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Fluctuating-rate model with multiple gene states
Journal of Mathematical Biology ( IF 1.9 ) Pub Date : 2020-09-30 , DOI: 10.1007/s00285-020-01538-2
Jingwei Li 1 , Hao Ge 2 , Yunxin Zhang 3
Affiliation  

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.



中文翻译:

多基因状态波动率模型

在存在正反馈的情况下,单个细胞的多种表型状态通常共存。单细胞中的随机基因状态转换和低拷贝数的蛋白质会导致相当大的波动。化学主方程 (CME) 是描述单个细胞动力学的强大工具,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) 模型来近似真实的基因状态转换中间区域中的完整 CME 模型。然而,只仔细分析了具有两个基因状态的场景。在本文中,我们将 FR 模型推广到具有多个基因状态的情况,其中数学推导变得更加复杂。每个表型状态周围波动的主要顺序,以及表型状态之间的转换速率,在中间基因状态转换区域中,其特征在于 Freidlin-Wentzell 型大偏差原理 (LDP) 中 FR 模型的平稳分布的速率函数。在某些合理的假设下,我们证明了速率函数的导数等于占优广义特征值问题的唯一非平凡解,从而产生了一种新的直接获得 LDP 速率函数的数值算法。此外,我们证明了对应确定性平均场动力学的速率函数的李雅普诺夫性质。最后,通过一个三稳态例子,我们表明,基因状态转换的中间和快速区域的局部波动(每个表型状态下平稳分布的渐近方差)是不同的。最后,构建了一个三稳态例子来说明我们理论的有效性。

更新日期:2020-10-02
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