Zero-eigenvalue bifurcations in Chemical Reaction Networks

• Applicant: Dr. Nicola Vassena
• Subject Area: Mathematics ; Bioinformatics and Theoretical Biology
• Term: from 2022 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 512355535

• Final Report Year: 2025

Final Report Abstract

Chemical reactions convert reactants into products, and interconnected reactions form a reaction network. The capacity of these networks to exhibit multistationarity and oscillations is fundamental to understanding their behavior. Multistationarity underlies epigenetic processes like cell differentiation, while oscillations regulate metabolic processes, circadian rhythms, and other core biological functions. This project developed efficient methods for detecting multistationarity and oscillations by investigating structural, network-based conditions enabling bifurcations that give rise to these phenomena. The focus was on equilibrium bifurcations characterized by a zero eigenvalue of the Jacobian, providing insights into the fundamental mechanisms governing the dynamics of these systems. To address the lack of precise quantitative data, the project introduced a general multiscale framework based on parameter-rich kinetics. This approach considers equilibrium values as parametrically independent of their linearization, an assumption satisfied by widely used kinetic models in biochemistry and metabolism, including Michaelis–Menten kinetics. Within this framework, I established structural network conditions for the existence of non-hyperbolic equilibria at both zero eigenvalues and purely imaginary eigenvalues. I also analyzed conditions governing the algebraic and geometric multiplicity of the eigenvalue zero. Notably, I identified a pathological case where a network Jacobian admits a zero eigenvalue, yet it is always a double eigenvalue zero, independent of parameter choices. This example illustrates how network structure alone can impose constraints on otherwise generic behavior. For a simple eigenvalue zero, I analyzed the nonlinear unfolding of a saddlenode bifurcation, offering sufficient conditions and identifying explicit bifurcation parameters. Interpreting these findings at an interdisciplinary level, we connected structural motifs to key biochemical concepts. Autocatalysis has long been associated with complex dynamics such as superlinear growth and periodic oscillations. Our results confirmed that autocatalysis is always sufficient for instability at equilibrium, yet explicit counterexamples demonstrated that it is not necessary. Additionally, we applied these methods to mathematical epidemiology, showing that nearly all epidemiological models can exhibit periodic oscillations under certain monotone interaction functions. The project also explored other independent directions. First, we described a novel bifurcation mechanism in which periodic orbits emerge in absence of a reference equilibrium: a drift is introduced along a line of equilibria with changing stability. Second, I derived new criteria for Hopf bifurcations and periodic oscillations in mass-action systems, which bypass the computationally demanding Routh–Hurwitz conditions.

Publications

• Structural Conditions for Saddle-Node Bifurcations in Chemical Reaction Networks. SIAM Journal on Applied Dynamical Systems, 22(3), 1639-1672.
  Vassena, Nicola
  
• Structural obstruction to the simplicity of the eigenvalue zero in chemical reaction networks. Mathematical Methods in the Applied Sciences, 47(4), 2993-3006.
  Vassena, Nicola
  
• Finding bifurcations in mathematical epidemiology via reaction network methods. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(12).
  Vassena, N.; Avram, F. & Adenane, R.
  
• Unstable cores are the source of instability in chemical reaction networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480(2285).
  Vassena, Nicola & Stadler, Peter F.
  
• Hybrid bifurcations: periodicity from eliminating a line of equilibria. Mathematische Annalen, 391(4), 6373-6399.
  López-Nieto, Alejandro; Lappicy, Phillipo; Vassena, Nicola; Stuke, Hannes & Dai, Jia-Yuan
  
• Mass Action Systems: Two Criteria for Hopf Bifurcation Without Hurwitz. SIAM Journal on Applied Mathematics, 85(3), 1046-1066.
  Vassena, Nicola
  
• Symbolic hunt of instabilities and bifurcations in reaction networks. Discrete and Continuous Dynamical Systems - B, 30(6), 2183-2208.
  Vassena, Nicola