Undulation interaction

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-... 

Walz, J. Y. Ruckenstein, F. Comparison of the van der Waals and Undulation Interactions between Uncharged Lipid Bilayers. J. Phys. Chem. B 1999, 103, 7461-7468. 

Non-DLVO colloidal interactions excluded volumes, undulation interactions, depletion forces and many-body effects... 

For typical membranes interacting via an exponential repulsion (hydration force) and a Van der Waals interaction, it was shown that at small separations the undulation interaction has an exponential behavior [16,22,23], However, the exponential repulsion cannot be valid at large separation, because it cannot predict the unbinding of the membranes (an exponential is always of shorter range that the van der Waals attraction). In general, the undulation repulsion is treated as an additive inter-... 

The force between neutral surfaces (with a surface dipole density) depends on the electrolyte concentrations, as shown in Fig. 3b, particularly at large separations. However, at small separations, the interaction appears to be well described by an exponential with a decay length AH. For neutral lipid bilayers, the equilibrium is reached at a distance of about 20 A, at which the attractive van der Waals interaction balances the repulsive hydration and thermal undulation interactions [43], The experiments regarding the forces between neutral lipid bilayers [11] sample the interactions at separations smaller than 20 A, for which the dependence on ionic strength is much weaker. By adding to the total pressure a typical van der Waals disjoining pressure [12] ... 

This interlayer interaction is known as the Helfrich undulation interaction. In a beautiful series of X-ray diffraction experiments, Safinya O and his co-workers have demonstrated that the undulation force is dominant where any electrostatic interactions are screened out, but are ineffective in the presence of unscreened double layer interactions. 

A nice example of double layer effects occurs with the lamellar phases discussed in Section II. We already mentioned the beautiful experiments of Safinya et al.20 where the Helfrich undulation force was clearly demonstrated, using electrostatic interactions between the layers as a control parameter. Let us try to understand how the electrostatic interlayer forces have impact upon the undulation interaction Recall (Eqn. III.3) that the counterion distribution in the neighborhood of a single charged surface falls off as x 2 for x A. Since the counterions may be approximately considered as an ideal gas,the double layer contribution to the disjoining pressure between two lamellae separated by a distance, h, is roughly... 

It is important to realize that this repulsive undulation interaction is long range, and as we shall see for flexible membranes with low rigidity k kgT, will dominate the attractive long range van der Waals interaction. 

Fig. 15. Power-law exponent i as a function of (1-S/d) for three dilution systems. The solid squares are for the DMPC-pentanol -water system while the open circles and squares are for the SDS-pentanol membranes along brine and oil dilution lines respectively. The solid line is the prediction of the Helfrich theory of undulation interactions.

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