Threshold-crossing time statistics for gene expression in growing cells

Many intracellular events are triggered by attaining critical concentrations of their corresponding regulatory proteins. How cells ensure precision in the timing of the protein accumulation is a fundamental problem, and contrasting predictions of different models can help us understand the mechanisms involved in such processes. Here, we formulate the timing of protein threshold-crossing as a first passage time (FPT) problem focusing on how the mean FPT and its fluctuations depend on the threshold protein concentration. First, we model the protein-crossing dynamics from the perspective of three classical models of gene expression that do not explicitly accounts for cell growth but consider the dilution as equivalent to degradation: (birth-death process, discrete birth with continuous deterministic degradation, and Fokker-Planck approximation). We compare the resulting FPT statistics with a fourth model proposed by us (growing cell) that comprises size-dependent expression in an exponentially growing cell. When proteins accumulate in growing cells, their concentration reaches a steady value. We observe that if dilution by cell growth is modeled as degradation, cells can reach concentrations higher than this steady-state level at a finite time. In the growing cell model, on the other hand, the FPT moments diverge if the threshold is higher than the steady-state level. This effect can be interpreted as a transition between noisy dynamics when cells are small to an almost deterministic behavior when cells grow enough. We finally study the mean FPT that optimizes the timing precision. The growing cell model predicts a higher optimal FPT and less variability than the classical models.