Conditions for extinction events in chemical reaction networks with discrete state spaces

  • Matthew D. Johnston
  • David F. Anderson
  • Gheorghe Craciun
  • Robert Brijder


We study chemical reaction networks with discrete state spaces and present sufficient conditions on the structure of the network that guarantee the system exhibits an extinction event. The conditions we derive involve creating a modified chemical reaction network called a domination-expanded reaction network and then checking properties of this network. Unlike previous results, our analysis allows algorithmic implementation via systems of equalities and inequalities and suggests sequences of reactions which may lead to extinction events. We apply the results to several networks including an EnvZ-OmpR signaling pathway in Escherichia coli.


Reaction network Reaction graph Extinction Stochastic process Petri net 

Mathematics Subject Classification

92C42 60J27 



MDJ and DFA were supported by Army Research Office Grant W911NF-14-1-0401. DFA was also supported by NSF-DMS-1318832 and MDJ was also supported by the Henry Woodward Fund. GC was supported by NSF-DMS-1412643. RB is a postdoctoral fellow of the Research Foundation—Flanders (FWO). The authors are also grateful to the anonymous referees whose suggestions have greatly improved the readability, clarity, and accuracy of the paper.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Matthew D. Johnston
    • 1
  • David F. Anderson
    • 2
  • Gheorghe Craciun
    • 2
    • 3
  • Robert Brijder
    • 4
  1. 1.Department of MathematicsSan José State UniversitySan JoseUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.Department of Biomolecular ChemistryUniversity of Wisconsin-MadisonMadisonUSA
  4. 4.Department WET-INFHasselt UniversityDiepenbeekBelgium

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