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.
Similar content being viewed by others
References
Agur, Z., Fraenkel, A.S., Klein, S.T., 1988. The number of fixed points of the majority rule. Discrete Math. 70, 295–302.
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.
Aracena, J., 2008. Maximum number of fixed points in regulatory Boolean networks. Bull. Math. Biol. 70, 1398–1409.
Aracena, J., Demongeot, J., Goles, E., 2004. Fixed points and maximal independent sets in AND–OR networks. Discrete Appl. Math. 138, 277–288.
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.
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.
Brualdi, R.A., Ryser, H.J., 1991. Combinatorial Matrix Theory. Encyclopedia of Mathematics and its Applications, vol. 39. Cambridge University Press, Cambridge.
Colón-Reyes, O., Laubenbacher, R., Pareigis, B., 2004. Boolean monomial dynamical systems. Ann. Comb. 8, 425–439.
Cull, P., 1971. Linear analysis of switching nets. Kybernetik 8, 31–39.
Davidich, M.I., Bornholdt, S., 2008. Boolean network model predicts cell cycle sequence of fission yeast. PLoS ONE 3, e1672.
Elspas, B., 1959. The theory of autonomous linear sequential networks. IRE Trans. Circuit Theory, 6, 45–60.
Gauzé, J., 1998. Positive and negative circuits in dynamical systems. J. Biol. Syst. 6, 11–15.
Goles, E., Hernández, G., 2000. Dynamical behavior of Kauffman networks with and-or gates. J. Biol. Syst. 8, 151–175.
Goles, E., Olivos, J., 1980. Periodic behaviour of generalized threshold functions. Discrete Math. 30, 187–189.
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.
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.
Hernández-Toledo, A., 2005. Linear finite dynamical systems. Commun. Algebra 33, 2977–2989.
Jarrah, A., Laubenbacher, R., Vastani, H. DVD: Discrete visual dynamics. World Wide Web. http://dvd.vbi.vt.edu
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).
Kauffman, S., 1969a. Homeostasis and differentiation in random genetic control networks. Nature 224, 177–178.
Kauffman, S., 1969b. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467.
Kauffman, S., 1993. Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, London.
Kauffman, S., Peterson, C., Samuelsson, B., Troein, C., 2003. Random Boolean network models and the yeast transcriptional network. PNAS 100, 14796–14799.
Kauffman, S., Peterson, C., Samuelsson, B., Troein, C., 2004. Genetic networks with canalyzing Boolean rules are always stable. PNAS 101, 17102–17107.
Kwon, Y.-K., Cho, K.-H., 2007. Boolean dynamics of biological networks with multiple coupled feedback loops. Biophys. J. 92, 2975–2981.
Lidl, R., Niederreiter, H., 1997. Finite Fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge.
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.
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.
Nikolajewaa, S., Friedela, M., Wilhelm, T., 2007. Boolean networks with biologically relevant rules show ordered behaviorstar, open. Biosystems 90, 40–47.
Plahte, E., Mestl, T., Omholt, S., 1995. Feedback loops, stability and multistationarity in dynamical systems. J. Biol. Syst. 3, 409–413.
Raeymaekers, L., 2002. Dynamics of boolean networks controlled by biologically meaningful functions. J. Theor. Biol. 218, 331–341.
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.
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.
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.
Soule, C., 2003. Graphic requirements for multistationarity. ComPlexUs 1, 123–133.
Stanley, R.P., 1986. Enumerative Combinatorics I. Translations of Mathematical Monographs. Wadsworth, Belmont.
Stigler, B., Veliz-Cuba, A., 2008. Network topology as a driver of bistability in the lac operon. http://arxiv.org/abs/0807.3995.
Thomas, R., D’Ari, R., 1990. Biological Feedback. CRC Press, Boca Raton.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported partially by NSF Grant DMS-0511441.
Rights and permissions
About this article
Cite this article
Jarrah, A.S., Laubenbacher, R. & Veliz-Cuba, A. The Dynamics of Conjunctive and Disjunctive Boolean Network Models. Bull. Math. Biol. 72, 1425–1447 (2010). https://doi.org/10.1007/s11538-010-9501-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-010-9501-z