Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China.
Beijing International Center for Mathematical Research (BICMR) and Biomedical Pioneering Innovation Center (BIOPIC), Peking University, Beijing, 100871, China.
J Math Biol. 2020 Nov;81(4-5):1099-1141. doi: 10.1007/s00285-020-01538-2. Epub 2020 Sep 30.
Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.
单细胞中常存在多个表型状态，这些表型状态受正反馈影响而共存。基因状态的随机转换和单细胞中蛋白质的低拷贝数会导致显著的波动。化学主方程（CME）是描述单细胞动力学的有力工具，但它可能过于复杂。在许多简化模型中，最近提出了一种波动率（FR）模型，用于在基因状态转换的实际中间区域中近似全 CME 模型。然而，这种模型仅在两种基因状态的情况下进行了仔细分析。在本文中，我们将 FR 模型推广到具有多个基因状态的情况，这使得数学推导变得更加复杂。在中间基因状态转换区域中，每个表型状态的波动主导阶以及表型状态之间的转换率，由 FR 模型在弗里德林-温策尔型大偏差原理（LDP）中的稳态分布的率函数来刻画。在某些合理的假设下，我们证明了率函数的导数等于一个主导广义特征值问题的唯一非平凡解，从而得到了一种直接获得 LDP 率函数的新数值算法。此外，我们证明了相应确定性平均场动力学的率函数的李雅普诺夫性质。最后，通过一个三稳态的例子，我们表明在基因状态转换的中间和快速区域中，局部波动（每个表型状态的稳态分布的渐近方差）是不同的。最后，构建了一个三稳态的例子来说明我们理论的有效性。
