Natural Computing

, Volume 16, Issue 2, pp 285–294 | Cite as

Dominance and deficiency for Petri nets and chemical reaction networks

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Abstract

Inspired by Anderson et al. (J R Soc Interface, 2014, doi: 10.1098/rsif.2013.0943) 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 any of 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.

Keywords

Chemical Reaction Network Incidence Matrix Outgoing Edge Terminal Vertex Mass Action Kinetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank David Anderson for kindly explaining his work during the Banff International Research Station (BIRS) workshop on CRNs (14w5167). Also, we thank the organizers of this workshop during which this research was initiated. We are indebted to Matthew Johnston for carefully reading an earlier version of this paper and for providing useful comments. And in particular for finding a counterexample to a conjecture in an earlier version of this paper. We also thank David Anderson, Gheorghe Craciun and Matthew Johnston for noticing an omission in the definition of \(\le _d\) in the conference version of this paper. We finally thank the referees for their useful comments, and in particular for noticing an omission in an earlier version of Lemma 3.3. R.B. is a postdoctoral fellow of the Research Foundation—Flanders (FWO).

References

  1. Anderson DF, Enciso GA, Johnston MD (2014) Stochastic analysis of biochemical reaction networks with absolute concentration robustness. J R Soc Interface. doi: 10.1098/rsif.2013.0943 Google Scholar
  2. Aris R (1965) Prolegomena to the rational analysis of systems of chemical reactions. Arch Ration Mech Anal 19:81–99. doi: 10.1007/BF00282276 MathSciNetCrossRefGoogle Scholar
  3. Boucherie RJ, Sereno M (1998) On closed support T-invariants and the traffc equations. J Appl Probab 35:473–481. doi: 10.1239/jap/1032192862 MathSciNetMATHGoogle Scholar
  4. Brijder R (2015) Dominance and T-invariants for Petri nets and chemical reaction networks. In: Phillips A, Yin P (eds) Proceedings of the 21st international conference on DNA computing and molecular programming (DNA 21), vol 9211. Lecture notes in computer science. Springer, pp. 1–15. doi: 10.1007/978-3-319-21999-8_1
  5. Chen H-L, Doty D, Soloveichik D (2012) Deterministic function computation with chemical reaction networks. In: Stefanovic D, Turberfeld AJ (eds) Proceedings of the 18th international conference on DNA computing and molecular programming (DNA 18), vol 7433. Lecture notes in computer science. Springer, pp. 25–42. doi: 10.1007/978-3-642-32208-2_3
  6. Cook M, Soloveichik D, Winfree E, Bruck J (2009) Programmability of chemical reaction networks. In: Condon A, Harel D, Kok JN, Salomaa A, Winfree E (eds) Algorithmic bioprocesses. Natural computing series. Springer, Berlin, pp. 543–584. doi: 10.1007/978-3-540-88869-7_27
  7. Cummings R, Doty D, Soloveichik D (2014) Probability 1 computation with chemical reaction networks. In: Murata S, Kobayashi S (eds) Proceedings of the 20th international conference on DNA computing and molecular programming (DNA 20), vol 8727. Lecture notes in computer science. Springer, pp. 37–52. doi: 10.1007/978-3-319-11295-4_3
  8. Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187–194. doi: 10.1007/BF00255665 MathSciNetCrossRefGoogle Scholar
  9. Feinberg M, Horn F (1977) Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch Ration Mech Anal 66:83–97. doi: 10.1007/BF00250853 MathSciNetCrossRefMATHGoogle Scholar
  10. Horn F (1972) Necessary and suffcient conditions for complex balancing in chemical kinetics. Arch Ration Mech Anal 49:172–186. doi: 10.1007/BF00255664 CrossRefGoogle Scholar
  11. Mairesse J, Nguyen H (2010) Defciency zero Petri nets and product form. Fundam Inform 105:237–261. doi: 10.3233/FI-2010-366 MATHGoogle Scholar
  12. Memmi G, Roucairol G (1975) Linear algebra in net theory. In: Brauer W (ed) Net theory and applications. Proceedings of the advanced course on general net theory of processes and systems, vol 84. Lecture notes in computer science. Springer, pp. 213–223. doi: 10.1007/3-540-10001-6_24
  13. Oxley J (2011) Matroid theory, 2nd edn. Oxford University Press, New York. doi: 10.1093/acprof:oso/9780198566946.001.0001 CrossRefMATHGoogle Scholar
  14. Paulevé L, Craciun G, Koeppl H (2014) Dynamical properties of discrete reaction networks. J Math Biol 69:55–72. doi: 10.1007/s00285-013-0686-2 MathSciNetCrossRefMATHGoogle Scholar
  15. Reisig W, Rozenberg G (eds) (1998) Lectures on Petri Nets I: Basic Models, vol 1491. Lecture notes in computer science. Springer. doi: 10.1007/3-540-65306-6
  16. Shinar G, Feinberg M (2010) Structural sources of robustness in biochemical reaction networks. Science 327:1389–1391. doi: 10.1126/science.1183372 CrossRefGoogle Scholar
  17. Soloveichik D, Cook M, Winfree E, Bruck J (2008) Computation with fnite stochastic chemical reaction networks. Nat Comput 7:615–633. doi: 10.1007/s11047-008-9067-y MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Hasselt UniversityDiepenbeekBelgium

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