International Workshop on Hybrid Systems Biology

Hybrid Systems Biology pp 58-74 | Cite as

High-Performance Discrete Bifurcation Analysis for Piecewise-Affine Dynamical Systems

  • Luboš Brim
  • Martin Demko
  • Samuel Pastva
  • David Šafránek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9271)

Abstract

Analysis of equilibria, their stability and instability, is an unavoidable ingredient of model analysis in systems biology. In particular, bifurcation analysis which focuses on behaviour of phase portraits under variations of parameters is of great importance. We propose a novel method for bifurcation analysis that employs coloured model checking to analyse phase portraits bifurcation in rectangular abstractions of piecewise-affine systems. The algorithm works on clusters of workstations and multi-core computers to allow scalability. We demonstrate the method on a repressilator genetic regulatory network.

References

  1. 1.
    Alon, U.: An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC Mathematical and Computational Biology, Boca Raton (2006)MATHGoogle Scholar
  2. 2.
    Anishchenko, V.S., Vadivasova, T.E., Strelkova, G.I.: Deterministic Nonlinear Systems: A Short Course. Springer, Heidelberg (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Bagley, R.J., Glass, L.: Counting and classifying attractors in high dimensional dynamical systems. J. Theor. Biol. 183(3), 269–284 (1996)CrossRefGoogle Scholar
  4. 4.
    Batt, G., Belta, C., Weiss, R.: Model checking genetic regulatory networks with parameter uncertainty. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 61–75. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Batt, G., Besson, B., Ciron, P.-E., de Jong, H., Dumas, E., Geiselmann, J., Monte, R., Monteiro, P.T., Page, M., Rechenmann, F., Ropers, D.: Genetic network analyzer: A tool for the qualitative modeling and simulation of bacterial regulatory networks. In: van Helden, J., Toussaint, A., Thieffry, D. (eds.) Bacterial Molecular Networks: Methods and Protocols. Methods in Molecular Biology, vol. 804, pp. 439–462. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Belta, C., Habets, L.C.G.J.M.: Controlling a class of nonlinear systems on rectangles. IEEE Trans. Automat. Contr. 51(11), 1749–1759 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brim, L., Češka, M., Demko, M., Pastva, S., Šafránek, D.: Parameter synthesis by parallel coloured CTL model checking. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 251–263. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  8. 8.
    Chaves, M., Tournier, L., Gouzé, J.-L.: Comparing boolean and piecewise affine differential models for genetic networks. Acta Biotherica 58(2–3), 217–232 (2010)CrossRefGoogle Scholar
  9. 9.
    Collins, P., Habets, L.C.G.J.M., van Schuppen, J.H., Černá, I., Fabriková, J., Šafránek, D.: Abstraction of biochemical reaction systems on polytopes. In: IFAC World Congress, pp. 14869–14875. IFAC (2011)Google Scholar
  10. 10.
    de Jong, H., Page, M.: Search for steady states of piecewise-linear differential equation models of genetic regulatory networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 5(2), 208–222 (2008)CrossRefGoogle Scholar
  11. 11.
    De Nicola, R., Vaandrager, F.: Three logics for branching bisimulation. In: Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, pp. 118–129 (1990)Google Scholar
  12. 12.
    Dvorak, P., Bidmanova, S., Damborsky, J., Prokop, Z.: Immobilized synthetic pathway for biodegradation of toxic recalcitrant pollutant 1,2,3-trichloropropane. Environ. Sci. Technol. 48(12), 6859–6866 (2014)CrossRefGoogle Scholar
  13. 13.
    Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)CrossRefGoogle Scholar
  14. 14.
    Glass, L., Kauffman, S.A.: The logical analysis of continuous, non-linear biochemical control networks. J. Theor. Biol. 39(1), 103–129 (1973)CrossRefGoogle Scholar
  15. 15.
    Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From cardiac cells to genetic regulatory networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Holzwarth, A.R., Müller, M.G., Reus, M., Nowaczyk, M., Sander, J., Rögner, M.: Kinetics and mechanism of electron transfer in intact photosystem II and in the isolated reaction center: Pheophytin is the primary electron acceptor. Proc. Nat. Acad. Sci. 103(18), 6895–6900 (2006)CrossRefGoogle Scholar
  17. 17.
    Jamshidi, S., Siebert, H., Bockmayr, A.: Comparing discrete and piecewise affine differential equation models of gene regulatory networks. In: Lones, M.A., Smith, S.L., Teichmann, S., Naef, F., Walker, J.A., Trefzer, M.A. (eds.) IPCAT 2012. LNCS, vol. 7223, pp. 17–24. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Jha, S., Shyamasundar, R.K.: Adapting biochemical Kripke structures for distributed model checking. In: Priami, C., Ingólfsdóttir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on computational systems biology vii. LNCS (LNBI), vol. 4230, pp. 107–122. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    De Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9, 67–103 (2002)CrossRefGoogle Scholar
  20. 20.
    Müller, S., Hofbauer, J., Endler, L., Flamm, C., Widder, S., Schuster, P.: A generalized model of the repressilator. J. Math. Biol. 53(6), 905–937 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Snoussi, E.H.: Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn. Stab. Syst. 4(3–4), 565–583 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Veliz-Cuba, A., Arthur, J., Hochstetler, L., Klomps, V., Korpi, E.: On the relationship of steady states of continuous and discrete models arising from biology. Bull. Math. Biol. 74(12), 2779–2792 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yordanov, B., Belta, C., Batt, G.: Model checking discrete time piecewise affine systems: application to gene networks. In: European Control Conference (ECC) (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Luboš Brim
    • 1
  • Martin Demko
    • 1
  • Samuel Pastva
    • 1
  • David Šafránek
    • 1
  1. 1.Systems Biology Laboratory, Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations