First passage time stochasticity in a gene network with feedback regulation

The inherent probabilistic nature of the biochemical reactions, and low copy number of species can lead to stochasticity in gene expression across identical cells. Interestingly, this would lead to stochasticity in the event timing as well. We investigate a gene expression model for stochasticity in event timing, specifically the events that occur when a critical protein threshold is achieved for the first time, called the first passage time (FPT). We first consider an unregulated gene expression model, and examine the conditions under which stochasticity in FPT will be minimum, given a fixed threshold and mean of FPT. We find that stochasticity in FPT decreases with a decrease in mean burst size (number of proteins produced from an mRNA before it degrades) in the gene expression. Further, we study effect of feedback regulation of transcription rate, a regulation motif known to affect stochasticity in gene expression, on the stochasticity of FPT. Our analysis reveals that such a regulation would rather result into higher stochasticity of FPT as compared with no regulation case. We relate the results obtained with time taken by λ phage-infected E. coli cells to lyse and infer the regulatory motif responsible for preciseness of the lysis time.