Stochastic synthesis-degradation processes: first-passage properties and connections with resetting

TL;DR

This paper investigates stochastic synthesis-degradation processes in biological systems, analyzing their first-passage times and optimal rates using resetting process theory, with implications for search efficiency and reaction kinetics.

Contribution

It introduces a novel application of resetting process theory to SSD systems, deriving optimal synthesis rates and first-passage properties under various conditions.

Findings

Equal synthesis and degradation rates minimize mean reaction time.

A universal relation determines the critical synthesis rate in bounded domains.

SSD can outperform non-degrading particles in search efficiency.

Abstract

Processes controlled by stochastic synthesis and degradation (SSD) are widespread in biology but their reaction kinetics are not well understood. Using methods borrowed from the theory of resetting processes, we determine the first-passage properties of a collection of independent particles that are synthesized and degraded at constant rates, and follow an arbitrary diffusive process in space. At equal synthesis and degradation rates, the mean reaction time with a target site can be minimized as in stochastic resetting, and a $CV$-criterion is derived. When the degradation rate is held fixed and the synthesis costs are taken into account, an optimal synthesis rate is obtained. In bounded domains, despite particle degradation, SSD improves the mean search time compared to a single non-degrading particle if the synthesis rate exceeds a critical value. The latter obeys a universal relation. We illustrate these findings with Brownian diffusion on the infinite line and in an interval.