International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 265
Section 2.4.2.2. Single isomorphous replacement method^{a}Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^{b}Raman Research Institute, Bangalore 560 080, India |
The number of replaceable (or `added') atoms is usually small and they generally have high atomic numbers. Their positions are often determined by a Patterson synthesis of one type or another (see Chapter 2.3 ). It will therefore be assumed in the following discussion that is known. Then it can be readily seen by referring to Fig. 2.4.2.2 that when is derived from its cosine function, it could obviously be positive or negative. Hence, there are two possible solutions for . These two solutions are distributed symmetrically about . One of these would correspond to the correct value of . Therefore, in general, the phase angle cannot be unambiguously determined using a pair of isomorphous crystals.
The twofold ambiguity in phase angle vanishes when the structures are centrosymmetric. and are all real in centric data and the corresponding phase angles are 0 or . From (2.4.2.4) The sign of is already known and the signs of and can be readily determined from (2.4.2.6) (Robertson & Woodward, 1937).
When the data are acentric, the best one can do is to use both the possible phase angles simultaneously in a Fourier synthesis (Bokhoven et al., 1951). This double-phased synthesis, which is equivalent to the isomorphous synthesis of Ramachandran & Raman (1959), contains the structure and its inverse when the replaceable atoms have a centrosymmetric distribution (Ramachandran & Srinivasan, 1970). When the distribution is noncentrosymmetric, however, the synthesis contains peaks corresponding to the structure and general background. Fourier syntheses computed using the single isomorphous replacement method of Blow & Rossmann (1961) and Kartha (1961) have the same properties. In this method, the phase angle is taken to be the average of the two possible solutions of , which is always or . Also, the Fourier coefficients are multiplied by cos , following arguments based on the Blow & Crick (1959) formulation of phase evaluation (see Section 2.4.4.4). Although Blow & Rossmann (1961) have shown that this method could yield interpretable protein Fourier maps, it is rarely used as such in protein crystallography as the Fourier maps computed using it usually have unacceptable background levels (Blundell & Johnson, 1976).
References
Blow, D. M. & Crick, F. H. C. (1959). The treatment of errors in the isomorphous replacement method. Acta Cryst. 12, 794–802.Google ScholarBlow, D. M. & Rossmann, M. G. (1961). The single isomorphous replacement method. Acta Cryst. 14, 1195–1202.Google Scholar
Blundell, T. L. & Johnson, L. N. (1976). Protein crystallography. London: Academic Press.Google Scholar
Bokhoven, C., Schoone, J. C. & Bijvoet, J. M. (1951). The Fourier synthesis of the crystal structure of strychnine sulphate pentahydrate. Acta Cryst. 4, 275–280.Google Scholar
Kartha, G. (1961). Isomorphous replacement in non-centrosymmetric structures. Acta Cryst. 14, 680–686.Google Scholar
Ramachandran, G. N. & Raman, S. (1959). Syntheses for the deconvolution of the Patterson function. Part I. General principles. Acta Cryst. 12, 957–964.Google Scholar
Ramachandran, G. N. & Srinivasan, R. (1970). Fourier methods in crystallography. New York: Wiley–Interscience.Google Scholar
Robertson, J. M. & Woodward, I. (1937). An X-ray study of the phthalocyanines. Part III. Quantitative structure determination of nickel phthalocyanine. J. Chem. Soc. pp. 219–230.Google Scholar