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  • 1
    Electronic Resource
    Electronic Resource
    On the equivalence of structure invariants (1966)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 20 (1966), S. 816-819 
    Details
    ISSN: 0001-5520
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Geosciences
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0365110X66001919
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  • 2
    Electronic Resource
    Electronic Resource
    On the theory and estimation of the cosine invariants cos(φl + φm + φn + φp) (1974)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 30 (1974), S. 822-829 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: For fixed h1 + h2 + h3 = 0, and uniformly distributed k, the conditional joint probability distribution of the pair of phases φk, φh1 + k, given |E-h3 + k|, |Ek|, |Eh1 + k| is found. If l + m + n + p = 0, this distribution leads, via a suitable sampling technique, to estimates having probabilistic validity for the cosine invariant cos (φl + φm + φn + φp) in terms of the seven magnitudes |El|, |Em|, |En|, |Ep|, |El + m|, |El + n|, |El + p|.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739474001926
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  • 3
    Electronic Resource
    Electronic Resource
    On integrating the techniques of direct methods and isomorphous replacement. II. The first applications (1982)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 38 (1982), S. 294-300 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: With error-free diffraction data from the protein cytochrome c550 from Paracoccus denitrificans, having molecular weight Mr ∼ 14 500, space group P212121, and a single PtCl2-4 derivative, estimates (0 or π) of the three-phase structure invariants are obtained by recently secured direct methods employing the six-magnitude first neighborhood [Hauptman (1982). Acta Cryst. A38, 289-294] and compared with the known values. The comparison shows that these methods are capable of estimating reliably several tens of thousands of those invariants having the extreme values 0 or π, approximately. It is therefore anticipated that direct methods will play an increasingly important role in the determination of macromolecular structures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739482000643
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  • 4
    Electronic Resource
    Electronic Resource
    On integrating the techniques of direct methods with anomalous dispersion: the one-phase structure seminvariants in the monoclinic and orthorhombic systems. III. Primitive non-centrosymmetric space groups of type 1P220. Erratum. (1990)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 46 (1990), S. 72-72 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: There are two errors in the short communication by Velmurugan & Hauptman [Acta Cryst. (1989). A45, 656]. On line 9 in the Abstract, Φ2h′2k′0 should read Φ2h2k0 and on line 3 in Summary of final results, Φ2h′2k′0 should read Φ2h,2k,0.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767389011852
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  • 5
    Electronic Resource
    Electronic Resource
    The role of molecular structure in the direct determination of phase (1964)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 17 (1964), S. 1421-1433 
    Details
    ISSN: 0001-5520
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Geosciences
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0365110X6400353X
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  • 6
    Electronic Resource
    Electronic Resource
    Ab Initio Structure Determination and Refinement of a Scorpion Protein Toxin (1997)
    Copenhagen : International Union of Crystallography (IUCr)
    Acta crystallographica 53 (1997), S. 551-557 
    Details
    ISSN: 1399-0047
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The structure of toxin II from the scorpion Androctonus australis Hector has been determined ab initio by direct methods using SnB at 0.96 Å resolution. For the purpose of this structure redetermination, undertaken as a test of the minimal function and the SnB program, the identity and sequence of the protein was withheld from part of the research team. A single solution obtained from 1 619 random atom trials was clearly revealed by the bimodal distribution of the final value of the minimal function associated with each individual trial. Five peptide fragments were identified from a conservative analysis of the initial E-map, and following several refinement cycles with X-PLOR, a model was built of the complete structure. At the end of the X-PLOR refinement, the sequence was compared with the published sequence and 57 of the 64 residues had been correctly identified. Two errors in sequence resulted from side chains with similar size while the rest of the errors were a result of severe disorder or high thermal motion in the side chains. Given the amino-acid sequence, it is estimated that the initial E-map could have produced a model containing 99% of all main-chain and 81% of side-chain atoms. The structure refinement was completed with PROFFT, including the contributions of protein H atoms, and converged at a residual of 0.158 for 30 609 data with F ≥ 2σ(F) in the resolution range 8.0–0.964 Å. The final model consisted of 518 non-H protein atoms (36 disordered), 407 H atoms, and 129 water molecules (43 with occupancies less than unity). This total of 647 non-H atoms represents the largest light-atom structure solved to date.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0907444997005386
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  • 7
    Electronic Resource
    Electronic Resource
    On the identity and estimation of those cosine invariants, cos(φl + φm + φn + φp), which are probably negative (1974)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 30 (1974), S. 472-476 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: If h1, h2, h3 are fixed reciprocal vectors which satisfy h1 + h2 + h3 = 0, and if k is the primitive, uniformly distributed random variable, then, under the assumption that each of |Eh1|, |E-h3 + k| is sufficiently small, the conditional probability distribution of the cosine invariant cos (φk + φ- h1 - k + φ- h2 + φ - h3), given |E - h3 + k|, |Ek|, Eh1 + k|, is found. The distribution leads to the surprising result that the conditional expected value of this cosine invariant is always negative and approaches - 1 with increasing |EkEh1 + kEh2Eh3|. If m, n, p, q are fixed reciprocal vectors satisfying m + n + p + q = 0, suitable sampling of reciprocal space then leads to a formula for the cosine invariant cos (φm + φn + φ p + φq) having probabilistic validity in the case that |Em + n|, |Em + p| and |Em + q| are sufficiently small. It follows, in particular, that under the stated conditions the value of the cosine is probably negative and the larger the value of |EmEnEpEq| the more negative the cosine is likely to be.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739474001136
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  • 8
    Electronic Resource
    Electronic Resource
    A new method in the probabilistic theory of the structure invariants (1975)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 31 (1975), S. 680-687 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is assumed that a crystal structure in P1 is fixed and that the seven non-negative numbers R1, R2, R3, R4, R12, R23, R31 are also specified. The random variables (vectors) h, k, l, m are assumed to be uniformly and independently distributed in the regions of reciprocal space defined by |Eh| = R1, |Ek| =R2, |El| = R3, |Em| = R4, (1) |Eh + k| = R12, |Ek + l| =R23, |El + h| = R31, (2) and h + k + l + m = 0. (3) Then the structure invariant, (φ = φh + φk + φl + φm, as a function of the primitive random variables h, k, l, m, is itself a random variable, and its conditional probability distribution, given (1) and (2), is derived and compared with the conditional distribution when only (1) is given. The distribution leads to improved estimates for cos φ in terms of the seven magnitudes (1) and (2). The results secured here suggest a generalization which is described in terms of the 'neighborhood concept'; and a 'principle of nested neighborhoods' is formulated. This terminology permits, in turn, an analogy with interpolation formulas.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739475001428
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  • 9
    Electronic Resource
    Electronic Resource
    Probabilistic theory of the structure invariants: extension to the unequal atom case with application to neutron diffraction (1976)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 32 (1976), S. 877-882 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Recent methods in the probabilistic theory of the structure invariants and seminvariants are here generalized to include the case that not all atoms in the unit cell are identical. The presence of unequal atoms, in particular a few heavy atoms, is thus clearly seen to enhance the power of the direct method. Since the method permits the presence of negative scatterers, the application to neutron diffraction is immediate. Only the conditional probability distributions associated with the first neighborhood of the three-phase structure invariant and the first two neighborhoods of the four-phase structure invariant in P1 and P{\bar 1} are treated here. However the methods are clearly sufficiently general to cope with structure invariants and seminvariants in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739476001757
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  • 10
    Electronic Resource
    Electronic Resource
    Pairs in P21: probability distributions which lead to unique estimates of the two-phase structure seminvariants in the interval (–π,π) (1978)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 34 (1978), S. 230-241 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Conditional probability distributions of the two-phase structure seminvariant φ12 = φh1kl1 - φh2kl2 in P21, given the values of one or more two-phase structure seminvariants, as well as appropriately chosen sets of structure factor magnitudes |E|, are derived. These distributions yield unique and reliable estimates, in the whole interval (-π,π), of the values of these seminvariants. In the case that a given structure seminvariant specifies the enantiomorph, the values of the remaining seminvariants are consistent with this choice of enantiomorph. In any event, the variance of any distribution of the kind described here is much smaller, in favorable cases, than that of a distribution which assumes as known magnitudes |E| alone, reflecting the fact that previously derived phase information severely limits the possible values of other phases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739478000455
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