Stochastic pH Oscillations in a Model of the Urea–Urease Reaction Confined to Lipid Vesicles

Abstract

The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.

Collective Behavior of Urease pH Clocks in Nano- and Microvesicles Controlled by Fast Ammonia Transport

“…Collective behavior has been mainly investigated with the inorganic Belousov–Zhabotinsky oscillating reaction in emulsion microdroplets or particles and vesicles. − The period of the reaction depended on the catalyst loading and the particle size, and the products diffused between compartments, synchronizing the oscillations or driving more complex responses. − In the encapsulation of more biologically relevant DNA and RNA transcriptional oscillators and protein oscillators, all the reactive species were confined to the microdroplets or vesicles; however, with urease-encapsulated vesicles, neutral acidic (CO 2 ) and basic products (NH 3 ) can diffuse into the surrounding solution. The methods for producing vesicles typically result in a distribution of sizes and enzyme content and so a variation in the pH clock time in individual vesicles might be expected in the absence of a collective response. , Theoretical work also suggested that autonomous pH oscillations may occur in urease vesicles providing there is the sufficiently fast transport of acid from the external solution ( P H+ > 10 –5 m s –1 ); to date, however, these have not been observed in experiments. − We show that the fast transport of ammonia controls the pH–time profile and synchronizes the pH change in the vesicles; here, the term synchronization is used to refer to a change in behavior (low to high pH) occurring at the same time in a heterogeneous population.…”

“…The reaction can be modeled by taking into account stochastic effects; however, in other work it was determined that population-level behavior was retained in the ODE models. 38,44 The rate of change of the concentration of a species A i in a vesicle was determined by the reaction and mass transfer rate as follows: 45−47…”

“… 34 , 35 Theoretical work also suggested that autonomous pH oscillations may occur in urease vesicles providing there is the sufficiently fast transport of acid from the external solution ( P H+ > 10 –5 m s –1 ); to date, however, these have not been observed in experiments. 36 − 38 We show that the fast transport of ammonia controls the pH–time profile and synchronizes the pH change in the vesicles; here, the term synchronization is used to refer to a change in behavior (low to high pH) occurring at the same time in a heterogeneous population.…”

“…Hence, some insight can be gained from simulations with a simple ODE model of the vesicles and the external solution (SI 3.4). The reaction can be modeled by taking into account stochastic effects; however, in other work it was determined that population-level behavior was retained in the ODE models. , The rate of change of the concentration of a species A i in a vesicle was determined by the reaction and mass transfer rate as follows: − where f ( A i ) contains the enzyme reaction and solution equilibria terms (SI 3.1–3.2) taken from earlier work, ,, A 0 is the concentration of the species in the outer solution, P i is the permeability coefficient of species i, and r is the radius of the vesicle. In the outer solution, the rate of change of the concentration of each species, A 0 , was given by the reaction rate ( g ( A 0 )) and the mass transfer rate (for identical vesicles) as follows: where ϕ = NV j / V 0 = the vesicle volume fraction, which takes into account dilution as a result of the volume change from vesicle to solution.…”

A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures

“…Mathematical modeling and numerical simulation of reaction-diffusion processes is a research topic of steady interest. Application areas include many kinds of cellular processes like gene expression [IP06,WS16], neurotransmission [ESSW22] or enzyme kinetics [SWSH21], but also social dynamics of interacting agents such as innovation spreading or epidemics within a human population [HDCD + 21, WZSDC21]. While, on the level of spatially well-mixed kinetics, the convergence of the stochastic jump process (characterized by the chemical master equation [Gil92]) to the corresponding deterministic limit given by an ordinary differential equation is fully understood [Kur70], an analogue analysis of the spatially resolved setting is not yet completed.…”