A reduced protein model with accurate native-structure identification ability

Generation of atomic four-body statistical potentials derived from the delaunay tessellation of protein structures

Delaunay tessellation of the atomic coordinates for a crystallographic protein structure yields an aggregate of non-overlapping and space-filling irregular tetrahedral simplices. The vertices of each simplex objectively identify a quadruplet of nearest neighbor atoms in the protein. Here we apply Delaunay tessellation to 1417 high-resolution structures of single chains that share low sequence identity, for the purpose of determining the relative frequencies of occurrence for all possible nearest neighbor atomic quadruplet types. Alternative distributions are explored by varying two fundamental parameters: atomic alphabet selection and cutoff length for admissible simplex edges. The distributions are then converted to four-body potential functions by implementing the inverted Boltzmann principle, which requires calculating the distribution of the reference state. Two alternative definitions for the reference state are presented, which introduces a third parameter, and we derive and compare an array of such potential functions. These knowledge-based statistical potentials based on higher-order interactions complement and generalize the more commonly encountered atom-pair potentials, for which a number of approaches are described in the literature.

Another look at the conditions for the extraction of protein knowledge-based potentials

Protein knowledge-based potentials are effective free energies obtained from databases of known protein structures. They are used to parameterize coarse-grained protein models in many folding simulation and structure prediction methods. Two common approaches are used in the derivation of knowledge-based potentials. One assumes that the energy parameters optimize the native structure stability. The other assumes that interaction events are related to their energies according to the Boltzmann distribution, and that they are distributed independently of other events, that is, the quasi-chemical approximation. Here, these assumptions are systematically tested by extracting contact energies from artificial databases of lattice proteins with predefined pairwise contact energies. Databases of protein sequences are designed to either satisfy the Boltzmann distribution at high or low temperatures, or to simultaneously optimize the native stability and folding kinetics. It is found that the quasi-chemical approximation, with the ideal reference state, accurately reproduce the true energies for high temperature Boltzmann distributed sequences (weakly interacting residues), but less accurately at low temperatures, where the sequences correspond to energy minima and the residues are strongly interacting. To overcome this problem, an iterative procedure for Boltzmann distributed sequences is introduced, which accounts for interacting residue correlations and eliminates the need for the quasi-chemical approximation. In this case, the energies are accurately reproduced at any ensemble temperature. However, when the database of sequences designed for optimal stability and kinetics is used, the energy correlation is less than optimal using either method, exhibiting random and systematic deviations from linearity. Therefore, the assumption that native structures are maximally stable or that sequences are determined according to the Boltzmann distribution seems to be inadequate for obtaining accurate energies. The limited number of sequences in the database and the inhomogeneous concentration of amino acids from one structure to another do not seem to be major obstacles for improving the quality of the extracted pairwise energies, with the exception of repulsive interactions.

Reduced C(beta) statistical potentials can outperform all-atom potentials in decoy identification

We developed a series of statistical potentials to recognize the native protein from decoys, particularly when using only a reduced representation in which each side chain is treated as a single C(beta) atom. Beginning with a highly successful all-atom statistical potential, the Discrete Optimized Protein Energy function (DOPE), we considered the implications of including additional information in the all-atom statistical potential and subsequently reducing to the C(beta) representation. One of the potentials includes interaction energies conditional on backbone geometries. A second potential separates sequence local from sequence nonlocal interactions and introduces a novel reference state for the sequence local interactions. The resultant potentials perform better than the original DOPE statistical potential in decoy identification. Moreover, even upon passing to a reduced C(beta) representation, these statistical potentials outscore the original (all-atom) DOPE potential in identifying native states for sets of decoys. Interestingly, the backbone-dependent statistical potential is shown to retain nearly all of the information content of the all-atom representation in the C(beta) representation. In addition, these new statistical potentials are combined with existing potentials to model hydrogen bonding, torsion energies, and solvation energies to produce even better performing potentials. The ability of the C(beta) statistical potentials to accurately represent protein interactions bodes well for computational efficiency in protein folding calculations using reduced backbone representations, while the extensions to DOPE illustrate general principles for improving knowledge-based potentials.

Scoring predictive models using a reduced representation of proteins: model and energy definition

Background

Reduced representations of proteins have been playing a keyrole in the study of protein folding. Many such models are available, with different representation detail. Although the usefulness of many such models for structural bioinformatics applications has been demonstrated in recent years, there are few intermediate resolution models endowed with an energy model capable, for instance, of detecting native or native-like structures among decoy sets. The aim of the present work is to provide a discrete empirical potential for a reduced protein model termed here PC2CA, because it employs a PseudoCovalent structure with only 2 Centers of interactions per Amino acid, suitable for protein model quality assessment.

Results

All protein structures in the set top500H have been converted in reduced form. The distribution of pseudobonds, pseudoangle, pseudodihedrals and distances between centers of interactions have been converted into potentials of mean force. A suitable reference distribution has been defined for non-bonded interactions which takes into account excluded volume effects and protein finite size. The correlation between adjacent main chain pseudodihedrals has been converted in an additional energetic term which is able to account for cooperative effects in secondary structure elements. Local energy surface exploration is performed in order to increase the robustness of the energy function.

Conclusion

The model and the energy definition proposed have been tested on all the multiple decoys' sets in the Decoys'R'us database. The energetic model is able to recognize, for almost all sets, native-like structures (RMSD less than 2.0 Å). These results and those obtained in the blind CASP7 quality assessment experiment suggest that the model compares well with scoring potentials with finer granularity and could be useful for fast exploration of conformational space. Parameters are available at the url: .

Empirical potential function for simplified protein models: combining contact and local sequence-structure descriptors

An effective potential function is critical for protein structure prediction and folding simulation. Simplified protein models such as those requiring only Calpha or backbone atoms are attractive because they enable efficient search of the conformational space. We show residue-specific reduced discrete-state models can represent the backbone conformations of proteins with small RMSD values. However, no potential functions exist that are designed for such simplified protein models. In this study, we develop optimal potential functions by combining contact interaction descriptors and local sequence-structure descriptors. The form of the potential function is a weighted linear sum of all descriptors, and the optimal weight coefficients are obtained through optimization using both native and decoy structures. The performance of the potential function in a test of discriminating native protein structures from decoys is evaluated using several benchmark decoy sets. Our potential function requiring only backbone atoms or Calpha atoms have comparable or better performance than several residue-based potential functions that require additional coordinates of side-chain centers or coordinates of all side-chain atoms. By reducing the residue alphabets down to size 10 for contact descriptors, the performance of the potential function can be further improved. Our results also suggest that local sequence-structure correlation may play important role in reducing the entropic cost of protein folding.

A multibody, whole-residue potential for protein structures, with testing by Monte Carlo simulated annealing

A new multibody, whole-residue potential for protein tertiary structure is described. The potential is based on the local environment surrounding each main-chain alpha carbon (CA), defined as the set of all residues whose CA coordinates lie within a spherical volume of set radius in 3-dimensional (3D) space surrounding that position. It is shown that the relative positions of the CAs in these local environments belong to a set of preferred templates. The templates are derived by cluster analysis of the presently available database of over 3000 protein chains (750,000 residues) having not more than 30% sequence similarity. For each template is derived also a set of residue propensities for each topological position in the template. Using lookup tables of these derived templates, it is then possible to calculate an energy for any conformation of a given protein sequence. The application of the potential to ab initio protein tertiary structure prediction is evaluated by performing Monte Carlo simulated annealing on test protein sequences.

Single nucleotide RNA choreography

New structural analysis methods, and a tree formalism re-define and expand the RNA motif concept, unifying what previously appeared to be disparate groups of structures. We find RNA tetraloops at high frequencies, in new contexts, with unexpected lengths, and in novel topologies. The results, with broad implications for RNA structure in general, show that even at this most elementary level of organization, RNA tolerates astounding variation in conformation, length, sequence and context. However the variation is not random; it is well-described by four distinct modes, which are 3-2 switches (backbone topology variations), insertions, deletions and strand clips.

Efficient Monte Carlo trial moves for polypeptide simulations

A new move set for the Monte Carlo simulations of polypeptide chains is introduced. It consists of a rigid rotation along the (C(alpha)) ends of an arbitrary long segment of the backbone in such a way that the atoms outside this segment remain fixed. This fixed end move, or FEM, alters only the backbone dihedral angles phi and psi and the C(alpha) bond angles of the segment ends. Rotations are restricted to those who keep the alpha bond angles within their maximum natural range of approximately +/-10 degrees. The equations for the angular intervals (tau) of the allowed rigid rotations and the equations required for satisfying the detailed balance condition are presented in detail. One appealing property of the FEM is that the required number of calculations is minimal, as it is evident from the simplicity of the equations. In addition, the moving backbone atoms undergo considerable but limited displacements of up to 3 A. These properties, combined with the small number of backbone angles changed, lead to high acceptance rates for the new conformations and make the algorithm very efficient for sampling the conformational space. The FEMs, combined with pivot moves, are used in a test to fold a group of coarse-grained proteins with lengths of up to 200 residues.

Development of novel statistical potentials for protein fold recognition

The need to perform large-scale studies of protein fold recognition, structure prediction and protein-protein interactions has led to novel developments of residue-level minimal models of proteins. A minimum requirement for useful protein force-fields is that they be successful in the recognition of native conformations. The balance between the level of detail in describing the specific interactions within proteins and the accuracy obtained using minimal protein models is the focus of many current protein studies. Recent results suggest that the introduction of explicit orientation dependence in a coarse-grained, residue-level model improves the ability of inter-residue potentials to recognize the native state. New statistical and optimization computational algorithms can be used to obtain accurate residue-dependent potentials for use in protein fold recognition and, more importantly, structure prediction.

Empirical force fields for biological macromolecules: overview and issues

Empirical force field-based studies of biological macromolecules are becoming a common tool for investigating their structure-activity relationships at an atomic level of detail. Such studies facilitate interpretation of experimental data and allow for information not readily accessible to experimental methods to be obtained. A large part of the success of empirical force field-based methods is the quality of the force fields combined with the algorithmic advances that allow for more accurate reproduction of experimental observables. Presented is an overview of the issues associated with the development and application of empirical force fields to biomolecular systems. This is followed by a summary of the force fields commonly applied to the different classes of biomolecules; proteins, nucleic acids, lipids, and carbohydrates. In addition, issues associated with computational studies on "heterogeneous" biomolecular systems and the transferability of force fields to a wide range of organic molecules of pharmacological interest are discussed.

Local Propensities and Statistical Potentials of Backbone Dihedral Angles in Proteins

The following three issues concerning the backbone dihedral angles of protein structures are presented. (1) How do the dihedral angles of the 20 amino acids depend on the identity and conformation of their nearest residues? (2) To what extent are the native dihedral angles determined by local (dihedral) potentials? (3) How to build a knowledge-based potential for a residue's dihedral angles, considering the identity and conformation of its nearest residues? We find that the dihedral angle distribution for a residue can significantly depend on the identity and conformation of its adjacent residues. These correlations are in sharp contrast to the Flory isolated-pair hypothesis. Statistical potentials are built for all combinations of residue triplets and depend on the dihedral angles between consecutive residues. First, a low-resolution potential is obtained, which only differentiates between the main populated basins in the dihedral angle density plots. Minimization of the dihedral potential for 125 test proteins reveals that most native alpha-helical residues (89%) and a large fraction of native beta-sheet residues (47%) adopt conformations close to their native one. For native loop residues, the percentage is 48%. It is also found that this fraction is higher for residues away from the ends of alpha or beta secondary structure elements. In addition, a higher resolution potential is built as a function of dihedral angles by a smoothing procedure and continuous functions interpolations. Monte Carlo energy minimization with this potential results in a lower fraction for native beta-sheet residues. Nevertheless, because of the higher flexibility and entropy of beta structures, they could be preferred under the influence of non-local interactions. In general, most alpha-helices and many beta-sheets are strongly determined by the local potential, while the conformations in loops and near the end of beta-sheets are more influenced by non-local interactions.