Distribution of first-reaction times with target regions on boundaries of shell-like domains

TL;DR

This paper derives exact and approximate probability density functions for the first-reaction times of diffusive molecules interacting with target regions within shell-like geometries, relevant to molecular signaling and escape problems.

Contribution

It provides a novel spectral and self-consistent approximation framework for modeling first-reaction times in complex shell geometries, including explicit solutions for concentric spheres.

Findings

Exact spectral form of the PDF derived

Approximate PDF calculated using self-consistent approximation

Explicit form and accuracy assessment for concentric spheres

Abstract

We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted "onion-shell" geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.