First-passage time calculations for a gene expression model

The stochastic nature of gene expression can lead to significant cell-to-cell variability in the time at which a certain protein level is attained. This is reflected in the timing of cellular events triggering at critical protein thresholds as well. A problem of interest is to understand how cells regulate gene expression to ensure precise timing of important events. To this end, we consider a gene expression model assuming constitutive expression in translation bursts. We also assume the proteins to be stable. The event timing is formulated as a first-passage time (FPT) problem and stochasticity in FPT for this model is quantified. We also investigate the effect of auto-regulation, a control mechanism often present in cells, on the stochasticity of FPT. In particular, we ask: given FPT threshold of proteins and mean FPT, what form of auto-regulation minimizes variance in FPT? Our results show that the objective is best achieved by having no auto-regulation. Moreover, a smaller mean burst size would result into lower stochasticity. We discuss our results in context of lysis time of E. coli cells infected by a λ phage virus. An optimal lysis time provides evolutionary advantage to λ phage, suggesting a possible regulation to minimize its stochasticity. Our results are consistent with previous studies showing there is no auto-regulation of the protein responsible for lysis. Moreover, congruent with experimental evidences, our analysis predicts that the expression of the lysis protein should have a small burst size.