Non-Markovian gene expression

TL;DR

This paper investigates non-Markovian gene expression models with fat-tailed waiting times, revealing two distinct dynamical regimes separated by the mean first passage time, affecting ergodicity and population variability.

Contribution

It introduces two non-Markovian gene expression models with fat-tailed waiting times and characterizes their scaling regimes and ergodic properties.

Findings

Two scaling regimes separated by MFPT

Stationary and ergodic dynamics at short times

Nonstationary and nonergodic behavior at long times

Abstract

We study two non-Markovian gene-expression models in which protein production is a stochastic process with a fat-tailed non-exponential waiting time distribution (WTD). For both models, we find two distinct scaling regimes separated by an exponentially long time, proportional to the mean first passage time (MFPT) to a ground state (with zero proteins) of the dynamics, from which the system can only exit via a non-exponential reaction. At times shorter than the MFPT the dynamics are stationary and ergodic, entailing similarity across different realizations of the same process, with an increased Fano factor of the protein distribution, even when the WTD has a finite cutoff. Notably, at times longer than the MFPT the dynamics are nonstationary and nonergodic, entailing significant variability across different realizations. The MFPT to the ground state is shown to directly affect the average population sizes and we postulate that the transition to nonergodicity is universal in such non-Markovian models.