Intrinsic limits of timekeeping precision in gene regulatory cascades.

Multiple cellular processes are triggered when the concentration of a regulatory protein reaches a critical threshold. Previous analyses have characterized timing statistics for single-gene systems. However, many biological timers are based on cascades of genes that activate each other sequentially. Here, we develop an analytical framework to describe the timing precision of such cascades using a burst-dilution hybrid stochastic model. We first revisit the single-gene case and recover the known result of an optimal activation threshold that minimizes first-passage-time (FPT) variability. Extending this concept to two-gene cascades, we identify three distinct optimization regimes determined by the ratio of intrinsic noise levels and the protein dilution rate, defining when coupling improves or worsens timing precision compared to a single-gene strategy. Generalizing to cascades of arbitrary gene length, we obtain a simple mathematical condition that determines when a new gene in the cascade can decrease the timing noise based on its intrinsic noise and protein dilution rate. In the specific case of a cascade of identical genes, our analytical results predict suppression of FPT noise with increasing cascade length and the existence of a mean time that decreases relative timing fluctuations. Together, these results define the intrinsic limits of timekeeping precision in gene regulatory cascades and provide a minimal analytical framework to explore timing control in biological systems.