Membranes elastic energy

If you blow up a balloon, energy is stored in it. There is the energy of the compressed gas in the balloon, and there is the elastic energy stored in the rubber membrane itself. As you increase the pressure, the total amount of elastic energy in the system increases. 

To make the flaw grow, say by 1 mm, we have to tear the rubber to create 1 mm of new crack surface, and this consumes energy the tear energy of the rubber per unit area X the area of surface torn. If the work done by the gas pressure inside the balloon, plus the release of elastic energy from the membrane itself, is less than this energy the tearing simply cannot take place - it would infringe the laws of thermodynamics. 

The first model of membrane electroporation was suggested by Crowley [1]. In Crowley s model the membrane is viewed as the isotropic elastic material. The necessary background for understanding its voltage-induced instability was discussed in Section II. Crowley s approximation for the elasticity energy term in Eq. (7) is... 

Typically, the insertion induces sharp variation of the membrane profile at the distances 0.5-1.0nm from the membrane-peptide interface [79-82]. The steepness of this perturbation indicates that the short-A, behavior of membrane moduli must be important in the estimates of the elastic energy. In addition, a peptide inserted in a membrane almost certainly perturbs the membrane s elastic moduli in the immediate vicinity of the inclusion. Both these effects, membrane nonlocality and nonuniform modification of elastic properties by insertions, might play an important role in resolving the contradiction between the local calculations [80] and the experimental data for the mean lifetime of a gramicidin channel [81,109,110]. ... 

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

The membrane bending energy in Eq. (2) is the sum of local elastic energies associated with deformations of individual membrane leaflets away from their spontaneous curvatures, as described by the Helfrich free energy ... 

The possibility of an exact hydrophobic match, Eq. (40), has been analyzed recently [89]. The elastic energy of matching should be compared with the free energy of hydro-phobic mismatch, i.e., the increase in surface energy due to the contact of the nonpolar lipid tails with water. While the elastic energy is proportional to mq, the mismatch energy is a linear function of mq. This means that for a sufficiently large value of mq the membrane... 

Being perturbed at the interface, the membrane profile u r) adjusts itself gradually to minimize the elastic energy. The corresponding free energy of membrane deformation can strongly affect both protein conformation and protein function. For the GA insertion considered below, these effects manifest themselves through the influence of membrane parameters (elastic constants, thickness of the bilayer) on the lifetime r of the ion channel,... 

Under this assumption, the bending energy Eg can be represented in terms of the membrane s curvature. For this reason. Eg is also referred to as the curvature elastic energy. The curvature of smooth surfaces is characterized by two functions that depend on the local canonical curvatures, h t) and h lr), in a surface element dS centered at r. These functions are the mean curvature, H = ( 1 + hi) , and the Gaussian curvature, K = hih2- In general, H and K change with the point r. 

To determine the valne of the adhesive fracture energy, it is necessary to decide the mode of deformation of the pressurized layer. In the case of a relatively thin blister, the mode of deformation is considered to be mainly that of tensile deformation of the blister, and the blister is then modelled as an elastic membrane. Alternatively, in the case of a relatively thick blister, the pressurized layer is considered to deform mainly by bending, and this is modelled as an elastic circular plate with a built-in edge constraint. A further contribution to the stored elastic energy, which is available to assist growth of a debond, arises from an internal stress inherent in the test specimen.Snch stresses may be inlrodnced during... 

Siegel DP (2008) The Gaussian curvature elastic energy of intermediates in membrane fusion. Biophys J 95(11) 5200-5215... 

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... 

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