Equilibrium probability distribution for number of bound receptor-ligand complexes

The phenomenon of molecular binding, where two molecules, referred to as a receptor and a ligand, bind together to form a ligand-receptor complex, is ubiquitous in biology and essential for the accurate functioning of all life-sustaining processes. The probability of a single receptor forming a complex with any one of L surrounding ligand molecules at thermal equilibrium can be derived from a partition function obtained from the Gibbs-Boltzmann distribution. We extend this approach to a system consisting of R receptors and L ligands to derive the probability density function p r ; R , L to find r bound receptor-ligand complexes at thermal equilibrium. This extension allows us to illustrate two aspects of this problem which are not apparent in the single receptor problem, namely, (a) a symmetry to be expected in the equilibrium distribution of the number of bound complexes under exchange of R and L and (b) the number of bound complexes obtained from chemical kinetic equations has an exact correspondence to the maximum probable value of r from the expression for p r ; R , L. We derive the number fluctuations of r and present a practically relevant molecular sensing application which benefits from the knowledge of p ( r ; R , L ).