Reaction Networks: Theory, Computation, and Applications

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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.