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Limit Cycle Analysis of a Class of Hybrid Gene Regulatory Networks

Part of the Lecture Notes in Computer Science book series (LNBI,volume 13447)

Abstract

Many gene regulatory networks have periodic behavior, for instance the cell cycle or the circadian clock. Therefore, the study of formal methods to analyze limit cycles in mathematical models of gene regulatory networks is of interest. In this work, we study a pre-existing hybrid modeling framework (HGRN) which extends René Thomas’ widespread discrete modeling. We propose a new formal method to find all limit cycles that are simple and deterministic, and analyze their stability, that is, the ability of the model to converge back to the cycle after a small perturbation. Up to now, only limit cycles in two dimensions (with two genes) have been studied; our work fills this gap by proposing a generic approach applicable in higher dimensions. For this, the hybrid states are abstracted to consider only their borders, in order to enumerate all simple abstract cycles containing possible concrete trajectories. Then, a Poincaré map is used, based on the notion of transition matrix of the concrete continuous dynamics inside these abstract paths. We successfully applied this method on existing models: three HGRNs of negative feedback loops with 3 components, and a HGRN of the cell cycle with 5 components.

Keywords

  • Hybrid modeling
  • Celerity
  • Transition matrix
  • Limit cycle
  • Gene regulatory networks
  • Poincaré map

Supported by China Scholarship Council.

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Notes

  1. 1.

    Computations were performed on a standard laptop computer, with an Intel Core I7-8550U 1.80 GHz processor and 16.0 GB RAM.

References

  1. Alur, R., Dang, T., Ivančić, F.: Counter-example guided predicate abstraction of hybrid systems. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 208–223. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36577-X_15

    CrossRef  Google Scholar 

  2. Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proc. IEEE 88(7), 971–984 (2000)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  5. Barik, D., Baumann, W.T., Paul, M.R., Novak, B., Tyson, J.J.: A model of yeast cell-cycle regulation based on multisite phosphorylation. Mol. Syst. Biol. 6(1), 405 (2010)

    CrossRef  Google Scholar 

  6. 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)

    Google Scholar 

  7. Behaegel, J., Comet, J.P., Folschette, M.: Constraint identification using modified Hoare logic on hybrid models of gene networks. In: 24th International Symposium on Temporal Representation and Reasoning (TIME 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  8. 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 

  9. Chaves, M., Gouzé, J.L.: Exact control of genetic networks in a qualitative framework: the bistable switch example. Automatica 47(6), 1105–1112 (2011)

    CrossRef  Google Scholar 

  10. Chaves, M., Preto, M.: Hierarchy of models: from qualitative to quantitative analysis of circadian rhythms in cyanobacteria. Chaos: Interdisc. J. Nonlinear Sci. 23(2), 025113 (2013)

    Google Scholar 

  11. Clark, W., Bloch, A.: A Poincaré-Bendixson theorem for hybrid dynamical systems on directed graphs. Math. Control Signals Syst. 32(1), 1–18 (2020)

    CrossRef  Google Scholar 

  12. Clark, W., Bloch, A., Colombo, L.: A Poincaré-Bendixson theorem for hybrid systems. Math. Control Relat. Fields 10(1), 27 (2020)

    CrossRef  Google Scholar 

  13. 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)

    CrossRef  Google Scholar 

  14. Comet, J.-P., Fromentin, J., Bernot, G., Roux, O.: A formal model for gene regulatory networks with time delays. In: Chan, J.H., Ong, Y.-S., Cho, S.-B. (eds.) CSBio 2010. CCIS, vol. 115, pp. 1–13. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16750-8_1

    CrossRef  Google Scholar 

  15. Cornillon, E., Comet, J.P., Bernot, G., Enée, G.: Hybrid gene networks: a new framework and a software environment. In: Advances in Systems and Synthetic Biology (2016)

    Google Scholar 

  16. Doyen, L., Frehse, G., Pappas, G.J., Platzer, A.: Verification of hybrid systems. In: Clarke, E., Henzinger, T., Veith, H., Bloem, R. (eds.) Handbook of Model Checking, pp. 1047–1110. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_30

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  18. Edwards, R., Glass, L.: A calculus for relating the dynamics and structure of complex biological networks. Adventures Chem. Phys.: A Special Volume of Advances in Chemical Physics 132, 151–178 (2005)

    Google Scholar 

  19. Farcot, E., Gouzé, J.L.: Periodic solutions of piecewise affine gene network models with non uniform decay rates: the case of a negative feedback loop. Acta. Biotheor. 57(4), 429–455 (2009)

    CrossRef  Google Scholar 

  20. 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)

    Google Scholar 

  21. 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)

    CrossRef  Google Scholar 

  22. Geva-Zatorsky, N., et al.: Oscillations and variability in the p53 system. Mol. Syst. Biol. 2(1), 2006–0033 (2006)

    Google Scholar 

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

    CrossRef  Google Scholar 

  24. Glass, L., Edwards, R.: Hybrid models of genetic networks: mathematical challenges and biological relevance. J. Theor. Biol. 458, 111–118 (2018)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  26. 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 

  27. Hiskens, I.A.: Stability of limit cycles in hybrid systems. In: Proceedings of the 34th Annual Hawaii International Conference on System Sciences, pp. 6-pp. IEEE (2001)

    Google Scholar 

  28. Karlebach, G., Shamir, R.: Modelling and analysis of gene regulatory networks. Nat. Rev. Mol. Cell Biol. 9(10), 770–780 (2008)

    CAS  CrossRef  Google Scholar 

  29. Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22(3), 437–467 (1969)

    CAS  CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

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

    CAS  CrossRef  Google Scholar 

  32. Prabhakar, P., Garcia Soto, M.: Abstraction based model-checking of stability of hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 280–295. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_20

    CrossRef  Google Scholar 

  33. Simic, S.N., Sastry, S., Johansson, K.H., Lygeros, J.: Hybrid limit cycles and hybrid Poincaré-Bendixson. IFAC Proc. Vol. 35(1), 197–202 (2002)

    CrossRef  Google Scholar 

  34. Sriram, K., Bernot, G., Képès, F.: Discrete delay model for the mammalian circadian clock. Complexus 3(4), 185–199 (2006)

    CrossRef  Google Scholar 

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

    CAS  CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

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

    Google Scholar 

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Acknowledgements

We would like to thank Gilles Bernot and Jean-Paul Comet for their fruitful discussions.

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Correspondence to Honglu Sun .

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Sun, H., Folschette, M., Magnin, M. (2022). Limit Cycle Analysis of a Class of Hybrid Gene Regulatory Networks. In: Petre, I., Păun, A. (eds) Computational Methods in Systems Biology. CMSB 2022. Lecture Notes in Computer Science(), vol 13447. Springer, Cham. https://doi.org/10.1007/978-3-031-15034-0_11

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  • DOI: https://doi.org/10.1007/978-3-031-15034-0_11

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