Gaussian curvature modulus

The resistance to bending is expressed in terms of two moduli the mean curvature modulus, kc, and the Gaussian curvature modulus. Kg, which are both defined via the energy, k, required for bending the sheet (43) ... 

Fig. 5.23, the mean curvature must be as close as possible to cq < 0, and indeed, it is symmetrically negative. When co < 0, the saddle surface is therefore preferred in comparison with the plane surface. These intuitive geometrical considerations can be re-expressed in terms of the Gaussian curvature modulus K of the bilayer. The curvature elasticity of the bilayer (Kbiiayer, biiayer and Co, bilayer = 0) Can bc simply calculated in terms of the curvature elasticity of the monolayers (i monoiayer) -f monoiayer and Cq). The problem is as simple as for a bimetallic strip. We obtain... 

Hu M, Briguglio JJ, Desemo M (2012) Determining the Gaussian curvature modulus of lipid membranes in simulations. Biophys J 102(6) 1403-1410... 

Templer RH, Khoo BJ, Seddon JM (1998) Gaussian curvature modulus of an amphiphile monolayer. Langmuir 14(26) 7427-7434... 

YatciUa et al. investigated the conversion of a mixtime of cationic and anionic surfactants to form vesicles. In this case, the reaction was veiy slow and a kinetic phase was associated with the evolution and growth of vesicles over a period of weeks (see Figure 6.12) to a final vesicular system, which, as already mentioned, was thought by the authors to be thermodynamically stable. A subsequent study explored the system CTAB mixed with sodium perfiuorooctanoate. Cylinders, disks, and spherical uni-lamellar vesicles were found to coexist at equilibrium by cryo-TEM. This observation confirms the importance of structural confirmation by cryo-TEM when this technique can be applied. Erom their analysis of the data, the mean curvature modulus, the Gaussian curvature modulus, and the spontaneous curvature could all be evaluated. 

Figure 4 The modified stalk mechanism of membrane fusion and inverted phase formation, (a) planar lamellar (La) phase bilayers (b) the stalk intermediate the stalk is cylindrically-symmetrical about the dashed vertical axis (c) the TMC (trans monolayer contact) or hemifusion structure the TMC can rupture to form a fusion pore, referred to as interlamellar attachment, ILA (d) (e) If ILAs accumulate in large numbers, they can rearrange to form Qn phases, (f) For systems close to the La/H phase boundary, TMCs can also aggregate to form H precursors and assemble Into H domains. The balance between Qn and H phase formation Is dictated by the value of the Gaussian curvature elastic modulus of the bIlayer (reproduced from (25) with permission of the Biophysical Society) The stalk in (b) is structural unit of the rhombohedral phase (b ) electron density distribution for the stalk fragment of the rhombohedral phase, along with a cartoon of a stalk with two lipid monolayers merged to form a hourglass structure (reproduced from (26) with permission of the Biophysical Society).

The modulus of Gaussian curvature has no effect on thermal undulations as its integral over a closed surface depends only on the genus of the latter. It has been determined indirectly from the shape of pierced unilamellar lipid vesicles [7] and, for monolayers, from phase equilibria between an interface and one or two bulk phases [8]. There are no measurements of the fourth order modulus c. ... 

Fig. 2 Electrical part of the modulus of Gaussian curvature versus surface charge density. The dashed and dashed-dotted lines represent the results of Debye-Huckel and Poisson-Boltzmann theory, respectively. The Debye lengths and other parameters are the same as in Fig. 1. The bilayer thickness was taken to be 4 nm. The results apply if the compensating mechanical tension resides in the interlace (and has no effect on the mechanical part of the beading rigidity). (From ref. [17].)...

As far as we are aware, the only system for which the volume fraction of the droplets, the average radius (R32), and the tension of the macroscopic oil/water interface have been measured is aWinsor II system composed of SDS, pentanol, cyclohexane, and 0.2 M NaCl with equal volumes of water and oil phases. This system was studied in Ref 55. The bending elastic modulus of this system was measured by ellipsometry [50], The results of Sec. V imply that when / and z are fixed, there are still two unknown parameters the Gaussian bending elastic modulus and the preferred curvature. Therefore, we choose to test the theory on consistency, that is, we fix z and / and fit the (R32, 4>) with Eqs. (55)-(58)—we choose... 

Here, R and R2 denote the local radii of curvature [30]. The bending modulus K, the Gaussian modulus ic, and the spontaneous curvature Co(T) are empirical material constants k and k describe the elastic energy needed to curve the interface away from its preferred curvature. According to Refs. 22 and 29, they take the values k = O.SkeT and ic = -0.4 k T. 

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