Regulatory strategies to schedule threshold crossing of protein levels at a prescribed time

The timing of diverse cellular processes is based on the instant when the concentration of regulatory proteins crosses a critical threshold level. Hence, noise mechanisms inherent to these protein synthesis pathways drive statistical fluctuations in such events timing. How to express proteins ensuring both the threshold crossing at a prescribed time and minimal timing fluctuations? To find this optimal strategy, we formulate a model where protein molecules are synthesized in random bursts of gene activity. The burst frequency depends on the protein level creating a feedback loop, and cellular growth dilutes protein concentration between consecutive bursts. Counterintuitively, our analysis shows that positive feedback in protein production is best for minimizing variability in threshold-crossing times. We analytically predict the optimal feedback strength in terms of the dilution rate. As a corollary to our result, a no-feedback strategy emerges as the optimal strategy in the absence of dilution. We further consider other noise sources, such as randomness in either the initial condition or the threshold level, and find that in many cases, we need either strongly negative or positive feedback for precise scheduling for events.