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Equations in Sections 2.3 and 2.4 of the article by Afonine et al. [Acta Cryst. (2013). D69, 625-634] are corrected.

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A fast and robust method for determining the parameters for a flat (mask-based) bulk-solvent model and overall scaling in macromolecular crystallographic structure refinement and other related calculations is described. This method uses analytical expressions for the determination of optimal values for various scale factors. The new approach was tested using nearly all entries in the PDB for which experimental structure factors are available. In general, the resulting R factors are improved compared with previously implemented approaches. In addition, the new procedure is two orders of magnitude faster, which has a significant impact on the overall runtime of refinement and other applications. An alternative function is also proposed for scaling the bulk-solvent model and it is shown that it outperforms the conventional exponential function. Similarly, alternative methods are presented for anisotropic scaling and their performance is analyzed. All methods are implemented in the Computational Crystallography Toolbox (cctbx) and are used in PHENIX programs.

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In the last decade, the progress of protein crystallography allowed several protein structures to be solved at a resolution higher than 0.9 A. Such studies provide researchers with important new information reflecting very fine structural details. The signal from these details is very weak with respect to that corresponding to the whole structure. Its analysis requires high-quality data, which previously were available only for crystals of small molecules, and a high accuracy of calculations. The calculation of structure factors using direct formulae, traditional for 'small-molecule' crystallography, allows a relatively simple accuracy control. For macromolecular crystals, diffraction data sets at a subatomic resolution contain hundreds of thousands of reflections, and the number of parameters used to describe the corresponding models may reach the same order. Therefore, the direct way of calculating structure factors becomes very time expensive when applied to large molecules. These problems of high accuracy and computational efficiency require a re-examination of computer tools and algorithms. The calculation of model structure factors through an intermediate generation of an electron density [Sayre (1951). Acta Cryst. 4, 362-367; Ten Eyck (1977). Acta Cryst. A33, 486-492] may be much more computationally efficient, but contains some parameters (grid step, 'effective' atom radii etc.) whose influence on the accuracy of the calculation is not straightforward. At the same time, the choice of parameters within safety margins that largely ensure a sufficient accuracy may result in a significant loss of the CPU time, making it close to the time for the direct-formulae calculations. The impact of the different parameters on the computer efficiency of structure-factor calculation is studied. It is shown that an appropriate choice of these parameters allows the structure factors to be obtained with a high accuracy and in a significantly shorter time than that required when using the direct formulae. Practical algorithms for the optimal choice of the parameters are suggested.

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In conventional structure refinement, the discrepancy between the calculated magnitudes and those observed in X-ray experiments is attributed to errors inherent in preliminary assigned values of the model parameters. However, the chosen set of model parameters may not be adequate to describe the structure factors precisely. For example, if some atoms are not included in the current model, then the structure factors calculated from such a partial model contain 'irremovable errors'. These errors cannot be eliminated by any choice of the parameters of the partial structure. Probabilistic modelling suggests a way to take irremovable errors into account. Every trial set of values of the model parameters is now associated with the joint probability distribution of the calculated magnitudes, rather than with a particular set of magnitudes. The new goal of the refinement is formulated as the search for the distribution that is the most consistent with the observed data. The statistical likelihood is a possible measure of the consistency. The suggested quadratic approximation of the likelihood function allows the likelihood-based refinement to be considered as a kind of least-squares refinement that uses appropriate weights and modified targets for the calculated magnitudes. This in turn enables the analysis of tendencies of the likelihood-based refinement in comparison with the classical least-squares refinement.

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Many crystallographic problems are reduced to the optimization of some functional. In most cases, this functional is expressed in terms of structure factors and depends on a large number of variables; an example is the refinement of atomic models. Calculation of the functional derivatives, necessary for different optimization methods, is a time-consuming procedure. Previously, a technique to calculate the exact gradient of any crystallographic functional for the time equal to that for a single-function-value calculation has been proposed [Lunin & Urzhumtsev (1985). Acta Cryst. A41, 327-333]. Currently, a similar scheme is proposed to calculate the exact matrix of the second derivatives of these functionals. The accuracy of this matrix is crucial for the calculation of the inverted matrix, which can be used in optimization methods of the second order.

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The knowledge of the molecular structure of LDL, a large lipoprotein complex, is of great interest for medical investigations. Currently available LDL crystals do not diffract to high resolution and do not allow the application of standard crystallographic techniques. Additional difficulties arise because of a very dense crystal packing and the presence of several components with quite different mean densities. Several ab initio phasing methods previously reported by the authors have been successfully applied to find a crystallographic image of LDL at a resolution of 27 A. The most promising results have been obtained using direct phasing with a connectivity analysis of the electron-density maps. The current image makes it possible to discern a single particle covered by a layer of relatively high density that is asymmetrically distributed on the particle surface. It shows a partition of high and low densities inside the particle and, in particular, strips of varying density in the lipid core.

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If only native amplitudes are used for structure determination, then additional 'theoretical' information is necessary to determine their phases. For use in a phasing procedure, this information can be formulated as a selection criterion (figure of merit) which assigns a reliability weight to every trial phase set and distinguishes the closest ones to the true phase set. Different types of additional information may be tested as a selection criterion: electron-density histograms, connectivity properties, statistical likelihood, atomicity etc. A common feature of such criteria is that they do not unambiguously judge the phase quality at low resolution. Nevertheless, the selection of the phase sets with best criterion values increases the ratio of good phase sets in the ensemble considered. An approximate solution of the phase problem may then be found by averaging the selected phase sets. Cluster analysis of the selected phase sets and averaging within clusters allow further improvement of this solution.

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Direct phasing needs additional information of a non-specific kind in order to select the correct phase set from all possible ones. This paper analyses the use of constraints which can be formulated in terms of electron-density values. One- and multi-dimensional histograms and connectivity properties are implemented as such constraints in density-modification procedures. These approaches usually cannot unambiguously select the best solution from a set of alternative phase variants. Nevertheless, they do allow the rejection of wrong solutions and the use of cluster analysis and averaging on the remaining variants provide a good starting point for further phase-refinement procedures.

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It is expected that a correctly phased low-resolution synthesis would show a set of isolated 'blobs' located near to the centres of the macromolecules if the corresponding cut-off level is chosen properly. This is not always the case when using experimentally measured structure-factor magnitudes. Nevertheless, this property can be efficiently used as a constraint in the low-resolution ab initio phasing of structure factors. The suggested procedure consists in generating a large number of random phase sets, selecting those that together with the observed magnitudes result in the desired number of blobs in Fourier syntheses, and averaging the selected phase sets. The current paper discusses the formal definitions, analysis of low-resolution syntheses, some phasing algorithms and their application to ab initio phasing.

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DNA-gyrase exhibits an unusual ATP-binding site that is formed as a result of gyrase B subunit dimerization, a structural transition that is also essential for DNA capture during the topoisomerization cycle. Previous structural studies on Escherichia coli DNA-gyrase B revealed that dimerization is the result of a polypeptidic exchange involving the N-terminal 14 amino acids. To provide experimental data that dimerization is critical for ATPase activity and enzyme turnover, we generated mutants with reduced dimerization by mutating the two most conserved residues of the GyrB N-terminal arm (Tyr-5 and Ile-10 residues). Our data demonstrate that the hydrophobic Ile-10 residue plays an important role in enzyme dimerization and the nucleotide-protein contact mediated by Tyr-5 side chain residue helps the dimerization process. Analysis of ATPase activities of mutant proteins provides evidence that dimerization enhances the ATP-hydrolysis turnover. The structure of the Y5S mutant of the N-terminal 43-kDa fragment of E. coli DNA GyrB subunit indicates that Tyr-5 residue provides a scaffold for the ATP-hydrolysis center. We describe a channel formed at the dimer interface that provides a structural mechanism to allow reactive water molecules to access the gamma-phosphate group of the bound ATP molecule. Together, these results demonstrate that dimerization strongly contributes to the folding and stability of the catalytic site for ATP hydrolysis. A role for the essential Mg(2+) ion for the orientation of the phosphate groups of the bound nucleotide inside the reactive pocket was also uncovered by superposition of the 5'-adenylyl beta-gamma-imidodiphosphate (ADPNP) wild-type structure to the salt-free ADPNP structure.

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The multisolution strategies for direct phasing at very low resolution, such as the few atoms model technique, result in a number of alternative phase sets, each of them arising from a cluster of closely related models. Use of a Monte-Carlo type computer procedure is suggested to choose between the possible phase sets. It consists of generating a large number of pseudo-atom models inside the mask defined by a trial phase set and the use of histograms of magnitude correlation to evaluate the masks. It is shown that the procedure may be considered as a generalization of the statistical maximum-likelihood principle and may be used as a powerful supplementary tool in the likelihood-based approaches to the phase problem solution.

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A multicopy protocol is proposed for modeling macromolecular hydration using diffraction experimental data (X-ray or neutron) to search for a better description of delocalized water sites than that given by point water models. The model consists of one macro-molecule and several copies of each water molecule, refined simultaneously against diffraction data using molecular dynamics techniques. The protocol was applied to BPTI and an RNA tetradecamer. The sites defined by the different copies range from very ordered ones to continuous channels; they fit the density maps and agree with the diffraction amplitudes with an accuracy comparable with usual crystallographic methods. The delocalization of water in channels agrees with the high mobility observed in NMR experiments.

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BACKGROUND: Aldose reductase (AR) is an NADPH-dependent enzyme implicated in long-term diabetic complications. Buried at the bottom of a deep hydrophobic cleft, the NADPH coenzyme is surrounded by the conserved hydrophilic residues of the AR active site. The existence of an anionic binding site near the NADP+ has been determined from the structures of the complexes of AR with citrate, cacodylate and glucose-6-phosphate. The inhibitor zopolrestat binds to this anionic site, and in the hydrophobic cleft, after a change of conformation which opens a 'specificity' pocket. RESULTS: The crystal structures of the porcine AR holoenzyme and its complexes with the inhibitors tolrestat and sorbinil have been solved; these structures are important as tolrestat and sorbinil are, pharmaceutically, the most well-studied AR inhibitors. The active site of the holoenzyme was analyzed, and binding of the inhibitors was found to involve two contact zones in the active site: first, a recognition region for hydrogen-bond acceptors near the coenzyme, with three centers, including the anionic site; and second, a hydrophobic contact zone in the active-site cleft, which in the case of tolrestat includes the specificity pocket. The conformational change leading to the opening of the specificity pocket upon tolrestat binding is different to the one seen upon zopolrestat binding; this pocket binds inhibitors that are more effective against AR than against aldehyde reductase. CONCLUSIONS: The active site of AR adapts itself to bind tightly to different inhibitors; this happens both upon binding to the inhibitor's hydrophilic heads, and at the hydrophobic and specificity pockets of AR, which can change their shape through different conformational changes of the same residues. This flexibility could explain the large variety of possible substrates of AR.

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A method is proposed for improvement of local portions of electron-density maps. The main difference between this method and several other dummy-atoms techniques is that the dummy atoms are placed independently of the initial weak density and are not biased by it. An example of an application of the method is given.

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Estimates for the phases of the X-ray diffraction data from the 50S ribosomal particle of Thermus thermophilus has been made to an effective resolution around 80 A using the few-atoms-modes ab initio technique [Lunin, Lunina, Petrova, Vernoslova, Urzhumtsev & Podjarny (1995). Acta Cryst. D51, 896-903]. This technique models the density with a small number of Gaussian spheres to generate a large number of possible phase sets and then uses clustering algorithms to identify the best ones. Independently, an envelope obtained from electron-micrograph image reconstruction [Yonath, Leonard & Wittmann (1987). Science, 236, 813-816] was oriented and positioned using the molecular-replacement technique, specially adapted to the very low resolution case [Urzhumtsev & Podjarny (1995). Acta Cryst. D51, 888-895]. The two methods show similar packing arrangements. The electron density calculated by the few-atoms-models technique without any assumption on the number of molecules in asymmetric unit or on their shape shows recognizable features of the particle.

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The applicability of the molecular-replacement (MR) method, implemented through the AMoRe package [Navaza (1994). Acta Cryst. A50, 157-163], is studied at very low resolution (d > 20 A) and for very large molecular complexes. Due to the nature of the low-resolution data, specific problems appear. In particular, rotation-function peaks are very broad and translation functions based on Patterson overlap show large spurious peaks. To solve these problems, the translation function is replaced by a search using amplitude correlation and a systematic three-dimensional angular search is performed around each rotation-function peak. Furthermore, these functions are applied in different resolution ranges during the same search. The corresponding algorithms are applied to two cases: the tRNA(Asp)-synthetase complex (neutron diffraction data) and a ribosome model crystal (calculated data). This new implementation is shown to solve the problem for a variety of search models, ranging from a detailed atomic model to a rough envelope.

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A method is proposed for the solution of the phase problem at very low resolution for macromolecules. It generates randomly a very large number of models, each consisting of a few (two to ten) pseudo-atoms. The corresponding amplitudes are used for selecting a subset of 'best' models by choosing those with the highest correlation with experimental values. The phases calculated from these 'best' models are analysed by a clusterization procedure leading to a few possible solutions, from which the correct one can be recognized by simple additional criteria. This method has been successfully applied to the neutron diffraction data of the AspRS-tRNA(Asp) complex at 50 A resolution and to data calculated from a model ribosome crystal at 60 A resolution.

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The crystal structure of gamma-crystallin IIIb (gamma C) from calf eye lens has been refined at 2.5 A resolution. The molecule of about 21 kDa consists of two similar domains. Each domain is composed of two motifs with the 'Greek key' topology which form a pair of four-stranded beta-sheets with an antiparallel packing. The molecule has three hydrophobic cores: one within each domain and one between them. Six of the eight functionally important cysteines are located within the N-domain, and only two in the C-domain. Several large clusters of charged residues are at the surface of the molecule. Surface residues Val 101, Met 103 and Leu 155 are important for packing of molecules in crystal medium and possibly in the lens. Features of the gamma-crystallin IIIb molecule which may be related to its function in the vertebrate eye lens are briefly discussed. An attempt has been made to correlate molecular characteristics with some general properties of the eye lens such as high density and refractive index gradients and strong stability of the lens during an organism's lifetime.

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