The enthalpy functional, F , in the spontaneous curvature model takes the following form [43] where C$_{1}$ and C$_{2}$ are the principal membrane curvatures, and Σ and P are Lagrange multipliers which constrain the surface area A and volume V (physically they correspond to the membrane tension and pressure difference across the interface). In Eq. (1), S ( t ) denotes the time-dependent surface boundary, and C$_{0}$ is the spontaneous curvature, which introduces an inherent mismatch in equilibrium preference of the membrane curvature. This quantity along with a fixed volume and surface area completely specifies the ensemble. Thus the spontaneous curvature model has area, volume, and integrated spontaneous curvature constrained, and we select as the control parameters the volume V and the spontaneous curvature C$_{0}$ (the fixed surface area merely selects the overall size of the vesicle). In contrast, in the bilayer coupling model, the enthalpy functional G assumes the area difference Δ A between the membrane monolayers to be constant. One possible representation of this area difference is in terms of the integrated mean curvature, Then the area difference is Δ A = 2 hM + O ( h $^{2}$/A ), where h is the distance between monolayers [43]. The enthalpy then takes the form where Σ $^{′}$, P , and Q are Lagrange multipliers associated with A (area), V (volume) and M (in- tegrated mean curvature) respectively. We select as control parameters the volume V and the integrated mean curvature M . It is important to note that the functionals F and G are related via a Legendre transform, (Σ $^{′}$,Q ) → ( Σ + κC 2 $_{0}$/ 2 , − 2 κC$_{0}$ ) , and thus describe the same system in a different ensemble. Phys- ically, the spontaneous curvature model corresponds to a bilayer in which the monolayer admits stretching or compression during bending, and thus finds an equilibrium distribution that has a preferred curvature. If the bilayer is composed of more than one species of lipid, each of which has a different preferred curvature ( i.e. radius of gyration), it is likely that the membrane will actually prefer to be in a non-flat state. Conversely, the bilayer coupling model corresponds to a system that enforces that both monolayers are incompressible. The area difference between monolayers stays approximately constant on the timescales relevant to our consideration, and as long as the G = κ 2 ∫ S ( t ) ( C$_{1}$ + C$_{2}$ ) $^{2}$dS + Σ $^{′}$A + P V + QM, (3) M = ∫ S ( t ) ( C$_{1}$ + C$_{2}$ ) dS. (2) 7 F = κ 2 ∫ S ( t ) ( C$_{1}$ + C$_{2}$ − C$_{0}$ ) $^{2}$dS + Σ A + P V, (1)