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Periodic Trajectories and Andronov-Hopf Bifurcations in Models of Gene Networks

  • V. P. GolubyatnikovEmail author
  • V. A. Likhoshvai
  • E. P. Volokitin
  • Yu. A. Gaidov
  • A. F. Osipov
Chapter

Abstract

Multistability is an important property of the gene network functioning. The estimate of possible numbers of limit cycles and stationary points of gene networks with various types of regulations is a fundamental problem of the applied mathematics, We prove existence of limit cycles for four classes of the gene network models where the control of protein concentrations is realized at the level of regulation of their stability. We show that the change in regulation from the stage of gene expression activation to the stage of degradation of the synthesis products does not affect the dynamical properties of the gene networks under consideration.

Key words

Dynamical systems Andronov-Hopf bifurcation fixed point theorem negative feedback regulation of degradation and synthesis of mRNA and proteins 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. P. Golubyatnikov
    • 1
    • 2
    Email author
  • V. A. Likhoshvai
    • 3
    • 4
  • E. P. Volokitin
    • 1
  • Yu. A. Gaidov
    • 5
  • A. F. Osipov
    • 5
  1. 1.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Weizmann Institute of SciencesRehovotIzrael
  3. 3.Institute of Cytology and GeneticsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  4. 4.Ugra Institute of InformaticsHanty-MansiiskRussia
  5. 5.Novosibirsk State UniversityNovosibirskRussia

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