Bulletin of Mathematical Biology

, Volume 72, Issue 6, pp 1425–1447 | Cite as

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

  • Abdul Salam Jarrah
  • Reinhard Laubenbacher
  • Alan Veliz-Cuba
Original Article
First Online:
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Accepted:

Abstract

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

Keywords

Gene regulatory network Conjunctive Disjunctive Boolean network model Feedback loop Network dynamics 
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References

  1. Agur, Z., Fraenkel, A.S., Klein, S.T., 1988. The number of fixed points of the majority rule. Discrete Math. 70, 295–302. MATHCrossRefMathSciNetGoogle Scholar
  2. Albert, R., Othmer, H., 2003. The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J. Theor. Biol. 223, 1–18. CrossRefMathSciNetGoogle Scholar
  3. Aracena, J., 2008. Maximum number of fixed points in regulatory Boolean networks. Bull. Math. Biol. 70, 1398–1409. MATHCrossRefMathSciNetGoogle Scholar
  4. Aracena, J., Demongeot, J., Goles, E., 2004. Fixed points and maximal independent sets in AND–OR networks. Discrete Appl. Math. 138, 277–288. MATHCrossRefMathSciNetGoogle Scholar
  5. Barrett, C.L., Chen, W.Y.C., Zheng, M.J., 2004. Discrete dynamical systems on graphs and boolean functions. Math. Comput. Simul. 66, 487–497. MATHCrossRefMathSciNetGoogle Scholar
  6. Berman, A., Plemmons, R.J., 1994. Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Revised reprint of the 1979 original. MATHGoogle Scholar
  7. Brualdi, R.A., Ryser, H.J., 1991. Combinatorial Matrix Theory. Encyclopedia of Mathematics and its Applications, vol. 39. Cambridge University Press, Cambridge. MATHGoogle Scholar
  8. Colón-Reyes, O., Laubenbacher, R., Pareigis, B., 2004. Boolean monomial dynamical systems. Ann. Comb. 8, 425–439. MATHCrossRefMathSciNetGoogle Scholar
  9. Cull, P., 1971. Linear analysis of switching nets. Kybernetik 8, 31–39. MATHCrossRefGoogle Scholar
  10. Davidich, M.I., Bornholdt, S., 2008. Boolean network model predicts cell cycle sequence of fission yeast. PLoS ONE 3, e1672. CrossRefGoogle Scholar
  11. Elspas, B., 1959. The theory of autonomous linear sequential networks. IRE Trans. Circuit Theory, 6, 45–60. Google Scholar
  12. Gauzé, J., 1998. Positive and negative circuits in dynamical systems. J. Biol. Syst. 6, 11–15. CrossRefGoogle Scholar
  13. Goles, E., Hernández, G., 2000. Dynamical behavior of Kauffman networks with and-or gates. J. Biol. Syst. 8, 151–175. Google Scholar
  14. Goles, E., Olivos, J., 1980. Periodic behaviour of generalized threshold functions. Discrete Math. 30, 187–189. MATHCrossRefMathSciNetGoogle Scholar
  15. Gummow, B., Sheys, J., Cancelli, V., Hammer, G., 2006. Reciprocal regulation of a glucocorticoid receptor-steroidogenic factor-1 transcription complex on the dax-1 promoter by glucocorticoids and adrenocorticotropic hormone in the adrenal cortex. Mol. Endocrinol. 20, 2711–2723. CrossRefGoogle Scholar
  16. Harris, S.E., Sawhill, B.K., Wuensche, A., Kauffman, S., 2002. A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity 7, 23–40. CrossRefGoogle Scholar
  17. Hernández-Toledo, A., 2005. Linear finite dynamical systems. Commun. Algebra 33, 2977–2989. MATHCrossRefGoogle Scholar
  18. Jarrah, A., Laubenbacher, R., Vastani, H. DVD: Discrete visual dynamics. World Wide Web. http://dvd.vbi.vt.edu
  19. Jarrah, A., Laubenbacher, R., Vera-Licona, M.S.P., 2008. An efficient algorithm for the phase space structure of linear dynamical systems over finite fields (submitted). Google Scholar
  20. Kauffman, S., 1969a. Homeostasis and differentiation in random genetic control networks. Nature 224, 177–178. CrossRefGoogle Scholar
  21. Kauffman, S., 1969b. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467. CrossRefMathSciNetGoogle Scholar
  22. Kauffman, S., 1993. Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, London. Google Scholar
  23. Kauffman, S., Peterson, C., Samuelsson, B., Troein, C., 2003. Random Boolean network models and the yeast transcriptional network. PNAS 100, 14796–14799. CrossRefGoogle Scholar
  24. Kauffman, S., Peterson, C., Samuelsson, B., Troein, C., 2004. Genetic networks with canalyzing Boolean rules are always stable. PNAS 101, 17102–17107. CrossRefGoogle Scholar
  25. Kwon, Y.-K., Cho, K.-H., 2007. Boolean dynamics of biological networks with multiple coupled feedback loops. Biophys. J. 92, 2975–2981. CrossRefGoogle Scholar
  26. Lidl, R., Niederreiter, H., 1997. Finite Fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge. Google Scholar
  27. Merika, M., Orkin, S., 1995. Functional synergy and physical interactions of the erythroid transcription factor gata-1 with the Krüppel family proteins sp1 and eklf. Mol. Cell. Biol. 15, 2437–2447. Google Scholar
  28. Nguyen, D.H., D’haeseleer, P., 2006. Deciphering principles of transcription regulation in eucaryotic genomes. Mol. Syst. Biol. 2, 2006.0012. doi: 10.1038/msb4100054. CrossRefGoogle Scholar
  29. Nikolajewaa, S., Friedela, M., Wilhelm, T., 2007. Boolean networks with biologically relevant rules show ordered behaviorstar, open. Biosystems 90, 40–47. CrossRefGoogle Scholar
  30. Plahte, E., Mestl, T., Omholt, S., 1995. Feedback loops, stability and multistationarity in dynamical systems. J. Biol. Syst. 3, 409–413. CrossRefGoogle Scholar
  31. Raeymaekers, L., 2002. Dynamics of boolean networks controlled by biologically meaningful functions. J. Theor. Biol. 218, 331–341. CrossRefMathSciNetGoogle Scholar
  32. Sachkov, V.N., Tarakanov, V.E., 2002. Combinatorics of Nonnegative Matrices. Translations of Mathematical Monographs, vol. 213. Am. Math. Soc., Providence. Translated from the 2000 Russian original by Valentin F. Kolchin. MATHGoogle Scholar
  33. Schutter, B.D., Moor, B.D., 2000. On the sequence of consecutive powers of a matrix in a Boolean algebra. SIAM J. Matrix Anal. Appl. 21, 328–354. CrossRefGoogle Scholar
  34. Sontag, E., Veliz-Cuba, A., Laubenabcher, R., Jarrah, A., 2008. The effect of negative feedback loops on the dynamics of Boolean networks. Biophys. J. 9(2), 518–526. doi: 10.1529/biophysj.107.125021. CrossRefGoogle Scholar
  35. Soule, C., 2003. Graphic requirements for multistationarity. ComPlexUs 1, 123–133. CrossRefGoogle Scholar
  36. Stanley, R.P., 1986. Enumerative Combinatorics I. Translations of Mathematical Monographs. Wadsworth, Belmont. MATHGoogle Scholar
  37. Stigler, B., Veliz-Cuba, A., 2008. Network topology as a driver of bistability in the lac operon. http://arxiv.org/abs/0807.3995.
  38. Thomas, R., D’Ari, R., 1990. Biological Feedback. CRC Press, Boca Raton. MATHGoogle Scholar
  39. Thomas, R., Kaufman, M., 2001. Multistationarity, the basis of cell differentiation and memory. II. Structural conditions of multistationarity and other nontrivial behavior. Chaos 11, 170–179. MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Abdul Salam Jarrah
    • 1
  • Reinhard Laubenbacher
    • 1
  • Alan Veliz-Cuba
    • 1
  1. 1.Virginia Bioinformatics InstituteVirginia TechBlacksburgUSA

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