Abstract
In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small ‘escape window’ in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container’s boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.
Highlights
The Narrow Escape Problem (NEP) describes the search by a diffusing particle for a small ‘escape window’ (EW) in the otherwise impermeable boundary of a finite domain [1, 2]
Nanomolar concentrations of specific signalling molecules in cells, the first-reaction time (FRT) probability density function (PDF) of a diffusing molecule with its target in the cell volume was shown to be strongly defocused and geometry-controlled [47, 51, 52]. These analyses demonstrate that the concept of mean FRTs is no longer adequate in such settings involving very low concentrations
One invariably needs knowledge of the full PDF to describe such systems faithfully. Complementary to this situation when a particle is released inside a finite volume and needs to react with a target somewhere else in this volume, we here analysed the important case when a diffusing particle is released inside a bounded volume and needs to react with a target on the boundary
Summary
In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and this work must maintain eventually leaves it through a small ‘escape window’ in the otherwise impermeable boundary, once it attribution to the author(s) and the title of arrives to this window and crosses an entropic barrier at the entrance to it. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions These results add new perspectives and offer a broad comprehension of various features of the by- classical NEP that are relevant for numerous biological and technological systems
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