Dynamic aspects of Chemical Reaction Network Theory

The project entitled Dynamic aspects of Chemical Reaction Network Theory deals with the qualitative properties of mass-action differential equations, probably the most common mathematical models in biochemistry, cell biology, and population dynamics. They represent a large class of polynomial dynamical systems that are very important both theoretically and from the point of view of applications. The main goal is a better understanding of the long-term behavior of these systems. We address stability properties of equilibrium points and periodic solutions, as well as the non-extinction of species. An important question is how much the qualitative behavior depends on the model parameters. Since the values of the parameters of the model are, in practice, impossible to measure precisely, it is desirable to find results that are robust against changes of the parameter values, or even completely independent of the precise values of the parameters. For example, if all the reactions in the network are reversible, the corresponding mass-action system admits a positive equilibrium, irrespectively of the precise values of the rate constants of the individual reactions. Our general aim is to provide similar conditions that guarantee certain robust dynamical behavior. The field of Chemical Reaction Network Theory is interdisciplinary not only within the natural sciences, but also within Mathematics. Since the objects to be studied are both of discrete and continuous nature, attacking the problems usually require various techniques from graph theory via linear and nonlinear algebra to analysis. There are two main aspects of mass-action systems, the algebraic and the dynamic. So far more attention has been paid to the algebraic aspects concerning the structure of equilibria. The present project aims to contribute mostly to the dynamic aspects, but of course heavily builds on the algebraic aspects, the two are closely tied to each other.

In this project, we investigated the qualitative properties of mass-action differential equations, probably the most common mathematical models in biochemistry, cell biology, epidemiology, and population dynamics. They represent a large class of polynomial dynamical systems that are very important both theoretically and from the point of view of applications. We contributed towards understanding the long-term behavior of these systems, with distinguished attention on oscillations. Using standard techniques from the theory of differential equations, we studied bifurcations of equilibria, and this allowed us to conclude the occurrence of nontrivial behaviors already in small networks (with only a few species, a few reactions, and low molecularity). We systematically addressed the question of which small bimolecular reaction networks endowed with mass-action kinetics are capable of Andronov-Hopf bifurcation. With the extensive help of computer algebra, we analyzed all 14670 three-species, four-reaction bimolecular networks and found that 138 of them admit an Andronov-Hopf bifurcation, and thereby periodic solutions. Further analysis revealed that 47 networks admit even bistability (namely, the coexistence of a stable equilibrium and a stable periodic solution), a phenomenon that is ubiquitous in nature. Our paper "The smallest bimolecular mass action networks admitting Andronov-Hopf bifurcation" has been included in IOP Publishing's celebratory collection of open-access articles published in 2023 by researchers in Austria, where the featured papers have been selected for the great impact they have achieved since publication. In another work, we classified all two-species, four-reaction bimolecular networks that admit multiple equilibria, in all cases, the two equilibria are born via a saddle-node bifurcation. By allowing the product complexes to be trimolecular, we identified 33 networks that admit not only saddle-node and Andronov-Hopf but even Bogdanov-Takens bifurcation. The presence of this codimension-two bifurcation in a system allows us to conclude the existence of a homoclinic solution. Analyzing larger systems directly could be complicated. Therefore, in the past 10 years, there has been a growing interest in developing tools that help drawing conclusions by finding specific motifs in larger networks. We introduced a new network enlargement, termed "adding a dependent species," and showed that this operation preserves the network's capacity for nondegenerate behaviors, including multistationarity and oscillations. In follow-up work, we proved that in the case of six network operations that are known to preserve the capacity for nondegenerate behaviors, even bifurcations are lifted from the small network to the larger one. This discovery has the potential to greatly improve our ability to analyze networks of size that are currently out of reach.