Model Checking of Biological Systems

  • Luboš Brim
  • Milan Češka
  • David Šafránek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7938)

Abstract

Model checking together with other formal methods and techniques is being adapted for applications to biological systems. We present a selection of approaches used for modeling biological systems and formalizing their interesting properties in temporal logics. We also give a brief account of high performance model checking techniques and add a few case studies that demonstrate the use of model checking in computational systems biology. The primary aim is to give a reference for further reading.

References

  1. 1.
    Allmaier, S., Dalibor, S., Kreische, D.: Parallel Graph Generation Algorithms for Shared and Distributed Memory Machines. In: Parallel Computing Conference (PARCO). LNCS, vol. 1253, pp. 207–218. Springer (1997)Google Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Dill, D.: Model-checking in dense real-time. Information and Computation 104, 2–34 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alur, R., Feder, T., Henzinger, T.A.: The benefits of relaxing punctuality. J. ACM 43(1), 116–146 (1996)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alur, R., Henzinger, T.A.: A really temporal logic. J. ACM 41(1), 181–203 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Antoniotti, M., Policriti, A., Ugel, N., Mishra, B.: Model building and model checking for biochemical processes. Cell Biochemistry and Biophysics 38, 271–286 (2003)CrossRefGoogle Scholar
  6. 6.
    Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  7. 7.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model Checking Continuous-Time Markov Chains by Transient Analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Ballarini, P., Forlin, M., Mazza, T., Prandi, D.: Efficient parallel statistical model checking of biochemical networks. In: Parallel and Distributed Methods in verifiCation (PDMC). EPTCS, vol. 14, pp. 47–61 (2009)Google Scholar
  9. 9.
    Ballarini, P., Guerriero, M.L.: Query-based verification of qualitative trends and oscillations in biochemical systems. Theor. Comput. Sci. 411(20), 2019–2036 (2010)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ballarini, P., Guido, R., Mazza, T., Prandi, D.: Taming the complexity of biological pathways through parallel computing. Briefings in Bioinformatics 10(3), 278–288 (2009)CrossRefGoogle Scholar
  11. 11.
    Barbuti, R., Caravagna, G., Maggiolo-Schettini, A., Milazzo, P., Tini, S.: Foundational aspects of multiscale modeling of biological systems with process algebras. Theor. Comput. Sci. 431, 96–116 (2012)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Barnat, J., Bauch, P., Brim, L., Češka, M.: Computing Strongly Connected Components in Parallel on CUDA. In: International Parallel & Distributed Processing Symposium (IPDPS), pp. 541–552. IEEE Computer Society (2011)Google Scholar
  13. 13.
    Barnat, J., Brim, L., Krejci, A., Streck, A., Safranek, D., Vejnar, M., Vejpustek, T.: On Parameter Synthesis by Parallel Model Checking. IEEE/ACM Transactions on Computational Biology and Bioinformatics 9(3), 693–705 (2012)CrossRefGoogle Scholar
  14. 14.
    Barnat, J., Brim, L., Ročkai, P.: Scalable Multi-core LTL Model-Checking. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 187–203. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Barnat, J., Brim, L., Ročkai, P.: A Time-Optimal On-the-Fly Parallel Algorithm for Model Checking of Weak LTL Properties. In: Breitman, K., Cavalcanti, A. (eds.) ICFEM 2009. LNCS, vol. 5885, pp. 407–425. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Barnat, J., Brim, L., Ročkai, P.: Parallel Partial Order Reduction with Topological Sort Proviso. In: Software Engineering and Formal Methods (SEFM), pp. 222–231. IEEE Computer Society (2010)Google Scholar
  17. 17.
    Barnat, J., Brim, L., Stříbrná, J.: Distributed LTL Model-Checking in SPIN. In: Dwyer, M.B. (ed.) SPIN 2001. LNCS, vol. 2057, pp. 200–216. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Barnat, J., Brim, L., Černá, I., Dražan, S., Fabriková, J., Láník, J., Šafránek, D., Ma, H.: BioDiVinE: A Framework for Parallel Analysis of Biological Models. In: Computational Models for Cell Processes (COMPMOD). EPTCS, vol. 6, pp. 31–45 (2009)Google Scholar
  19. 19.
    Barnat, J., Brim, L., Černá, I., Moravec, P., Ročkai, P., Šimeček, P.: DiVinE – A Tool for Distributed Verification. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 278–281. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Barnat, J., Brim, L., Šafránek, D.: High-Performance Analysis of Biological Systems Dynamics with the DiVinE Model Checker. Briefings in Bioinformatics 11(3), 301–312 (2010)CrossRefGoogle Scholar
  21. 21.
    Barnat, J., Brim, L., Šafránek, D., Vejnár, M.: Parameter Scanning by Parallel Model Checking with Applications in Systems Biology. In: Parallel and Distributed Methods in Verification and High Performance Computational Systems Biology (HiBi/PDMC 2010), pp. 95–104. IEEE Computer Society (2010)Google Scholar
  22. 22.
    Barnat, J., Ročkai, P.: Shared Hash Tables in Parallel Model Checking. In: Parallel and Distributed Methods in verifiCation (PDMC). ENTCS, vol. 198, pp. 79–91 (2008)Google Scholar
  23. 23.
    Barnat, J., Bauch, P., Brim, L., Češka, M.: Designing fast LTL model checking algorithms for many-core GPUs. Journal of Parallel and Distributed Computing 72(9), 1083–1097 (2012)CrossRefGoogle Scholar
  24. 24.
    Bartocci, E., Corradini, F., Merelli, E., Tesei, L.: Detecting synchronisation of biological oscillators by model checking. Theoretical Computer Science 411(20), 1999–2018 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Batt, G., Page, M., Cantone, I., Goessler, G., Monteiro, P., de Jong, H.: Efficient parameter search for qualitative models of regulatory networks using symbolic model checking. Bioinformatics 26(18), 603–610 (2010)CrossRefGoogle Scholar
  27. 27.
    Batt, G., Ben Salah, R., Maler, O.: On timed models of gene networks. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 38–52. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  28. 28.
    Behrmann, G., Hune, T., Vaandrager, F.: Distributed Timed Model Checking — How the Search Order Matters. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 216–231. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  29. 29.
    Behrmann, G., David, A., Larsen, K.G.: A tutorial on uppaal. In: Bernardo, M., Corradini, F. (eds.) SFM-RT 2004. LNCS, vol. 3185, pp. 200–236. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  30. 30.
    Belta, C., Habets, L.: Controlling a class of nonlinear systems on rectangles. IEEE Transactions on Automatic Control 51(11), 1749–1759 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Bernot, G., Comet, J.P., Richard, A., Guespin, J.: Application of formal methods to biological regulatory networks: extending thomas’ asynchronous logical approach with temporal logic. Journal of Theoretical Biology 229(3), 339–347 (2004)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Bonzanni, N., Krepska, E., Feenstra, K.A., Fokkink, W., Kielmann, T., Bal, H.E., Heringa, J.: Executing multicellular differentiation: quantitative predictive modelling of C.elegans vulval development. Bioinformatics 25(16), 2049–2056 (2009)CrossRefGoogle Scholar
  33. 33.
    Bortolussi, L., Hillston, J.: Fluid model checking. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 333–347. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  34. 34.
    Bortolussi, L., Policriti, A.: Hybrid systems and biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 424–448. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  35. 35.
    Bošnački, D., Edelkamp, S., Sulewski, D.: Efficient Probabilistic Model Checking on General Purpose Graphics Processors. In: Păsăreanu, C.S. (ed.) SPIN 2009. LNCS, vol. 5578, pp. 32–49. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  36. 36.
    Bošnački, D., ten Eikelder, H.M.M., Steijaert, M.N., de Vink, E.P.: Stochastic analysis of amino acid substitution in protein synthesis. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 367–386. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  37. 37.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM (1987)Google Scholar
  38. 38.
    Brim, L., Černá, I., Moravec, P., Šimša, J.: Accepting predecessors are better than back edges in distributed LTL model-checking. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 352–366. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  39. 39.
    Brim, L., Barnat, J.: Platform Dependent Verification: On Engineering Verification Tools for 21st Century. In: Parallel and Distributed Methods in verifiCation (PDMC). EPTCS, vol. 72, pp. 1–12 (2011)Google Scholar
  40. 40.
    Brim, L., Česka, M., Dražan, S., Šafránek, D.: Exploring parameter space of stochastic biochemical systems using quantitative model checking. Tech. rep., Faculty of Informatics, Masaryk University (2013), http://sybila.fi.muni.cz/TR-01-2013.pdf
  41. 41.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Information and Computation 98(2), 142–170 (1992)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  43. 43.
    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
  44. 44.
    Campagna, D., Piazza, C.: Hybrid automata in systems biology: How far can we go? In: From Biology to Concurrency and Back (FBTC). ENTCS, vol. 229, pp. 93–108 (2009)Google Scholar
  45. 45.
    Caravagna, G., Hillston, J.: Modeling biological systems with delays in Bio-PEPA. In: Proceedings Fourth Workshop on Membrane Computing and Biologically Inspired Process Calculi 2010. EPTCS, vol. 40, pp. 85–101 (2010)Google Scholar
  46. 46.
    Carrillo, M., Góngora, P.A., Rosenblueth, D.A.: An overview of existing modeling tools making use of model checking in the analysis of biochemical networks. Front Plant Sci. 3(155), 1–13 (2012)Google Scholar
  47. 47.
    Caselli, S., Conte, G., Marenzoni, P.: Parallel state space exploration for GSPN models. In: DeMichelis, G., Díaz, M. (eds.) ICATPN 1995. LNCS, vol. 935, pp. 181–200. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  48. 48.
    Černá, I., Pelánek, R.: Distributed explicit fair cycle detection (Set based approach). In: Ball, T., Rajamani, S.K. (eds.) SPIN 2003. LNCS, vol. 2648, pp. 49–73. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  49. 49.
    Chaouiya, C.: Petri net modelling of biological networks. Briefings in Bioinformatics 8(4), 210–219 (2007)CrossRefGoogle Scholar
  50. 50.
    Chaouiya, C., Remy, E., Mossé, B., Thieffry, D.: Qualitative analysis of regulatory graphs: A computational tool based on a discrete formal framework. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems. LNCIS, vol. 294, pp. 830–832. Springer, Heidelberg (2003)Google Scholar
  51. 51.
    Che, S., Li, J., Sheaffer, J., Skadron, K., Lach, J.: Accelerating Compute-Intensive Applications with GPUs and FPGAs. In: IEEE Symposium on Application Specific Processors (SASP), pp. 101–107. IEEE Computer Society (2008)Google Scholar
  52. 52.
    Ciardo, G., Gluckman, J., Nicol, D.: Distributed state-space generation of discrete-state stochastic models. INFORMS J. Comp. 10(1), 82–93 (1998)CrossRefGoogle Scholar
  53. 53.
    Ciardo, G.: Automated parallelization of discrete state-space generation. J. Parallel Distrib. Comput. 47, 153–167 (1997)CrossRefGoogle Scholar
  54. 54.
    Cimatti, A., Clarke, E., Giunchiglia, F., Roveri, M.: NuSMV: a new symbolic model checker. J. Softw. Tools Technol. Transf. 2, 410–425 (2000)MATHCrossRefGoogle Scholar
  55. 55.
    Cimatti, A., Clarke, E., Giunchiglia, E., Giunchiglia, F., Pistore, M., Roveri, M., Sebastiani, R., Tacchella, A.: NuSMV 2: An OpenSource Tool for Symbolic Model Checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 359–364. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  56. 56.
    Ciocchetta, F., Hillston, J.: Bio-PEPA: A framework for the modelling and analysis of biological systems. Theor. Comput. Sci. 410(33-34), 3065–3084 (2009)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (1986)MATHCrossRefGoogle Scholar
  58. 58.
    Clarke, E.M., Enders, R., Filkorn, T., Jha, S.: Exploiting symmetry in temporal logic model checking. Form. Methods Syst. Des. 9(1-2), 77–104 (1996)CrossRefGoogle Scholar
  59. 59.
    Clarke Jr., E.M., Grumberg, O., Peled, D.A.: Model checking. MIT Press (1999)Google Scholar
  60. 60.
    Clarke, E.M., Zuliani, P.: Statistical Model Checking for Cyber-Physical Systems. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 1–12. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  61. 61.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Progress on the State Explosion Problem in Model Checking. In: Wilhelm, R. (ed.) Informatics: 10 Years Back, 10 Years Ahead. LNCS, vol. 2000, pp. 176–194. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  62. 62.
    Collins, P., Habets, L.C., van Schuppen, J.H., Černá, I., Fabriková, J., Šafránek, D.: Abstraction of biochemical reaction systems on polytopes. In: Proceedings of the 18th IFAC World Congress, vol. 18, pp. 14869–14875 (2011)Google Scholar
  63. 63.
    Courcoubetis, C., Vardi, M., Wolper, P., Yannakakis, M.: Memory-Efficient Algorithms for the Verification of Temporal Properties. Formal Methods in System Design 1, 275–288 (1992)MATHCrossRefGoogle Scholar
  64. 64.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Crudu, A., Debussche, A., Radulescu, O.: Hybrid stochastic simplifications for multiscale gene networks. BMC Systems Biology 3(1), 89 (2009)CrossRefGoogle Scholar
  66. 66.
    NVIDIA CUDA Compute Unified Device Architecture - Programming Guide Version 2.0, (2009), http://www.nvidia.com/object/cuda_develop.html
  67. 67.
    Dang, T., Guernic, C.L., Maler, O.: Computing reachable states for nonlinear biological models. Theor. Comput. Sci. 412(21), 2095–2107 (2011)MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Danos, V., Laneve, C.: Formal molecular biology. Theor. Comput. Sci. 325(1), 69–110 (2004)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Darling, R., Norris, J.: Differential equation approximations for markov chains. Probab. Surveys 5, 37–79 (2008)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    David, A., Du, D., Larsen, K.G., Legay, A., Mikucionis, M., Poulsen, D.B., Sedwards, S.: Statistical model checking for stochastic hybrid systems. In: Hybrid Systems and Biology (HSB). EPTCS, vol. 92, pp. 122–136 (2012)Google Scholar
  71. 71.
    Derman, C.: Finite State Markovian Decision Processes. Academic Press, Inc., Orlando (1970)MATHGoogle Scholar
  72. 72.
    Didier, F., Henzinger, T.A., Mateescu, M., Wolf, V.: Fast Adaptive Uniformization for the Chemical Master Equation. In: Parallel and Distributed Methods in Verification and High Performance Computational Systems Biology (HiBi/PDMC 2009), pp. 118–127. IEEE Computer Society (2009)Google Scholar
  73. 73.
    Didier, F., Henzinger, T.A., Mateescu, M., Wolf, V.: Sabre: A tool for stochastic analysis of biochemical reaction networks. CoRR abs/1005.2819 (2010)Google Scholar
  74. 74.
    Dluhoš, P., Brim, L., Šafránek, D.: On expressing and monitoring oscillatory dynamics. In: Hybrid Systems and Biology (HSB). EPTCS, vol. 92, pp. 73–87 (2012)Google Scholar
  75. 75.
    Doi, A., Fujita, S., Matsuno, H., Nagasaki, M., Miyano, S.: Constructing Biological Pathway Models with Hybrid Functional Petri Nets. In Silico Biology 4(3), 271–291 (2004)Google Scholar
  76. 76.
    Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 167–170. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  77. 77.
    Donzé, A., Clermont, G., Langmead, C.J.: Parameter synthesis in nonlinear dynamical systems: Application to systems biology. Journal of Computational Biology 17(3), 325–336 (2010)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Edelkamp, S., Sulewski, D.: Parallel State Space Search on the GPU (2009), symposium on Combinatorial Search (SoCS)Google Scholar
  79. 79.
    Eker, S., Knapp, M., Laderoute, K., Lincoln, P., Meseguer, J., Sonmez, K.: Pathway logic: Symbolic analysis of biological signaling. In: Pacific Symposium on Biocomputing, pp. 400–412 (2002)Google Scholar
  80. 80.
    El Samad, H., Khammash, M., Petzold, L., Gillespie, D.: Stochastic Modelling of Gene Regulatory Networks. Int. J. of Robust and Nonlinear Control 15(15), 691–711 (2005)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Emerson, E.A., Sistla, A.P.: Symmetry and model checking. Form. Methods Syst. Des. 9(1-2), 105–131 (1996)CrossRefGoogle Scholar
  82. 82.
    Engl, H.W., Flamm, C., Kügler, P., Lu, J., Müller, S., Schuster, P.: Inverse problems in systems biology. Inverse Problems 25(12), 123014 (2009)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Ezekiel, J., Lüttgen, G., Ciardo, G.: Parallelising symbolic state-space generators. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 268–280. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  84. 84.
    Fages, F., Soliman, S.: Formal cell biology in Biocham. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 54–80. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  85. 85.
    Fages, F., Soliman, S., Rivier, C.N.: Modelling and querying interaction networks in the biochemical abstract machine BIOCHAM. Journal of Biological Physics and Chemistry 4(2), 64–73 (2004)CrossRefGoogle Scholar
  86. 86.
    Fages, F., Rizk, A.: On temporal logic constraint solving for analyzing numerical data time series. Theor. Comput. Sci. 408(1), 55–65 (2008)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Fisher, J., Henzinger, T.A.: Executable cell biology. Nature Biotechnology 25(11), 1239–1249 (2007)CrossRefGoogle Scholar
  88. 88.
    Forejt, V., Kwiatkowska, M., Norman, G., Parker, D.: Automated verification techniques for probabilistic systems. In: Bernardo, M., Issarny, V. (eds.) SFM 2011. LNCS, vol. 6659, pp. 53–113. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  89. 89.
    Fromentin, J., Eveillard, D., Roux, O.: Hybrid modeling of biological networks: mixing temporal and qualitative biological properties. BMC Systems Biology 4(1), 79 (2010)CrossRefGoogle Scholar
  90. 90.
    Galpin, V., Hillston, J., Bortolussi, L.: HYPE Applied to the Modelling of Hybrid Biological Systems. ENTCS 218, 33–51 (2008)MATHGoogle Scholar
  91. 91.
    Garavel, H., Mateescu, R., Lang, F., Serwe, W.: CADP 2006: A toolbox for the construction and analysis of distributed processes. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 158–163. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  92. 92.
    Garavel, H., Mateescu, R., Smarandache, I.: Parallel State Space Construction for Model-Checking. In: Dwyer, M.B. (ed.) SPIN 2001. LNCS, vol. 2057, pp. 217–234. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  93. 93.
    Geldenhuys, J., de Villiers, P.J.A.: Runtime efficient state compaction in SPIN. In: Dams, D.R., Gerth, R., Leue, S., Massink, M. (eds.) SPIN 1999. LNCS, vol. 1680, pp. 12–21. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  94. 94.
    Gillespie, D.T.: Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry 81(25), 2340–2381 (1977)CrossRefGoogle Scholar
  95. 95.
    Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A: Statistical Mechanics and its Applications 188(1-3), 404–425 (1992)CrossRefGoogle Scholar
  96. 96.
    Gillespie, D.T.: Stochastic Simulation of Chemical Kinetics. Annual Review of Physical Chemistry 58(1), 35–55 (2007)MathSciNetCrossRefGoogle Scholar
  97. 97.
    Goethem, S.V., Jacquet, J.M., Brim, L., Šafránek, D.: Timed modelling of gene networks with arbitrary expression level discretization. In: Interactions between Computer Science and Biology. ENTCS. Elsevier (in press, 2013)Google Scholar
  98. 98.
    Grumberg, O., Heyman, T., Schuster, A.: A work-efficient distributed algorithm for reachability analysis. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 54–66. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  99. 99.
    Habets, L., van Schuppen, J.H.: A control problem for affine dynamical systems on a full-dimensional polytope. Automatica 40(1), 21–35 (2004)MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 512–535 (1994)MATHCrossRefGoogle Scholar
  101. 101.
    Haverkort, B.R., Bell, A., Bohnenkamp, H.C.: On the efficient sequential and distributed generation of very large Markov chains from stochastic Petri nets. In: Petri Net and Performance Models (PNPM), pp. 12–21. IEEE Computer Society Press (1999)Google Scholar
  102. 102.
    Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. Theoretical Computer Science 319(3), 239–257 (2008)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  104. 104.
    Heljanko, K., Khomenko, V., Koutny, M.: Parallelisation of the Petri Net Unfolding Algorithm. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 371–385. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  105. 105.
    Henzinger, T.: The theory of hybrid automata. In: Logic in Computer Science (LICS), pp. 278 –292. IEEE Computer Society (1996)Google Scholar
  106. 106.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pp. 373–382. ACM (1995)Google Scholar
  107. 107.
    Holzmann, G.J.: The Spin Model Checker: Primer and Reference Manual. Addison-Wesley (2003)Google Scholar
  108. 108.
    Holzmann, G.J.: A Stack-Slicing Algorithm for Multi-Core Model Checking. Electonic Notes in Theoretical Computer Science 198(1), 3–16 (2008)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Holzmann, G.J., Bosnacki, D.: The design of a multicore extension of the spin model checker. IEEE Trans. Software Eng. 33(10), 659–674 (2007)CrossRefGoogle Scholar
  110. 110.
    Horn, F., Jackson, R.: General mass action kinetics. Archive for Rational Mechanics and Analysis 47, 81–116 (1972), doi:10.1007/BF00251225MathSciNetCrossRefGoogle Scholar
  111. 111.
    Inggs, C.P., Barringer, H.: CTL* Model Checking on a Shared-Memory Architecture. Electronic Notes in Theoretical Computer Science 128(3), 107–123 (2005)MATHCrossRefGoogle Scholar
  112. 112.
    Iyengar, M.S.: Symbolic Systems Biology: Theory and Methods. Jones & Bartlett Publishers (2010)Google Scholar
  113. 113.
    Jayachandran, G., Vishal, V., Pande, V.S.: Using massively parallel simulations and Markovian models to study protein folding: examining the villin head-piece. Journal of Chemical Physics 124(6), 903–914 (2006)Google Scholar
  114. 114.
    Jha, S.K., Clarke, E.M., Langmead, C.J., Legay, A., Platzer, A., Zuliani, P.: A bayesian approach to model checking biological systems. In: Degano, P., Gorrieri, R. (eds.) CMSB 2009. LNCS, vol. 5688, pp. 218–234. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  115. 115.
    de Jong, H.: Modeling and Simulation of Genetic Regulatory Systems: A Literature Review. Journal of Computational Biology 9(1), 67–103 (2002)CrossRefGoogle Scholar
  116. 116.
    de Jong, H., Gouzé, J., Hernandez, C., Page, M., Sari, T., Geiselmann, J.: Qualitative simulations of genetic regulatory networks using piecewise linear models. Bull. Math. Biol. 66, 301–340 (2004)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962)MATHCrossRefGoogle Scholar
  118. 118.
    Keener, J.P., Sneyd, J.: Mathematical Physiology. Springer (1998)Google Scholar
  119. 119.
    Khademi, S., O’Connell III, J., Remis, J., Robles-Colmenares, Y., Miercke, L., Stroud, R.: Mechanism of ammonia transport by Amt/MEP/Rh: Structure of AmtB at 1.35. Science 305(5690), 1587–1594 (2004)CrossRefGoogle Scholar
  120. 120.
    Kholodenko, B.N.: Cell-signalling dynamics in time and space. Nature Molecular Cell Biology 7, 165–176 (2006)CrossRefGoogle Scholar
  121. 121.
    Klarner, H., Streck, A., Šafránek, D., Kolčák, J., Siebert, H.: Parameter identification and model ranking of thomas networks. In: Gilbert, D., Heiner, M. (eds.) CMSB 2012. LNCS, vol. 7605, pp. 207–226. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  122. 122.
    Knottenbelt, W., Mestern, M., Harrison, P., Kritzinger, P.: Probability, parallelism and the state space exploration problem. In: Puigjaner, R., Savino, N.N., Serra, B. (eds.) TOOLS 1998. LNCS, vol. 1469, pp. 165–179. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  123. 123.
    Koymans, R.: Specifying real-time properties with metric temporal logic. Real-Time Systems 2, 255–299 (1990)CrossRefGoogle Scholar
  124. 124.
    Kumar, R., Mercer, E.G.: Load Balancing Parallel Explicit State Model Checking. In: Parallel and Distributed Methods in Verification (PDMC). ENTCS, vol. 128, pp. 19–34. Elsevier (2005)Google Scholar
  125. 125.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: Verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  126. 126.
    Kwiatkowska, M., Norman, G., Parker, D.: Stochastic model checking. In: Bernardo, M., Hillston, J. (eds.) SFM 2007. LNCS, vol. 4486, pp. 220–270. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  127. 127.
    Kwiatkowska, M.Z., Norman, G., Parker, D.: Using probabilistic model checking in systems biology. SIGMETRICS Performance Evaluation Review 35(4), 14–21 (2008)CrossRefGoogle Scholar
  128. 128.
    Laarman, A., van de Pol, J., Weber, M.: Boosting Multi-Core Reachability Performance with Shared Hash Tables. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 247–255. IEEE Computer Science (2010)Google Scholar
  129. 129.
    Lerda, F., Sisto, R.: Distributed-memory Model Checking with SPIN. In: Dams, D.R., Gerth, R., Leue, S., Massink, M. (eds.) SPIN 1999. LNCS, vol. 1680, pp. 22–39. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  130. 130.
    Lerda, F., Visser, W.: Addressing Dynamic Issues of Program Model Checking. In: Dwyer, M.B. (ed.) SPIN 2001. LNCS, vol. 2057, pp. 80–102. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  131. 131.
    Ma, H., Boogerd, F., Goryanin, I.: Modelling nitrogen assimilation of Escherichia coli at low ammonium concentration. Journal of Biotechnology 144(3), 175–183 (2009)CrossRefGoogle Scholar
  132. 132.
    Madsen, C., Myers, C., Roehner, N., Winstead, C., Zhang, Z.: Utilizing Stochastic Model Checking to Analyze Genetic Circuits. In: IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB), pp. 379–386 (2012)Google Scholar
  133. 133.
    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)CrossRefGoogle Scholar
  134. 134.
    Maler, O., Nickovic, D.: Monitoring temporal properties of continuous signals. In: Lakhnech, Y., Yovine, S. (eds.) FORMATS 2004 and FTRTFT 2004. LNCS, vol. 3253, pp. 152–166. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  135. 135.
    Maler, O., Nickovic, D., Pnueli, A.: Checking temporal properties of discrete, timed and continuous behaviors. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 475–505. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  136. 136.
    Mateescu, R., Monteiro, P.T., Dumas, E., de Jong, H.: CTRL: Extension of CTL with regular expressions and fairness operators to verify genetic regulatory networks. Theoretical Computer Science 412(26), 2854–2883 (2011)MathSciNetMATHCrossRefGoogle Scholar
  137. 137.
    Melham, T., Bard, J., Werner, E., Noble, D.: Conceptual foundations of systems biology. Prog. Biophys. Mol. Biol. (2012)Google Scholar
  138. 138.
    Merrill, D., Garland, M., Grimshaw, A.: Scalable GPU Graph Traversal. In: ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPoPP), pp. 117–128. ACM (2012)Google Scholar
  139. 139.
    Pelánek, R.: Fighting State Space Explosion: Review and Evaluation. In: Cofer, D., Fantechi, A. (eds.) FMICS 2008. LNCS, vol. 5596, pp. 37–52. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  140. 140.
    Peled, D.: Ten years of partial order reduction. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 17–28. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  141. 141.
    Phillips, A., Cardelli, L.: Efficient, correct simulation of biological processes in the stochastic π-calculus. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 184–199. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  142. 142.
    Pnueli, A.: The temporal semantics of concurrent programs. Theoretical Computer Science 13(1), 45–60 (1981)MathSciNetMATHCrossRefGoogle Scholar
  143. 143.
    Popova-Zeugmann, L., Heiner, M., Koch, I.: Time Petri Nets for Modelling and Analysis of Biochemical Networks. Fundam. Inform. 67(1-3), 149–162 (2005)MathSciNetMATHGoogle Scholar
  144. 144.
    Priami, C.: Algorithmic systems biology. Commun. ACM 52(5), 80–88 (2009)CrossRefGoogle Scholar
  145. 145.
    Regev, A., Silverman, W., Shapiro, E.Y.: Representation and Simulation of Biochemical Processes Using the π-Calculus Process Algebra. In: Pacific Symposium on Biocomputing, pp. 459–470 (2001)Google Scholar
  146. 146.
    Reif, J.: Depth-first Search is Inherently Sequential. Information Proccesing Letters 20(5), 229–234 (1985)MathSciNetMATHCrossRefGoogle Scholar
  147. 147.
    Rizk, A., Batt, G., Fages, F., Soliman, S.: A general computational method for robustness analysis with applications to synthetic gene networks. Bioinformatics 25(12) (2009)Google Scholar
  148. 148.
    Satish, N., Harris, M., Garland, M.: Designing efficient sorting algorithms for manycore gpus. In: IEEE International Parallel & Distributed Processing Symposium (IPDPS), pp. 1–10. IEEE Computer Society (2009)Google Scholar
  149. 149.
    Schaub, M., Henzinger, T., Fisher, J.: Qualitative networks: a symbolic approach to analyze biological signaling networks. BMC Systems Biology 1(1), 4 (2007)CrossRefGoogle Scholar
  150. 150.
    Schivo, D.S., Scholma, J., Wanders, B., Urquidi Camacho, R., van der Vet, P., Karperien, H., Langerak, R., van de Pol, J., Post, J.: Modelling biological pathway dynamics with timed automata. In: IEEE International Conference on Bioinformatics and Bioengineering (ICBB), pp. 447–453. IEEE Computer Society (2012)Google Scholar
  151. 151.
    Schwarick, M., Heiner, M.: CSL model checking of biochemical networks with interval decision diagrams. In: Degano, P., Gorrieri, R. (eds.) CMSB 2009. LNCS, vol. 5688, pp. 296–312. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  152. 152.
    Schwarick, M., Rohr, C., Heiner, M.: MARCIE - Model checking and Reachability analysis done effiCIEntly. In: Quantitative Evaluation of SysTems (QEST 2011), pp. 91–100. IEEE Computer Society (2011)Google Scholar
  153. 153.
    Siebert, H., Bockmayr, A.: Incorporating time delays into the logical analysis of gene regulatory networks. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 169–183. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  154. 154.
    Singh, A., Hespanha, J.P.: Stochastic hybrid systems for studying biochemical processes. Physical and Engineering Sciences 368(1930), 4995–5011 (2010)MathSciNetMATHCrossRefGoogle Scholar
  155. 155.
    Stern, U., Dill, D.L.: Parallelizing the murϕ verifier. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 256–267. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  156. 156.
    Stern, U., Dill, D.L.: Using Magnetic Disk Instead of Main Memory in the Murϕ Verifier. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 172–183. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  157. 157.
    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
  158. 158.
    Tarjan, R.: Depth First Search and Linear Graph Algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)MathSciNetMATHCrossRefGoogle Scholar
  159. 159.
    Thomas, R.: Regulatory networks seen as asynchronous automata: A logical description. Journal of Theoretical Biology 153(1), 1–23 (1991)CrossRefGoogle Scholar
  160. 160.
    Vardi, M.Y., Wolper, P.: An Automata-Theoretic Approach to Automatic Program Verification. In: IEEE Symposium on Logic in Computer Science (LICS), pp. 332–344. IEEE Computer Society Press (1986)Google Scholar
  161. 161.
    Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press (1995)Google Scholar
  162. 162.
    Yang, E., van Nimwegen, E., Zavolan, M., Rajewsky, N., Schroeder, M., Magnasco, M., Darnell, J.E.: Decay Rates of Human mRNAs: Correlation With Functional Characteristics and Sequence Attributes. Genome Research 13(8), 1863–1872 (2003)Google Scholar
  163. 163.
    Yang, H.T., Ko, M.S.H.: Stochastic modeling for the expression of a gene regulated by competing transcription factors. PLoS ONE 7(3), e32376 (2012)Google Scholar
  164. 164.
    Yovine, S.: Kronos: a verification tool for real-time systems. International Journal on Software Tools for Technology Transfer 1, 123–133 (1997)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luboš Brim
    • 1
  • Milan Češka
    • 1
  • David Šafránek
    • 1
  1. 1.Systems Biology Laboratory at Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations