Markov Chain Aggregation and Its Application to Rule-Based Modelling

  • Tatjana PetrovEmail author
Part of the Methods in Molecular Biology book series (MIMB, volume 1945)


Rule-based modelling allows to represent molecular interactions in a compact and natural way. The underlying molecular dynamics, by the laws of stochastic chemical kinetics, behaves as a continuous-time Markov chain. However, this Markov chain enumerates all possible reaction mixtures, rendering the analysis of the chain computationally demanding and often prohibitive in practice. We here describe how it is possible to efficiently find a smaller, aggregate chain, which preserves certain properties of the original one. Formal methods and lumpability notions are used to define algorithms for automated and efficient construction of such smaller chains (without ever constructing the original ones). We here illustrate the method on an example and we discuss the applicability of the method in the context of modelling large signaling pathways.

Key words

Markov chain aggregation Lumpability Bisimulation Rule-based modelling 



Tatjana Petrov’s research was supported by the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and by a Swiss National Science Foundation Advanced Postdoc.Mobility Fellowship (grant number P300P2_161067).

Supplementary material (1 kb)
Supplementary File 1 (ZIP 1 kb)


  1. 1.
    Blinov ML, Faeder JR, Goldstein B, Hlavacek WS (2006) A network model of early events in epidermal growth factor receptor signaling that accounts for combinatorial complexity. BioSystems 83:136–151CrossRefGoogle Scholar
  2. 2.
    Fisher J, Henzinger TA (2007) Executable cell biology. Nat Biotechnol 25:1239–1249CrossRefGoogle Scholar
  3. 3.
    Danos V, Laneve C (2004) Formal molecular biology. Theor Comput Sci 325:69–110CrossRefGoogle Scholar
  4. 4.
    Blinov ML, Faeder JR, Hlavacek WS (2004) BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains. Bioinformatics 20:3289–3291CrossRefGoogle Scholar
  5. 5.
    Danos V, Feret J, Fontana W, Krivine J (2007) Scalable simulation of cellular signaling networks. Lect Notes Comput Sci 4807:139–157CrossRefGoogle Scholar
  6. 6.
    Hlavacek WS, Faeder JR, Blinov ML, Perelson AS, Goldstein B (2003) The complexity of complexes in signal transduction. Biotechnol Bioeng 84:783–794CrossRefGoogle Scholar
  7. 7.
    Aldridge BB, Burke JM, Lauffenburger DA, Sorger PK (2006) Physicochemical modelling of cell signalling pathways. Nat Cell Biol 8:1195–1203CrossRefGoogle Scholar
  8. 8.
    Petrov T, Feret J, Koeppl H (2012) Reconstructing species-based dynamics from reduced stochastic rule-based models. In: Laroque C, Himmelspach J, Pasupathy R, Rose O, Uhrmacher AM (eds) Proceedings of the 2012 Winter Simulation Conference (WSC). IEEE, Los AlamitosGoogle Scholar
  9. 9.
    Petrov T, Koeppl H (2013) Approximate reductions of rule-based models. In: 2013 European Control Conference (ECC), pp 4172–4177Google Scholar
  10. 10.
    Feret J, Koeppl H, Petrov T (2013) Stochastic fragments: a framework for the exact reduction of the stochastic semantics of rule-based models. Int J Softw Info 7:527–604Google Scholar
  11. 11.
    Feret J, Henzinger T, Koeppl H, Petrov T (2012) Lumpability abstractions of rule-based systems. Theor Comput Sci 431:137–164CrossRefGoogle Scholar
  12. 12.
    McAdams HH, Arkin A (1999) It’s a noisy business! Genetic regulation at the nanomolar scale. Trends Genet 15:65–69CrossRefGoogle Scholar
  13. 13.
    Gillespie DT (1992) Markov processes: an introduction for physical scientists. Academic, San DiegoGoogle Scholar
  14. 14.
    Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar
  15. 15.
    Anderson DF, Kurtz TG (2010) Continuous time Markov chain models for chemical reaction networks. In: Koeppl H, Setti G, di Bernardo M, Densmore D (eds) Design and analysis of biomolecular circuits. Springer, New York, pp 3–42Google Scholar
  16. 16.
    Rao CV, Arkin AP (2003) Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J Chem Phys 118:4999–5010CrossRefGoogle Scholar
  17. 17.
    Kang HW, Kurtz TG (2013) Separation of time-scales and model reduction for stochastic reaction networks. Ann Appl Probab 23:529–583CrossRefGoogle Scholar
  18. 18.
    Gorban AN, Radulescu O (2007) Dynamical robustness of biological networks with hierarchical distribution of time scales. IET Syst Biol 1:238–246CrossRefGoogle Scholar
  19. 19.
    Conzelmann H, Fey D, Gilles ED (2008) Exact model reduction of combinatorial reaction networks. BMC Syst Biol 2:78CrossRefGoogle Scholar
  20. 20.
    Borisov NM, Chistopolsky AS, Faeder JR, Kholodenko BN (2008) Domain-oriented reduction of rule-based network models. IET Syst Biol 2:342–351CrossRefGoogle Scholar
  21. 21.
    Feret J, Danos V, Krivine J, Harmer R, Fontana W (2009) Internal coarse-graining of molecular systems. Proc Natl Acad Sci USA 106:6453–6458CrossRefGoogle Scholar
  22. 22.
    Petrov T (2013) Formal reductions of stochastic rule-based models of biochemical systems. PhD thesis, ETH ZürichGoogle Scholar
  23. 23.
    Ganguly A, Petrov T, Koeppl H (2014) Markov chain aggregation and its applications to combinatorial reaction networks. J Math Biol 69:767–797CrossRefGoogle Scholar
  24. 24.
    Danos V, Feret J, Fontana W, Harmer R, Krivine J (2010) Abstracting the differential semantics of rule-based models: exact and automated model reduction. In: 25th annual IEEE symposium on Logic in Computer Science (LICS 2010). IEEE, Los AlamitosGoogle Scholar
  25. 25.
    Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 8:344–356CrossRefGoogle Scholar
  26. 26.
    Deeds EJ, Krivine J, Feret J, Danos V, Fontana W (2012) Combinatorial complexity and compositional drift in protein interaction networks. PLoS One 7:e32032CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer and Information SciencesUniversity of KonstanzKonstanzGermany
  2. 2.Centre for the Advanced Study of Collective BehaviourUniversity of KonstanzKonstanzGermany

Personalised recommendations