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| Gene regulatory networks lie at the heart of many important intracellular signal transduction processes. In this paper we analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA. The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour. 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.
Applying linearized stability analysis we study the stability of a spatially inhomogeneous steady state of the model and prove the existence of two Hopf bifurcation point. The local stability of periodic solutions, bifurcating from the steady state, is shown using a weakly nonlinear analysis and normal form theory. |
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