Combined use of proton-proton Overhauser enhancements and a distance geometry algorithm for determination of polypeptide conformations. Application to micelle-bound glucagon

In a new approach for the determination of polypeptide conformation, experimental data on intramolecular distances between pairs of hydrogen atoms obtained from nuclear Overhauser enhancement studies are used as input for a distance geometry algorithm. The algorithm determines the limits of the conformation space occupied by the polypeptide chain. The experimental data are used in such a way that the real conformation should in all cases be within these limits. Two important features of the method are that the results do not depend critically on the accuracy of the distance measurements by nuclear Overhauser enhancement studies and that internal mobility of the polypeptide conformation is explicitly taken into consideration. The use of this new procedure is illustrated with a structural study of the region 19-27 of glucagon bound to perdeuterated dodecylphosphocholine micelles.     

Limited sampling of conformational space by the distance geometry algorithm: implications for structures generated from NMR data

Calculations with a metric matrix distance geometry algorithm were performed that show that the standard implementation of the algorithm generally samples a very limited region of conformational space. This problem is most severe when only a small amount of distance information is used as input for the algorithm. Control calculations were performed on linear peptides, disulfide-linked peptides, and a double-stranded DNA decamer where only distances defining the covalent structures of the molecules (as well as the hydrogen bonds for the base pairs in the DNA) were included as input. Since the distance geometry algorithm is commonly used to generate structures of biopolymers from distance data obtained from NMR experiments, simulations were performed on the small globular protein basic pancreatic trypsin inhibitor (BPTI) that mimic calculations performed with actual NMR data. The results on BPTI and on the control peptides indicate that the standard implementation of the algorithm has two main problems: first, that it generates extended structures; second, that it has a tendency to consistently produce similar structures instead of sampling all structures consistent with the input distance information. These results also show that use of a simple root-mean-square deviation for evaluating the quality of the structures generated from NMR data may not be generally appropriate. The main sources of these problems are identified, and our results indicate that the problems are not a fundamental property of the distance geometry algorithm but arise from the implementations presently used to generate structures from NMR data. Several possible methods for alleviating these problems are discussed.     

Application of a genetic algorithm in the conformational analysis of methylene-acetal-linked thymine dimers in DNA: Comparison with distance geometry calculations

The three-dimensional spatial structure of a methylene-acetal-linked thymine dimer presentin a 10 base-pair (bp) sense–antisense DNA duplex was studied with a geneticalgorithm designed to interpret NOE distance restraints. Trial solutions were represented bytorsion angles. This means that bond angles for the dimer trial structures are kept fixed duringthe genetic algorithm optimization. Bond angle values were extracted from a 10 bpsense–antisense duplex model that was subjected to energy minimization by means ofa modified AMBER force field. A set of 63 proton–proton distance restraints definingthe methylene-acetal-linked thymine dimer was available. The genetic algorithm minimizesthe difference between distances in the trial structures and distance restraints. A largeconformational search space could be covered in the genetic algorithm optimization byallowing a wide range of torsion angles. The genetic algorithm optimization in all cases ledto one family of structures. This family of the methylene-acetal-linked thymine dimer in theduplex differs from the family that was suggested from distance geometry calculations. It isdemonstrated that the bond angle geometry around the methylene-acetal linkage plays animportant role in the optimization.     

Constrained global optimization for estimating molecular structure from atomic distances.

Finding optimal three-dimensional molecular configurations based on a limited amount of experimental and/or theoretical data requires efficient nonlinear optimization algorithms. Optimization methods must be able to find atomic configurations that are close to the absolute, or global, minimum error and also satisfy known physical constraints such as minimum separation distances between atoms (based on van der Waals interactions). The most difficult obstacles in these types of problems are that 1) using a limited amount of input data leads to many possible local optima and 2) introducing physical constraints, such as minimum separation distances, helps to limit the search space but often makes convergence to a global minimum more difficult. We introduce a constrained global optimization algorithm that is robust and efficient in yielding near-optimal three-dimensional configurations that are guaranteed to satisfy known separation constraints. The algorithm uses an atom-based approach that reduces the dimensionality and allows for tractable enforcement of constraints while maintaining good global convergence properties. We evaluate the new optimization algorithm using synthetic data from the yeast phenylalanine tRNA and several proteins, all with known crystal structure taken from the Protein Data Bank. We compare the results to commonly applied optimization methods, such as distance geometry, simulated annealing, continuation, and smoothing. We show that compared to other optimization approaches, our algorithm is able combine sparse input data with physical constraints in an efficient manner to yield structures with lower root mean squared deviation.     

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.     

Distances as degrees of freedom

Tertiary contact distance information of varying resolution for large biological molecules abounds in the literature. The results provided herein develop a framework by which information of this type can be used to reduce the allowable configuration space of a macromolecule. The approach combines graph theory and distance geometry. Large molecules are represented as simple, undirected graphs, with atoms, or groups, as vertices, and distances between them as edges. It is shown that determination of the exact structure of a molecule in three dimensions only requires the specification of all the distances in a single tetrahedron, and four distances to every other atom. This is 4N-10 distances which is a subset of the total N(N-1)/2 unique distances in a molecule consisting of N atoms. This requirement for only 4N-10 distances has serious implications for distance geometry implementations in which all N(N-1)/2 distances are specified by bounded random numbers. Such distance matrices represent overspecified systems which when solved lead to non-obvious distribution of any error caused by inherent contradictions in the input data. It is also shown that numerous valid subsets of 4N-10 distances can be constructed. It is thus possible to tailor a subset of distances using all known distances as degrees of freedom, and thereby reduce the configuration space of the molecule. Simple algebraic relationships are derived that relate sets of distances, and complicated rotations are avoided. These relationships are used to construct minimum, complete sets of distances necessary to specify the exact structure of the entire molecule in three dimensions from incomplete distance information, and to identify sets of inconsistent distances. The method is illustrated for the flexible structural types present in large ribosomal RNAs: 1.) A five-membered ring; 2.) a chemically bonded chain with its ends in contact (i.e., a hairpin loop); 3.) the spatial orientation of two separate molecules, and; 4.) an RNA helix that can have variation in individual base pairs, giving rise to global deviation from standardized helical forms.     

Distances as Degrees of Freedom

Abstract

Tertiary contact distance information of varying resolution for large biological molecules abounds in the literature. The results provided herein develop a framework by which information of this type can be used to reduce the allowable configuration space of a macromolecule. The approach combines graph theory and distance geometry. Large molecules are represented as simple, undirected graphs, with atoms, or groups, as vertices, and distances between them as edges. It is shown that determination of the exact structure of a molecule in three dimensions only requires the specification of all the distances in a single tetrahedron, and four distances to every other atom. This is 4N-10 distances which is a subset of the total N(N-l)/2 unique distances in a molecule consisting of N atoms. This requirement for only 4N-10 distances has serious implications for distance geometry implementations in which all N(N-l)/2 distances are specified by bounded random numbers. Such distance matrices represent overspecified systems which when solved lead to non-obvious distribution of any error caused by inherent contradictions in the input data. It is also shown that numerous valid subsets of 4N-10 distances can be constructed. It is thus possible to tailor a subset of distances using all known distances as degrees of freedom, and thereby reduce the configuration space of the molecule. Simple algebraic relationships are derived that relate sets of distances, and complicated rotations are avoided. These relationships are used to construct minimum, complete sets of distances necessary to specify the exact structure of the entire molecule in three dimensions from incomplete distance information, and to identify sets of inconsistent distances. The method is illustrated for the flexible structural types present in large ribosomal RNAs: 1.) A five-membered ring; 2.) a chemically bonded chain with its ends in contact (i.e., a hairpin loop); 3.) the spatial orientation of two separate molecules, and; 4.) an RNA helix that can have variation in individual base pairs, giving rise to global deviation from standardized helical forms.     

An approach to the multiple-minima problem in protein folding by relaxing dimensionality. Tests on enkephalin

An algorithm for locating the region in conformational space containing the global energy minimum of a polypeptide is described. Distances are used as the primary variables in the minimization of an objective function that incorporates both energetic and distance-geometric terms. The latter are obtained from geometry and energy functions, rather than nuclear magnetic resonance experiments, although the algorithm can incorporate distances from nuclear magnetic resonance data if desired. The polypeptide is generated originally in a space of high dimensionality. This has two important consequences. First, all interatomic distances are initially at their energetically most favorable values; i.e. the polypeptide is initially at a global minimum-energy conformation, albeit a high-dimensional one. Second, the relaxation of dimensionality constraints in the early stages of the minimization removes many potential energy barriers that exist in three dimensions, thereby allowing a means of escaping from three-dimensional local minima. These features are used in an algorithm that produces short trajectories of three-dimensional minimum-energy conformations. A conformation in the trajectory is generated by allowing the previous conformation in the trajectory to evolve in a high-dimensional space before returning to three dimensions. The resulting three-dimensional structure is taken to be the next conformation in the trajectory, and the process is iterated. This sequence of conformations results in a limited but efficient sampling of conformational space. Results for test calculations on Met-enkephalin, a pentapeptide with the amino acid sequence H-Tyr-Gly-Gly-Phe-Met-OH, are presented. A tight cluster of conformations (in three-dimensional space) is found with ECEPP energies (Empirical Conformational Energy Program for Peptides) lower than any previously reported. This cluster of conformations defines a region in conformational space in which the global-minimum-energy conformation of enkephalin appears to lie.     

Refinement of the NMR solution structure of a protein to remove distortions arising from neglect of internal motion.

The effect of internal motion on the quality of a protein structure derived from nuclear magnetic resonance (NMR) cross relaxation has been investigated experimentally. Internal rotation of the tyrosine-31 ring of turkey ovomucoid third domain was found to mediate magnetization transfer; the effect led to underestimation of proton-proton distances in its immediate neighborhood. Experimental methods that distinguish pure cross relaxation from chemical exchange mediated cross relaxation were used to separate true distances from distorted ones. Uncorrected and corrected sets of distances, where the corrections took internal motion into account, each were used as input to a distance geometry program for structural modeling. Each set of distances yielded a family of similar (converged) structures. The two families of structures differed considerably (2 A) in the region of tyrosine-31. In addition, differences as large as 1 A were observed at other positions throughout the structure. These results emphasize the importance of analyzing the effects of internal motions in order to obtain more accurate NMR solution structures.     

NMR studies of defensin antimicrobial peptides. 1. Resonance assignment and secondary structure determination of rabbit NP-2 and human HNP-1.

Two-dimensional nuclear magnetic resonance spectroscopy has been used to make resonance assignments of the proton spectra of two defensin antimicrobial peptides, human neutrophil peptide HNP-1 and rabbit neutrophil peptide NP-2. The secondary structures of these peptides were determined from analysis of the proton-proton NOEs and from the positions of slowly exchanging amide protons. Both peptides contain a long stretch of a double-stranded antiparallel beta-sheet in a hairpin conformation that contains a beta-bulge, a short region of triple-stranded beta-sheet, and several tight turns. The NMR results clearly show that HNP-1 forms a dimer or higher order aggregate in solution and that Pro8 exists as a cis peptide bond. The NMR data on these peptides are compared with NMR data for a homologous peptide NP-5 [Bach, A. C., Selsted, M. E., & Pardi, A. (1987) Biochemistry 26, 4389-4397]. Analysis of the conformation-dependent proton chemical shifts shows that it is not possible to confidently judge the structural similarity of the three defensins from chemical shift data alone. However, comparison of the 3JHN alpha coupling constants in NP-2 and NP-5 indicates that the backbone conformations for these peptides are very similar. A more detailed comparison of the solution conformations of the defensins peptides is made in the following paper in this issue where the NMR data are used as input for distance geometry and molecular dynamics calculations to determine the three-dimensional structures of HNP-1 and NP-2.     

Solution structures of proteins from NMR data and modeling: alternative folds for neutrophil peptide 5

The structure of neutrophil peptide 5 in solution has recently been reported (Pardi et al., 1988). The structure determination was accomplished by using a distance geometry algorithm and 107 interproton distance constraints obtained from 2D NMR data. In each of the eight independent solutions to the distance geometry equations, the overall fold of the polypeptide backbone was identical and the root mean square (rms) deviation between backbone atoms of the superimposed structures was small (approximately 2.4 A). In this paper we report additional NP-5 structures obtained by using a new structure generation algorithm: a Monte Carlo search in torsion angle space. These structures have a large rms backbone deviation from the distance geometry structures (approximately 5.0 A). The backbone topologies differ in significant respects from the distance geometry structures and from each other. Structures are found that are pseudo mirror images of part or all of the fold corresponding to that first obtained with the distance geometry procedure. For small proteins, the problem of distinguishing the correct structure among pseudo mirror images is likely to be greater than previously recognized. When a set of test distance constraints constructed from a novel Monte Carlo structure is used as input in the distance geometry algorithm, the fold of the resulting structure does not correspond to that of the target. The results also demonstrate that the previously accepted criteria (the magnitude of the rms deviation between multiple solutions of the distance geometry equations) for defining the accuracy and precision of a peptide structure generated from NMR data are inadequate. An energetic analysis of structures corresponding to the different folding topologies has been carried out. The molecular mechanics energies obtained by minimization and molecular dynamics refinement provide sufficient information to eliminate certain alternative structures. On the basis of a careful comparison of the different trial structures with the experimental data, it is concluded that the NP-5 peptide fold which was originally reported is most consistent with the data. An alternative fold corresponding to structures with low energies and small total distance violations is ruled out because for this fold predicted NOEs are not observed experimentally.     

Explicit distance geometry: identification of all the degrees of freedom in a large RNA molecule.

An alternative approach to distance geometry ("explicit" distance geometry) is being developed for problems, such as the modeling of RNA folding in the ribosome, where relatively few distances are known. The approach explicitly identifies minimal sets of additional distances that can be added to a distance matrix in order to calculate structures that are consistent with all the known information without distorting the original input data. These additional distances are bounded to the extent possible by the known distances. These explicitly added distances can be treated as degrees of freedom and used to explore the full range of alternative foldings consistent with the original input in an organized way. The present paper establishes that it is practical to explicitly determine such degrees of freedom for even very large RNAs. To demonstrate the feasibility of the approach tRNA was represented as a simple undirected graph containing all relevant information represented in the usual cloverleaf secondary structure and nine base-base tertiary interactions. Using a three atom representation for each residue a total of 206 degrees of freedom are explicitly identified. To accomplish this a graph theoretic approach was used in which a minimal covering cycle basis was determined.     

Three-dimensional structure of acyl carrier protein determined by NMR pseudoenergy and distance geometry calculations

Distance constraints from two-dimensional NMR cross-relaxation data are used to derive a three-dimensional structure for acyl carrier protein from Escherichia coli. Several approaches to structure determination are explored. The most successful proves to be an approach that combines the early stages of a distance geometry program with energy minimization in the presence of NMR constraints represented as pseudopotentials. Approximately 450 proton to proton distance constraints including 50 long-range constraints were included in these programs. Starting structures were generated at random by the distance geometry program and energies minimized by a molecular mechanics module to give final structures. Seven of the structures were deemed acceptable on the basis of agreement with experimentally determined distances. Root-mean-square deviations from the mean of these structures for backbone atoms range from 2 to 3 A. All structures show three roughly parallel helices with hydrophobic residues facing inward and hydrophilic residues facing outward. A hydrophobic cleft is recognizable and is identified as a likely site for acyl chain binding.     

A comparison of the restrained molecular dynamics and distance geometry methods for determining three-dimensional structures of proteins on the basis of interproton distances

A direct comparison of the metric matrix distance geometry and restrained molecular dynamics methods for determining three-dimensional structures of proteins on the basis of interproton distances is presented using crambin as a model system. It is shown that both methods reproduce the overall features of the secondary and tertiary structure (shape and polypeptide fold). The region of conformational space sampled by the converged structures generated by the two methods is similar in size, and in both cases the converged structures are distributed about mean structures which are closer to the X-ray structure than any of the individual structures. The restrained molecular dynamics structures are superior to those obtained from distance geometry as regards local backbone conformation, side chain positions and non-bonding energies.     

Comparison of the accuracy of protein solution structures derived from conventional and network-edited NOESY data.

Network-editing experiments are variants of the basic NOESY experiment that allow more accurate direct measurement of interproton distances in macromolecules by defeating specific spin-diffusion pathways. Two network-editing approaches, block-decoupled NOESY and complementary-block-decoupled-NOESY, were applied as three-dimensional, heteronuclear-edited experiments to distance measurement in a small protein, turkey ovomucoid third domain (OMTKY3). Two-hundred and twelve of the original 655 distance constraints observed in this molecule (Krezel AM et al., 1994, J Mol Biol 242:203-214) were improved by their replacement by distances derived from network-edited spectra, and distance geometry/simulated annealing solution structure calculations were performed from both the unimproved and improved distance sets. The resulting two families of structures were found to differ significantly, the most important differences being the hinge angle of a beta-turn and an expansion of the sampled conformation space in the region of the reactive-site loop. The structures calculated from network-editing data are interpreted as a more accurate model of the solution conformation of OMTKY3.     

Nuclear magnetic resonance determination of metal-proton distances in a synthetic calcium binding site of rabbit skeletal troponin C

The binding of gadolinium to a synthetic peptide of 13 amino acid residues representing the calcium binding loop of site 3 of rabbit skeletal troponin C [AcSTnC(103-115)amide] has been studied by using proton nuclear magnetic resonance (1H NMR) spectroscopy. In particular, the proton line broadening and enhanced spin-lattice relaxation have been used to determine proton-metal ion distances for several assigned nuclei in the peptide-metal ion complex. These distances have been used in conjunction with other constraints and a distance algorithm procedure to demonstrate that the structure of the peptide-metal complex as shown by 1H NMR is consistent with the structure of the EF calcium binding loop in the X-ray structure of parvalbumin but that the available 1H NMR distances do not uniquely define the solution structure.     

Extrahelical adenosine stacks into right-handed DNA: solution conformation of the d(C-G-C-A-G-A-G-C-T-C-G-C-G) duplex deduced from distance geometry analysis of nuclear Overhauser effect spectra

This paper reports on features of the three-dimensional structure of the d(C-G-C-A-G-A-G-C-T-C-G-C-G) self-complementary duplex (designated adenosine 13-mer), which contains symmetrical extrahelical adenosines in the interior of the helix. The majority of the protons have been assigned from two-dimensional nuclear Overhauser effect (NOESY) spectra of the adenosine 13-mer in H2O and D2O solution. The measurement of NOESY cross-peak volume integrals as a function of mixing time has yielded a set of 96 short (less than 4.5-A) proton-proton distances defined by lower and upper bounds, which have served as input parameters for a distance geometry analysis of one symmetric half of the adenosine 13-mer duplex. We demonstrate that the extrahelical adenosine stacks into the duplex for all refined structures without disruption of base pairing on either side of the modification site. The distance geometry refinement yields two classes of conformations consistent with distance measurements but which differ in orientation of the stacked extrahelical adenosine at the modification site.     

A Geometric Buildup Algorithm for the Solution of the Distance Geometry Problem Using Least-Squares Approximation

We propose a new geometric buildup algorithm for the solution of the distance geometry problem in protein modeling, which can prevent the accumulation of the rounding errors in the buildup calculations successfully and also tolerate small errors in given distances. In this algorithm, we use all instead of a subset of available distances for the determination of each unknown atom and obtain the position of the atom by using a least-squares approximation instead of an exact solution to the system of distance equations. We show that the least-squares approximation can be obtained by using a special singular value decomposition method, which not only tolerates and minimizes small distance errors, but also prevents the rounding errors from propagation effectively, especially when the distance data is sparse. We describe the least-squares formulations and their solution methods, and present the test results from applying the new algorithm for the determination of a set of protein structures with varying degrees of availability and accuracy of the distances. We show that the new development of the algorithm increases the modeling ability, and improves stability and robustness of the geometric buildup approach significantly from both theoretical and practical points of view.     

The NMR structure of cyclosporin A bound to cyclophilin in aqueous solution

Cyclosporin A bound to the presumed receptor protein cyclophilin was studied in aqueous solution at pH 6.0 by nuclear magnetic resonance spectroscopy using uniform 15N- or 13C-labeling of cyclosporin A and heteronuclear spectral editing techniques. Sequence-specific assignments were obtained for all but one of the cyclosporin A proton resonances. With an input of 108 intramolecular NOEs and four vicinal 3JHN alpha coupling constants, the three-dimensional structure of cyclosporin A bound to cyclophilin was calculated with the distance geometry program DISMAN, and the structures resulting from 181 converged calculations were energy refined with the program FANTOM. A group of 120 conformers was selected on the basis of the residual constraint violations and energy criteria to represent the solution structure. The average of the pairwise root-mean-square distances calculated for the backbone atoms of the 120 structures was 0.58 A. The structure represents a novel conformation of cyclosporin A, for which the backbone conformation is significantly different from the previously reported structures in single crystals and in chloroform solution. The structure has all peptide bonds in the trans form, contains no elements of regular secondary structure and no intramolecular hydrogen bonds, and exposes nearly all polar groups to its environment. The root-mean-square distance between the backbone atoms of the crystal structure of cyclosporin A and the mean of the 120 conformers representing the NMR structure of cyclosporin A bound to cyclophilin is 2.5 A.