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Sharp equilibrium and stable-state capacities of multisite phosphorylation

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by Michael Mislan ↗

Version 1 / Oct 17, 2026 / CC BY 4.0

Abstract

For the sequential distributive nn-site phosphorylation cycle with one kinase and one phosphatase, the maximal number of positive equilibria is 2n−12n-1 and the maximal number of locally asymptotically stable equilibria is nn, for every n≥1n\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: nn sinks alternate with n−1n-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.

Provenance statement

The text is primarily AI-generated. AI systems assisted the mathematical development, computations and Lean formalization under human direction; human understanding covers some parts of the work.

Formalizations

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Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 17, 2026