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]/(Kd + [L]). Hence, by considering all n binding sites, there will

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

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

## Problems with the method

The Scatchard method is rarely used because it is prone to error. As with the Lineweaver-Burk method, inferring ligand affinity using the Scatchard equation requires using the reciprocal of the free ligand concentration on the y-axis, which compounds small errors in measurement. A modern alternative is to use surface plasmon resonance, which has the added benefit of being able to measure the on-rate and off-rate of ligand-receptor binding.

## References

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