Narrow escape through a funnel and effective diffusion on a crowded membrane

Particles diffusing on a membrane crowded with obstacles have to squeeze between them through funnel-shaped narrow straits. The computation of the mean passage time through the straits is a new narrow escape problem that gives rise to new, hitherto unknown, behavior that we communicate here. The motion through the straits on the coarse scale of the narrow escape time is an effective diffusion with coefficient that varies nonlinearly with the density of obstacles. We calculate the coarse-grained diffusion coefficient on a planar lattice of circular obstacles and use it to estimate the density of obstacles on a neuronal membrane and in a model of a cytoplasm crowded by identical parallel circular rods.