We have studied an enzymatic pH oscillator based on the urea–urease reaction confined to a lipid vesicle. 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 have analysed the structure of the limit cycle, which controls the dynamics for giant vesicles and dominates the strongly stochastic oscillations in lipid nanoparticles. We have derived reduced two-variable models, amenable to analytic treatments, and have shown that the accuracy of predictions, including the period of oscillations, is highly sensitive to the choice of the reduction scheme. The accurate description of a single pH oscillator is crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.

Main results

Starting from a four-variable model, we have reduced it to a new accurate two-variable model. Elimination of one product variable can be done reliably within a standard quasi-steady state approximation (QSSA). The latter presents a constraint that is in contradiction with the analogous QSSA for another product variable. Instead, we suggest a virtually exact reduction resulting in a simple two-variable model. Further, we have shown that

• the constraint explains why the structure of the limit cycle is best unveiled in terms of logarithmic scales, in contrast to conventional oscillator models; 
• the naive QSSA elimination of variables results in degenerate behaviour in the base region of the phase plane and it predicts a qualitatively incorrect form of the limit cycle;
• the water ionisation is irrelavant for the oscillation dynamics and accounting for it explicity does not change the dynamcis.

Finally, we have performed a stability analysis of the fixed point that determines the existence of oscillatory regimes for experimentally relevant parameter ranges.

Publication:
A. V. Straube, S. Winkelmann, and F. Höfling,
Accurate reduced models for the pH oscillations in the urea−urease reaction confined to giant lipid vesicles,
J. Phys. Chem. B 127, 2955 (2023).