ψ r z GLYPH<246> n s ( r ( s ) ↪ z ( s )) GLYPH<246> t ψ FIG. 2: Parameterization o f an axisymmetric bilayer vesicle. We assume axisymmet ry about the z-axis. The surface is described by x = ( r ( s, t ) , z ( s, t )) in cylindrical coordinates, with s an arc-length parameter, ˆ t the unit tangent vector, ˆ n the outward pointing normal vector, and ψ the angle between the x-axis and ˆ t . distance between layers remains very small this implies that the integrated mean curvature also remains constant. Assuming an axisymmetric vesicle shape, the body surface S ( t ) is parameterized at each time t as illustrated in Fig. 2. The arc-length measured along the surface in the x - z plane is denoted by s ∈ [0 , L ], with ˆ t the unit tangent vector, ˆ n the outward pointing normal vector, and ψ the angle between the x-axis and ˆ t . The body surface is represented in cylindrical polar coordinates, where φ ∈ [0 , 2 π ) is the azimuthal angle, the surface ˜ x is taken to have its center of volume at the origin, and z$_{0}$ ( t ) is a translation of that center of volume which depends upon the fluid in- teraction. Under this parameterization, the principal membrane curvatures are C$_{1}$ = ∂ψ/∂s and C$_{2}$ = sin ψ/r . Upon insertion into either of the enthalpy functionals F or G , and performing a vari- ational extremization, we obtain the following system of first-order ordinary differential equations to describe the energetically stationary vesicle shapes at time t [43] B. Determination of the vesicle shape x ( s, φ, t ) = ˜ x ( s, φ, t ) + z$_{0}$ ( t ) ˆ z = ( r ( s, t ) cos( φ ) , r ( s, t ) sin( φ ) , z ( s, t ) + z$_{0}$ ( t )) , (4) ψ$_{s}$ = K, (5) K$_{s}$ = − K r cos ψ + γ r sin ψ + cos ψ sin ψ r 2 + 1 2 P r cos ψ, (6) γ$_{s}$ = ( K − C$_{0}$ ) 2 2 − sin 2 ψ 2 r 2 + P r sin ψ + Σ , (7) r$_{s}$ = cos ψ. (8) Here K is an auxiliary function used to make the system of equations first-order (physically it corresponds to the curvature), γ is the Lagrange multiplier that enforces the interdependence of ψ 8