For parameter variation significantly slower than the membrane relaxation rate, i.e. for a cycle time scale t$_{cycle}$ ≫ t$_{rel}$ , then we operate safely within the decoupled regime. In this case, we may thus assume that there are no hydrodynamically induced shape changes, and that the shapes are determined quasi-statically in equilibrium. Using this time scale t$_{rel}$ , we can also set a maximum swimming velocity scale, U$_{c}$ = κ/µr 2 $_{0}$. Similar scaling arguments have been made in Refs. [35, 36]. For biologically relevant systems in water, κ ≈ 100 k$_{B}$T, µ ≈ 10 − $^{3}$Pa s, ρ = 1 g/cm $^{3}$, and r$_{0}$ ≲ 1 − 10 µ m, leading to L$_{c}$ ≈ 1 − 10 µ m, U$_{c}$ ≈ 1 − 10 µ m/s, t$_{rel}$ ≈ 0 . 01 − 1 s and Re ≈ 10 − $^{4}$. For a vesicle with length scale L$_{c}$ = 10 µm , diffusive time scales are approximately 10 $^{4}$s , and thus negligible for the time being. In addition, we neglect thermal fluctuations in the determination of the vesicle shape, as they come in as a perturbation about the mean equilibrium shape of order ( k$_{B}$T/κ ) $^{2}$, which is very small under most conditions [7]. 

There are a number of different physical means by which a vesicle shape can be changed in a controlled fashion, and the methods could be different depending on the type of vesicle considered. We will consider two such means, internal volume changes and local membrane compositional changes. 

One experimentally feasible example of a possible volume-changing mechanism is a light-induced osmotic change. In an ordinary biological membrane the bilayer is embedded with numerous proteins, many of which are sensitive to mechanical forces, chemical gradients, or light. The protein bacteriorhodopsin, for example, is sensitive to green light, and in response to a signal the protein opens and closes like a valve [37]. The presence of such ion channels or active proteins on the surface of a membrane can cause osmotic changes of the fluid volume contained within the vesicle [19]. Recently, vesicle volume control was demonstrated via pH modulation of block copolymer networks along the surface of membrane [38]. The vesicles in this study were well separated from regimes associated with morphological transition, and thus changes in osmotic pressure induced only a volume change, leading to a “breathing” vesicle. 

Adjusting the membrane composition requires a more indirect experimental approach. Some bilayers are composed of different species of constituent parts, leading to an inherent mismatch between the intrinsic curvatures. In other words, there is an intrinsic curvature that would develop across the bilayer in the absence of other considerations. Because of the inherent difficulty in measuring these quantities it is likely to be more difficult to specify an exact change from one value of intrinsic curvature to another. However, the actual process of changing the intrinsic curvature can be achieved through inducing chemical changes of the lipid constituents of the membrane [38], or by conformational changes of polymers grafted to the surface of the vesicle [39]. 

