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  • 1
    Electronic Resource
    Electronic Resource
    Bulk-solvent correction in direct-methods phasing (2000)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 56 (2000), S. 451-457 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is shown that for crystals of large proteins at low diffraction resolution, with N ∼ 10 000 independent non-H protein atoms and dmin ∼ 8 Å, a simple bulk-solvent correction yields the Sayre equation in its classical form, F _{\bf h} = q \textstyle \sum_{\bf k} F_{\bf k} F _{{\bf h} - {\bf k}}. In the low-resolution protein case, the proportionality factor becomes q = 1/[(\langle \rho_P \rangle - \rho_{S })V] , where V is the unit-cell volume, ρS is the assumed constant electron density in the solvent regions of the crystal and 〈ρP〉 is the average electron density in the protein regions. The classical form of the tangent formula follows from the bulk-solvent-corrected Sayre equation and its validity at low resolution is verified in empirical calculations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444900000573
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  • 2
    Electronic Resource
    Electronic Resource
    Bulk-solvent correction in direct-methods phasing. Erratum (2000)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 56 (2000), S. 936-936 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Equation (12) of the paper by Guo et al. [(2000), Acta Cryst. D56, 451–457] was printed incorrectly. The factor (4/3)\pi R_s^3 should be omitted. The correct equation is as follows f_S =\rho_s\textstyle\int\limits_{V_P/N_P}\exp (2\pi i {\bf h\cdot r}) {\rm d}V \simeq Z_s \Phi(2\pi |{\bf h}|R_s). \eqno (12)
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444900008453
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  • 3
    Electronic Resource
    Electronic Resource
    Globbic approximation in low-resolution direct-methods phasing (2000)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 56 (2000), S. 1148-1155 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Probabilistic direct-methods phasing theory, originally based on a uniform atomic distribution hypothesis, is shown to be adaptable to a non-uniform bulk-solvent-compensated globbic approximation for protein crystals at low resolution. The effective number ng of non-H protein atoms per polyatomic glob increases with decreasing resolution; low-resolution phases depend on the positions of only Ng = Na/ng globs rather than Na atoms. Test calculations were performed with measured structure-factor data and the refined structural parameters from a protein crystal with ∼10 000 non-H protein atoms per molecule and ∼60% solvent volume. Low-resolution data sets with dmin ranging from 15 to 5 Å gave ng = admin + b, with a = 1.0 Å−1 and b = −1.9 for the test case. Results of tangent-formula phase-estimation trials emphasize that completeness of the low-resolution data is critically important for probabilistic phasing.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444900008362
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  • 4
    Electronic Resource
    Electronic Resource
    Statistical Expectation Value of the Debye–Waller Factor and E(hkl) Values for Macromolecular Crystals (1996)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 52 (1996), S. 257-266 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: If the unit-cell distribution of atomic mean-square displacement parameters B = 8π2〈u2〉 is assumed to be normal, with mean µ = 〈B〉 and variance σ2 = \langle (B-\langle B \rangle ) ^2 \rangle, the statistical expectation value of the Debye-Waller factor W2 = exp(−2Bs2), where s = (sin θ)/λ, is 〈W2〉 = exp[−2(µ − σ2s2)s2]. This result has been incorporated into procedures for scaling and normalizing measured Bragg intensities to their Wilson expectation values. The procedures can determine both isotropic µ(B) and σ(B) and anisotropic µ(Uij) and σ(Uij distribution parameters. Tests with experimental data and refined structural models for several protein crystals show that the procedures yield reliable normalized structure-factor amplitudes for direct-methods applications, with values of R = \sum {_h} || E _o | - | E _c||/\sum {_h} | E _o| averaging \sim5%.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444995014053
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  • 5
    Electronic Resource
    Electronic Resource
    On `globbicity' of low-resolution protein structures (1999)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 55 (1999), S. 230-237 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Using Harker's [Harker (1953). Acta Cryst. 6, 731–736] idea of spherically averaged polyatomic groups or `globs' as the units of structure suitable for analyzing low-resolution diffraction data from protein crystals, `globbic' scattering factors have been calculated for main-chain peptide units and amino-acid side-chain groups to 3 Å resolution via Debye's [Debye (1915). Ann. Phys. (Leipzig), 46, 809–823] scattering formula. It is shown that the scattering factors are insensitive to intra-globbic conformational variation and can be approximated fairly well by a single-Gaussian formula, i.e. fg(s) = Zg exp(−1.7Zgs2), where s = (sinθ)/λ and Zg is the total electron count for the atoms of the glob. Phase errors due to the globbic approximation and their effect on electron-density maps at 3.5 Å resolution have been assessed via calculations for the crambin structure; this analysis indicates that the globbic scattering factors will be useful in efforts to develop procedures for direct-methods phasing of diffraction data to ∼3.5 Å resolution from protein crystals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444998008208
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  • 6
    Electronic Resource
    Electronic Resource
    On integrating the techniques of direct methods with anomalous dispersion. II. Statistical properties of the two-phase structure invariants (1991)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 47 (1991), S. 340-345 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Results of a statistical study of probabilistic estimates of two-phase structure invariants (TPSI) for Friedel pairs in the case of single-wavelength anomalous scattering are reported. Numerical analysis of the TPSI sign, magnitude and error distributions shows that the concise formula for TPSI by probability theory [Hauptman (1982). Acta Cryst. A38, 632–641; Giacovazzo (1983). Acta Cryst. A39, 585–592] has desirable statistical properties. Computational results for the known structures of cocaine methiodide (N-methylcocaine iodide) and of cytochrome c550 and its PtCl42− derivative show that when |E| values are large most of the signs of the TPSI are correctly determined – for |E| 〉 1.0, 90% or more of the TPSI signs are positive as predicted – and the errors in the estimated TPSI magnitudes do not exceed 1̃0% for |E| 〉 1.0 in the small-molecule case or 5̃0% for |E| 〉 1.5 in the macromolecular case. These results suggest that the theory will be useful for estimating the TPSI for unknown structures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767391000892
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  • 7
    Electronic Resource
    Electronic Resource
    On Integrating the Techniques of Direct Methods with Anomalous Dispersion. IV. A Simplified Perturbation Treatment for SAS Phasing (1997)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 53 (1997), S. 74-83 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Results from probabilistic theory for the single-wavelength anomalous-scattering (SAS) Friedel pair, two-phase structure invariants, \psiH = \varphiH + \varphi− H, are used to show that the SAS three-phase structure invariants, \psiHK = \varphiH + \varphiK + \varphi− H−K, tend to positive values that are easily estimated. Appropriate averages of the estimates provide SAS perturbation corrections in the form of positive origin shifts for the probability distribution of \psiHK values and for the tangent formula. The theoretical probabilistic results are verified by empirical statistical analyses of model-calculated phases and experimentally measured structure-factor magnitudes for a small-molecule and a protein crystal structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767396011993
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  • 8
    Electronic Resource
    Electronic Resource
    Sign distribution of two-phase structure invariants (1990)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 46 (1990), S. 942-944 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: An empirical investigation of the distribution of Friedel-pair two-phase structure invariants has been reported by Guo & Hauptman [Chin. Sci. Bull. (1989), 34, 137-141]. In the present paper their sign distributions are calculated for some small molecules and for a protein. The statistical figures show that there exists a strong tendency towards positive values for the signs of the two-phase structure invariants. It is anticipated that Hauptman's formula [Acta Cryst. (1982), A38, 632-641] may be good enough to estimate statistically the signs of the two-phase structure invariants for these model crystals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767390006699
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  • 9
    Electronic Resource
    Electronic Resource
    On integrating the techniques of direct methods with anomalous dispersion. III. Estimation of two-wavelength two-phase structure invariants (1994)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 50 (1994), S. 307-311 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: For diffraction data at two wavelengths from a crystal with anomalous scatterers, there are six types of two-phase structure invariants for Friedel pairs. Two of the six are single-wavelength invariants; the other four are mixed-wavelength invariants. It is shown that the latter can be estimated by a straightforward extension of results from the probabilistic direct-methods theory for the single-wavelength anomalous scattering case described in paper I [Hauptman (1982). Acta Cryst. A38, 632–641]. Statistical tests of the mixed-wavelength estimates are reported for small-molecule and macromolecular examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767393009778
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  • 10
    Electronic Resource
    Electronic Resource
    TDSIR phasing: direct use of phase-invariant distributions in macromolecular crystallography (1995)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 51 (1995), S. 535-542 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A new strategy for employing three phase triples invariant estimates from Hauptman's single isomorphous replacement (SIR) and anomalous dispersion (SAS) joint probability distribution formulae is outlined which produces a single unique phase-invariant solution in the case where the positions of the heavy-atom scatterers is known. A similar but non-identical result is obtained for the phase invariants of a structure for which a molecular-replacement solution has been obtained. It is important to note that the values of the individual native/derivative phases can be determined directly from the probability distribution formulae without having to utilize the phase-invariant estimates in an active way. Elimination of the multisolution aspect of utilizing phase-invariant estimates should have important repercussions with regard to phasing macromolecular sets of derivatized data. Trial calculations based on experimentally measured 2.5 Å data for three derivatives of cytochrome c550 are encouraging. The average of the three SIR maps resolves a number of structural ambiguities seen in the published multiple isomorphous replacement (MIR) map obtained from eight derivatives.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767395000870
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