Spontaneous curvature model

An alternative approach is the so-called spontaneous curvature model. 1 2,183 Here, the asymmetry between the two layers is taken into account as a modification on the mean curvature. In this case, the new bending energy is derived from Eq. [27] by changing H into hi + where the new... 

A subtle refutation of the simple spontaneous curvature model without the bilayer aspect follows from the observation of vesicles of non-spherical topology, i.e. vesicles with holes (like anchor rings) or with handles. For vesicles with at least two holes or handles, the shape of lowest total energy is not unique but rather one fold continuously degenerate due to conformal invariance. This theoretical finding led to the prediction that such vesicles should permanently change their shape, a phenomenon termed conformal diffusion [21], This was indeed verified experimentally somewhat later [22]. Apart from the esthetic pleasure when visualizing conformal transformations under the microscope, this observation also shows the spontaneous curvature model to be incomplete with out the additional term. 

More recently, Smith et al. have developed another model based on spontaneous curvature.163 Their analysis is motivated by a remarkable experimental study of the elastic properties of individual helical ribbons formed in model biles. As mentioned in Section 5.2, they measure the change in pitch angle and radius for helical ribbons stretched between a rigid rod and a movable cantilever. They find that the results are inconsistent with the following set of three assumptions (a) The helix is in equilibrium, so that the number of helical turns between the contacts is free to relax, (b) The tilt direction is uniform, as will be discussed below in Section 6.3. (c) The free energy is given by the chiral model of Eq. (5). For that reason, they eliminate assumption (c) and consider an alternative model in which the curvature is favored not by a chiral asymmetry but by an asymmetry between the two sides of the bilayer membrane, that is, by a spontaneous curvature of the bilayer. With this assumption, they are able to explain the measurements of elastic properties. 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

This may not be the case when the model system is a small unilamellar vesicle in which the monolayer curvatures are significantly different for the inner and outer monolayer. If these vesicles are prepared from a mixture of lipids, the inner monolayer may be expected preferentially to contain that lipid species that has a more negative spontaneous curvature at the bilayer water interface. 

In order to explore whether insertions of the BAR dimer s N-helices can enhance membrane curvature, various penetration depths of N-helices were examined, and the results are illustrated in Figure 3. We observe larger membrane deformations upon deeper insertion of N-helices (represented in the model by increasing the local spontaneous curvature). By performing quantitative analysis on binding... 

The quantity cq is the spontaneous curvature of the membrane, which this model endows with a simple physical meaning When the imposed head area, Lq is larger than the optimal area, vo/tg, dictated by the chain packing, the preferred curvature is negative the system prefers to pack with the heads on the outside . Note that the free energy of the curved interface is lower than that of the flat interface the system accommodates part of the strain induced by the mismatch between the heads and chains by bending. 

For small curvatures, Eq. (6.15) shows that the curvature energy of a thin film is characterized by the three parameters k, k, and cq. The qualitative behavior of any system, including such properties such as the equilibrium shape, magnitude of thermal fluctuations, and any phase transitions, can of course be calculated as a function of these constants. However, the physics of the system can be radically different depending on the physical parameters e.g., a change in cq can induce shape changes in the system. It is thus of interest to relate the bending elastic moduli and the spontaneous curvature to the physics of the particular system of interest. This section first shows how these parameters are related to the pressure distribution in the membrane and then presents a simple but instructive microscopic model that relates k, and Co to more molecular properties. 

Clearly, the siuface tension has a minimum when the spontaneous curvature of the surfactant film equals the mean ciuvature of the interface. The mean curvature for a flat interface is zero, larger than zero for an interface curving towards the oil (oil-in-water emulsions), and smaller than zero for a water-in-oil emulsion. Hence, a large positive spontaneous monolayer curvature, as for a strongly hydro-philic surfactant, favors oil-in-water emulsions and vice versa. The Kabalnov-Wennerstrom model also allows the thickness of the film to vary in order to minimize the free energy of hole formation, i.e., the mean curvature of... 

As said before, when entangled worm-like micelles are formed, viscoelastic behavior only follows Cates model at low and intermediate frequencies, indicating that other fast relaxation processes exist (see Figure 12.5). Moreover, at cosurfactant-surfactant ratios above the viscosity maximum, further addition of cosurfactant decreases viscosity and viscoelastic functions up to phase separation, where lamellar liquid crystal appears. As said above, this has been related to the fact that, after the maximum in viscosity, the decrease in spontaneous curvature produces branching of micelles before phase separation [9, 10). 

In 1995, the high-density structures above the cmc were imaged directly by atomic force microscopy (AFM) for both hydrophilic and hydrophobic surfaces. These results showed for the first time that interfacial micelles existed in well-defined shapes and sizes, that they generally possessed spherical or cylindrical curvature (in contrast to the standard models), and that this curvature was a compromise between the spontaneous curvature of bulk micelles and the constraints imposed by the fiat surface. 

Yaghmur, A., Sartori, B. Rappolt, M. (2011). The role of calcium in membrane condensation and spontaneous curvature variations in model lipidic systems. Physical Chemistry Chemical Physics, 13(8), 3115-25. 

On our experimental model that signified the formation of a cylindrical membrane tube. By increasing the pressure in the outside section it could be made to collapse with the formation of two planar bilayers. Thus the cycle is completed and can be repeated endlessly. The theory of the collapse of the membrane tube made it possible to develop a new method for measuring surface tension, which is particularly promising for dry membranes. The performance of the entire cycle of fusion with membranes formed from the lipids with a different molecular geometry (i.e. different spontaneous curvature of bilayers) make it possible to prove[37] the Justice of the stalk mechanism. 

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