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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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Statistical likelihood criteria were tested to select the true (or closest to true) structure-factor phases from an ensemble of phase sets. To define the criterion value for a given trial phase set, the trial 'molecular region' is defined as a region consisting of the points with the highest values in the Fourier synthesis calculated with the observed magnitudes and the trial set of phases. The structure studied is considered as composed of atoms randomly placed inside the trial molecular region. The figure of merit is defined as the likelihood corresponding to this hypothesis, i.e. the probability that the structure-factor magnitudes calculated (from the positions of atoms randomly placed into the trial region) are equal to the observed magnitudes. The concept of generalized likelihood is introduced to make the calculations more straightforward. The tests performed for known structures with the use of experimentally observed magnitudes show that in general it is impossible to unambiguously determine the best phases among a 'population' of trial phase sets. Nevertheless, the random generation of a great number of phase sets and the selection of phase sets with high likelihood values give a collection of variants with a higher concentration of 'good' phase sets than those found in the original population. Averaging the selected phase sets gives a starting solution of the low-resolution phase problem.

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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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For a chosen unit cell, a function defined in real space (a standard signal) is considered as a crystallographic wavelet-type function if it is localized in a small region of the real space, if its Fourier transform is likewise localized in reciprocal space, and if it is a periodical function which possesses a symmetry. The fixed-scale analysis consists in the decomposition of a studied distribution into a sum of copies of the same standard signal, but shifted into nodes of a grid in the unit cell. For a specified standard signal and grid of the permitted shifts in the unit cell, the following questions are discussed: whether an arbitrary function may be represented as the sum of the shifted standard signals; how the coefficients in the decomposition are calculated; what is the best fixed-scale approximation in the case that the exact decomposition does not exist. The interrelations between the fixed-scale decomposition and the phase problem, automatic map interpretation and density-modification methods are pointed out.

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The three-dimensional crystal structure of Serratia marcescens endonuclease has been refined at 1.1 A resolution to an R factor of 12.9% and an R(free) of 15.6% with the use of anisotropic temperature factors. The model contains 3694 non-H atoms, 715 water molecules, four sulfate ions and two Mg(2+)-binding sites at the active sites of the homodimeric protein. It is shown that the magnesium ion linked to the active-site Asn119 of each monomer is surrounded by five water molecules and shows an octahedral coordination geometry. The temperature factors for the bound Mg(2+) ions in the A and B subunits are 7.08 and 4.60 A(2), respectively, and the average temperature factors for the surrounding water molecules are 12.13 and 10.3 A(2), respectively. In comparison with earlier structures, alternative side-chain conformations are defined for 51 residues of the dimer, including the essential active-site residue Arg57. A plausible mechanism of enzyme function is proposed based on the high-resolution S. marcescens nuclease structure, the functional characteristics of the natural and mutational forms of the enzyme and consideration of its structural analogy with homing endo-nuclease I-PpoI.

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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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The three-dimensional crystal structure of Serratia marcescens (Sm) nuclease has been refined at 1.7 A resolution to the R-factor of 17.3% and R-free of 22.2%. The final model consists of 3678 non-hydrogen atoms and 443 water molecules. The analysis of the secondary and the tertiary structures of the Sm nuclease suggests a topology which reveals essential inner symmetry in all the three layers forming the monomer. We propose the plausible mechanism of its action based on a concerted participation of the catalytically important amino acid residues of the enzyme active site.

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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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In judging the effectiveness of methods of solving crystal structures, or in phase refinement and development, two criteria are commonly used. The first is the mean phase error, which may be weighted in some way, and the second is the map correlation coefficient which describes the similarity of a map with estimated phases to that with true phases. It is shown that these two measures are directly related and that given the individual phase errors the map correlation coefficient may be found without the need to calculate a map. Various aspects of this connection are examined, including the map correlation coefficient when weights are used for calculating maps and the conditions under which phase extension leads to maps with a higher map correlation coefficient - which involves a balance between the advantage of employing more data and the disadvantage that the extra data may have a higher average phase error.

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The specific properties of the range of possible electron-density values may serve as useful additional information for the determination and refinement of structure-factor phases. Fourier synthesis histograms (showing the spectra of frequencies of different possible values) produce the most adequate representation of these properties. The mathematical background and practical uses of these histograms are discussed. The investigation provides new information on some traditional methods of phase refinement, including density-modification procedures, maximization of Cochran's integral value and other techniques.

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