Intermediates, Catalysts, Persistence, and Boundary Steady States

TL;DR

This paper introduces graphical methods to simplify chemical reaction networks, making it easier to determine persistence and boundary steady states by removing intermediates and catalysts, especially in monomolecular and certain post-translational modification networks.

Contribution

It presents easy-to-apply graphical procedures for simplifying reaction networks without losing key persistence conditions, and characterizes these conditions via graph connectivity for specific network classes.

Findings

Simplification procedures preserve persistence conditions.

Conditions for persistence are equivalent in certain network classes.

Graph connectivity characterizes persistence and boundary steady states.

Abstract

For dynamical systems arising from chemical reaction networks, persistence is the property that each species concentration remains positively bounded away from zero, as long as species concentrations were all positive in the beginning. We describe two graphical procedures for simplifying reaction networks without breaking known necessary or sufficient conditions for persistence, by iteratively removing so-called intermediates and catalysts from the network. The procedures are easy to apply and, in many cases, lead to highly simplified network structures, such as monomolecular networks. For specific classes of reaction networks, we show that these conditions for persistence are equivalent to one another. Furthermore, they can also be characterized by easily checkable strong connectivity properties of a related graph. In particular, this is the case for (conservative) monomolecular networks, as well as cascades of a large class of post-translational modification systems (of which the MAPK cascade and the $n$-site futile cycle are prominent examples). Since one of the aforementioned sufficient conditions for persistence precludes the existence of boundary steady states, our method also provides a graphical tool to check for that.