Intermediates, Catalysts, Persistence, and Boundary Steady States

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 socalled
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.