Event timing precision in stochastic gene expression with bursty production

The timing of many cellular events is controlled by the accumulation of a regulatory molecule to a threshold level. This timing can be characterized by the first-passage time (FPT), defined as the random time required to reach the threshold for the first time. In stochastic gene expression, fluctuations in molecular production and degradation lead to variability in event timing. Feedback regulation of the production rate is a natural mechanism for reducing FPT variability. Here, we first investigate how degradation reshapes the optimal feedback law for minimizing timing variability across low- and high-threshold regimes. Using backward equations and Lagrange multiplier formulation, we derive the corresponding optimality conditions under a fixed mean FPT constraint. Exact results for the low-threshold case and asymptotic analysis in the high-threshold, small-degradation regime show that the optimal production rates increase approximately linearly with molecule number before decreasing near the threshold. For larger degradation rates, the optimality conditions develop multiple solution branches and fold behavior, and the optimal production rate structure can become nonmonotonic. We then extend the framework to bursty production and numerically characterize the optimal feedback in both low- and high-threshold regimes. The results show how degradation and bursty synthesis qualitatively reshape the structure of the optimal feedback for stochastic cellular event timing.