An Algebraic Algorithm for the Identification of Glass Networks with Periodic Orbits Along Cyclic Attractors

  • Igor Zinovik
  • Daniel Kroening
  • Yury Chebiryak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4545)


Glass piecewise linear ODE models are frequently used for simulation of neural and gene regulatory networks. Efficient computational tools for automatic synthesis of such models are highly desirable. However, the existing algorithms for the identification of desired models are limited to four-dimensional networks, and rely on numerical solutions of eigenvalue problems. We suggest a novel algebraic criterion to detect the type of the phase flow along network cyclic attractors that is based on a corollary of the Perron-Frobenius theorem. We show an application of the criterion to the analysis of bifurcations in the networks. We propose to encode the identification of models with periodic orbits along cyclic attractors as a propositional formula, and solving it using state-of-the-art SAT-based tools for real linear arithmetic. New lower bounds for the number of equivalence classes are calculated for cyclic attractors in six-dimensional networks. Experimental results indicate that the run-time of our algorithm increases slower than the size of the search space of the problem.


Periodic Orbit Gene Regulatory Network Conjunctive Normal Form Glass Network Propositional Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Glass, L., Kaufmann, S.: The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103–129 (1973)CrossRefGoogle Scholar
  2. 2.
    Glass, L.: Combinatorial aspects of dynamics in biological systems. In: Stat. Mech Stat. Methods in Theory and Application, pp. 585–611. Plenum Press, New York (1976)Google Scholar
  3. 3.
    Edwards, R.: Symbolic dynamics and computation in model gene networks. Chaos 11, 160–169 (2001)CrossRefGoogle Scholar
  4. 4.
    Ghosh, R., Tiwari, A., Tomlin, C.: Automated symbolic reachability analysis; with application to delta-notch signalic automata. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 233–248. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Mason, J., Linsay, P., Collins, J., Glass, L.: Evolving complex dynamics in electronic models of genetic networks. Chaos 14, 707–715 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Batt, G., Ropers, D., de Jong, H., Geiselmann, J., Mateescu, R., Page, M., Schneider, D.: Analysis and verification of qualitative models of genetic regulatory networks: A model-checking approach. In: 19th Int Joint Conference on Artificial Intelligence, pp. 370–375 (2005)Google Scholar
  7. 7.
    Gedeon, T.: Global dynamics of neural nets with infinite gain. Physica D: Nonlinear Phenomena 146, 200–212 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gedeon, T.: Attractors in continuous time switching networks. Communications on Pure and Applied Analysis 2, 187–209 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    de Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. J. Comp. Biol. 9, 67–103 (2002)CrossRefGoogle Scholar
  10. 10.
    Casey, R., de Jong, H., Gouze, J.L.: Piecewise-liner models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52, 27–56 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Farcot, E.: Geometric properties of a class of piecewise affine biological network models. J. Math. Biol. 52, 373–418 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Edwards, R.: Analysis of continuous-time switching networks. Physica D 146, 165–199 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Glass, L., Pasternack, J.: Stable oscillations in mathematical models of biological control systems. J. Math. Biol. 6, 207–223 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mestl, T., Plahte, E., Omholt, S.: Periodic solutions in systems of piecewise-linear differential equations. Dynam. Stabil. Syst. 10, 179–193 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Edwards, R., Glass, L.: Combinatorial explosion in model gene networks. Chaos 10, 691–704 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mestl, T., Lemay, C., Glass, L.: Chaos in high-dimensional neural and gene networks. Physica D 98, 33–52 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Thomas, R., Kaufman, M.: Multistationarity, the basis of cell differentiation and memory. Chaos 11, 170–195 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kauffman, S.: A proposal for using the ensemble approach to understand genetic regulatory networks. Theor. Biol. 230, 581–590 (2004)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Glass, L.: Global analysis of nonlinear chemical kinetics. Statistical mechanics, part B: time dependent processes, 311–349 (1977)Google Scholar
  20. 20.
    Glass, L., Pasternack, J.: Prediction of limit cycles in mathematical models of biological oscillations. Bull. Math. Biol. 40, 27–44 (1978)MathSciNetGoogle Scholar
  21. 21.
    Laubenbacher, R., Stigler, B.: A computational algebra approach to the reverse engineering of gene regulatory networks. J. Theor. Biol. 229, 523–537 (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Gantmacher, F.: The Theory of Matrices. vol. 2. Chelsea (1974)Google Scholar
  23. 23.
    Zinovik, I., Kroening, D., Chebiryak, Y.: An algebraic algorithm for the identification of Glass networks with periodic orbits along cyclic attractors. Technical Report 557, ETH Zurich, Computer Science Department (2007)Google Scholar
  24. 24.
    Rajan, D., Shende, A.: Maximal and reversible snakes in hypercubes. In: 24th Annual Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (1999)Google Scholar
  25. 25.
    Casella, W.P.D.: Using evolutionary techniques to hunt for the snakes and coils. In: The 2005 IEEE Congress on Evolutionary Computation, vol. 3, pp. 2499–2505. IEEE Press, NJ (2005)CrossRefGoogle Scholar
  26. 26.
    Plaisted, D., Biere, A., Zhu, Y.: A satisfiability tester for quantified boolean formulae. J. Discrete Appl. Math. 130, 291–328 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Eén, N., Sörensson, N.: An extendable SAT-solver. Theory and Applications of Satisfiability Testing 2919, 502–518 (2004)Google Scholar
  28. 28.
    Sheini, H., Sakallah, K.: From propositional satisfiability to satisfiability modulo theories. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 1–9. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  29. 29.
    Stump, A., Barrett, C., Dill, D.: CVC: a cooperating validity checker. In: 14th Int. Conf. on Computer-Aided Verification (CAV), pp. 87–105. Springer, Heidelberg (2002)Google Scholar
  30. 30.
    Dutertre, B., de Moura, L.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Calzone, L., Chabrier-Rivier, N., Fages, F., Soliman, S.: Machine learning biochemical networks from temporal logic properties. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 68–94. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  32. 32.
    Piazza, C., Antoniotti, M., Mysore, V., Policriti, A., Winkler, F., Mishra, B.: Algorithmic algebraic model checking I: Challenges from systems biology. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 5–19. Springer, Heidelberg (2005)Google Scholar
  33. 33.
    Henzinger, T., Preussig, J., Wong-Toi, H.: Some lessons from the HyTech experience. In: Proc of the 40th Annual Conference on Decision and Control (CDC), pp. 2887–2892. IEEE Press, NJ (2001)Google Scholar
  34. 34.
    Frehse, G.: Phaver: Algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005)Google Scholar
  35. 35.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symb. Comp. 11, 1–26 (1996)Google Scholar
  36. 36.
    Wang, D.: Elimination theory, methods, and practice. In: Mathematics, and Mathematics-Mech, pp. 91–137. Shandong Education Publishing House, Jinan (2001)Google Scholar
  37. 37.
    Wang, D., Xia, B.: Stability analysis of biological systems with real solution classification. In: ISSAC 2005, vol. 3414, pp. 354–361. ACM, New York (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Igor Zinovik
    • 1
  • Daniel Kroening
    • 1
  • Yury Chebiryak
    • 1
  1. 1.Computer Systems Institute, ETH Zurich, 8092 ZurichSwitzerland

Personalised recommendations