Bulletin of Mathematical Biology

, Volume 75, Issue 6, pp 920–938 | Cite as

Analysis and Characterization of Asynchronous State Transition Graphs Using Extremal States

  • Therese Lorenz
  • Heike Siebert
  • Alexander Bockmayr
Original Article

Abstract

Logical modeling of biological regulatory networks gives rise to a representation of the system’s dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multivalued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step toward meaningful network reduction as well as model refinement methods.

Keywords

Regulatory networks Logical modeling Thomas formalism Model reduction Finite dynamical systems Systems biology 

References

  1. Abou-Jaoudé, W., Ouattara, D. A., & Kaufman, M. (2009). From structure to dynamics: frequency tuning in the p53-Mdm2 network: I. Logical approach. J. Theor. Biol., 258(4), 561–577. CrossRefGoogle Scholar
  2. Aracena, J. (2008). Maximum number of fixed points in regulatory boolean networks. Bull. Math. Biol., 70(5), 1398–1409. . MathSciNetMATHCrossRefGoogle Scholar
  3. Bernot, G., & Tahi, F. (2009). Behaviour preservation of a biological regulatory network when embedded into a larger network. Fundam. Inform., 91(3–4), 463–485. MathSciNetMATHGoogle Scholar
  4. Didier, G., Remy, E., & Chaouiya, C. (2011). Mapping multivalued onto boolean dynamics. J. Theor. Biol., 270(1), 177–184. MathSciNetCrossRefGoogle Scholar
  5. Mabrouki, M., Aiguier, M., Comet, J. P., Gall, P. L., & Richard, A. (2011). Embedding of biological regulatory networks and properties preservation. Math. Comput. Sci., 5(3), 263–288. MathSciNetCrossRefGoogle Scholar
  6. Naldi, A., Carneiro, J., Chaouiya, C., & Thieffry, D. (2010). Diversity and plasticity of Th cell types predicted from regulatory network modelling. PLoS Comput. Biol., 6(9), e1000912. CrossRefGoogle Scholar
  7. Naldi, A., Remy, E., Thieffry, D., & Chaouiya, C. (2011). Dynamically consistent reduction of logical regulatory graphs. Theor. Comput. Sci., 412(21), 2207–2218. MathSciNetMATHCrossRefGoogle Scholar
  8. Remy, E., & Ruet, P. (2008). From minimal signed circuits to the dynamics of boolean regulatory networks. Bioinformatics, 24(16), i200–i226. CrossRefGoogle Scholar
  9. Remy, E., Ruet, P., & Thieffry, D. (2008). Graphic requirements for multistability and attractive cycles in a boolean dynamical framework. Adv. Appl. Math., 41(3), 335–350. MathSciNetMATHCrossRefGoogle Scholar
  10. Sánchez, L., Chaouiya, C., & Thieffry, D. (2008). Segmenting the fly embryo: logical analysis of the role of the segment polarity cross-regulatory module. Int. J. Dev. Biol., 52, 1059–1075. CrossRefGoogle Scholar
  11. Siebert, H. (2011). Analysis of discrete bioregulatory networks using symbolic steady states. Bull. Math. Biol., 73(4), 873–898. MathSciNetMATHCrossRefGoogle Scholar
  12. Thomas, R. (1973). Boolean formalization of genetic control circuits. J. Theor. Biol., 42(3), 565–583. CrossRefGoogle Scholar
  13. Thomas, R., & d’Ari, R. (1990). Biological feedback. Boca Raton: CRC Press. MATHGoogle Scholar
  14. Thomas, R., & Kaufman, M. (2001). Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits. Chaos, 11, 180–195. MathSciNetMATHCrossRefGoogle Scholar
  15. Tournier, L., & Chaves, M. (2009). Uncovering operational interactions in genetic networks using asynchronous boolean dynamics. J. Theor. Biol., 260(2), 196–209. MathSciNetCrossRefGoogle Scholar
  16. Veliz-Cuba, A. (2011). Reduction of boolean network models. J. Theor. Biol., 289, 167–172. MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Therese Lorenz
    • 1
  • Heike Siebert
    • 1
  • Alexander Bockmayr
    • 1
  1. 1.DFG Research Center MatheonFreie Universität BerlinBerlinGermany

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