Saddle-shaped membranes

Going in the opposite direction, i.e. when we consider the membrane stability with increasing ionic strength, we notice the approach of k towards zero. Going towards this value, the tendency of the bilayers to form saddle-shaped connections (also called stalks ) between bilayers increases. Saddle-shaped membrane structures also occur in processes like vesicle fusion, endo and exocytosis. The SCF predictions thus indicate that these events will occur with more ease at high ionic strength than at very low ionic strength. 

It is now well established that proteins can induce phase transitions in lipid membranes, resulting in new structures not found in pure lipid-water systems (c/. section 5.1). However, this property is not peculiar to proteins the same effect can be induced by virtually any amphiphilic molecule. Depending on the structure and nature of proteins, their interactions with lipid bilayers can be manifested in very different ways. We may further assume that the role of proteins in the biogenesis of cubic membranes is analogous to that in condensed systems, and lipids are necessary for the formation of a cubic membrane. This assumption is supported by studies of membrane oxidation, which induce a structure-less proteinaceous mass [113]. However, the existence of a lipid bilayer by itself does not guarantee the formation of a cubic membrane, as proteins may also play an essential role in setting the membrane curvature. In this context, note that the presence of chiral components e.g. proteins) may induce saddle-shaped structures characteristic of cubic membranes. (This feature of chiral packings has been discussed briefly in section 4.14)... 

The infinite membrane without edge has complex topology. It is everywhere curved into a saddle shape in order to multiconnect with itself throughout the sample in the three dimensions of space. 

Fig. 5.23. (a) Two monolayers of negative spontaneous curvature cq. (b) When they are stuck together to form a plane bilayer, there is an antagonism between them. This antagonism can be removed by spontaneous deformation of the bilayer into a saddle shape. represents the plane of symmetry of the membrane. and 2 represent interfaces between hydrophobic interiors and polar head groups... 

FIGURE 3.10 These snacks demonstrate the geometry of two different types of membrane curvature. On the left, the bowl-shaped chip has two different positive curvatures, C, and Cj, whereas on the right, the saddle-shaped chip has a positive curvature C, and a negative curvature Cj. 

A curved liquidus surface more complex than that shown in Figure 3.8 can be conveniently visualized with the help of a (hypothetical) infinitely flexible membrane stretched between points Ta-Tb-Tc and weighted down by small point-like spheres of different density at the positions of the eutectic points. The downward deflection of the membrane caused by these balls is proportional to their density, so that larger deflections correspond to lower binary and ternary eutectic temperatures. Here, the physical model ends as the membrane must have a peculiar property of being able to bulge between points of deflection to account for the convex curvature of the Hquidus surface and also the occurrence of saddles (cols), ridges, and dome-shaped surface areas, the positions of which are defined by the AUcemade fines (see above). 

Figure 17.2 Manifestations of superstmctural aspects of DOPC bilayers, observed by cryo-TEM on surfaces of unilamellar vesicles that were prepared in pure water [26]. Left column linear isolated membrane bends or folds. They are found as deep notches as in heart-shaped vesicles and also as extended overhangs on flat membranes. (The cushion-like contours indicate the presence of passages (overpasses) on the membrane.) The constrictions on the branched tube may be envisaged as lines of saddles. Right column two-dimensional grainy membrane textures. The graininess was observed on small vesicles as well as on planar membranes and may even coexist with smooth bilayer portions.

Any curvature of a 2D membrane at a point can be defined by a combination of these parameters. If Q and Cj are both positive, the membrane will have a concave-like cup shape. If one curvature is negative, this will produce a saddle-like shape as demonstrated in Figure 3.10. In the case of a sphere, Q = Cj everywhere on the surface for a cylinder, we can let one of the principle curvatures equal zero. 

Cq is the membrane intrinsic curvature, that is, the curvature of the membrane with zero deformation (e.g., a single layer of cylindrical molecules should have an intrinsic curvature of zero, but cone-shaped molecules will pack two dimensionally to have a curvature with no energy cost, k is the bending modulus, and is known as the saddle-splay modulus. As we can see in Figure 3.10, it is quite possible to have a curved membrane with a mean curvature of zero in the case of saddle-like deformations. Therefore, it becomes clear that to describe the energy cost of a saddle deformation the extra terms are necessary. 

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