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

Abstract

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)

Direct methods beyond small‐molecule crystallography

“…Similar investigations were reported by different authors during 1960s and 1970s 33–38. From the early 1980s to the early 2000s, the combination of direct methods with SAD/SIR data was emphasized in direct‐method research worldwide 39–63. The first application of direct‐method SAD phasing in solving originally unknown protein structures was reported in 1999 56 with data at 2.1 Å resolution.…”

Combining direct methods with isomorphous replacement or anomalous scattering data. VIII. Phasing experimental SIR data with the replacing atoms in a centrosymmetric arrangement

“…A number of procedures have been proposed to solve this problem. These include the solventattening method (Wang, 1981(Wang, , 1985 and various kinds of direct methods (Coulter, 1965;Fan, 1965;Karle, 1966;Hendrickson, 1971;Hauptman, 1982;Fan & Gu, 1985;Fortier et al, 1985;Klop et al, 1987;Giacovazzo et al, 1988;Kyriakidis et al, 1993). In practice, the solvent-¯attening method has been the most successful; however, it is impossible even in theory to resolve the phase ambiguity when the replacing atoms are in a centrosymmetric arrangement.…”

On Integrating Direct Methods and Isomorphous Replacement Techniques. I. A Distribution Function for Quartet Invariants

“…The probabilistic theory of quartet invariants when isomorphous data are available has not been settled so far. In order to derive the joint probability distribution of isomorphous data sets, Kiriakidis, Peschar & Schenk (1993) have applied a technique that relies on the use of the so-called single difference between two isomorphous structure factors as a variable, instead of using the individual structure factors separately as done by Hauptman and by GCZ. While this technique succeeded in reproducing triplet distributions, it did not provide accurate probability distributions for quartet invariants (Kiriakidis, 1993).…”