Gaussian curvature and the equilibrium among bilayer cylinders, spheres, and discs

Abstract

In mixtures of cetyltrimethylammonium bromide (CTAB) and sodium perfluorooctanoate (FC 7) in aqueous solution, novel bilayer cylinders with hemispherical end caps and open, flat discs coexist with spherical unilamellar vesicles, apparently at equilibrium. Such equilibrium among bilayer cylinders, spheres, and discs is only possible for systems with a spontaneous curvature, R o, and a positive Gaussian curvature modulus, . We have measured the size distributions of the spherical vesicles, cylinders, and discs by using cryo-electron microscopy; a simple analysis of this length distribution allows us to independently determine that the mean curvature modulus, Ϸ 5 ؎ 1 kBT and Ϸ 2 ؎ 1 kBT. This is one of the few situations in which R o, , and can be determined from the same experiment. From a similar analysis of the disk size distribution, we find that the edges of the discs are likely stabilized by excess CTAB. The fraction of discs, spherical vesicles, and cylinders depends on the CTAB͞FC 7 mole ratio: increasing CTAB favors discs, while decreasing CTAB favors cylinders. This control over aggregate shape with surfactant concentration may be useful for the design of templates for polymerization, mesoporous silicates, etc.cryogenic transmission electron microscopy ͉ surfactants ͉ vesicles T he starting point for the description of bilayer organization in solution is the harmonic approximation to the bending free energy (1):R 1 and R 2 are the principle radii of curvature of the structures, R o is the spontaneous radius of curvature, and are the mean and Gaussian curvature elastic constants, respectively, and A is the area of the bilayer membrane. The harmonic approximation is appropriate when the membrane thickness (1-3) [here Ϸ3 nm (4)] and the Debye length for ionic surfactants (2, 5, 6) [also Ϸ1-3 nm (4)] are small compared with R 1 and R 2 (Ϸ20-30 nm, see Fig. 1). The differences of the bending free energy, F B , of different aggregate geometries can often be of the order of k B T, leading to the possibility of multiple structures in equilibrium.The two elastic constants, and , play very different roles in determining bilayer organization. The magnitude of reflects the energy needed to bend the bilayer away from its spontaneous radius of curvature, R o . For ϳ k B T, thermal fluctuations give rise to significant curvature fluctuations, which lead to a net repulsive interaction between bilayers at short distances. This steric repulsion can stabilize unilamellar vesicles over multilamellar liposomes (3,4,7,8). Larger values of (ӷk B T), combined with a spontaneous curvature that picks out a particular vesicle radius, lead to unilamellar vesicles as the curvature variations inherent to multilamellar structures are energetically prohibited (3,4,9). A spontaneous bilayer curvature (1͞R o 0) is only possible when nonideal surfactant mixing causes the interior and exterior monolayers of the vesicle bilayer to have different compositions or environments (3, 10, 11).influences only the topology (and hence the number) of t...

Phase Behavior and Microstructure in Aqueous Mixtures of Cationic and Anionic Surfactants

“…Work in our laboratories has demonstrated the existence of spontaneous, apparently equilibrium vesicles with both narrow [25][26][27] and broad size distributions [6,25,[28][29][30][31][32] in aqueous mixtures of a wide range of surfactants. The most common cationic surfactant used has been an alkytrimethylammonium bromide or tosylate, e.g., cetyltrimethylammonium bromide (CTAB) or tosylate (CTAT) as well as cetylpryidinium chloride (CPCl), while the common anionic surfactants are the sodium alkylsulfates, e.g., sodium octyl (SOS) or decyl sulfate (SDS), or dodecylbenzene sulfonate (SDBS), which may have either a branched or comb structure [6,33].…”

“…To distinguish between these two mechanisms of vesicle stability, the bending elasticity and bilayer spontaneous curvature has been measured for a number of different systems of spontaneous, unilamellar vesicles [26,27,32] by analysis of the vesicle size distribution using cryo-microscopy and freezefracture replication. The size distribution is calculated as follows.…”

“…For " > 0, infinite films are unstable towards formation of a surface with many handles and in this case cubic or sponge-like phases are predicted to be the stable bilayer configuration. For sufficiently dilute systems with finite bilayer fragments, vesicles are still possible [27] as systems with only negative Gaussian curvature cannot close on themselves, which results in bilayers with energetically unfavorable edges in contact with water. The equilibrium size distribution of a population of vesicles is determined by a subtle competition between the entropy of mixing and the curvature elasticity of the bilayers.…”

“…4380-Abe elastic constant, and the formation of two-bilayer vesicles shows that the CTAB/FC 7 vesicles are stabilized by spontaneous curvature. Interesting other shapes (rods and disks) can also be analyzed quantitatively [27].…”

Templated growth of platinum nanowheels using the inhomogeneous reaction environment of bicelles

“…[2][3][4][5] A great deal of attention has been paid to the alignment of bicelles in an external magnetic field and their use as biomimetic membrane models in biophysical and biochemical studies, especially for reconstitution of membrane proteins. [6][7][8][9][10] From the materials synthesis point of view, bicelles can offer a unique inhomogeneous reaction environment in which the central flat bilayer region and the peripheral curved region coexist within a single molecular assembly.…”