Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator

TL;DR

This paper analyzes the complex pH oscillations in the urea-urease reaction within vesicles by applying geometric singular perturbation theory to understand multiple timescales and limit cycle behavior.

Contribution

It introduces a novel geometric analysis framework combining multiple coordinate scalings and GSPT extensions to study oscillations in a biochemical system with well-separated timescales.

Findings

Resolved local dynamics near critical fold points.

Derived closed-form estimates for oscillation periods.

Linked oscillation existence to biochemical asymmetry.

Abstract

We study a two-variable dynamical system modeling pH oscillations in the urea-urease reaction within giant lipid vesicles -- a problem that intrinsically contains multiple, well-separated timescales. Building on an existing, deterministic formulation via ordinary differential equations, we resolve different orders of magnitude within a small parameter and analyze the system's limit cycle behavior using geometric singular perturbation theory (GSPT). By introducing two different coordinate scalings -- each valid in a distinct region of the phase space -- we resolve the local dynamics near critical fold points, using the extension of GSPT through such singular points due to Krupa and Szmolyan. This framework enables a geometric decomposition of the periodic orbits into slow and fast segments and yields closed-form estimates for the period of oscillation. In particular, we link the existence of such oscillations to an underlying biochemical asymmetry, namely, the differential transport across the vesicle membrane.