Lipid curvature elasticity

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... 

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).

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. 

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. 

A.G. Petrov and M.M. Kozlov, Curvature elasticity and passage formation in lipid bilayers. Lattice of passages, Compt. Rend. Acad. Bulg. Sci. 37, 1191, (1984). 

Keywords Coarse graining Curvature elasticity Lipid bilayer Mediated interactions MultiscaUng Simulation... 

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],... 

Since the solubility of lipids in water is very low, the number of lipid molecules in a membrane is essentially constant over typical experimental time scales. Also, the osmotic pressure generated by a small number of ions or macromolecules in solution, which cannot penetrate the lipid bUayer, keeps the internal volume essentially constant. The shape of fluid vesicles [176] is therefore determined by the competition of the curvature elasticity of the membrane, and the constraints of constant volume V and constant surface area S. In the simplest case of vanishing spontaneous curvature, the curvature elasticity is given by (98). In this case, the vesicle shape in the absence of thermal fluctuations depends on a single dimensionless parameter, the reduced volume V = V/Vo, where Vb = (47t/3)1 o nd Ro = (5/4 r) are the volume and radius of a sphere of the same surface area S, respectively. The calculated vesicle shapes are shown in Fig. 23. There are three phases. For reduced volumes not too far from the sphere, elongated prolate shapes are stable. In a small range of reduced volumes of V e [0.592,0.651], oblate discocyte shapes have the lowest curvature energy. Finally, at very low reduced volumes, cup-like stomatocyte shapes are found. 

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-... 

Marsh D. Elastic curvature constants of lipid monolayers and bilayers. Chem. Phys. Lipids 2006 144 146-159. 

Keywords cell signaling lipid rafts BAR domains membrane curvature membrane elasticity PIP2 diffusion mean-field model coarse-grained theory Poisson-Boltzmann theory Cahn-Hilliard equations... 

These fields differ quite substantially in their theoretical description concentrations are scalar variables, orientations are vectors, and differential geometry is at heart a tensor theory but, aU of them are known to mediate interactions. For instance, the fact that proteins might prefer one lipid composition over another and thus aggregate [217-220] is central to an important mechanism attributed to lipid rafts. Tilt-mediated protein interactions have also been studied in multiple contexts [32, 33, 159, 221-223]. It is even possible to describe all these phenomena within a common language [224], using the framework of covariant surface stresses [154, 155, 157-161]. However, in the present review we will restrict the discussion to only two examples, both related to membrane elasticity in Sect. 3.1 we will discuss interactions due to hydrophobic mismatch, and in Sect. 3.2 we will look at interactions mediated by the large-scale curvature deformation of the membrane. 

The ratio of the elastic moduli a is given by the lipid system studied and is essentially fixed for each set of experiments. Although a is important for the topology of the phase diagram, existing shapes and shape classes, in fact, do not depend on it. Thus, it is convenient to define the effective spontaneous curvature [22,32]... 

---