Thermodynamic Graph-Rewriting

  • Vincent Danos
  • Russ Harmer
  • Ricardo Honorato-Zimmer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8052)


We develop a new ‘thermodynamic’ approach to stochastic graph-rewriting. The ingredients are a finite set of reversible graph-rewriting rules \({\mathcal{G}}\) (called generating rules), a finite set of connected graphs \({\mathcal{P}}\) (called energy patterns), and an energy cost function \(\epsilon:{\mathcal{P}}\to{\mathbb{R}}\). The idea is that \({\mathcal{G}}\) defines the qualitative dynamics by showing which transformations are possible, while \({\mathcal{P}}\) and ε specify the long-term probability π of any graph reachable under \({\mathcal{G}}\). Given \({\mathcal{G}}, {\mathcal{P}}\), we construct a finite set of rules \({\mathcal{G}}_{\mathcal{P}}\) which (i) has the same qualitative transition system as \({\mathcal{G}}\), and (ii) when equipped with suitable rates, defines a continuous-time Markov chain of which π is the unique fixed point. The construction relies on the use of site graphs and a technique of ‘growth policy’ for quantitative rule refinement which is of independent interest. The ‘division of labour’ between the qualitative and the long-term quantitative aspects of the dynamics leads to intuitive and concise descriptions for realistic models (see the example in §4). It also guarantees thermodynamical consistency (aka detailed balance), otherwise known to be undecidable, which is important for some applications. Finally, it leads to parsimonious parameterizations of models, again an important point in some applications.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bachman, J.A., Sorger, P.: New approaches to modeling complex biochemistry. Nature Methods 8(2), 130 (2011)CrossRefGoogle Scholar
  2. 2.
    Bai, F., Branch, R.W., Nicolau Jr., D.V., Pilizota, T., Steel, B.C., Maini, P.K., Berry, R.M.: Conformational spread as a mechanism for cooperativity in the bacterial flagellar switch. Science 327(5966), 685–689 (2010)CrossRefGoogle Scholar
  3. 3.
    Bournez, O., Côme, G.-M., Conraud, V., Kirchner, H., Ibanescu, L.: A rule-based approach for automated generation of kinetic chemical mechanisms. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 30–45. Springer, Heidelberg (2003)Google Scholar
  4. 4.
    Danos, V., Feret, J., Fontana, W., Krivine, J.: Scalable simulation of cellular signaling networks. In: Asian Symposium on Programming Languages and Systems, pp. 139–157 (2007)Google Scholar
  5. 5.
    Danos, V., Harmer, R., Winskel, G.: Constraining rule-based dynamics with types. Mathematical Structures in Computer Science 23(2), 272–289 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Danos, V., Oury, N.: Equilibrium and termination II: the case of Petri Nets. Mathematical Structures in Computer Science 23(2), 290–307 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Danos, V.: Agile modelling of cellular signalling. SOS 2008 Invited paper, Electronic Notes in Theoretical Computer Science 229(4), 3–10 (2009)CrossRefGoogle Scholar
  8. 8.
    Danos, V., Oury, N.: Equilibrium and termination. In: Barry Cooper, S., Panangaden, P., Kashefi, E. (eds.) Proceedings Sixth Workshop on Developments in Computational Models: Causality, Computation, and Physics. EPTCS, vol. 26, pp. 75–84 (2010)Google Scholar
  9. 9.
    Diers, Y.: Familles universelles de morphismes. Tech. report, Université des Sciences et Techniques de Lille I (1978)Google Scholar
  10. 10.
    Dixon, L., Kissinger, A.: Open graphs and monoidal theories. arXiv:1011.4114 (2010)Google Scholar
  11. 11.
    Ehrig, H.: Handbook of graph grammars and computing by graph transformation: Applications, Languages and Tools, vol. 2. World Scientific Publishing Company (1999)Google Scholar
  12. 12.
    Faeder, J.R., Blinov, M.L., Hlavacek, W.S.: Rule-based modeling of biochemical systems with BioNetGen. Methods Mol. Biol. 500, 113–167 (2009)CrossRefGoogle Scholar
  13. 13.
    Gross, T., Sayama, H.: Adaptive networks. Springer (2009)Google Scholar
  14. 14.
    Hayman, J., Heindel, T.: Pattern graphs and rule-based models: The semantics of kappa. In: Pfenning, F. (ed.) FOSSACS 2013. LNCS, vol. 7794, pp. 1–16. Springer, Heidelberg (2013)Google Scholar
  15. 15.
    Heckel, R.: DPO transformation with open maps. Graph Transformations, 203–217 (2012)Google Scholar
  16. 16.
    Heckel, R.: Dpo transformation with open maps. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2012. LNCS, vol. 7562, pp. 203–217. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Hlavacek, W.S., Faeder, J.R., Blinov, M.L., Posner, R.G., Hucka, M., Fontana, W.: Rules for modeling signal-transduction systems. Science Signalling 2006(344) (2006)Google Scholar
  18. 18.
    Krivine, J., Milner, R., Troina, A.: Stochastic bigraphs. Electronic Notes in Theoretical Computer Science 218, 73–96 (2008)CrossRefGoogle Scholar
  19. 19.
    Lack, S., Sobociński, P.: Adhesive categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 273–288. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Lopez, C.F., Muhlich, J.L., Bachman, J.A., Sorger, P.K.: Programming biological models in python using pysb. Molecular Systems Biology 9(1) (2013)Google Scholar
  21. 21.
    Lynch, J.: A logical characterization of individual-based models. In: Proceedings of Logic in Computer Science, pp. 203–217 (2008)Google Scholar
  22. 22.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E., et al.: Equation of state calculations by fast computing machines. The Journal of Chemical Physics 21(6), 1087 (1953)CrossRefGoogle Scholar
  23. 23.
    Murphy, E., Danos, V., Feret, J., Harmer, R., Krivine, J.: Rule-based modelling and model resolution. In: Lohdi, H., Muggleton, S. (eds.) Elements of Computational Systems Biology. Wiley (2010)Google Scholar
  24. 24.
    Tiger, C.-F., Krause, F., Cedersund, G., Palmér, R., Klipp, E., Hohmann, S., Kitano, H., Krantz, M.: A framework for mapping, visualisation and automatic model creation of signal-transduction networks. Molecular Systems Biology 8(1) (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Vincent Danos
    • 1
  • Russ Harmer
    • 2
  • Ricardo Honorato-Zimmer
    • 1
  1. 1.School of InformaticsUniversity of EdinburghUK
  2. 2.CNRS & Université Paris-DiderotFrance

Personalised recommendations