Controlling Noise in the Timing of Intracellular Events: A First-Passage Time Approach

Abstract

In the noisy cellular environment, gene products are subject to inherent random fluctuations in copy numbers over time. How cells ensure precision in the timing of key intracellular events, in spite of such stochasticity is an intriguing fundamental problem. We formulate event timing as a first-passage time problem, where an event is triggered when the level of a protein crosses a critical threshold for the first time. Novel analytical calculations are preformed for the first-passage time distribution in stochastic models of gene expression, including models with feedback regulation.Derivation of these formulas motivates an interesting question: is there an optimal feedback strategy to regulate the synthesis of a protein to ensure that an event will occur at a precise time, while minimizing deviations or noise about the mean. Counter-intuitively, results show that for a stable long-lived protein, the optimal strategy is to express the protein at a constant rate without any feedback regulation, and any form of feedback (positive, negative or any combination of them) will always amplify noise in event timing. In contrast, a positive feedback mechanism provides the highest precision in timing for an unstable protein. These theoretical results explain recent experimental observations of single-cell lysis times in bacteriophage λ. Here, lysis of an infected bacterial cell is orchestrated by the expression and accumulation of a stable λ protein up to a threshold, and precision in timing is achieved via feedforward, rather than feedback control. Our results have broad implications for diverse cellular processes that rely on precise temporal triggering of events.

Effect of gene-expression bursts on stochastic timing of cellular events

“…where µ is the parameter of geometric distribution [35]. Interestingly, here except for the first transcription rate (when the protein level is zero), the other transcription rates are equal, and this optimal feedback strategy is quite similar to a no feedback mechanism.…”

“…First-passage time is defined as the first time at which a random walker reached a certain critical level, and has been used in several fields to study threshold crossing phenomena [20][21][22][23][24][25][26][27][28][29][30][31][32]. Particularly, in the context of stochastic gene expression models, previous works have developed exact analytical formulas for FPT moments when gene expresses in translation bursts [32][33][34][35]. In these models, one usually assumes a geometrically distributed translation burst distribution which arises from the assumption that both mRNA translation and mRNA degradation are one step processes occurring at exponentially distributed times [36].…”