Diffusion in Cells with Stochastically Gated Gap Junctions

Abstract

We analyze a one-dimensional (1D) model of molecules diffusing along a line of $N$ cells that are connected via stochastically gated gap junctions. Each gate switches between an open $(n=0)$ and a closed $(n=1$) state according to a two-state Markov process, and the gates are treated as statistically independent. We proceed by spatially discretizing the stochastic diffusion equation using finite differences and constructing the Chapman--Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We thus generate a hierarchy of equations for the $r$th-order moments of the stochastic concentration, which in the continuum limit take the form of $r$-dimensional parabolic PDEs. We explicitly solve the first-order moment equations for $N=2$ and calculate the effective permeability of the gap junction. For $N>2$ (more than one gap junction), we show that the $N$-cell network has a completely different effective single-gate permeability when each particle (rather than each gate) independently switches between two conformational states $n=0,1$ and can only pass through a gate when $n=0$. This difference is due to the fact that for switching gates, all particles diffuse in the same random environment, resulting in nontrivial statistical correlations. In both cases, the effective single-gate permeability has a nontrivial dependence on the number of cells $N$.