On the relation between the diffraction ratio, the doublet values and the reliability of the triplet estimates

Peschar²,

Schenk³

1993

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An efficient procedure is presented for the derivation of joint probability distributions of isomorphous data sets. The new technique is based on the use of the differences of isomorphous structure factors as random variables. It will be shown that the usual probabilistic techniques, applied to these random variables, finally result in the joint probability distribution of three single differences of isomorphous structure factors comprising three doublet and eight triplet phase sums. An advantage of the new technique is that the inherent correlation between the isomorphous data sets is removed if a probabilistic procedure is set up for the small difference itself. In this way, an enormous mathematical simplification is obtained while the final results are much better than those obtainable by previous probabilistic expressions. The final triplet distribution seems to be of sufficient quality to be used in a normal directmethods procedure. In contrast to usual approaches, the heavy-atom substructure need not be solved first. The probabilistic expression will be explained in detail for one and three single differences. Applications for the cases of single anomalous scattering, two different wavelengths and single isomorphous replacement (excluding anomalous-scattering effects)

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Section: Introductionmentioning

confidence: 99%

On direct-methods phase information from differences between isomorphous structure factors

Peschar²,

Schenk³

1993

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“…In the PDB release of July 1991, this structure is referred to as 155C. Tables 2 to 11 employed Doublet estimation using probabilistic technique (mode) Doublet estimation using probabilistic technique (numerical) Doublet estimation using algebraic technique Doublet estimation using difference Patterson synthesis The doublet estimates are equal to zero The doublet estimates are equal to the true doublet values Reflection with indices hkl Diffraction ratio (Kyriakidis, Peschar & Schenk, 1993) E values from the first data set E values from the second data set Reliability factor of the distribution (Peschar & Schenk, 1991) Number of the triplets involved in the statistics* Mean absolute doublet (triplet) error in mc [equations (22a), (23a)] Mean doublet (triplet) error in mc [equations (22b), (23b)] * Instead of one triplet (Cochran distribution), eight isomorphous triplets exist because of the distribution involving two isomorphous data sets. Hence, the real number of triplets involved in the statistics is eight times the NTR.…”

Section: Test Results and Discussionmentioning

confidence: 99%

“…The first column indicates the relevant technique (SAS, SIRNAS, SIRAS and 2DW). The second column lists the theoretical DR (Kyriakidis, Peschar & Schenk, 1993) and columns 3 to 12 show the error in mc (1000mc= 2~ rad) of the mean absolute difference, AER, AER = (I I ~,7"ltru¢-I ~,7"l~s, [>, (22a) and of the mean difference, ERR, Table 5 illustrate the strength of ALG compared with the JPD estimation. The improved algebraic estimation PAT gives the same results as the normal algebraic estimation ALG since the two data sets differ only in one pair of non-identical anomalous scatterers so the double summation in (14) is not expected to contribute considerably.…”

Section: Test Results and Discussionmentioning

confidence: 99%

“…A recent investigation of two probabilistic formulae for the ab initio determination of protein structures (Peschar & Schenk, 1991) has revealed that correct doublet phase sums [which are of order O(N°)] are important for a correct triplet-phase-sum evaluation. For this purpose, a new diffraction ratio (DR) was developed which shows a linear relationship with the ideal doublet phase sum as calculated from the atomic coordinates (Kyriakidis, Peschar & Schenk, 1993).…”

Section: Introductionmentioning

confidence: 99%

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On the doublet phase sums of isomorphous data sets

Peschar²,

Schenk³

1993

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The role of the doublet phase sum present among isomorphous data sets is investigated in connection with the triplet-phase-sum statistics. Several probabilistic and algebraic techniques are discussed to estimate the doublets. The combination of an algebraic estimation technique and a new difference Patterson synthesis, the maxima of which are used to improve iteratively the doublet phase sums, is shown to be successful. Test results for large model structures and idealized protein data show that this technique reduces the triplet-phase-sum errors to a level small enough for ab initio direct-methods applications.

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“…An additional advantage of the difference-structure-factor approach is that the mathematical calculations are simplified. It has been shown for calculated structure-factor data that, by taking the F~ a as a random variable, reliable estimates of the triplet phase sums present among isomorphous data sets can be obtained, even if the diffraction ratio is small (Kyriakidis, Peschar & Schenk, 1993a). An additional improvement is achieved by supplementing the estimation of the doublet phase sums with vectors from a difference Patterson synthesis (Kyriakidis, Peschar & Schenk, 1993b,c).…”

Section: Introductionmentioning

confidence: 99%

The Estimation of Four-Phase Structure Invariants Using the Single Difference of Isomorphous Structure Factors

Peschar²,

Schenk³

1996

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