Threshold-crossing time statistics for size-dependent gene expression in growing cells

Many intracellular events trigger upon attaining critical concentrations of their corresponding regulatory proteins. How cells ensure precision in the timing of these events is a fundamental problem. Models used to study protein dynamics often do not explicitly account for cell growth and consider the dilution by the growth as equivalent to degradation. Here, we formulate the timing of protein threshold-crossing as a first-passage time (FPT) problem. We study the FPT statistics of three classical models of gene expression that do not explicitly consider cell growth: (birth-death process, discrete birth with continuous deterministic degradation, and Fokker-Planck approximation). These results were compared with a fourth model (growing cell) that comprises size-dependent expression in an exponentially elongating cell. This article specifically focuses on how the mean FPT and its fluctuations depend on the threshold concentration. The first three models offer similar FPT statistics, while the fourth one shows a transition between noisy dynamics in small sizes to an almost deterministic behavior when cells grow enough. While in the first three models, cells can reach concentrations higher than the steady-state concentration. In the growing cell model, although some cells can reach steady level at small sizes, the mean FPT diverges if the threshold concentration is higher than the steady one. We use this threshold as parameter to estimate the mean time that minimizes the FPT variability. The fourth model predicts higher FPT that minimizes the noise and fewer fluctuations on the FPTs than the other three models. We discuss the origin of these divergences and some applications of this model when cell growth is the leading cause of the decreasing molecule concentration within a cell.