math.DS — Dynamical Systems

Sharp equilibrium and stable-state capacities of multisite phosphorylation

Contributed by Michael Mislan

For the sequential distributive $n$-site phosphorylation cycle with one kinase and one phosphatase, the maximal number of positive equilibria is $2n-1$ and the maximal number of locally asymptotically stable equilibria is $n$, for every $n\ge1$. The stable-state ceiling includes nonhyperbolic attractors. Both bounds are attained simultaneously at common conserved totals by rational positive rate constants and rational hyperbolic equilibria: $n$ sinks alternate with $n-1$ saddles, each with one unstable direction. This configuration persists on a nonempty open subset of the full parameter space. An explicit polynomial recurrence and interlacing construct the equilibria. Stability follows by separating equilibrium geometry from kinetic time scales, retaining enzyme-bound substrate in the slow variables, and controlling a path matrix with positive rank-one feedback. Further results give a diagonal Lyapunov certificate, a quantitative eigenvalue splitting law, exact finite-size witnesses and a three-site robustness and recovery certificate. An accompanying registered Lean theorem establishes equilibrium attainment; the broader capacity results are proved here by conventional arguments.