Undulation force

A typical force curve showing the specific avidin-biotin interaction is depicted in figure Bl.20.10. The SFA revealed the strong influence of hydration forces and membrane undulation forces on the specific binding of proteins to membrane-bound receptors [81]. 

The forth issue is the increase in the repulsion between bilayers at short distances. In Fig. 1, the osmotic pressure is plotted as a function of separation distance (data from Ref. [13]) for no added salt, for l M KC1 and for 1 M KBr. They reveal an increase in repulsion at short separation distances upon addition of salt. While the relatively small difference between 1 M KC1 and 1 M KBr can be attributed to the charging of the neutral lipid bilayers by the binding of Br (but not C.1-) [14], the relatively large difference between no salt and 1 M KCl is more difficult to explain. Even a zero value for the Hamaker constant (continuous line (2) in Fig. 1), in the 1 M KCl case, is not enough to explain the increase in repulsion, determined experimentally. The screening of the van der Waals interaction, at distances of the order of three Debye-Hiickel lengths (about 10 A) should lead, according to Petrache et al. calculations, to a decrease of only about 30% of the Hamaker constant (from 1.2kT to about 0.8kT, see Fig. 5C of Ref. [14]). Therefore, an additional mechanism to increase the hydration repulsion or the undulation force (or both) upon addition of salt should exist to explain the experiments. 

Fig. 1. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars water circles 1 M KCl triangles 1 M KBr) are compared with calculations based on the simple equations ((1), (3) and (9)), with parameters reported in Ref. [13] (note that in Ref. [13] it was suggested that hydration interaction increases upon addition of salt) Ah = 1.6 x 10s N/m2, H = 2.1 A, H = 9.2 x KT21 J. b = 39 A, Kc = 5.8 x 10 2(1 J, An = 1.06 A 2, >.fl = 6.0 A (Line 1). Even for H = 0 and the rest of the parameters as before (Line 2), the repulsion at short separations is weaker than in the experiment. This points out that either hydration and/or undulation forces must increase upon addition of electrolyte.

More importantly, the undulation force cannot be treated as an additive interaction [16],... 

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. 

The undulation force arises from the configurational confinement related to the bending mode of deformation of two fluid bilayers. This mode consists in undulation of the bilayer at constant bilayer area and thickness (Figure 5.30a). Helfrich et al. doi established that two such bilayers, apart at a mean distance h, experience a repulsive disjoining pressure given by the expression ... 

The effect of glycerol on planar La-phase has been discussed in detail in our previous paper. For nonclosed planar structures, the bilayers have a well-defined spacing that is the result of equilibrium between attractive van der Waals forces and repulsive undulation forces. Upon substitution of the solvent water gradually by glycerol-water mixed solvent, the attractive van der Waals forces approach zero. 

To bring the opposing layers of membranes closer together one has to overcome the undulation forces [29], the forces due to the restructuring of water (true hydration forces) and forces due to steric repulsions. What are the relative contributions of these forces into the total force is very hard to establish from the simulations. These contributions are hard to separate in experiment also, although McIntosh and Simon, based on a series of experiments, proposed that the repulsive force which is acting over distances 0.8-1.5 nm is mostly determined by the undulation force [30]. The force that is observed to act at distances below 0.8 nm is due to contributions from a true hydration force and steric repulsion. At this point we want to remind the reader that the contribution of undulations into membrane dynamics is impossible to obtain from the simulations of the type described here, since the present simulations are limited in time and size. The present simulations were done to establish the role of water in the repulsive force and in the stability of membrane/water interface. 

Many authors have described structural transitions in dilute lamellar phases under the influence of shear. For example, Roux and co-workers (19) studied different dilute lamellar phases which were stabilized by undulation forces and contained flat bilayers with defects, at rest. With increasing shear rates, these bilayers undergo a transition into relatively monodisperse multilamellar vesicles above a characteristic shear rate. The size of the formed vesicles is indirectly proportional to the shear rate. Beyond a second characteristic shear rate, the vesicles are again transformed into flat oriented bilayers. These results were explained in terms of a balance between shear stress and elastic forces which come from the bending and the Gaussian moduli of the bilayers. The same authors observed a similar sequence with increasing shear rate for other lamellar phases. It was found... 

Figure 30 Surface forces due to configurational confinement of thermally excited modes into a narrow region between two approaching interfaces, (a) Fluctuating protrusion of adsorbed amphiphilic molecules gives rise to the protrusion surface force (b) bending mode of membrane fluctuations gives rise to the undulation force and (c) squeezing (peristaltic) mode of membrane fluctuations gives rise to the peristaltic force.

Similar to the undulation force, a peristaltic force [317] can appear between two surfactant lamellas or lipid bilayers. It originates from the configurational confinement related to the peristaltic (squeezing) mode of deformation of a fluid bilayer (Fig. 30c). This mode of deformation consists in fluctuation of the bilayer thickness without bending of the bilayer midsurface. The peristaltic deformation is accompanied with extension of the bilayer surfaces. Israelachvili and Wennerstrom [317] demonstrated that the peristaltic disjoining pressure is related to the stretching modulus, k of the bilayer ... 

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... 

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. 

Third, dilute phases can be regarded as one-dimensional colloidal suspensions of sheets in analogy to the familiar three-dimensional suspensions of charged polystyrene spheres (e.g. polyballs). We shall see that the Poisson-Boltzmann equation in one-dimension accurately describes the intermembrane interactions for phases where the dilution is a consequence of long range electrostatic repulsion (rather than undulation forces). This happens when charged sheets are separated by water containing only the counter-ions. 

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