Autocatalysis and Unstable Cores in Parameter-Rich Chemical Reaction Networks

Abstract: Parameter-rich CRNs are built upon kinetic laws that have sufficient flexibility to allow the independent choice of fixed-point coordinates and the absolute numerical values of the non-zero entries of the Jacobian. Although
this seems to be a tall order, Michaelis-Menten and generalized mass action kinetics are of this type. In this setting, the existence of choices of parameters that make any given given inner fixed point Hurwitz is determined entirely by the stoichiometric matrix, i.e., by structure of the chemical reaction network. More precisely, the existence of certain minimal quadratic sub-matrices, which we call unstable cores, is sufficient for dynamic instability. Autocatalytic cores (in the sense of Nghe) turn out to be unstable cores with a particular structure known as Metzler matrices. Autocatalysis, therefore can always act as source of instability given suitable choices of parameters, but there are also other causes of dynamic instability, entirely unrelated to autocatalysis.