Graph-theoretic approaches to biochemical reaction networks

Abstract

This dissertation examines ways in which graph theory can unravel biochemical reaction networks. The many processes that occur in cells contribute to an overwhelming degree of molecular complexity. Representing biomolecular processes with graphs provides us with mathematical tools to reduce this complexity. At the same time, focusing on the graph's structure in terms of topology and geometry reveals unexpected behaviors that emerge from the underlying biological system. In this dissertation, we examine this interplay through three stand-alone chapters. The first two chapters utilize the linear framework, a graph-theoretic approach to time-scale separation in biochemical systems. The third chapter uses chemical reaction network theory (CRNT), which is based on systems of nonlinear ODEs derived from chemical reactions. While the linear framework and CRNT are both based on finite, directed graphs, the two methods are distinct, as discussed in the Introduction (Chapter 0).

A major success of the linear framework has been in its thermodynamic interpretation. Chapter 1 explores this area by updating and analyzing the method for detecting departure from thermodynamic equilibrium proposed by I.Z. Steinberg. The findings presented here demonstrate the signature’s anomalous behavior as a 3-vertex system is driven from thermodynamic equilibrium, particularly in the asymptotic analysis of the spectrum of its Laplacian matrix. Chapter 2 extends the mathematical foundations of the linear framework by introducing a novel graph-theoretic construction. This construction mimics what would happen if a single parameter in a graph is taken to infinity, producing an asymptotic graph. Characterizing this construction offers a potential formalization for a biological process that could happen instantaneously. Chapter 3 shifts gears from the linear framework and uses the machinery of CRNT. This work focuses on disguised topic dynamical systems, which are systems which exhibit a particularly stable dynamics but lack the typical structural features or algebraic constraints on the rate constants. This work, broadly speaking, interrogates how network geometry and topology influence qualitative dynamics of biochemical systems under the assumption of mass-action kinetics.