Narrow escape problem in two-shell spherical domains

TL;DR

This study investigates the narrow escape problem in two-shell spherical domains, analyzing how passive Brownian particles' mean first passage time depends on diffusion constants, potential barriers, and shell width, with implications for cellular transport optimization.

Contribution

It provides asymptotic and numerical analysis of the MFPT for passive particles in two-shell domains, revealing conditions for optimal escape times based on system parameters.

Findings

MFPT depends monotonically on diffusion constants and potential barrier height.

MFPT can have a minimum with respect to outer shell width under certain conditions.

Analytical expressions match numerical simulations for optimal parameters.

Abstract

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.