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Combinatorics and Stochasticity for Chemical Reaction Networks


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


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Generating functions, chemical reaction networks, universality, combinatorics, molecular assembly
Degree Grantor:California Institute of Technology
Division:Biology and Biological Engineering
Major Option:Systems Biology
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Winfree, Erik (advisor)
  • Fontana, Walter (co-advisor)
Thesis Committee:
  • Pierce, Niles Niles (chair)
  • Bois, Justin S.
  • Phillips, Robert B.
  • Winfree, Erik
Defense Date:12 November 2021
Record Number:CaltechTHESIS:04182022-222416729
Persistent URL:
Related URLs:
URLURL TypeDescription related to ch. 1 related to ch. 2 related to ch. 3
Ortiz-Muñoz, Andrés0000-0003-1824-3230
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14552
Deposited By: Andrés Ortiz-Muñoz
Deposited On:20 Apr 2022 19:42
Last Modified:08 Nov 2023 00:23

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