Direct phase determination of triple products from Bijvoet inequalities

The recently secured mathematical formalism of direct methods is here generalized to the case that the atomic scattering factors are arbitrary complex numbers, thus including the special case that one or more anomalous scatterers are present. Once again the neighborhood concept plays the central role. Final results from the probabilistic theory of the two-and three-phase structure invariants are briefly summarized. In particular, the conditional probability distribution of the three-phase structure invariant, given the six magnitudes IEI in its first neighborhood, is described. The distribution yields an estimate for the three-phase structure invariant which is particularly good in the favorable case that the variance of the distribution happens to be small (the neighborhood principle). Particularly noteworthy is the fact that, in sharp contrast to all earlier work, the estimate is unique in the whole range 0 to 2n. An example shows that the method is capable of yielding unique estimates for tens of thousands of three-phase structure invariants with unprecedented accuracy, even in the macromolecular case. The clear implication is that the fusion of the traditional techniques of direct methods with anomalous dispersion, which is described here, will facilitate the solution of those crystal structures which contain one or more anomalous scatterers.

Section: Introductionmentioning

confidence: 91%

On integrating the techniques of direct methods with anomalous dispersion. I. The theoretical basis

Hauptman¹

1982

“…If H consists of three reflexions forming a triplet, the joint probability distribution ~SP(F*) will yield as an approximation when F* -~ 0 all the probabilistic formulae recently derived by Hauptman and others (Hauptman, 1982;Giacovazzo, 1983a;Cascarano & Giacovazzo, 1985;Giacovazzb, 1987;Klop, Krabbendam & Kroon, 1987) and cast in the form of inference rules by Karle (1983Karle ( , 1984Karle ( , 1985Karle ( , 1986, following earlier work by K.roon, Spek & Krabbendam (1977) and Heinerman, Krabbendam, Kroon & Spek (1978). These developments, however, suffer from certain limitations, which are overcome in the present multichannel approach.…”

Section: Application To Mir and Mwas Methodsmentioning

confidence: 99%

A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors

Bricogne¹

1988

150

In this first of three papers on a full Bayesian theory of crystal structure determination, it is shown that all currently used sources of phase information can be represented and combined through a universal expression for the joint probability distribution of structure factors. Particular attention is given to situations arising in macromolecular crystallography, where the proper treatment of non-uniform distributions of atoms is absolutely essential. A procedure is presented, in stages of gradually increasing complexity, for constructing the joint probability distribution of an arbitrary collection of structure factors. These structure factors may be gathered from one or several crystal forms of an unknown molecule, each comprising one or several isomorphous structures related by substitution operations, possibly containing solvent regions and known fragments, and/or obeying a set of non-crystallographic symmetries. This universal joint probability distribution can be effectively approximated by the saddlepoint method, using maximum-entropy distributions of atoms [Bricogne (1984). Acta Cryst. A40, 410-445] and a generalization of structure-factor algebra. Atomic scattering factors may assume arbitrary complex values, so that this formalism applies to neutron as well as to X-ray diffraction methods. This unified procedure will later be extended by the construction of conditional distributions allowing phase extension, and of likelihood functions capable of detecting and characterizing all potential sources of phase information considered so far, thus completing the formulation of a full Bayesian inference scheme for crystal structure determination.

“…~fter the initial algebraically oriented approach of Kroon, Spek & Krabbendam (1977), the probabilistic integration of DM with the techniques to solve protein structures was undertaken. Expressions for the SIR-NAS* case have been derived by Hauptman (1982a) and Giacovazzo, Cascarano & Zheng (1988), those for the single-wavelength anomalous-scattering (SAS) case by Hauptman (1982b) and Giacovazzo (1983).…”

Section: Introductionmentioning

confidence: 99%

On the doublet phase sums of isomorphous data sets

Kyriakidis¹,

Peschar²,

Schenk³

1993

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.