Normal mode refinement: crystallographic refinement of protein dynamic structure. I. Theory and test by simulated diffraction data.

A dynamic structure refinement method for X-ray crystallography, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the low-frequency normal modes whose amplitudes and eigenvectors are experimentally optimized in the process of the crystallographic refinement. In this model, the atomic fluctuations are treated as anisotropic and concerted. The normal modes of the external motion (TLS model) are also introduced to cover the factors other than the internal fluctuations, such as the lattice disorder and diffusion. A program for the normal mode refinement (NM-REF) has been developed. The method has first been tested against simulated diffraction data for human lysozyme calculated by a Monte Carlo simulation. Applications of the method have demonstrated that the normal mode refinement has: (1) improved the fitting to the diffraction data, even with fewer adjustable parameters; (2) distinguished internal fluctuations from external ones; (3) determined anisotropic thermal factors; and (4) identified concerted fluctuations in the protein molecule.     

Normal mode refinement: crystallographic refinement of protein dynamic structure applied to human lysozyme.

A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.     

Response of dynamic structure to removal of a disulfide bond: normal mode refinement of C77A/C95A mutant of human lysozyme.

In order to investigate the response of dynamic structure to removal of a disulfide bond, the dynamic structure of human lysozyme has been compared to its C77A/C95A mutant. The dynamic structures of the wild type and mutant are determined by normal mode refinement of 1.5-A-resolution X-ray data. The C77A/C95A mutant shows an increase in apparent fluctuations at most residues. However, most of the change originates from an increase in the external fluctuations, reflecting the effect of the mutation on the quality of crystals. The effects of disulfide bond removal on the internal fluctuations are almost exclusively limited to the mutation site at residue 77. No significant change in the correlation of the internal fluctuations is found in either the overall or local dynamics. This indicates that the disulfide bond does not have any substantial role to play in the dynamic structure. A comparison of the wild-type and mutant coordinates suggests that the disulfide bond does not prevent the 2 domains from parting from each other. Instead, the structural changes are characteristic of a cavity-creating mutation, where atoms surrounding the mutation site move cooperatively toward the space created by the smaller alanine side chain. Although this produces tighter packing, more than half of the cavity volume remains unoccupied, thus destabilizing the native state.     

Full-matrix least-squares refinement of lysozymes and analysis of anisotropic thermal motion

Crystal structures of turkey egg lysozyme (TEL) and human lysozyme (HL) were refined by full-matrix least-squares method using anisotropic temperature factors. The refinement converged at the conventional R-values of 0.104 (TEL) and 0.115 (HL) for reflections with F_o > 0 to the resolution of 1.12 Å and 1.15 Å, respectively. The estimated r.m.s. coordinate errors for protein atoms were 0.031 Å (TEL) and 0.034 Å (HL). The introduction of anisotropic temperature factors markedly reduced the R-value but did not significantly affect the main chain coordinates. The degree of anisotropy of atomic thermal motion has strong positive correlation with the square of distance from the molecular centroid. The ratio of the radial component of thermal ellipsoid to the r.m.s. magnitude of three principal components has negative correlation with the distance from the molecular centroid, suggesting the domination of libration rather than breathing motion. The TLS model was applied to elucidate the characteristics of the rigid-body motion. The TLS tensors were determined by the least-squares fit to observed temperature factors. The profile of the magnitude of reproduced temperature factors by the TLS method well fitted to that of observed B_eqv. However, considerable disagreement was observed in the shape and orientation of thermal ellipsoid for atoms with large temperature factors, indicating the large contribution of local motion. The upper estimate of the external motion, 67% (TEL) and 61% (HL) of B_eqv, was deduced from the plot of the magnitude of TLS tensors determined for main chain atoms which were grouped into shells according to the distance from the center of libration. In the external motion, the translational portion is predominant and the contribution of libration and screw motion is relatively small. The internal motion, estimated by subtracting the upper estimate of the external motion from the observed temperature factor, is very similar between TEL and HL in spite of the difference in 54 of 130 amino acid residues and in crystal packing, being suggested to reflect the intrinsic internal motion of chicken-type lysozymes. Proteins 30:232–243, 1998. © 1998 Wiley-Liss, Inc.     

Effect of anisotropy and anharmonicity on protein crystallographic refinement. An evaluation by molecular dynamics

Molecular dynamics simulations are employed to determine the errors introduced by anharmonicity and anisotropy in the structure and temperature factors obtained for proteins by refinement of X-ray diffraction data. Simulations (25 ps and 300 ps) of metmyoglobin are used to generate time-averaged diffraction data at 1.5 A resolution. The crystallographic restrained-parameter least-squares refinement program PROLSQ is used to refine models against these simulated data. The resulting atomic positions and isotropic temperature factors are compared with the average structure and fluctuations calculated directly from the simulations. It is found that significant errors in the atomic positions and fluctuations are introduced by the refinement, and that the errors increase with the magnitude of the atomic fluctuations. Of particular interest is the fact that the refinement generally underestimates the atomic motions. Moreover, while the actual fluctuations go up to a mean-square value of about 5 A2, the X-ray results never go above approximately 2 A2. This systematic deviation in the motional parameters appears to be due to the use of a single-site isotropic model for the atomic fluctuations. Many atoms have multiple peaks in their probability distribution functions. For some atoms, the multiple peaks are seen in difference electron density maps and it is possible to include these in the refinement as disordered residues. However, for most atoms the refinement fits only one peak and neglects the rest, leading to the observed errors in position and temperature factor. The use of strict stereochemical restraints is inconsistent with the average dynamical structure; nevertheless, refinement with tight restraints results in structures that are comparable to those obtained with loose restraints and better than those obtained with no restraints. The results support the use of tight stereochemical restraints, but indicate that restraints on the variation of temperature factors are too restrictive.     

Crystallographic refinement by knowledge-based exploration of complex energy landscapes

Although X-ray crystallography remains the most versatile method to determine the three-dimensional atomic structure of proteins and much progress has been made in model building and refinement techniques, it remains a challenge to elucidate accurately the structure of proteins in medium-resolution crystals. This is largely due to the difficulty of exploring an immense conformational space to identify the set of conformers that collectively best fits the experimental diffraction pattern. We show here that combining knowledge-based conformational sampling in RAPPER with molecular dynamics/simulated annealing (MD/SA) vastly improves the quality and power of refinement compared to MD/SA alone. The utility of this approach is highlighted by the automated determination of a lysozyme mutant from a molecular replacement solution that is in congruence with a model prepared independently by crystallographers. Finally, we discuss the implications of this work on structure determination in particular and conformational sampling and energy minimization in general.     

Dynamic information from protein crystallography. An analysis of temperature factors from refinement of the hen egg-white lysozyme structure.

A preliminary analysis is presented of whether the isotropic temperature factors derived from refinement of the tetragonal crystal structure of hen egg-white lysozyme at 2 Å resolution can be interpreted in terms of molecular motion. If the contributions to the temperature factors from experimental errors, crystal disorder and imperfections of the molecular model are neglected, the apparent thermal motion is found to be compatible with the pair of molecules that have the strongest interactions in the crystal either moving as a rigid-body with libration about a common axis, or vibrating in an intramolecular mode with the principal amplitudes radial to this axis. There is only weak evidence for hingebending, i.e. that the two lobes of lysozyme vibrate so that their separation varies. Side-chains that are exposed to the solvent and some segments of the main polypeptide chain (including residues 70–73, 82–84, 101–103) have greater apparent thermal motion than the remainder of the molecule.Although this analysis at a single temperature cannot establish whether thermal motion or static disorder (including conformational variability) underlies the observed effects, it suggests that the accurate determination of temperature factors will be useful in detailed studies of the dynamic properties of macromolecules.     

Deriving correlated motions in proteins from X-ray structure refinement by using TLS parameters

Dynamic information in proteins may provide valuable information for understanding allosteric regulation of protein complexes or long-range effects of the mutations on enzyme activity. Experimental data such as X-ray B-factors or NMR order parameters provide a convenient estimate of atomic fluctuations (or atomic auto-correlated motions) in proteins. However, it is not as straightforward to obtain atomic cross-correlated motions in proteins — one usually resorts to more sophisticated computational methods such as Molecular Dynamics, normal mode analysis or atomic network models. In this report, we show that atomic cross-correlations can be reliably obtained directly from protein structure using X-ray refinement data. We have derived an analytic form of atomic correlated motions in terms of the original TLS parameters used to refine the B-factors of X-ray structures. The correlated maps computed using this equation are well correlated with those of the method based on a mechanical model (the correlation coefficient is 0.75) for a non-homologous dataset comprising 100 structures. We have developed an approach to compute atomic cross-correlations directly from X-ray protein structure. Being in analytic form, it is fast and provides a feasible way to compute correlated motions in proteins in a high throughput way. In addition, avoiding sophisticated computational operations; it provides a quick, reliable way, especially for non-computational biologists, to obtain dynamics information directly from protein structure relevant to its function.     

Dynamical transition of myoglobin in a crystal: comparative studies of X-ray crystallography and Mössbauer spectroscopy

The crystallographic normal mode refinements of myoglobin at a wide range of temperature from 40 K to 300 K were carried out to study the temperature dependence of the internal atomic fluctuations. The refinement method decomposes the mean square displacement from the average position, (deltar2), into the contributions from the internal degrees of freedom and those from the external degrees of freedom. The internal displacements show linear temperature dependence as (deltar2)=alphaT+beta, throughout the temperature range measured here, and exhibit no obvious change in the slope alpha at the dynamical transition temperature (Tc=ca. 180 K). The slope alpha is practically the same as the value predicted theoretically by normal mode analysis. Such linear dependence is considered to be due to the following reason. The crystallographic Debye-Waller factor represents the static distribution caused by convolution of temperature-dependent normal mode motions and a temperature-independent set of the conformational substates. In contrast, M?ssbauer absorption spectroscopy shows a clear increase in the gradient alpha at Tc. This difference from X-ray diffraction originates from the incoherent nature of the M?ssbauer effect together with its high-energy resolution, which yields the self-correlation, and the temporal behavior of individual Fe atoms in the myoglobin crystal.     

Refined structure of baboon alpha-lactalbumin at 1.7 A resolution. Comparison with C-type lysozyme

The solution of the structure of alpha-lactalbumin from baboon milk (Papio cynocephalus) at 4.5 A resolution using the isomorphous replacement method has been reported previously. Initial refinement on the basis of these low-resolution studies was not successful because of the poor isomorphism of the best heavy-atom derivative. Because of the striking similarity between the structure of lysozyme and alpha-lactalbumin, a more cautious molecular replacement approach was tried to refine the model. Using hen egg-white lysozyme as the starting model, preliminary refinement was performed using heavily constrained least-squares minimization in reciprocal space. The model was further refined using stereochemical restraints at 1.7 A resolution to a conventional crystallographic residual of 0.22 for 1141 protein atoms. In the final model, the root-mean-square deviation from ideality for bond distances is 0.015 A, and for angle distances it is 0.027 A. The refinement was carried out using the human alpha-lactalbumin sequence and "omit maps" calculated during the course of refinement indicated eight possible sequence changes in the baboon alpha-lactalbumin X-ray sequence. During the refinement, a tightly bound calcium ion and 150 water molecules, of which four are internal, have been located. Some of the water molecules were modelled for disordered side-chains. The co-ordination around the calcium is a slightly distorted pentagonal bipyramid. The Ca-O distances vary from 2.2 A to 2.6 A, representing a tight calcium-binding loop in the structure. The calcium-binding fold only superficially resembles the "EF-hand" and presumably has no evolutionary relationship with other EF-hand structures. The overall structure of alpha-lactalbumin is very similar to that of lysozyme. All large deviations occur in the loops where all sequence deletions and insertions are found. The C terminus appears to be rather flexible in alpha-lactalbumin compared to lysozyme. The experimental evidence supports the earlier predictions for the alpha-lactalbumin structure that were based upon the assumption that alpha-lactalbumin and lysozyme have similar three-dimensional structures, with minimal deletions and insertions. A detailed comparison of the two structures shows striking features as well as throwing some light on the evolution of these two proteins from a common precursor.     

Normal-mode refinement of anisotropic thermal parameters for potassium channel KcsA at 3.2 A crystallographic resolution

We report a normal-mode method for anisotropic refinement of membrane-protein structures, based on a hypothesis that the global near-native-state disordering of membrane proteins in crystals follows low-frequency normal modes. Thus, a small set of modes is sufficient to represent the anisotropic thermal motions in X-ray crystallographic refinement. By applying the method to potassium channel KcsA at 3.2 A, we obtained a structural model with an improved fit with the diffraction data. Moreover, the improved electron density maps allowed for large structural adjustments for 12 residues in each subunit, including the rebuilding of 3 missing side chains. Overall, the anisotropic KcsA structure at 3.2 A was systematically closer to a 2.0 A KcsA structure, especially in the selectivity filter. Furthermore, the anisotropic thermal ellipsoids from the refinement revealed functionally relevant structural flexibility. We expect this method to be a valuable tool for structural refinement of many membrane proteins with moderate-resolution diffraction data.     

Temperature-dependent molecular dynamics and restrained X-ray refinement simulations of a Z-DNA hexamer

Molecular dynamics simulations of the Z-DNA hexamer 5BrdC-dG-5BrdC-dG-5BrdC-dG were performed at several temperatures between 100 K and 300 K. Above 250 K, a strong sequence-dependent flexibility in the nucleic acid is observed, with the guanine sugar and the phosphate of GpC sequences much more mobile than the cytosine sugar and phosphate of CpG sequences. At 300 K, the hexamer is in dynamic equilibrium between several Z forms, including the crystallographically determined ZI and ZII forms. The local base-pair geometry, however, is not very variable, except for the roll of the base-pairs. The hexamer molecular dynamics trajectories have been used to test the restrained parameter crystallographic refinement model for nucleic acids. X-ray diffraction intensities corresponding to observed diffraction data were computed. The average structures obtained from the simulations were then refined against the calculated intensities, using a restrained least-squares program developed for nucleic acids in order to analyse the effects of the refinement model on the derived quantities. In general, the temperature dependence of the atomic fluctuations determined directly from the refined Debye-Waller factors is in reasonably good agreement with the results obtained by calculating the atomic fluctuations directly from the Z-DNA molecular dynamics trajectories. The agreement is best for refinement of temperature factors without restraints. At the highest temperature studied (300 K), the effect of the refinement on the most mobile atoms (phosphates) is to significantly reduce the mean-square atomic fluctuations estimated from the refined Debye-Waller factors below the actual values (less than (delta r)2 greater than congruent to 0.5 A2). Analysis of the temperature-dependence of the mean-square atomic fluctuations provides information concerning the conformational potential within which the atoms move. The calculated temperature-dependence and anharmonicity of the Z-DNA helix are compared with the results observed for proteins. The average structures from the simulations were refined against the experimental X-ray intensities. It is found that low-temperature molecular dynamics simulations provide a useful tool for optimizing the refinement of X-ray structures.     

A molecular dynamics analysis of protein structural elements

The relation between protein secondary structure and internal motions was examined by using molecular dynamics to calculate positional fluctuations of individual helix, beta-sheet, and loop structural elements in free and substrate-bound hen egg-white lysozyme. The time development of the fluctuations revealed a general correspondence between structure and dynamics; the fluctuations of the helices and beta-sheets converged within the 101 psec period of the simulation and were lower than average in magnitude, while the fluctuations of the loop regions were not converged and were mostly larger than average in magnitude. Notable exceptions to this pattern occurred in the substrate-bound simulation. A loop region (residues 101-107) of the active site cleft had significantly reduced motion due to interactions with the substrate. Moreover, part of a loop and a 3(10) helix (residues of 67-88) not in contact with the substrate showed a marked increase in fluctuations. That these differences in dynamics of free and substrate-bound lysozyme did not result simply from sampling errors was established by an analysis of the variations in the fluctuations of the two halves of the 101 psec simulation of free lysozyme. Concerted transitions of four to five mainchain phi and psi angles between dihedral wells were shown to be responsible for large coordinate shifts in the loops. These transitions displaced six or fewer residues and took place either abruptly, in 1 psec or less, or with a diffusive character over 5-10 psec. Displacements of rigid secondary structures involved longer timescale motions in bound lysozyme; a 0.5 A rms change in the position of a helix occurred over the 55 psec simulation period. This helix reorientation within the protein appears to be a response to substrate binding. There was little correlation between the solvent accessible surface area and the dynamics of the different structural elements.     

Influence of Structural Symmetry on Protein Dynamics

Structural symmetry in homooligomeric proteins has intrigued many researchers over the past several decades. However, the implication of protein symmetry is still not well understood. In this study, we performed molecular dynamics (MD) simulations of two forms of trp RNA binding attenuation protein (TRAP), the wild-type 11-mer and an engineered 12-mer, having two different levels of circular symmetry. The results of the simulations showed that the inter-subunit fluctuations in the 11-mer TRAP were significantly smaller than the fluctuations in the 12-mer TRAP while the internal fluctuations were larger in the 11-mer than in the 12-mer. These differences in thermal fluctuations were interpreted by normal mode analysis and group theory. For the 12-mer TRAP, the wave nodes of the normal modes existed at the flexible interface between the subunits, while the 11-mer TRAP had its nodes within the subunits. The principal components derived from the MD simulations showed similar mode structures. These results demonstrated that the structural symmetry was an important determinant of protein dynamics in circularly symmetric homooligomeric proteins.     

Structure and internal mobility of proteins: a molecular dynamics study of hen egg white lysozyme

The structure and internal motions of the protein hen egg white lysozyme are studied by analysis of simulation and experimental data. A molecular dynamics simulation and an energy minimization of the protein in vacuum have been made and the results compared with high-resolution structures and temperature factors of hen egg white lysozyme in two different crystal forms and of the homologous protein human lysozyme. The structures obtained from molecular dynamics and energy minimization have root-mean-square deviations for backbone atoms of 2.3 Å and 1.1–1.3 Å, respectively, relative to the crystal structures; the different crystal structures have root-mean-square deviations of 0.73–0.81 Å for the backbone atoms. In comparing the backbone dihedral angles, the difference between the dynamics and the crystal structure on which it is based is the same as that between any two crystal structures. The internal fluctuations of atomic positions calculated from the molecular dynamics trajectory agree well with the temperature factors from the three structures. Simulation and crystal results both show that there are large motions for residues involved in exposed turns of the backbone chain, relatively smaller motions for residues involved in the middle of helices or β-sheet structures, and relatively small motions of residues near disulfide bridges. Also, both the simulation and crystal data show that side-chain atoms have larger fluctuations than main-chain atoms. Moreover, the regions that have large deviations among the x-ray crystal structures, which indicates flexibility, are found to have large fluctuations in the simulation.     

Structure of the complex formed by bovine trypsin and bovine pancreatic trypsin inhibitor. II. Crystallographic refinement at 1.9 A resolution

The crystal structure of the complex formed by bovine trypsin and bovine pancreatic trypsin inhibitor has been refined with data to 1.9 Å resolution, using a procedure described by Deisenhofer &; Steigemann (1974) in their refinement of the crystal structure of the free inhibitor. This procedure involves cycles consisting of phase calculation using the current atomic model, Fourier synthesis using these phases and the observed structure factor amplitudes and Diamond's real-space refinement (Diamond, 1971,1974). At various stages, difference Fourier syntheses are calculated to detect and correct gross errors in the model and to localize solvent molecules.The refinement progressed smoothly, starting with the model obtained from the isomorphous Fourier map at 2.6 Å resolution. The R-factor is 0.23 for 20,500 significantly measured reflections to 1.9 Å resolution, using an over-all temperature factor of 20 Ų. The estimated standard deviation of atomic positions is 0.09 Å.An objective assessment of the upper limit of the error in the atomic coordinates of the final model is possible by comparing the inhibitor component in the model of the complex with the refined structure of the free inhibitor (Deisenhofer &; Steigemann, 1974). The mean deviation of main-chain atoms of the two molecular models in internal segments is 0.25 Å, of main-chain dihedral angles 5.1 ° and side-chain dihedral angles 6.5 °.A comparison of the trypsin component with α-chymotrypsin (Birktoft &; Blow, 1972) showed a mean deviation of main-chain atoms of 0.75 Å. The structures are closely similar and the various deletions and insertions cause local structural differences only.     

Structure refinement of membrane proteins via molecular dynamics simulations

A refinement protocol based on physics‐based techniques established for water soluble proteins is tested for membrane protein structures. Initial structures were generated by homology modeling and sampled via molecular dynamics simulations in explicit lipid bilayer and aqueous solvent systems. Snapshots from the simulations were selected based on scoring with either knowledge‐based or implicit membrane‐based scoring functions and averaged to obtain refined models. The protocol resulted in consistent and significant refinement of the membrane protein structures similar to the performance of refinement methods for soluble proteins. Refinement success was similar between sampling in the presence of lipid bilayers and aqueous solvent but the presence of lipid bilayers may benefit the improvement of lipid‐facing residues. Scoring with knowledge‐based functions (DFIRE and RWplus) was found to be as good as scoring using implicit membrane‐based scoring functions suggesting that differences in internal packing is more important than orientations relative to the membrane during the refinement of membrane protein homology models.     

Dynamic light-scattering studies of DNA. I. The coupling of internal modes with anisotropic translational diffusion in congested solutions

A model for the coupling between internal modes, or molecular rotation, and anisotropic translational diffusion in congested solutions is proposed to account for the anomalously slow component that has appeared ubiquitously in reported autocorrelation functions of Rayleigh scattered light from solutions of DNA's with molecular weights greater than about 10⁷. The predicted existence of an anomalously slow mode in addition to a faster “normal” mode, as well as the predicted relative amplitudes of both fast and slow components, are qualitatively in agreement with the observations. For sufficiently long-wavelength fluctuations all of the amplitude appears in the slower mode, which then exhibits an appropriately averaged translational diffusion coefficient. In support of the model it is shown in the Appendix that nonideal central interactions between macromolecules are by themselves insufficient to generate isolated internal mode relaxation terms in the autocorrelation function, unless translational ordering of the macromolecules extends over the illuminated observation region.     

Normal modes as refinement parameters for the F-actin model.

The slow normal modes of G-actin were used as structural parameters to refine the F-actin model against 8-A resolution x-ray fiber diffraction data. The slowest frequency normal modes of G-actin pertain to collective rearrangements of domains, motions that are characterized by correlation lengths on the order of the resolution of the fiber diffraction data. Using a small number of normal mode degrees of freedom (< or = 12) improved the fit to the data significantly. The refined model of F-actin shows that the nucleotide binding cleft has narrowed and that the DNase I binding loop has twisted to a lower radius, consistent with other refinement techniques and electron microscopy data. The methodology of a normal mode refinement is described, and the results, as applied to actin, are detailed.