Analytically defined surfaces to analyze molecular interaction properties

The Area Law of Molecular Entropy: Moving beyond Harmonic Approximation

This article shows that the gas-phase entropy of molecules is proportional to the area of the molecules, with corrections for the different curvatures of the molecular surface. The ability to estimate gas-phase entropy by the area law also allows us to calculate molecular entropy faster and more accurately than currently popular methods of estimating molecular entropy with harmonic oscillator approximation. The speed and accuracy of our method will open up new possibilities for the explicit inclusion of entropy in various computational biology methods.

Correlation of the Molecular Cross-Sectional Area of Organic Monofunctional Compounds with Topological Descriptors

The molecular cross-sectional area (σ) has proved to be an interesting molecular measure not only in the field of adsorption phenomena on solids but also in biochemistry, physiology, or surfactant chemistry. The existing methods to estimate the cross-sectional areas are either not readily applicable or can only be applied to a limited number of compounds. The aim of this work was to describe a method, as general as possible, quick and easy to perform. To that end, the molecular cross-sectional areas were correlated with topological indices. The Emmett-Brunauer formula was used to calculate the reference cross-sectional areas (σEB) of 431 compounds. The correlations of the Wiener (W), hyper-Wiener (WW), Balaban (J), and Randić (χ) indices with σEB were compared for n-alkanes as well as branched and cyclic alkanes. Only the Wiener and hyper-Wiener indices correlated well with σEB, with the data being best fitted by power law regression curves. The lower degeneracy of the hyper-Wiener index did not translate into any significant gain of performance when correlated with σEB. Following the parsimony principle, the less complex Wiener index was thus selected to correlate with the σEB of compounds representing 31 other monofunctional and structural families. The integration of all the compound families into a single curve allowed a quick rough estimation of the cross-sectional areas. The specific reference equations σEB = qWp were determined for the 34 selected families, allowing the fast and reliable calculation of the cross-sectional area of any monofunctional compound.

CavVis-A Field-of-View Geometric Algorithm for Protein Cavity Detection

Several geometric-based methods have been developed for the last two to three decades to detect and identify cavities (i.e., putative binding sites) on proteins, as needed to study protein-ligand interactions and protein docking. This paper introduces a new protein cavity method, called CavVis, which combines voxelization (i.e., a grid of voxels) and an analytic formulation of Gaussian surfaces that approximates the solvent-excluded surface. This method builds upon visibility of points on protein surface to find its cavities. Specifically, the visibility criterion combines three concepts we borrow from computer graphics, the field-of-view of each surface point, voxel ray casting, and back-face culling.

GPU-based detection of protein cavities using Gaussian surfaces

Background

Protein cavities play a key role in biomolecular recognition and function, particularly in protein-ligand interactions, as usual in drug discovery and design. Grid-based cavity detection methods aim at finding cavities as aggregates of grid nodes outside the molecule, under the condition that such cavities are bracketed by nodes on the molecule surface along a set of directions (not necessarily aligned with coordinate axes). Therefore, these methods are sensitive to scanning directions, a problem that we call cavity ground-and-walls ambiguity, i.e., they depend on the position and orientation of the protein in the discretized domain. Also, it is hard to distinguish grid nodes belonging to protein cavities amongst all those outside the protein, a problem that we call cavity ceiling ambiguity.

Results

We solve those two ambiguity problems using two implicit isosurfaces of the protein, the protein surface itself (called inner isosurface) that excludes all its interior nodes from any cavity, and the outer isosurface that excludes most of its exterior nodes from any cavity. Summing up, the cavities are formed from nodes located between these two isosurfaces. It is worth noting that these two surfaces do not need to be evaluated (i.e., sampled), triangulated, and rendered on the screen to find the cavities in between; their defining analytic functions are enough to determine which grid nodes are in the empty space between them.

Conclusion

This article introduces a novel geometric algorithm to detect cavities on the protein surface that takes advantage of the real analytic functions describing two Gaussian surfaces of a given protein.

High-order fractional partial differential equation transform for molecular surface construction

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

F2Dock: fast Fourier protein-protein docking

The functions of proteins are often realized through their mutual interactions. Determining a relative transformation for a pair of proteins and their conformations which form a stable complex, reproducible in nature, is known as docking. It is an important step in drug design, structure determination, and understanding function and structure relationships. In this paper, we extend our nonuniform fast Fourier transform-based docking algorithm to include an adaptive search phase (both translational and rotational) and thereby speed up its execution. We have also implemented a multithreaded version of the adaptive docking algorithm for even faster execution on multicore machines. We call this protein-protein docking code F2Dock (F2 = Fast Fourier). We have calibrated F2Dock based on an extensive experimental study on a list of benchmark complexes and conclude that F2Dock works very well in practice. Though all docking results reported in this paper use shape complementarity and Coulombic-potential-based scores only, F2Dock is structured to incorporate Lennard-Jones potential and reranking docking solutions based on desolvation energy .

The tumour necrosis factor as a mediator of vessel inflammation: importance of exposed receptor residues for its neutralization

BACKGROUND

The analysis of the macromolecular tumour necrosis factor (TNF)-receptor interface helps to understand the antigenicity of this inflammatory protein.

METHOD

The calculations are based on structural data from the protein database. The residues of the macromolecular interface are identified in the interface contact matrix, a plot of pair-wise interactions between adjacent residues in the TNF-receptor complex. Starting from the matrix elements, the most exposed residues of the receptor, together with their relative contribution to the interface, are determined. This is done by Voronoi tessellation, a unique and well defined partition of the protein into polyhedral cells defining the proprietary space of the associated amino acid and its contact faces with neighboured residue cells.

RESULTS

Several interfacial receptor residues, contributing with a total amount of 63% to the macromolecular interface, could be identified.

CONCLUSION

Based on the assumption that residues with higher interfacial exposure values are playing the most important role in TNF-receptor complex, they will be the original material for further developments in engineering more efficient TNF blockers.

Shape complementarity of protein-protein complexes at multiple resolutions

Biological complexes typically exhibit intermolecular interfaces of high shape complementarity. Many computational docking approaches use this surface complementarity as a guide in the search for predicting the structures of protein-protein complexes. Proteins often undergo conformational changes to create a highly complementary interface when associating. These conformational changes are a major cause of failure for automated docking procedures when predicting binding modes between proteins using their unbound conformations. Low resolution surfaces in which high frequency geometric details are omitted have been used to address this problem. These smoothed, or blurred, surfaces are expected to minimize the differences between free and bound structures, especially those that are due to side chain conformations or small backbone deviations. Despite the fact that this approach has been used in many docking protocols, there has yet to be a systematic study of the effects of such surface smoothing on the shape complementarity of the resulting interfaces. Here we investigate this question by computing shape complementarity of a set of 66 protein-protein complexes represented by multiresolution blurred surfaces. Complexed and unbound structures are available for these protein-protein complexes. They are a subset of complexes from a nonredundant docking benchmark selected for rigidity (i.e. the proteins undergo limited conformational changes between their bound and unbound states). In this work, we construct the surfaces by isocontouring a density map obtained by accumulating the densities of Gaussian functions placed at all atom centers of the molecule. The smoothness or resolution is specified by a Gaussian fall-off coefficient, termed "blobbyness." Shape complementarity is quantified using a histogram of the shortest distances between two proteins' surface mesh vertices for both the crystallographic complexes and the complexes built using the protein structures in their unbound conformation. The histograms calculated for the bound complex structures demonstrate that medium resolution smoothing (blobbyness = -0.9) can reproduce about 88% of the shape complementarity of atomic resolution surfaces. Complexes formed from the free component structures show a partial loss of shape complementarity (more overlaps and gaps) with the atomic resolution surfaces. For surfaces smoothed to low resolution (blobbyness = -0.3), we find more consistency of shape complementarity between the complexed and free cases. To further reduce bad contacts without significantly impacting the good contacts we introduce another blurred surface, in which the Gaussian densities of flexible atoms are reduced. From these results we discuss the use of shape complementarity in protein-protein docking.

Protein-protein interfaces: properties, preferences, and projections

Herein, we study the interfaces of a set of 146 transient protein-protein interfaces in order to better understand the principles of their interactions. We define and generate the protein interface using tools from computational geometry and topology and then apply statistical analysis to its residue composition. In addition to counting individual occurrences, we evaluate pairing preferences, both across and as neighbors on one side of an interface. Likelihood correction emphasizes novel and unexpected pairs, such as the His-Cys pair found in most complexes of serine proteases with their diverse inhibitors and the Met-Met neighbor pair found in unrelated protein interfaces. We also present a visualization of the protein interface that allows for facile identification of residue-residue contacts and other biochemical properties.

Quality Meshing of Implicit Solvation Models of Biomolecular Structures

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and binding rate constants.

What role do surfaces play in GB models? A new-generation of surface-generalized born model based on a novel gaussian surface for biomolecules

We have developed a version of our surface generalized Born (SGB) model that employs a Gaussian surface, as opposed to the van der Waals surface used previously. The Gaussian surface is smooth and its properties are analytically differentiable with respect to the positions of atoms. A significant advantage of a solvent model based on this analytically differentiable surface is the availability of analytical gradients of the surface and solvation forces. An efficient and robust algorithm is designed to construct and triangulate the Gaussian surface for large biomolecules with arbitrary shapes, and to compute the various terms required for energy gradients. The Gaussian surface is shown to better mimic the boundary between the solute and solvent by properly addressing solvent accessibility, as is demonstrated by comparisons with standard Poisson-Boltzmann calculations for proteins of different sizes. These results also demonstrate that surface definition is a dominant contribution to differences between GB and PB calculations, especially if the system is large. Application of the new surface to prediction of long loop regions is presented, and significant improvement in the energetics is seen compared with results obtained using the van der Waals surface, even in the absence of optimized empirical correction terms that were used in the latter calculations.

Intersurf: dynamic interface between proteins

Protein docking is a fundamental biological process that links two proteins. This link is typically defined by an interaction between two large zones of the protein boundaries. Visualizing such an interface is useful to understand the process thanks to 3D protein structures, to estimate the quality of docking simulation results, and to classify interactions in order to predict docking affinity between classes of interacting zones. Since the interface may be defined by a surface that separates the two proteins, it is possible to create a map of interaction that allows comparisons to be performed in 2D. This paper presents a very fast algorithm that extracts an interface surface and creates a valid and low-distorted interaction map. Another benefit of our approach is that a pre-computed part of the algorithm enables the surface to be updated in real-time while residues are moved.

Real spherical harmonic expansion coefficients as 3D shape descriptors for protein binding pocket and ligand comparisons

MOTIVATION

An increasing number of protein structures are being determined for which no biochemical characterization is available. The analysis of protein structure and function assignment is becoming an unexpected challenge and a major bottleneck towards the goal of well-annotated genomes. As shape plays a crucial role in biomolecular recognition and function, the examination and development of shape description and comparison techniques is likely to be of prime importance for understanding protein structure-function relationships.

RESULTS

A novel technique is presented for the comparison of protein binding pockets. The method uses the coefficients of a real spherical harmonics expansion to describe the shape of a protein's binding pocket. Shape similarity is computed as the L2 distance in coefficient space. Such comparisons in several thousands per second can be carried out on a standard linux PC. Other properties such as the electrostatic potential fit seamlessly into the same framework. The method can also be used directly for describing the shape of proteins and other molecules.

AVAILABILITY

A limited version of the software for the real spherical harmonics expansion of a set of points in PDB format is freely available upon request from the authors. Binding pocket comparisons and ligand prediction will be made available through the protein structure annotation pipeline Profunc (written by Roman Laskowski) which will be accessible from the EBI website shortly.

e-molecular shapes and properties

Due to recent computer technology advances, shape analysis has gained importance in all domains. In drug design and proteomics, molecular surfaces (van der Waals surface, solvent accessible surface, solvent excluded surface, polar surface area, electron density surface, separating surface, etc.), buried surfaces (gap, cleft, cavity, etc.) as well as shape properties of these surfaces, can be easily computed and visualized via the Internet. Freely available resources from the Internet for academic use, are reviewed.

MolSurfer: A macromolecular interface navigator

We describe the current status of the Java molecular graphics tool, MolSurfer. MolSurfer has been designed to assist the analysis of the structures and physico-chemical properties of macromolecular interfaces. MolSurfer provides a coupled display of two-dimensional (2D) maps of the interfaces generated with the ADS software and a three-dimensional (3D) view of the macromolecular structure in the Java PDB viewer, WebMol. The interfaces are analytically defined and properties such as electrostatic potential or hydrophobicity are projected on to them. MolSurfer has been applied previously to analyze a set of 39 protein-protein complexes, with structures available from the Protein Data Bank (PDB). A new application, described here, is the visualization of 75 interfaces in structures of protein-DNA and protein-RNA complexes. Another new feature is that the MolSurfer web server is now able to compute and map Poisson-Boltzmann electrostatic potentials of macromolecules onto interfaces. The MolSurfer web server is available at http://projects.villa-bosch.de/mcm/software/molsurfer.

Application of the frozen atom approximation to the GB/SA continuum model for solvation free energy

The generalized Born/surface area (GB/SA) continuum model for solvation free energy is a fast and accurate alternative to using discrete water molecules in molecular simulations of solvated systems. However, computational studies of large solvated molecular systems such as enzyme-ligand complexes can still be computationally expensive even with continuum solvation methods simply because of the large number of atoms in the solute molecules. Because in such systems often only a relatively small portion of the system such as the ligand binding site is under study, it becomes less attractive to calculate energies and derivatives for all atoms in the system. To curtail computation while still maintaining high energetic accuracy, atoms distant from the site of interest are often frozen; that is, their coordinates are made invariant. Such frozen atoms do not require energetic and derivative updates during the course of a simulation. Herein we describe methodology and results for applying the frozen atom approach to both the generalized Born (GB) and the solvent accessible surface area (SASA) parts of the GB/SA continuum model for solvation free energy. For strictly pairwise energetic terms, such as the Coulombic and van-der-Waals energies, contributions from pairs of frozen atoms can be ignored. This leaves energetic differences unaffected for conformations that vary only in the positions of nonfrozen atoms. Due to the nonlocal nature of the GB analytical form, however, excluding such pairs from a GB calculation leads to unacceptable inaccuracies. To apply a frozen-atom scheme to GB calculations, a buffer region within the frozen-atom zone is generated based on a user-definable cutoff distance from the nonfrozen atoms. Certain pairwise interactions between frozen atoms in the buffer region are retained in the GB computation. This allows high accuracy in conformational GB comparisons to be maintained while achieving significant savings in computational time compared to the full (nonfrozen) calculation. A similar approach for using a buffer region of frozen atoms is taken for the SASA calculation. The SASA calculation is local in nature, and thus exact SASA energies are maintained. With a buffer region of 8 A for the frozen-atom cases, excellent agreement in differences in energies for three different conformations of cytochrome P450 with a bound camphor ligand are obtained with respect to the nonfrozen cases. For various minimization protocols, simulations run 2 to 10.5 times faster and memory usage is reduced by a factor of 1.5 to 5. Application of the frozen atom method for GB/SA calculations thus can render computationally tractable biologically and medically important simulations such as those used to study ligand-receptor binding conformations and energies in a solvated environment.

Approximate solvent-accessible surface areas from tetrahedrally directed neighbor densities

A fast analytical formula (TDND) has been derived for the calculation of approximate atomic and molecular solvent-accessible surface areas (SASA), as well as the first and second derivatives of these quantities with respect to atomic coordinates. Extending the work of Stouten et al. (Molecular Simulation, 1993, Vol. 10, pp. 97-120), as well as our own (Journal of Computational Chemistry, 1999, Vol. 20, pp. 586-596), the method makes use of a Gaussian function to calculate the neighbor density in four tetrahedral directions in three-dimensional space, sometimes twice with different orientations. SASA and first derivatives of the 2366 heavy atoms of penicillopepsin are computed in 0.13 s on an SGI R10000/194 MHz processor. When second derivatives are computed as well, the total time is 0.23 s. This is considerably faster than timings reported previously for other algorithms. Based on a parameterization set of nineteen compounds of different size (11-4346 atoms) and class (organics, proteins, DNA, and various complexes) consisting of a total 23,197 atoms, the method exhibits relative errors in the range 0.2-12.6% for total molecular surface areas and average absolute atomic surface area deviations in the range 1.7 to 3.6 A(2).

Classification of auxin plant hormones by interaction property similarity indices

Although auxins were the first type of plant hormone to be identified, little is known about the molecular mechanism of this important class of plant hormones. We present a classification of a set of about 50 compounds with measured auxin activities, according to their interaction properties. Four classes of compounds were defined: strongly active, weakly active with weak antiauxin behaviour, inactive and inhibitory. All compounds were modeled in two low-energy conformations, 'P' and 'T', so as to obtain the best match to the 'planar' and 'tilted' conformations, respectively, of indole 3-acetic acid. Each set of conformers was superimposed separately using several different alignment schemes. Molecular interaction energy fields were computed for each molecule with five different chemical probes and then compared by computing similarity indices. Similarity analysis showed that the classes are on average distinguishable, with better differentiation achieved for the T conformers than the P conformers. This indicates that the T conformation might be the active one. Further, a screening was developed which could distinguish compounds with auxin activity from inactive compounds and most antiauxins using the T conformers. The classifications rationalize ambiguities in activity data found in the literature and should be of value in predicting the activities of new plant growth substances and herbicides.