Analytical results for non-Markovian models of bursty gene expression

Modeling stochastic gene expression has long relied on Markovian hypothesis. In recent years, however, this hypothesis is challenged by the increasing availability of time-resolved data. Correspondingly, there is considerable interest in understanding how non-Markovian reaction kinetics of gene expression impact protein variations across a population of genetically identical cells. Here, we analyze a stochastic model of gene expression with arbitrary waiting-time distributions, which includes existing gene models as its special cases. We find that stationary probabilistic behavior of this non-Markovian system is exactly the same as that of an equivalent Markovian system with the same substrates. Based on this fact, we derive analytical results, which provide insight into the roles of feedback regulation and molecular memory in controlling the protein noise and properties of the steady states, which are inaccessible via existing methodology. Our results also provide quantitative insight into diverse cellular processes involving stochastic sources of gene expression and molecular memory.