A translation function combining packing and diffraction information: an application to lysozyme (high-temperature form)

Abstract

A new translation function T(t) is defined, whose main peak is expected to give directly the absolute position t = t o of a known molecular configuration (the isostructural model) of known orientation in an unknown crystal structure. T(t) combines all required information: crystal symmetry, molecular steric properties of the model and experimental data (observed structure factors F o) leading to more reliable results: T(t) = TO(t)/O(t), where TO(t) measures the degree of agreement between F o and Fc(0, while O(t) is a function measuring intermolecular overlap. TO and O can both be computed efficiently by the possibility of their expansion in Fourier series, and the use of FFT. A theoretical study followed by application to a simple synthetic structure cast light upon ambiguities and failures [e.g. Bott & Sarma (1976). J. Mol. Biol. 106, 1037-1046 encountered in using translation functions previously described [e.g. Crowther & Blow (1967). Acta Cryst. 23,[544][545][546][547][548]. The crystal structure of the high-temperature form of hen-egg-white lysozyme was determined at 6 A resolution by the application of this fast translation function and compared with the equivalent structure obtained by the isomorphous replacement method.

Experiences with a new translation-function program

“…The calculation of this function is time consuming as it is not in a form suitable for the use of FFT and the correlation coeffÉcient for each trial position must be evaluated separately. Using Parseval's theorem (Bracewell, 1965), one can show that C' defined by Harada et al (1981) is equal to the corresponding quantity for Patterson functions,…”

“…Crowther & Blow (1967) R factor Harada et al (1981) Tonin 1 st 4-6 5" 1 4-0 3.6 6.3 2nd 3"8 3-4 3"5 2" 1 3"8 S/N 1"2 1"5 1-1 1"7 1"7 Pepsinogen 1st 5-2 3"8 5.3 2nd 3"9 2.4 2.5 S/N 1.3 1.6 2"1 SGT Orientation 1 t I st 3"8* 3" 1 * 1.5" 1.5" 2-9* 2nd 3.7 3"1 1"5 1"5 2.5 S/N 1.0…”

“…We were inspired by the work of Harada et al (1981) to use the correlation coefficient for the solution of the translation problem. We chose to work with the standard linear correlation coefficient defined as…”

“…There are numerous translation functions in the literature (Tollin, 1966;Crowther & Blow, 1967;Hendrickson & Ward, 1976;Harada, Lifchitz, Berthou & Jolles, 1981;Langs, 1985). The function of Crowlher & Blow (1967) is often used.…”

New probabilistic formulas for finding the positions of correctly oriented atomic groups

“…The position of the fragment with respect to the symmetry elements of the space group is usually achieved by: (a) translation functions in the vector space (Braun, Hornstra & Leenhouts, 1969;Huber, 1965;Nordman & Nakatsu, 1963) or in the reciprocal space (Tollin, 1966;Crowther & Blow, 1967;Karle, 1972;Langs, 1975;Harada, Lifchitz, Berthou & Jolles, 1981); or (b) special direct-methods procedures. Among these it is worth noting: (bl) a modified tangent formula (Karle, 1968) is used to recycle in P1 phases derived from the known fragment; (b2) reflection data are expanded in the space group P1 (Doesburg & Beurskens, 1983;Bruins Slot & Beurskens, 1984) and coefficients for Fourier synthesis are obtained by direct methods on difference structure factors.…”