Journal of Mathematical Biology

, Volume 74, Issue 4, pp 887–932 | Cite as

Intermediates, catalysts, persistence, and boundary steady states

  • Michael Marcondes de Freitas
  • Elisenda Feliu
  • Carsten Wiuf
Article

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.

Keywords

Reaction network theory Model reduction Persistence Boundary steady states Intermediates Catalysts Post-translational modification 

Mathematics Subject Classification

34C99 80A30 92C42 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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