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A G Urzhumtsev

Publications and source records attributed to A G Urzhumtsev.

10 recordsLinked to original sources
PubMed2002

Likelihood-based refinement. I. Irremovable model errors.

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.

Journal Article↗
PubMed2001

Fast differentiation algorithm and efficient calculation of the exact matrix of second derivatives.

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.

Journal Article↗
PubMed2000

Connectivity properties of high-density regions and ab initio phasing at low resolution.

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.

Chemical Phenomena↗
PubMed2000

Density constraints and low-resolution phasing.

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.

Crystallography, X-Ray↗
PubMed2000

Low-resolution ab initio phasing: problems and advances.

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.

Cluster Analysis↗
PubMed1998

On the ab initio solution of the phase problem or macromolecules at very low resolution. II. Generalized likelihood based approach to cluster discrimination.

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.

Cluster Analysis↗
PubMed1997

Local improvement of electron-density maps.

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.

Journal Article↗
PubMed1996

Approaches to very low resolution phasing of the ribosome 50S particle from Thermus thermophilus by the few-atoms-models and molecular-replacement methods.

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.

Journal Article↗
PubMed1995

On the ab initio solution of the phase problem for macromolecules at very low resolution: the few atoms model method.

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.

Journal Article↗