Autocatalysis in Reaction Networks

Abstract

The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of "critical siphon." We explore the combinatorics of critical siphons, with a view toward the persistence conjecture. We introduce the notions of "drainable" and "self-replicable" (or autocatalytic) siphons. We show that: Every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and nonautocatalytic weakly reversible networks are persistent. Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons.

Computational complexity of atomic chemical reaction networks

“…In their work on autocatalysis of reaction networks [15], Abhishek and Manoj showed that a consistent reaction network is self-replicable if and only if it is critical. Since weak-reversibility implies consistency and our definition of reversibility implies weak-reversibility, we obtain the following equivalence: let a chemical reaction network C be reversible.…”

Intermediates, catalysts, persistence, and boundary steady states

“…We say that a reaction network has the siphon/P-semiflow property if every siphon contains the support of a P-semiflow. △ Nonempty sets of species not containing the support of a P-semiflow are also known in the literature as critical [9]. So, a reaction network has the siphon/P-semiflow property if, and only if every siphon is noncritical.…”

“…Since the seminal works of Horn, Jackson and Feinberg in the 70's ( [11,17,18], and references therein), chemical reaction network theory (CRNT) has provided a fruitful framework to study the dynamical systems describing how the concentrations of the involved chemical species evolve over time. Of great interest has been the long-term behavior of these systems, for example, whether they may exhibit oscillatory behavior [12], local asymptotic stability [2,3,12,16,26], or persistence [4,6,8,9,[14][15][16].…”

“…A recent contribution was given by Angeli, De Leenheer and Sontag [4], who provided two checkable conditions, one sufficient, and the other one necessary, for the persistence of conservative reaction networks. Their sufficient conditions were further developed and relaxed by Deshpande and Gopalkrishnan in [9]. These criteria work under fairly general assumptions on the reaction kinetics.…”

“…Theorem 1. The conditions for persistence of reaction networks in [4] and [9] are invariant under the removal of intermediate species.…”