Vesicles 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 ... 

The total curvature energy of a spherical vesicle is given by 4tt(2/cc + k). As all experimental data on phospholipids indicate that kc is not small, one is inclined to conclude that the vesicles are thermodynamically unstable the reduction of the number of vesicles, e.g. by vesicle fusion or by Ostwald ripening, will reduce the overall curvature energy. However, such lines of thought overlook the possibility that k is sufficiently negative to allow the overall curvature free energy of vesicles to remain small. 

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 ... 

A vesicle phase will form instead of the L3 phase when the elastic constant k (which is controlled by the allQ l chain length of the alcohol) [134] and k (which is controlled by the alcohol concentration in the interface) [134] are chosen so that the curvature energy required to create a sphere is small [225]. The phases formed reflect the competition between entropic contributions (which favor smaller aggregates that allow for a more homogeneous distribution of the free counterions in space) [85] and elastic contributions to the free energy [223,225]. Decreasing surfactant concentration at constant k leads first to a highly swollen L,... 

J (7,53) one finds at room temperature Ac 1 mmol/L. Thus, only minute concentration differences between interior and exterior solution can be balanced by the curvature energy. Large osmotic pressure differences will instead lead to the inward/outward permeation of water and concomitantly an increase/decrease of the vesicle volume until the concentration difference is Ac = 0 and thus V = Vo- In practice, this allows one to control the vesicle volume by the molar concentration ( osmolarity ) difference of the inner and outer solutions. 

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 vesicle shapes can be obtained by minimizing the appropriate curvature energy subject to the geometric constraints. Video microscopy reveals that these shapes typically exhibit thermal fluctuations. How can these fluctuations be described ... 

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 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. 

We first analyse the free energy of a one-component spherical vesicle bilayer (fig. 11), and investigate its stability in the absence of packing restrictions. It should be emphasised that without packing no sensible results emerge unless we include curvature corrections. Subsequently, when packing is built into the model, curvature corrections become of secondary importance, so that the principal results do not... 

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