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 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.     

Sequence Structure of van der Waals Locks in Proteins

Abstract

Recent works has suggested that proteins in early evolution have gone through a stage of closed loop elements with a typical contour size of 25–35 residues. These closed loops are still the elementary protein units to these days, and can be used to spell out protein sequence/structure relationship through a relatively small number of protein prototypes. In this study we aimed to identify the sequences that are used to lock the loop ends to one another, and to show how an extensive dictionary of such locking pairs can be created using positional correlation data from a large proteome database, and structural data from PDB databases. Such a dictionary can be used in reconstructing the evolutionary pathway the modern proteins have gone through, and in identifying closed loop elements in modern proteins with yet unknown 3D structure.     

Sequence structure of van der Waals locks in proteins

Recent works has suggested that proteins in early evolution have gone through a stage of closed loop elements with a typical contour size of 25-35 residues. These closed loops are still the elementary protein units to these days, and can be used to spell out protein sequence/structure relationship through a relatively small number of protein prototypes. In this study we aimed to identify the sequences that are used to lock the loop ends to one another, and to show how an extensive dictionary of such locking pairs can be created using positional correlation data from a large proteome database, and structural data from PDB databases. Such a dictionary can be used in reconstructing the evolutionary pathway the modern proteins have gone through, and in identifying closed loop elements in modern proteins with yet unknown 3D structure.     

Statistical mechanics approach to inhomogeneous van der Waals fluids

A general methodology is proposed to formulate density functional approximation (DFA) for inhomogeneous van der Waals fluids. The present methodology needs as input only a hard sphere DFA, second order direct correlation function (DCF) and pressure of coexistence bulk fluid, and therefore can be applicable to both supercitical and subcritical temperature regions. As illustrating example, the present report combines a recently proposed hard sphere “Formally exact truncated non-uniform excess Helmholtz free energy density functional approximation” with the present methodology, and applies the resultant DFA to calculate density profile of the inhomogeneous Lennard-Jones (LJ) fluid in coexistence with a bulk LJ fluid being situated at “dangerous” regions, i.e. the coexistence bulk state is near the critical temperature or the gas-liquid coexistence line. The theoretical predictions are in very good agreement with the recent simulation results, it is concluded that the present DFA is a globally excellent one. A discussion is given why the present methodology can lead to so excellent DFA.     

A new approach for the calculation of the energy of van der Waals interactions in macromolecules of globular proteins

Van der Waals interaction energy in globular proteins is presented by the interaction energies between regions of protein spatial structure with homogenous medium density distribution. We introduce a notion of the local medium permittivity as a function of absorptance of molecular groups with particular conformation. Proposed theory avoids shortcomings which are typical for the calculations on the basis of the pairwise additive approximation. The approach takes into account local peculiarities of protein spatial structure and physical-chemical characteristics of amino acid residues and molecular groups.     

Calibration of effective van der Waals atomic contact radii for proteins and peptides

Effective van der Waals radii were calibrated in such a way that molecular models built from standard bond lengths and bond angles reproduced the amino acid conformations observed by crystallography in proteins and peptides. The calibrations were based on the comparison of the Ramachandran plots prepared from high-resolution X-ray data of proteins and peptides with the allowed phi, psi torsional angle space for the dipeptide molecular models. The calibrated radii are useful as criteria with which to filter energetically improbable conformations in molecular modeling studies of proteins and peptides.     

Discrete structure of van der Waals domains in globular proteins

Most globular proteins are divisible by domains, distinct substructures of the globule. The notion of hierarchy of the domains was introduced earlier via van der Waals energy profiles that allow one to subdivide the proteins into domains (subdomains). The question remains open as to what is the possible structural connection of the energy profiles. The recent discovery of the loop-n-lock elements in the globular proteins suggests such a structural connection. A direct comparison of the segmentation by van der Waals energy criteria with the maps of the locked loops of nearly standard size reveals a striking correlation: domains in general appear to consist of one to several such loops. In addition, it was demonstrated that a variety of subdivisions of the same protein into domains is just a regrouping of the loop-n-lock elements.     

Display of quantum mechanical properties on van der Waals surfaces

A technique is described for comparing graphically the electrostatic properties of molecules. Results from quantum mechanical calculations are mapped onto the van der Waals surface. The surface is defined by CSG (constructive solid geometry), a technique used in CAD (computer-aided design). Colour is used to represent the electrostatic properties and the results are displayed on a high resolution raster device as a ‘realistic’ model. This technique is expected to be of value in discriminating qualitatively between compounds of different biological activities.     

A model for the recognition of protein kinases based on the entropy of 3D van der Waals interactions

The study and prediction of kinase function (kinomics) is of major importance for proteome research due to the widespread distribution of kinases. However, the prediction of protein function based on the similarity between a functionally annotated 3D template and a query structure may fail, for instance, if a similar protein structure cannot be identified. Alternatively, function can be assigned using 3D-structural empirical parameters. In previous studies, we introduced parameters based on electrostatic entropy (Proteins 2004, 56, 715) and molecular vibration entropy (Bioinformatics 2003, 19, 2079) but ignored other important factors such as van der Waals (vdw) interactions. In the work described here, we define 3D-vdw entropies (degrees theta(k)) and use them for the first time to derive a classifier for protein kinases. The model classifies correctly 88.0% of proteins in training and more than 85.0% of proteins in validation studies. Principal components analysis of heterogeneous proteins demonstrated that degrees theta(k) codify information that is different to that described by other bulk or folding parameters. In additional validation experiments, the model recognized 129 out of 142 kinases (90.8%) and 592 out of 677 non-kinases (87.4%) not used above. This study provides a basis for further consideration of degrees theta(k) as parameters for the empirical search for structure-function relationships.     

ProteinVolume: calculating molecular van der Waals and void volumes in proteins

Background

Voids and cavities in the native protein structure determine the pressure unfolding of proteins. In addition, the volume changes due to the interaction of newly exposed atoms with solvent upon protein unfolding also contribute to the pressure unfolding of proteins. Quantitative understanding of these effects is important for predicting and designing proteins with predefined response to changes in hydrostatic pressure using computational approaches. The molecular surface volume is a useful metric that describes contribution of geometrical volume, which includes van der Waals volume and volume of the voids, to the total volume of a protein in solution, thus isolating the effects of hydration for separate calculations.

Results

We developed ProteinVolume, a highly robust and easy-to-use tool to compute geometric volumes of proteins. ProteinVolume generates the molecular surface of a protein and uses an innovative flood-fill algorithm to calculate the individual components of the molecular surface volume, van der Waals and intramolecular void volumes. ProteinVolume is user friendly and is available as a web-server or a platform-independent command-line version.

Conclusions

ProteinVolume is a highly accurate and fast application to interrogate geometric volumes of proteins. ProteinVolume is a free web server available on http://gmlab.bio.rpi.edu. Free-standing platform-independent Java-based ProteinVolume executable is also freely available at this web site.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-015-0531-2) contains supplementary material, which is available to authorized users.     

Many-atom van der Waals interactions lead to direction-sensitive interactions of covalent bonds

Strict physical theory and numerical calculations show that a specific coupling of many-atom van der Waals interactions with covalent bonding can significantly (half as much) increase the strength of attractive dispersion interactions when the direction of interaction coincides with the direction of the covalent bond, and decrease this strength when the direction of interaction is perpendicular to the direction of the covalent bond. The energy effect is comparable to that caused by the replacement of atoms (e.g. N by C or O) in conventional pairwise van der Waals interactions. Analysis of protein structures shows that they bear an imprint of this effect. This means that many-atom van der Waals interactions cannot be ignored in refinement of protein structures, in simulations of their folding, and in prediction of their binding affinities.     

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.     

Computational protein design with side-chain conformational entropy

Recent advances in modeling protein structures at the atomic level have made it possible to tackle "de novo" computational protein design. Most procedures are based on combinatorial optimization using a scoring function that estimates the folding free energy of a protein sequence on a given main-chain structure. However, the computation of the conformational entropy in the folded state is generally an intractable problem, and its contribution to the free energy is not properly evaluated. In this article, we propose a new automated protein design methodology that incorporates such conformational entropy based on statistical mechanics principles. We define the free energy of a protein sequence by the corresponding partition function over rotamer states. The free energy is written in variational form in a pairwise approximation and minimized using the Belief Propagation algorithm. In this way, a free energy is associated to each amino acid sequence: we use this insight to rescore the results obtained with a standard minimization method, with the energy as the cost function. Then, we set up a design method that directly uses the free energy as a cost function in combination with a stochastic search in the sequence space. We validate the methods on the design of three superficial sites of a small SH3 domain, and then apply them to the complete redesign of 27 proteins. Our results indicate that accounting for entropic contribution in the score function affects the outcome in a highly nontrivial way, and might improve current computational design techniques based on protein stability.     

van der Waals forces between cylinders: Arrays of thin cylinders and three body forces

We study the van der Waals energy of interaction of an array of parallel dielectric cylinders immersed in a dielectric medium. We consider only “thin” cylinders which have radius small compared to the separation of the cylinders. The energy is calculated as a sum of two body plus three body interactions. The case of hexagonally close packed cylinders is studied in more detail. Some biophysical applications are discussed and in particular the van der Waals energy of the myosin lattice in striated muscle is examined.     

A new approach to calculation of the energy from van der Waals interactions in macromolecular proteins. Dielectric permeability as a physical parameter for calculations

The problem in the calculation of Van der Waals interactions in protein globules based on the theory of condensed media was considered. The Van der Waals interactions are represented as energies of interaction of regions with a uniform density distribution. A definition of the local dielectric constant as a function of coefficients of absorption of molecular groups with a particular conformation was introduced. The applicability of this approach was estimated. The theory enables one to circumvent the problems arising in calculations based on pairwise additive approximation. The methods provides a high accuracy in determining the local features of spatial structures of globular proteins and physicochemical characteristics of their constituent amino acids and molecular groups.