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.     

Determination of interspin distances between spin labels attached to insulin: comparison of electron paramagnetic resonance data with the X-ray structure

A method was developed to determine the interspin distances of two or more nitroxide spin labels attached to specific sites in proteins. This method was applied to different conformations of spin-labeled insulins. The electron paramagnetic resonance (EPR) line broadening due to dipolar interaction is determined by fitting simulated EPR powder spectra to experimental data, measured at temperatures below 200 K to freeze the protein motion. The experimental spectra are composed of species with different relative nitroxide orientations and interspin distances because of the flexibility of the spin label side chain and the variety of conformational substates of proteins in frozen solution. Values for the average interspin distance and for the distance distribution width can be determined from the characteristics of the dipolar broadened line shape. The resulting interspin distances determined for crystallized insulins in the R6 and T6 structure agree nicely with structural data obtained by x-ray crystallography and by modeling of the spin-labeled samples. The EPR experiments reveal slight differences between crystal and frozen solution structures of the B-chain amino termini in the R6 and T6 states of hexameric insulins. The study of interspin distances between attached spin labels can be applied to obtain structural information on proteins under conditions where other methods like two-dimensional nuclear magnetic resonance spectroscopy or x-ray crystallography are not applicable.     

Effects of limited input distance constraints upon the distance geometry algorithm.

In this paper we examine the distance geometry (DG) algorithm in the form used to determine the structure of proteins. We focus on three aspects of the algorithm: bound smoothing with the triangle inequality, the random selection of distances within the bounds, and the number of distances needed to specify a structure. Computational experiments are performed using simulated and real data for basic pancreatic trypsin inhibitor (BPTI) from nmr and crystallographic measurements. We find that the upper bounds determined by bound smoothing to be a linear function of the true crystal distance. A simple model that describes the results obtained with randomly selected trial distances is proposed. Using this representation of the trial distances, we show that BPTI DG structures are more compact than the true crystal structure. We also show that the DG-generated structures no longer resemble test structures when the number of these interresidue distance constraints is less than the number of degrees of freedom of the protein backbone. While the actual model will be sensitive the way distances are chosen, our conclusions are likely to apply to other versions of the DG algorithm.     

Nanometer distance measurements in RNA using site-directed spin labeling

The method of site-directed spin labeling (SDSL) utilizes a stable nitroxide radical to obtain structural and dynamic information on biomolecules. Measuring dipolar interactions between pairs of nitroxides yields internitroxide distances, from which quantitative structural information can be derived. This study evaluates SDSL distance measurements in RNA using a nitroxide probe, designated as R5, which is attached in an efficient and cost-effective manner to backbone phosphorothioate sites that are chemically substituted in arbitrary sequences. It is shown that R5 does not perturb the global structure of the A-form RNA helix. Six sets of internitroxide distances, ranging from 20 to 50 A, were measured on an RNA duplex with a known X-ray crystal structure. The measured distances strongly correlate (R(2) = 0.97) with those predicted using an efficient algorithm for determining the expected internitroxide distances from the parent RNA structure. The results enable future studies of global RNA structures for which high-resolution structural data are absent.     

Error distribution derived NOE distance restraints

Errors and imprecisions in distance restraints derived from NOESY peak volumes are usually accounted for by generous lower and upper bounds on the distances. In this paper, we propose a new form of distance restraints, replacing the subjective bounds by a potential function obtained from the error distribution of the distances. We derived the shape of the potential from molecular dynamics calculations and by comparison of NMR data with X-ray crystal structures. We used complete cross-validation to derive the optimal weight for the data in the calculation. In a model system with synthetic restraints, the accuracy of the structures improved significantly compared to calculations with the usual form of restraints. For experimental data sets, the structures systematically approach the X-ray crystal structures of the same protein. Also standard quality indicators improve compared to standard calculations. The results did not depend critically on the exact shape of the potential. The new approach is less subjective and uses fewer assumptions in the interpretation of NOESY peak volumes as distance restraints than the usual approach. Figures of merit for the structures, such as the RMS difference from the average structure or the RMS difference from the data, are therefore less biased and more meaningful measures of structure quality than with the usual form of restraints.     

A molecular ruler for measuring quantitative distance distributions

We report a novel molecular ruler for measurement of distances and distance distributions with accurate external calibration. Using solution X-ray scattering we determine the scattering interference between two gold nanocrystal probes attached site-specifically to a macromolecule of interest. Fourier transformation of the interference pattern provides a model-independent probability distribution for the distances between the probe centers-of-mass. To test the approach, we measure end-to-end distances for a variety of DNA structures. We demonstrate that measurements with independently prepared samples and using different X-ray sources are highly reproducible, we demonstrate the quantitative accuracy of the first and second moments of the distance distributions, and we demonstrate that the technique recovers complex distribution shapes. Distances measured with the solution scattering-interference ruler match the corresponding crystallographic values, but differ from distances measured previously with alternate ruler techniques. The X-ray scattering interference ruler should be a powerful tool for relating crystal structures to solution structures and for studying molecular fluctuations.     

Distance restraints from crosslinking mass spectrometry: Mining a molecular dynamics simulation database to evaluate lysine–lysine distances

Integrative structural biology attempts to model the structures of protein complexes that are challenging or intractable by classical structural methods (due to size, dynamics, or heterogeneity) by combining computational structural modeling with data from experimental methods. One such experimental method is chemical crosslinking mass spectrometry (XL‐MS), in which protein complexes are crosslinked and characterized using liquid chromatography‐mass spectrometry to pinpoint specific amino acid residues in close structural proximity. The commonly used lysine‐reactive N‐hydroxysuccinimide ester reagents disuccinimidylsuberate (DSS) and bis(sulfosuccinimidyl)suberate (BS³) have a linker arm that is 11.4 Å long when fully extended, allowing C_α (alpha carbon of protein backbone) atoms of crosslinked lysine residues to be up to ～24 Å apart. However, XL‐MS studies on proteins of known structure frequently report crosslinks that exceed this distance. Typically, a tolerance of ～3 Å is added to the theoretical maximum to account for this observation, with limited justification for the chosen value. We used the Dynameomics database, a repository of high‐quality molecular dynamics simulations of 807 proteins representative of diverse protein folds, to investigate the relationship between lysine–lysine distances in experimental starting structures and in simulation ensembles. We conclude that for DSS/BS³, a distance constraint of 26–30 Å between C_α atoms is appropriate. This analysis provides a theoretical basis for the widespread practice of adding a tolerance to the crosslinker length when comparing XL‐MS results to structures or in modeling. We also discuss the comparison of XL‐MS results to MD simulations and known structures as a means to test and validate experimental XL‐MS methods.     

Structural characterization of manganese(II)-nucleotide complexes bound to yeast 3-phosphoglycerate kinase: 13C relaxation measurements using [U-13C]ATP and [U-13C]ADP

A complete characterization of the conformations of Mn.ADP and Mn.ATP bound to the active site of yeast 3-P-glycerate kinase is presented. These conformations have been deduced on the basis of paramagnetic effects on 13C spin-lattice relaxation rates in [U-13C]nucleotides due to Mn(II), used as a substituent activating cation. The 13C relaxation measurements were performed on exclusively enzyme-bound complexes E.Mn.[U-13C]ATP and E.Mn.[U-13C]ADP at three distinct 13C NMR frequencies: 75.4, 125.7, and 181 MHz. The frequency dependence of the relaxation data has been analyzed in an effort to evaluate distances from the cation for all 10 13C nuclei in the adenosine moieties of E.Mn.ATP and E.Mn.ADP. These distance data, taken along with previously published cation-31P distances, have been used as constraints in the molecular modeling program Quanta, in which molecular dynamics simulations and energy minimization have been performed to determine the conformations that are compatible with the distance data. It was possible to model the distances on the basis of a single enzyme-bound conformation for each of the nucleotides. The details of the enzyme-bound Mn.ATP and Mn.ADP conformations are distinguishably different from each other, indicating that structural alterations occur in the enzyme-bound reaction complex as the enzyme turns over. For example, when the adenosine moieties in the bound structures of Mn.ATP and Mn.ADP are superposed, the cation is found to be displaced by approximately 2.4 A between the two conformations, suggesting that these structural changes may involve movements with significant amplitudes. Furthermore, the NMR-determined structures of enzyme-bound Mn.ATP and Mn.ADP are significantly different from those in published X-ray crystal structures of the enzyme-nucleotide complexes.     

Fitting low-resolution cryo-EM maps of proteins using constrained geometric simulations

Recent experimental advances in producing density maps from cryo-electron microscopy (cryo-EM) have challenged theorists to develop improved techniques to provide structural models that are consistent with the data and that preserve all the local stereochemistry associated with the biomolecule. We develop a new technique that maintains the local geometry and chemistry at each stage of the fitting procedure. A geometric simulation is used to drive the structure from some appropriate starting point (a nearby experimental structure or a modeled structure) toward the experimental density, via a set of small incremental motions. Structural motifs such as α-helices can be held rigid during the fitting procedure as the starting structure is brought into alignment with the experimental density. After validating this procedure on simulated data for adenylate kinase and lactoferrin, we show how cryo-EM data for two different GroEL structures can be fit using a starting x-ray crystal structure. We show that by incorporating the correct local stereochemistry in the modeling, structures can be obtained with effective resolution that is significantly higher than might be expected from the nominal cryo-EM resolution.     

Molecular flexibility demonstrated by paramagnetic enhancements of nuclear relaxation. Application to alamethicin: a voltage-gated peptide channel.

A nitroxide spin label attached to the C-terminus of the channel forming peptide alamethicin produces an enhancement of the nuclear spin-lattice relaxation rates of peptide protons as a result of both intermolecular and intramolecular magnetic dipole-dipole interactions. The intermolecular contribution provides evidence that alamethicin monomers collide preferentially in a C-terminal-to-N-terminal configuration in methanol. From the intramolecular paramagnetic enhancement of nuclear spin-lattice relaxation times, effective distances between the unpaired electron on the nitroxide at the C-terminus of alamethicin and protons along the peptide backbone were calculated. These distances are much shorter than distances based on the reported crystal structure of alamethicin, and cannot be accounted for by motion in the bonds that attach the nitroxide to the peptide. In addition, the differences between distances deduced from the nuclear spin relaxation and the distances seen in the crystal structure increase toward the N-terminal end of the peptide. The simplest explanation for these data is that the alamethicin backbone suffers large structural fluctuations that yield shorter effective distances between the C-terminus and positions along the backbone. This finding can be interpreted in terms of a molecular mechanism for the voltage-gating of the alamethicin channel. When the distances between a paramagnetic center and a nucleus fluctuate, paramagnetic enhancements are expected to yield distances that are weighted by r-6, and distances calculated using the Solomon-Bloembergen equations may more nearly represent a distance of closest approach than a time average distance.(ABSTRACT TRUNCATED AT 250 WORDS)     

On the status of ferryl protonation

We examine the issue of ferryl protonation in heme proteins. An analysis of the results obtained from X-ray crystallography, resonance Raman spectroscopy, and extended X-ray absorption spectroscopy (EXAFS) is presented. Fe-O bond distances obtained from all three techniques are compared using Badger's rule. The long Fe-O bond lengths found in the ferryl crystal structures of myoglobin, cytochrome c peroxidase, horseradish peroxidase, and catalase deviate substantially from the values predict by Badger's rule, while the oxo-like distances obtained from EXAFS measurements are in good agreement with the empirical formula. Density functional calculations, which suggest that M?ssbauer spectroscopy can be used to determine ferryl protonation states, are presented. Our calculations indicate that the quadrupole splitting (DeltaE(Q)) changes significantly upon ferryl protonation. New resonance Raman data for horse-heart myoglobin compound II (Mb-II, pH 4.5) are also presented. An Fe-O stretching frequency of 790cm(-1) (shifting to 754cm(-1) with (18)O substitution) was obtained. This frequency provides a Badger distance of r(Fe-O)=1.66A. This distance is in agreement with the 1.69A Fe-O bond distance obtained from EXAFS measurements but is significantly shorter than the 1.93A bond found in the crystal structure of Mb-II (pH 5.2). In light of the available evidence, we conclude that the ferryl forms of myoglobin (pKa4), horseradish peroxidase (pKa4), cytochrome c peroxidase (pKa4), and catalase (pKa7) are not basic. They are authentic Fe(IV)oxos with Fe-O bonds on the order of 1.65A.     

A distance geometry program for determining the structures of small proteins and other macromolecules from nuclear magnetic resonance measurements of intramolecular1H−1H proximities in solution

DISGEO is a new implementation of a distance geometry algorithm which has been specialized for the calculation of macromolecular conformation from distance measurements obtained by two-dimensional nuclear Overhauser enhancement spectroscopy. The improvements include (1) a decomposition of the complete embedding process into two successive, more tractable calculations by the use of “substructures”, (2) a compact data structure for storing incomplete distance information on a structure, (3) a more efficient shortest-path algorithm for computing the triangle inequality limits on all distances from this information, (4) a new algorithm for selecting random metric spaces from within these limits, (5) the use of chirality constraints to obtain good covalent geometry without the use ofad hoc weights or excessive optimization. The utility of the resultant program with nuclear magnetic resonance data is demonstrated by embedding complete spatial structures for the protein basic pancreatic trypsin inhibitor vs all 508 intramolecular, interresidue proton-proton contacts shorter than 4.0 Å that were present in its crystal structure. The crystal structure could be reproduced from this data set to within 1.3 Å minimum root mean square coordinate difference between the backbone atoms. We conclude that the information potentially available from nuclear magnetic resonance experiments in solution is sufficient to define the spatial structure of small proteins.     

Structural dynamics of calmodulin and troponin C

We present the results of computational simulation studies of the structures of calmodulin (CAM) and troponin C (TNC). Possible differences between the structures of these molecules in the crystal and in solution were suggested by results from some recent experimental studies, which implied that their conformations in solution may be more compacted than the characteristic dumbbell shape observed in the crystal. The molecular dynamics simulations were carried out with the CHARMM system of programs, and the environment was modeled with a distance-dependent dielectric permittivity and discrete water molecules surrounding the proteins at starting positions identified in the crystals of CAM and TNC. Methods of macromolecular structure analysis, including linear distance plots, distance matrices and a matrix representation of hydrogen bonding, were used to analyze the nature, the extent and the source of structural differences between the computed structures of the molecules and their conformations in the crystal. Following the longest simulation, in which intradomain structure was conserved, the crystallographically observed dumbbell structure of the molecule changed due to a kinking or bending in the region of the central tether helix connecting the two Ca(2+)-binding domains which moved into close proximity. The resulting structure correlates with experimental observations of complexes between CAM and peptides such as melittin and mastoparan. Analysis of the corresponding pair distance distribution functions in comparison to experimental results suggests the dynamic existence of a non-negligible fraction of the compacted structure in aqueous solutions of CAM. In this more nearly globular shape, CAM reveals to the environment two interior pockets that contain a number of hydrophobic residues, in agreement with NMR data suggesting involvement of such residues in the binding of inhibitors and proteins to CAM.     

Comprehensive modeling and functional analysis of Toll‐like receptor ligand‐recognition domains

Toll‐like receptors (TLRs) are innate immune pattern‐recognition receptors endowed with the capacity to detect microbial pathogens based on pathogen‐associated molecular patterns. The understanding of the molecular principles of ligand recognition by TLRs has been greatly accelerated by recent structural information, in particular the crystal structures of leucine‐rich repeat‐containing ectodomains of TLR2, 3, and 4 in complex with their cognate ligands. Unfortunately, for other family members such as TLR7, 8, and 9, no experimental structural information is currently available. Methods such as X‐ray crystallography or nuclear magnetic resonance are not applicable to all proteins. Homology modeling in combination with molecular dynamics may provide a straightforward yet powerful alternative to obtain structural information in the absence of experimental (structural) data, provided that the generated three‐dimensional models adequately approximate what is found in nature. Here, we report the development of modeling procedures tailored to the structural analysis of the extracellular domains of TLRs. We comprehensively compared secondary structure, torsion angles, accessibility for glycosylation, surface charge, and solvent accessibility between published crystal structures and independently built TLR2, 3, and 4 homology models. Finding that models and crystal structures were in good agreement, we extended our modeling approach to the remaining members of the TLR family from human and mouse, including TLR7, 8, and 9.     

Structural templates for modeling homodimers

Oligomeric proteins are more abundant in nature than monomeric proteins, and involved in all biological processes. In the absence of an experimental structure, their subunits can be modeled from their sequence like monomeric proteins, but reliable procedures to build the oligomeric assembly are scarce. Template‐based methods, which start from known protein structures, are commonly applied to model subunits. We present a method to model homodimers that relies on a structural alignment of the subunits, and test it on a set of 511 target structures recently released by the Protein Data Bank, taking as templates the earlier released structures of 3108 homodimeric proteins (H‐set), and 2691 monomeric proteins that form dimer‐like assemblies in crystals (M‐set). The structural alignment identifies a H‐set template for 97% of the targets, and in half of the cases, it yields a correct model of the dimer geometry and residue–residue contacts in the target. It also identifies a M‐set template for most of the targets, and some of the crystal dimers are very similar to the target homodimers. The procedure efficiently detects homology at low levels of sequence identities, and points to erroneous quaternary structures in the Protein Data Bank. The high coverage of the target set suggests that the content of the Protein Data Bank already approaches the structural diversity of protein assemblies in nature, and that template‐based methods should become the choice method for modeling oligomeric as well as monomeric proteins.     

Solution- and Adsorbed-State Structural Ensembles Predicted for the Statherin-Hydroxyapatite System

We have developed a multiscale structure prediction technique to study solution- and adsorbed-state ensembles of biomineralization proteins. The algorithm employs a Metropolis Monte Carlo-plus-minimization strategy that varies all torsional and rigid-body protein degrees of freedom. We applied the technique to fold statherin, starting from a fully extended peptide chain in solution, in the presence of hydroxyapatite (HAp) (001), (010), and (100) monoclinic crystals. Blind (unbiased) predictions capture experimentally observed macroscopic and high-resolution structural features and show minimal statherin structural change upon adsorption. The dominant structural difference between solution and adsorbed states is an experimentally observed folding event in statherin's helical binding domain. Whereas predicted statherin conformers vary slightly at three different HAp crystal faces, geometric and chemical similarities of the surfaces allow structurally promiscuous binding. Finally, we compare blind predictions with those obtained from simulation biased to satisfy all previously published solid-state NMR (ssNMR) distance and angle measurements (acquired from HAp-adsorbed statherin). Atomic clashes in these structures suggest a plausible, alternative interpretation of some ssNMR measurements as intermolecular rather than intramolecular. This work demonstrates that a combination of ssNMR and structure prediction could effectively determine high-resolution protein structures at biomineral interfaces.     

Identification and analysis of long-range electrostatic effects in proteins by computer modeling:Aspartate transcarbamylase

While ion pairs are readily identified in crystal structures, longer range electrostatic interactions cannot be identified from the three-dimensional structure alone. These interactions are likely to be important in large, multisubunit proteins that are regulated by allosteric interactions. In this paper, we show that these interactions are readily detected by electrostatic modeling, using, as an example, unliganded Escherichia coli aspartate transcarbamylase, a widely studied allosteric enzyme with 12 subunits and a molecular weight of 310 kD. The Born, dipolar, and site-site interaction terms of the free energy of protonation of the 810 titratable sites in the holoenzyme were calculated using the multigrid solution of the nonlinear Poisson-Boltzmann equation. Calculated titration curves are in good agreement with experimental titration curves, and the structural asymmetry observed in the crystal structure is readily apparent in the calculated free energies and pK_1/2 values. Most of the residues with pK_1/2 values that differ substantially from those of model compounds are buried in the low dielectric medium of the protein, particularly at the intersubunit interfaces. The dependence of the site-site interaction free energies on distance is complex, with a steep dependence at distances less than 5 Å and a more shallow dependence at longer distances. Interactions over distances of 6 to 15 Å require a bridging residue and are often not apparent in the structure. The network of interactions between ionizable groups extends across and between subunits and provides a potential mechanism for transmitting long-range structural effects and allosteric signals. © 1996 Wiley-Liss, Inc.     

Crystallography of polysaccharides: Current state and challenges

Polysaccharides are the most abundant class of biopolymers, holding an important place in biological systems and sustainable material development. Their spatial organization and intra- and intermolecular interactions are thus of great interest. However, conventional single crystal crystallography is not applicable since polysaccharides crystallize only into tiny crystals. Several crystallographic methods have been developed to extract atomic-resolution structural information from polysaccharide crystals. Small-probe single crystal diffractometry, high-resolution fiber diffraction and powder diffraction combined with molecular modeling brought new insights from various types of polysaccharide crystals, and led to many high-resolution crystal structures over the past two decades. Current challenges lie in the analysis of disorder and defects by further integrating molecular modeling methods for low-resolution diffraction data.     

Application of molecular dynamics with interproton distance restraints to three-dimensional protein structure determination. A model study of crambin

The applicability of restrained molecular dynamics for the determination of three-dimensional protein structures on the basis of short interproton distances (less than 4 A) that can be realistically determined from nuclear magnetic resonance measurements in solution is assessed. The model system used is the 1.2 A resolution crystal structure of the 46 residue protein crambin, from which a set of 240 approximate distance restraints, divided into three ranges (2.5 +/- 0.5, 3.0+0.5(-1.0) and 4 +/- 1 A), is derived. This interproton distance set comprises 159 short-range ([i-j] less than or equal to 5) and 56 ([i-j] greater than 5) long-range inter-residue distances and 25 intra-residue distances. Restrained molecular dynamics are carried out using a number of different protocols starting from two initial structures: a completely extended beta-strand; and an extended structure with two alpha-helices in the same positions as in the crystal structure (residues 7 to 19, and 23 to 30) and all other residues in the form of extended beta-strands. The root-mean-square (r.m.s.) atomic differences between these two initial structures and the crystal structure are 43 A and 23 A, respectively. It is shown that, provided protocols are used that permit the secondary structure elements to form at least partially prior to folding into a tertiary structure, convergence to the correct final structure, both globally and locally, is achieved. The r.m.s. atomic differences between the converged restrained dynamics structures and the crystal structure range from 1.5 to 2.2 A for the backbone atoms and from 2.0 to 2.8 A for all atoms. The r.m.s. atomic difference between the X-ray structure and the structure obtained by first averaging the co-ordinates of the converged restrained dynamics structures is even smaller: 1.0 A for the backbone atoms and 1.6 A for all atoms. These results provide a measure with which to judge future experimental results on proteins whose crystal structures are unknown. In addition, from an examination of the dynamics trajectories, it is shown that the convergence pathways followed by the various simulations are different.     

Statistical measures on residue-level protein structural properties

The atomic-level structural properties of proteins, such as bond lengths, bond angles, and torsion angles, have been well studied and understood based on either chemistry knowledge or statistical analysis. Similar properties on the residue-level, such as the distances between two residues and the angles formed by short sequences of residues, can be equally important for structural analysis and modeling, but these have not been examined and documented on a similar scale. While these properties are difficult to measure experimentally, they can be statistically estimated in meaningful ways based on their distributions in known proteins structures. Residue-level structural properties including various types of residue distances and angles are estimated statistically. A software package is built to provide direct access to the statistical data for the properties including some important correlations not previously investigated. The distributions of residue distances and angles may vary with varying sequences, but in most cases, are concentrated in some high probability ranges, corresponding to their frequent occurrences in either α-helices or β-sheets. Strong correlations among neighboring residue angles, similar to those between neighboring torsion angles at the atomic-level, are revealed based on their statistical measures. Residue-level statistical potentials can be defined using the statistical distributions and correlations of the residue distances and angles. Ramachandran-like plots for strongly correlated residue angles are plotted and analyzed. Their applications to structural evaluation and refinement are demonstrated. With the increase in both number and quality of known protein structures, many structural properties can be derived from sets of protein structures by statistical analysis and data mining, and these can even be used as a supplement to the experimental data for structure determinations. Indeed, the statistical measures on various types of residue distances and angles provide more systematic and quantitative assessments on these properties, which can otherwise be estimated only individually and qualitatively. Their distributions and correlations in known protein structures show their importance for providing insights into how proteins may fold naturally to various residue-level structures.