Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles

TL;DR

This study develops and analyzes reduced mathematical models of pH oscillations in urea-urease reactions within giant lipid vesicles, providing insights into their dynamics and parameter sensitivities.

Contribution

It introduces an accurate two-variable model for pH oscillations, demonstrating its equivalence to a three-variable model and enhancing understanding of vesicle communication.

Findings

Derived reduced models for pH oscillations

Predicted oscillation period and amplitude

Identified parameter domain for sustained oscillations

Abstract

This theoretical study concerns a pH oscillator based on the urea-urease reaction confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the phase flow and of the limit cycle, which controls the dynamics for giant vesicles and dominates the pronouncedly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, which are amenable to analytic treatments that are complemented by numerical solutions, and obtain the period and amplitude of the oscillations as well as the parameter domain, where oscillatory behavior persists. We show that the accuracy of these predictions is highly sensitive to the employed reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The faithful modeling of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.