The protein hourglass: First passage time distributions for protein thresholds

Stochastic protein accumulation, until reaching a specific threshold, governs the temporal regulation of numerous cellular physiological processes. In our study, we derive the exact probability distribution of threshold crossing times for protein levels in both Laplace and time domains, along with their associated statistical moments: mean, variance, and skewness. The resulting distribution exhibits asymmetry, with skewness showing a non-monotonic dependence on the threshold value. We investigate the lysis durations of E. coli cells pertaining to holin gene mutants of bacteriophage-λ and observe a favorable agreement with theoretical predictions. Mutants requiring higher holin thresholds demonstrate more pronounced asymmetry in their lysis time distributions, aligning with our theory. Additionally, our theoretical framework predicts a linear relationship between infection delay time and host doubling time for lytic viruses, a phenomenon that has recently been experimentally observed.