Computing Extinction Barriers in a Quorum-Sensing Reaction Network

Abstract

We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $\Delta V(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{N\Delta V(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we check it against mean extinction times obtained without the action solver, i.e., from the backward equation of the jump process on a truncated lattice and from exact stochastic simulation. Over a range of $r$, the minimum-action barriers satisfy $\Delta V(r)=\Delta V_{\infty }+O(1/r)$, where $\Delta V_{\infty }$ is obtained by eliminating the signal first, and at $r=1$ the barrier is 55% larger than $\Delta V_{\infty }$. A formal first-order expansion attributes this increase to the lag of the signal, of which the fluctuations of the signal remove a quarter. The saddle barrier $\Delta V(r)$ is also the least action needed to enter the basin of extinction, whereas the density threshold can be crossed more cheaply, by paths which keep the signal high and are usually followed by recovery. The mean extinction times of the backward equation, which include failed attempts, have a prefactor which tends to a constant, and they grow at a rate within $10^{-5}$ of $\Delta V(r)$ at every tested rate.

arXiv comment28 pages, 5 figures, 3 tables. Submitted to the MMSC 2026 proceedings (EPiC Series in Computing). Code and data: https://doi.org/10.5281/zenodo.23348482