Lateral tension

In the case of solids, there is no doubt that a lateral tension (which may be anisotropic) can exist between molecules on the surface and can be related to actual stretching or compression of the surface region. This is possible because of the immobility of solid surfaces. Similarly, with thin soap films, whose thickness can be as little as 100 A, stretching or extension of the film may involve a corresponding variation in intermolecular distances and an actual tension between molecules. 

Again, using a lateral dimension of a site, d = 0.2 nm, and the lattice site area as = 3d2, means that y 1 corresponds with about 33mNm 1 lateral tension. In other words, one needs to apply a lateral tension of order 40mNm 1 to double the membrane area. This prediction seems to be a factor of about six lower than estimates that were recently reported by Evans and co-workers [107], These authors use micropipettes to pressurise giant vesicles and obtain values of the order Ka = 8y/Sinn = 230mNm. There are also some data on the compressibility modulus, as found by MD simulations on primitive surfactants [62] Ka = 400 mN m 1. In a molecular detailed simulation study on DPPC lipids, Feller and Pastor [40] report a KA value of about 140 mNm 1. 

Zipping causes curvature and lateral tension on bilayers, favoring hemifusion between outer leaflets and causing formation of an energetically unfavorable void space. 

Fig. 8 Left The phase behavior of amphiphiles as observed with the model of [114,115], is shown in the main panel, plotted as a function of rescaled temperature kgT/e and attraction width w,. ja at zero lateral tension. Each symbol corresponds to one simulation and identifies different bilayer phases. Crosses denote the gel phase, solid circles mark fluid bilayers, and vertical crosses indicate the region where bilayers are unstable. The dashed lines are merely guides to the eye. The inset shows the pair potential between tail beads (solid line) and the purely repulsive head-head and head-tail interaction (dashed line). Reprinted with permission from Ref. 114. Copyright (2005) by the American Physical Society. Right Phase separation and budding sequence for a vesicle containing a 50 50 mixture of two lipids. The vesicle is in equilibrium with a very dilute vapor of amphiphiles (i.e., the lipids seen floating in the exterior volume). From [114]...

Before proteins are considered, it is important to explicitly state some limitations of the use of the spontaneous curvature. The spontaneous curvature is a result of the distribution of lateral tensions, s(z), as a function of depth, z9 through the monolayer. For a monolayer to remain flat, as in the case of a bilayer leaflet, one must have (17)... 

While the quantitative details will be explained in more detail in later chapters, it should be intuitively clear that atoms or molecules at a surface will experience a net positive inward (i.e., into the bulk phase) attraction normal to the surface, the resultant of which will be a state of lateral tension along the surface, giving rise to the concept of surface tension. For a flat surface, the surface tension may be defined as a force acting parallel to the surface and perpendicular to a line of unit length anywhere in the surface (Fig. 2.2). The definition for a curved surface is somewhat more complex, but the difference becomes significant only for a surface of very small radius of curvature. 

If lipid membranes are subjected to lateral tension, they typically rupture at stresses of several millinewtons per meter (mN/m), with a remarkably low rupture strain of only a few percent [3]. At large scales and moderate tensions it is hence an excellent approximation to consider membranes as largely unstretchable two-dimensional surfaces. Their dominant soft modes are not associated with stretching but with bending [4-6]. Within the well-established mathematical framework developed by Helfrich [5], the energy of a membrane patch 5, amended by a contribution due to its boundary dt [7], is expressible as ... 

Weikl TR, Kozlov MM, Helfrich W (1998) Interaction of conical membrane inclusions effect of lateral tension. Phys Rev E 57 6988-6995. doi 10.1103/PhysRevE.57.6988... 

Integrating the lateral stress [Pg.55]

However, the neutral surface cannot be the surface of equal molecular density if the membrane curvature is nonuniform [6]. This is because in mechanical equilibrium the monolayer lateral tension contains the bending energy density extra... 

To calculate the position of the surface of equal molecular density relative to the neutral surface, consider again cylindrical curvature C] and assume 7m = for C = 0. Accordingly, the change in area of the neutral surface due to the lateral tension of bending, —k cqc, is... 

In general, it is sufficient to know that there is a surface in the monolayer for which stretching and bending deformations are in effect decoupled. This will be the classical neutral surface (for Cq = 0) or a new surface of equal molecular density at a slightly shifted position (for Cq 0). In the latter case, nonunifoim curvature c, + C2 is associated with an inherent lateral tension that is proportional to it Henceforth, the term bending surface is used in both cases. 

In the flat state (or a flat region) the torque per unit length is simply —kcq b- In the case considered, it is due to a dilation of one bilayer and an equal compression of the other, if the bilayer as a whole has no lateral tension. Employing the homogeneous plate model for the monolayers (cq = 0), the line torque balance may be written as... 

As the monolayer dilations of the outer and inner monolayers need not cancel each other, this formula allows for nonvanishing bilayer lateral tension. The height z is the distance of the monolayer neutral surfaces from the bilayer middle. It enters as the average lever arm of the elastic forces due to monolayer dilation and compression. In the closed bilayer of a vesicle, the difference of the monolayer dilations can change with the shape of the vesicle. Simple geometric arguments lead, in this particular model, to... 

The stress profile expected for a flat monolayer of zero lateral tension is sketched in Figure 6.3, with a positive stress at the water-hydrocarbon interface and a compensatory negative stress in the region of the hydrocarbon chains. In the case of charged membranes, there will be additional negative stresses in the water at z >h. 

These considerations are easily extended to bilayers of zero lateral tension (y = 0), resulting in... 

The formula applies to symmetric and asymmetric bilayers, including the case of equal but opposite lateral tensions of the monolayers (y m = —0). An abbreviated version of it has been used above to calculate in the homogeneous plate model the spontaneous curvature induced by lipid imbalance. For the... 

The emphasis of this review of membrane elasticity is on the fluid plate model of the amphiphilic monolayer. The homogeneous fluid plate is used for some elementary considerations and estimates. Plates modeling actual monolayers will possess a nonvanishing stress profile even when undeformed, i.e. in the flat state at zero lateral tension. 

If the lateral tension X vanishes, the second bracket becomes unity. This form corresponds to the famous Helfrich potential [53], The numerical prefactor 6fe2 0.1 can be estimated by Monte-Carlo simulations [54]. In the presence of a lateral tension Z, which pulls at the membrane, the power law decay of the first bracket is cutoff by the second factor at a length scale l = (4br/Z) [55,52]. The best choice for b in this case is 6 = 1 jin. The form of eq. (7) covering the full range of tension and separation values has been derived using a simple self-consistent picture [50]. 

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