基因调控系统的非高斯随机动力学分析

Stochastic fluctuations play an important role in inducing phenotypic diversity in gene expression process. A large number of protein models have been developed for assumption of Gaussian distribution. However, the production of proteins occurs in an intermittent burst and heavy tail distribution which are suitable for non-Gaussian Levy noise. Due to lack of technology for non-Gaussian dynamics, the project have developed the deterministic quantities, the global stability and the stochastic bifurcation for non-Gaussian dynamics. These approaches can characterize transition behavior, and establish the index for global stability independent of linearization to quantify how stable the state is. The project will focus on developing a accurate numerical algorithm to solve the nonlocal elliptic equation with directional boundary condition defined by fractional Laplace operator, studying the sensitivity of parameters and relativity analysis of escape criterion for global stability on a genetic regulatory system. In this way, the developed stochastic dynamics will provide the new insights into non-Gaussian genetic regulatory systems.

基因表达过程中随机噪声是诱发表型多样性的重要因素，高斯噪声模拟蛋白质生成过程研究已有丰富的理论。然而实验表明蛋白质生成过程中表现出的随机跳跃以及“重尾”分布特性更适用于非高斯Levy噪声模拟。基因调控系统高斯动力学的现有理论不能直接用于非高斯的研究，本项目通过非高斯随机动力系统的确定性量化、全局稳定性以及随机分岔等理论工具，刻画非高斯Levy噪声影响下基因调控系统的转移行为，构建系统全局稳定性的量化指标，从而不依赖于线性化的方法充分地评估基因调控系统的稳定程度。基于由分数阶Laplace算子定义的具有逃逸方向性边值条件的非局部椭圆方程的数值计算，揭示全局稳定性对基因调控系统参数的敏感依赖性，分析全局稳定性和逃逸指标的相关性，为基因调控系统的非高斯随机动力学研究提供新方法和新工具。

随机动力系统是对随机因素影响下复杂系统的数学建模，能够描述一些确定性系统所不能描述的随机现象。本项目通过随机动力系统的确定性量化：平均首次逃逸时间，首次逃逸概率刻画了在非高斯Levy噪声影响下基因调控系统在多个亚稳态之间的迁移行为,将随机动力系统和非局部偏微分方程建立了联系，从而提出了一种研究非高斯Lévy过程驱动的随机动力系统状态间转移的新思路。基于非局部Fokker-Planck方程以及Onsager-Machlup作用泛函，构建了最有可能转移路径，用以度量非高斯随机基因调控系统状态的稳定性。这个理论为研究基因调控系统中“birst-like”事件，气候系统中突然气候突变等现象的发生机制提供理论依据。同时，通过最小化参数空间上的Hellinger测度的方法进行系统中的参数识别，为由数据驱动的随机动力系统的建模提供理论和技术的支持。.

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