Analysis of stochastic timing of intracellular events with gene switching

An important step in execution of several cellular processes is accumulation of a regulatory protein up to a specific threshold level. Since production of a protein is inherently stochastic, the time at which its level crosses a threshold exhibits cell-to-cell variation. A problem of interest is to characterize how the statistics of event timing is affected by various steps of protein expression. Our previous work studied this problem by considering a gene expression model where gene was always active. Here we extend our analysis to a scenario where gene stochastically switches between active and inactive states. We formulate event timing as the first-passage time for a protein's level to cross a threshold and investigate how the rates of gene activation/inactivation affect the distribution and moments of the first-passage time. Our results show that both the time-scale of gene switching with respect to the protein degradation rate as well as the ratio of the gene inactivation to gene activation rates are important parameters in shaping the event-timing distribution.