Bending models of lipid bilayer membranes: Spontaneous curvature and area-difference elasticity

Abstract

We preset a computational study of bending models for the curvature elasticity of lipid bilayer membranes that are relevant for simulations of vesicles and red blood cells. We compute bending energy and forces on triangulated meshes and evaluate and extend four well established schemes for their approximation: Kantor and Nelson [1], Jülicher [2], Gompper and Kroll [3] and Meyer et. al. [4], termed A, B, C, D. We present a comparative study of these four schemes on the minimal bending model and propose extensions for schemes B, C and D. These extensions incorporate the reference state and non-local energy to account for the spontaneous curvature, bilayer coupling, and area-difference elasticity models. Our results indicate that the proposed extensions enhance the models to account for shape transformation including budding/vesiculation as well as for non-axisymmetric shapes. We find that the extended scheme B is superior to the rest in terms of accuracy, and robustness as well as simplicity of implementation. We demonstrate the capabilities of this scheme on several benchmark problems including the budding-vesiculating process and the reproduction of the phase diagram of vesicles.

Buckling, crumpling, and tumbling of semiflexible sheets in simple shear flow

“…where κ is the bending rigidity andn i andn j are (consistently oriented) triangle normals 55,[57][58][59] . Note that such a model of bending may not be generally applicable for capturing bending forces of real materials (e.g., those that are nonlinear or plastically deform) or other models (e.g., the Helfrich-Canham model) at large curvatures.…”

Effect of the Membrane Composition of Giant Unilamellar Vesicles on Their Budding Probability: A Trade-Off between Elasticity and Preferred Area Difference

“…In the area difference elasticity model [ 40 , 43 ] a vesicle is considered as an elastic membrane and its equilibrium shape can be found with the minimization of the Helfrich energy [ 48 , 49 ] ( W ). In this model not just the local principal curvatures ( and ) can differ from their equilibrium value, i.e., from the local spontaneous curvature ( ), but an elastic membrane is taken into account, so the area difference between the inner and outer leaflets ( ) can be different from its equilibrium value (from the preferred area difference ) as well.…”

“…In particular, the urea–urease enzymatic reaction [ 32 , 33 , 34 ] inside the lumen of the vesicles, coupled to the cross-membrane transport of urea, can trigger the shape deformation of the pH-sensitive mixed membrane by promoting the solubilization of oleate molecules in the inner water phase. In the framework of the Area Difference Elasticity (ADE) theory [ 23 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 ] we demonstrated that the shape deformation is driven by the synergic action of osmosis, acting on the volume of the vesicle, and pH change, the latter responsible for the increase of the preferred area difference of the bilayer leaflets. However, the ADE model predicts that the equilibrium shape of an elastic membrane undergoing a transformation, also depends on a third factor: the mechanical properties of the bilayer accounted by the ratio of the local (intrinsic fluidity of the membrane) and the non-local (tension generated by the different curvature of the two leaflets) bending moduli.…”

Shape changes and budding of giant vesicles induced by an internal chemical trigger: an interplay between osmosis and pH change

“…The software Surface Evolver 2.70 42 was used to compute the relaxed vesicle shape by minimizing the energy functional W, i.e. the Helfrich energy, 43,44 complemented with a term that, according to the ADE theory, accounts for the area difference between the outer and inner leaflets of the lipid membrane, 45,46 expressed by…”