standard continuation scheme in order to interpolate solutions from one point in the parameter space ( v, c$_{0}$ ) or ( v, Δ a ) to neighboring points. By extremizing the enthalpies F or G , the shape equations give only stationary solutions, not necessarily the lowest energy solutions. A numerically determined shape may correspond to an energy saddle point, maximum, or minimum. Although it is possible that the lowest energy state may not be achievable for a non-equilibrium shape change, for our purposes we will examine the minimum energy shapes, and thus a “phase diagram” for the possible shapes is of great use. Just as in a more conventional phase transition, shape transformations correspond to transitions between different symmetry states. Since we consider only axisymmetric shapes here, spherical solutions have the highest symmetry state. For small perturbations around spherical shapes, the solution can be represented as r ( s, t ) = R$_{0}$ ( 1 + ∞ ∑ ℓ =0 B$_{ℓ}$$_{0}$Y 0 ℓ ( θ ( s ) , φ = 0 , t ) , (12) where the functions Y 0 ℓ are the spherical harmonics, and the constants B$_{ℓ}$$_{0}$ can generate symmetry breaking. Because we consider only axisymmetric vesicles, only the m = 0 spherical harmonics (of the Y m ℓ ) contribute to the sum, and the angle θ is given by tan θ = r/z . While it is not possible to produce an analytical solution using this formulation, it is useful for understanding the morphological transitions in terms of symmetry breaking. For example, breaking ℓ = 2 symmetry ( B$_{20}$ = 0) leads to a prolate or oblate shape, while breaking ℓ > 2 symmetry can give more complicated shapes, such as the so-called “pear” or “stomatocyte” shapes [7]. In our numerical investigation, symmetry is frequently exploited in order to efficiently compute the equilibrium shape. In regions of multiple stability the solution branches that correspond to lowest energy shapes must be chosen, and by inserting numerically an initial symmetry breaking the algorithm used can more readily converge upon the appropriate solution. Modulation of the dimensionless parameter set ( v, c$_{0}$ ) or ( v, Δ a ) generates quasi-static defor- mations which in turn lead to motion in the surrounding fluid medium. Given that the Reynolds number is small, the dynamics of the fluid surrounding the vesicle is effectively governed by viscous dissipation and is well modeled by the incompressible Stokes equations, C. Fluid-body interaction ∇ · σ = 0 , ∇ · u = 0 , (13) 10