Dynamical Phase Transitions in Models of Stochastic Gene Expression Conditioned on Large Deviations

Dynamical phase transitions (DPTs) represent finite-size crossovers between distinct steady-state fluctuation regimes. We investigate DPTs in a general class of stochastic models of gene expression with reducible promoter dynamics conditioned on large deviations. Although DPTs have been observed in many stochastic systems, characterizing their corresponding conditioned dynamics remains an open problem. Here, we analytically characterize the large deviation functions for the class of models under consideration and identify regions in the parameter space that give rise to DPTs. Developing a framework for characterizing the conditioned dynamics, we find that the optimal fluctuation regime acquires a deterministic component, losing the original structure of a Markovian arrival process. Remarkably, this process still captures the most probable stochastic trajectories corresponding to DPTs, which we confirm using numerical simulations. Our results give insight into how DPTs control the likelihood of rare events in gene expression and the techniques used in our analysis can be readily extended to further gain novel insights on DPTs in other systems of interest.