Narrow Escape and Narrow Capture Problems in Three Dimensions: Solutions and the Optimization of Trap Configurations

Narrow escape (NE) problems are concerned with the calculation of the mean first passage time (MFPT) for a Brownian particle to escape a domain whose boundary contains N small windows (traps). NE problems arise in escape kinetics modeling in chemistry and cell biology, including receptor trafficking in synaptic membranes and RNA transport through nuclear pores. The related Narrow capture (NC) problems are characterized by the presence of small traps within the domain volume; such traps may be fully absorbing, or have absorbing and reflecting boundary parts. The MFPT of Brownian particles traveling in domains with traps is commonly modeled using a linear Poisson problem with Dirichlet-Neumann boundary conditions. We provide an overview of recent analytical and numerical work pertaining to the understanding and solution of different variants of NE and NC problems in three dimensions. The discussion includes asymptotic MFPT expressions in in the limit of small trap sizes, the cases of spherical and non-spherical domains, same and different trap sizes, the dilute trap fraction limit and MFPT scaling laws for N 1 traps, and the global optimization of trap positions to seek globally and locally optimal MFPT-minimizing trap arrangements. We also present recent comparisons of asymptotic and numerical solutions of NE problems to results of full numerical Brownian motion simulations, in the usual case of constant diffusivity, as well as considering more realistic anisotropic diffusion near the domain boundary.