Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations

Mark A. J. Chaplain, Mariya Ptashnyk, Marc Sturrock

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a one-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NF-κB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.

Original languageEnglish
Pages (from-to)1179-1215
Number of pages37
JournalMathematical Models and Methods in Applied Sciences
Volume25
Issue number6
DOIs
Publication statusPublished - 15 Jun 2015

Keywords

  • center manifold and normal form
  • Gene regulatory network
  • Hopf bifurcation
  • negative feedback loop
  • oscillations
  • transcription factor
  • weakly nonlinear analysis

ASJC Scopus subject areas

  • Applied Mathematics
  • Modelling and Simulation

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