Analytical time-dependent distributions for gene expression models with complex promoter switching mechanisms

Abstract

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 non-exponential 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 solutions obtained in [SIAM J. Appl. Math. 72, 789-818 (2012)] and [SIAM J. Appl. Math. 79, 1007-1029 (2019)]. 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 the transient and stationary bimodality of the copy number distribution, while molecular memory in the active gene state enhances such bimodality.

Modulation of nuclear and cytoplasmic mRNA fluctuations by time-dependent stimuli: analytical distributions

“…We note that other theoretical studies have sought to derive closed-form time-dependent mRNA distributions for various models of gene expression. These can be classified as follows: (i) those which do not consider a time-varying stimulus [11,[57][58][59][60][61], in that they study how gene expression approaches steady state given a perturbation that is applied at a point in time, e.g. with the initial condition given by mRNA numbers following cell division -in that case, the kinetic rates do not vary with time; (ii) those which consider a stimulus that varies with time [36][37][38][39].…”

Coupling gene expression dynamics to cell size dynamics and cell cycle events: exact and approximate solutions of the extended telegraph model

“…Progress in this direction will be reported in a separate paper. We also anticipate that the results of the present paper can be generalized to include more than two gene states [16–18].…”

“…The chemical master equation (CME) describing the telegraph model can be exactly solved in steady-state, as well as in time [11][12][13]. Extensions of this model to include more than two gene states have also been considered [14][15][16].…”

“…The chemical master equation (CME) describing the telegraph model can be exactly solved in steady-state, as well as in time [12][13][14][15]. Extensions of this model to include more than two gene states have also been considered [16][17][18]. A substantial number of genes are inactive most of the time and in the brief time that they are active, a large number of mRNA molecules are transcribed but not degraded [19].…”

Coupling gene expression dynamics to cell size dynamics and cell cycle events: Exact and approximate solutions of the extended telegraph model

“… 12 , 13 , 14 , 15 Extensions of this model to include more than two gene states have also been considered. 16 , 17 , 18 …”