Rigorous proof of oscillation phase separation (OPS) in nonlinear biochemical oscillators

Prove rigorously that oscillation phase separation (OPS)—the decomposition of the limit-cycle trajectory into distinct fast and slow phases—occurs in the nonlinear regime far from the Hopf bifurcation onset for each of the biochemical oscillator models studied, including the Van der Pol oscillator, the irreversible and reversible Brusselator, the Tyson model for the Drosophila circadian clock, and the van Zon–Hatakeyama model of the KaiABC system, thereby validating the generality of OPS as a mechanism enabling temperature compensation through large positive period sensitivities.

Background

The paper proposes a general mechanism for temperature compensation (TC) in biochemical oscillators based on kinetic regulation that produces large positive period sensitivities. This mechanism relies on oscillation phase separation (OPS), where the limit-cycle trajectory splits into slow and fast phases; in the dominant slow phase, the amplitude increases with certain reaction rates while the progression speed is controlled by other rates. The authors demonstrate OPS and its role in creating period-lengthening reactions across four oscillator models: Van der Pol, Brusselator, a Drosophila circadian model (Tyson), and the Kai system model (van Zon–Hatakeyama).

Because the central TC mechanism in this work depends on OPS being a generic feature of oscillators operating far from onset, a rigorous proof establishing OPS in the nonlinear regime for these biochemical oscillator models is crucial. The authors note that, despite empirical and numerical evidence, they cannot provide a rigorous proof, leaving the mathematical validation and characterization of OPS across these systems as an explicit open problem.