Curvature elasticity

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

Splay ( bending or curvature ) defined by a splay constant K, or by a curvature elastic modulus = K h [76],... 

Lekkerkerker, H. N. W. (1990). The electric contribution to the curvature elastic moduli of charged fluid interfaces, Physica A, 167, 384—394. 

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).

Gruner SM. Coupling between bilayer curvature elasticity and membrane-protein activity. In Biomembrane Electrochemistry, Volume 235. Blank M, Vodyanoy I, eds. 1994. American Chemical Society, Washington, DC. pp. 129-149. 

Honger T, Mortensen K, Ipsen JH, Lemmich I, Bauer R, Mouritsen OG. Anomalous swelling of multilamellar lipid bilayers in the transition region by renormalization of curvature elasticity. Phys. Rev. Lett. 1994 72 3911-3914. 

A general theory of curvature-elasticity in the molecularly uniaxial liquid crystals, similar to that of Oseen, is established on a revised basis. There are certain significant differences in particular one of his coefficients is shown to be zero in the classical liquid crystals. Another, which he did not recognize, does not interfere with the determination of the three principal coefficients. The way is therefore open for exact experimental determination of these coefficients, giving unusually direct information regarding the mutual orienting effect of molecules. 

Taking k as 10 6 dyne, and E as lO o dyne/cm, this ratio is unity when a is about 3-5 X 10 8 cm, and negligible for beams of thickness as large as a micron. Thus, on the visible scale, the curvature-elastic constants are always negligible compared with the ordinary elastic constants, unless the latter are zero. 

An application of the optical microscopy to the detennination of the curvature elastic-modulus of biological and model membranes. Journal of Physics, 48 (5). 855-867. 

Safran, S.A. andThisty, T. (1996) Curvature elasticity models ofmicroemulsions. Ber. Bunsenges. Phys. Chem., 100,252-263. 

Coupling between Bilayer Curvature Elasticity and Membrane Protein Activity... 

The purpose of this chapter is to summarize some recent developments in the physics of lipid bilayers that demonstrate the existence of curvature-elastic stresses in bilayers and to review mechanisms whereby the resultant forces may couple to membrane protein conformations (see also references 1-3 for reviews). A consequence of these forces is that membrane proteins may have mechanistic themes that are qualitatively different from themes operative in aqueous proteins. Moreover, because these forces are directed generally parallel to the membrane surface, the actual conformational motions to which the forces couple may ultimately be simpler to understand than the complex conformations of aqueous proteins. 

Electric or magnetic fields acting on the anisotropy of the electric or magnetic susceptibility exert torques within a liquid crystal which may compete with the elastic torques determining its internal structure (55). Equations w ich describe the liquid crystalline structure can be derived from molecularly uniaxial liquid crystals on the basis of the curvature-elasticity theory (54). In doing so, tl structure is determined so as to minimize the total free energy of the system, and this method is applied to the cholesteric structure (55, 55). 

Under this assumption, the bending energy Eg can be represented in terms of the membrane s curvature. For this reason. Eg is also referred to as the curvature elastic energy. The curvature of smooth surfaces is characterized by two functions that depend on the local canonical curvatures, h t) and h lr), in a surface element dS centered at r. These functions are the mean curvature, H = ( 1 + hi) , and the Gaussian curvature, K = hih2- In general, H and K change with the point r. 

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