Effective Permeability of a Gap Junction with Age-Structured Switching

We analyze the diffusion equation in a bounded interval with a stochastically gated interior barrier at the center of the domain. This represents a stochastically gated gap junction linking a pair of identical cells. Previous work has modeled the switching of the gate as a two-state Markov process and used the theory of diffusion in randomly switching environments to derive an expression for the effective permeability of the gap junction. In this paper we extend the analysis of gap junction permeability to the case of a gate with age-structured switching. The latter could reflect the existence of a set of hidden internal states such that the statistics of the non-Markovian two-state model matches the statistics of a higher-dimensional Markov process. Using a combination of the method of characteristics and transform methods, we solve the partial differential equations for the expectations of the stochastic concentration, conditioned on the state of the gate and after integrating out the residence time of the age-structured process. This allows us to determine the jump discontinuity of the concentration at the gap junction and thus the effective permeability. We then use stochastic analysis to show that the solution to the stochastic PDE is a certain statistic of a single Brownian particle diffusing in a stochastically fluctuating environment. In addition to providing a simple probabilistic interpretation of the stochastic PDE, this representation enables an efficient numerical approximation of the solution of the PDE by Monte Carlo simulations of a single diffusing particle. The latter is used to establish that our analytical results match those obtained from Monte Carlo simulations for a variety of age-structured distributions.