A Quasi-Stationary Distribution Bound for Fault Analysis in Gene Regulatory Networks

The inherent stochastic fluctuations in signaling 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 due 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 to analyze these multi-stable systems using continuous-time Markov chains (CTMCs) and quasi-stationary distributions. This framework is broadly applicable, requiring only that the state space is connected, making it applicable to a variety of systems. We then apply the framework to current examples from the literature and conclude that our method provides quantitative design principles for toggle switch design that match current experimental insights, identifying parameter thresholds where systems transition from frequent stochastic switching (hours) to stable operation (years to decades) and demonstrate upper bound calculations for false positive/negative rates in population-level biosensor dynamics.