Combinatorics and Stochasticity for Chemical Reaction Networks

Citation

Ortiz-Muñoz, Andrés (2022) Combinatorics and Stochasticity for Chemical Reaction Networks. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/s9mc-3d59. https://resolver.caltech.edu/CaltechTHESIS:04182022-222416729

Abstract

Stochastic chemical reaction networks (SCRNs) are a mathematical model which serves as a first approximation to ensembles of interacting molecules. SCRNs approximate such mixtures as always being well-mixed and consisting of a finite number of molecules, and describe their probabilistic evolution according to the law of mass-action. In this thesis, we attempt to develop a mathematical formalism based on formal power series for defining and analyzing SCRNs that was inspired by two different questions. The first question relates to the equilibrium states of systems of polymerization. Formal power series methods in this case allow us to tame the combinatorial complexity of polymer configurations as well as the infinite state space of possible mixture states. Chapter 1 presents an application of these methods to a model of polymerizing scaffolds. The second question relates to the expressive power of SCRNs as generators of stochasticity. In Chapter 2, we show that SCRNs are universal approximators of discrete distributions, even when only allowing for systems with detailed-balance. We further show that SCRNs can exactly simulate Boltzmann machines. In Chapter 3, we develop a formalism for defining the semantics of SCRNs in terms of formal power series which grew as a result of work included in the previous chapters. We use that formulation to derive expressions for the dynamics and stationary states of SCRNs. Finally, we focus on systems that satisfy complex balance and conservation of mass and derive a general expressions for their factorial moments using generating function methods.

Thesis Files