Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

Propensity functions of the Hill-type are commonly used to model transcriptional regulation in stochastic models of gene expression. This leads to an effective reduced master equation for the mRNA and protein dynamics only. Based on deterministic considerations, it is often stated or tacitly assumed that such models are valid in the limit of rapid promoter switching. Here, starting from the chemical master equation describing promoter-protein interactions, mRNA transcription, protein translation and decay, we prove that in the limit of fast promoter switching, the distribution of protein numbers is different than that given by standard stochastic models with Hill-type propensities. We show the differences are pronounced whenever the protein-DNA binding rate is much larger than the unbinding rate, a special case of fast promoter switching. Furthermore we show using both theory and simulations that use of the standard stochastic models leads to drastically incorrect predictions for the switching properties of positive feedback loops and that these differences decrease with increasing mean protein burst size. Our results confirm that commonly used stochastic models of gene regulatory networks are only accurate in a subset of the parameter space consistent with rapid promoter switching.\n\nStatement of SignificanceA large number of models of gene regulatory networks in the literature assume that since promoter switching is fast then transcriptional regulation can be effectively modeled using Hill functions. While this approach can be rigorously justified for deterministic models, it is presently unclear if it is also the case for stochastic models. In this article we prove that this is not the case, i.e. stochastic models of gene regulatory systems, namely those with feedback loops, describing transcriptional regulation using Hill functions are only valid in a subset of parameter conditions consistent with fast promoter switching. We identify parameter regimes where these models are correct and where their predictions cannot be trusted.