Bulletin of Mathematical Biology

, Volume 66, Issue 2, pp 301–340 | Cite as

Qualitative simulation of genetic regulatory networks using piecewise-linear models

  • Hidde de Jong
  • Jean-Luc Gouzé
  • Céline Hernandez
  • Michel Page
  • Tewfik Sari
  • Johannes Geiselmann
Article

Abstract

In order to cope with the large amounts of data that have become available in genomics, mathematical tools for the analysis of networks of interactions between genes, proteins, and other molecules are indispensable. We present a method for the qualitative simulation of genetic regulatory networks, based on a class of piecewise-linear (PL) differential equations that has been well-studied in mathematical biology. The simulation method is well-adapted to state-of-the-art measurement techniques in genomics, which often provide qualitative and coarse-grained descriptions of genetic regulatory networks. Given a qualitative model of a genetic regulatory network, consisting of a system of PL differential equations and inequality constraints on the parameter values, the method produces a graph of qualitative states and transitions between qualitative states, summarizing the qualitative dynamics of the system. The qualitative simulation method has been implemented in Java in the computer tool Genetic Network Analyzer.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alur, R., C. Belta, F. Ivančíc V. Kumar, M. Mintz, G. J. Pappas, H. Rubin and J. Schlug (2001). Hybrid modeling and simulation of biomolecular networks, in Hybrid Systems: Computation and Control (HSCC 2001), Lecture Notes in Computer Science 2034, M. D. Di Benedetto and A. Sangiovanni-Vincentelli (Eds), Berlin: Springer, pp. 19–32.Google Scholar
  2. Alves, R. and M. A. Savageau (2000a). Comparing systemic properties of ensembles of biological networks by graphical and statistical methods. Bioinformatics 16, 527–533.CrossRefGoogle Scholar
  3. Alves, R. and M. A. Savageau (2000b). Systemic properties of ensembles of metabolic networks: application of graphical and statistical methods to simple unbranched pathways. Bioinformatics 16, 534–547.CrossRefGoogle Scholar
  4. Barkai, N. and S. Leibler (1997). Robustness in simple biochemical networks. Nature 387, 913–917.CrossRefGoogle Scholar
  5. Becskei, A. and L. Serrano (2000). Engineering stability in gene networks by autoregulation. Nature 405, 590–591.CrossRefGoogle Scholar
  6. Ben-Hur, A. and H. T. Siegelmann (2001). Computation in gene networks, in Machines, Computations, and Universality: Third International Conference (MCU 2001), Lecture Notes in Computer Science 2055, M. Margenstern and Y. Rogozhin (Eds), Berlin: Springer, pp. 11–24.Google Scholar
  7. Cherry, J. L. and F. R. Adler (2000). How to make a biological switch. J. Theor. Biol. 203, 117–133.CrossRefGoogle Scholar
  8. de Jong, H. (2002). Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9, 69–105.Google Scholar
  9. de Jong, H., M. Page, C. Hernandez and J. Geiselmann (2001). Qualitative simulation of genetic regulatory networks: method and application, in Proc. Seventeenth Int. Joint Conf. Artif. Intell., IJCAI-01, B. Nebel (Ed.), San Mateo, CA: Morgan Kaufmann, pp. 67–73.Google Scholar
  10. de Jong, H., J.-L. Gouzé, C. Hernandez, M. Page, T. Sari and H. Geiselmann (2002). Qualitative simulation of genetic regulatory networks using piecewise-linear models. Technical Report RR-4407, INRIA.Google Scholar
  11. de Jong, H., J. Geiselmann, G. Batt, C. Hernandez and M. Page (2003a). Qualitative simulation of the initiation of sporulation in Bacillus subtilis. Bull. Math. Biol. (this volume).Google Scholar
  12. de Jong, H., J. Geiselmann, C. Hernandez and M. Page (2003b). Genetic Network Analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19, 336–344.CrossRefGoogle Scholar
  13. Edwards, R. (2000). Analysis of continuous-time switching networks. Physica D 146, 165–199.MATHMathSciNetCrossRefGoogle Scholar
  14. Edwards, R. and L. Glass (2000). Combinatorial explosion in model gene networks. Chaos 10, 691–704.MathSciNetCrossRefMATHGoogle Scholar
  15. Edwards, C. and S. K. Spurgeon (1998). Sliding Mode Control: Theory and Applications, London: Taylor & Francis.Google Scholar
  16. Edwards, R., H. T. Siegelmann, K. Aziza and L. Glass (2001). Symbolic dynamics and computation in model gene networks. Chaos 11, 160–169.CrossRefGoogle Scholar
  17. Elowitz, M. B. and S. Leibler (2000). A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338.CrossRefGoogle Scholar
  18. Endy, D. and R. Brent (2001). Modelling cellular behavior. Nature 409, 391–395.CrossRefGoogle Scholar
  19. Filippov, A. F. (1988). Differential Equations with Discontinuous Righthand Sides, Dordrecht: Kluwer Academic Publishers.Google Scholar
  20. Forbus, K. D. (1984). Qualitative process theory. Artif. Intell. 24, 85–168.CrossRefGoogle Scholar
  21. Gardner, T. S., C. R. Cantor and J. J. Collins (2000). Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339–342.CrossRefGoogle Scholar
  22. Ghosh, R., A. Tiwari and C. L. Tomlin (2003). Automated symbolic reachability analysis with application to Delta-Notch signaling automata, in Hybrid Systems: Computation and Control (HSCC 2003), Lecture Notes in Computer Science 2623, A. Pnueli and O. Maler (Eds), Berlin: Springer, pp. 233–248.Google Scholar
  23. Ghosh, R. and C. J. Tomlin (2001). Lateral inhibition through Delta-Notch signaling: a piecewise affine hybrid model, in Hybrid Systems: Computation and Control (HSCC 2001), Lecture Notes in Computer Science 2034, M. D. Di Benedetto and A. Sangiovanni-Vincentelli (Eds), Berlin: Springer, pp. 232–246.Google Scholar
  24. Glass, L. (1975a). Classification of biological networks by their qualitative dynamics. J. Theor. Biol. 54, 85–107.Google Scholar
  25. Glass, L. (1975b). Combinatorial and topological methods in nonlinear chemical kinetics. J. Chem. Phys. 63, 1325–1335.CrossRefGoogle Scholar
  26. Glass, L. (1977). Global analysis of nonlinear chemical kinetics, in Statistical Mechanics, Part B: Time Dependent Processes, B. Berne (Ed.), New York: Plenum Press, pp. 311–349.Google Scholar
  27. Glass, L. and S. A. Kauffman (1973). The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103–129.CrossRefGoogle Scholar
  28. Glass, L. and J. S. Pasternack (1978a). Prediction of limit cycles in mathematical models of biological oscillations. Bull. Math. Biol. 40, 27–44.MathSciNetCrossRefGoogle Scholar
  29. Glass, L. and J. S. Pasternack (1978b). Stable oscillations in mathematical models of biological control systems. J. Math. Biol. 6, 207–223.MathSciNetMATHGoogle Scholar
  30. Goodwin, B. C. (1963). Temporal Organization in Cells, New York: Academic Press.Google Scholar
  31. Gouzé, J.-L. and T. Sari (2003). A class of piecewise linear differential equations arising in biological models. Dyn. Syst. 17, 299–316.CrossRefGoogle Scholar
  32. Heidtke, K. R. and S. Schulze-Kremer (1998). Design and implementation of a qualitative simulation model of λ phage infection. Bioinformatics 14, 81–91.CrossRefGoogle Scholar
  33. Karp, P. D. (1993). Design methods for scientific hypothesis formation and their application to molecular biology. Mach. Learn. 12, 89–116.Google Scholar
  34. Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution, New York: Oxford University Press.Google Scholar
  35. Keller, A. D. (1994). Specifying epigenetic states with autoregulatory transcription factors. J. Theor. Biol. 170, 175–181.CrossRefGoogle Scholar
  36. Keller, A. D. (1995). Model genetic circuits encoding autoregulatory transcription factors. J. Theor. Biol. 172, 169–185.CrossRefGoogle Scholar
  37. Kelley, J. L. (1969). General Topology, New York: Van Nostrand.Google Scholar
  38. Kohn, K. W. (2001). Molecular interaction maps as information organizers and simulation guides. Chaos 11, 1–14.MATHMathSciNetCrossRefGoogle Scholar
  39. Kuipers, B. (1994). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge, Boston, MA: MIT Press.Google Scholar
  40. Kuipers, B. J. (1989). Qualitative reasoning: modeling and simulation with incomplete knowledge. Automatica 25, 571–585.CrossRefGoogle Scholar
  41. Lewis, J. E. and L. Glass (1991). Steady states, limit cycles, and chaos in models of complex biological networks. Int. J. Bifurcat. Chaos 1, 477–483.MathSciNetCrossRefMATHGoogle Scholar
  42. Lewis, J. E. and L. Glass (1992). Nonlinear dynamics and symbolic dynamics of neural networks. Neural Comput. 4, 621–642.Google Scholar
  43. Lockhart, D. J. and E. A. Winzeler (2000). Genomics, gene expression and DNA arrays. Nature 405, 827–836.CrossRefGoogle Scholar
  44. McAdams, H. H. and A. Arkin (1998). Simulation of prokaryotic genetic circuits. Ann. Rev. Biophys. Biomol. Struct. 27, 199–224.CrossRefGoogle Scholar
  45. McAdams, H. H. and L. Shapiro (1995). Circuit simulation of genetic networks. Science 269, 650–656.Google Scholar
  46. Mendoza, L., D. Thieffry and E. R. Alvarez-Buylla (1999). Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis. Bioinformatics 15, 593–606.CrossRefGoogle Scholar
  47. Mestl, T., E. Plahte and S. W. Omholt (1995a). A mathematical framework for describing and analysing gene regulatory networks. J. Theor. Biol. 176, 291–300.CrossRefGoogle Scholar
  48. Mestl, T., E. Plahte and S. W. Omholt (1995b). Periodic solutions in systems of piecewise-linear differential equations. Dyn. Stabil. Syst. 10, 179–193.MathSciNetMATHGoogle Scholar
  49. Morohashi, M., A. E. Winn, M. T. Borisuk, H. Bolouri, J. Doyle and H. Kitano (2002). Robustness as a measure of plausibility in models of biochemical networks. J. Theor. Biol. 216, 19–30.MathSciNetCrossRefGoogle Scholar
  50. Omholt, S. W., X. Kefang, Ø. Andersen and E. Plahte (1998). Description and analysis of switchlike regulatory networks exemplified by a model of cellular iron homeostasis. J. Theor. Biol. 195, 339–350.CrossRefGoogle Scholar
  51. Pandey, A. and M. Mann (2000). Proteomics to study genes and genomes. Nature 405, 837–846.CrossRefGoogle Scholar
  52. Plahte, E., T. Mestl and S. W. Omholt (1994). Global analysis of steady points for systems of differential equations with sigmoid interactions. Dyn. Stabil. Syst. 9, 275–291.MathSciNetMATHGoogle Scholar
  53. Plahte, E., T. Mestl and S. W. Omholt (1995). Stationary states in food web models with threshold relationships. J. Biol. Syst. 3, 569–577.CrossRefGoogle Scholar
  54. Plahte, E., T. Mestl and S. W. Omholt (1998). A methodological basis for description and analysis of systems with complex switch-like interactions. J. Math. Biol. 36, 321–348.MathSciNetCrossRefMATHGoogle Scholar
  55. Prokudina, E. I., R. Y. Valeev and R. N. Tchuraev (1991). A new method for the analysis of the dynamics of the molecular genetic control systems. II. Application of the method of generalized threshold models in the investigation of concrete genetic systems. J. Theor. Biol. 151, 89–110.Google Scholar
  56. Ptashne, M. (1992). A Genetic Switch: Phage λ and Higher Organisms, 2nd edn, Cambridge, MA: Cell Press & Blackwell Science.Google Scholar
  57. Ren, B. et al. (2000). Genome-wide location and function of DNA binding proteins. Science 290, 2306–2309.CrossRefGoogle Scholar
  58. Sánchez, L. and D. Thieffry (2001). A logical analysis of the Drosophila gap genes. J. Theor. Biol. 211, 115–141.CrossRefGoogle Scholar
  59. Santillán, M. and M. C. Mackey (2001). Dynamic regulation of the tryptophan operon: a modeling study and comparison with experimental data. Proc. Natl Acad. Sci. USA 98, 1364–1369.CrossRefGoogle Scholar
  60. Snoussi, E. H. (1989). Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn. Stabil. Syst. 4, 189–207.MATHMathSciNetGoogle Scholar
  61. Snoussi, E. H. and R. Thomas (1993). Logical identification of all steady states: the concept of feedback loop characteristic states. Bull. Math. Biol. 55, 973–991.CrossRefMATHGoogle Scholar
  62. Somogyi, R. and C. A. Sniegoski (1996). Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation. Complexity 1, 45–63.MathSciNetGoogle Scholar
  63. Tchuraev, R. N. and V. A. Ratner (1983). A continuous approach with threshold characteristics for simulation of gene expression, in Molecular Genetic Information Systems: Modelling and Simulation, K. Bellmann (Ed.), Berlin: Akademie-Verlag, pp. 64–80.Google Scholar
  64. Thieffry, D., A. M. Huerta, E. Pérez-Rueda and J. Collado-Vides (1998). From specific gene regulation to genomic networks: a global analysis of transcriptional regulation in Escherichia coli. BioEssays 20, 433–440.CrossRefGoogle Scholar
  65. Thieffry, D. and R. Thomas (1995). Dynamical behaviour of biological networks: II. Immunity control in bacteriophage lambda. Bull. Math. Biol. 57, 277–297.CrossRefMATHGoogle Scholar
  66. Thomas, R. and R. d’Ari (1990). Biological Feedback, Boca Raton, FL: CRC Press.MATHGoogle Scholar
  67. Thomas, R., D. Thieffry and M. Kaufman (1995). Dynamical behaviour of biological regulatory networks: I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol. 57, 247–276.CrossRefMATHGoogle Scholar
  68. Trelease, R. B., R. A. Henderson and J. B. Park (1999). A qualitative process system for modelingNF-κB and AP-1 gene regulation in immune cell biology research. Artif. Intell. Med. 17, 303–321.CrossRefGoogle Scholar
  69. von Dassow, G., E. Meir, E. M. Munro and G. M. Odell (2000). The segment polarity network is a robust developmental module. Nature 406, 188–192.CrossRefGoogle Scholar
  70. Wolf, D. M. and F. H. Eeckman (1998). On the relationship between genomic regulatory element organization and gene regulatory dynamics. J. Theor. Biol. 195, 167–186.CrossRefGoogle Scholar
  71. Yagil, G. and E. Yagil (1971). On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J. 11, 11–27.CrossRefGoogle Scholar
  72. Zhu, H. and M. Snyder (2001). Protein arrays and microarrays. Curr. Opin. Chem. Biol. 5, 40–45.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2004

Authors and Affiliations

  • Hidde de Jong
    • 1
  • Jean-Luc Gouzé
    • 2
  • Céline Hernandez
    • 3
  • Michel Page
    • 1
    • 4
  • Tewfik Sari
    • 5
  • Johannes Geiselmann
    • 6
  1. 1.Institut National de Recherche en Informatique et en Automatique (INRIA)Unité de recherche Rhône-AlpesSaint Ismier CedexFrance
  2. 2.Institut National de Recherche en Informatique et en Automatique (INRIA)Unité de recherche Sophia AntipolisSophia AntipolisFrance
  3. 3.Swiss Institute of Bioinformatics (SIB)Geneva 4Switzerland
  4. 4.École Supérieure des AffairesUniversité Pierre Mendès FranceGrenobleFrance
  5. 5.Laboratoire de MathématiquesUniversité de Haute AlsaceMulhouseFrance
  6. 6.Laboratoire Adaptation et Pathogénie des Microorganismes (CNRS UMR 5163)Université Joseph FourierGrenoble Cedex 9France

Personalised recommendations