Sampling and efficiency of metric matrix distance geometry: A novel partial metrization algorithm

Summary In this paper, we present a reassessment of the sampling properties of the metric matrix distance geometry algorithm, which is in wide-spread use in the determination of three-dimensional structures from nuclear magnetic resonance (NMR) data. To this end, we compare the conformational space sampled by structures generated with a variety of metric matrix distance geometry protocols. As test systems we use an unconstrained polypeptide, and a small protein (rabbit neutrophil defensin peptide 5) for which only few tertiary distances had been derived from the NMR data, allowing several possible folds of the polypeptide chain. A process called metrization in the preparation of a trial distance matrix has a very large effect on the sampling properties of the algorithm. It is shown that, depending on the metrization protocol used, metric matrix distance geometry can have very good sampling properties'indeed, both for the unconstrained model system and the NMR-structure case. We show that the sampling properties are to a great degree determined by the way in which the first few distances are chosen within their bounds. Further, we present a new protocol (partial metrization) that is computationally more efficient but has the same excellent sampling properties. This novel protocol has been implemented in an expanded new release of the program X-PLOR with distance geometry capabilities.     

Trajectory analysis of NMR structure calculations

Summary NMR as well as X-ray crystallography are used to determine the three-dimensional structures of macromolecules at atomic resolution. Structure calculation generates coordinates that are compatible with NMR data from randomly generated initial structures. We analyzed the trajectory taken by structures during NMR structure calculation in conformational space, assuming that the distance between two structures in conformational space is the root-mean-square deviation between the two structures. The coordinates of a structure in conformational space were obtained by applying the metric multidimensional scaling method. As an example, we used a 22-residue peptide, -Conotoxin GIIIA, and a simulated annealing protocol of XPLOR. We found that the three-dimensional solution of the multidimensional scaling analysis is sufficient to describe the overall configuration of the trajectories in conformational space. By comparing the trajectories of the entire calculation with those of the converged calculation, random sampling of conformational space is readily discernible. Trajectory analysis can also be used for optimization of protocols of NMR structure calculation, by examining individual trajectories.Abbreviations MD molecular dynamics - MDS multidimensional scaling - rmsd root-mean-square deviation - armsd angular rmsd - R multiple correlation coefficient - YASAP yet another simulated annealing protocol - PCA principal component analysis     

GENFOLD: a genetic algorithm for folding protein structures using NMR restraints.

We report the development and validation of the program GENFOLD, a genetic algorithm that calculates protein structures using restraints obtained from NMR, such as distances derived from nuclear Overhauser effects, and dihedral angles derived from coupling constants. The program has been tested on three proteins: the POU domain (a small three-helix DNA-binding protein), bovine pancreatic trypsin inhibitor (BPTI), and the starch-binding domain from Aspergillus niger glucoamylase I, a 108-residue beta-sheet protein. Structures were calculated for each protein using published NMR restraints. In addition, structures were calculated for BPTI using artificial restraints generated from a high-resolution crystal structure. In all cases the fittest calculated structures were close to the target structure, and could be refined to structures indistinguishable from the target structures by means of a low-temperature simulated annealing refinement. The effectiveness of the program is similar to that of distance geometry and simulated annealing methods, and it is capable of using a very wide range of restraints as input. It can thus be readily extended to the calculation of structures of large proteins, for which few NOE restraints may be available.     

A computer-based protocol for semiautomated assignments and 3D structure determination of proteins

Summary A strategy is presented for the semiautomated assignment and 3D structure determination of proteins from heteronuclear multidimensional Nuclear Magnetic Resonance (NMR) data. This approach involves the computer-based assignment of the NMR signals, identification of distance restraints from nuclear Overhauser effects, and generation of 3D structures by using the NMR-derived restraints. The protocol is described in detail and illustrated on a resonance assignment and structure determination of the FK506 binding protein (FKBP, 107 amino acids) complexed to the immunosuppressant, ascomycin. The 3D structures produced from this automated protocol attained backbone and heavy atom rmsd of 1.17 and 1.69 Å, respectively. Although more highly resolved structures of the complex have been obtained by standard interpretation of NMR data (Meadows et al. (1993) Biochemistry, 32, 754–765), the structures generated with this automated protocol required minimal manual intervention during the spectral assignment and 3D structure calculations stages. Thus, the protocol may yield an approximate order of magnitude reduction in the time required for the generation of 3D structures of proteins from NMR data.     

Simulated annealing with restrained molecular dynamics using CONGEN: energy refinement of the NMR solution structures of epidermal and type-alpha transforming growth factors.

The new functionality of the program CONGEN (Bruccoleri RE, Karplus M, 1987, Biopolymers 26:137-168; Bassolino-Klimas D et al., 1996, Protein Sci 5:593-603) has been applied for energy refinement of two previously determined solution NMR structures, murine epidermal growth factor (mEGF) and human type-alpha transforming growth factor (hTGF alpha). A summary of considerations used in converting experimental NMR data into distance constraints for CONGEN is presented. A general protocol for simulated annealing with restrained molecular dynamics is applied to generate NMR solution structures using CONGEN together with real experimental NMR data. A total of 730 NMR-derived constraints for mEGF and 424 NMR-derived constraints for hTGF alpha were used in these energy-refinement calculations. Different weighting schemes and starting conformations were studied to check and/or improve the sampling of the low-energy conformational space that is consistent with all constraints. The results demonstrate that loosened (i.e., "relaxed") sets of the EGF and hTGF alpha internuclear distance constraints allow molecules to overcome local minima in the search for a global minimum with respect to both distance restraints and conformational energy. The resulting energy-refined structures of mEGF and hTGF alpha are compared with structures determined previously and with structures of homologous proteins determined by NMR and X-ray crystallography.     

Simultaneous determination of disulphide bridge topology and three-dimensional structure using ambiguous intersulphur distance restraints: Possibilities and limitations

Knowledge of the native disulphide bridge topology allows the introduction of conformational restraints between remote parts of the peptide chain. This information is therefore of great importance for the successful determination of the three-dimensional structure of cysteine-rich proteins by NMR spectroscopy. In this paper we investigate the limitations of using ambiguous intersulphur restraints [Nilges, M. (1995) J. Mol. Biol., 245, 645–660] associated with NMR experimental information to determine the native disulphide bridge pattern. Using these restraints in a simulated annealing protocol we have determined the correct topology of numerous examples, including a protein with seven disulphide bridges (phospholipase A₂) and a protein in which 25% of the total number of residues are cysteines (-conotoxin GIIIB). We have also characterised the behaviour of the method when only limited experimental data is available, and find that the proposed protocol permits disulphide bridge determination even with a small number of restraints (around 5 NOEs – including a long-range restraint – per residue). In addition, we have shown that under these conditions the use of a reduced penalty function allows the identification of misassigned NOE restraints. These results indicate that the use of ambiguous intersulphur distances with the proposed simulated annealing protocol is a general method for the determination of disulphide bridge topology, particularly interesting in the first steps of NMR study of cysteine-rich proteins. Comparison with previously proposed protocols indicates that the presented method is more reliable and the interpretation of results is straightforward.     

Exhaustive enumeration of protein conformations using experimental restraints.

We present an efficient new algorithm that enumerates all possible conformations of a protein that satisfy a given set of distance restraints. Rapid growth of all possible self-avoiding conformations on the diamond lattice provides construction of alpha-carbon representations of a protein fold. We investigated the dependence of the number of conformations on pairwise distance restraints for the proteins crambin, pancreatic trypsin inhibitor, and ubiquitin. Knowledge of between one and two contacts per monomer is shown to be sufficient to restrict the number of candidate structures to approximately 1,000 conformations. Pairwise RMS deviations of atomic position comparisons between pairs of these 1,000 structures revealed that these conformations can be grouped into about 25 families of structures. These results suggest a new approach to assessing alternative protein folds given a very limited number of distance restraints. Such restraints are available from several experimental techniques such as NMR, NOESY, energy transfer fluorescence spectroscopy, and crosslinking experiments. This work focuses on exhaustive enumeration of protein structures with emphasis on the possible use of NOESY-determined distance restraints.     

Methods of NMR structure refinement: molecular dynamics simulations improve the agreement with measured NMR data of a C-terminal peptide of GCN4-p1

The C-terminal trigger sequence is essential in the coiled-coil formation of GCN4-p1; its conformational properties are thus of importance for understanding this process at the atomic level. A solution NMR model structure of a peptide, GCN4p16–31, encompassing the GCN4-p1 trigger sequence was proposed a few years ago. Derived using a standard single-structure refinement protocol based on 172 nuclear Overhauser effect (NOE) distance restraints, 14 hydrogen-bond and 11 ϕ torsional-angle restraints, the resulting set of 20 NMR model structures exhibits regular α-helical structure. However, the set slightly violates some measured NOE bounds and does not reproduce all 15 measured ³J(H_N-H_Cα)-coupling constants, indicating that different conformers of GCN4p16–31 might be present in solution. With the aim to resolve structures compatible with all NOE upper distance bounds and ³J-coupling constants, we executed several structure refinement protocols employing unrestrained and restrained molecular dynamics (MD) simulations with two force fields. We find that only configurational ensembles obtained by applying simultaneously time-averaged NOE distance and ³J-coupling constant restraining with either force field reproduce all the experimental data. Additionally, analyses of the simulated ensembles show that the conformational variability of GCN4p16–31 in solution admitted by the available set of 187 measured NMR data is larger than represented by the set of the NMR model structures. The conformations of GCN4p16–31 in solution differ in the orientation not only of the side-chains but also of the backbone. The inconsistencies between the NMR model structures and the measured NMR data are due to the neglect of averaging effects and the inclusion of hydrogen-bond and torsional-angle restraints that have little basis in the primary, i.e. measured NMR data.     

Complete relaxation matrix refinement of NMR structures of proteins using analytically calculated dihedral angle derivatives of NOE intensities

Summary A new method for refining three-dimensional (3D) NMR structures of proteins is described, which takes account of the complete relaxation pathways. Derivatives of the NOE intensities with respect to the dihedral angles are analytically calculated, and efficiently evaluated with the use of a filter technique for identifying the dominant terms of these derivatives. This new method was implemented in the distance geometry program DIANA. As an initial test, we refined 30 rigid distorted helical structures, using a simulated data set of NOE distance constraints for a rigid standard -helix. The final root-mean-square deviations of the refined structures relative to the standard helix were less than 0.1 Å, and the R-factors dropped from values between 7% and 32% to values of less than 0.5% in all cases, which compares favorably with the results from distance geometry calculations. In particular, because spin diffusion was not explicitly considered in the evaluation of exact¹H–¹H distances corresponding to the simulated NOE intensities, a group of nearly identical distance geometry structures was obtained which had about 0.5 Å root-mean-square deviation from the standard -helix. Further test calculations using an experimental NOE data set recorded for the protein trypsin inhibitor K showed that the complete relaxation matrix refinement procedure in the DIANA program is functional also with systems of practical interest.Abbreviations RMSD root-mean-square deviation - NOE nuclear Overhauser enhancement - NOESY 2-dimensional nuclear Overhauser enhancement spectroscopy - CPU central processing unit     

A novel method for the biosynthesis of deuterated proteins with selective protonation at the aromatic rings of Phe,Tyr and Trp

A novel biosynthetic strategy is described for the preparation of deuterated proteins containing protons at the ring carbons of Phe, Tyr and Trp, using the aromatic amino acid precursor shikimic acid. Specific protonation at aromatic side chains, with complete deuteration at C^/positions was achieved in proteins overexpressed in bacteria grown in shikimate-supplemented D₂O medium. Co-expression of a shikimate transporter in prototrophic bacteria resulted in protonation levels of 62–79%, whereas complete labeling was accomplished using shikimate auxotrophic bacteria. Our labeling protocol permits the measurement of important aromatic side chain derived distance restraints in perdeuterated proteins that could be utilized to enhance the accuracy of NMR structures calculated using low densities of NOEs from methyl selectively protonated samples.     

Capturing Conformational States in Proteins Using Sparse Paramagnetic NMR Data

Capturing conformational changes in proteins or protein-protein complexes is a challenge for both experimentalists and computational biologists. Solution nuclear magnetic resonance (NMR) is unique in that it permits structural studies of proteins under greatly varying conditions, and thus allows us to monitor induced structural changes. Paramagnetic effects are increasingly used to study protein structures as they give ready access to rich structural information of orientation and long-range distance restraints from the NMR signals of backbone amides, and reliable methods have become available to tag proteins with paramagnetic metal ions site-specifically and at multiple sites. In this study, we show how sparse pseudocontact shift (PCS) data can be used to computationally model conformational states in a protein system, by first identifying core structural elements that are not affected by the environmental change, and then computationally completing the remaining structure based on experimental restraints from PCS. The approach is demonstrated on a 27 kDa two-domain NS2B-NS3 protease system of the dengue virus serotype 2, for which distinct closed and open conformational states have been observed in crystal structures. By changing the input PCS data, the observed conformational states in the dengue virus protease are reproduced without modifying the computational procedure. This data driven Rosetta protocol enables identification of conformational states of a protein system, which are otherwise difficult to obtain either experimentally or computationally.     

NMR-validated structural model for oxidized<Emphasis Type="Italic"> Rhodopseudomonas palustris</Emphasis> cytochrome<Emphasis Type="Italic"> c</Emphasis><Subscript>556</Subscript>

The structure of oxidized Rhodopseudomonas palustris cytochrome c ₅₅₆ has been modeled after that of high-spin cytochrome c from the same bacterium, the latter being the protein with the greatest sequence identity (35%) among all sequenced proteins in the genomes. The two proteins differ in the number of ligands to iron and in spin state, the former being six-coordinate low-spin and the latter five-coordinate high-spin. In order to validate this modeled structure, several structural restraints were obtained by performing a restricted set of NMR experiments, without performing a complete assignment of the protein signals. The aim was to exploit the special restraints arising from the paramagnetism of the metal ion. A total of 43 residual-dipolar-coupling and 74 pseudocontact-shift restraints, which together sampled all regions of the protein, were used in conjunction with over 40 routinely obtained NOE distance restraints. A calculation procedure was undertaken combining the program MODELLER and the solution structure determination program PARAMAGNETIC DYANA, which includes paramagnetism-based restraints. The directions and magnitude of the magnetic susceptibility anisotropy tensor were also calculated. The approach readily provides useful results, especially for paramagnetic metalloproteins of moderate to large dimensions.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00775-003-0511-2     

NMR three-dimensional solution structure of the serine protease inhibitor cyclotheonamide A

The nmr solution conformation of cyclotheonamide A (CtA) was determined in aqueous media. The data produced 15 distance and 10 torsional constraints which were used to generate conformations using restrained simulated annealing (SA) and distance geometry/simulated annealing (DG/SA) calculations. Two different calculation protocols were performed to ensure proper sampling of conformational space and even though the torsional restraints were input differently, both calculation methods yielded the same conformation of CtA. In the structure calculations, all solutions of the Karplus equation were sampled simultaneously using the restrained SA protocol and large ranges were used for the dihedral restraints in the DG/SA protocol because all solutions to the Karplus equation could not be sampled simultaneously. The solution conformation was also compared to the solid state x-ray conformations of CtA bound to thrombin and trypsin. The conformation of the residues important for active site binding (d -Phe, h-Arg, and Pro) are nearly identical in aqueous solution and solid state with largest differences at the a-Ala and v-Tyr residues. CtA appears to be preordered in structure and does not undergo a significant conformational change upon binding to the enzyme active site. © 1997 John Wiley & Sons, Inc.     

Determination of three-dimensional structures of proteins by simulated annealing with interproton distance restraints. Application to crambin, potato carboxypeptidase inhibitor and barley serine proteinase inhibitor 2

An automated method, based on the principle of simulated annealing, is presented for determining the three-dimensional structures of proteins on the basis of short (less than 5 A) interproton distance data derived from nuclear Overhauser enhancement (NOE) measurements. The method makes use of Newton's equations of motion to increase temporarily the temperature of the system in order to search for the global minimum region of a target function comprising purely geometric restraints. These consist of interproton distances supplemented by bond lengths, bond angles, planes and soft van der Waals repulsion terms. The latter replace the dihedral, van der Waals, electrostatic and hydrogen-bonding potentials of the empirical energy function used in molecular dynamics simulations. The method presented involves the implementation of a number of innovations over our previous restrained molecular dynamics approach [Clore, G.M., Brünger, A.T., Karplus, M. and Gronenborn, A.M. (1986) J. Mol. Biol., 191, 523-551]. These include the development of a new effective potential for the interproton distance restraints whose functional form is dependent on the magnitude of the difference between calculated and target values, and the design and implementation of robust and fully automatic protocol. The method is tested on three systems: the model system crambin (46 residues) using X-ray structure derived interproton distance restraints, and potato carboxypeptidase inhibitor (CPI; 39 residues) and barley serine proteinase inhibitor 2 (BSPI-2; 64 residues) using experimentally derived interproton distance restraints. Calculations were carried out starting from the extended strands which had atomic r.m.s. differences of 57, 38 and 33 A with respect to the crystal structures of BSPI-2, crambin and CPI respectively. Unbiased sampling of the conformational space consistent with the restraints was achieved by varying the random number seed used to assign the initial velocities. This ensures that the different trajectories diverge during the early stages of the simulations and only converge later as more and more interproton distance restraints are satisfied. The average backbone atomic r.m.s. difference between the converged structures is 2.2 +/- 0.3 A for crambin (nine structures), 2.4 +/- 0.3 A for CPI (eight structures) and 2.5 +/- 0.2 A for BSPI-2 (five structures). The backbone atomic r.m.s. difference between the mean structures derived by averaging the coordinates of the converged structures and the corresponding X-ray structures is 1.2 A for crambin, 1.6 A for CPI and 1.7 A for BSPI-2.     

1H 2D NMR and distance geometry study of the folding of Ecballium elaterium trypsin inhibitor, a member of the squash inhibitors family

The solution conformation of synthetic Ecballium elaterium trypsin inhibitor II, a 28-residue peptide with 3 disulfide bridges, has been studied by 1H 2D NMR measurements. Secondary structure elements were determined: a miniantiparallel beta-sheet Met 7-Cys 9 and Gly 25-Cys 27, a beta-hairpin 20-28 with beta-turn 22-25, and two tight turns Asp 12-Cys 15 and Leu 16-Cys 19. A set of interproton distance restraints deduced from two-dimensional nuclear Overhauser enhancement spectra and 13 phi backbone torsion angles restraints were used as the basis of three-dimensional structure computations including disulfide bridges arrangement by using distance geometry calculations. Computations for the 15 possible S-S linkage combinations lead to the proposal of the array 2-19, 9-21, 15-27 as the most probable structure for EETI II.     

Protein structures in solution by nuclear magnetic resonance and distance geometry. The polypeptide fold of the basic pancreatic trypsin inhibitor determined using two different algorithms, DISGEO and DISMAN

A set of conformational restraints derived from nuclear magnetic resonance (n.m.r.) measurements on solutions of the basic pancreatic trypsin inhibitor (BPTI) was used as input for distance geometry calculations with the programs DISGEO and DISMAN. Five structures obtained with each of these algorithms were systematically compared among themselves and with the crystal structure of BPTI. It is clear that the protein architecture observed in single crystals of BPTI is largely preserved in aqueous solution, with local structural differences mainly confined to the protein surface. The results confirm that protein conformations determined in solution by combined use of n.m.r. and distance geometry are a consequence of the experimental data and do not depend significantly on the algorithm used for the structure determination. The data obtained further provide an illustration that long intramolecular distances in proteins, which are comparable with the radius of gyration, are defined with high precision by relatively imprecise nuclear Overhauser enhancement measurements of a large number of much shorter distances.     

Optimisation of metric matrix embedding by genetic algorithms

Summary To improve the convergence properties of embedding distance geometry, a new approach was developed by combining the distance-geometry methodology with a genetic algorithm. This new approach is called DG-OMEGA (DG, optimised metric matrix embedding by genetic algorithms). The genetic algorithm was used to combine well-defined parts of individual structures generated by the distance-geometry program, and to identify new lower and upper distance bounds within the original experimental restraints in order to restrict the sampling of the metrisation algorithm to promising regions of the conformational space. The algorithm was tested on cyclosporin A, which is notorious for its intrinsic difficult sampling properties. A set of 58 distance restraints was employed. It was shown that DG resulted in an improvement of convergence behaviour as well as sampling properties with respect to the standard distance-geometry protocol.     

Molecular dynamics and accuracy of NMR structures: effects of error bounds and data removal

The effect of internal dynamics on the accuracy of nuclear magnetic resonance (NMR) structures was studied in detail using model distance restraint sets (DRS) generated from a 6.6 nanosecond molecular dynamics trajectory of bovine pancreatic trypsin inhibitor. The model data included the effects of internal dynamics in a very realistic way. Structure calculations using different error estimates were performed with iterative removal of systematically violated restraints. The accuracy of each calculated structure was measured as the atomic root mean square (RMS) difference to the optimized average structure derived from the trajectory by structure factors refinement. Many of the distance restraints were derived from NOEs that were significantly affected by internal dynamics. Depending on the error bounds used, these distance restraints seriously distorted the structure, leading to deviations from the coordinate average of the dynamics trajectory even in rigid regions. Increasing error bounds uniformly for all distance restraints relieved the strain on the structures. However, the accuracy did not improve. Significant improvement of accuracy was obtained by identifying inconsistent restraints with violation analysis, and excluding them from the calculation. The highest accuracy was obtained by setting bounds rather tightly, and removing about a third of the restraints. The limiting accuracy for all backbone atoms was between 0.6 and 0.7 A. Also, the precision of the structures increased with removal of inconsistent restraints, indicating that a high precision is not simply the consequence of tight error bounds but of the consistency of the DRS. The precision consistently overestimated the accuracy.     

Application of sparse NMR restraints to large-scale protein structure prediction

The protein structure prediction algorithm TOUCHSTONEX that uses sparse distance restraints derived from NMR nuclear Overhauser enhancement (NOE) data to predict protein structures at low-to-medium resolution was evaluated as follows: First, a representative benchmark set of the Protein Data Bank library consisting of 1365 proteins up to 200 residues was employed. Using N/8 simulated long-range restraints, where N is the number of residues, 1023 (75%) proteins were folded to a C(alpha) root-mean-square deviation (RMSD) from native <6.5 A in one of the top five models. The average RMSD of the models for all 1365 proteins is 5.0 A. Using N/4 simulated restraints, 1206 (88%) proteins were folded to a RMSD <6.5 A and the average RMSD improved to 4.1 A. Then, 69 proteins with experimental NMR data were used. Using long-range NOE-derived restraints, 47 proteins were folded to a RMSD <6.5 A with N/8 restraints and 61 proteins were folded to a RMSD <6.5 A with N/4 restraints. Thus, TOUCHSTONEX can be a tool for NMR-based rapid structure determination, as well as used in other experimental methods that can provide tertiary restraint information.     

Solution structure of neurotoxin I from the sea anemone Stichodactyla helianthus. A nuclear magnetic resonance, distance geometry, and restrained molecular dynamics study

The three-dimensional structure of the sea anemone polypeptide Stichodactyla helianthus neurotoxin I in aqueous solution has been determined using distance geometry and restrained molecular dynamics simulations based on NMR data acquired at 500 MHz. A set of 470 nuclear Overhauser enhancement values was measured, of which 216 were used as distance restraints in the structure determination along with 15 dihedral angles derived from coupling constants. After restrained molecular dynamics refinement, the eight structures that best fit the input data form a closely related family. They describe a structure that consists of a core of twisted, four-stranded, antiparallel beta-sheet encompassing residues 1-3, 19-24, 29-34, and 40-47, joined by three loops, two of which are well defined by the NMR data. The third loop, encompassing residues 7-16, is poorly defined by the data and is assumed to undergo conformational averaging in solution. Pairwise root mean square displacement values for the backbone heavy atoms of the eight best structures are 1.3 +/- 0.2A when the poorly defined loop is excluded and 3.6 +/- 1.0A for all backbone atoms. Refinement using restrained molecular dynamics improved the quality of the structures generated by distance geometry calculations with respect to the number of nuclear Overhauser enhancements violated, the size of the total distance violations and the total potential energies of the structures. The family of structures for S. heliathus neurotoxin I is compared with structures of related sea anemone proteins that also bind to the voltage-gated sodium channel.