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In this paper we analyze the relaxation to steady-state of intracellulardiffusion in a pair of cells with gap-junction coupling. Gap junctions areprevalent in most animal organs and tissues, providing a direct diffusionpathway for both electrical and chemical communication between cells. Mostanalytical models of gap junctions focus on the steady-state diffusive flux andthe associated effective diffusivity. Here we investigate the relaxation tosteady state in terms of the so-called local accumulation time. The latter iscommonly used to estimate the time to form a protein concentration gradientduring morphogenesis. The basic idea is to treat the fractional deviation fromthe steady-state concentration as a cumulative distribution for the localaccumulation time. One of the useful features of the local accumulation time isthat it takes into account the fact that different spatial regions can relax atdifferent rates. We consider both static and dynamic gap junction models. Theformer treats the gap junction as a resistive channel with effectivepermeability $\mu$, whereas the latter represents the gap junction as astochastic gate that randomly switches between an open and closed state. Thelocal accumulation time is calculated by solving the diffusion equation inLaplace space and then taking the small-$s$ limit. We show that theaccumulation time is a monotonically increasing function of spatial position,with a jump discontinuity at the gap junction. This discontinuity vanishes inthe limit $\mu \rightarrow \infty$ for a static junction and $\beta \rightarrow0$ for a stochastically-gated junction, where $\beta$ is the rate at which thegate closes.  Finally, our results are generalized to the case of a linear array of cellswith nearest neighbor gap junction coupling.

Local accumulation time for diffusion in cells with gap junction coupling