Modeling the Stochastic Dynamics of Gene Expression in Single Cells: A Birth and Death Markov Chain Analysis

Fluctuations in protein number (noise) caused by the stochasticity in gene expression plays a central role in the dynamic behavior of cellular pathways. Deterministic models capture average cell population behavior and are limited in their relevance in modeling stochastic deviations of gene expression in single cells. In this paper, we develop a birth and death Markov chain model to capture the discrete molecular events of transcription and translation in prokaryotic cells. We derive mathematical models for the expression `burst frequency´ distribution as well as the number of protein molecules per burst. We validate our stochastic models with recent single cell experiments on the lacZ gene in Escherichia Coli. Further, we build a discrete-event stochastic simulation system to study the transient dynamics of lacZ gene expression, quantifying the role of promoters in controlling the `burstiness´ of protein synthesis.