A Stochastic Model of Gene Expression with Polymerase Recruitment and Pause Release

Transcriptional bursting is a major source of noise in gene expression. The telegraph model of gene expression, whereby transcription switches between on and off states, is the dominant model for bursting. Recently, it was shown that the telegraph model cannot explain a number of experimental observations from perturbation data. Here, we study an alternative model that is consistent with the data and which explicitly describes RNA polymerase recruitment and polymerase pause release, two steps necessary for messenger RNA (mRNA) production. We derive the exact steady-state distribution of mRNA numbers and an approximate steady-state distribution of protein numbers, which are given by generalized hypergeometric functions. The theory is used to calculate the relative sensitivity of the coefficient of variation of mRNA fluctuations for thousands of genes in mouse fibroblasts. This indicates that the size of fluctuations is mostly sensitive to the rate of burst initiation and the mRNA degradation rate. Furthermore, we show that 1) the time-dependent distribution of mRNA numbers is accurately approximated by a modified telegraph model with a Michaelis-Menten like dependence of the effective transcription rate on RNA polymerase abundance, and 2) the model predicts that if the polymerase recruitment rate is comparable or less than the pause release rate, then upon gene replication, the mean number of RNA per cell remains approximately constant. This gene dosage compensation property has been experimentally observed and cannot be explained by the telegraph model with constant rates.