Engineering Enzyme-Driven Dynamic Behaviour in Lipid Vesicles

Abstract

The urea–urease system is a pH dependent enzymatic reaction that was proposed as a convenient model to study pH oscillations in vitro; here, in order to determine the best conditions for oscillations, a two-variable model is used in which acid and substrate, urea, are supplied at rates $k_h$ and $k_s$ from an external medium to an enzyme-containing compartment. Oscillations were observed between pH 4 and 8. Thus the reaction appears a good candidate for the observation of oscillations in experiments, providing the necessary condition that $k_h > k_s$ is met. In order to match these conditions, we devised an experimental system where we can ensure the fast transport of acid to the encapsulated urease, compared to that of urea. In particular, by means of the droplet transfer method, we encapsulate the enzyme, together with a suitable pH indicator, in a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC) lipid membrane, where differential diffusion of H$^+$ and urea is ensured by the different permeability ($P_m$) of membranes to the two species. Here we present preliminary tests for the stability of the enzymatic reaction in the presence of lipids and also the successful encapsulation of the enzyme into lipid vesicles.

Appendix

Introducing $\mathrm {s}=\mathrm {[S]}/\mathrm {K_m}$, $\mathrm {h}=\mathrm {[H^+]}/\mathrm {K_{ES1}}$, $\tau =t\mathrm {k_{E}}\mathrm {[E]}/\mathrm {K_m}$, $\kappa =\mathrm {K_m}/\mathrm {K_{ES1}}$, $\kappa _{\mathrm {es}}=\mathrm {K_{ES2}}/\mathrm {K_{ES1}}$, $\kappa _{\mathrm {w}}=\mathrm {K_{w}}/\mathrm {K_{ES1}^2}$, $\kappa _{\mathrm {s}}=\mathrm {k_{S}}\mathrm {K_m}/(\mathrm {k_{E}}\mathrm {[E]})$ and $\kappa _{\mathrm {h}}=\mathrm {k_{H}}\mathrm {K_m}/(\mathrm {k_{E}}\mathrm {[E]})$ Eq. (2) become

$$\begin{aligned} \begin{aligned} \frac{\partial \mathrm {s}}{\partial \tau }&=\kappa _{\mathrm {s}}(\mathrm {s}_0-\mathrm {s})-\mathrm {r}\\ \frac{\partial \mathrm {h}}{\partial \tau }&=\left( \kappa _{\mathrm {h}}\left( \mathrm {h}_0-\dfrac{\kappa _{\mathrm {w}}}{\mathrm {h}_0}-\mathrm {h}+\dfrac{\kappa _{\mathrm {w}}}{\mathrm {h}}\right) -2\kappa \mathrm {r}\right) \left( 1+\dfrac{\kappa _{\mathrm {w}}}{\mathrm {h}^2}\right) ^{-1} \end{aligned} \end{aligned}$$

(A.1)

where

$$ \mathrm {r}=\dfrac{\mathrm {s}}{(1+\mathrm {s})\left( 1+\kappa _{\mathrm {es}}/\mathrm {h}+\mathrm {h}\right) }. $$