Modeling of Resilience Properties in Oscillatory Biological Systems Using Parametric Time Petri Nets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9308)


Automated verification of living organism models allows us to gain previously unknown knowledge about underlying biological processes. In this paper, we show the benefits to use parametric time Petri nets in order to analyze precisely the dynamic behavior of biological oscillatory systems. In particular, we focus on the resilience properties of such systems. This notion is crucial to understand the behavior of biological systems (e.g. the mammalian circadian rhythm) that are reactive and adaptive enough to endorse major changes in their environment (e.g. jet-lags, day-night alternating work-time). We formalize these properties through parametric TCTL and demonstrate how changes of the environmental conditions can be tackled to guarantee the resilience of living organisms. In particular, we are able to discuss the influence of various perturbations, e.g. artificial jet-lag or components knock-out, with regard to quantitative delays. This analysis is crucial when it comes to model elicitation for dynamic biological systems. We demonstrate the applicability of this technique using a simplified model of circadian clock.


Parametric time Petri net Resilience Biological oscillators Model checking 


  1. 1.
    Ahmad, J., Bernot, G., Comet, J.P., Lime, D., Roux, O.: Hybrid modelling and dynamical analysis of gene regulatory networks with delays. ComPlexUs 3(4), 231–251 (2007)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Dill, D.: Model-checking for real-time systems. In: 1990 Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, LICS 1990, pp. 414–425. IEEE (1990)Google Scholar
  3. 3.
    Andreychenko, A., Magnin, M., Inoue, K.: Modeling of resilience properties in oscillatory biological systems using parametric time petri nets, supplementary information (2015). arXiv preprint arXiv:1506.06299 [cs.LO]
  4. 4.
    Ballarini, P.: Analysing oscillatory trends of discrete-state stochastic processes through hasl statistical model checking (2014). arXiv preprint arXiv:1410.4027
  5. 5.
    Ballarini, P., Mardare, R., Mura, I.: Analysing biochemical oscillation through probabilistic model checking. Electron. Notes Theoret. Comput. Sci. 229(1), 3–19 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bérard, B., Cassez, F., Haddad, S., Lime, D., Roux, O.H.: The expressive power of time Petri nets. Theoret. Comput. Sci. 474, 1–20 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boucheneb, H., Gardey, G., Roux, O.H.: TCTL model checking of time Petri nets. J. Logic Comput. 19(6), 1509–1540 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chaouiya, C., Remy, É., Thieffry, D.: Qualitative Petri net modelling of genetic networks. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 95–112. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  9. 9.
    Chaouiya, C., Remy, E., Thieffry, D.: Petri net modelling of biological regulatory networks. J. Discrete Algorithms 6(2), 165–177 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Comet, J.P., Bernot, G., Das, A., Diener, F., Massot, C., Cessieux, A.: Simplified models for the mammalian circadian clock. Procedia Comput. Sci. 11, 127–138 (2012)CrossRefGoogle Scholar
  12. 12.
    Comet, J.-P., Klaudel, H., Liauzu, S.: Modeling multi-valued genetic regulatory networks using high-level Petri nets. In: Ciardo, G., Darondeau, P. (eds.) ICATPN 2005. LNCS, vol. 3536, pp. 208–227. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  13. 13.
    Edery, I.: Circadian rhythms in a nutshell. Physiol. Genomics 3(2), 59–74 (2000)Google Scholar
  14. 14.
    Gilbert, D., Heiner, M., Lehrack, S.: A unifying framework for modelling and analysing biochemical pathways using Petri nets. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 200–216. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  15. 15.
    Golombek, D.A., Rosenstein, R.E.: Physiology of circadian entrainment. Physiol. Rev. 90(3), 1063–1102 (2010)CrossRefGoogle Scholar
  16. 16.
    Grimm, V., Calabrese, J.M.: What is resilience? A short introduction. In: Deffuant, D., Gilbert, N. (eds.) Viability and Resilience of Complex Systems, pp. 3–13. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  17. 17.
    Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  18. 18.
    Holling, C.S.: Resilience and stability of ecological systems. Annu. Rev. Ecol. Syst. 4, 1–23 (1973)CrossRefGoogle Scholar
  19. 19.
    Jovanović, A., Lime, D., Roux, O.H.: Integer parameter synthesis for timed automata. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 401–415. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  20. 20.
    Koch, I., Heiner, M.: Petri Nets. In: Junker, B.H., Schreiber, F. (eds.) Biological Network Analysis, 7. Wiley Book Series on Bioinformatik, pp. 139–179. Wiley, New York (2008)CrossRefGoogle Scholar
  21. 21.
    Larsen, K.G., Pettersson, P., Yi, W.: Model-checking for real-time systems. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 62–88. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  22. 22.
    Leloup, J.C., Goldbeter, A.: Toward a detailed computational model for the mammalian circadian clock. Proc. Natl. Acad. Sci. 100(12), 7051–7056 (2003)CrossRefGoogle Scholar
  23. 23.
    Leveson, N., Dulac, N., Zipkin, D., Cutcher-Gershenfeld, J., Carroll, J., Barrett, B.: Engineering resilience into safety-critical systemsGoogle Scholar
  24. 24.
    Lime, D., Roux, O.H., Seidner, C., Traonouez, L.-M.: Romeo: a parametric model-checker for Petri nets with stopwatches. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 54–57. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  25. 25.
    Maruyama, H., Legaspi, R., Minami, K., Yamagata, Y.: General resilience: taxonomy and strategies. In: 2014 International Conference and Utility Exhibition on Green Energy for Sustainable Development (ICUE), pp. 1–8 (2014)Google Scholar
  26. 26.
    Matsuno, H., Inouye, S.I.T., Okitsu, Y., Fujii, Y., Miyano, S.: A new regulatory interaction suggested by simulations for circadian genetic control mechanism in mammals. J. Bioinform. Comput. Biol. 4(01), 139–153 (2006)CrossRefGoogle Scholar
  27. 27.
    Merlin, P.M., Farber, D.J.: Recoverability of communication protocols-implications of a theoretical study. IEEE Trans. Commun. 24(9), 1036–1043 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Oster, H., Yasui, A., van der Horst, G.T.J., Albrecht, U.: Disruption of mCry2 restores circadian rhythmicity in mPer2 mutant mice. Genes Dev. 16(20), 2633–2638 (2002)CrossRefGoogle Scholar
  29. 29.
    Pnueli, A.: The temporal logic of programs. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, SFCS 1977, pp. 46–57. IEEE Computer Society, Washington, DC, USA (1977)Google Scholar
  30. 30.
    Rudell, R.L., Sangiovanni-Vincentelli, A.: Multiple-valued minimization for PLA optimization. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 6(5), 727–750 (1987)CrossRefGoogle Scholar
  31. 31.
    Schwind, N., Okimoto, T., Inoue, K., Chan, H., Ribeiro, T., Minami, K., Maruyama, H.: Systems resilience: a challenge problem for dynamic constraint-based agent systems. In: Proceedings of the 2013 International Conference on Autonomous Agents and Multi-agent Systems. pp. 785–788. International Foundation for Autonomous Agents and Multiagent Systems (2013)Google Scholar
  32. 32.
    Spieler, D.: Characterizing oscillatory and noisy periodic behavior in Markov population models. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 106–122. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  33. 33.
    Tavana, M., Busch, T.E., Davis, E.L.: Modeling operational robustness and resiliency with high-level Petri nets. Technical report, DTIC Document (2012)Google Scholar
  34. 34.
    Toussaint, J., Simonot-Lion, F., Thomesse, J.P.: Time constraint verifications methods based time Petri nets. In: 6th Workshop on Future Trends in Distributed Computing Systems (FTDCS 1997), pp. 262–267, Tunis, Tunisia (1997)Google Scholar
  35. 35.
    Traonouez, L.M., Lime, D., Roux, O.H.: Parametric model-checking of stopwatch Petri nets. J. Univers. Comput. Sci. 15(17), 3273–3304 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Saarland UniversitySaarbruckenGermany
  2. 2.National Institute of InformaticsChiyoda-ku, TokyoJapan
  3. 3.IRCCyN UMR CNRS 6597 (Institut de Recherche en Communications et Cybernétique de Nantes)LUNAM Université, École Centrale de NantesNantes Cedex 3France

Personalised recommendations