Dissecting the roles of local packing density and longer‐range effects in protein sequence evolution

What are the structural determinants of protein sequence evolution? A number of site‐specific structural characteristics have been proposed, most of which are broadly related to either the density of contacts or the solvent accessibility of individual residues. Most importantly, there has been disagreement in the literature over the relative importance of solvent accessibility and local packing density for explaining site‐specific sequence variability in proteins. We show that this discussion has been confounded by the definition of local packing density. The most commonly used measures of local packing, such as contact number and the weighted contact number, represent the combined effects of local packing density and longer‐range effects. As an alternative, we propose a truly local measure of packing density around a single residue, based on the Voronoi cell volume. We show that the Voronoi cell volume, when calculated relative to the geometric center of amino‐acid side chains, behaves nearly identically to the relative solvent accessibility, and each individually can explain, on average, approximately 34% of the site‐specific variation in evolutionary rate in a data set of 209 enzymes. An additional 10% of variation can be explained by nonlocal effects that are captured in the weighted contact number. Consequently, evolutionary variation at a site is determined by the combined effects of the immediate amino‐acid neighbors of that site and effects mediated by more distant amino acids. We conclude that instead of contrasting solvent accessibility and local packing density, future research should emphasize on the relative importance of immediate contacts and longer‐range effects on evolutionary variation. Proteins 2016; 84:841–854. © 2016 Wiley Periodicals, Inc.     

Calculation of standard atomic volumes for RNA and comparison with proteins: RNA is packed more tightly

Traditionally, for biomolecular packing calculations research has focused on proteins. Besides proteins, RNA is the other large biomolecule that has tertiary structure interactions and complex packing. No one has yet quantitatively investigated RNA packing or compared its packing to that of proteins because, until recently, there were no large RNA structures. Here we address this question in detail, using Voronoi volume calculations on a set of high-resolution RNA crystal structures. We do a careful parameterization, taking into account many factors such as atomic radii, crystal packing, structural complexity, solvent, and associated protein to obtain a self-consistent, universal set of volumes that can be applied to both RNA and protein. We report this set of volumes, which we call the NucProt parameter set. Our measured values are consistent across the many different RNA structures and packing environments. When common atom types are compared between proteins and RNA, nine of 12 types show that RNA has a smaller volume and packs more tightly than protein, suggesting that close-packing may be as important for the folding of RNAs as it is for proteins. Moreover, calculated partial specific volumes show that RNA bases pack more densely than corresponding aromatic residues from proteins. Finally, we find that RNA bases have similar packing volumes to DNA bases, despite the absence of tertiary contacts in DNA. Programs, parameter sets and raw data are available online at.     

Determining the minimum number of types necessary to represent the sizes of protein atoms.

MOTIVATION: Traditionally, for packing calculations people have collected atoms together into a number of distinct 'types'. These, in fact, often represent a heavy atom and its associated hydrogens (i.e. a united atom). Also, atom typing is usually done according to basic chemistry, giving rise to 20-30 protein atom types, such as carbonyl carbons, methyl groups, and hydroxyl groups. No one has yet investigated how similar in packing these chemically derived types are. Here we address this question in detail, using Voronoi volume calculations on a set of high-resolution crystal structures. RESULTS: We perform a rigorous clustering analysis with cross-validation on tens of thousands of atom volumes and attempt to compile them into types based purely on packing. From our analysis, we are able to determine a 'minimal' set of 18 atom types that most efficiently represent the spectrum of packing in proteins. Furthermore, we are able to uncover a number of inconsistencies in traditional chemical typing schemes, where differently typed atoms have almost the same effective size. In particular, we find that tetrahedral carbons with two hydrogens are almost identical in size to many aromatic carbons with a single hydrogen. AVAILABILITY: Programs available from http://geometry.molmovdb.org. CONTACT: JerryTsai@TAMU.edu; neil.voss@yale.edu; Mark.Gerstein@yale.edu SUPPLEMENTARY INFORMATION: Available at http://geometry.molmovdb.org.     

Repulsive interactions between polar and apolar atoms in globular proteins

A significant number of tetrahedral carbon-backbone nitrogen pairs in ferrocytochrome c have interatomic distances that are more than 0.4 Å shorter than the sum of the atomic van der Waals' radii. The non-bonded repulsions of these pairs destabilize the native structure by as much as 50 to 60 kcal/mol. A detailed examination of the close contacts suggests that these result from packing forces associated with the formation of secondary and tertiary structure in the protein.     

Atomic contacts in protein structures. A detailed analysis of atomic radii, packing, and overlaps

A rigorous quantitative assessment of atomic contacts and packing in native protein structures is presented. The analysis is based on optimized atomic radii derived from a set of high-resolution protein structures and reveals that the distribution of atomic contacts and overlaps is a structural constraint in proteins, irrespective of structural or functional classification and size. Furthermore, a newly developed method for calculating packing properties is introduced and applied to sets of protein structures at different levels of resolution. The results show that limited resolution yields decreasing packing quality, which underscores the relevance of packing considerations for structure prediction, design, dynamics, and docking.     

Geometrical analysis of protein-DNA interactions on the basis of the Voronoĭ-Delaune tessellation

A new method for the rigorous analysis of DNA-protein contacts has been developed on the basis of Voronoi tessellation. This method permits one to determine close neighbors on the atomic level and compute the area of contact by edges of Voronoi polyhedra. Based on the results of the study of 1109 protein-DNA complexes from PDB, it was demonstrated that about one third of the contacts are the contacts with positively charged Arg and Lys. There is distinct amino acid prevalence for nucleotides: for A - Pro; for T - His; for G - Asp; for C - Trp, Asp, Glu. Therefore, GC pairs prefer to interact with negatively charged residues, and alanine and methionine, whereas AT pairs prefer to interact with histidine and unpolar residues.     

Assessing methods for identifying pair-wise atomic contacts across binding interfaces

An essential step in understanding the molecular basis of protein-protein interactions is the accurate identification of inter-protein contacts. We evaluate a number of common methods used in analyzing protein-protein interfaces: a Voronoi polyhedra-based approach, changes in solvent accessible surface area (DeltaSASA) and various radial cutoffs (closest atom, Cbeta, and centroid). First, we compared the Voronoi polyhedra-based analysis to the DeltaSASA and show that using Voronoi polyhedra finds knob-in-hole contacts. To assess the accuracy between the Voronoi polyhedra-based approach and the various radial cutoff methods, two sets of data were used: a small set of 75 experimental mutants and a larger one of 592 structures of protein-protein interfaces. In an assessment using the small set, the Voronoi polyhedra-based methods, a solvent accessible surface area method, and the closest atom radial method identified 100% of the direct contacts defined by mutagenesis data, but only the Voronoi polyhedra-based method found no false positives. The other radial methods were not able to find all of the direct contacts even using a cutoff of 9A. With the larger set of structures, we compared the overall number contacts using the Voronoi polyhedra-based method as a standard. All the radial methods using a 6-A cutoff identified more interactions, but these putative contacts included many false positives as well as missed many false negatives. While radial cutoffs are quicker to calculate as well as to implement, this result highlights why radial cutoff methods do not have the proper resolution to detail the non-homogeneous packing within protein interfaces, and suggests an inappropriate bias in pair-wise contact potentials. Of the radial cutoff methods, using the closest atom approach exhibits the best approximation to the more intensive Voronoi calculation. Our version of the Voronoi polyhedra-based method QContacts is available at .     

Nonatomic solvent-driven Voronoi tessellation of proteins: an open tool to analyze protein folds

A three-dimensional Voronoi tessellation of folded proteins is used to analyze geometrical and topological properties of a set of proteins. To each amino acid is associated a central point surrounded by a Voronoi cell. Voronoi cells describe the packing of the amino acids. Special attention is given to reproduction of the protein surface. Once the Voronoi cells are built, a lot of tools from geometrical analysis can be applied to investigate the protein structure; volume of cells, number of faces per cell, and number of sides per face are the usual signatures of the protein structure. A distinct difference between faces related to primary, secondary, and tertiary structures has been observed. Faces threaded by the main-chain have on average more than six edges, whereas those related to helical packing of the amino acid chain have less than five edges. The faces on the protein surface have on average five edges within 1% error. The average number of faces on the protein surface for a given type of amino acid brings a new point of view in the characterization of the exposition to the solvent and the classification of amino acid as hydrophilic or hydrophobic. It may be a convenient tool for model validation.     

Understanding the recognition of protein structural classes by amino acid composition

Knowledge of amino acid composition, alone, is verified here to be sufficient for recognizing the structural class, α, β, α+β, or α/β of a given protein with an accuracy of 81%. This is supported by results from exhaustive enumerations of all conformations for all sequences of simple, compact lattice models consisting of two types (hydrophobic and polar) of residues. Different compositions exhibit strong affinities for certain folds. Within the limits of validity of the lattice models, two factors appear to determine the choice of particular folds: 1) the coordination numbers of individual sites and 2) the size and geometry of non-bonded clusters. These two properties, collectively termed the distribution of non-bonded contacts, are quantitatively assessed by an eigenvalue analysis of the so-called Kirchhoff or adjacency matrices obtained by considering the non-bonded interactions on a lattice. The analysis permits the identification of conformations that possess the same distribution of non-bonded contacts. Furthermore, some distributions of non-bonded contacts are favored entropically, due to their high degeneracies. Thus, a competition between enthalpic and entropic effects is effective in determining the choice of a distribution for a given composition. Based on these findings, an analysis of non-bonded contacts in protein structures was made. The analysis shows that proteins belonging to the four distinct folding classes exhibit significant differences in their distributions of non-bonded contacts, which more directly explains the success in predicting structural class from amino acid composition. Proteins 29:172–185, 1997. Published 1997 Wiley-Liss, Inc.

1 This article is a US Goverment work and, as such, is in the public domain in the United States of America.

     

Molecular packing and packing defects in helical membrane proteins

The packing of helices spanning lipid bilayers is crucial for the stability and function of alpha-helical membrane proteins. Using a modified Voronoi procedure, we calculated packing densities for helix-helix contacts in membrane spanning domains. Our results show that the transmembrane helices of protein channels and transporters are significantly more loosely packed compared with helices in globular proteins. The observed packing deficiencies of these membrane proteins are also reflected by a higher amount of cavities at functionally important sites. The cavities positioned along the gated pores of membrane channels and transporters are noticeably lined by polar amino acids that should be exposed to the aqueous medium when the protein is in the open state. In contrast, nonpolar amino acids surround the cavities in those protein regions where large rearrangements are supposed to take place, as near the hinge regions of transporters or at restriction sites of protein channels. We presume that the observed deficiencies of helix-helix packing are essential for the helical mobility that sustains the function of many membrane protein channels and transporters.     

New techniques for three-dimensional data analysis in histopathology.

This paper describes two three-dimensional (3D) analytical techniques based on 3D mathematical morphology that have been found useful in quantifying the 3D spatial distribution of S-phase cells in a tubular tumor of the human breast. One technique is based on determining the normalized radial distribution of the S-phase cells with respect to the central axis of the tumor. The other technique is a novel extension of the polyhedra of Voronoi to quantify the distribution. The Voronoi polyhedron of a given S-phase cell nucleus is that polyhedron of minimal volume defined by planes all of which are perpendicular bisectors of the vectors extending from the given cell to all other S-phase cells in the tumor. Methods are demonstrated for generating these polyhedra and for histogramming their volumes. An illustration is given of using the histogram to sort the S-phase cells according to their 3D positional relationships. Displays showing the sorted cells in 3D and their associated Voronoi polyhedra are provided.     

Inhomogeneous molecular density: reference packing densities and distribution of cavities within proteins

MOTIVATION: There is no consensus in the literature about how the deepest portions of protein structures are packed. Using an improved Voronoi procedure, we calculate reference packing densities for different regions in the protein interior. Furthermore, we want to clarify where cavities are located. RESULTS: Sets of reference packing densities are provided for regions in proteins that differ in their distance to the surface and to internal cavities, supplementing previous data. Packing in the protein interior is tight but generally inhomogeneous. There are about 4.4 cavities per 100 amino acids in protein structures, they occur in all regions, most frequently in a depth of 2.5-3.6 A underneath the Connolly surface. However, the deepest protein regions have a lower mean packing density than circumjacent regions, because more contacts to cavities occur in the core. AVAILABILITY/SUPPLEMENTARY INFORMATION: Calculation software and detailed packing data are available on request.     

Predicting Transmembrane Helix Packing Arrangements using Residue Contacts and a Force-Directed Algorithm

Alpha-helical transmembrane proteins constitute roughly 30% of a typical genome and are involved in a wide variety of important biological processes including cell signalling, transport of membrane-impermeable molecules and cell recognition. Despite significant efforts to predict transmembrane protein topology, comparatively little attention has been directed toward developing a method to pack the helices together. Here, we present a novel approach to predict lipid exposure, residue contacts, helix-helix interactions and finally the optimal helical packing arrangement of transmembrane proteins. Using molecular dynamics data, we have trained and cross-validated a support vector machine (SVM) classifier to predict per residue lipid exposure with 69% accuracy. This information is combined with additional features to train a second SVM to predict residue contacts which are then used to determine helix-helix interaction with up to 65% accuracy under stringent cross-validation on a non-redundant test set. Our method is also able to discriminate native from decoy helical packing arrangements with up to 70% accuracy. Finally, we employ a force-directed algorithm to construct the optimal helical packing arrangement which demonstrates success for proteins containing up to 13 transmembrane helices. This software is freely available as source code from http://bioinf.cs.ucl.ac.uk/memsat/mempack/.     

Are proteins well-packed?

The average packing density inside proteins is as high as in crystalline solids. Does this mean proteins are well-packed? We go beyond average densities, and look at the full distribution functions of free volumes inside proteins. Using a new and rigorous Delaunay triangulation method for parsing space into empty and filled regions, we introduce formal definitions of interior and surface packing densities. Although proteins look like organic crystals by the criterion of average density, they look more like liquids and glasses by the criterion of their free volume distributions. The distributions are broad, and the scalings of volume-to-surface, volume-to-cluster-radius, and numbers of void versus volume show that the interiors of proteins are more like randomly packed spheres near their percolation threshold than like jigsaw puzzles. We find that larger proteins are packed more loosely than smaller proteins. And we find that the enthalpies of folding (per amino acid) are independent of the packing density of a protein, indicating that van der Waals interactions are not a dominant component of the folding forces.     

Cavities and Atomic Packing in Protein Structures and Interfaces

A comparative analysis of cavities enclosed in a tertiary structure of proteins and interfaces formed by the interaction of two protein subunits in obligate and non-obligate categories (represented by homodimeric molecules and heterocomplexes, respectively) is presented. The total volume of cavities increases with the size of the protein (or the interface), though the exact relationship may vary in different cases. Likewise, for individual cavities also there is quantitative dependence of the volume on the number of atoms (or residues) lining the cavity. The larger cavities tend to be less spherical, solvated, and the interfaces are enriched in these. On average 15 ų of cavity volume is found to accommodate single water, with another 40–45 ų needed for each additional solvent molecule. Polar atoms/residues have a higher propensity to line solvated cavities. Relative to the frequency of occurrence in the whole structure (or interface), residues in β-strands are found more often lining the cavities, and those in turn and loop the least. Any depression in one chain not complemented by a protrusion in the other results in a cavity in the protein–protein interface. Through the use of the Voronoi volume, the packing of residues involved in protein–protein interaction has been compared to that in the protein interior. For a comparable number of atoms the interface has about twice the number of cavities relative to the tertiary structure.     

Protein packing: dependence on protein size, secondary structure and amino acid composition

We have used the occluded surface algorithm to estimate the packing of both buried and exposed amino acid residues in protein structures. This method works equally well for buried residues and solvent-exposed residues in contrast to the commonly used Voronoi method that works directly only on buried residues. The atomic packing of individual globular proteins may vary significantly from the average packing of a large data set of globular proteins. Here, we demonstrate that these variations in protein packing are due to a complex combination of protein size, secondary structure composition and amino acid composition. Differences in protein packing are conserved in protein families of similar structure despite significant sequence differences. This conclusion indicates that quality assessments of packing in protein structures should include a consideration of various parameters including the packing of known homologous proteins. Also, modeling of protein structures based on homologous templates should take into account the packing of the template protein structure.     

Quantification of protein surfaces,volumes and atom-atom contacts using a constrained Voronoi procedure

MOTIVATION: Geometric representations of proteins and ligands, including atom volumes, atom-atom contacts and solvent accessible surfaces, can be used to characterize interactions between and within proteins, ligands and solvent. Voronoi algorithms permit quantification of these properties by dividing structures into cells with a one-to-one correspondence with constituent atoms. As there is no generally accepted measure of atom-atom contacts, a continuous analytical representation of inter-atomic contacts will be useful. Improved geometric algorithms will also be helpful in increasing the speed and accuracy of iterative modeling algorithms. RESULTS: We present computational methods based on the Voronoi procedure that provide rapid and exact solutions to solvent accessible surfaces, volumes, and atom contacts within macromolecules. Furthermore, we define a measure of atom-atom contact that is consistent with the calculation of solvent accessible surfaces, allowing the integration of solvent accessibility and inter-atomic contacts into a continuous measure. The speed and accuracy of the algorithm is compared to existing methods for calculating solvent accessible surfaces and volumes. The presented algorithm has a reduced execution time and greater accuracy compared to numerical and approximate analytical surface calculation algorithms, and a reduced execution time and similar accuracy to existing Voronoi procedures for calculating atomic surfaces and volumes.