  Intracellular transport in living cells is often spatially inhomogeneous with
an accelerated effective diffusion close to the cell membrane and a ballistic
motion away from the centrosome due to active transport along actin filaments
and microtubules, respectively. Recently it was reported that the mean first
passage time (MFPT) for transport to a specific area on the cell membrane is
minimal for an optimal actin cortex width. In this paper we ask whether this
optimization in a two-compartment domain can also be achieved by passive
Brownian particles. We consider a Brownian motion with different diffusion
constants in the two shells and a potential barrier between the two and
investigate the narrow escape problem by calculating the MFPT for Brownian
particles to reach a small window on the external boundary. In two and three
dimensions, we derive asymptotic expressions for the MFPT in the thin cortex
and small escape region limits confirmed by numerical calculations of the MFPT
using the finite element method and stochastic simulations. From this
analytical and numeric analysis we finally extract the dependence of the MFPT
on the ratio of diffusion constants, the potential barrier height and the width
of the outer shell. The first two are monotonous whereas the last one may have
a minimum for a sufficiently attractive cortex, for which we propose an
analytical expression of the potential barrier height matching very well the
numerical predictions.
