Unstable Cores are the source of instability in chemical reaction networks

Abstract

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behavior. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.

arXiv comment42 pages

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Cited by in corpus (2)

• Advancing Mathematical Epidemiology and Chemical Reaction Network Theory via Synergies Between Them
• Enumeration of Autocatalytic Subsystems in Large Chemical Reaction Networks