[SS04] Dynamics and Structures of Biological Networks I: Chemical Reaction Networks

[SS04-03] Parameter-rich kinetics: a framework for bifurcation analysis in reaction networks

*Nicola Vassena¹ (1. Leipzig University (Germany))

Keywords:

Bifurcation Theory,Parameter-rich Kinetics,Qualitative Analysis,Autocatalysis

A primary difficulty in the analysis of reaction networks is the pervasive lack of precise quantitative information. Most existing work in the literature focuses on mass-action kinetics, often viewed as the simplest baseline case. A recurring question in the literature is whether specific parameter values can induce interesting dynamics, such as multistability or oscillations. Bifurcation theory is a natural tool for this purpose. However, the simplicity of mass-action kinetics hinders any bifurcation analysis, as the scarcity of parameters (one per reaction) imposes strong limitations: the partial derivatives of reaction rates at steady-states cannot be chosen independently of the steady-state values themselves. Specifically, the steady-state flux cone constrains its linearization, and it obfuscates the structural relationship between the stoichiometry and the spectrum of the Jacobian. To address these limitations, I will present an alternative framework, "parameter-rich kinetics”, which includes reaction schemes such as Michaelis-Menten, Hill kinetics, and Generalized Mass Action. This broader and more flexible approach enables the desired parametric independence between steady-state values and their linearization. As applications of this framework, I will demonstrate that autocatalysis is always sufficient to ensure the existence of an unstable steady-state. Finally, I will comment on stoichiometric conditions for multistationarity and periodic oscillations.