Bilayers curvature energy

For lipid bilayers, equation (4) can be simplified. Above we have seen that the flat unsupported bilayer is without tension, i.e. y(0, 0) = 0, and therefore the first two terms must cancel y0 = — kcj. As argued above, JQ = 0, and thus also the third term drops out. The remaining two terms are proportional to the curvature to the power two. For a cylindrical geometry only, the term proportional to J2 is present. For spherical vesicles, the two combine into one ( kc + k)J2. The curvature energy of a homogeneously curved bilayer is found by integrating the surface tension over the available area ... 

Derive an expression for the curvature elastic free energy of a bilayer in terms of the curvature elastic constants of the monolayer. Treat the case where the two monolayers are equivalent and noninterpenetrating, so that one adds their curvature energies, but note that the monolayers each have a finite thickness, which makes their curvatures inequivalent. Compare with the Helfrich form and comment on the effective saddle curvature as a function of the spontaneous curvature of each monolayer. 

We consider the curvature energy of spherical vesicles. As a first approximation, we assume that the area per molecule is the same in each of the two monolayers that compose the vesicle bilayer we therefore just add the bending energies of each monolayer. The total bending energy per unit area of the midplane between the two monolayers that compose the bilayer can be obtained from Eq. (6.15). Noting that for spheres, the two curvatures k = K2 — c, we write ... 

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... 

The competition between the polar and steric dipoles of molecules may also lead to internal frustration. In this case, the local energetically ideal configuration cannot be extended to the whole space, but tends to be accomodated by the appearance of a periodic array of defects. For example, the presence of the strong steric dipole at the head of a molecule forming bilayers will induce local curvature. As the size of the curved areas increases, an increase in the corresponding elastic energy makes energetically preferable the... 

The experiments discussed in this chapter have shown that a variety of chiral molecules self-assemble into cylindrical tubules and helical ribbons. These are indeed surprising structures because of their high curvature. One would normally expect the lowest energy state of a bilayer membrane to be flat or to have the minimum curvature needed to close off the edges of the membrane. By contrast, these structures have a high curvature, with a characteristic radius that depends on the material but is always fairly small compared with vesicles or other membrane structures. Thus, the key issue in understanding the formation of tubules and helical ribbons is how to explain the morphology with a characteristic radius. 

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. 

The thermodynamic analysis of curved bilayers is typically done on very weakly curved objects (much less curved then the example shown in Figure 23). This is to guarantee that the free energy of curvature remains quadratic in J and K. 

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