Abstract
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.
Highlights
Protein phosphorylation cycles consist of three proteins, a substrate S and two enzymes K and F
While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations 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
Summary
Protein phosphorylation cycles consist of three proteins, a substrate S and two enzymes K and F. Such a study of the behavior of an important biochemical module is of particular interest in the light of studies elucidating the complex behavior of signaling pathways composed of such modules [2]. The mass action model of the sequential and distributive phosphorylation cycle depicted in Figure 1a is arguably one of the – mathematically – best studied and challenging systems of post translational modification: both multistationarity (the existence of at least two positive steady states) and bistability (the existence of two locally stable positive steady states) have been established (cf for example [6, 7] for multistationarity, [8] for bistability). To the best of our knowledge, for the mass action model of the phosphorylation cycle in Figure 1a nor simplifications of it, neither Hopf bifurcations nor oscillations have been reported to date.
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