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Analysis of the behaviour of Kauffman binary networks—I. State space description and the distribution of limit cycle lengths

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Abstract

The basis of an analytical description of the behaviour of large random nets of binary elements of the type first investigated in detail by S. A. Kauffman is presented. It is shown that information about the network dynamics can be deduced from quite general considerations of the properties of the state transition graph and matrix. An expression for the matrix elements of the state transition matrix in terms of the Boolean function specification of the net is derived. Using these ideas the distribution of limit cycle lengthsl for a completely random net is calculated and shown to bex 1/l, a result which agrees well with experimental data.

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Literature

  • Abramowitz, M. and I. A. Segun. 1968.Handbook of Mathematical Functions. New York: Dover.

    Google Scholar 

  • Aleksander, I. 1973. “Random Logic Nets: Stability and Adaptation.”Int. J. Man-Mach. Stud.,5, 115–131.

    Google Scholar 

  • — and P. Atlas. 1973. “Cyclic Activity in Nature.”Int. J. Neurosci.,6, 45–50.

    Article  Google Scholar 

  • Andrews, G. E. 1976.The Theory of Partitions, Vol. 2 ofEncyclopedia of Mathematics and its Applications. Reading, Mass: Addison-Wesley.

    Google Scholar 

  • Cavender, J. A. 1977. “Kauffman's Square-root Law: Possible Correlatives.”J. Theor. Biol.,65, 791–793.

    Article  Google Scholar 

  • Glass, L. and S. A. Kauffman. 1973. “The Logical Analysis of Continuous Non-linear Biochemical Control Networks.”J. Theor. Biol.,39, 103–129.

    Article  Google Scholar 

  • Goodwin, B. C. 1976.Analytical Physiology of Cells and Developing Organisms pp. 53–56. London: Academic Press.

    Google Scholar 

  • Jacob, F. and J. Monod. 1961. “Genetic Regulatory Mechanisms in the Synthesis of Proteins.”J. Mol. Biol.,3, 318–356.

    Article  Google Scholar 

  • Kauffman, S. A. 1969a. “Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets.”J. Theor. Biol.,22, 437–467.

    Article  MathSciNet  Google Scholar 

  • — 1969b. “Homeostasis and Differentiation in Random Genetic Networks.”Nature,224, 177–178.

    Article  Google Scholar 

  • — 1970a. “Behaviour of Randomly Constructed Genetic Nets: Binary Element Nets.” InTowards a Theoretical Biology 3: Drafts, Ed. C. H. Waddington, pp. 18–37. Edinburgh University Press.

    Google Scholar 

  • — 1970b. “Behaviour of Randomly Constructed Genetic Nets: Continuous Element Nets.” InTowards a Theoretical Biology 3: Drafts. Ed. C. H. Waddington, pp. 38–46. Edinburgh: Edinburgh University Press.

    Google Scholar 

  • — 1971. “Gene Regulation Networks: A Theory for their Global Structure and Behaviour.”Curr. Topics Dev. Biol.,6, 145–182.

    Google Scholar 

  • — 1973. “Control Circuits for Determination and Transdetermination.”Science,181, 310–318.

    Google Scholar 

  • Pattee, H. H. 1969. “How Does a Molecule Become a Message?”Dev. Biol. Suppl.,3, 1–16.

    Google Scholar 

  • Torng, H. C. 1972.Switching Circuits: Theory and Logic Design. Reading, Mass: Addison Wesley.

    MATH  Google Scholar 

  • Wilson, R. J. 1972.Introduction to Graph Theory. Glasgow: Bell & Bain.

    MATH  Google Scholar 

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Sherlock, R.A. Analysis of the behaviour of Kauffman binary networks—I. State space description and the distribution of limit cycle lengths. Bltn Mathcal Biology 41, 687–705 (1979). https://doi.org/10.1007/BF02462422

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  • DOI: https://doi.org/10.1007/BF02462422

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