Helfrich interaction

Consider a stack of membranes with an average spacing with only entropic (Helfrich) interactions. What will happen if more solvent is added to the system Now, consider the same stack, but with an additional attractive interaction between the membranes. What do you expect to happen if the attraction is strong enough and how might this affect what happens if one now tries to dilute the membranes by adding more solvent. (See Ref. 13 for a review.)... 

When membranes fuse, the so-called stalk hypothesis suggests that the intermediate hemifusion state (Fig. 6.4c) comprises a structure in which proximal monolayers layers are connected by a bent stalk and the distal layers are pulled towards each other, thus forming a dimple (see also Fig. 6.5) The stalk model has been supported by theoretical and experimental observations. The fusion of model membranes appears to occur via the same series of fusion intermediates as those in vivo, although the approach of membranes is not Rab/SNARE mediated but is driven by reduced bilayer repulsion forces arising from hydration, electrostatic interactions, thermal fluctuations (Helfrich interaction) or osmotic stress. Membrane fusion is also promoted by defects introduced into the membrane by lateral phase separation (for example of lipid rafts, see above), high spontaneous membrane curvature, or addition of macromolecules or proteins into the membrane. 

If K = T, h 1, which gives /h < 1. Thus, the mean square amplitude is much less than the interplane separation, h. The double layer interaction has completely screened out the undulation force In this case, the Helfrich interaction is not only much weaker than the double layer disjoining pressure (Eqn. III.ll), but is exponentially screened as well. However, with added electrolyte, as soon as the Debye screening length becomes significantly smaller than the interplanar spacing, the Helfrich interaction switches back on. It is precisely this effect which Safinya et al. used to pin down the Helfrich force. 

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-... 

Helfrich-Forster C, Winter C, Hofbauer A, Hall JC, Stanewsky R 2001 The circadian clock of fruit flies is blind after elimination of all known photoreceptors. Neuron 30 249-261 Lin FJ, Song W, Meyer-Bernstein E, Naidoo N, Sehgal A 2001 Photic signaling by cryptochrome in the Drosophilacitcidiaa system. Mol Cell Biol 21 7287-7294 Mas P, Devlin PF, Panda S, Kay SA 2000 Functional Interaction of phytochrome B and cryptochrome 2. Nature 408 207-211... 

In the case of a linear interaction between neighboring lipid bilayers, Helfrich has demonstrated that the repulsive free energy due to confinement is inversely proportional to (7b2. While this result is strictly valid for a harmonic interaction potential (linear force), we assume that it can be extended to any interaction. We will examine later under what conditions this approximation is accurate. 

The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments G-7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3 8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric 9 the root-mean-square fluctuations obtained from experiment were also shown to be in disagreement with this theory.10... 

A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,18 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent pieces , whose area is related to the root mean square fluctuations of the positions of the undulatingbilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. 

The Helfrich theory is valid for hard-wall interaction between membranes. However, at small separation additional interactions have to be taken into account. 

Helfrich original theory assumed that the membranes interact only with a hard-wall potential, but when interactions become longer range, they affect themself the undulation of the membranes, contributing to their confinement. In this case, there is a mutual interdependence between the thermal fluctuations and the interaction potentials, which cannot be any longer assumed independent of each other, hence they cannot be simply additive. 

The entropic term of the free energy (pci unit area) due to the confinement is obtained by subtracting the interaction energy per unit area from Eq. (B.10), a result which is essentially due to Helfrich [4] ... 

It turns out that cjeff is a function of the flexibility of the bilayers and their mutual interaction (Van der Linden and Droge 1993). These two parameters are the mesosopic determinants. In a special case of AOT bilayers in brine, one can deduce from theory as well as from experiment (Nallet et al. 1989 Nallet 1991 Helfrich 1978), making use of the theoretical expression in Van der Linden and Droge 1993), that... 

Having established that bilayer flexibility and bilayer interaction are the mesoscopic determinants, the next question is whether these determinants can be coupled to molecular parameters. In fact, this has been done to quite some extent. In general, bilayer flexibility can be shown (both experimentally as well as theoretically by simulation methods) to be directly related to bilayer thickness, lateral interaction between heads and tails of the surfactants, type of head group (ethoxylate, sugar, etc.), type of tail (saturated, unsaturated) and specific molecular mixes (e.g. SDS with or without pen-tanol). The bilayer interaction is known to be related to characteristics such as classical electrostatics. Van der Waals, Helfrich undulation forces (stemming from shape fluctuations), steric hindrance, number, density of bilayers, ionic strength, and type of salt. Two examples will be dicussed. 

Helfrich, W. (1978). Steric interaction of fluid membranes in multilayer systems. Z. Natur-forsch., A33, 305-315. 

W. Helfrich and R.-M. Servuss, Nuotfo Cimento D, 3, 137 (1984). Undulations, Steric Interaction, and Cohesion of Fluid Membranes. 

Following Helfrich we assume that the net result of these collisions is that each membrane experiences an effective interaction with its nearest neighbors which has as its lowest energy state, the periodic configuration, where... 

It is important to note that the lamellar phase is thus stabilized by the balance of a negative interfacial tension (of the free oil/water interface covered by an amphiphilic monolayer), which tends to increase the internal area, and a repulsive interaction between interfaces. The result, Eq. (48), indicates that the scattering intensity in a lamellar phase, with wave vector q parallel to the membranes, should have a peak at nonzero q for d > d due to the negative coefficient of the q term in the spectrum of Eq. (40). just as in the microemulsion phase. This effect should be very small for strongly swollen lamellar phases (in coexistence with excess oil and excess water), as both very small [96]. Very similar behavior has been observed in smectic liquid crystals (Helfrich-Hurault effect) [122]. Experimentally, the lamellar phase under an external tension can be studied with the surface-force apparatus [123,124] simultaneous scattering experiments have to be performed to detect the undulation modes. 

This remarkable paradox was resolved by the theoretical work of W. Helfrich [5.10], who calculated the steric interaction between membranes. A simple intuitive idea can be obtained as follows. Consider first a single flexible membrane. 

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