  In this paper we analyze the relaxation to steady-state of intracellular
diffusion in a pair of cells with gap-junction coupling. Gap junctions are
prevalent in most animal organs and tissues, providing a direct diffusion
pathway for both electrical and chemical communication between cells. Most
analytical models of gap junctions focus on the steady-state diffusive flux and
the associated effective diffusivity. Here we investigate the relaxation to
steady state in terms of the so-called local accumulation time. The latter is
commonly used to estimate the time to form a protein concentration gradient
during morphogenesis. The basic idea is to treat the fractional deviation from
the steady-state concentration as a cumulative distribution for the local
accumulation time. One of the useful features of the local accumulation time is
that it takes into account the fact that different spatial regions can relax at
different rates. We consider both static and dynamic gap junction models. The
former treats the gap junction as a resistive channel with effective
permeability $\mu$, whereas the latter represents the gap junction as a
stochastic gate that randomly switches between an open and closed state. The
local accumulation time is calculated by solving the diffusion equation in
Laplace space and then taking the small-$s$ limit. We show that the
accumulation time is a monotonically increasing function of spatial position,
with a jump discontinuity at the gap junction. This discontinuity vanishes in
the limit $\mu \rightarrow \infty$ for a static junction and $\beta \rightarrow
0$ for a stochastically-gated junction, where $\beta$ is the rate at which the
gate closes.
  Finally, our results are generalized to the case of a linear array of cells
with nearest neighbor gap junction coupling.
