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and r , and the subscript s denotes a derivative with respect to the arclength. The vesicle shape at time t is set by Eqs. (5-8), subject to the four boundary conditions r (0 , t ) = r ( L, t ) = ψ (0 , t ) = 0 and ψ ( L, t ) = π . Once the angle ψ is determined from the above, z ( s, t ) is set by an integration of z$_{s}$ = sin( ψ ), where the constant of integration is chosen such that the center of volume of the surface ˜ x is at the origin. The vertical position z$_{0}$ ( t ) has no bearing on the vesicle shape determination, and we hold off further discussion on its dynamics until the following section. 

For the spontaneous curvature model, constraints on the unknown integration length L , the surface area A , the volume V , and the two constant Lagrange multipliers P and Σ are imposed as 

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Defining R$_{0}$ as the radius of the sphere with surface area A , the boundary conditions for the five constraint equations above are A (0) = V (0) = 0, A ( L ) = 4 πR 2 $_{0}$, V ( L ) = 4 πR 3 $_{0}$v/ 3, where v is a dimensionless “reduced volume.” Due to the Lagrange function being independent of the arc- length s , the “Hamiltonian” is a conserved quantity and we have γ (0) = 0 (see Refs. [43, 44]). Also defining a reduced spontaneous curvature c$_{0}$ = C$_{0}$R$_{0}$ , we finally obtain the vesicle morphology as set by the two parameters ( v, c$_{0}$ ). 

In the bilayer coupling model, Eqs. (5-10) are solved with two additional constraints. First, the integrated mean curvature M is controlled, M$_{s}$ = π ( rK + sin ψ ), and second, a new Lagrangian constraint enters, Q$_{s}$ = 0. The system is now closed with boundary conditions on the integrated mean curvature: M (0) = 0 and M ( L ) = 4 πR$_{0}$ Δ a , where Δ a is the reduced surface area difference between monolayers, Δ a = Δ A/ 8 πR$_{0}$h . In this case the vesicle morphology is set by the two parameters ( v, Δ a ), and the reduced spontaneous curvature c$_{0}$ has been removed from the shape equations via the Legendre transform given above. 

Equations (5-10) are solved numerically. Due to coordinate singularities in the derivatives of r and z at the poles, the shape is determined on the contracted interval s ∈ [ L δ, L (1 − δ )] for ( L δ ) ≪ 1, and Taylor-expanded versions of the boundary conditions are applied. For example, 

To compute the shapes using either model, the arc-length is discretized using m uniformly spaced grid points, s$_{i}$ , with s$_{1}$ = Lδ and s$_{m}$ = L (1 − δ ). A collocation method is then applied in a formulation and implementation similar to that recently used by Jiang et al. [44]. We employ a 

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