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Gene regulatory networks, i.e. DNA segments in a cell which interact witheach other indirectly through their RNA and protein products, lie at the heartof many important intracellular signal transduction processes. In this paper weanalyse a mathematical model of a canonical gene regulatory network consistingof a single negative feedback loop between a protein and its mRNA (e.g. theHes1 transcription factor system). The model consists of two partialdifferential equations describing the spatio-temporal interactions between theprotein and its mRNA in a 1-dimensional domain. Such intracellular negativefeedback systems are known to exhibit oscillatory behaviour and this is thecase for our model, shown initially via computational simulations. In order toinvestigate this behaviour more deeply, we next solve our system using Green'sfunctions and then undertake a linearized stability analysis of the steadystates of the model. Our results show that the diffusion coefficient of theprotein/mRNA acts as a bifurcation parameter and gives rise to a Hopfbifurcation. This shows that the spatial movement of the mRNA and proteinmolecules alone is sufficient to cause the oscillations. This has implicationsfor transcription factors such as p53, NF-$\kappa$B and heat shock proteinswhich are involved in regulating important cellular processes such asinflammation, meiosis, apoptosis and the heat shock response, and are linked todiseases such as arthritis and cancer.

Hopf Bifurcation in a Gene Regulatory Network Model: Molecular Movement Causes Oscillations