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

Abstract

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.

Combining direct methods with isomorphous replacement or anomalous scattering data. VII.Ab initiophasing of one-wavelength anomalous scattering data from a small protein

“…Attempts have been made to resolve the phase ambiguity arising from the OAS technique by direct methods since the 1960's (Fan, 1965;Karle, 1966). More recently, Hauptman (1982), Giacovazzo (1983) and Karle (1984) have succeeded in deriving a large number of three-phase structure invariants * Supported in part by the National Natural Science Foundation of China.…”

Rules for estimating the values of triplet phase invariants in multiwavelength anomalous dispersion experiments

“…These include the many established procedures that have been used for years (Ramaseshan & Abrahams, 1975), recent developments in extending the range of applications (Hendrickson & Teeter, 1981) and new ones essentially untested with respect to experimental data that derive from an exact algebraic analysis (Karle, 1980), applications in probability theory (Heinerman, Krabbendam, Kroon & Spek, 1978;Hauptman, 1982;Giacovazzo, 1983) and analyses of single-wavelength experiments (Karle, 1984b). Optimal strategies for the use of the various techniques await future investigations.…”

TELS: least-squares solution of the structure-invariant equations

“…Three-phase structure invariants were generated and estimated assuming anomalous-scattering data for reflections in group one (Hauptman, 1982). Table 1 shows that the quality of the estimation is extremely dependent on the A value and the errors are unbiased about the true values.…”