Scatchard equation

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 (k_on) and off-rate (k_off) related to the dissociation constant through K_d=k_off/k_on. 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.

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.

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

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

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