On the equivalence of local and global area-constraint formulations for lipid bilayer vesicles

Abstract

Abstract. Lipid bilayer membranes are commonly modeled as area-preserving fluid surfaces that resist bending. There appear to be two schools of thought in the literature concerning the actual area constraint. In some works the total or global area (GA) of the vesicle is a prescribed constant, while in others the local area ratio is assigned to unity. In this work we demonstrate the equivalence of these ostensibly distinct approaches in the specific case when the equilibrium configuration is a smooth, closed surface of genus zero. We accomplish this in the context of the Euler-Lagrange equilibrium equations, constraint equations and the second-variation with admissibility conditions, for a broad class of models -including the phase-field type.

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“…We adopt the conventional assumption [Evans and Skalak 1980] that deformations of the bilayer/cytoskeleton system conserve local surface area. This assumption is invoked for both the bilayer and cytoskeleton separately.…”

“…The presumed inextensibility of the lipids -expressed by the condition |n| = 1 -then implies areal incompressibility. For the cytoskeleton it is justified by empirical evidence [Evans and Skalak 1980] indicating that areal compressibility of the bilayer/cytoskeleton system is typically negligible; areal incompressibility, in the case of a convecting cytoskeleton, then follows from that of the bilayer. Here we impose areal incompressibility of the cytoskeleton whether or not it convects with the bilayer (for a discussion of this issue, see [Krishnaswamy 1996]).…”

“…According to prevailing opinion [Evans and Skalak 1980;Tartibi et al 2015], the cytoskeletal membrane exhibits response that is characteristic of an isotropic Figure 2. Hexagonal substructure of the cytoskeletal network.…”

“…Thus, the view expressed in the literature is consistent with the substructure of the cytoskeletal network if terms through quadratic order in are retained in the strain-energy function. Indeed, quadratic-order energies figure prominently in Evans' and Skalak's extensive treatment [Evans and Skalak 1980] of cytoskeletal membranes in which isotropy is assumed at the outset.…”

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

“…In our recent work [10], we also consider the model (2.1) with (2.2)1 replaced by the localarea constraint 1, J ≡ i.e., the sixth term in (2.6) is replaced by…”