Statistically based reduced representation of amino acid side chains

Preferred conformations of amino acid side chains have been well established through statistically obtained rotamer libraries. Typically, these provide bond torsion angles allowing a side chain to be traced atom by atom. In cases where it is desirable to reduce the complexity of a protein representation or prediction, fixing all side-chain atoms may prove unwieldy. Therefore, we introduce a general parametrization to allow positions of representative atoms (in the present study, these are terminal atoms) to be predicted directly given backbone atom coordinates. Using a large, culled data set of amino acid residues from high-resolution protein crystal structures, anywhere from 1 to 7 preferred conformations were observed for each terminal atom of the non-glycine residues. Side-chain length from the backbone C(alpha) is one of the parameters determined for each conformation, which should itself be useful. Prediction of terminal atoms was then carried out for a second, nonredundant set of protein structures to validate the data set. Using four simple probabilistic approaches, the Monte Carlo style prediction of terminal atom locations given only backbone coordinates produced an average root mean-square deviation (RMSD) of approximately 3 A from the experimentally determined terminal atom positions. With prediction using conditional probabilities based on the side-chain chi(1) rotamer, this average RMSD was improved to 1.74 A. The observed terminal atom conformations therefore provide reasonable and potentially highly accurate representations of side-chain conformation, offering a viable alternative to existing all-atom rotamers for any case where reduction in protein model complexity, or in the amount of data to be handled, is desired. One application of this representation with strong potential is the prediction of charge density in proteins. This would likely be especially valuable on protein surfaces, where side chains are much less likely to be fixed in single rotamers. Prediction of ensembles of structures provides a method to determine the probability density of charge and atom location; such a prediction is demonstrated graphically.     

Spinor product computations for protein conformations

Spinor operators in geometric algebra (GA) can efficiently describe conformational changes of proteins by ordered products that act on individual bonds and represent their net rotations. Backward propagation through the protein backbone yields all rotational spinor axes in advance allowing the efficient computation of atomic coordinates from internal coordinates. The introduced mathematical framework enables to efficiently manipulate and generate protein conformations to any arbitrary degree. Moreover, several new formulations in the context of rigid body motions are added. Emphasis is placed on the intimate relationship between spinors and quaternions, which can be recovered from within the GA approach. The spinor methodology is implemented and tested versus the state of the art algorithms for both protein construction and coordinate updating. Spinor calculations have a smaller computational cost and turn out to be slightly faster than current alternatives. © 2012 Wiley Periodicals, Inc.     

A self-organizing algorithm for molecular alignment and pharmacophore development

We present a method for simultaneous three-dimensional (3D) structure generation and pharmacophore-based alignment using a self-organizing algorithm called Stochastic Proximity Embedding (SPE). Current flexible molecular alignment methods either start from a single low-energy structure for each molecule and tweak bonds or torsion angles, or choose from multiple conformations of each molecule. Methods that generate structures and align them iteratively (e.g., genetic algorithms) are often slow. In earlier work, we used SPE to generate good-quality 3D conformations by iteratively adjusting pairwise distances between atoms based on a set of geometric rules, and showed that it samples conformational space better and runs faster than earlier programs. In this work, we run SPE on the entire ensemble of molecules to be aligned. Additional information about which atoms or groups of atoms in each molecule correspond to points in the pharmacophore can come from an automatically generated hypothesis or be specified manually. We add distance terms to SPE to bring pharmacophore points from different molecules closer in space, and also to line up normal/direction vectors associated with these points. We also permit pharmacophore points to be constrained to lie near external coordinates from a binding site. The aligned 3D molecular structures are nearly correct if the pharmacophore hypothesis is chemically feasible; postprocessing by minimization of suitable distance and energy functions further improves the structures and weeds out infeasible hypotheses. The method can be used to test 3D pharmacophores for a diverse set of active ligands, starting from only a hypothesis about corresponding atoms or groups.     

Efficient algorithm for the reconstruction of a protein backbone from the α-carbon coordinates

We present an analytical method for generating a whole protein backbone structure from the coordinates of the α-carbons. The procedure begins by automatically positioning the β-carbons for every residue, and then the positions of the carbonyl groups and the amide nitrogens are also computed. The method is based upon the simultaneous minimization of a number of geometrical constraints that appear in real proteins and that can be very easily formulated as a set of trigonometric relations between the coordinates of the atoms involved in the backbone reconstruction. The resulting algorithm has been tested for proteins of very different sizes and topologies, and can advantageously compete with other methods proposed for this goal both in accuracy and in computational requirements. Possible ways of further refinement of the resulting structures are discussed.     

Protein structure prediction from inaccurate and sparse NMR data using an enhanced genetic algorithm

Nuclear Magnetic Resonance Spectroscopy (most commonly known as NMR Spectroscopy) is used to generate approximate and partial distances between pairs of atoms of the native structure of a protein. To predict protein structure from these partial distances by solving the Euclidean distance geometry problem from the partial distances obtained from NMR Spectroscopy, we can predict three-dimensional (3D) structure of a protein. In this paper, a new genetic algorithm is proposed to efficiently address the Euclidean distance geometry problem towards building 3D structure of a given protein applying NMR's sparse data. Our genetic algorithm uses (i) a greedy mutation and crossover operator to intensify the search; (ii) a twin removal technique for diversification in the population; (iii) a random restart method to recover from stagnation; and (iv) a compaction factor to reduce the search space. Reducing the search space drastically, our approach improves the quality of the search. We tested our algorithms on a set of standard benchmarks. Experimentally, we show that our enhanced genetic algorithms significantly outperforms the traditional genetic algorithms and a previously proposed state-of-the-art method. Our method is capable of producing structures that are very close to the native structures and hence, the experimental biologists could adopt it to determine more accurate protein structures from NMR data.     

Direct use of unassigned resonances in NMR structure calculations with proxy residues

We present a method that significantly enhances the robustness of (automated) NMR structure determination by allowing the NOE data corresponding to unassigned NMR resonances to be used directly in the calculations. The unassigned resonances are represented by additional atoms or groups of atoms that have no interaction with the regular protein atoms except through distance restraints. These so-called "proxy" residues can be used to generate NOE-based distance restraints in a similar fashion as for the assigned part of the protein. If sufficient NOE information is available, the restraints are expected to place the proxies at positions close to the correct atoms for the unassigned resonance, which can facilitate subsequent assignment. Convergence can be further improved by supplying additional information about the possible identities of the unassigned resonances. We have implemented this approach in the widely used automated assignment and structure calculation protocols ARIA and CANDID. We find that it significantly increases the robustness of structure calculations with regard to missing assignments and yields structures of higher quality. Our approach is still able to find correctly folded structures with up to 30% randomly missing resonance assignments, and even when only backbone and beta resonances are present! This should be of significant value to NMR-based structural proteomics initiatives.     

GRID: A high‐resolution protein structure refinement algorithm

The energy‐based refinement of protein structures generated by fold prediction algorithms to atomic‐level accuracy remains a major challenge in structural biology. Energy‐based refinement is mainly dependent on two components: (1) sufficiently accurate force fields, and (2) efficient conformational space search algorithms. Focusing on the latter, we developed a high‐resolution refinement algorithm called GRID. It takes a three‐dimensional protein structure as input and, using an all‐atom force field, attempts to improve the energy of the structure by systematically perturbing backbone dihedrals and side‐chain rotamer conformations. We compare GRID to Backrub, a stochastic algorithm that has been shown to predict a significant fraction of the conformational changes that occur with point mutations. We applied GRID and Backrub to 10 high‐resolution (≤ 2.8 Å) crystal structures from the Protein Data Bank and measured the energy improvements obtained and the computation times required to achieve them. GRID resulted in energy improvements that were significantly better than those attained by Backrub while expending about the same amount of computational resources. GRID resulted in relaxed structures that had slightly higher backbone RMSDs compared to Backrub relative to the starting crystal structures. The average RMSD was 0.25 ± 0.02 Å for GRID versus 0.14 ± 0.04 Å for Backrub. These relatively minor deviations indicate that both algorithms generate structures that retain their original topologies, as expected given the nature of the algorithms. © 2012 Wiley Periodicals, Inc.     

Performance analysis of the double-iterated Kalman filter for molecular structure estimation

A possible application of a novel double-iterated Kalman filter (DIKF) as an algorithm for molecular structure determination is investigated in this work. Unlike traditional optimization algorithms, the DIKF does not exploit experimental nuclear magnetic resonance (NMR) constraints in a penalty function to be minimized but used them to filter the atomic coordinates. Furthermore, it is a nonlinear Bayesian estimator able to handle the uncertainty in the experimental data and in the computed structures, represented as covariance matrices. The algorithm presented applies all constraints simultaneously, in contrast with DIKF algorithms for structure determination found in literature, which apply the constraints one at a time. The performances of both paradigms are tested and compared with those obtained by a commonly used optimization algorithm (based on the conjugate gradient method). Besides providing estimates of the conformational uncertainty directly in the final covariance matrix, DIKF algorithms appear to generate structures with a better stereochemistry and be able to work with realistically imprecise constraints, while time performances are strongly affected by the heavy matricial calculations they require. © 1996 by John Wiley & Sons, Inc.     

Reconstruction of the 3D coordinates of α-carbon atoms of proteins from a pair of stereographic figures

Summary In an attempt to promptly use the experimentally determined structures of proteins in modeling studies, we have developed the program ReconstC to generate the 3D coordinates of -carbon atoms from a pair of stereographic figures. Calculations of the 3D coordinates were performed by estimating the stereo parameters systematically. Geometrical features of C traces were used to evaluate the integrity of the calculated structure. The program was applied to four kinds of protein structures to examine the performance. It was found that the root-mean-square deviation of atomic positions between constructed and reference crystal structures ranged from 0.36 to 0.78 Å. The range represents a reasonable accuracy and its automatic feature suggests that our approach would be expedient for providing initial structures for protein modeling studies.     

On the perception of molecules from 3D atomic coordinates

A method is presented for perceiving chemical types of atoms in molecules given 3D atomic coordinates and element identities. The method assigns hybridizations, bond orders, and formal charges for structures whether hydrogen atoms are present. The Maximum Weighted Matching algorithm for nonbipartite graphs is used to assign bond orders with weights derived from statistics of a large collection of organic molecules. Results form tests on a collection of functional groups, heterocycles, entries from the Protein Data Bank, and Cambridge Structural Database as well as a comparison to other methods, are presented and discussed.     

PROGEN: An automated modelling algorithm for the generation of complete protein structures from the α-carbon atomic coordinates

Summary A modelling algorithm (PROGEN) for the generation of complete protein atomic coordinates from only the -carbon coordinates is described. PROGEN utilizes an optimal geometry parameter (OGP) database for the positioning of atoms for each amino acid of the polypeptide model. The OGP database was established by examining the statistical correlations between 23 different intra-peptide and inter-peptide geometric parameters relative to the -carbon distances for each amino acid in a library of 19 known proteins from the Brookhaven Protein Database (BPDB). The OGP files for specific amino acids and peptides were used to generate the atomic positions, with respect to -carbons, for main-chain and side-chain atoms in the modelled structure. Refinement of the initial model was accomplished using energy minimization (EM) and molecular dynamics techniques. PROGEN was tested using 60 known proteins in the BPDB, representing a wide spectrum of primary and secondary structures. Comparison between PROGEN models and BPDB crystal reference structures gave r.m.s.d. values for peptide main-chain atoms between 0.29 and 0.76 Å, with a grand average of 0.53 Å for all 60 models. The r.m.s.d. for all non-hydrogen atoms ranged between 1.44 and 1.93 Å for the 60 polypeptide models. PROGEN was also able to make the correct assignment of cis- or trans-proline configurations in the protein structures examined. PROGEN offers a fully automatic building and refinement procedure and requires no special or specific structural considerations for the protein to be modelled.     

From secondary structure to three-dimensional structure: Improved dihedral angle probability distribution function for use with energy searches for native structures of polypeptides and proteins

An improved scheme to help in the prediction of protein structure is presented. This procedure generates improved starting conformations of a protein suitable for energy minimization. Trivariate gaussian distribution functions for the π, ψ, and χ¹ dihedral angles have been derived, using conformational data from high resolution protein structures selected from the Protein Data Bank (PDB). These trivariate probability functions generate initial values for the π, ψ, and χ¹ dihedral angles which reflect the experimental values found in the PDB. These starting structures speed the search of the conformational space by focusing the search mainly in the regions of native proteins. The efficiency of the new trivariate probability distributions is demonstrated by comparing the results for the α-class polypeptide fragment, the mutant Antennapedia (C39 → S) homeodomain (2HOA), with those from two reference probability functions. The first reference probability function is a uniform or flat probability function and the second is a bivariate probability function for π and ψ. The trivariate gaussian probability functions are shown to search the conformational space more efficiently than the other two probability functions. The trivariate gaussian probability functions are also tested on the binding domain of Streptococcal protein G (2GB1), an α/β class protein. Since presently available energy functions are not accurate enough to identify the most native-like energy-minimized structures, three selection criteria were used to identify a native-like structure with a 1.90-Å rmsd from the NMR structure as the best structure for the Antennapedia fragment. Each individual selection criterion (ECEPP/3 energy, ECEPP/3 energy-plus-free energy of hydration, or a knowledge-based mean field method) was unable to identify a native-like structure, but simultaneous application of more than one selection criterion resulted in a successful identification of a native-like structure for the Antennapedia fragment. In addition to these tests, structure predictions are made for the Antennapedia polypeptide, using a Pattern Recognition-based Importance-Sampling Minimization (PRISM) procedure to predict the backbone conformational state of the mutant Antennapedia homeodomain. The ten most probable backbone conformational state predictions were used with the trivariate and bivariate gaussian dihedral angle probability distributions to generate starting structures (i.e., dihedral angles) suitable for energy minimization. The final energy-minimized structures show that neither the trivariate nor the bivariate gaussian probability distributions are able to overcome the inaccuracies in the backbone conformational state predictions to produce a native-like structure. Until highly accurate predictions of the backbone conformational states become available, application of these dihedral angle probability distributions must be limited to problems, such as homology modeling, in which only a limited portion of the backbone (e.g., surface loops) must be explored. © 1996 John Wiley & Sons, Inc.     

Protein folding using fragment assembly and physical energy function

We perform a systematic study of the effects of sequence-independent backbone interactions and sequence-dependent side-chain interactions on protein folding using fragment assembly and physical energy function. Structures for ten proteins belonging to various structural classes are predicted only with Lennard-Jones interaction between backbone atoms. We find nativelike structures for beta proteins, suggesting that for proteins in this class, the global tertiary structures can be determined mainly by sequence-independent backbone interactions. On the other hand, for alpha proteins, nonlocal hydrophobic side-chain interaction is also required to obtain nativelike structures.     

Backbone building from quadrilaterals: a fast and accurate algorithm for protein backbone reconstruction from alpha carbon coordinates

In this contribution, we present an algorithm for protein backbone reconstruction that comprises very high computational efficiency with high accuracy. Reconstruction of the main chain atomic coordinates from the alpha carbon trace is a common task in protein modeling, including de novo structure prediction, comparative modeling, and processing experimental data. The method employed in this work follows the main idea of some earlier approaches to the problem. The details and careful design of the present approach are new and lead to the algorithm that outperforms all commonly used earlier applications. BBQ (Backbone Building from Quadrilaterals) program has been extensively tested both on native structures as well as on near-native decoy models and compared with the different available existing methods. Obtained results provide a comprehensive benchmark of existing tools and evaluate their applicability to a large scale modeling using a reduced representation of protein conformational space. The BBQ package is available for downloading from our website at http://biocomp.chem.uw.edu.pl/services/BBQ/. This webpage also provides a user manual that describes BBQ functions in detail.     

Determination of the major solution conformation of tyrocidine A,using molecular mechanics energy minimization and NMR-derived distance and torsion angle constraints

Molecular mechanics energy calculations coupled with nuclear magnetic resonance-determined distance and torsion angle constraints have been used to determine the three-dimensional structure of tyrocidine A, a cyclic decapeptide which exists largely as a single conformation in solution. Two open-chain polyalanine models were used to represent separate halves of the peptide backbone and a combinatorial method of searching conformation space used to generate candidate structures consistent with experimental distance constraints. These structures were energy-minimized using the AMBER molecular mechanics forcefield and the resulting conformations classified by factor analysis of their Cartesian coordinates. Representative low-energy conformers of the two halves of the backbone were fused together and two candidate conformations of the completed backbone refined by further minimization using both distance and torsional constraints. Side chains were then added as their experimentally preferred rotamers and the whole molecule minimized without constraints to give the final model structure. This shows type II' and III ß turns at residues 4–5 and 9–10, respectively, coupled by twisted antiparallel strands which show hydrogen bonds between all four pairs of opposing peptide groups. The backbone conformation of residues 2–6 closely resembles that found in the crystal structure of gramicidin S.     

Scaffold mining of kinase hinge binders in crystal structure database

Protein kinases are the second most prominent group of drug targets, after G-protein-coupled receptors. Despite their distinct inhibition mechanisms, the majority of kinase inhibitors engage the conserved hydrogen bond interactions with the backbone of hinge residues. We mined Pfizer internal crystal structure database (CSDb) comprising of several thousand of public as well as internal X-ray binary complexes to compile an inclusive list of hinge binding scaffolds. The minimum ring scaffolds with directly attached hetero-atoms and functional groups were extracted from the full compounds by applying a rule-based filtering procedure employing a comprehensive annotation of ATP-binding site of the human kinase complements. The results indicated large number of kinase inhibitors of diverse chemical structures are derived from a relatively small number of common scaffolds, which serve as the critical recognition elements for protein kinase interaction. Out of the nearly 4,000 kinase-inhibitor complexes in the CSDb we identified approximately 600 unique scaffolds. Hinge scaffolds are overwhelmingly flat with very little sp3 characteristics, and are less lipophilic than their corresponding parent compounds. Examples of the most common as well as the uncommon hinge scaffolds are presented. Although the most common scaffolds are found in complex with multiple kinase targets, a large number of them are uniquely bound to a specific kinase, suggesting certain scaffolds could be more promiscuous than the others. The compiled collection of hinge scaffolds along with their three-dimensional binding coordinates could serve as basis set for hinge hopping, a practice frequently employed to generate novel invention as well as to optimize existing leads in medicinal chemistry.     

Peptide backbone reconstruction using dead-end elimination and a knowledge-based forcefield

A novel, yet simple and automated, protocol for reconstruction of complete peptide backbones from C(alpha) coordinates only is described, validated, and benchmarked. The described method collates a set of possible backbone conformations for each set of residue triads from a structural library derived from the PDB. The optimal permutation of these three residue segments of backbone conformations is determined using the dead-end elimination (DEE) algorithm. Putative conformations are evaluated using a pairwise-additive knowledge-based forcefield term and a fragment overlap term. The protocol described in this report is able to restore the full backbone coordinates to within 0.2-0.6 A of the actual crystal structure from C(alpha) coordinates only. In addition, it is insensitive to errors in the input C(alpha) coordinates with RMSDs of 3.0 A, and this is illustrated through application to deliberately distorted C(alpha) traces. The entire process, as described, is rapid, requiring of the order of a few minutes for a typical protein on a typical desktop PC. Approximations enable this to be reduced to a few seconds, although this is at the expense of prediction accuracy. This compares very favorably to previously published methods, being sufficiently fast for general use and being one of the most accurate methods. Because the method is not restricted to the reconstruction from only C(alpha) coordinates, reconstruction based on C(beta) coordinates is also demonstrated.     

Prediction of NMR order parameters in proteins using weighted protein contact-number model

In the NMR experiment, the protein backbone motion can be described by the N–H order parameters. Though protein dynamics is determined by a complex network of atomic interactions, we show that the order parameter of residues can be determined using a very simple method, the weighted protein contact number model. We computed for each Cα atom the number of neighboring Cα atoms weighted by the inverse distance squared between them. We show that the weighted contact number of each residue is directly related to its order parameter. Despite the simplicity of this model, it performs better than the other method. Since we can compute the order parameters directly from the topological properties (such as protein contact number) of protein structures, our study underscores a very direct link between protein topological structure and its dynamics.     

A method for the calculation of normal-mode vibrational frequencies using symmetry coordinates. Application to the calculation of secondary deuterium isotope effects on carbocations

A method is introduced for the calculation of normal-mode vibrational frequencies of polyatomic molecules based on numerical differencing of analytical gradients in symmetry coordinates. This procedure requires a number of gradient evaluations equal to the largest number of symmetry coordinates belonging to any single irreducible representation of the molecular point group (plus a single gradient evaluation at the equilibrium configuration), which is fewer than the 3N-6 (N atoms) gradient evaluations needed for schemes based on Cartesian or internal coordinates. While the proposed method will not generally be as efficient as procedures which involve the direct calculation of energy second derivatives analytically (as are now available for single-determinant wavefunctions) it appears to be equally accurate, and it should be the method of choice for frequency calculations involving multideterminant wavefunctions for which analytical second-derivative algorithms have yet to be developed. The method is illustrated by the calculation of equilibrium secondary deuterium-isotope effects on a number of reactions involving simple carbocations.