Liu, S.; Wu, B.; Jia, C.

In this study, we obtain the exact switching time distributions between the free and bound gene states for a detailed stochastic model of an autoregulatory genetic feedback loop with transcription, translation, mRNA and protein decay, as well as protein-gene interactions. The analytical solution generalizes and corrects the previous ones obtained in [Phys. Rev. Lett. 101, 118104 (2008)] and [Nat. Commun. 9, 3305 (2018)] for a reduced model of an autoregulatory loop that ignores the mRNA dynamics. We find that when the mRNA dynamics is modelled explicitly, the holding time in the free gene state can produce three shapes of steady-state distributions (decaying, bell-shaped, and bimodal). In particular, the detailed model with both mRNA and protein descriptions can produce a distribution shape that the reduced model fails to capture : the detailed model of a negative (positive) feedback loop can display a bimodal (bell-shaped) holding time distribution, while the reduced model cannot. Interestingly, we also find that an autoregulatory loop can produce a heavy-tailed holding time distribution and the origin of this heavy-tailed phenomenon is clarified using our analytical solution. Finally, we investigate how the distribution shape is affected by the type of feedback, the binding and unbinding rates, and the transcription rates.