Skip to main content
SpringerLink
Log in

Identification of all steady states in large networks by logical analysis

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bacchus, F. and P. van Beek (1998). On the conversion between non-binary and binary constraint satisfaction problems, in Proceedings of the Fifteenth National Conference on Artificial Intelligence and Tenth Innovative Applications of Artificial Intelligence Conference, AAAI 98, IAAI 98, Madison, Wisconsin: AAAI Press/The MIT Press, pp. 310–318.

    Google Scholar 

  • Barták, R. (2001). Theory and practice of constraint propagation, in Proceedings of 3rd Workshop on Constraint Programming for Decision and Control (CPDC2001), Gliwice, Poland: Wydavnictvo Pracovni Komputerowej, pp. 7–14.

    Google Scholar 

  • Bhalla, U. S. and R. Iyengar (1999). Emergent properties of networks of biological signalling pathways. Science 283, 381–387.

    Article  Google Scholar 

  • Cornish-Bowden, A. (1995). Fundamentals of Enzyme Kinetics, revised edition, London: Portland Press.

    Google Scholar 

  • Covert, M. W., C. H. Schilling and B. Palsson (2001). Regulation of gene expression in flux balance models of metabolism. J. Theor. Biol. 213, 73–88.

    Article  Google Scholar 

  • de Jong, H. (2002). Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9, 69–105.

    Google Scholar 

  • de Jong, H., J.-L. Gouzé, C. Hernandez, M. Page, T. Sari and J. Geiselmann (2002). Qualitative simulation of genetic regulatory networks using piecewise-linear models. Report of Research 4407, INRIA.

  • Demongeot, J. (1998). Multistationarity and cell differentiation. J. Biol. Syst. 6, 1–2.

    Article  Google Scholar 

  • Devloo, V. (2003). Développement d’outils théoriques et logicielspour la modélisation du réseau des processus cellulaires, PhD thesis, Université Libre de Bruxelles.

  • Glass, L. (1975). Classification of biological networks by their qualitative dynamics. J. Theor. Biol. 54, 85–107.

    Google Scholar 

  • Glass, L. and S. A. Kauffman (1972). Co-operative components, spatial localization and oscillatory cellular dynamics. J. Theor. Biol. 34, 219–237.

    Article  Google Scholar 

  • Glass, L. and S. A. Kauffman (1973). The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103–129.

    Article  Google Scholar 

  • Gouzé, J.-L. (1998). Positive and negative circuits in dynamical systems. J. Biol. Syst. 6, 11–15.

    Article  MATH  Google Scholar 

  • Haralick, R. and G. Elliott (1980). Increasing tree search efficiency for constraint satisfaction problems. Artif. Intell. 14, 263–313.

    Article  Google Scholar 

  • Heinrich, R. and S. Schuster (1996). The Regulation of Cellular Systems, New York: Chapman and Hall.

    Google Scholar 

  • Hofmeyr, J. H. S. and H. V. Westerhoff (2001). Building the cellular puzzle. Control in multi-level reaction networks. J. Theor. Biol. 208, 261–285.

    Article  Google Scholar 

  • Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution, New York: Oxford University Press.

    Google Scholar 

  • Kuipers, B. (1994). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge, Cambridge, MA: MIT Press.

    Google Scholar 

  • Kumar, V. (1992). Algorithms for constraint satisfaction problems: A survey. AI Magazine 13, 32–44.

    Google Scholar 

  • Levin, J. Z. and E. M. Meyerowitz (1995). UFO: an Arabidopsis gene involved in both floral meristem and floral organ development. Plant Cell 7, 529–548.

    Article  Google Scholar 

  • Lewis, J. E. and L. Glass (1991). Steady states, limit cycles, and chaos in models of complex biological networks. Int. J. Bifurcation Chaos 1, 477–483.

    Article  MathSciNet  Google Scholar 

  • Liu, Z. and E. M. Meyerowitz (1995). LEUNIG regulates AGAMOUS expression in Arabidopsis flowers. Development 121, 975–991.

    Google Scholar 

  • Mackworth, A. K. (1977). Consistency in networks of relations. Artif. Intell. 8, 99–118.

    Article  MATH  Google Scholar 

  • Marriott, K. and P. Stuckey (1998). Programming with Constraints: An Introduction, Cambridge, MA: MIT Press.

    Google Scholar 

  • McAdams, H. M. and A. Arkin (1997). Stochastic mechanisms in gene expression. Proc. Natl Acad. Sci. USA 94, 814–819.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Nilsson, N. J. (1980). Principles of Artificial Intelligence, Palo Alto: Tioga.

    Google Scholar 

  • Plahte, E., T. Mestl and S. W. Omholt (1994). Global analysis of steady points for systems of differential equations with sigmoid interactions. Dyn. Stab. Syst. 9, 275–291.

    MathSciNet  Google Scholar 

  • Plahte, E., T. Mestl and S. W. Omholt (1995). Feedback loops, stability and multistationarity in dynamical systems. J. Biol. Syst. 3, 569–577.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Sakai, H., L. J. Medrano and E. M. Meyerowitz (1995). Role of SUPERMAN in maintaining Arabidopsis floral whorl boundaries. Nature 378, 199–203.

    Article  Google Scholar 

  • Sánchez, L. and D. Thieffry (2001). A logical analysis of the Drosophila gap genes. J. Theor. Biol. 189, 377–389.

    Article  Google Scholar 

  • Sánchez, L., J. van Helden and D. Thieffry (1997). Establishment of the dorso-ventral pattern during embryonic development of Drosophila melanogaster: a logical analysis. J. Theor. Biol. 189, 377–389.

    Article  Google Scholar 

  • Shen-Orr, S. S., R. Milo, S. Mangan and U. Alon (2002). Networks motifs in the transcriptional regulation network of Escherichia coli. Nat. Genetics 31, 64–68.

    Article  Google Scholar 

  • Smith, B. M., S. C. Brailsford, P. M. Hubbard and H. P. Williams (1996). The progressive party problem: integer linear programming and constraint programming compared. Constraints 1, 119–138.

    Article  MathSciNet  Google Scholar 

  • Smolen, P., D. A. Baxter and J. H. Byrne (2000). Modeling transcriptional control in gene networks: methods, recent results, and future directions. Bull. Math. Biol. 62, 247–292.

    Article  Google Scholar 

  • Snoussi, E. H. (1989). Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn. Stab. Syst. 4, 189–207.

    MATH  MathSciNet  Google Scholar 

  • Snoussi, E. H. (1998). Necessary conditions for multistationarity and stable periodicity. J. Biol. Syst. 6, 3–9.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Somogyi, R. and C. A. Sniegoski (1996). Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation. Complexity 1, 45–63.

    MathSciNet  Google Scholar 

  • Sugita, M. (1963). Functional analysis of chemical systems in vivo using a logical circuit equivalent: II. The idea of a molecular automaton. J. Theor. Biol. 4, 179–192.

    Article  Google Scholar 

  • Thieffry, D., M. Colet and R. Thomas (1993). Formalization of regulatory nets: a logical method and its automatization. Math. Modelling Sci. Comput. 2, 144–151.

    Google Scholar 

  • Thieffry, D. and R. Thomas (1995). Dynamical behaviour of biological networks: II. Immunity control in bacteriophage lambda. Bull. Math. Biol. 57, 277–297.

    Article  Google Scholar 

  • Thomas, R. (ed.) (1979). Kinetic Logic: A Boolean Approach to the Analysis of Complex Regulatory Systems, Lecture Notes in Biomathematics 29, Berlin: Springer.

    Google Scholar 

  • Thomas, R. (1991). Regulatory networks seen as asynchronous automata: a logical description. J. Theor. Biol. 153, 1–23.

    Google Scholar 

  • Thomas, R. (1994). The role of feedback circuits: positive feedback circuits are a necessary condition for positive real eigenvalues of the Jacobian matrix. Ber. Bunsenges. Phys. Chem. 98, 1148–1151.

    Google Scholar 

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

    Google Scholar 

  • Thomas, R., A.-M. Gathoye and L. Lambert (1976). A complex control circuit: regulation of immunity in temperate bacteriophages. Eur. J. Biochem. 71, 211–227.

    Article  Google Scholar 

  • Thomas, R. and M. Kaufman (2001). Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits. Chaos 11, 180–195.

    Article  MathSciNet  Google Scholar 

  • Thomas, R. and P. van Ham (1974). Analyse formelle de circuits de régulation génétique: le contrôle de l’immunitéchez les bactériophages lambdoides. Biochimie 56, 1529–1547.

    Google Scholar 

  • Tsang, E. (1993). Foundations of Constraint Satisfaction, London and San Diego: Academic Press.

    Google Scholar 

  • Tyson, J. J. and H. G. Othmer (1978). The dynamics of feedback control circuits in biochemical pathways. Prog. Theor. Biol. 5, 1–62.

    Google Scholar 

  • van Ham, P. (1979). How to deal with more than two levels, in Kinetic Logic: A Boolean Approach to the Analysis of Complex Regulatory Systems, Lecture Notes in Bioinformatics 29, R. Thomas (Ed.), Berlin: Springer, pp. 326–343.

    Google Scholar 

  • van Hentenryck, P. (1989). Constraint Satisfaction in Logic Programming, Cambridge, MA: MIT Press.

    Google Scholar 

  • van Hentenryck, P. (1999). The OPL Optimization Programming Language, Cambridge, MA: MIT Press.

    Google Scholar 

  • Voit, E. O. (2000). Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists, Cambridge: Cambridge University Press.

    Google Scholar 

  • Yang, C. H., L. J. Chen and Z. R. Sung (1995). Genetic regulation of shoot development in Arabidopsis: role of the EMF genes. Dev. Biol. 169, 421–435.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SCMBB, CP 263, Université Libre de Bruxelles, Boulevard du Triomphe, 1050, Brussels, Belgium

    Vincent Devloo

  2. GERAD and HEC, Montréal, Canada

    Pierre Hansen

  3. Optimization, ISRO, Université Libre de Bruxelles, Av. F. Roosevelt 50, 1050, Brussels, Belgium

    Martine Labbé

Authors
  1. Vincent Devloo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martine Labbé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Devloo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Devloo, V., Hansen, P. & Labbé, M. Identification of all steady states in large networks by logical analysis. Bull. Math. Biol. 65, 1025–1051 (2003). https://doi.org/10.1016/S0092-8240(03)00061-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1016/S0092-8240(03)00061-2

Keywords

Search

Navigation