Efficient substructure RMSD query algorithms.

Protein structure analysis is a very important research topic in the molecular biology of the post-genomic era. The root mean square deviation (RMSD) is the most frequently used measure for comparing two protein three-dimensional (3-D) structures. In this paper, we deal with two fundamental problems related to the RMSD. We first deal with a problem called the "range RMSD query" problem. Given an aligned pair of structures, the problem is to compute the RMSD between two aligned substructures of them without gaps. This problem has many applications in protein structure analysis. We propose a linear-time preprocessing algorithm that enables constant-time RMSD computation. Next, we consider a problem called the "substructure RMSD query" problem, which is a generalization of the above range RMSD query problem. It is a problem to compute the RMSD between any substructures of two unaligned structures without gaps. Based on the algorithm for the range RMSD problem, we propose an O(nm) preprocessing algorithm that enables constant-time RMSD computation, where n and m are the lengths of the given structures. Moreover, we propose O(nm log r/r)-time and O(nm/r)-space preprocessing algorithm that enables O(r) query, where r is an arbitrary integer such that 1 < or = r < or = min(n, m). We also show that our strategy also works for another measure called the unit-vector root mean square deviation (URMSD), which is a variant of the RMSD.     

RNA loop structure prediction via bond scaling and relaxation

We have developed a method for predicting the structure of small RNA loops that can be used to augment already existing RNA modeling techniques. The method requires no input constraints on loop configuration other than end-to-end distance. Initial loop structures are generated by randomizing the torsion angles, beginning at one end of the polynucleotide chain and correlating each successive angle with the previous. The bond lengths of these structures are then scaled to fit within the known end constraints and the equilibrium bond lengths of the potential energy function are scaled accordingly. Through a series of rescaling and minimization steps the structures are allowed to relax to lower energy configurations with standard bond lengths and reduced van der Waals clashes. This algorithm has been tested on the variable loops of yeast tRNA-Asp and yeast tRNA-Phe, as well as the sarcin-ricin tetraloop and the anticodon loop of yeast tRNA-Phe. The results indicate good correlation between potential energy and the loop structure predictions that are closest to the variable loop crystal structures, but poorer correlation for the more isolated stem loops. The number of stacking interactions has proven to be a good objective measure of the best loop predictions. Selecting on the basis of energy and stacking, we obtain two structures with 0.65 and 0.75 Å all-atom rms deviations (RMSD) from the crystal structure for the tRNA-Asp variable loop. The best structure prediction for the tRNA-Phe variable loop has an all-atom RMSD of 2.2 Å and a backbone RMSD of 1.6 Å, with a single base responsible for most of the deviation. For the sarcin-ricin loop from 28S ribosomal RNA, the predicted structure's all-atom RMSD from the nmr structure is 1.0 Å. We obtain a 1.8 Å RMSD structure for the tRNA-Phe anticodon loop. © 1996 John Wiley & Sons, Inc.     

Defining and computing optimum RMSD for gapped and weighted multiple-structure alignment

Pairwise structure alignment commonly uses root mean square deviation (RMSD) to measure the structural similarity, and methods for optimizing RMSD are well established. We extend RMSD to weighted RMSD for multiple structures. By using multiplicative weights, we show that weighted RMSD for all pairs is the same as weighted RMSD to an average of the structures. Thus, using RMSD or weighted RMSD implies that the average is a consensus structure. Although we show that in general, the two tasks of finding the optimal translations and rotations for minimizing weighted RMSD cannot be separated for multiple structures like they can for pairs, an inherent difficulty and a fact ignored by previous work, we develop a near-linear iterative algorithm to converge weighted RMSD to a local minimum. 10,000 experiments of gapped alignment done on each of 23 protein families from HOMSTRAD (where each structure starts with a random translation and rotation) converge rapidly to the same minimum. Finally we propose a heuristic method to iteratively remove the effect of outliers and find well-aligned positions that determine the structural conserved region by modeling B-factors and deviations from the average positions as weights and iteratively assigning higher weights to better aligned atoms.     

Size-independent comparison of protein three-dimensional structures

Protein structures are routinely compared by their root-mean-square deviation (RMSD) in atomic coordinates after optimal rigid body superposition. What is not so clear is the significance of different RMSD values, particularly above the customary arbitrary cutoff for obvious similarity of 2–3 Å. Our earlier work argued for an intrinsic cutoff for protein similarity that varied with the number of residues in the polypeptide chains being compared. Here we introduce a new measure, ρ, of structural similarity based on RMSD that is independent of the sizes of the molecules involved, or of any other special properties of molecules. When ρ is less than 0.4–0.5, protein structures are visually recognized to be obviously similar, but the mathematically pleasing intrinsic cutoff of ρ>1.0 corresponds to overall similarity in folding motif at a level not usually recognized until smoothing of the polypeptide chain path makes it striking. When the structures are scaled to unit radius of gyration and equal principle moments of inertia, the comparisons are even more universal, since they are no longer obscured by differences in overall size and ellipticity. With increasing chain length, the distribution of ρ for pairs of random structures is skewed to higher values, but the value for the best 1% of the comparisons rises only slowly with the number of residues. This level is close to an intrinsic cutoff between similar and dissimilar comparisons, namely the maximal scaled ρ possible for the two structures to be more similar to each other than one is to the other's mirror image. The intrinsic cutoff is independent of the number of residues or points being compared. For proteins having fewer than 100 residues, the 1% ρ falls below the intrinsic cutoff, so that for very small proteins, geometrically significant similarity can often occur by chance. We believe these ideas will be helpful in judging success in NMR structure determination and protein folding modeling. © 1995 Wiley-Liss, Inc.     

NMR of hydrogen bonding in cold-shock protein A and an analysis of the influence of crystallographic resolution on comparisons of hydrogen bond lengths

Hydrogen bonding in cold-shock protein A of Escherichia coli has been investigated using long-range HNCO spectroscopy. Nearly half of the amide protons involved in hydrogen bonds in solution show no measurable protection from exchange in water, cautioning against a direct correspondence between hydrogen bonding and hydrogen exchange protection. The N to O atom distance across a hydrogen bond, R(NO), is related to the size of the (3h)J(NC') trans hydrogen bond coupling constant and the amide proton chemical shift. Both NMR parameters show poorer agreement with the 2.0-A resolution X-ray structure of the cold-shock protein studied by NMR than with a 1.2-A resolution X-ray structure of a homologous cold-shock protein from the thermophile B. caldolyticus. The influence of crystallographic resolution on comparisons of hydrogen bond lengths was further investigated using a database of 33 X-ray structures of ribonuclease A. For highly similar structures, both hydrogen bond R(NO) distance and Calpha coordinate root mean square deviations (RMSD) show systematic increases as the resolution of the X-ray structure used for comparison decreases. As structures diverge, the effects of coordinate errors on R(NO) distance and Calpha coordinate root mean square deviations become progressively smaller. The results of this study are discussed with regard to the influence of data precision on establishing structure similarity relationships between proteins.     

A systematic study of the vibrational free energies of polypeptides in folded and random states

Molecular vibrations, especially low frequency motions, may be used as an indication of the rigidity or the flatness of the protein folding energy landscape. We have studied the vibrational properties of native folded as well as random coil structures of more than 60 polypeptides. The picture we obtain allows us to perceive how and why the energy landscape progressively rigidifies while still allowing potential flexibility. Compared with random coil structures, both alpha-helices and beta-hairpins are vibrationally more flexible. The vibrational properties of loop structures are similar to those of the corresponding random coil structures. Inclusion of an alpha-helix tends to rigidify peptides and so-called building blocks of the structure, whereas the addition of a beta-structure has less effect. When small building blocks coalesce to form larger domains, the protein rigidifies. However, some folded native conformations are still found to be vibrationally more flexible than random coil structures, for example, beta(2)-microglobulin and the SH3 domain. Vibrational free energy contributes significantly to the thermodynamics of protein folding and affects the distribution of the conformational substates. We found a weak correlation between the vibrational folding energy and the protein size, consistent with both previous experimental estimates and theoretical partition of the heat capacity change in protein folding.     

TOUCHSTONE II: a new approach to ab initio protein structure prediction

We have developed a new combined approach for ab initio protein structure prediction. The protein conformation is described as a lattice chain connecting C(alpha) atoms, with attached C(beta) atoms and side-chain centers of mass. The model force field includes various short-range and long-range knowledge-based potentials derived from a statistical analysis of the regularities of protein structures. The combination of these energy terms is optimized through the maximization of correlation for 30 x 60,000 decoys between the root mean square deviation (RMSD) to native and energies, as well as the energy gap between native and the decoy ensemble. To accelerate the conformational search, a newly developed parallel hyperbolic sampling algorithm with a composite movement set is used in the Monte Carlo simulation processes. We exploit this strategy to successfully fold 41/100 small proteins (36 approximately 120 residues) with predicted structures having a RMSD from native below 6.5 A in the top five cluster centroids. To fold larger-size proteins as well as to improve the folding yield of small proteins, we incorporate into the basic force field side-chain contact predictions from our threading program PROSPECTOR where homologous proteins were excluded from the data base. With these threading-based restraints, the program can fold 83/125 test proteins (36 approximately 174 residues) with structures having a RMSD to native below 6.5 A in the top five cluster centroids. This shows the significant improvement of folding by using predicted tertiary restraints, especially when the accuracy of side-chain contact prediction is >20%. For native fold selection, we introduce quantities dependent on the cluster density and the combination of energy and free energy, which show a higher discriminative power to select the native structure than the previously used cluster energy or cluster size, and which can be used in native structure identification in blind simulations. These procedures are readily automated and are being implemented on a genomic scale.     

Statistics of random protein superpositions: p-values for pairwise structure alignment.

Quantification of statistical significance is essential for the interpretation of protein structural similarity. To address this, a random model for protein structure comparison was developed. Novelty of the model is threefold. First, a sample of random structure comparisons is restricted to molecules of the same size and shape as the superposition of interest. Second, careful selection of the sample and accurate modeling of shape allows approximation of the root mean square deviation (RMSD) distribution of random comparisons with a Nakagami probability density function. Third, through convolution, a second probability density function is obtained that describes the coordinate difference vector projections underlying the random distribution of RMSD. This last feature allows sampling random distributions of not only RMSD, but also any similarity score that depends on difference vector projections, such as GDT_TS score, TM score, and LiveBench 3D score. Probabilities estimated from the method correlate well with common measures of structural similarity, such as the Dali Z-score and the GDT_TS score. As a result, the p-value for a given superposition can be calculated using simple formulae depending on RMSD, radius of gyration, and thinnest molecular dimension. In addition to scoring structural similarity, p-values computed by this method can be applied to evaluation of homology modeling techniques, providing a statistically sound alternative to scores used in reference-independent evaluation of alignment quality.     

Ab initio construction of polypeptide fragments: Accuracy of loop decoy discrimination by an all-atom statistical potential and the AMBER force field with the Generalized Born solvation model

The accuracy of model selection from decoy ensembles of protein loop conformations was explored by comparing the performance of the Samudrala-Moult all-atom statistical potential (RAPDF) and the AMBER molecular mechanics force field, including the Generalized Born/surface area solvation model. Large ensembles of consistent loop conformations, represented at atomic detail with idealized geometry, were generated for a large test set of protein loops of 2 to 12 residues long by a novel ab initio method called RAPPER that relies on fine-grained residue-specific phi/psi propensity tables for conformational sampling. Ranking the conformers on the basis of RAPDF scores resulted in selected conformers that had an average global, non-superimposed RMSD for all heavy mainchain atoms ranging from 1.2 A for 4-mers to 2.9 A for 8-mers to 6.2 A for 12-mers. After filtering on the basis of anchor geometry and RAPDF scores, ranking by energy minimization of the AMBER/GBSA potential energy function selected conformers that had global RMSD values of 0.5 A for 4-mers, 2.3 A for 8-mers, and 5.0 A for 12-mers. Minimized fragments had, on average, consistently lower RMSD values (by 0.1 A) than their initial conformations. The importance of the Generalized Born solvation energy term is reflected by the observation that the average RMSD accuracy for all loop lengths was worse when this term is omitted. There are, however, still many cases where the AMBER gas-phase minimization selected conformers of lower RMSD than the AMBER/GBSA minimization. The AMBER/GBSA energy function had better correlation with RMSD to native than the RAPDF. When the ensembles were supplemented with conformations extracted from experimental structures, a dramatic improvement in selection accuracy was observed at longer lengths (average RMSD of 1.3 A for 8-mers) when scoring with the AMBER/GBSA force field. This work provides the basis for a promising hybrid approach of ab initio and knowledge-based methods for loop modeling.     

Efficient methods for filtering and ranking fragments for the prediction of structurally variable regions in proteins

The prediction of protein 3D structures close to insertions and deletions or, more generally, loop prediction, is still one of the major challenges in homology modeling projects. In this article, we developed ranking criteria and selection filters to improve knowledge-based loop predictions. These criteria were developed and optimized for a test data set containing 678 insertions and deletions. The examples are, in principle, predictable from the used loop database with an RMSD < 1 A and represent realistic modeling situations. Four noncorrelated criteria for the selection of fragments are evaluated. A fast prefilter compares the distance between the anchor groups in the template protein with the stems of the fragments. The RMSD of the anchor groups is used for fitting and ranking of the selected loop candidates. After fitting, repulsive close contacts of loop candidates with the template protein are used for filtering, and fragments with backbone torsion angles, which are unfavorable according to a knowledge-based potential, are eliminated. By the combined application of these filter criteria to the test set, it was possible to increase the percentage of predictions with a global RMSD < 1 A to over 50% among the first five ranks, with average global RMSD values for the first rank candidate that are between 1.3 and 2.2 A for different loop lengths. Compared to other examples described in the literature, our large numbers of test cases are not self-predictions, where loops are placed in a protein after a peptide loop has been cut out, but are attempts to predict structural changes that occur in evolution when a protein is affected by insertions and deletions.     

SAS-Pro: simultaneous residue assignment and structure superposition for protein structure alignment

Protein structure alignment is the problem of determining an assignment between the amino-acid residues of two given proteins in a way that maximizes a measure of similarity between the two superimposed protein structures. By identifying geometric similarities, structure alignment algorithms provide critical insights into protein functional similarities. Existing structure alignment tools adopt a two-stage approach to structure alignment by decoupling and iterating between the assignment evaluation and structure superposition problems. We introduce a novel approach, SAS-Pro, which addresses the assignment evaluation and structure superposition simultaneously by formulating the alignment problem as a single bilevel optimization problem. The new formulation does not require the sequentiality constraints, thus generalizing the scope of the alignment methodology to include non-sequential protein alignments. We employ derivative-free optimization methodologies for searching for the global optimum of the highly nonlinear and non-differentiable RMSD function encountered in the proposed model. Alignments obtained with SAS-Pro have better RMSD values and larger lengths than those obtained from other alignment tools. For non-sequential alignment problems, SAS-Pro leads to alignments with high degree of similarity with known reference alignments. The source code of SAS-Pro is available for download at http://eudoxus.cheme.cmu.edu/saspro/SAS-Pro.html.     

Electrostatic pKa computations in proteins: role of internal cavities

The solvent accessible surface area (SASA) algorithm is conventionally used to characterize protein surfaces in electrostatic energy computations of proteins. Unfortunately, it often fails to find narrow cavities inside a protein. As a consequence pK(a) computations based on this algorithm perform badly. In this study a new cavity-algorithm is introduced, which solves this problem and provides improved pK(a) values. The procedure is applied to 20 pK(a) values of titratable groups introduced as point mutations in SNase variants, where crystal structures are available. The computations of these pK(a)s are particular challenging, since they are placed in a rather hydrophobic environment. For nine mutants, where the titratable residue is in contact with a large cavity, the RMSD(pKa) between computed and measured pK(a) values is 2.04, which is a considerable improvement as compared to the original results obtained with Karlsberg(+) (http://agknapp.chemie.fu-berlin.de/karlsberg/) that yielded an RMSD(pKa) of 8.8. However, for 11 titratable residues the agreement with experiments remains poor (RMSD(pKa) = 6.01). Considering 15 pK(a)s of SNase, which are in a more conventional less hydrophobic protein environment, the RMSD(pKa) is 2.1 using the SASA-algorithm and 1.7 using the new cavity-algorithm. The agreement is reasonable but less good than what one would expect from the general performance of Karlsberg(+) indicating that SNase belongs to the more difficult proteins with respect to pK(a) computations. We discuss the possible reasons for the remaining discrepancies between computed and measured pK(a)s.     

Structural effects of point mutations in proteins

A structural database of 11 families of chains differing by a single amino acid substitution has been built. Another structural dataset of 5 families with identical sequences has been used for comparison. The RMSD computed after a global superimposition of the mutated protein on each native one is smaller than the RMSD calculated among proteins of identical sequences. The effect of the perturbation is very local, and not necessarily the highest at the position of the mutation. A RMSD between mutated and native proteins is computed over a 3‐residue or a 7‐residue window at each position. To separate the effects of structural fluctuations due to point mutations from other sources, pair RMSD have been translated into P values which themselves are included in a score called P‐RANK. This score allows highlighting small backbone distortions by comparing these RMSD between mutated and native positions to the RMSD at the same positions in the absence of a mutation. It results from the P‐RANK that 38% of all mutations produce a significant effect on the displacement. When compared with a random distribution of RMSD at un‐mutated positions, we show that, even if the RMSD is greater when the mutation is in loops than in regular secondary structure, the relative effect is more important for regular secondary structures and for buried positions. We confirm the absence of correlation between RMSD and the predicted variation of free energy of folding but we found a small correlation between high RMSD and the error in the prediction of ΔΔG.     

The size of the intermolecular energy funnel in protein-protein interactions

Revealing the fundamental principles of protein interactions is essential for the basic knowledge of molecular processes and designing better predictive tools. Protein docking procedures allow systematic sampling of intermolecular energy landscapes, revealing the distribution of energy basins and their characteristics. A systematic search docking procedure GRAMM-X was applied to a comprehensive nonredundant database of nonobligate protein-protein complexes to determine the size of the intermolecular energy funnel. The unbound structures were simulated using rotamer library. The procedure generated grid-based matches, based on a smoothed Lennard-Jones potential, and minimized them off the grid with the same potential. The minimization generated a distribution of distances, based on a variety of metrics, between the grid-based and the minimized matches. The metric selected for the analysis, ligand interface RMSD, provided three independent estimates of the funnel size: based on the distribution amplitude for the near-native matches, deviation from random, and correlation with the energy values. The three methods converge to similar estimates of approximately 6-8 A ligand interface RMSD. The results indicated dependence of the funnel size on the type of the complex (smaller for antigen-antibody, medium for enzyme-inhibitor, and larger for the rest of the complexes) and the funnel size correlation with the size of the interface. Guidelines for the optimal sampling of docking coordinates, based on the funnel size estimates, were explored.     

Structural alignment of proteins by a novel TOPOFIT method, as a superimposition of common volumes at a topomax point

Similarity of protein structures has been analyzed using three-dimensional Delaunay triangulation patterns derived from the backbone representation. It has been found that structurally related proteins have a common spatial invariant part, a set of tetrahedrons, mathematically described as a common spatial subgraph volume of the three-dimensional contact graph derived from Delaunay tessellation (DT). Based on this property of protein structures, we present a novel common volume superimposition (TOPOFIT) method to produce structural alignments. Structural alignments usually evaluated by a number of equivalent (aligned) positions (N(e)) with corresponding root mean square deviation (RMSD). The superimposition of the DT patterns allows one to uniquely identify a maximal common number of equivalent residues in the structural alignment. In other words, TOPOFIT identifies a feature point on the RMSD N(e) curve, a topomax point, until which the topologies of two structures correspond to each other, including backbone and interresidue contacts, whereas the growing number of mismatches between the DT patterns occurs at larger RMSD (N(e)) after the topomax point. It has been found that the topomax point is present in all alignments from different protein structural classes; therefore, the TOPOFIT method identifies common, invariant structural parts between proteins. The alignments produced by the TOPOFIT method have a good correlation with alignments produced by other current methods. This novel method opens new opportunities for the comparative analysis of protein structures and for more detailed studies on understanding the molecular principles of tertiary structure organization and functionality. The TOPOFIT method also helps to detect conformational changes, topological differences in variable parts, which are particularly important for studies of variations in active/ binding sites and protein classification.     

Tertiary structure predictions on a comprehensive benchmark of medium to large size proteins

We evaluate tertiary structure predictions on medium to large size proteins by TASSER, a new algorithm that assembles protein structures through rearranging the rigid fragments from threading templates guided by a reduced Calpha and side-chain based potential consistent with threading based tertiary restraints. Predictions were generated for 745 proteins 201-300 residues in length that cover the Protein Data Bank (PDB) at the level of 35% sequence identity. With homologous proteins excluded, in 365 cases, the templates identified by our threading program, PROSPECTOR_3, have a root-mean-square deviation (RMSD) to native < 6.5 angstroms, with >70% alignment coverage. After TASSER assembly, in 408 cases the best of the top five full-length models has a RMSD < 6.5 angstroms. Among the 745 targets are 18 membrane proteins, with one-third having a predicted RMSD < 5.5 A. For all representative proteins less than or equal to 300 residues that have corresponding multiple NMR structures in the Protein Data Bank, approximately 20% of the models generated by TASSER are closer to the NMR structure centroid than the farthest individual NMR model. These results suggest that reasonable structure predictions for nonhomologous large size proteins can be automatically generated on a proteomic scale, and the application of this approach to structural as well as functional genomics represent promising applications of TASSER.     

TASSER-based refinement of NMR structures

The TASSER structure prediction algorithm is employed to investigate whether NMR structures can be moved closer to their corresponding X-ray counterparts by automatic refinement procedures. The benchmark protein dataset includes 61 nonhomologous proteins whose structures have been determined by both NMR and X-ray experiments. Interestingly, by starting from NMR structures, the majority (79%) of TASSER refined models show a structural shift toward their X-ray structures. On average, the TASSER refined models have a root-mean-square-deviation (RMSD) from the X-ray structure of 1.785 A (1.556 A) over the entire chain (aligned region), while the average RMSD between NMR and X-ray structures (RMSD(NMR_X-ray)) is 2.080 A (1.731 A). For all proteins having a RMSD(NMR_X-ray) >2 A, the TASSER refined structures show consistent improvement. However, for the 34 proteins with a RMSD(NMR_X-ray) <2 A, there are only 21 cases (60%) where the TASSER model is closer to the X-ray structure than NMR, which may be due to the inherent resolution of TASSER. We also compare the TASSER models with 12 NMR models in the RECOORD database that have been recalculated recently by Nederveen et al. from original NMR restraints using the newest molecular dynamics tools. In 8 of 12 cases, TASSER models show a smaller RMSD to X-ray structures; in 3 of 12 cases, where RMSD(NMR_X-ray) <1 A, RECOORD does better than TASSER. These results suggest that TASSER can be a useful tool to improve the quality of NMR structures.     

TOUCHSTONEX: protein structure prediction with sparse NMR data

TOUCHSTONEX, a new method for folding proteins that uses a small number of long-range contact restraints derived from NMR experimental NOE (nuclear Overhauser enhancement) data, is described. The method employs a new lattice-based, reduced model of proteins that explicitly represents C(alpha), C(beta), and the sidechain centers of mass. The force field consists of knowledge-based terms to produce protein-like behavior, including various short-range interactions, hydrogen bonding, and one-body, pairwise, and multibody long-range interactions. Contact restraints were incorporated into the force field as an NOE-specific pairwise potential. We evaluated the algorithm using a set of 125 proteins of various secondary structure types and lengths up to 174 residues. Using N/8 simulated, long-range sidechain contact restraints, where N is the number of residues, 108 proteins were folded to a C(alpha)-root-mean-square deviation (RMSD) from native below 6.5 A. The average RMSD of the lowest RMSD structures for all 125 proteins (folded and unfolded) was 4.4 A. The algorithm was also applied to limited experimental NOE data generated for three proteins. Using very few experimental sidechain contact restraints, and a small number of sidechain-main chain and main chain-main chain contact restraints, we folded all three proteins to low-to-medium resolution structures. The algorithm can be applied to the NMR structure determination process or other experimental methods that can provide tertiary restraint information, especially in the early stage of structure determination, when only limited data are available.     

A new Hybrid Monte Carlo algorithm for protein potential function test and structure refinement

A new Hybrid Monte Carlo (HMC) algorithm has been developed to test protein potential functions and, ultimately, refine protein structures. The main principle of this algorithm is, in each cycle, a new trial conformation is generated by carrying out a short period of molecular dynamics (MD) iterations with a set of random parameters (including the MD time step, the number of MD steps, the MD temperature, and the seed for initial MD velocity assignment); then to accept or reject the new conformation on the basis of the Metropolis criterion. The novelty in this paper is that the potential in MD iterations is different from that in the MC step. In the former, it is a molecular mechanics potential, in the latter it is a knowledge-based potential (KBP). Directed by the KBP, the MD iteration is used to search conformational space for realistic conformations with low KBP energy. It circumvents the difficulty in using KBP functions directly in MD simulation, as KBP functions are typically incomplete, and do not always have continuous derivatives required for the calculation of the forces. The new algorithm has been tested in explorations of conformational space. In these test calculations the KBP energy was found to drop below the value for the native conformation, and the correlation between the root mean square deviation (RMSD) and the KBP energy was shown to be different from the test results in other references. At the present time, the algorithm is useful for testing new KBP functions. Furthermore, if a KBP function can be found for which the native conformation has the lowest energy and the energy/RMSD correlation is good, then this new algorithm also will be a tool for refinement of the theory-based structural models.     

Superimposition of protein structures with dynamically weighted RMSD

In protein modeling, one often needs to superimpose a group of structures for a protein. A common way to do this is to translate and rotate the structures so that the square root of the sum of squares of coordinate differences of the atoms in the structures, called the root-mean-square deviation (RMSD) of the structures, is minimized. While it has provided a general way of aligning a group of structures, this approach has not taken into account the fact that different atoms may have different properties and they should be compared differently. For this reason, when superimposed with RMSD, the coordinate differences of different atoms should be evaluated with different weights. The resulting RMSD is called the weighted RMSD (wRMSD). Here we investigate the use of a special wRMSD for superimposing a group of structures with weights assigned to the atoms according to certain thermal motions of the atoms. We call such an RMSD the dynamically weighted RMSD (dRMSD). We show that the thermal motions of the atoms can be obtained from several sources such as the mean-square fluctuations that can be estimated by Gaussian network model analysis. We show that the superimposition of structures with dRMSD can successfully identify protein domains and protein motions, and that it has important implications in practice, e.g., in aligning the ensemble of structures determined by nuclear magnetic resonance.