A quasi-stationary distribution bound for fault analysis in gene regulatory networks.

The inherent stochastic fluctuations in signalling molecules of gene regulatory networks (GRNs) add unpredictability, complicating the design of robust synthetic GRNs that must function within precise ranges. Multi-stable GRNs, such as toggle switches, are central to systems like biosensors and logic gates, but can fail owing to unintended transitions between stable states caused by the fluctuations. Despite their importance, tools to characterize the probability distributions around stable states remain limited. We present a mathematical framework for analysing these multi-stable systems using continuous-time Markov chains (CTMCs) and quasi-stationary distributions (QSDs). This framework is broadly applicable, requiring only that the state space is connected, making it applicable to a variety of systems. We apply the framework to a toggle-switch design from the literature, identifying parameter thresholds at which systems transition from frequent stochastic switching (hours) to stable operation (years to decades), consistent with current experimental insights. We further present upper-bound calculations of false-positive/false-negative rates for population-level biosensor dynamics as supplementary results.