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.
Several biological functions are carried out via complexes that are formed via multimerization of either a single species (homomers) or multiple species (heteromers). Given functional relevance of these complexes, it is arguably desired to maintain their level at a set point and minimize fluctuations around it. Here we consider two simple models of complex formation -one for homomer and another for heteromer of two species -and analyze how important model parameters affect the noise in complex level. In particular, we study effects of (i) sensitivity of the complex formation rate with respect to constituting species' abundance, and (ii) relative stability of the complex as compared with that of the constituents. By employing an approximate moment analysis, we find that for a given steady state level, there is an optimal sensitivity that minimizes noise (quantified by fano-factor; variance/mean) in the complex level. Furthermore, the noise becomes smaller if the complex is less stable than its constituents.Finally, for the heteromer case, our findings show that noise is enhanced if the complex is comparatively more sensitive to one constituent. We briefly discuss implications of our result for general complex formation processes.
Stochastic variation in the level of a protein among cells of the same population is ubiquitous across cell types and organisms. These random variations are a consequence of lowcopy numbers, amplified by the characteristically probabilistic nature of biochemical reactions associated with gene-expression. We systematically compare and contrast negative feedback architectures in their ability to regulate random fluctuations in protein levels. Our stochastic model consists of gene synthesizing pre-mRNAs in transcriptional bursts. Each pre-mRNA transcript is exported to the cytoplasm and is subsequently translated into protein molecules. In this setup, three feedbacks architectures are implemented: protein inhibiting transcription of its own gene (I), protein enhancing the nuclear pre-mRNA decay rate (II), and protein inhibiting the export of pre-mRNAs (III). Explicit analytic expressions are developed to quantify the protein noise levels for each feedback strategy. Mathematically controlled comparisons provide insights into the noisesuppression properties of these feedbacks. For example, when the protein half-life is long, or the pre-mRNA decay is fast, then feedback architecture I provides the best noise attenuation. In contrast, when the timescales of export, mRNA, and protein turnover are similar, then III is superior to both II and I. We finally discuss biological relevance of these findings in context of noise suppression in an HIV cell-fate decision circuit.
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