Minimal diffusion formulation of Markov chain ensembles and its application to ion channel clusters

We study ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. A diffusion approximation, composed of two specifically coupled Ornstein-Uhlenbeck processes in stochastic differential equation representation, is formulated to deduce how the number of chains in a given particular state evolves in time. This particular form of the formulation builds upon a theoretical argument adduced here. The formulation is minimal in the sense that it is always a two-dimensional stochastic process regardless of the state space size or the transition matrix density, and that it requires no matrix square root operations. A set of criteria, put forward here as to be necessarily captured by any consistent approximation scheme, is used together with the master equation to determine uniquely the parameter values and noise variances in the formulation. The model is applied to the gating dynamics in ion channel clusters.