Oscillations in Chemical Reaction Networks

Parameter-rich Reaction Networks 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 a given inner fixed point unstable is determined entirely by the stoichiometric matrix, i.e., the topological structure of the network. More precisely, the existence of certain quadratic sub-matrices provides sufficient conditions of the existence of instability. We call these unstable cores. Autocatalytic cores (in the sense of Nghe) turn out to be unstable cores that have a specia structure. 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. The notion of cores is surprisingly versatile and can be extended to other forms of dynamical behavior that is determined by spectral properties. In particular, it pertains to the search for oscillatory behavior. Although oscillatory cores are not strictly necessary, they form a large class of substructures that guarantees the existence of non-trival periodic behavior.