Bilayer curvature elasticity, membrane

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. 

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

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 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. 

The experiments by Koltover et al. claim that isotropic colloids on membranes experience a surface-mediated (presumably, curvature-mediated) attraction. All theories we have discussed so far claim that the force is repulsive, unless one goes to large detachment angles. Can simulaticMis shed more light onto the problem If so, it will not be necessary to represent the bilayer in any greater detail because only fluid curvature elasticity needs to be captured. 

In order to simulate larger systems, such as giant unilamellar vesicles (GUV) or red blood cells, which have a radius on the order of several micrometers, a different approach is required. It has been shown that in this limit the properties of lipid bilayer membranes are described very well by modeling the membrane as a two-dimensional manifold embedded in three-dimensional space, with the shape and fluctuations conffoUed by the curvature elasticity [165],... 

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. 

A biologic reason for the abundance of nonlamellar lipids in membranes is that they possess the ability to modulate the activities of membrane proteins (15, 16). It has been recognized that membranes exist in a state of curvature frustration, which may be sufficiently large to have significant effect on certain protein conformations (17). Many examples show that the lipid bilayer elastic curvature stress indeed couples to conformational changes of membrane proteins (15, 18, 19). Protein kinase C is one such example of an enzyme activated by lipids that exhibit a propensity for nonlamellar phase formation (20). The activity of Ca " -ATPase from sarcoplasmic reticulum membranes also strongly correlates with the occurrence of nonbilayer lipids in the membrane and increases with the increase of their amount. It is noteworthy that the protein activity does not depend on the chemical structure of the lipids but only on their phase propensity thus specific binding interactions are ruled out. The list of proteins with activities that depend on the phase properties... 

Membrane Elasticity. For a vesicle with fixed volume V = Vo. area A = Aq, and genus the curvature energy of the bilayer membrane reduces to the sum of the remaining two terms of equation 9... 

In order to define the curvatures of a membrane, it is convenient and often sufficient to think of a mathematical surface. In a purely formal way, a formula for the Hookean bending energy of such a surface is derived, after finding suitable expressions that are linear or quadratic in the principal curvatures. These preparations are followed by three-dimensional descriptions of monolayers and bilayers. Subsequently, stress profiles and thermal undulations are discussed in terms of Hookean bending elasticity. 

The previous section shows that there are two different elastic parameters that eontrol the curvature of a vesicle membrane. These parameters, the spontaneous curvature Cq and the preferred differential area Aqq, depend on the properties of the membrane molecules and the surrounding aqueous medium. Any asymmetry across the membrane will lead to a curved bilayer state characterized by an effective spontaneous curvature. A technique for measurement of this quantity has recently been developed [22,41]. Figure 10.2 shows several situations leading to membrane curvature. The following discussion gives a brief overview of these mechanisms and the techniques to characterize them. 

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