First-passage times of multiple diffusing particles with reversible target-binding kinetics

TL;DR

This paper develops an approximate renewal-based method to analyze the distribution of reaction times for multiple diffusing particles with reversible target-binding, highlighting how parameters influence mean reaction times.

Contribution

It introduces a novel renewal-based approximation for reaction time distribution in reversible binding scenarios, extending prior irreversible models.

Findings

Approximate probability density for reaction times is highly accurate across parameters.

Mean reaction time depends significantly on binding/unbinding rates, number of particles, and binding threshold.

Reversible binding significantly alters reaction time distribution compared to irreversible cases.

Abstract

We investigate a class of diffusion-controlled reactions that are initiated at the time instance when a prescribed number $K$ among $N$ particles independently diffusing in a solvent are simultaneously bound to a target region. In the irreversible target-binding setting, the particles that bind to the target stay there forever, and the reaction time is the $K$-th fastest first-passage time to the target, whose distribution is well-known. In turn, reversible binding, which is common for most applications, renders theoretical analysis much more challenging and drastically changes the distribution of reaction times. We develop a renewal-based approach to derive an approximate solution for the probability density of the reaction time. This approximation turns out to be remarkably accurate for a broad range of parameters. We also analyze the dependence of the mean reaction time or, equivalently, the inverse reaction rate, on the main parameters such as $K$, $N$, and binding/unbinding constants. Some biophysical applications and further perspectives are briefly discussed.