Physically, η$_{H}$ measures the proportion of work done by the vesicle against the surrounding fluid which is used for swimming purposes, and is typically on the order of 1% for biological cells. Note that the swimming efficiency only measures the hydrodynamic efficiency, not a total efficiency. For example, the bending energy of the vesicle is not captured in this measure. The inclusion of bending costs into swimming efficiency measures has recently been proposed to study optimal locomotion strategies in flagellated cells, but presents an avenue of inquiry beyond the scope of this paper [46]. As stated in the introduction, due to the linearity and time-reversibility of Eqs. (13), any time-reversible geometrical surface deformations cannot result in a net locomotion. This result is known as the Scallop theorem, in reference to the sole, time-reversible motions available to a small scallop (opening and closing) [1]. As a consequence of this constraint, a single degree of freedom is insufficient for swimming. Two degrees of freedom are however sufficient to generate a swimming motion, as first described in Ref. [1], and as we shall show presently for the systems of interest. We begin by presenting a characteristic shape cycle that can be generated by adjusting the reduced volume and spontaneous curvature, ( v, c$_{0}$ ), in a periodic fashion. By selecting a specific elliptical path in the ( v, c$_{0}$ ) parameter space, namely v ( t ) = 0 . 425 + 0 . 125 cos(2 πt ), c$_{0}$ ( t ) = − 0 . 1 + 0 . 3 sin(2 πt ), the resulting shape cycle is not time-reversible; hence, the constraints of the Scallop theorem are bypassed, and locomotion may be achieved. For these parameters the vesicle shapes are always stomatocytes, and the neck separating the internal sphere of fluid from the external fluid is very small. Figure 3 shows the corresponding minimal energy vesicle shapes at four times, along with the vorticity generated in the surrounding fluid by the body deformation, ω = ∇ × u . Positive vorticity, corresponding to counter-clockwise rotation, is shown in red, and negative vorticity, corresponding to clockwise rotation, is shown in blue. Hollow arrows indicate the instantaneous swimming velocity of the vesicle, while the plain arrows indicate the direction of time. At zero Reynolds number the swimming velocity, external flow, and swimming efficiency are determined uniquely by the time-dependent surface geometry and surface deformation velocity, so we need not consider the internal flow dynamics (which may in general depend upon the means of modulating IV. VESICLE LOCOMOTION BY SHAPE-CHANGE A. Spontaneous curvature model 12