Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T17:20:53.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium and termination II: the case of Petri nets

Published online by Cambridge University Press:  28 February 2013

VINCENT DANOS and
NICOLAS OURY
VINCENT DANOS
Affiliation:
School of Informatics, University of Edinburgh, Edinburgh, United Kingdom Email: vdanos@inf.ed.ac.uk; Nicolas.Oury@gmail.com
NICOLAS OURY
Affiliation:
School of Informatics, University of Edinburgh, Edinburgh, United Kingdom Email: vdanos@inf.ed.ac.uk; Nicolas.Oury@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper is concerned with the asymptotic properties of a restricted class of Petri nets equipped with stochastic mass-action semantics. We establish a simple algebraic criterion for the existence of an equilibrium, that is to say, an invariant probability that satisfies the detailed balance condition familiar from the thermodynamics of reaction networks. We also find that when such a probability exists, it can be described by a free energy function that combines an internal energy term and an entropy term. Under strong additional conditions, we show how the entropy term can be deconstructed using the finer-grained individual-token semantics of Petri nets.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Special Issue 2: Developments In Computational Models 2010 , April 2013, pp. 290 - 307
Copyright
Copyright © Cambridge University Press 2013

References

Anderson, D. F., Craciun, G. and Kurtz, T. G. (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bulletin of Mathematical Biology 72 (8)1947–1970.CrossRefGoogle ScholarPubMed
Bacci, G., Danos, V. and Kammar, O. (2011) On the statistical thermodynamics of reversible communicating processes. In: Corradini, A., Klin, B. and Cîrstea, C. (eds.) Proceedings 4th International Conference, CALCO 2011. Springer-Verlag Lecture Notes in Computer Science 6859 1–18.CrossRefGoogle Scholar
Bruni, R., Meseguer, J., Montanari, U. and Sassone, V. (1999) Functorial semantics for Petri nets under the individual token philosophy. In: Proceedings of CTCS'99.CrossRefGoogle Scholar
Chaouiya, C. (2007) Petri net modelling of biological networks. Briefings in Bioinformatics 8 (4)210–219.CrossRefGoogle ScholarPubMed
Danos, V. (2009) Agile modelling of cellular signalling. Electronic Notes in Theoretical Computer Science 229 (4)3–10.CrossRefGoogle Scholar
Danos, V. and Laneve, C. (2004) Formal molecular biology. Theoretical Computer Science 325 (1)69–110.CrossRefGoogle Scholar
Danos, V. and Oury, N. (2010) Equilibrium and termination. CoRR, abs/1006.1430.CrossRefGoogle Scholar
Danos, V., Feret, J., Fontana, W., Harmer, R. and Krivine, J. (2007) Rule-based modelling of cellular signalling. In: Caires, L. and Vasconcelos, V. (eds.) Proceedings CONCUR 2007. Springer-Verlag Lecture Notes in Computer Science 4703 17–41.CrossRefGoogle Scholar
Danos, V., Feret, J., Fontana, W., Harmer, R. and Krivine, J. (2008a) Rule-based modelling, symmetries, refinements. In: Fisher, J. (ed.) Proceedings FMSB 2008. Springer-Verlag Lecture Notes in Computer Science 5054 103–122.CrossRefGoogle Scholar
Danos, V., Feret, J., Fontana, W. and Krivine, J. (2008b) Abstract interpretation of cellular signalling networks. In: Logozzo, F., Peled, D. and Zuck, L. D. (eds.) Proceedings VMCAI 2008. Springer-Verlag Lecture Notes in Computer Science 4905 83–97.CrossRefGoogle Scholar
Danos, V., Feret, J., Fontana, W., Harmer, R. and Krivine, J. (2010) Abstracting the differential semantics of rule-based models: Exact and automated model reduction. In: Proceedings LICS 2010, IEEE Computer Society 362–381.Google Scholar
Darling, R. and Norris, J. R. (2008) Differential equation approximations for Markov chains. Probability surveys 5 37–79.CrossRefGoogle Scholar
Delzanno, G., Giusto, C. D., Gabbrielli, M., Laneve, C. and Zavattaro, G. (2009) The Kappa-Lattice: Decidability Boundaries for Qualitative Analysis in Biological Languages. In: Degano, P. and Gorrieri, R. (eds.) Computational Methods in Systems Biology, Proceedings CMSB 2009. Springer-Verlag Lecture Notes in Computer Science 5688 158–172.CrossRefGoogle Scholar
Desharnais, J. and Panangaden, P. (2003) Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes. Journal of Logic and Algebraic Programming 56 (1-2)99–115.CrossRefGoogle Scholar
Easley, D. and Kleinberg, J. (2010) Networks, crowds, and markets: Reasoning about a highly connected world, Cambridge University Press.CrossRefGoogle Scholar
Ederer, M. and Gilles, E.-D. (2008) Thermodynamic constraints in kinetic modeling: thermodynamic-kinetic modeling in comparison to other approaches. Engineering in Life Sciences 8 (5)467–476.CrossRefGoogle Scholar
Ederer, M. and Gilles, E.-D. (2007) Thermodynamically feasible kinetic models of reaction networks. Biophysical Journal 92 (6)1846–1857.CrossRefGoogle ScholarPubMed
Ehrig, H., Habel, A., Padberg, J. and Prange, U. (2004) Adhesive high-level replacement categories and systems. In: Ehrig, H., Engels, G., Parisi-Presicce, F. and Rozenberg, G. (eds.) Proceedings Second International Conference, ICGT 2004. Springer-Verlag Lecture Notes in Computer Science 3256 144–160.CrossRefGoogle Scholar
Epstein, J. M. (1999) Agent-based computational models and generative social science. Complexity 4 (5)41–60.3.0.CO;2-F>CrossRefGoogle Scholar
Faeder, J. R., Blinov, M. L. and Hlavacek, W. S. (2009) Rule-based modeling of biochemical systems with BioNetGen. Methods in Molecular Biology 500 113–167.CrossRefGoogle ScholarPubMed
Feinberg, M. (1987) Chemical reaction network structure and the stability of complex isothermal reactors–I. The deficiency zero and deficiency one theorems. Chemical Engineering Science 42 (10)2229–2268.CrossRefGoogle Scholar
Feret, J., Danos, V., Krivine, J., Harmer, R. and Fontana, W. (2009) Internal coarse-graining of molecular systems. Proceedings of the National Academy of Sciences 106 (16)6453.CrossRefGoogle ScholarPubMed
Goss, P. J. E. and Peccoud, J. (1998) Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proceedings of the National Academy of Sciences 95 (12)6750.CrossRefGoogle ScholarPubMed
Harmer, R., Danos, V., Feret, J., Krivine, J. and Fontana, W. (2010) Intrinsic Information carriers in combinatorial dynamical systems. Chaos 2950 (3)037108.CrossRefGoogle Scholar
Karp, R. M. and Miller, R. E. (1969) Parallel Program Schemata. Journal of Computer and System Sciences 3 147–195.CrossRefGoogle Scholar
Kimura, H., Okano, H. and Tanaka, R. J. (2007) Stochastic approach to molecular interactions and computational theory of metabolic and genetic regulations. Journal of Theoretical Biology 248 (4)590–607.CrossRefGoogle ScholarPubMed
Lack, S. and Sobociński, P. (2004) Adhesive categories. In: Walukiewicz, I. (ed.) Foundations of Software Science and Computation Structures, Proceedings 7th International Conference, FOSSACS 2004. Springer-Verlag Lecture Notes in Computer Science 2987 273–288.CrossRefGoogle Scholar
Lack, S. and Sobociński, P. (2005) Adhesive and quasiadhesive categories. Theoretical Informatics and Applications 39 (3)511–545.CrossRefGoogle Scholar
Marsan, M. (1990) Stochastic Petri nets: an elementary introduction. In: Rozenberg, G. (ed.) Advances in Petri Nets 1989. Springer-Verlag Lecture Notes in Computer Science 424 1–29.CrossRefGoogle Scholar
Norris, J. R. (1998) Markov chains, Cambridge University Press.Google Scholar
Ollivier, J. F., Shahrezaei, V. and Swain, P. S. (2010) Scalable Rule-Based Modelling of Allosteric Proteins and Biochemical Networks. PLOS Computational Biology 6 (11)6750.CrossRefGoogle ScholarPubMed
Pedersen, M. (2008) Compositional definitions of minimal flows in Petri nets. In: Heiner, M. and Uhrmacher, A. M. (eds.) Computational Methods in Systems Biology – Proceedings CMSB 2008. Springer-Verlag Lecture Notes in Computer Science 5307 288–307.CrossRefGoogle Scholar
Regev, A., Silverman, W. and Shapiro, E. (2001) Representation and simulation of biochemical processes using the π-calculus process algebra. In: Pacific symposium on biocomputing 6 459–470.Google Scholar
Schuster, S. and Schuster, R. (1989) A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation. Journal of Mathematical Chemistry 3 (1)25–42.CrossRefGoogle Scholar
Yang, J., Bruno, W. J., Hlavacek, W. S. and Pearson, J. E. (2008) On imposing detailed balance in complex reaction mechanisms. Biophysical Journal 91 (3)1136–1141.CrossRefGoogle Scholar