Optimal auto-regulation to minimize first-passage time variability in protein level

The timing of cellular events is inherently random because of the probabilistic nature of gene expression. Yet cells manage to have precise timing of important events. Here, we study how gene expression could possibly be regulated to precisely schedule timing of an event around a given time. Event timing is modeled as the first-passage time (FPT) for a protein´s level to cross a critical threshold. Considering auto-regulation as a possible regulatory mechanism, we investigate what form of auto-regulation would lead to minimum stochasticity in FPT around a fixed time. We formulate a stochastic gene expression model and show that under certain assumptions, it reduces to a birth-death process. Our results show that when the death rate is zero, the objective is best achieved when all of the birth rates are equal. On the contrary, when the death rate is non-zero, the optimal birth rates are not equal. In terms of the gene expression model, these results illustrate that when protein does not degrade, stochasticity in FPT around a given time is minimized when there is no auto-regulation of its expression. However, when the protein degrades, some form of auto-regulation is required to achieve this. These results are consistent with experimental findings for the lysis time stochasticity in λ phage.