Unstable Cores are the source of instability in chemical reaction networks

TL;DR

This paper identifies unstable cores within biochemical reaction networks as key sources of dynamical instability, independent of autocatalysis, and demonstrates how to design networks exhibiting superlinear growth and oscillations.

Contribution

It reveals that unstable cores, not just autocatalysis, cause instability in biochemical networks and provides methods to design networks with complex dynamics.

Findings

Unstable cores are sufficient for instability in reaction networks.

Autocatalytic cores are a subset of unstable-positive feedbacks.

Designed non-autocatalytic networks with oscillations and superlinear growth.

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.