A fourier fingerprint-based method for protein surface representation

A crucial enabling technology for structural genomics is the development of algorithms that can predict the putative function of novel protein structures: the proposed functions can subsequently be experimentally tested by functional studies. Testable assignments of function can be made if it is possible to attribute a putative, or indeed probable, function on the basis of the shapes of the binding sites on the surface of a protein structure. However the comparison of the surfaces of 3D protein structures is a computationally demanding task. Here we present four surface representations that can be used locally to describe the global shape of specifically bounded local region models. The most successful of these representations is obtained by a Fourier analysis of the distribution of surface curvature on concentric spheres around a surface point and summarizes a 24 A diameter spherically clipped region of protein surface by a fingerprint of 18 Fourier amplitude values. Searching experiments using these fingerprints on a set of 366 proteins demonstrate that this provides an effective and an efficient technique for the matching of protein surfaces.     

New approach for representation of molecular surface

A new algorithm is proposed for approximation to the molecular surface. It starts with a triangular mesh built on an ellipsoid embracing the whole molecular surface. The triangular mesh is obtained from an icosahedron subdivision sphere with highly uniform vertex distribution, and the embracing surface is deflated stepwise to the best adherence of its triangles onto the surface of the molecule. The deflating direction of each vertex of a triangle is defined by the vector normal at this point to the previous deflated embracing surface. Our results show that the speed of the triangulation embracing ellipsoid method and the quality of the surface obtained by the method are faster and better than the method that starts with a quadrilateral mesh built from meridian and parallel representations on an embracing sphere to get the molecular surface. Furthermore, the surface obtained by the method can be used directly to approximate the molecular surface by spherical harmonic expansions. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1805–1815, 1998     

Triangulation algorithms for the representation of molecular surface properties

Summary A triangulation algorithm for a dotted surface (i.e. a surface defined by point coordinates in three dimensions) is given. The individual triangles are generated on the basis of a hierarchy of strategies according to increasing surface complexity. While for small molecules an elementary algorithm is sufficient to triangulate the surface, large molecules-like proteins-generally need all steps of the hierarchy. Although this program has been developed with the aim of triangulating molecular surfaces, it can in principle be applied to any surface defined by 3D point coordinates.     

Parametrization strategy for the MolFESD concept: quantitative surface representation of local hydrophobicity

We derive a new model for the established concept of the molecular free energy surface density (MolFESD) yielding a more rigorous representation of local surface contributions to the overall hydrophobicity of a molecule. The model parametrization makes efficient use of both local and global information about solvation thermodynamics, as formulated earlier for the problem of predicting free energies of hydration. The free energy of transfer is separated into an interaction contribution and a term related to the cavity formation. Interaction and cavity components are obtained from the statistical three-dimensional (3D) free energy density and a linear combination of surface and volume terms, respectively. An appropriate molecular interaction field generated by the program Grid is used as an approximate representation of the interaction part of the 3D free energy density. We further compress the 3D density by means of a linear combination of localized surface functions allowing for the derivation of local hydrophobic contributions in the form of a free energy surface density. For a set of 400 compounds our model yields significant correlation (R(2) = 0.95, sigma = 0.57) between experimental and calculated log P values. The final model is applied to establish a correlation between partial free energies of transfer for a series of sucrose derivatives and their relative sweetness, as studied earlier in the group of the authors. We find considerable improvement regarding the rms error of the regression thus validating the presented approach.     

Van der Waals surface graphs and molecular shape

A van der Waals surface graph is the graph defined on a van der Waals surface by the intersections of the atomic van der Waals spheres. A van der Waals shape graph has a vertex for each atom with a visible face on the van der Waals surface, and edges between vertices representing atoms with adjacent faces on the van der Waals surface. These are discrete invariants of three‐dimensional molecular shape. Some basic properties of van der Waals surface graphs are studied, including their relationship with the Voronoi diagram of the atom centres, and a class of molecular embeddings is identified for which the dual of the van der Waals surface graph coincides with the van der Waals shape graph. This revised version was published online in July 2006 with corrections to the Cover Date.     

Molecular mechanics and molecular shape. V. on the computation of the bare surface area of molecules

This article examines the numerical estimation of molecular surface areas within the model of overlapping atomic spheres. One has the choice of either basing the estimate on all elements that contribute to the surface, or of ignoring systematically some elements in the interatomic clefts. It is argued that the second choice, even though more approximate, implicitly improves on the model and is to be preferred. Since surface areas are not measurable, the demonstration is unavoidably roundabout, relying mostly on correlation analysis. Among the regressors occur two compounded parameters. One, ratio of the surface area of the equivalent sphere to the surface area, is interpreted as a measure of molecular globularity. It reflects the molecular axis-ratio and surface convolution. The other, ratio of the surface area to the volume, is interpreted as a measure of the global congestion of a chemical residue. Together with a measure of the local congestion at the point of attachment, it affects the steric hindrance that a residue offers. The relation between the surface area and the number of valence electrons is also discussed.     

A novel method of aligning molecules by local surface shape similarity

A novel shape-based method has been developed for overlaying a series of molecule surfaces into a common reference frame. The surfaces are represented by a set of circular patches of approximately constant curvature. Two molecules are overlaid using a clique-detection algorithm to find a set of patches in the two surfaces that correspond, and overlaying the molecules so that the similar patches on the two surfaces are coincident. The method is thus able to detect areas of local, rather than global, similarity. A consensus overlay for a group of molecules is performed by examining the scores of all pairwise overlays and performing a set of overlays with the highest scores. The utility of the method has been examined by comparing the overlaid and experimental configurations of 4 sets of molecules for which there are X-ray crystal structures of the molecules bound to a protein active site. Results for the overlays are generally encouraging. Of particular note is the correct prediction of the `reverse orientation' for ligands binding to human rhinovirus coat protein HRV14.     

Calculating the electric potential of macromolecules: A simple method for molecular surface triangulation

An improved version of the “marching-cube” method¹ is proposed for molecular surface triangulation. This new algorithm involves fewer and simpler basic building blocks and avoids the artificial gaps of the original one. Moreover, to make it applicable to the boundary element method, the procedures for the protein cavity identification and triangle reduction are also presented. The triangulation procedure was tested by incorporating it into the boundary element method (BEM) to estimate the pK_a values of subtilisin BPN′ and bovine trypsin inhibitor (BPTI). © 1995 by John Wiley & Sons, Inc.     

Potential energy surface exploration with equilibrial paths. Part I: Theory

The equilibrial path concept is further developed. Special attention is spent the symmetry conservation along equilibrial paths and symmetry-breaking. Symmetry-breaking can occur only at singular points. The simple singular points of an equilibrial path are valley–ridge inflection points. In contrast to the intrinsic reaction paths and the gradient extremal paths, the equilibrial paths enable to describe the branching of reaction channels.     

Determination of molecular properties by the method of moments. I

The method of moments appears to have some significant advantages over other variational methods of quantum chemistry if complicated molecular systems are investigated and elaborate variational wave functions are applied. The main advantages are (a) the possibility of avoiding the worst difficulties of integration or reducing the number of integrals (b) the possibility of improving the accuracy of the wave function in physically important regions of the configurational space and (c) the possibility of obtaining some random sampling type error estimates. The aim of the series of papers starting with the present one is to collect experience in applying the method of moments to problems of quantum chemistry.

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Zusammenfassung Die Methode der Momente scheint gegenüber den üblichen Variationsmethoden in der Quantenchemie einige bedeutende Vorteile zu besitzen, besonders wenn es sich um Untersuchungen komplizierter molekularer Systeme handelt, die nur mit umfangreichen Variationswellenfunktionen behandelt werden können. Die Hauptvorteile der neuen Methode sind: (a) Die bei der Integration auftretenden großen Schwierigkeiten bzw. die Anzahl der zu berechnenden Integrale können erheblich herabgesetzt werden. (b) In den physikalisch besonders wichtigen Bereichen des Konfigurationsraumes besteht die Möglichkeit, die Wellenfunktionen beträchtlich zu verbessern. (c) Es können Fehlerabschätzungen gemacht werden. Ziel dieser Arbeit ist es, erste Erfahrungen in der Anwendung der neuen Rechenmethode auf quantenchemische Probleme zu sammeln.

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Résumé La méthode des moments parait posséder des avantages nets sur les autres méthodes variationnelles de la chimie quantique si l'on étudie des systèmes moléculaires complexes et si l'on utilise des fonctions d'onde variationnelles perfectionnées. Les avantages principaux sont a) la possibilité d'éviter les difficultés les plus gênantes de l'intégration et de réduire le nombre d'intégrales; b) la possibilité d'améliorer la précision de la fonction d'onde dans les régions de l'espace de configuration les plus importantes pour une observable, et c) la possibilité d'obtenir des évaluations du type «échantillonnage au hasard» des erreurs commises. Le but de la série d'articles qu'inaugure celui-ci est de présenter une information sur l'application de la méthode des moments aux problèmes de la chimie quantique.

     

Numerical calculation of molecular surface area. I. Assessment of errors

Several points distributions have been used to calculate van der Waals surface areas of a set of molecules. It is shown that there is no strict correlation between the global statistical characteristics of the points distribution, such as deviation and standard deviation, and the accuracy of the calculation of molecular surface. Information about details of the points distribution is needed for predicting the precision of the results. The results show that points distributions produced by optimization of the U function of Le Grand and Merz [J. Comput. Chem., 14 , 349 (1993)] give the most accurate estimation of the molecular surface in numerical calculations. The precision of the numerical evaluation of the van der Waals surface areas has been assessed for 256, 512, 1024, and 2048 points on a single sphere. © 1996 by John Wiley & Sons, Inc.     

A test of the method of images at the surface of molecular materials

The method of images is tested by comparing two ways of calculating the polarization energy in crystalline fullerene C60 and in bulk amorphous polyethylene (PE): (i) treating the whole molecular material microscopically, and (ii) replacing part of the material by a uniform dielectric continuum of the same relative permittivity. The method of images is accurate to within 5% once the distance of the charge from the surface of the dielectric continuum exceeds about twice the average spacing between the polarizable units in the molecular material. For C60 crystals the method of images always overestimates the magnitude of the polarization energy, partly because it ignores the reduction in the relative permittivity of the dielectric continuum near its surface. For amorphous PE the method of images can overestimate or underestimate the true result, depending on the local density around the charge.     

Non-Hamiltonian molecular dynamics implementation of the Gibbs ensemble method. I. Algorithm

In this paper (paper I) and the following paper (paper II) [C. Bratschi, H. Huber, and D. J. Searles, J. Chem. Phys. 126, 164105 (2007)], a new molecular dynamics algorithm implementing the Gibbs ensemble will be presented and then on applied to the liquid-vapor coexistence curve for two ab initio CO2 potentials. In paper I, the Gibbs ensemble molecular dynamics algorithm using non-Hamiltonian molecular dynamics techniques is introduced. It is shown that states of the correct probability density function are sampled and the correct exchange probability is generated. The extended system Nose-Hoover formalism is used to generate a constant temperature ensemble with equal pressures in the subsystems, combined with single particle transfers between the subsystems, over several time steps, to get equal chemical potentials.     

Orientation and molecular dynamics in uniaxial polymers. I. Theory of fluorescence polarization

A theory of polarized fluorescence is developed for uniaxial physical systems in which micro-Brownian motion is not negligible. All experimental information is shown to reduce, with reasonable assumptions, to four quantities characterizing molecular orientation and reorientational molecular mobility. The geometrical properties of the mobility in uniaxial systems are studied and methods are given for correcting for the effects of the birefringence and for a possible delocalization of the fluorescence transition moments.