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

The dynamic structure of a protein, human lysozyme, is determined by the normal mode refinement of X-ray crystal structure. This method uses the normal modes of both internal and external motions to distinguish the real internal dynamics from the external terms such as lattice disorder, and gives an anisotropic and concerted picture of atomic fluctuations. The refinement is carried out with diffraction data of 5.0 to 1.8 A resolution, which are collected on an imaging plate. The results of the refinement show: (1) Debye-Waller factor consists of two parts, highly anisotropic internal fluctuations and almost isotropic external terms. The former is smaller than the latter by a factor of 0.72 in the scale of B-factor. Therefore, the internal dynamics cannot be recognized directly from the apparent electron density distribution. (2) The internal fluctuations show basically similar features as those predicted by the normal mode analysis, with almost the same amplitude and a similar level of anisotropy. (3) Correlations of fluctuations are detected between two lobes forming the active site cleft, which move simultaneously in opposite directions. This corresponds to the hinge-bending motion of lysozyme.     

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.     

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.     

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.     

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.     

Ensemble refinement of protein crystal structures: validation and application

X-ray crystallography typically uses a single set of coordinates and B factors to describe macromolecular conformations. Refinement of multiple copies of the entire structure has been previously used in specific cases as an alternative means of representing structural flexibility. Here, we systematically validate this method by using simulated diffraction data, and we find that ensemble refinement produces better representations of the distributions of atomic positions in the simulated structures than single-conformer refinements. Comparison of principal components calculated from the refined ensembles and simulations shows that concerted motions are captured locally, but that correlations dissipate over long distances. Ensemble refinement is also used on 50 experimental structures of varying resolution and leads to decreases in R(free) values, implying that improvements in the representation of flexibility observed for the simulated structures may apply to real structures. These gains are essentially independent of resolution or data-to-parameter ratio, suggesting that even structures at moderate resolution can benefit from ensemble refinement.     

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.     

A target function for quaternary structural refinement from small angle scattering and NMR orientational restraints

We present a novel target function based on atomic coordinates that permits quaternary structural refinement of multi-domain protein–protein or protein–RNA complexes. It requires that the high-resolution structures of the individual domains are known and that small angle scattering (SAS) data as well as NMR orientational restraints from residual dipolar couplings (RDCs) of the complex are available. We show that, when used in combination, the translational and rotational restraints contained in SAS intensities and RDCs, respectively, define a target potential function that permits to determine the overall topology of complexes made up of domains with low internal symmetry. We apply the target function on a modestly anisotropic model system, the Barnase/Barstar complex, and discuss factors that influence the structural refinement such as data errors and the geometrical properties of the individual domains.     

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.     

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.     

Models with energy penalty on interresidue rotation address insufficiencies of conventional elastic network models

In this study, I present a new elastic network model, to our knowledge, that addresses insufficiencies of two conventional models—the Gaussian network model (GNM) and the anisotropic network model (ANM). It has been shown previously that the GNM is not rotation-invariant due to its energy, which penalizes rigid-body rotation (external rotation). As a result, GNM models are found contaminated with rigid-body rotation, especially in the most collective ones. A new model (EPIRM) is proposed to remove such external component in modes. The extracted internal motions result from a potential that penalizes interresidue stretching and rotation in a protein. The new model is shown to pertinently describe crystallographic temperature factors (B-factors) and protein open↔closed transitions. Also, the capability of separating internal and external motions in GNM slow modes permits reexamining important mechanochemical properties in enzyme active sites. The results suggest that catalytic residues stay closer to rigid-body rotation axes than their immediate backbone neighbors. I show that the cumulative density of states for EPIRM and ANM follow different power laws as functions of low-mode frequencies. When using a cutoff distance of 7.5 Å, The cumulative density of states of EPIRM scales faster than that of all-atom normal mode analysis and slower than that of simple lattices.     

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.     

Refinement of F-Actin Model against Fiber Diffraction Data by Long-Range Normal Modes

The atomic model of F-actin was refined against fiber diffraction data using long-range normal modes as adjustable parameters to account for the collective long-range filamentous deformations. To determine the effect of long-range deformations on the refinement, each of the four domains of G-actin was treated as a rigid body. It was found that among all modes, the bending modes make the most significant contributions to the improvement of the refinement. Inclusion of only 7–9 bending modes as adjustable parameters yielded a lowest R-factor of 6.3%. These results demonstrate that employing normal modes as refinement parameters has the advantage of using a small number of adjustable parameters to achieve a good fitting efficiency. Such a refinement procedure may therefore prevent the refinement from overfitting the structural model. More importantly, the results of this study demonstrate that, for any fiber diffraction data, a substantial amount of refinement error is due to long-range deformations, especially the bending, of the filaments. The effects of these intrinsic deformations cannot be easily compensated for by adjusting local structural parameters, and must be properly accounted for in the refinement to achieve improved fit of refined models with experimental diffraction data.     

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.     

Automated protein model building combined with iterative structure refinement.

In protein crystallography, much time and effort are often required to trace an initial model from an interpretable electron density map and to refine it until it best agrees with the crystallographic data. Here, we present a method to build and refine a protein model automatically and without user intervention, starting from diffraction data extending to resolution higher than 2.3 A and reasonable estimates of crystallographic phases. The method is based on an iterative procedure that describes the electron density map as a set of unconnected atoms and then searches for protein-like patterns. Automatic pattern recognition (model building) combined with refinement, allows a structural model to be obtained reliably within a few CPU hours. We demonstrate the power of the method with examples of a few recently solved structures.     

Characterization of protein matrix motions in the Rb. sphaeroides photosynthetic reaction center

We use Normal Mode Analysis to investigate motions in the photosynthetic reaction center (RC) protein. We identify the regions involved in concerted fluctuations of the protein matrix and analyze the normalized amplitudes and the directionality of the first few dominant modes. We also seek to quantify the coupling of normal modes to long-range electron transfer (ET). We find that a quasi-continuous spectrum of protein motions rather than one individual mode contributes to light-driven electron transfer. This is consistent with existing theoretical models (e.g. the spin-boson/dispersed polaron model) for the coupling of the protein and solvent "bath" to charge separation events. [Figure: see text].