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.Classical gene expression models assume exponential switching time distributions between the active and inactive promoter states. However, recent experiments have shown that many genes in mammalian cells may produce nonexponential switching time distributions, implying the existence of multiple promoter states and molecular memory in the promoter switching dynamics. Here we analytically solve a gene expression model with random bursting and complex promoter switching, and derive the time-dependent distributions of the mRNA and protein copy numbers, generalizing the steady-state solution obtained in T. Zhou and J. Zhang, SIAM J. Appl. Math., 72 (2012), pp. 789–818 and U. Herbach, SIAM J. Appl. Math., 79 (2019), pp. 1007–1029. Using multiscale simplification techniques, we find that molecular memory has no influence on the time-dependent distribution when promoter switching is very fast or very slow, while it significantly affects the distribution when promoter switching is neither too fast nor too slow. By analyzing the dynamical phase diagram of the system, we also find that molecular memory in the inactive gene state weakens transient and stationary bimodality of the copy number distribution, while molecular memory in the active gene state enhances such bimodality.Keywordstransient distributionburstingmolecular memorymultiscale simplificationMSC codes60J2760J2892C4078A7092B05
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