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On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle

1 Life Science Engineering, HTW Berlin, Wilhelminenhoftstr. 75, 10459 Berlin, Germany
2 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
3 Department of Mathematical Sciences, Northern Illinois University, 1425 W. Lincoln Hwy., DeKalb IL 60115, USA

Special Issues: Mathematical analysis of reaction networks: theoretical advances and applications

Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parametrize the steady state and Hurwitz matrices.
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