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 that benefits from the knowledge of p(r;R,L).

中文翻译：

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结合受体-配体复合物数量的平衡概率分布

分子结合现象，其中两个分子，称为受体和配体，结合在一起形成配体 - 受体复合物，在生物学中无处不在，对于所有维持生命的过程的准确运作至关重要。单个受体在热平衡时与 L 个周围配体分子中的任何一个形成复合物的概率可以从由 Gibbs-Boltzmann 分布获得的分配函数推导出来。我们将此方法扩展到由 R 受体和 L 配体组成的系统，以推导出概率密度函数 p(r;R,L) 以找到处于热平衡的 r 结合受体-配体复合物。这个扩展使我们能够说明这个问题的两个方面，这些方面在单受体问题中并不明显，即，a) 在 R 和 L 交换下结合复合物数量的平衡分布中预期的对称性和 b) 从化学动力学方程获得的结合复合物数量与表达式中 r 的最大可能值精确对应对于 p(r;R,L)。我们推导出了 r 的数量波动，并提出了一个实际相关的分子传感应用，该应用受益于 p(r;R,L) 的知识。