First-passage-time problems in gene expression: the role of extrinsic noise

Several cellular behaviours are triggered by the accumulation of specific proteins to a threshold concentration. Fluctuations in protein levels can considerably affect the timing of this threshold crossing, and thus the precision of many cellular processes, such as cell cycle or circadian clocks. Cell-to-cell variability in protein concentration is driven by intrinsic and extrinsic causes. Intrinsic noise arises from the inherent stochastic nature of the molecular reactions involved in transcription, translation and degradation of mRNAs and proteins. On the other hand, extrinsic noise is caused by fluctuations in global cellular factors such as cell volume or the concentration of key enzymes of the expression process. Therefore, extrinsic noise can be represented as parameter fluctuations with a specific time-scale in a stochastic model of gene expression. The main focus of this work is to analyse the role of extrinsic noise on the timing precision of a threshold crossing problem in gene expression. This is essentially a first-passage-time problem in presence of coloured noise on the process parameters. We used an extension of the classic Gillespie simulation algorithm to include extrinsic fluctuations, in order to characterize the role of both variance and autocorrelation time of the extrinsic noise on the first-passage time distribution. Specifically, we identified the relevant parameters that shape the profile of crossing-time fluctuations for an unregulated gene. In order to clearly link our results to an experimentally testable setting, we developed a timing noise decomposition that can be compared with recent two-color reporter experiments.