Reaction Networks: Theory, Computation, and Applications

生物系统是非常复杂的，它们的突现或系统行为是由大量的分子相互作用决定的。直到今天，在生物系统中发现的复杂的相互作用如何产生它们的突现特性和行为仍然是难以捉摸的（这被认为是生物学的重大挑战之一）。理论数学提供了一条可能的前进之路，它迟早会对生物学产生深远的影响。该研究项目旨在突破生物模型的复杂性，阐明决定细胞行为的机制。此外，该项目为化学实现的神经网络的算法构建开发了一个数学框架，这是执行机器学习和“人工智能”的流行手段。最后，该项目旨在开发新的计算方法，以解决目前与生物过程的长期行为相关的不可行的问题。该项目不仅将极大地增强我们对生物系统的理解，而且还将为下一代数学和生物学交叉领域的科学家提供肥沃的训练基地。重点是建立包括教师、研究生和本科生在内的研究团队，特别强调招募女性和代表性不足的少数民族。离散空间、连续时间马尔可夫链模型通常用于模拟生物相互作用网络，包括基因调控网络、病毒感染、信号系统、神经网络等。这些模型可以通过反应图来描述，反应图是模型组成分子之间相互作用的图形表示。交互网络可能非常复杂；例如，人类基因组中有超过20,000个基因，它们编码的蛋白质可以通过无数种方式进行修饰。此外，细胞系统通常有不同的子系统，这些子系统在多个不同的尺度上运行（时间上和拷贝数上），在一个尺度上运行的物种极大地影响了在不同尺度上运行的物种。在这种复杂性背后往往隐藏着一些潜在的结构，如果适当地量化，就可以深入了解系统的动态或静态行为。该项目的第一部分旨在发展数学理论，将这些系统的紧急性质与相关反应图及其子图的易于检查的性质联系起来。第二部分着重于实现神经网络和机器学习算法的生化反应网络的发展。这里的目标不仅植根于这种网络的算法构建，而且还在于为这个研究领域开发一个适当的数学框架。最后一部分着重于算法开发（和分析），用于生物相互作用网络中常用的随机模型的平稳分布的无偏估计。研究者和他的学生使用的主要工具和方法来自概率论、随机分析、动力系统理论、化学反应网络理论和计算数学。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Biological systems are extraordinarily complex, with their emergent, or system, behavior determined though a vast number of molecular interactions. To this day, how the complex interactions found in biological systems produce their emergent properties and behaviors remains elusive (and is considered one of the grand challenges of biology). Theoretical mathematics offers a possible route forward, and one that could, in time, have a profound influence on biology. This research project aims to cut through the complexity of biological models and elucidate the mechanisms that determine cellular behavior. Further, this project develops a mathematical framework for the algorithmic construction of chemically implemented neural networks, which are a popular means of performing machine learning and "artificial intelligence." Finally, this project aims to develop new computational methods that can address currently infeasible problems related to the long-term behavior of biological processes. The project will not only greatly enhance our understanding of biological systems but will also serve as a fertile training ground for the next generation of scientists at the intersection of mathematics and biology. A key focus is on building research teams that involve faculty, graduate students, and undergraduates, with a special emphasis towards the recruitment of women and underrepresented minorities. Discrete-space, continuous-time Markov chain models are commonly used to model biological interaction networks, including gene regulatory networks, viral infections, signaling systems, neuronal networks, etc. These models can be depicted via a reaction graph, which is a graphical representation of the interactions between the constituent molecules of the model. Interaction networks can be extraordinarily complex; for example, there are over 20,000 genes in the human genome and the proteins they encode may be modified in myriad ways. Further, cellular systems often have different sub-systems that operate on multiple different scales (both temporally and in terms of copy numbers), with the species operating at one scale greatly influencing those at a different scale. Hidden within this complexity there are often underlying structures that, if properly quantified, give great insight into the dynamical or stationary behavior of the system. The first part of this project aims to develop mathematical theory that relates the emergent properties of these systems with easily checked properties of the associated reaction graphs, and their sub-graphs. A second part focuses on the development of biochemical reaction networks that implement neural networks and machine learning algorithms. Here the goal is not solely rooted in the algorithmic construction of such networks, but also in developing a proper mathematical framework for this research area. A final part focuses on algorithm development (and analysis) for the unbiased estimation of stationary distributions for the stochastic models commonly utilized for biological interaction networks. The primary tools and methods utilized by the investigator and his students are from probability theory, stochastic analysis, dynamical systems theory, chemical reaction network theory, and computational mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.