Complete classification of one-reaction atoms of multistationarity
Badal Joshi · arXiv (Cornell University) · 2012
This article characterizes certain small multistationary chemical reaction networks. Specifically we identify the `smallest CFSTR atoms of multistationarity', namely those containing one non-flow reaction, which may be irreversible or reversible. We will refer to such atoms as one-reaction atoms of multistationarity. CFSTR atoms of multistationarity were introduced in the recent work of Joshi and Shiu. We recall that a fully open network (alternatively a fully open CFSTR) is a chemical reaction network where all chemical species participate in the inflow and the outflow; and a CFSTR atom of multistationarity N is a multistationary fully open network such that the operation of `removing reactions' or `removing species' destroys the multistationarity of N. The class of CFSTR atoms of multistationarity determines the entire class of multistationary fully open networks since possessing a CFSTR atom of multistationarity as an `embedded network' (N is embedded in G if N can be obtained from G by removing a finite subset of reactions and species) was proven recently to be a sufficient condition for a fully open network to admit multistationarity. We find that there are infinitely many one-reaction atoms, however the set of such atoms can be characterized using two types, each type containing two parameters. The first type contains one chemical species while the second type contains two chemical species; while both types contain one irreversible reaction. We identify both types with the chemical process of autocatalysis. Furthermore, we give a complete classification by multistationarity of all fully open networks which contain one non-flow (possibly reversible) reaction. Moreover, we obtain new sufficient conditions for establishing multistationarity of certain fully open networks beyond the one-reaction setting.