Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources

Vijayan, M.; Ramaseshan, S.

doi:10.1107/97809553602060000557

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 274 | 1 | 2 |

Section 2.4.4.7. Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources

M. Vijayan^a ^* and S. Ramaseshan^b ^‡

^a Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^bRaman Research Institute, Bangalore 560 080, India
Correspondence e-mail: mv@mbu.iisc.ernet.in

2.4.4.7. Suggested modifications to Blow and Crick formulation and the inclusion of phase information from other sources

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Modifications to the Blow and Crick procedure of phase evaluation have been suggested by several workers, although none represent a fundamental departure from the essential features of their formulation. In one of the modifications (Cullis et al., 1961a; Ashida, 1976), all $[E_{i}]$ 's are assumed to be the same, but the lack-of-closure error $[\xi_{Hi}]$ for the ith derivative is measured as the distance from the mean of all intersections between phase circles to the point of intersection of the phase circle of that derivative with the phase circle of the native protein. Alternatively, individual values of $[E_{i}]$ are retained, but the lack of closure is measured from the weighted mean of all intersections (Ashida, 1976). This is obviously designed to undo the effects of the unduly high weight given to $[F_{N}]$ in the Blow and Crick formulation. In another modification (Raiz & Andreeva, 1970; Einstein, 1977), suggested for the same purpose, the $[F_{N}]$ and $[F_{NHi}]$ circles are treated as circular bands, the width of each band being related to the error in the appropriate structure factor. A comprehensive set of modifications suggested by Green (1979) treats different types of errors separately. In particular, errors arising from imperfect isomorphism are treated in a comprehensive manner.

Although the isomorphous replacement method still remains the method of choice for the ab initio determination of protein structures, additional items of phase information from other sources are increasingly being used to replace, supplement, or extend the information obtained through the application of the isomorphous replacement. Methods have been developed for the routine refinement of protein structures (Watenpaugh et al., 1973; Huber et al., 1974; Sussman et al., 1977; Jack & Levitt, 1978; Isaacs & Agarwal, 1978; Hendrickson & Konnert, 1980) and they provide a rich source of phase information. However, the nature of the problem and the inherent limitations of the Fourier technique are such that the possibility of refinement yielding misleading results exists (Vijayan, 1980a,b). It is therefore sometimes desirable to combine the phases obtained during refinement with the original isomorphous replacement phases. The other sources of phase information include molecular replacement (see Chapter 2.3 ), direct methods (Hendrickson & Karle, 1973; Sayre, 1974; de Rango et al., 1975; see also Chapter 2.2 ) and different types of electron-density modifications (Hoppe & Gassmann, 1968; Collins, 1975; Schevitz et al., 1981; Bhat & Blow, 1982; Agard & Stroud, 1982; Cannillo et al., 1983; Raghavan & Tulinsky, 1979; Wang, 1985).

The problem of combining isomorphous replacement phases with those obtained by other methods was first addressed by Rossmann & Blow (1961). The problem was subsequently examined by Hendrickson & Lattman (1970) and their method, which involves a modification of the Blow and Crick formulation, is perhaps the most widely used for combining phase information from different sources.

The Blow and Crick procedure is based on an assumed Gaussian `lumped' error in $[F_{NHi}]$ which leads to a lack of closure, $[\xi_{Hi}(\alpha)]$ , in $[F_{NHi}]$ defined by (2.4.4.20). Hendrickson and Lattman make an equally legitimate assumption that the lumped error, again assumed to be Gaussian, is associated with $[F_{NHi}^{2}]$ . Then, as in (2.4.4.20), we have $[\xi''_{Hi}(\alpha) = F_{NHi}^{2} - D_{Hi}^{2}(\alpha), \eqno(2.4.4.33)]$ where $[\xi''_{Hi}(\alpha)]$ is the lack of closure associated with $[F_{NHi}^{2}]$ for an assumed protein phase angle $[\alpha]$ . Then the probability for $[\alpha]$ being the correct phase angle can be expressed as $[\hfil\qquad P_{i} (\alpha) = N_{i} \exp [-\xi''^{2}_{Hi} (\alpha)/2E''^{2}_{i}],\eqno(2.4.4.34)]$ where $[E''_{i}]$ is the r.m.s. error in $[F^{2}_{NHi}]$ , which can be evaluated using methods similar to those employed for evaluating $[E_{i}]$ . Hendrickson and Lattman have shown that the exponent in the probability expression (2.4.4.34) can be readily expressed as a linear combination of five terms in the following manner. $[\eqalignno{ -\xi''^{2}_{Hi} (\alpha)/2E''^{2}_{i} &= K_{i} + A_{i} \cos \alpha + B_{i} \sin \alpha + C_{i} \cos 2\alpha\cr &\quad + D_{i} \sin 2\alpha, &(2.4.4.35)}]$ where $[K_{i}, A_{i}, B_{i}, C_{i}]$ and $[D_{i}]$ are constants dependent on $[F_{N}, F_{Hi}, F_{NHi}]$ and $[E''_{i}]$ . Thus, five constants are enough to store the complete probability distribution of any reflection. Expressions for the five constants have been derived for phase information from anomalous scattering, tangent formula, partial structure and molecular replacement. The combination of the phase information from all sources can then be achieved by simply taking the total value of each constant. Thus, the total probability of the protein phase angle being $[\alpha]$ is given by $[\eqalignno{ P(\alpha) = \textstyle\prod P_{s}(\alpha) &= N \exp \left({\textstyle\sum\limits_{s}} K_{s} + {\textstyle\sum\limits_{s}} A_{s} \cos \alpha + {\textstyle\sum\limits_{s}} B_{s} \sin \alpha \right.\cr &\quad \left. + {\textstyle\sum\limits_{s}} C_{s} \cos 2\alpha + {\textstyle\sum\limits_{s}} D_{s} \sin 2\alpha\right),\cr & &(2.4.4.36)}]$ where $[K_{s}, A_{s}]$ etc. are the constants appropriate for the sth source and N is the normalization constant.

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International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 274