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Nested Canalyzing Depth and Network Stability


We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.

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  • 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.

    MathSciNet  Article  Google Scholar 

  • Balleza, E., Alvarez-Buylla, E., Chaos, A., Kauffman, S. A., Shmulevich, I., & Aldana, M. (2008). Critical dynamics in genetic regulatory networks: Examples from four kingdoms. PLoS ONE, 3(6), e2456.

    Article  Google Scholar 

  • Derrida, B., & Pomeau, Y. (1986). Random networks of automata: a simple annealed approximation. Europhys. Lett., 1, 45–49.

    Article  Google Scholar 

  • Drossel, B. (2009). Random Boolean networks, Chap. 3, pp. 69–110. Weinheim: Wiley-VCH Verlag GmbH & Co.

    Google Scholar 

  • Gambin, A., Lasota, S., & Rutkowski, M. (2006). Analyzing stationary states of gene regulatory network using petri nets. Silico Biol., 6, 93–109.

    Google Scholar 

  • Jarrah, A. S., Raposa, B., & Laubenbacher, R. (2007). Nested canalyzing, unate cascade, and polynomial functions. Physica D, 233, 167–174.

    MathSciNet  MATH  Article  Google Scholar 

  • Karlssona, F., & Hörnquist, M. (2007). Order or chaos in Boolean gene networks depends on the mean fraction of canalyzing functions. Physica A, 384, 747–757.

    Article  Google Scholar 

  • Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol., 22(3), 437–467.

    Article  Google Scholar 

  • Kauffman, S. A. (1993). The origins of order: self-organization and selection in evolution. London: Oxford University Press.

    Google Scholar 

  • Kauffman, S. A., Peterson, C., Samuelsson, B., & Troein, C. (2003). Random Boolean network models and the yeast transcriptional network. Proc. Natl. Acad. Sci., 100(25), 14796–14799.

    Article  Google Scholar 

  • Kauffman, S. A., Peterson, C., Samuelsson, B., & Troein, C. (2004). Genetic networks with canalyzing Boolean rules are always stable. Proc. Natl. Acad. Sci., 101(49), 17102–17107.

    Article  Google Scholar 

  • Li, F., Long, T., Lu, Y., Ouyang, Q., & Tang, C. (2004). The yeast cell-cycle network is robustly designed. Proc. Natl. Acad. Sci., 11, 4781–4786.

    Article  Google Scholar 

  • Nikolajewa, S., Friedel, M., & Wilhelm, T. (2006). Boolean networks with biologically relevant rules show ordered behavior. Biosystems, 90(1), 40–47.

    Article  Google Scholar 

  • Nykter, M., Price, N. D., Aldana, M., Ramsey, S. A., Kauffman, S. A., Hood, L. E., Yli-Harja, O., & Shmulevich, I. (2008a). Gene expression dynamics in the macrophage exhibit criticality. Proc. Natl. Acad. Sci., 105, 1897–1900.

    Article  Google Scholar 

  • Nykter, M., Price, N. D., Larjo, A., Aho, T., Kauffman, S. A., Yli-Harja, O., & Shmulevich, I. (2008b). Critical networks exhibit maximal information diversity in structure-dynamics relationships. Phys. Rev. Lett., 100, 058702.

    Article  Google Scholar 

  • Peixoto, T. P. (2010). The phase diagram of random Boolean networks with nested canalizing functions. Eur. Phys. J. B, 78(2), 187–192.

    Article  Google Scholar 

  • Saez-Rodriguez, J., Simeoni, L., Lindquist, J., Hemenway, R., Bommhardt, U., Arndt, B., Haus, U., Weismantel, R., Gilles, E., Klamt, S., & Schraven, B. (2007). A logical model provides insights into T cell receptor signaling. PLoS Comput. Biol., 3, e163.

    MathSciNet  Article  Google Scholar 

  • Shmulevich, I., & Kauffman, S. A. (2004). Activities and sensitivities in Boolean network models. Phys. Rev. Lett., 93(4), 048701.

    Article  Google Scholar 

  • Shmulevich, I., Kauffman, S. A., & Aldana, M. (2005). Eukaryotic cells are dynamically ordered or critical but not chaotic. Proc. Natl. Acad. Sci., 102, 13439–13444.

    Article  Google Scholar 

  • Waddington, C. H. (1942). Canalisation of development and the inheritance of acquired characters. Nature, 150, 563–564.

    Article  Google Scholar 

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Correspondence to Lori Layne.

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Layne, L., Dimitrova, E. & Macauley, M. Nested Canalyzing Depth and Network Stability. Bull Math Biol 74, 422–433 (2012).

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  • Boolean networks
  • Stability
  • Gene networks
  • Canalyzation