Optimizing Precision in Cellular Clocks Through Self-Regulated Accumulation Of Molecules

The timing of cellular events is often determined by the instant when the level of a chemical species crosses a threshold from an initial value of zero. To attenuate the inherent random fluctuations in this first-passage time, the production rate can be a function of the current species level by implementing a feedback or self-regulation. In this study, we aim to identify the self-regulation strategy that minimizes timing noise around a specified mean first-passage time. Previous results showed that no-feedback is the optimal strategy in the absence of molecule degradation. Based on this result, we study the case where the species may undergo degradation. First, we set the feedback to be a linear function of the number of molecules. We derive analytical formulas concluding that the optimal strategy involves positive feedback, whereby the production rate increases with the number of molecules. When feedback constraints are removed, our numerical methods suggest that a mixed self-regulation scheme is optimal. This strategy involves an initial positive feedback phase, followed by a sharp transition to a negative feedback scheme in proximity to the threshold. In summary, this research quantifies the fundamental limits of timing noise reduction, considering different strategies of self-regulation in situations where time-keeping species may be subjected to degradation.