Fluctuating environments pose tremendous challenges to bacterial populations. It is widely observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes  to survive in rapidly changing environments. This  phenotypic heterogeneity with stochastic phenotypic switching is generally assumed to be an adaptive bet-hedging strategy. To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed . Traditional deterministic models cannot provide a correct description of stochastic phenotype switching , and besides, recent research has demonstrated that cellular processes during gene expression are inherently stochastic . In this article, we proposed a unified nonlinear stochastic model of multistable bacterial systems at the molecular level. We presented a mathematical explanation of phenotypic heterogeneity, stochastic phenotype switching , and bet-hedging within isogenic bacterial populations, and thus provided a theoretical framework for the analysis of experiment dataat the cellular or molecular level . In addition, we also provided a quantitative characterization of the critical state during the transition among multiple phenotypes .
Fluctuating environments pose tremendous challenges to bacterial populations. It is observed in numerous bacterial species that individual cells can stochastically switch among multiple phenotypes for the population to survive in rapidly changing environments. This kind of phenotypic heterogeneity with stochastic phenotype switching is generally understood to be an adaptive bet-hedging strategy. Mathematical models are essential to gain a deeper insight into the principle behind bet-hedging and the pattern behind experimental data . Traditional deterministic models cannot provide a correct description of stochastic phenotype switching and bet-hedging, and traditional Markov chain models at the cellular level fail to explain their underlying molecular mechanisms . In this paper, we propose a nonlinear stochastic model of multistable bacterial systems at the molecular level. It turns out that our model not only provides a clear description of stochastic phenotype switching and bet-hedging within isogenic bacterial populations, but also provides a deeper insight into the analysis of multidimensional experimental data. Moreover, we use some deep mathematical theories to show that our stochastic model and traditional Markov chain models are essentially consistent and reflect the dynamic behavior of the bacterial system at two different time scales . In addition, we provide a quantitative characterization of the critical state of multistable bacterial systems and develop an effective data-driven method to identify the critical state without resorting to specific mathematical models .
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