Dominance and T-Invariants for Petri Nets and Chemical Reaction Networks

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9211)


Inspired by Anderson et al. [J. R. Soc. Interface, 2014] we study the long-term behavior of discrete chemical reaction networks (CRNs). In particular, using techniques from both Petri net theory and CRN theory, we provide a powerful sufficient condition for a structurally-bounded CRN to have the property that none of the non-terminal reactions can fire for all its recurrent configurations. We compare this result and its proof with a related result of Anderson et al. and show its consequences for the case of CRNs with deficiency one.


  1. 1.
    Anderson, D.F., Enciso, G.A., Johnston, M.D.: Stochastic analysis of biochemical reaction networks with absolute concentration robustness. J. R. Soc. Interface 11(93), 20130943 (2014)CrossRefGoogle Scholar
  2. 2.
    Aris, R.: Prolegomena to the rational analysis of systems of chemical reactions. Arch. Ration. Mech. Anal. 19(2), 81–99 (1965)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boucherie, R.J., Sereno, M.: On closed support T-invariants and the traffic equations. J. Appl. Probab. 35(2), 473–481 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brijder, R.: Dominance and deficiency for Petri nets and chemical reaction networks. arXiv preprint, arXiv:1503.04005 (2015)
  5. 5.
    Chen, H.-L., Doty, D., Soloveichik, D.: Deterministic function computation with chemical reaction networks. In: Stefanovic, D., Turberfield, A. (eds.) DNA 2012. LNCS, vol. 7433, pp. 25–42. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  6. 6.
    Cook, M., Soloveichik, D., Winfree, E., Bruck, J.: Programmability of chemical reaction networks. In: Condon, A., Harel, D., Kok, J.N., Salomaa, A., Winfree, E. (eds.) Algorithmic Bioprocesses. Natural Computing Series, pp. 543–584. Springer, Berlin Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Cummings, R., Doty, D., Soloveichik, D.: Probability 1 computation with chemical reaction networks. In: Murata, S., Kobayashi, S. (eds.) DNA 2014. LNCS, vol. 8727, pp. 37–52. Springer, Heidelberg (2014) Google Scholar
  8. 8.
    Feinberg, M.: Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49(3), 187–194 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Feinberg, M., Horn, F.: Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch. Ration. Mech. Anal. 66(1), 83–97 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Horn, F.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49(3), 172–186 (1972)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mairesse, J., Nguyen, H.: Deficiency zero Petri nets and product form. Fundam. Inf. 105(3), 237–261 (2010)MathSciNetGoogle Scholar
  12. 12.
    Memmi, G., Roucairol, G.: Linear algebra in net theory. In: Brauer, W. (ed.) Net Theory and Applications. Lecture Notes in Computer Science, vol. 84, pp. 213–223. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  13. 13.
    Oxley, J.: Matroid theory, 2nd edn. Oxford University Press, Oxford (2011) CrossRefGoogle Scholar
  14. 14.
    Paulevé, L., Craciun, G., Koeppl, H.: Dynamical properties of discrete reaction networks. J. Math. Biol. 69(1), 55–72 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Reisig, W., Rozenberg, G. (eds.): APN 1998. LNCS, vol. 1491. Springer, Heidelberg (1998) Google Scholar
  16. 16.
    Shinar, G., Feinberg, M.: Structural sources of robustness in biochemical reaction networks. Science 327(5971), 1389–1391 (2010)CrossRefGoogle Scholar
  17. 17.
    Soloveichik, D., Cook, M., Winfree, E., Bruck, J.: Computation with finite stochastic chemical reaction networks. Nat. Comput. 7(4), 615–633 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Hasselt UniversityHasseltBelgium
  2. 2.Transnational University of LimburgHasseltBelgium

Personalised recommendations