Mixed-Feedback Architectures for Precise Event Timing Through Stochastic Accumulation of Biomolecules

Abstract

The timing of biochemical events is often determined by the accumulation of a protein or chemical species to a critical threshold level. In a stochastic model, we define event timing as the first-passage time for the level to cross the threshold from zero or random initial conditions. This first-passage time can be modulated by implementing feedback in synthesis, that is, making the production rate an arbitrary function of the current species level. We aim to find the optimal feedback strategy that reduces the timing noise around a given mean first-passage time. Previous results have shown that while a no-feedback strategy (i.e., an independent constant production rate) is optimal in the absence of degradation and zero-molecules initial condition, a negative feedback is optimal when the process starts at random initial conditions. We show that when the species can be degraded and the synthesis rates are set to depend linearly on the number of molecules, a positive feedback strategy (the production rate increases with the level of the molecule) minimizes timing noise. However, if no constraints on the feedback are imposed, the optimal strategy involves a mixed feedback approach, which consists of an initial positive feedback followed by a sharp negative feedback (the production rate decreases with the level) near the threshold. Finally, we quantify the fundamental limits of timing noise reduction with and without feedback control when time-keeping species are subject to degradation.