Van der Waals forces. Special characteristics in lipid-water systems and a general method of calculation based on the Lifshitz theory

A practical method for examining and calculating van der Waals forces is derived from Lifshitz'' theory. Rather than treat the total van der Waals energy as a sum of pairwise interactions between atoms, the Lifshitz theory treats component materials as continua in which there are electromagnetic fluctuations at all frequencies over the entire body. It is necessary in principle to use total macroscopic dielectric data from component substances to analyze the permitted fluctuations; in practice it is possible to use only partial information to perform satisfactory calculations. The biologically interesting case of lipid-water systems is considered in detail for illustration. The method gives good agreement with measured van der Waals energy of interaction across a lipid film. It appears that fluctuations at infrared frequencies and microwave frequencies are very important although these are usually ignored in preference to UV contributions. “Retardation effects” are such as to damp out high frequency fluctuation contributions; if interaction specificity is due to UV spectra, this will be revealed only at interactions across <200 angstrom (A). Dependence of van der Waals forces on material electric properties is discussed in terms of illustrative numerical calculations.     

Temperature-dependent van der Waals forces

Biological systems can experience a strong van der Waals interaction involving electromagnetic fluctuations at the low frequency limit. In lipid-water mixtures the free energy of this interaction is proportional to temperature, primarily involves an entropy change, and has qualitative features of a “hydrophobic bond.” Protein-protein attraction in dilute solution is due as much to low frequency proton fluctuation (Kirkwood-Shumaker forces) and permanent dipole forces as to high frequency (infrared and UV) van der Waals intreactions. These conclusions are described in terms of numerical calculations via the Lifshitz theory of van der Waals forces.     

van der Waals Forces between Cylinders: II. Retarded Interaction between Thin Isotropic Rods

We derive the retarded van der Waals interaction between two long thin parallel dielectric cylinders immersed in a solvent. The result shows that the ultraviolet interactions which may be responsible for the specificity of macromolecular interactions do not operate strongly over distances R ≥ 50 Å. For greater distances the contribution of these frequencies ξ is damped by a factor ∞ e^-ξR/c.     

Direct measurements of forces between phosphatidylcholine and phosphatidylethanolamine bilayers in aqueous electrolyte solutions

We report direct measurements of the full interbilayer force laws (force vs. distance) between bilayers of various phosphatidylcholines and phosphatidylethanolamine in aqueous solutions. Bilayers were first deposited on molecularly smooth (mica) surfaces and the interbilayer forces then measured at a resolution of 1 A. Three types of forces were identified: attractive van der Waals forces, repulsive electrostatic (double-layer) forces, and (at short range) repulsive steric hydration forces. Double-layer forces, which arise from ion binding, were insignificant in monovalent salt solutions, e.g., NaCl up to 1 M, but were already present in solutions containing millimolar levels of CaCl2 and MgCl2, giving rise to forces in excellent agreement with theory. Ca2+ binds more strongly than Mg2+, and both bind less to lecithin bilayers in the fluid state (T greater than Tc). The plane of charge coincides with the location of the negative phosphate groups, while the effective plane of origin of the van der Waals force is 4-5 A farther out. In water, the adhesion energies are in the range 0.10-0.15 erg/cm2 for lecithins and approximately 0.8 erg/cm2 for phosphatidylethanolamine. The adhesion energies vary on addition of salt due to changes in the repulsive double-layer and hydration forces rather than to a change in the attractive van der Waals force. The short-range repulsive forces which balance the van der Waals force at separations of 10-30 A are due to a combination of hydration and steric repulsions, the latter arising from thermal motions of head groups and thickness fluctuations of fluid bilayers (above Tc). It is also concluded that bilayer fusion is not simply related to the interbilayer force law.     

Protein docking using continuum electrostatics and geometric fit

The computer program DOT quickly finds low-energy docked structures for two proteins by performing a systematic search over six degrees of freedom. A novel feature of DOT is its energy function, which is the sum of both a Poisson-Boltzmann electrostatic energy and a van der Waals energy, each represented as a grid-based correlation function. DOT evaluates the energy of interaction for many orientations of the moving molecule and maintains separate lists scored by either the electrostatic energy, the van der Waals energy or the composite sum of both. The free energy is obtained by summing the Boltzmann factor over all rotations at each grid point. Three important findings are presented. First, for a wide variety of protein-protein interactions, the composite-energy function is shown to produce larger clusters of correct answers than found by scoring with either van der Waals energy (geometric fit) or electrostatic energy alone. Second, free-energy clusters are demonstrated to be indicators of binding sites. Third, the contributions of electrostatic and attractive van der Waals energies to the total energy term appropriately reflect the nature of the various types of protein-protein interactions studied.     

Poisson–Boltzmann model analysis of binding mRNA cap analogues to the translation initiation factor eIF4E

The electrostatic free energy of binding of two analogues of the 5′-mRNA cap, differing in size and electric charge, to the wild type and mutated eukaryotic initiation factor eIF4E was computed using the finite difference solutions to the Poisson–Boltzmann equation. Two definitions of the solute–solvent dielectric boundary were used: van der Waals model, solvent exclusion (SE) model. The computed electrostatic energies were supplemented by estimations of the non polar and entropic contributions. A comparison with experimental data for the investigated systems was done. It appears that the SE model with additional contribution fits experimental findings better than the van der Waals model does.     

The non-polar solvent potential of mean force for the dimerization of alanine dipeptide: the role of solute-solvent van der Waals interactions

The non-polar component of the potential of mean force of dimerization of alanine dipeptide has been calculated in explicit solvent by free energy perturbation. We observe that the calculated PMF is inconsistent with a non-polar hydration free energy model based solely on the solute surface area. The non-linear behavior of the solute-solvent van der Waals energy is primarily responsible for the non-linear dependence of the potential of mean force with respect to the surface area. The calculated potential of mean force is reproduced by an implicit solvent model based on a solvent continuum model for the solute-solvent van der Waals interaction energy and the surface area for the work of forming the solute cavity.     

Van der Waals forces involving thin rods

The Susceptibility theory of van der Waals dispersion forces is used to calculate the non-retarded dispersion free energy between two parallel anisotropic elliptical rods, between a rod and an ellipsoid, and between a rod and a wall, when the dimensions of the bodies (semiaxes) are small compared to their separation. The results show that the dispersion energy is orientation dependent, so that in general the bodies will experience, in addition to a force, a torque which will tend to align them relative to each other so as to minimize the free energy. The results should prove useful in calculating the dispersion interactions in biological systems involving chainlike molecules.     

Computing van der Waals energies in the context of the rotamer approximation

The rotamer approximation states that protein side-chain conformations can be described well using a finite set of rotational isomers. This approximation is often applied in the context of computational protein design and structure prediction to reduce the complexity of structural sampling. It is an effective way of reducing the structure space to the most relevant conformations. However, the appropriateness of rotamers for sampling structure space does not imply that a rotamer-based energy landscape preserves any of the properties of the true continuous energy landscape. Specifically, because the energy of a van der Waals interaction can be very sensitive to small changes in atomic separation, meaningful van der Waals energies are particularly difficult to calculate from rotamer-based structures. This presents a problem for computational protein design, where the total energy of a given structure is often represented as a sum of precalculated rigid rotamer self and pair contributions. A common way of addressing this issue is to modify the van der Waals function to reduce its sensitivity to atomic position, but excessive modification may result in a strongly nonphysical potential. Although many different van der Waals modifications have been used in protein design, little is known about which performs best, and why. In this paper, we study 10 ways of computing van der Waals energies under the rotamer approximation, representing four general classes, and compare their performance using a variety of metrics relevant to protein design and native-sequence repacking calculations. Scaling van der Waals radii by anywhere from 85 to 95% gives the best performance. Linearizing and capping the repulsive portion of the potential can give additional improvement, which comes primarily from getting rid of unrealistically large clash energies. On the other hand, continuously minimizing individual rotamer pairs prior to evaluating their interaction works acceptably in native-sequence repacking, but fails in protein design. Additionally, we show that the problem of predicting relevant van der Waals energies from rotamer-based structures is strongly nonpairwise decomposable and hence further modifications of the potential are unlikely to give significant improvement.     

Van der Waals locks: loop-n-lock structure of globular proteins

In a globular protein the polypeptide chain returns to itself many times, making numerous chain-to-chain contacts. The stability of these contacts is maintained primarily by van der Waals interactions. In this work we isolated and analysed van der Waals contacts that stabilise spatial structures of nine major folds. We suggest a specific way to identify the tightest contacts of prime importance for the stability of a given crystallized protein and introduce the notion of the van der Waals lock. The loops closed by the van der Waals interactions provide a basically novel view of protein globule organization: the loop-n-lock structure. This opens a new perspective in understanding protein folding as well: the consecutive looping of the polypeptide chain and the locking of the loop ends by tight van der Waals interactions.     

Novel homodimer model of the β-adrenergic receptor in complex with free fatty acids and cholesterol: first-principles calculation studies

We propose a theoretical novel homodimer model of the β- adrenergic receptor (βAR) in complex with a heterogeneous mixture of free fatty acids (FFAs) and cholesterol based on first-principles calculations. We used the density-functional-based tight binding with dispersion (DFTB-D) method, which accurately evaluates van der Waals interactions between FFAs and amino acid residues in the receptor. The calculations suggest that a stable homodimer of bAR can form a complex with FFAs and cholesterol by extensive van der Waals interactions in the cell membrane, and that the heterogeneous composition of the FFAs is important for the stability of the homodimer complex. The stable van der Waals interactions propagate from one of the bAR to the other through the cholesterol and FFAs in the homodimer complex. The energy propagation in the complex has the potential to enhance molecular signaling in adipocytes, because the stability of the complex can influence anti-adiposity effects after oral treatment of the FFA components.     

van der Waals forces influencing adhesion of cells

Adhesion molecules, often thought to be acting by a ‘lock and key’ mechanism, have been thought to control the adhesion of cells. While there is no doubt that a coating of adhesion molecules such as fibronectin on a surface affects cell adhesion, this paper aims to show that such surface contamination is only one factor in the equation. Starting from the baseline idea that van der Waals force is a ubiquitous attraction between all molecules, and thereby must contribute to cell adhesion, it is clear that effects from geometry, elasticity and surface molecules must all add on to the basic cell attractive force. These effects of geometry, elasticity and surface molecules are analysed. The adhesion force measured between macroscopic polymer spheres was found to be strongest when the surfaces were absolutely smooth and clean, with no projecting protruberances. Values of the measured surface energy were then about 35 mJ m⁻², as expected for van der Waals attractions between the non-polar molecules. Surface projections such as abrasion roughness or dust reduced the molecular adhesion substantially. Water cut the measured surface energy to 3.4 mJ m⁻². Surface active molecules lowered the adhesion still further to less than 0.3 mJ m⁻². These observations do not support the lock and key concept.     

Interrod forces in aqueous gels of tobacco mosaic virus.

The lateral separation of virus rod particles of tobacco mosaic virus has been studied as a function of externally applied osmotic pressure using an osmotic stress technique. The results have been used to test the assumption that lattice equilibrium in such gels results from a balance between repulsive (electrostatic) and attractive (van der Waals and osmotic) forces. Results have been obtained at different ionic strengths (0.001 to 1.0 M) and pH's (5.0 to 7.2) and compared with calculated curves for electrostatic nad van der Waals pressure. Under all conditions studied, interrod spacing decreased with increasing applied pressure, the spacings being smaller at higher ionic strengths. Only small differences were seen when the pH was changed. At ionic strengths near 0.1 M, agreement between theory and experiment is good, but the theory appears to underestimate electrostatic forces at high ionic strengths and to underestimate attractive forces at large interrod spacings (low ionic strengths). It is concluded that an electrostatic-van der Waals force balance can explain stability in tobacco mosaic virus gels near physiological conditions and can provide a good first approximation elsewhere.     

On the Role of Density Fluctuations in the Equation of State of a Simple Fluid

Abstract

We address the wellknown problems intorduced into the theory of fluids by density fluctuations in the form of van der Waals loops and nonclassical critical phenomena. A clean separation of long and short range density fluctuations is achieved by use of cell-constrained models which display well-defined van der Waals loops and classical behaviour around the critical point. For a pure Lennard-Jones fluid with occupancy restricted to 1 or 8 particles per cell, the phase diagram is determined by Monte Carlo simulation. By considering the deviations from the normal simulations without cell constraint, the effects of longer range density fluctuations are exposed. The system size dependence of the van der Waals loops present in all simulations of fluids is analyzed in terms of the GvdW free energy density functional theory, which is formuiated on the basis of the cell concept. The loops are found to gradually disappear either with greater cel occupancy or increasing total particle number in the simulation box.     

Direct measurement of the intermolecular forces between counterion-condensed DNA double helices. Evidence for long range attractive hydration forces.

Rather than acting by modifying van der Waals or electrostatic double layer interactions or by directly bridging neighboring molecules, polyvalent ligands bound to DNA double helices appear to act by reconfiguring the water between macromolecular surfaces to create attractive long range hydration forces. We have reached this conclusion by directly measuring the repulsive forces between parallel B-form DNA double helices pushed together from the separations at which they have self organized into hexagonal arrays of parallel rods. For all of the wide variety of "condensing agents" from divalent Mn to polymeric protamines, the resulting intermolecular force varies exponentially with a decay rate of 1.4-1.5 A, exactly one-half that seen previously for hydration repulsion. Such behavior qualitatively contradicts the predictions of all electrostatic double layer and van der Waals force potentials previously suggested. It fits remarkably well with the idea, developed and tested here, that multivalent counterion adsorption reorganizes the water at discrete sites complementary to unadsorbed sites on the apposing surface. The measured strength and range of these attractive forces together with their apparent specificity suggest the presence of a previously unexpected force in molecular organization.     

Van der Waals Forces Between Plasmonic Nanoparticles

We propose a new approach to calculate van der Waals forces between nanoparticles where the van der Waals energy can be reduced to the energy of localized plasmons in nanoparticles. The general theory is applied to describe the interaction between two metallic nanoparticles and between a nanoparticle and a perfectly conducting plane. Our results could be used to prove experimentally the existence of new, recently predicted type of plasmon oscillation (Klimov and Guzatov, Phys Rev B 75:024303, 2007a; Klimov and Guzatov, Quantum Electron 37:209, 2007b) and to elaborate new control mechanisms for the adherence of nanoparticles between each other or onto surfaces.     

Long range intermolecular forces between macroscopic bodies: macroscopic and microscopic approaches.

Van der Waals energies of interaction are calculated by two methods, the macroscopic method of Lifshitz and the microscopic method of London-Casimir and Polder-Hamaker for the case of two semi-infinite slabs separated by a thin film. When retardation effects may be neglected, the London-Hamaker approach yields values of dispersion interactions which almost coincide with those of the Lifshitz approach, the magnitude of the former values being larger by approximately 10–25%, which is attributed to the effect of the molecular environment in condensed media. At 50–100 Å film thicknesses where retardation effects are small, dispersion terms are generally the major part of van der Waals forces in the Lifshitz formulation. Hence, for 50–100 Å film thicknesses the Hamaker approach, which only includes dispersion interactions is generally adequate. By accounting for retardation effects, which significantly reduce the magnitude of dispersion interactions at several hundred Å, there is a reasonable agreement between the values obtained by the macroscopic and microscopic approaches. When polar substances are present and for film thicknesses of several hundred Å, where dispersion interactions are significantly reduced, the major contribution to van der Waals forces may arise from orientation and induction terms. For such cases the Hamaker approach may lead to critical underestimates of the calculated magnitude of van der Waals forces. An ad hoc way to overcome this difficulty which is applicable to any geometry is proposed. This study presents a simple procedure for the determination of free energies of interaction between macroscopic bodies of various shapes. The procedure, which is applicable when the molecules of bodies and surrounding medium are isotropic, yields results which closely approximate those obtained with the Lifshitz theory.     

Electrostatic contribution to the binding stability of protein-protein complexes

To investigate roles of electrostatic interactions in protein binding stability, electrostatic calculations were carried out on a set of 64 mutations over six protein-protein complexes. These mutations alter polar interactions across the interface and were selected for putative dominance of electrostatic contributions to the binding stability. Three protocols of implementing the Poisson-Boltzmann model were tested. In vdW4 the dielectric boundary between the protein low dielectric and the solvent high dielectric is defined as the protein van der Waals surface and the protein dielectric constant is set to 4. In SE4 and SE20, the dielectric boundary is defined as the surface of the protein interior inaccessible to a 1.4-A solvent probe, and the protein dielectric constant is set to 4 and 20, respectively. In line with earlier studies on the barnase-barstar complex, the vdW4 results on the large set of mutations showed the closest agreement with experimental data. The agreement between vdW4 and experiment supports the contention of dominant electrostatic contributions for the mutations, but their differences also suggest van der Waals and hydrophobic contributions. The results presented here will serve as a guide for future refinement in electrostatic calculation and inclusion of nonelectrostatic effects.     

Numerical Analysis of the Adhesive Forces in Nano-Scale Structure

Nanohairs, which can be found on the epidermis of Tokay gecko＇s toes, contribute to the adhesion by means of van der Waals force, capillary force, etc. This structure has inspired many researchers to fabricate the attachable nano-scale structures. However, the efficiency of artificial nano-scale structures is not reliable sufficiently. Moreover, the mechanical parameters related to the nano-hair attachment are not yet revealed qualitatively. The mechanical parameters which have influence on the ability of adhesive nano-hairs were investigated through numerical simulation in which only van der Waals force was considered. For the numerical analysis, finite element method was utilized and van der Waals force, assumed as 12-6 Lennard-Jones potential, was implemented as the body force term in the finite element formulation.     

Free energy potential for aggregation of giant, neutral lipid bilayer vesicles by Van der Waals attraction.

Here, we report the first direct observation of Van der Waals' attraction between biomembrane capsules using measurements of the free energy reduction per unit area of membrane-membrane contact formation. In these studies, the membrane capsules were reconstituted neutral (egg phosphatidylcholine) lipid bilayers of giant (greater than 10(-3) cm diam) vesicles. Micromanipulation methods were used to select and maneuver two vesicles into proximity for contact; after adhesion was allowed to occur, the extent of contact formation was regulated through the vesicle membrane tensions that were controlled by micropipette suction. The free energy reduction per unit area of contact formation was proportional to the membrane tension multiplied by a simple function of the pipette and vesicle dimensions. The free energy potential for Van der Waals attraction between the neutral bilayers in 120 mM NaCl solutions was 1.5 X 10(-2) ergs/cm2. Also, when human serum albumin was added to the medium in the range of 0-1 mg/ml, the free energy potential for bilayer-bilayer adhesion was not affected. Using published values for equilibrium spacing between lipid bilayers in multilamellar lipid-water dispersions and the theoretical equation for van der Waals attraction between continuous dielectric layers, we calculated the value for the Hamaker coefficient of the Van der Waals attraction to be 5.8 X 10(-14) ergs.