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  • 1
    Electronic Resource
    Electronic Resource
    Bayesian Difference Refinement (1996)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 52 (1996), S. 1004-1011 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Interest in a pair of highly isomorphous structures often focuses on the differences between them. In cases where substantial correlated model errors exist or where there are differences in the quality of the two experimental data sets (cases quite common in macromolecular crystallography), independent refinement of the two structures does not lead to the most accurate estimate of the differences between them. An alternative procedure that has proven effective in some such cases is difference refinement, in which the residual between observed and calculated differences in structure-factor amplitudes between the two structures is minimized. A Bayesian approach has been used to extend the range of applicability of difference refinement to cases where there is only partial correlation in model errors and where the overlap between the data sets is limited. The resulting method, Bayesian difference refinement, uses residuals to be minimized that vary smoothly between difference refinement and independent refinement. When the errors in the two structural models are very similar, difference refinement is used; when they are very different, independent refinement is used; and when they are partially correlated, a combination of the two is used. The procedure is very simple to apply and does not significantly increase the computational demands of refinement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444996006725
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  • 2
    Electronic Resource
    Electronic Resource
    Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions (1983)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 39 (1983), S. 813-817 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A procedure is presented for three-dimensional refinement of heavy-atom parameters without the use of phase information in the methods of single and multiple isomorphous replacement. This procedure is based on the Patterson-function correlation method of Rossmann [Acta Cryst. (1960), 13, 221-226] except that the origins of the Patterson functions are now removed from this correlation and centric and acentric reflections are treated separately. The resulting procedure is shown theoretically and by application to three test cases to yield accurate and essentially unbiased estimates of the occupancies, thermal parameters, and positions of heavy-atom sites.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767383001592
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  • 3
    Electronic Resource
    Electronic Resource
    Difference refinement: obtaining differences between two related structures (1995)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 51 (1995), S. 609-618 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: There are many examples in macromolecular crystallography where interest focuses on the differences between a previously determined `native' structure and a nearly isomorphous `variant'. In such cases, a useful approach to atomic refinement of the variant structure is through weighted least-squares minimization of the residual between the observed and calculated differences in amplitudes of structure factors, a strategy first used in the refinement of deoxycobalt hemoglobin [Fermi, Perutz, Dickinson & Chien (1982). J. Mol. Biol. 155, 495–505] and termed `difference refinement'. For cases in which the modeling errors for the native and variant structures are correlated, theoretical arguments indicate that difference refinement should lead to improved estimates of structural differences when compared with conventional independent refinement. Tests employing simulated peptide data sets and real data from a wild-type protein and a mutant show that difference refinement can substantially reduce errors in the differences between structures when compared with independent refinement. The algorithm is very easy to implement and does not increase the computational demands of refinement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444994013247
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  • 4
    Electronic Resource
    Electronic Resource
    Correlated Phasing of Multiple Isomorphous Replacement Data (1996)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 52 (1996), S. 749-757 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Substantial highly correlated differences sometimes exist between a series of heavy-atom derivatives of a macromolecule and the native structure. Use of such a series of derivatives for phase determination by multiple isomorphous replacement (MIR) has been difficult because MIR analysis has treated errors as independent. A simple Bayesian approach has been used to derive probability distributions for the phase in the case where a group of MIR derivatives have correlated errors. The utility of the resulting `correlated-phasing' method has been examined by applying it to both simulated and real MIR data sets that contain sizeable correlated errors and it has been found that it can dramatically improve MIR phase estimates in these cases. Correlated phasing is applicable to situations where derivatives exhibit substantial correlated changes in protein conformation or crystal packing or where correlated errors in heavy-atom models are large. Correlated phasing does not substantially increase the complexity of phase computation and is suitable for routine use.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444996000832
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  • 5
    Electronic Resource
    Electronic Resource
    Bayesian Weighting for Macromolecular Crystallographic Refinement (1996)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 52 (1996), S. 743-748 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A simple weighting scheme for atomic refinement is discussed. The approach, called `Bayesian weighting', is designed to be robust with respect to the bias that arises from the incomplete nature of the atomic model, which in macromolecular crystallography is typically quite serious. Bayesian weights are based on the mean-squared residual errors over shells of resolution, with centric and acentric reflections considered separately and with allowances made for experimental uncertainties. Use of Bayesian weighting is shown in test cases typical for macromolecular crystallography to improve the accuracy of the refined coordinates when compared with schemes employing unit weights or experimental variances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444996001473
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  • 6
    Electronic Resource
    Electronic Resource
    Bayesian Correlated MAD Phasing (1997)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 53 (1997), S. 571-579 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A Bayesian treatment for phase calculation in the multiwavelength anomalous diffraction (MAD) technique is presented. This approach explicitly treats effects of errors correlated among measurements at different wavelengths and between Bijvoet pairs. The resulting method, which is called Bayesian correlated MAD phasing, gives proper statistical consideration to all data and does not give special treatment to data from a particular wavelength. Results obtained using Bayesian correlated MAD phasing and two other strategies on both a model test case and on data obtained in two actual MAD experiments are compared. Although all procedures performed well when the completeness of the data was high, it is shown that Bayesian correlated MAD phasing is more robust with respect to incompleteness of data than the other methods are. At 60% completeness the improvement over other methods for the examples given was nearly 50% in the correlation coefficients, and made a substantial difference in the interpretability of an electron-density map.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444997005398
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  • 7
    Electronic Resource
    Electronic Resource
    MAD phasing: Bayesian estimates of FA (1994)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 50 (1994), S. 11-16 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A Bayesian approach is applied to the calculation of Patterson functions and cross-Fourier maps in the analysis of multi-wavelength anomalous-diffraction (MAD) data. This procedure explicitly incorporates information available a priori on the likely magnitudes of partial structure factors (FA) corresponding to the anomalously scattering atoms, uses weighted-average estimates of FA, and incorporates estimates of errors in the data that are not represented in the instrumental uncertainties. The method is demonstrated by application to MAD data collected on selenomethionine-containing gene V protein.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444993008224
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  • 8
    Electronic Resource
    Electronic Resource
    Isomorphous replacement: effects of errors on the phase probability distribution (1987)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 43 (1987), S. 6-13 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A simple framework for the analysis of the effects of errors in the method of isomorphous replacement is developed. This framework is used to derive phase probability distributions similar to those given by Blow & Crick [Acta Cryst. (1959). 12, 794-802]. The present analysis suggests an improved means of calculating the mean-square 'lack-of-closure' residuals and shows that they differ by a factor of two for centric and acentric reflections. It is also shown that the lack-of-closure residuals have a straightforward interpretation and that they may be used to estimate the error in the heavy-atom model and the degree of isomorphism between native and derivative structures if anomalous differences have been measured.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767387099987
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  • 9
    Electronic Resource
    Electronic Resource
    Generalized method of determining heavy-atom positions using the difference Patterson function (1987)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 43 (1987), S. 1-5 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: An automated procedure for locating the positions of heavy atoms in crystals of macromolecules has been developed. The method is simple to apply, is independent of space group, and permits inclusion of non-crystallographic symmetry. The procedure is a search of the difference Patterson function; a trial solution consisting of a set of heavy-atom sites is considered likely to be correct only if the corresponding 'minimum function' (the minimum value of the difference Patterson function at the self- and cross- vector positions for this group of sites) is large. Although the method may be used to search for 'single-site' solutions to the difference Patterson function, it is more effectively used to search for pairs of sites unrelated by crystallographic symmetry. In the latter case the number of predicted cross vectors for each trial solution is larger, and correct solutions may be more readily distinguished from incorrect ones. Because of noise in difference Patterson functions, it is helpful in evaluating the solutions obtained to calculate the probability that a given value of the minimum function might occur by chance. The method has been applied to nine difference Patterson functions for which solutions were known. In eight of these cases, including several which had resisted earlier attempts at interpretation, this procedure yielded at least half of the known sites.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767387099999
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  • 10
    Electronic Resource
    Electronic Resource
    MAD phasing: treatment of dispersive differences as isomorphous replacement information (1994)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 50 (1994), S. 17-23 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A framework analogous to that used for the analysis of data in the method of isomorphous replacement is applied to the analysis of multi-wavelength anomalous-diffraction (MAD) data. The present method is advantageous in that refinement of parameters describing the anomalously scattering atoms in the structure can be carried out using nearly all the data, and in that phase information can be readily combined. The procedure described here involves an approximation in which it is assumed that the magnitude of the structure factor corresponding to anomalously scattering atoms in the unit cell is small compared to that corresponding to all other atoms in the structure. A model calculation is applied to a protein crystal with 682 non-H atoms in the asymmetric unit and two Se atoms as the anomalous scatterers. It is shown using this model calculation that the approximation used in this analysis does not substantially affect the accuracy of phase calculations for this MAD data. The method is demonstrated by application to MAD data collected on gene V protein.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444993008236
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