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 $p(r;R,L)$⁠.