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Computing Reachable States for Nonlinear Biological Models

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Computational Methods in Systems Biology (CMSB 2009)

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Abstract

In this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models. We then develop a novel algorithm for computing reachable states for nonlinear systems and report experimental results obtained using a prototype implementation. We believe these results constitute a promising contribution to the analysis of complex models of biological systems.

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References

  1. Althoff, M., Stursberg, O., Buss, M.: Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization. In: CDC 2008 (2008)

    Google Scholar 

  2. Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The Algorithmic Analysis of Hybrid Systems. Theoretical Computer Science 138, 3–34 (1995)

    Article  Google Scholar 

  3. Alur, R., Dang, T., Ivancic, F.: Counterexample-guided Predicate Abstraction of Hybrid Systems. Theoretical Computer Science 354, 250–271 (2006)

    Article  Google Scholar 

  4. Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate Reachability Analysis of Piecewise Linear Dynamical Systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 21–31. Springer, Heidelberg (2000)

    Google Scholar 

  5. Asarin, E., Dang, T.: Abstraction by Projection and Application to Multi-affine Systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 32–47. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  6. Asarin, E., Dang, T., Girard, A.: Hybridization Methods for the Analysis of Nonlinear Systems. Acta Informatica 43, 451–476 (2007)

    Article  Google Scholar 

  7. Aubin, J.-P., Cellina, A.: Differential Inclusions. Springer, Heidelberg (1984)

    Book  Google Scholar 

  8. 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)

    Chapter  Google Scholar 

  9. Ben Salah, R., Bozga, M., Maler, O.: On Interleaving in Timed Automata. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 465–476. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  10. Botchkarev, O., Tripakis, S.: Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 73–88. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  11. Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 76–90. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  12. Chutinan, A., Krogh, B.H.: Computational Techniques for Hybrid System Verification. IEEE Trans. on Automatic Control 48, 64–75 (2003)

    Article  Google Scholar 

  13. Dang, T.: Verification and Synthesis of Hybrid Systems, PhD thesis, Institut National Polytechnique de Grenoble, Laboratoire Verimag (2000)

    Google Scholar 

  14. Dang, T.: Approximate Reachability Computation for Polynomial Systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 138–152. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. Dang, T., Maler, O.: Reachability Analysis via Face Lifting. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 96–109. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  16. Donze, A., Clermont, G., Legay, A., Langmead, C.J.: Parameter Synthesis in Nonlinear Dynamical Systems: Application to Systems Biology. In: RECOMB 2009, pp. 155–169 (2009)

    Google Scholar 

  17. 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)

    Chapter  Google Scholar 

  18. Han, Z., Krogh, B.H.: Reachability Analysis of Nonlinear Systems using Trajec- tory Piecewise Linearized Models. In: American Control Conference, pp. 1505–1510 (2006)

    Google Scholar 

  19. Halasz, A., Kumar, V., Imielinski, M., Belta, C., Sokolsky, O., Pathak, S.: Analysis of Lactose Metabolism in E.coli using Reachability Analysis of Hybrid Systems. IEE Proceedings - Systems Biology 21, 130–148 (2007)

    Article  Google Scholar 

  20. Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Algorithmic Analysis of Nonlinear Hybrid Systems. IEEE Trans. on Automatic Control 43, 540–554 (1998)

    Article  Google Scholar 

  21. de Jong, H., Page, M., Hernandez, C., Geiselmann, J.: Qualitative Simulation of Genetic Regulatory Networks: Method and Application. In: IJCAI 2001, pp. 67–73 (2001)

    Google Scholar 

  22. Gillespie, D.T.: Stochastic Simulation of Chemical Kinetics. Annual Review of Physical Chemistry 58, 35–55 (2007)

    Article  CAS  PubMed  Google Scholar 

  23. Girard, A.: Reachability of Uncertain Linear Systems using Zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  24. Girard, A., Le Guernic, C., Maler, O.: Efficient Computation of Reachable Sets of Linear Time-invariant Systems with Inputs. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 257–271. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  25. Greenstreet, M.R.: Verifying Safety Properties of Differential Equations. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 277–287. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  26. Greenstreet, M.R., Mitchell, I.: Reachability Analysis Using Polygonal Projections. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 103–116. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  27. Le Guernic, C.: Calcul efficace de l’ensemble atteignable des systémes linaires avec incertitudes, Master’s thesis, Université Paris 7 (2005)

    Google Scholar 

  28. Hirsch, M., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, London (1974)

    Google Scholar 

  29. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, Heidelberg (2001)

    Book  Google Scholar 

  30. de Jong, H., Page, M., Hernandez, C., Geiselmann, J.: Qualitative Simulation of Genetic Regulatory Networks: Method and Application. In: IJCAI 2001, pp. 67–73 (2001)

    Google Scholar 

  31. Klipp, E., Herwig, R., Kowald, A., Wierling, C., Lehrach, H.: Systems Biology in Practice: Concepts, Implementation and Application. Wiley, Chichester (2005)

    Book  Google Scholar 

  32. Kurzhanskiy, A., Varaiya, P.: Ellipsoidal Techniques for Reachability Analysis of Discrete-time Linear Systems. IEEE Trans. Automatic Control 52, 26–38 (2007)

    Article  Google Scholar 

  33. Kurzhanski, A., Varaiya, P.: Ellipsoidal tehcniques for reachability analysis. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 202. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  34. Maler, O.: A Unified Approach for Studying Discrete and Continuous Dynamical Systems. In: CDC 1998, pp. 2083–2088 (1998)

    Google Scholar 

  35. Maler, O.: Control from Computer Science. Ann. Rev. in Control 26, 175–187 (2002)

    Article  Google Scholar 

  36. Maler, O., Batt, G.: Approximating Continuous Systems by Timed Automata. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 77–89. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  37. Mitchell, I., Tomlin, C.J.: Level Set Methods for Computation in Hybrid Systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 310–323. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  38. Sastry, S.: Nonlinear systems. Analysis, Stability and Control. Springer, Heidelberg (1999)

    Google Scholar 

  39. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)

    Google Scholar 

  40. Thomas, R., D’Ari, R.: Biological Feedback. CRC Press, Boca Raton (1990)

    Google Scholar 

  41. Varaiya, P.: Reach Set computation using Optimal Control. In: KIT Workshop, pp. 377–383. Verimag, Grenoble (1998)

    Google Scholar 

  42. Zeigler, G.M.: Lectures on Polytpoes. Springer, Heidelberg (1995)

    Book  Google Scholar 

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Dang, T., Le Guernic, C., Maler, O. (2009). Computing Reachable States for Nonlinear Biological Models. In: Degano, P., Gorrieri, R. (eds) Computational Methods in Systems Biology. CMSB 2009. Lecture Notes in Computer Science(), vol 5688. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03845-7_9

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  • DOI: https://doi.org/10.1007/978-3-642-03845-7_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03844-0

  • Online ISBN: 978-3-642-03845-7

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