School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China.
Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing, 100193, China.
J Math Biol. 2020 Mar;80(4):959-994. doi: 10.1007/s00285-019-01445-1. Epub 2019 Nov 21.
Stochastic gene regulatory networks with bursting dynamics can be modeled mesocopically as a generalized density-dependent Markov chain (GDDMC) or macroscopically as a piecewise deterministic Markov process (PDMP). Here we prove a limit theorem showing that each family of GDDMCs will converge to a PDMP as the system size tends to infinity. Moreover, under a simple dissipative condition, we prove the existence and uniqueness of the stationary distribution and the exponential ergodicity for the PDMP limit via the coupling method. Further extensions and applications to single-cell stochastic gene expression kinetics and bursty stochastic gene regulatory networks are also discussed and the convergence of the stationary distribution of the GDDMC model to that of the PDMP model is also proved.
具有爆发动力学的随机基因调控网络可以在介观尺度上建模为广义密度依赖马尔可夫链（GDDMC），或者在宏观尺度上建模为分段确定性马尔可夫过程（PDMP）。在这里，我们证明了一个极限定理，表明随着系统尺寸趋于无穷大，每个 GDDMC 族将收敛到 PDMP。此外，在一个简单的耗散条件下，我们通过耦合方法证明了 PDMP 极限的平稳分布的存在性和唯一性，以及指数遍历性。还讨论了对单细胞随机基因表达动力学和爆发性随机基因调控网络的进一步扩展和应用，并证明了 GDDMC 模型的平稳分布收敛到 PDMP 模型的平稳分布。
