Detecting Attractors in Biological Models with Uncertain Parameters

  • Jiří Barnat
  • Nikola BenešEmail author
  • Luboš Brim
  • Martin Demko
  • Matej Hajnal
  • Samuel Pastva
  • David Šafránek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10545)


Complex behaviour arising in biological systems is typically characterised by various kinds of attractors. An important problem in this area is to determine these attractors. Biological systems are usually described by highly parametrised dynamical models that can be represented as parametrised graphs typically constructed as discrete abstractions of continuous-time models. In such models, attractors are observed in the form of terminal strongly connected components (tSCCs). In this paper, we introduce a novel method for detecting tSCCs in parametrised graphs. The method is supplied with a parallel algorithm and evaluated on discrete abstractions of several non-linear biological models.


  1. 1.
    Barnat, J., Chaloupka, J., Van De Pol, J.: Distributed algorithms for SCC decomposition. J. Logic Comput. 21(1), 23–44 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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). doi: 10.1007/978-3-540-71493-4_8 CrossRefGoogle Scholar
  3. 3.
    Batt, G., Yordanov, B., Weiss, R., Belta, C.: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 23(18), 2415–2422 (2007)CrossRefGoogle Scholar
  4. 4.
    Beneš, N., Brim, L., Demko, M., Pastva, S., Šafránek, D.: A model checking approach to discrete bifurcation analysis. In: Fitzgerald, J., Heitmeyer, C., Gnesi, S., Philippou, A. (eds.) FM 2016. LNCS, vol. 9995, pp. 85–101. Springer, Cham (2016). doi: 10.1007/978-3-319-48989-6_6 Google Scholar
  5. 5.
    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, Cham (2015). doi: 10.1007/978-3-319-23401-4_21 CrossRefGoogle Scholar
  6. 6.
    Brim, L., Demko, M., Pastva, S., Šafránek, D.: High-performance discrete bifurcation analysis for piecewise-affine dynamical systems. In: Abate, A., Šafránek, D. (eds.) HSB 2015. LNCS, vol. 9271, pp. 58–74. Springer, Cham (2015). doi: 10.1007/978-3-319-26916-0_4 CrossRefGoogle Scholar
  7. 7.
    Chandy, K.M., Misra, J.: Distributed computation on graphs: shortest path algorithms. Commun. ACM 25(11), 833–837 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chatain, T., Haar, S., Jezequel, L., Paulevé, L., Schwoon, S.: Characterization of reachable attractors using petri net unfoldings. In: Mendes, P., Dada, J.O., Smallbone, K. (eds.) CMSB 2014. LNCS, vol. 8859, pp. 129–142. Springer, Cham (2014). doi: 10.1007/978-3-319-12982-2_10 Google Scholar
  9. 9.
    Choo, S.M., Cho, K.H.: An efficient algorithm for identifying primary phenotype attractors of a large-scale boolean network. BMC Syst. Biol. 10(1), 95 (2016)CrossRefGoogle Scholar
  10. 10.
    Coayla-Teran, E.A., Mohammed, S.E.A., Ruffino, P.R.C.: Hartman-grobman theorems along hyperbolic stationary trajectories. Discret. Contin. Dyn. Syst. 17(2), 281–292 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Collins, P., Habets, L., van Schuppen, J., Černá, I., Fabriková, J., Šafránek, D.: Abstraction of biochemical reaction systems on polytopes. In: IFAC World Congress, pp. 14869–14875. IFAC (2011)Google Scholar
  12. 12.
    Demko, M., Beneš, N., Brim, L., Pastva, S., Šafránek, D.: High-performance symbolic parameter synthesis of biological models: a case study. In: Bartocci, E., Lio, P., Paoletti, N. (eds.) CMSB 2016. LNCS, vol. 9859, pp. 82–97. Springer, Cham (2016). doi: 10.1007/978-3-319-45177-0_6 CrossRefGoogle Scholar
  13. 13.
    Dijkstra, E.W.: In reaction to Ernest Chang’s “Deadlock Detection” (1979).
  14. 14.
    Dilão, R.: The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models. J. Theor. Biol. 340, 199–208 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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). doi: 10.1007/978-3-642-22110-1_31 CrossRefGoogle Scholar
  17. 17.
    Guo, W., Yang, G., Wu, W., He, L., Sun, M.: A parallel attractor finding algorithm based on boolean satisfiability for genetic regulatory networks. PLOS ONE 9(4), 1–10 (2014)Google Scholar
  18. 18.
    MacArthur, B.D., Ma’ayan, A., Lemischka, I.R.: Systems biology of stem cell fate and cellular reprogramming. Nat. Rev. Mol. Cell Biol. 10(10), 672–681 (2009)Google Scholar
  19. 19.
    McLendon III, W., Hendrickson, B., Plimpton, S.J., Rauchwerger, L.: Finding strongly connected components in distributed graphs. J. Parallel Distrib. Comput. 65(8), 901–910 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99(2), 177–195 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Reif, J.H.: Depth-first search is inherently sequential. Inf. Process. Lett. 20(5), 229–234 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sullivan, D., Williams, R.: On the homology of attractors. Topology 15(3), 259–262 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Swat, M., Kel, A., Herzel, H.: Bifurcation analysis of the regulatory modules of the mammalian G1/S transition. Bioinformatics 20(10), 1506–1511 (2004)CrossRefGoogle Scholar
  25. 25.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jiří Barnat
    • 1
  • Nikola Beneš
    • 1
    Email author
  • Luboš Brim
    • 1
  • Martin Demko
    • 1
  • Matej Hajnal
    • 1
  • Samuel Pastva
    • 1
  • David Šafránek
    • 1
  1. 1.Systems Biology Laboratory, Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations