  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 four
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 parameterize the steady state and Hurwitz matrices.
