Hyperbolic bilayers

In glycerol monooleate/decane bilayers we find the steady-state conductance at zero current to be proportional to the first power of the ion concentration and to the second power of the ionophore concentration, as illustrated in Fig. 1. (The current-voltage characteristic is hyperbolic for all ionic species indicating that this molecule is in the equilibrium domain for the interfacial reactions, with the rate-limiting step being the ion translocation across the membrane interior.) The conductance selectivity sequence is seen to be Na>K>Rb>Cs, Li. 

The new large-aperture MCM-41 family of zeolites mentioned earlier have FD s far below those expected from this analysis. This is due to the novel structural type of these frameworks. They consist of a silica bilayer wrapped onto hyperbolic surfaces (or cylinders), rather than the hyperbolic silica monolayers characteristic of conventional zeolites. The structures of MCM-41 zeolites are closer to those of the double-layer sheet silicates [20] than zeolites. In these cases, the connectivity within each layer is three (the fourth link fuses the two layers), and pores of infinite radius are apparently realisable, e.g. CaAl2Si208, which is a flat bilayer ... 

We have already argued that a self-assembled bilayer composed of equivalent surfactant "blocks" should form a homogeneous minimal surface, tracing the mid-surface of the bilayer. Within the constraints of this simple geometric model, we thus expect the formation of hyperbolic bUayers, wrapped onto three-periodic minimal surfaces of genus three or four per unit cell. 

The details of the calculations for both reversed and normal bilayers (for which the tunnels are filled with water and surfactant respectively) are given elsewhere [7-9]. We characterise the concentration of the surfactant by the volume fraction of the hydrophobic region, chains- The relation between composition and molecular shape for hyperbolic bilayers is ... 

A similar relation can be derived between the local and global geometry of hyperbolic reversed bilayers, for which v/al varies between and... 

The regions within the local/global "phase diagram" for which these hyperbolic bilayer structures can be realised within a surfactant-water mixture are plotted in Fig. 4.8. 

In the previous section, it has been shown that a surfactant bilayer is constrained to adopt a hyperbolic (or planar) geometry if the constituent monolayers have identical molecular shape (characterised by the surfactant parameter). In the case of monolayers, all three geometries - elliptic. 

The resulting local/global "phase diagram" for elliptic, parabolic and hyperbolic monolayers, together with bilayers (flat, as in the classical lamellar La phase, and hyperbolic) is shown in fig. 4.11. To generate specific data, die value of the tail length, I, is set to 14A, which is characteristic of a molten 12-carbon tail, for which the fully stretched length is about IbA. 

Thus, these mesostructures are predominantly lamellar, and identified as conventional (parabolic) lamellar phases, although they may in fact be hyperbolic. Indeed, unless v/al is exactly unity, a planar interface (lamellar mesophase) incurs a bending energy cost hyperbolic sponge monolayers or bilayers or mesh monolayer mesophases are favoured if v/al differs from unity. It is likely then that many "lamellar"" phases in fact adopt a hyperbolic geometry. Careful neutron-scattering studies of a lamellar phase have revealed the presence of a large number of hyperbolic "defects" (pores within the bilayers) in one case [16]. (An example of this mis-identification of hyperbolic phases in block copolymers is discussed in section 4.10.)... 

Detailed data are available for some cubic phases, which offer a good test of the model of hyperbolic bilayers wrapped onto IPMS. In these cases, the results suggest that these phases consist of bilayers of cubic S3onmetry, whose mid-surfaces closely follow IPMS [25,26]. 

The relative stability of mesh and IPMS structures is still unclear. For example, the Ri mesophase (of rhombohedral symmetry) in the SDS-water system transforms continuously into the neighbouring bicontinuous cubic phase (Fig. 4.14) [20]. This suggests that this mesophase is a hyperbolic (reversed) bilayer Ijring on a rhombohedral IPMS. Indeed, the rhombohedral rPD surface is only marginally less homogeneous than its cubic counterparts, the P- and D-svu-faces. 

It is clear from the universal diagram (Fig. 4.11) that a variety of bilayer phases can form only if die surfactant parameter is between about 0.5 and 1.7. For higher values of the surfactant parameter, steric constraints e.g. head-group crowding) preclude the formation of curved hyperbolic bilayers or monolayers. The opportunity for bilayer pol)rmorphism exists for surfactant parameters l3dng between 1.0 and about 1.5. Iliese bilayer phases are expected to adopt cubic or rhombohedral symmetries, corresponding to the most homogeneous three-periodic minimal surfaces. 

This (local) double twist configuration clearly involves a hyperbolic deformation of the imaginary layers. In contrast to the hyperbolic layers found in bicontinuous bilayer lyotropic mesophases, the molecules within these chiral thermotropic mesophases are oriented parallel to the layers, to achieve nonzero average twist. The magnitude of this twist is deternuned by the direction along which the molecules lie (relative to the principal directions on the surface), and a function of the local curvatures of the layers (K1-K2), cf. eq. 1.4. Just as the molecular shape of (achiral) surfactant molecules determines the membrane curvatures, the chirality of these molecules induces a preferred curvature-orientation relation, via the geodesic torsion of the layer. 

This membrane fusion process (outside the brain) is known to involve thousands of single membrane units, previously thought of as vesicles, assembled into units that have been termed "boutons". We have examined the EM texture of the boutons and found that they are in fact a cubic phase. The synaptic signal transmission can take place as frequently as hundreds of times per second. A fusion process involving a hyperbolic membrane can be well controlled, and the calcium ion influx - which induces fusion - is expected to change the conformation of the cubosome surface membrane from its planar bilayer conformation to the fusogenic saddle-saddle conformation. (It is known phase transitions of membrane lipids can occur when exposed to calcium, e.g. [40]). 

We propose that these triglycerides-enriched domains consist of folded bilayer units in the form of a hyperbolic bilayer. Evidence for such a structure comes from lipid phase studies of transitions from the La phase to a cubic phase accompan3dng solubilisation of triglycerides into the La phase [56]. A h3 erbolic bilayer domain of this type is consistent with the high-resolution NMR signals. 

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