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A Formal Approach for Tuning Stochastic Oscillators

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Computational Methods in Systems Biology (CMSB 2023)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 14137))

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Abstract

Periodic recurrence is a prominent behavioural of many biological phenomena, including cell cycle and circadian rhythms. Although deterministic models are commonly used to represent the dynamics of periodic phenomena, it is known that they are little appropriate in the case of systems in which stochastic noise induced by small population numbers is actually responsible for periodicity. Within the stochastic modelling settings automata-based model checking approaches have proven an effective means for the analysis of oscillatory dynamics, the main idea being that of coupling a period detector automaton with a continuous-time Markov chain model of an alleged oscillator. In this paper we address a complementary aspect, i.e. that of assessing the dependency of oscillation related measure (period and amplitude) against the parameters of a stochastic oscillator. To this aim we introduce a framework which, by combining an Approximate Bayesian Computation scheme with a hybrid automata capable of quantifying how distant an instance of a stochastic oscillator is from matching a desired (average) period, leads us to identify regions of the parameter space in which oscillation with given period are highly likely. The method is demonstrated through a couple of case studies, including a model of the popular Repressilator circuit.

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Notes

  1. 1.

    The population of species i is \(k\in \mathbb {N}\).

  2. 2.

    \(t_{p_k}\) could alternatively be defined as \(t_{p_k}\!=\! min(T_{(k\!+\!1)\uparrow })- min(T_{k\uparrow })\), that is, w.r.t. crossing into the high region, rather than into the low region. It is straightforward to show that both definitions are semantically equivalent, i.e., the average value of \(t_{p_k}\) measured along a trace is equivalent with both definitions.

References

  1. Andreychenko, A., Krüger, T., Spieler, D.: Analyzing oscillatory behavior with formal methods. In: Remke, A., Stoelinga, M. (eds.) Stochastic Model Checking. Rigorous Dependability Analysis Using Model Checking Techniques for Stochastic Systems. LNCS, vol. 8453, pp. 1–25. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45489-3_1

    Chapter  Google Scholar 

  2. Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61474-5_75

    Chapter  Google Scholar 

  3. Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Software Eng. 29, 524–541 (2003)

    Article  Google Scholar 

  4. Ballarini, P., Barbot, B., Duflot, M., Haddad, S., Pekergin, N.: HASL: a new approach for performance evaluation and model checking from concepts to experimentation. Perform. Eval. 90, 53–77 (2015)

    Article  Google Scholar 

  5. Ballarini, P., Guerriero, M.L.: Query-based verification of qualitative trends and oscillations in biochemical systems. Theoret. Comput. Sci. 411(20), 2019–2036 (2010)

    Article  Google Scholar 

  6. Ballarini, P.: Analysing oscillatory trends of discrete-state stochastic processes through HASL statistical model checking. Int. J. Softw. Tools Technol. Transf. 17(4), 505–526 (2015)

    Article  Google Scholar 

  7. Ballarini, P., Duflot, M.: Applications of an expressive statistical model checking approach to the analysis of genetic circuits. Theor. Comput. Sci. 599, 4–33 (2015)

    Article  Google Scholar 

  8. Ballarini, P., Mardare, R., Mura, I.: Analysing biochemical oscillation through probabilistic model checking. Electron. Notes Theor. Comput. Sci. 229(1), 3–19 (2009)

    Article  Google Scholar 

  9. Beaumont, M.A., Cornuet, J.-M., Marin, J.-M., Robert, C.P.: Adaptive approximate Bayesian computation. Biometrika 96(4), 983–990 (2009)

    Article  Google Scholar 

  10. Bentriou, M.: Statistical inference and verification of chemical reaction networks. Ph.D. thesis, École doctorale Interfaces, University Paris Saclay (2021)

    Google Scholar 

  11. Bentriou, M., Ballarini, P., Cournède, P.-H.: Reachability design through approximate Bayesian computation. In: Bortolussi, L., Sanguinetti, G. (eds.) CMSB 2019. LNCS, vol. 11773, pp. 207–223. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31304-3_11

    Chapter  Google Scholar 

  12. Bentriou, M., Ballarini, P., Cournède, P.-H.: Automaton-ABC: a statistical method to estimate the probability of spatio-temporal properties for parametric Markov population models. Theor. Comput. Sci. 893, 191–219 (2021)

    Article  Google Scholar 

  13. Bentriou, M., Boatto, S., Viaud, G., Bonnet, C., Cournède, P.-H.: Assimilation de données par filtrage particulaire régularisé dans un modèle d’épidémiologie, pp. 1–6 (2008)

    Google Scholar 

  14. Bortolussi, L., Milios, D., Sanguinetti, G.: Smoothed model checking for uncertain continuous-time Markov chains. Inf. Comput. 247, 235–253 (2016)

    Article  Google Scholar 

  15. Brim, L., Češka, M., Dražan, S., Šafránek, D.: Exploring parameter space of stochastic biochemical systems using quantitative model checking. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 107–123. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_7

    Chapter  Google Scholar 

  16. Cardelli, L.: Artificial biochemistry. In: Condon, A., Harel, D., Kok, J., Salomaa, A., Winfree, E. (eds.) Algorithmic Bioprocesses, pp. 429–462. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-88869-7_22

    Chapter  Google Scholar 

  17. Češka, M., Dannenberg, F., Kwiatkowska, M., Paoletti, N.: Precise parameter synthesis for stochastic biochemical systems. In: Mendes, P., Dada, J.O., Smallbone, K. (eds.) CMSB 2014. LNCS, vol. 8859, pp. 86–98. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-12982-2_7

    Chapter  Google Scholar 

  18. Chabrier-Rivier, N., Chiaverini, M., Danos, V., Fages, F., Schächter, V.: Modeling and querying biomolecular interaction networks. Theor. Comput. Sci. 325, 25–44 (2004)

    Article  Google Scholar 

  19. Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite state concurrent systems using temporal logic specifications: a practical approach. In: Wright, J.R., Landweber, L., Demers, A.J., Teitelbaum, T. (eds.) Conference Record of the Tenth Annual ACM Symposium on Principles of Programming Languages, Austin, Texas, USA, January 1983, pp. 117–126. ACM Press (1983)

    Google Scholar 

  20. Del Moral, P., Doucet, A., Jasra, A.: An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. 22(5), 1009–1020 (2012)

    Article  Google Scholar 

  21. Elowitz, M., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(335), 335–338 (2000)

    Article  CAS  PubMed  Google Scholar 

  22. Goldbeter, A.: Computational approaches to cellular rhythms. Nature 420, 238–245 (2002)

    Article  CAS  PubMed  Google Scholar 

  23. Han, T., Katoen, J.P., Mereacre, A.: Approximate parameter synthesis for probabilistic time-bounded reachability. In: 2008 Real-Time Systems Symposium, pp. 173–182 (2008)

    Google Scholar 

  24. Thomas, P.J., Lindner, B., MacLaurin, J., Fellous, J.M.: Stochastic oscillators in biology: introduction to the special issue. Biol. Cybern. 116(2), 119–120 (2022)

    Article  PubMed  PubMed Central  Google Scholar 

  25. Marin, J.-M., Pudlo, P., Robert, C.P., Ryder, R.J.: Approximate Bayesian computational methods. Stat. Comput. 22(6), 1167–1180 (2012)

    Article  Google Scholar 

  26. Molyneux, G.W., Abate, A.: ABC(SMC)\(^2\): simultaneous inference and model checking of chemical reaction networks. In: Abate, A., Petrov, T., Wolf, V. (eds.) CMSB 2020. LNCS, vol. 12314, pp. 255–279. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60327-4_14

    Chapter  Google Scholar 

  27. Molyneux, G.W., Wijesuriya, V.B., Abate, A.: Bayesian verification of chemical reaction networks. In: Sekerinski, E., et al. (eds.) FM 2019. LNCS, vol. 12233, pp. 461–479. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-54997-8_29

    Chapter  Google Scholar 

  28. Pnueli, A.: The temporal logic of programs. In: 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October–1 November 1977, pp. 46–57. IEEE Computer Society (1977)

    Google Scholar 

  29. Sisson, S.A., Fan, Y., Beaumont, M.: Handbook of Approximate Bayesian Computation. Chapman and Hall/CRC (2018)

    Google Scholar 

  30. Sneyd, J., Tsaneva-Atanasova, K., Reznikov, V., Bai, Y., Sanderson, M.J., Yule, D.I.: A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations. Proc. Natl. Acad. Sci. 103(6), 1675–1680 (2006)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  31. Spieler, D.: Characterizing oscillatory and noisy periodic behavior in Markov population models. In: Proceedings of QEST 2013 (2013)

    Google Scholar 

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Authors and Affiliations

  1. MICS, CentraleSupélec, Université Paris-Saclay, Paris, France

    Paolo Ballarini, Mahmoud Bentriou & Paul-Henry Cournède

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  1. Paolo Ballarini
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  2. Mahmoud Bentriou
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  3. Paul-Henry Cournède
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Correspondence to Paolo Ballarini .

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Editors and Affiliations

  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Jun Pang

  2. Inria Lille, Villeneuve d’Ascq, France

    Joachim Niehren

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Ballarini, P., Bentriou, M., Cournède, PH. (2023). A Formal Approach for Tuning Stochastic Oscillators. In: Pang, J., Niehren, J. (eds) Computational Methods in Systems Biology. CMSB 2023. Lecture Notes in Computer Science(), vol 14137. Springer, Cham. https://doi.org/10.1007/978-3-031-42697-1_1

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  • DOI: https://doi.org/10.1007/978-3-031-42697-1_1

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