基因表达随机调控的数学模型及动态分析

Gene expression is fundamentally a discontinuous and stochastic process. This notion has been accepted only recently facilitated by various advanced methods in the early 2000s. The revolution of our basic views motivates us to rethink the natural of life, which has made the stochastic gene expression immediately a hot and vigorous research area. The probability distribution of the expression product number provides the complete characterization for the statistics and dynamics of random gene expression. However, the calculation of the probability distribution has remained elusive, which has been a bottleneck impeding the advancement of this field. In this program, combined with the classic models of gene expression, we will continue exploring the method for the calculation of exact forms of the expression product number distribution, as well as the analytical technique for the dynamic property of its profiles; Find the invariable system parameters and the mechanism of the modulation by exploring the correlations between the probability distribution and the other measurable indices such as the mean and the noise. By developing our data-driven mathematical models, we try to understand the nonlinear dynamics for the expression of anti-microbial peptide genes modulated by signal transduction networks in the immune system of mosquitoes; Investigate more real mechanism of mRNA stochastic degradation and its influence on the gene expression. Finally, we hope our work could provide innovative inspiration for the molecular biology research area, and improve the application of the theories of differential equations, stochastic processes in the real word through self-contained mathematical method.

得益20世纪初诸多检测技术的发展，人们发现基因表达实质上是一个不连续的、随机的过程。观念上的颠覆促使我们对生命本质的重新思考，基因表达的随机性也迅速成为一个极其热门却又充满朝气的研究领域。表达产物数量的概率分布能最全面刻画基因表达的随机性。然而概率分布的计算极为困难，这种困难已严重阻碍这一领域研究的向前发展。我们将以经典随机基因表达模型为基础，继续探讨表达产物数量分布精确表达式的计算方法及其形态动态特征的分析技巧；探索概率分布与均值、噪声等量化指标的内在联系，寻找隐藏在差异表象下不变的系统参数和随机机制。通过建立实验驱动的数学模型，理解蚊子免疫系统中抗菌肽基因的表达在信号转导网络调控下所展现的非线性动力学行为；探讨更实际的mRNA随机降解机制及其对基因表达的影响。我们一方面希望能为分子生物学家提供新的研究思路，另一方面也希望通过建立新的数学方法来促进微分方程、随机过程等理论在实际中的应用。

本项目主要以两状态模型和交互式信号转导路径模型为基础，探索随机基因表达中 mRNA 数量分布精确表达式的计算方法及其形态特征的分析技巧；探讨信号调控网络结构对基因表达随机行为的影响，以帮助人们理解物种通过信号转导系统对病原体等外界刺激的识别原理和反应机制。为此，本项目针对以下具体问题进行了深入探讨，相关结果发表在《SIAM J. Appl. Math.》，《Discrete Contin. Dyn. Syst. Ser. B》等期刊上，包括：（1）解决了经典两状态模型中 mRNA 数量分布形态的三个公开问题，证明两状态模型能且仅能产生递减、单峰、双峰三种分布类型，并给出了分布形态与参数之间的对应法则；（2）探讨了随机信号条件下基因转录的随机动力学行为，通过建立带有竞争路径的两状态模型来回答随机的外部噪声信号是否会引起比确定的外部信号更大的转录噪声；（3）通过引入压力反应路径，探讨了压力反应基因在信号刺激下的转录分布特征；（4）建立了一个基因转录模型来获得 mRNA分布的精确波动并分析多重路径对转录噪声的影响，揭示了交互式路径在转录噪声调节和优化中的作用。