where σ = − p I + 2 µ E is the Newtonian stress tensor with p the pressure, u the fluid velocity, and E the symmetric rate-of-strain tensor, E = 1 $_{2}$( ∇ u + ( ∇ u ) $^{T}$). The fluid equations are made dimensionless by scaling velocities upon U$_{c}$ , lengths upon L$_{c}$ , and time upon t$_{rel}$ = L$_{c}$/U$_{c}$ . Since the surface area A = 4 πR 2 0 is constant, we define the characteristic length scale by this radius, i.e. L$_{c}$ = R$_{0}$ . Henceforth, the swimming velocity is understood to be dimensionless, and each shape cycle occurs over a unit in dimensionless time. A no-slip condition is applied on the body surface. For a given path through the parameter space ( v, c$_{0}$ ) or ( v, Δ a ), the resulting sequence of instantaneously determined shapes set uniquely the “surface deformation velocity” u d ( x , t ); namely, u d ( x ( s, φ, t ) , t ) = ∂ ˜ x ∂t ( s, φ, t ) . (14) In addition, the surface moves as a rigid body along the ˆ z direction due to axisymmetry, with velocity U = U ˆ z = z ′ $_{0}$( t ) ˆ z . The no-slip condition is thus written as u ( x , t ) = U ˆ z + u d ( x , t ). To close the system of equations describing the fluid-body interaction, we assume that no external forces are acting upon the vesicle, and thus force and torque balance give ∫ S ( t ) σ ( x ) · ˆ n ( x ) dS = 0 , ∫ S ( t ) x × [ σ ( x ) · ˆ n ( x )] dS = 0 . (15) The computation of the swimming velocity is performed using a standard double-layer boundary integral formulation of the Stokes equations. The details of this formulation and numerical method are presented in the appendix. In addition to computing the swimming velocity, we consider a possibly more important quan- tity, the hydrodynamic efficiency. This swimming efficiency is defined as (see Ref. [45]) η$_{H}$ = 〈 U · F 〉 〈 ∫ S ( t ) ( U + u d ) · f dS 〉 = 〈 U · F 〉 〈 ∫ S ( t ) u d · f dS 〉 , (16) where f = − σ · ˆ n is the force density acting on the fluid at the body surface, 〈·〉 denotes a time- average over a full shape cycle, and F = 6 πµ a U ˆ z is the force required to move a sphere of radius a at a speed U . At each time we use the maximum vesicle radius, a ( t ) = ‖ r ( s, t ) ‖$_{∞}$ . The first term in the denominator of Eq. (16) integrates to zero due to the zero-net force condition (Eq. (15)). The computation of the fluid stress σ is significantly more involved than the computation of the swimming velocity. We employ a numerical method for computing σ based on the evaluation of a hypersingular integral which may be derived from the double-layer formulation of the fluid velocity. The framework and numerical approach are described in the appendix, and a more detailed description of the method and examples of its use will be featured in a subsequent paper. 11