Distribution Modes and Their Corresponding Parameter Regions in Stochastic Gene Transcription

Abstract

Gene transcription is a stochastic process, manifested by heterogeneous distribution of Messenger RNA (mRNA) copy numbers in a population of isogenic cells. The most panoramic information for the stochasticity is captured by the mass function $P_m$, defined as the probability that there are $m$ mRNA molecules for a gene of interest in one cell. Although the two-state model of stochastic gene transcription has been the most prevailing mathematical framework to interpret gene expression data and gain mechanistic insights into the process of transcription, several fundamental questions on $P_m$ remain open, including these: What types of the distribution can the two-state model generate? Given the values of the system parameters, can we predict the distribution type? Our major aim in this paper is to tackle these important questions analytically. We prove that the two-state model can generate, and generate only, the decaying, unimodal, or bimodal steady-state distributions. We also decompose the system parameter space into simply connected regions, each of which corresponds to exactly one of the three distributions.