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Gene transcription is a highly stochastic and dynamic process. As a result,the mRNA copy number of a given gene is heterogeneous both between cells andacross time. We present a framework to model gene transcription in populationsof cells with time-varying (stochastic or deterministic) transcription anddegradation rates. Such rates can be understood as upstream cellular drivesrepresenting the effect of different aspects of the cellular environment. Weshow that the full solution of the master equation contains two components: amodel-specific, upstream effective drive, which encapsulates the effect ofcellular drives (e.g., entrainment, periodicity or promoter randomness), and adownstream transcriptional Poissonian part, which is common to all models. Ouranalytical framework treats cell-to-cell and dynamic variability consistently,unifying several approaches in the literature. We apply the obtained solutionto characterise different models of experimental relevance, and to explain theinfluence on gene transcription of synchrony, stationarity, ergodicity, as wellas the effect of time-scales and other dynamic characteristics of drives. Wealso show how the solution can be applied to the analysis of noise sources insingle-cell data, and to reduce the computational cost of stochasticsimulations.

Stochastic models of gene transcription with upstream drives: exact solution and sample path characterization