Attracting oscillations in sequential distributive multisite phosphorylation
The sequential distributive $n$-site phosphorylation cycle with one kinase and one phosphatase admits attracting positive periodic orbits under mass-action kinetics for every $n\ge3$. An explicit three-site family has a fixed positive equilibrium, a transversal imaginary-eigenvalue crossing, a stable complementary spectrum and a negative first Lyapunov coefficient. A site-addition argument extends the supercritical Hopf bifurcation to every larger site count. Together with the one-site calculation and Vassena's two-site exclusion, this makes three the exact threshold for Hopf bifurcation in…
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