Diffusion in Cells with Stochastically Gated Gap Junctions

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 rth-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 .