# Binding constant

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The binding constant is a special case of the equilibrium constant ${\displaystyle K}$. It is associated with the binding and unbinding reaction of receptor (R) and ligand (L) molecules, which is formalized as:

${\displaystyle {\rm {R}}+{\rm {L}}\rightleftharpoons {\rm {RL}}}$.

The reaction is characterized by the on-rate constant ${\displaystyle k_{\rm {on}}}$ and the off-rate constant ${\displaystyle k_{\rm {off}}}$, which have units of 1/(concentration time) and 1/time, respectively. In equilibrium, the forward binding transition ${\displaystyle {\rm {R}}+{\rm {L}}\to {\rm {RL}}}$ should be balanced by the backward unbinding transition ${\displaystyle {\rm {RL}}\to {\rm {R}}+{\rm {L}}}$. That is,

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

where ${\displaystyle [{\rm {R}}]}$, ${\displaystyle [{\rm {L}}]}$ and ${\displaystyle [{\rm {RL}}]}$ represent the concentration of unbound free receptors, the concentration of unbound free ligand and the concentration of receptor-ligand complexes. The binding constant, or the association constant ${\displaystyle K_{\rm {a}}}$ is defined by

${\displaystyle K_{\rm {a}}={k_{\rm {on}} \over k_{\rm {off}}}={[{\rm {RL}}] \over {[{\rm {R}}]\,[{\rm {L}}]}}}$.

An often considered quantity is the dissociation constant ${\displaystyle K_{\rm {d}}\equiv 1/K_{\rm {a}}}$, which has the unit of concentration, despite the fact that strictly speaking, all association constants are unitless values. The inclusion of units arises from the simplification that such constants are calculated solely from concentrations, which is not the case. Once chemical activity is factored into the correct form of the equation, a dimensionless value is obtained. For the binding of receptor and ligand molecules in solution, the molar Gibbs free energy ${\displaystyle \Delta G}$, or the binding affinity is related to the dissociation constant ${\displaystyle K_{\rm {d}}}$ via

in which ${\displaystyle R}$ is the ideal gas constant, ${\displaystyle T}$ temperature and the standard reference concentration ${\displaystyle c^{\ominus }}$ = 1 mol / L.