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arXiv · 2508.07275

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

Arthur V. Straube·Guillermo Olicón Méndez·Stefanie Winkelmann·Felix Höfling·Maximilian Engel
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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.

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Arthur V. Straube, Guillermo Olicón Méndez, Stefanie Winkelmann, Felix Höfling, Maximilian Engel. 2025-08-10. Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator. https://arxiv.org/abs/2508.07275

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