Introduction

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 231 | 1 | 2 |

Section 2.2.10.1. Introduction

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.10.1. Introduction

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Protein structures cannot be solved ab initio by traditional direct methods (i.e., by application of the tangent formula alone). Accordingly, the first applications were focused on two tasks:

(a) improvement of the accuracy of the available phases (refinement process);
(b) extension of phases from lower to higher resolution (phase-extension process).

The application of standard tangent techniques to (a) and (b) has not been found to be very satisfactory (Coulter & Dewar, 1971; Hendrickson et al., 1973; Weinzierl et al., 1969). Tangent methods, in fact, require atomicity and non-negativity of the electron density. Both these properties are not satisfied if data do not extend to atomic resolution $[(d\gt 2\;\hbox{\AA})]$ . Because of series termination and other errors the electron-density map at $[d\gt 2\;\hbox{\AA}]$ presents large negative regions which will appear as false peaks in the squared structure. However, tangent methods use only a part of the information given by the Sayre equation (2.2.6.5). In fact, (2.2.6.5) express two equations relating the radial and angular parts of the two sides, so obtaining a large degree of overdetermination of the phases. To achieve this Sayre (1972) [see also Sayre & Toupin (1975)] suggested minimizing (2.2.10.1) by least squares as a function of the phases: $[\textstyle\sum\limits_{\bf h} \left|a_{\bf h} F_{\bf h} - \textstyle\sum\limits_{\bf k} F_{\bf k} F_{{\bf h}-{\bf k}}\right|^{2}. \eqno(2.2.10.1)]$ Even if tests on rubredoxin (extensions of phases from 2.5 to 1.5 Å resolution) and insulin (Cutfield et al., 1975) (from 1.9 to 1.5 Å resolution) were successful, the limitations of the method are its high cost and, especially, the higher efficiency of the least-squares method. Equivalent considerations hold for the application of determinantal methods to proteins [see Podjarny et al. (1981); de Rango et al. (1985) and literature cited therein].

A question now arises: why is the tangent formula unable to solve protein structures? Fan et al. (1991) considered the question from a first-principle approach and concluded that:

(1) the triplet phase probability distribution is very flat for proteins (N is very large) and close to the uniform distribution;
(2) low-resolution data create additional problems for direct methods since the number of available phase relationships per reflection is small.

Sheldrick (1990) suggested that direct methods are not expected to succeed if fewer than half of the reflections in the range 1.1–1.2 Å are observed with $[|F|\gt 4\sigma(|F|)]$ (a condition seldom satisfied by protein data).

The most complete analysis of the problem has been made by Giacovazzo, Guagliardi et al. (1994). They observed that the expected value of α (see Section 2.2.7) suggested by the tangent formula for proteins is comparable with the variance of the α parameter. In other words, for proteins the signal determining the phase is comparable with the noise, and therefore the phase indication is expected to be unreliable.

References

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Fan, H. F., Hao, Q. & Woolfson, M. M. (1991). Proteins and direct methods. Z. Kristallogr. 197, 197–208.Google Scholar

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 231