Exact distribution of threshold-crossing times for protein concentrations: Implication for biological timekeeping

Stochastic transcription and translation dynamics of protein accumulation up to some concentration threshold sets the timing of many cellular physiological processes. Here we obtain the exact distribution of first threshold-crossing times of protein concentration, in either Laplace or time domain, and its associated cumulants: mean, variance and skewness. The distribution is asymmetric and its skewness non-monotonically varies with the threshold. We study lysis times of E-coli cells for holin gene mutants of bacteriophage-λ and find a good match with theory. Mutants requiring higher holin thresholds show more skewed lysis time distributions as predicted.