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Reachability Analysis of a Class of Hybrid Gene Regulatory Networks

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Reachability Problems (RP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14235))

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Abstract

In this work, we study the reachability analysis method of a class of hybrid systems called HGRN which is a special case of hybrid automata. The reachability problem concerned in this work is, given a singular state and a region (a set of states), to determine whether the trajectory from this singular state can reach this region. This problem is undecidable for general hybrid automata, and is decidable only for a restricted class of hybrid automata, but this restricted class does not include HGRNs. A priori, reachability in HGRNs is not decidable; however, we show in this paper that it is decidable in certain cases, more precisely if there is no chaos. Based on this fact, the main idea of this work is that if the decidable cases can be determined automatically, then the reachability problem can be solved partially. The two major contributions are the following: firstly, we classify trajectories into different classes and provide theoretical results about decidability; then based on these theoretical results, we propose a reachability analysis algorithm which always stops in finite time and answers the reachability problem partially (meaning that it can stop with the inconclusive result, for example with the presence of chaos).

Supported by China Scholarship Council.

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References

  1. Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. Technical report, Cornell University (1993)

    Google Scholar 

  2. Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoret. Comput. Sci. 138(1), 35–65 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asarin, E., Mysore, V.P., Pnueli, A., Schneider, G.: Low dimensional hybrid systems-decidable, undecidable, don’t know. Inf. Comput. 211, 138–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Behaegel, J., Comet, J.P., Bernot, G., Cornillon, E., Delaunay, F.: A hybrid model of cell cycle in mammals. J. Bioinform. Comput. Biol. 14(01), 1640001 (2016)

    Article  Google Scholar 

  5. Belgacem, I., Gouzé, J.L., Edwards, R.: Control of negative feedback loops in genetic networks. In: 2020 59th IEEE Conference on Decision and Control (CDC), pp. 5098–5105. IEEE (2020)

    Google Scholar 

  6. Chai, X., Ribeiro, T., Magnin, M., Roux, O., Inoue, K.: Static analysis and stochastic search for reachability problem. Electron. Notes Theor. Comput. Sci. 350, 139–158 (2020)

    Article  MATH  Google Scholar 

  7. Cornillon, E., Comet, J.P., Bernot, G., Enée, G.: Hybrid gene networks: a new framework and a software environment. Adv. Syst. Synthetic Biol., 57–84 (2016)

    Google Scholar 

  8. Dang, T., Testylier, R.: Reachability analysis for polynomial dynamical systems using the Bernstein expansion. Reliab. Comput. 17(2), 128–152 (2012)

    MathSciNet  Google Scholar 

  9. Edwards, R., Glass, L.: A calculus for relating the dynamics and structure of complex biological networks. Adv. Chem. Phys. 132, 151–178 (2006)

    Google Scholar 

  10. Edwards, R.: Analysis of continuous-time switching networks. Phys. D 146(1–4), 165–199 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Edwards, R., Glass, L.: A calculus for relating the dynamics and structure of complex biological networks. Adv. Chem. Phys.: Spec. Vol. Adv. Chem. Phys. 132, 151–178 (2005)

    Google Scholar 

  12. Firippi, E., Chaves, M.: Topology-induced dynamics in a network of synthetic oscillators with piecewise affine approximation. Chaos Interdisc. J. Nonlinear Sci. 30(11), 113128 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flieller, D., Riedinger, P., Louis, J.P.: Computation and stability of limit cycles in hybrid systems. Nonlinear Anal.: Theory Methods Appl. 64(2), 352–367 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Folschette, M., Paulevé, L., Magnin, M., Roux, O.: Sufficient conditions for reachability in automata networks with priorities. Theoret. Comput. Sci. 608, 66–83 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_30

    Chapter  Google Scholar 

  16. Girard, A.: Computation and stability analysis of limit cycles in piecewise linear hybrid systems. IFAC Proc. Vol. 36(6), 181–186 (2003)

    Article  Google Scholar 

  17. Gouzé, J.L., Sari, T.: A class of piecewise linear differential equations arising in biological models. Dyn. Syst. 17(4), 299–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hamatani, S., Tsubone, T.: Analysis of a 3-dimensional piecewise-constant chaos generator without constraint. IEICE Proc. Ser. 48(A2L-B-3), 11–14 (2016)

    Google Scholar 

  19. Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pp. 373–382 (1995)

    Google Scholar 

  20. Hiskens, I.A.: Stability of hybrid system limit cycles: application to the compass gait biped robot. In: Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No. 01CH37228), vol. 1, pp. 774–779. IEEE (2001)

    Google Scholar 

  21. Maler, O., Pnueli, A.: Reachability analysis of planar multi-linear systems. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 194–209. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56922-7_17

    Chapter  Google Scholar 

  22. Mestl, T., Lemay, C., Glass, L.: Chaos in high-dimensional neural and gene networks. Phys. D 98(1), 33–52 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paulevé, L.: Reduction of qualitative models of biological networks for transient dynamics analysis. IEEE/ACM Trans. Comput. Biol. Bioinf. 15(4), 1167–1179 (2017)

    Article  Google Scholar 

  24. Plahte, E., Kjøglum, S.: Analysis and generic properties of gene regulatory networks with graded response functions. Phys. D 201(1–2), 150–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sandler, A., Tveretina, O.: Deciding reachability for piecewise constant derivative systems on orientable manifolds. In: Filiot, E., Jungers, R., Potapov, I. (eds.) RP 2019. LNCS, vol. 11674, pp. 178–192. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30806-3_14

    Chapter  MATH  Google Scholar 

  26. Sun, H., Folschette, M., Magnin, M.: Limit cycle analysis of a class of hybrid gene regulatory networks. In: Petre, I., Păun, A. (eds.) CMSB 2022. LNCS, vol. 13447, pp. 217–236. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15034-0_11

    Chapter  Google Scholar 

  27. Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42(3), 563–585 (1973)

    Article  Google Scholar 

  28. Thomas, R.: Regulatory networks seen as asynchronous automata: a logical description. J. Theor. Biol. 153(1), 1–23 (1991)

    Article  MathSciNet  Google Scholar 

  29. Znegui, W., Gritli, H., Belghith, S.: Design of an explicit expression of the poincaré map for the passive dynamic walking of the compass-gait biped model. Chaos Solitons Fractals 130, 109436 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, 44000, Nantes, France

    Honglu Sun & Morgan Magnin

  2. Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, 59000, Lille, France

    Maxime Folschette

Authors
  1. Honglu Sun
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  2. Maxime Folschette
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  3. Morgan Magnin
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Corresponding author

Correspondence to Honglu Sun .

Editor information

Editors and Affiliations

  1. École Polytechnique, Paris, France

    Olivier Bournez

  2. Université Côte d’Azur, Nice, France

    Enrico Formenti

  3. University of Liverpool, Liverpool, UK

    Igor Potapov

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Additional Information

Link to the code: https://github.com/Honglu42/Reachability_HGRN/. Link to the Appendix: https://hal.science/hal-04182253.

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Sun, H., Folschette, M., Magnin, M. (2023). Reachability Analysis of a Class of Hybrid Gene Regulatory Networks. In: Bournez, O., Formenti, E., Potapov, I. (eds) Reachability Problems. RP 2023. Lecture Notes in Computer Science, vol 14235. Springer, Cham. https://doi.org/10.1007/978-3-031-45286-4_5

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  • DOI: https://doi.org/10.1007/978-3-031-45286-4_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-45285-7

  • Online ISBN: 978-3-031-45286-4

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