Relating Attractors and Singular Steady States in the Logical Analysis of Bioregulatory Networks

  • Heike Siebert
  • Alexander Bockmayr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4545)

Abstract

In 1973 R. Thomas introduced a logical approach to modeling and analysis of bioregulatory networks. Given a set of Boolean functions describing the regulatory interactions, a state transition graph is constructed that captures the dynamics of the system. In the late eighties, Snoussi and Thomas extended the original framework by including singular values corresponding to interaction thresholds. They showed that these are needed for a refined understanding of the network dynamics. In this paper, we study systematically singular steady states, which are characteristic of feedback circuits in the interaction graph, and relate them to the type, number and cardinality of attractors in the state transition graph. In particular, we derive sufficient conditions for regulatory networks to exhibit multistationarity or oscillatory behavior, thus giving a partial converse to the well-known Thomas conjectures.

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References

  1. 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. J. Theor. Biol. 229, 339–347 (2004)CrossRefMathSciNetGoogle Scholar
  2. Remy, É., Mossé, B., Chaouiya, C., Thieffry, D.: A description of dynamical graphs associated to elementary regulatory circuits. Bioinform. 19, 172–178 (2003)Google Scholar
  3. Remy, É., Ruet, P., Thieffry, D.: Graphic requirements for multistability and attractive cycles in a boolean dynamical framework. (prépublication 2005)Google Scholar
  4. Remy, É., Ruet, P., Thieffry, D.: Positive or negative regulatory circuit inference from multilevel dynamics. In: Positive Systems: Theory and Applications. LNCIS, vol. 341, pp. 263–270. Springer, Heidelberg (2006)Google Scholar
  5. Richard, A., Comet, J.-P.: Necessary conditions for multistationarity in discrete dynamical systems. Rapport de Recherche (2005)Google Scholar
  6. Richard, A., Comet, J.-P., Bernot, G., Thomas, R.: Modeling of biological regulatory networks: introduction of singular states in the qualitative dynamics. Fundamenta Informaticae 65, 373–392 (2005)MATHMathSciNetGoogle Scholar
  7. Siebert, H., Bockmayr, A.: Relating attractors and singular steady states in the logical analysis of bioregulatory networks. preprint 373, DFG Research Center MATHEON (2007)Google Scholar
  8. Snoussi, E.H., Thomas, R.: Logical identification of all steady states: the concept of feedback loop characteristic states. Bull. Math. Biol. 55, 973–991 (1993)MATHGoogle Scholar
  9. Soulé, C.: Graphical requirements for multistationarity. ComPlexUs 1, 123–133 (2003)CrossRefGoogle Scholar
  10. Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585 (1973)CrossRefGoogle Scholar
  11. Thomas, R., d’Ari, R.: Biological Feedback. CRC Press (1990)Google Scholar
  12. Thomas, R., Kaufman, M.: Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits. Chaos 11, 180–195 (2001)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Heike Siebert
    • 1
  • Alexander Bockmayr
    • 1
  1. 1.DFG Research Center MATHEON, Freie Universität Berlin, Arnimallee 3, D-14195 BerlinGermany

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