In this paper we analyze the relaxation to steady state of intracellular diffusion in a pair of cells with gapjunction 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 μ, 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 μ → ∞ for a static junction and β → 0 for a stochastically
gated junction, where β 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.