AMC-SS: Analysis and Computation of Multi-Scale Stochastic Chemical Kinetic Systems with Application to Genetic Regulatory Networks

蛋白质在生物体的发育和功能中起着至关重要的作用。 细胞蛋白质的合成是活细胞内的多步骤过程。 储存在DNA中的遗传信息通过称为基因表达的生化过程指导蛋白质的产生。 当特定基因表达时，其DNA首先转录成mRNA的单链序列。 mRNA序列然后在蛋白质形成时被翻译成氨基酸序列。 这些步骤形成基因表达的三个水平：转录、翻译和翻译后修饰。 基因调控网络（Genetic Regulatory Networks，GRNs）是由基因、蛋白质、小分子以及它们之间的相互作用组成的，在一定的物理、化学和生物刺激下，调控基因的表达，产生蛋白质。 GRNs只能有效地建模为随机分子波动的随机系统，而不是确定性系统。 GRN通常在多个稳态、反应速率和分子浓度方面是多尺度的，使得标准模拟算法效率低下。 简化和减少复杂的化学和生物模型，使更容易的概念化和解释和有效的模拟固有的随机系统。 拟议的研究项目的主要目标是提供简化的动力学，并设计有效的数值方案，表现出多个稳态和多个时间和浓度尺度的复杂随机化学动力学系统。 应用将强调遗传调控网络，其中只有随机建模纳入随机分子波动已被证明是成功的。 在数学方面，利用渐近分析和概率论，将研究具有多个定态的化学动力学系统的跃迁速率和跃迁途径，并将研究具有多个良好分离的时间和浓度尺度的随机化学动力学系统的有效动力学。 在数值方面，将开发计算方法来模拟从数学工作中获得的简化动态。 将采用基于计算机的优化方法来寻找多稳定系统的转换速率和路径。 收敛性和误差估计所提出的数值方案将证明数学。 还将研究计算方案的效率、鲁棒性、自适应性和并行性等问题。 拟议的研究将推进应用数学的前沿，通过显着的概括和随机和多尺度系统的建模，分析和计算技术的发展。 这将有助于在系统水平上理解化学反应网络的功能问题，这正在成为基因组研究的新焦点。 随机和多尺度分析的思想可以应用于涉及多尺度建模的生物，化学，物理和其他科学问题的频谱，并可以拓宽相关领域的研究课题的范围。 研究项目还将促进应用数学家，化学家和生物学家之间的跨学科互动。

Proteins play an essential role in the development and functioning of organisms. The synthesis of cellular proteins is a multi-step process inside living cells. The genetic information stored in DNA directs the production of proteins through a bio-chemical process called gene expression. When a specific gene is expressed, its DNA is first transcribed into a single stranded sequence of mRNA. The mRNA sequence is then translated into a sequence of amino acids as the protein is formed. These steps form the three levels of gene expression: transcription, translation, and post-translational modification. Genetic Regulatory Networks (GRNs), consisting of genes, proteins, small molecules within cells in relatively low concentrations, and their interactions, function to regulate gene expression process for the production of proteins in response to certain physical, chemical, and biological stimuli. GRNs can only be effectively modeled as stochastic systems with random molecular fluctuations instead of deterministic systems. Very often GRNs are multi-scale in terms of multiple steady states, reaction rates, and molecular concentrations, making standard simulation algorithms inefficient. Simplifying and reducing complex chemical and biological models enables easier conceptualization and interpretation and efficient simulations of inherently stochastic systems. The main objective of the proposed research project is to provide simplified dynamics and to design efficient numerical schemes for complex stochastic chemical kinetic systems exhibiting multiple steady states and multiple time and concentration scales. Applications will be emphasized on Genetic Regulatory Networks, for which only stochastic modeling incorporating random molecular fluctuations has proved to be successful. On the mathematical side, using asymptotic analysis and probability theory, the transition rates and transition pathways of chemical kinetic systems with multiple steady states will be investigated, and the effective dynamics for stochastic chemical kinetic systems with multiple well-separated time and concentration scales will be studied. On the numerical side, computational methods will be developed to simulate the reduced dynamics obtained from the mathematical work. Computer-based optimization methods will be adopted to find the transition rates and pathways of multi-stable systems. Convergence and error estimates for the proposed numerical schemes will be proved mathematically. Issues like efficiency, robustness, adaptivity, and parallelism of the computing schemes will also be studied. The proposed research will advance the frontiers of Applied Mathematics through significant generalizations and developments of the modeling, analytical, and computational techniques for stochastic and multi-scale systems. It will help to understand functional issues of chemically reacting networks at the system level, which is becoming the new focus of genomic research. The ideas from the stochastic and multi-scale analysis can be applied to a spectrum of biological, chemical, physical, and other scientific problems involving multi-scale modeling and can broaden the scope of research topics in the related fields. The research projects will also promote interdisciplinary interactions between applied mathematicians, chemists, and biologists.