Abstract: A biochemical reaction network is based on a discrete object: a directed graph whose edges are labeled with reaction rates. At the same time, it is governed by continuous dynamics, typically expressed as a nonlinear system of polynomial ODEs. These two descriptions encode the same system but have fundamentally different mathematical structures. This mathematical tension raises the question: how does the discrete structure of a biochemical network govern its continuous dynamics? In this talk, I will explore this question through several examples at the intersection of graph theory, dynamical systems, and applied algebraic geometry. In particular, I will focus on singular limits in which reaction-rate parameters tend to infinity. These limits motivate a graph operation that resembles an edge contraction, and I will examine their effects on graph invariants expressed as rational functions of the model parameters. I will then extend this construction to structural identifiability and discuss how model reduction alters the geometry of parameter space. Finally, I will turn to a stochastic interpretation of this graph-theoretic framework and describe how asymmetry in the spectral decomposition of a finite-state continuous-time Markov process gives rise to a signature of departure from thermodynamic equilibrium, with unexpectedly nonmonotone behavior even on the simplest networks. I will conclude with a discussion of broader future directions focused on understanding how network operations may affect qualitative dynamical properties of the system. Together, these examples illustrate how biologically inspired networks generate subtle structural questions and new mathematics at the interface of discrete and continuous systems.

Contact: Zhiyan Ding