Curvature of membranes

In 1980 it was pointed out that the prolamellar body is a perfect example of a Cp structure [4]. (Later, more detailed, analyses have revealed that it may also be a Cd structure cf. section 7.2.) Following work on the structure of cubic phases, it was also realised that two-dimensional analogues are possible. This in turn suggested that a phase transition involving changes in the intrinsic curvature of membranes might be possible [29, 30]. Such a mechanism has far reaching implications. Clear evidence for such transitions between bilayer conformations has been reported [9]. This membrane bilayer model will be described below. 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

FIG. 22-81 Permeant -concentration profile in a pervaporation membrane. 1— Upstream side (swollen). 2—Convex curvature due to concentration-dependent permeant diffiisivity. 3—Downstream concentration gradient. 4—Exit surface of membrane, depleted of permeant, thus unswollen. (Couttesy Elseoier )... 

Baumgartner and coworkers [145,146] study lipid-protein interactions in lipid bilayers. The lipids are modeled as chains of hard spheres with heads tethered to two virtual surfaces, representing the two sides of the bilayer. Within this model, Baumgartner [145] has investigated the influence of membrane curvature on the conformations of a long embedded chain (a protein ). He predicts that the protein spontaneously localizes on the inner side of the membrane, due to the larger fluctuations of lipid density there. Sintes and Baumgartner [146] have calculated the lipid-mediated interactions between cylindrical inclusions ( proteins ). Apart from the... 

The main problem with this explanation for tubule formation arises from its prediction for the dimensions of the cylinder. In this model, the energy of a cylinder comes from two sources the curvature of the entire membrane and the edge energy of the membrane edge exposed to the solvent. These mechanisms give... 

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. 

Concerning the nature and structure of such amyloid peptide or protein channels, oligomers with annular morphologies have in fact been observed by EM for a-synuclein (Lashuel et al., 2002) and equine lysozyme (Malisauskas et al., 2003) even in the absence of any lipids or membranes. Channel-like structures have also been reconstituted in liposomes and observed by SFM for A/ i 4o, A/ j 42, human amylin, a-synuclein, ABri, ADan, and serum amyloid A (Fig. 5A Lin et al., 2001 Quist et al., 2005). Doughnut-shaped structures with a diameter of 10-12 nm and a central hole size of 1-2 nm (Fig. 5B) were imaged on top of lipid membranes (Quist et al., 2005). However, the radius of curvature of the SFM tips meant that it is not possible to say whether the pores were really traversing the lipid bilayer. 

Lipids also show asymmetrical distributions between the inner and outer leaflets of the bilayer. In the erythrocyte plasma membrane, most of the phosphatidylethanolamine and phosphatidylserine are in the inner leaflet, whereas the phosphatidylcholine and sphingomyelin are located mainly in the outer leaflet. A similar asymmetry is seen even in artificial liposomes prepared from mixtures of phospholipids. In liposomes containing a mixture of phosphatidylethanolamine and phosphatidylcholine, phosphatidylethanolamine localizes preferentially in the inner leaflet, and phosphatidylcholine in the outer. For the most part, the asymmetrical distributions of lipids probably reflect packing forces determined by the different curvatures of the inner and outer surfaces of the bilayer. By contrast, the disposition of membrane proteins reflects the mechanism of protein synthesis and insertion into the membrane. We return to this topic in chapter 29. 

Fig. 5 The snake PLA2 neurotoxin is depicted here as a snake, which binds to an active zone, i.e., a synaptic vesicle (SV) release site, and hydrolyses the phospholipids of the external layer of the presynaptic membrane (green) with formation of the inverted-cone shaped lysophospholipid (yellow) and the cone-shaped fatty acid (dark blue). Fatty acids rapidly equilibrate by trans-bilayer movement among the two layers of the presynaptic membrane. In such a way lysophospholipids, which induce a positive curvature of the membrane, are present in trans and fatty acid, which induce a negative curvature, are present also in cis, with respect to the fusion site. This membrane conformation facilitates the transition from a hemifusion intermediate to a pore. Thus, the action of the toxin promotes exocytosis of neurotransmitter (NT) (from the left to the right panel) and, for the same membrane topological reason, it inhibits the opposite process, i.e., the fission of the synaptic vesicle.

Amphiphilic compounds are also known as potent modifiers of the bilayer intrinsic radius of curvature and utilize this property to act as a non-specific perturbator of membrane protein function [27]. Catamphiphilic drugs that can interact with the head groups or with the scramblases or flippases can change cell functioning. 

Bozic and Svetina [36] analysed a different situation, where addition of membrane constituents happens from the external milieu, and there is no metabolism inside, but there is limited permeability. They supposed that the membrane assumes spontaneous membrane curvature. This is non-zero if the properties of the inside and outside solutions differ, or if the two layers of a bilayer membrane differ in composition, or if some membrane-embedded constituents are asymmetrically shaped. They were able to show that under these assumptions membrane division is possible provided TLkC4 > 1.85, where T is the time taken to double the membrane area, L is the hydraulic permeability of the membrane, k is the bending modulus, and C is the spontaneous membrane curvature. In this model growing vesicles first retain spherical shape, then are distorted to a dumbbell, then to a pair of asymmetric vesicles coupled by a narrow neck, and finally to a pair of spherical vesicles linked by a narrow neck. Separation of the two daughter vesicles occurs as a result of mechanical agitation in the solution. 

Heterogeneity of membrane constituents may also play an important role in the stabilisation of vesicles. Amphiphiles with cationic and anionic head groups can assemble into vesicles that are stable over a year [37]. This effect may be explained by assuming an asymmetric distribution of the two constituents between the two layers. Note that the two layers have curvatures of equal magnitude but opposite sign. How such an asymmetric membrane structure would be maintained through generation of protocells is not obvious, however. 

Asymmetry potential — In case of any membrane it happens that the potential drop between the solution and either inner side of the - membrane is not completely identical so that a nonzero net potential drop arises across the entire membrane. This is best known for - glass electrodes and other - ion-selective electrodes. The reasons of asymmetry potentials are chemical or physical differences between each side of a membrane, in particular an inhomogeneous membrane structure resulting from fabrication conditions and/or curvature. Asymmetry p. can change in the course of membrane ageing. To measure asymmetry p. one should use a symmetrical cell with identical solutions and -> reference electrodes on each side of the membrane. 

Curved space elements. Membranes, micelles, helices. Higher structures by curvature of lower structures... 

Consequently, larger spheres require a smaller pressure differential to maintain a given membrane tension than smaller spheres, and vice versa. The magnitudes of the pressure differential required to generate specific levels of membrane tension can be estimated from the following considerations. For a membrane with a radius of curvature = 3 pm (typical of the dimensions in a patch clamp experiment), application of 0.1 atm pressure (where 1 atm = 760 mm Hg = 1.013 x 106 dyn cm-2) corresponds to a = 15 dyn cm-1. The maximum applied pressure depends on the limit at which the patches break ( 20-30 dyn cm-1) and the curvature of the membrane, but is typically 0.2 atm. 

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