Narrow escape in composite domains forming heterogeneous networks

Abstract

Cellular networks are often composed of thin tubules connecting much larger node compartments. These structures serve for active or diffusion transport of proteins. Examples are glial networks in the brain, the endoplasmic reticulum in cells or dendritic spines. In this latter case, a large ball forming the head is connected to the dendrite by a narrow passage. In all cases, how the transport of molecules, ions or proteins is regulated determines the time scale of chemical reactions or signal transduction. In the present study, based on modeling diffusion in three dimensions, we compute the mean time for a Brownian particle to reach a narrow target inside such a composite network made of tubules connected to spherical nodes. We derive asymptotic formulas by solving a mixed Neumann–Dirichlet boundary value problem with small Dirichlet part. We first consider the general case of a network domain organized in a 2-D lattice structure that consists of spherical ball compartments connected via narrow cylindrical passages. For a single target located on the boundary of one of the spherical domains, we derive a sparse linear system of equations for each Mean First Passage Time (MFPT) averaged over the different compartments. We then consider a composite domain consisting of a spherical head-like domain, with narrow absorbing targets on its boundary, that is connected to a large cylinder via a narrow cylindrical neck. For Brownian particles starting within the neck, we derive an asymptotic formula for the MFPT. When diffusing particles can be absorbed upon hitting additional absorbing boundaries of the large cylindrical compartment, we derive asymptotic formulas for the splitting probability and conditional MFPT to reach a target on the spherical head. We compare these formulas with numerical solutions of the mixed boundary value problem and with Brownian simulations, allowing to explore geometrical parameter ranges. To conclude, the present analysis reveals that the mean arrival time, driven by diffusion in heterogeneous networks, is controlled by the targets and tubules sizes, as well as the size of the nodal compartments.