Isomorphism and size of the heavy-atom substitution

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 245-246 | 1 | 2 |

Section 2.3.3.7. Isomorphism and size of the heavy-atom substitution

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.3.7. Isomorphism and size of the heavy-atom substitution

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It is insufficient to discuss Patterson techniques for locating heavy-atom substitutions without also considering errors of all kinds. First, it must be recognized that most heavy-atom labels are not a single atom but a small compound containing one or more heavy atoms. The compound itself will displace water or ions and locally alter the conformation of the protein or nucleic acid. Hence, a simple Gaussian approximation will suffice to represent individual heavy-atom scatterers responsible for the difference between native and heavy-atom derivatives. Furthermore, the heavy-atom compound often introduces small global structural changes which can be detected only at higher resolution. These problems were considered with some rigour by Crick & Magdoff (1956). In general, lack of isomorphism is exhibited by an increase in the size of the isomorphous differences with increasing resolution (Fig. 2.3.3.6).

Figure 2.3.3.6 | top | pdf |

A plot of mean isomorphous differences as a function of resolution. (a) The theoretical size of mean differences following roughly a Gaussian distribution. (b) The observed size of differences for a good isomorphous derivative where the smaller higher-order differences have been largely masked by the error of measurement. (c) Observed differences where `lack of isomorphism' dominates beyond approximately 5 Å resolution.

Crick & Magdoff (1956) also derived the approximate expression $[\sqrt{{2N_{H} \over N_{P}}} \cdot {f_{H} \over f_{P}}]$ to estimate the r.m.s. fractional change in intensity as a function of heavy-atom substitution. Here, $[N_{H}]$ represents the number of heavy atoms attached to a protein (or other large molecule) which contains $[N_{P}]$ light atoms. $[f_{H}]$ and $[f_{P}]$ are the scattering powers of the average heavy and protein atom, respectively. This function was tabulated by Eisenberg (1970) as a function of molecular weight (proportional to $[N_{P}]$ ). For instance, for a single, fully substituted, Hg atom the formula predicts an r.m.s. intensity change of around 25% in a molecule of 100 000 Da. However, the error of measurement of a reflection intensity is likely to be arround 10% of I, implying perhaps an error of around 14% of I on a difference measurement. Thus, the isomorphous replacement difference measurement for almost half the reflections will be buried in error for this case.

Scaling of the different heavy-atom-derivative data sets onto a common relative scale is clearly important if error is to be reduced. Blundell & Johnson (1976, pp. 333–336) give a careful discussion of this subject. Suffice it to say here only that a linear scale factor is seldom acceptable as the heavy-atom-derivative crystals frequently suffer from greater disorder than the native crystals. The heavy-atom derivative should, in general, have a slightly larger mean value for the structure factors on account of the additional heavy atoms (Green et al., 1954). The usual effect is to make $[{\textstyle\sum} |{\bf F}_{NH}|^{2}/{\textstyle\sum} |{\bf F}_{N}|^{2} \simeq 1.05]$ (Phillips, 1966).

As the amount of heavy atom is usually unknown in a yet unsolved heavy-atom derivative, it is usual practice either to apply a scale factor of the form $[k \exp [- B(\sin \theta/\lambda)^{2}]]$ or, more generally, to use local scaling (Matthews & Czerwinski, 1975). The latter has the advantage of not making any assumption about the physical nature of the relative intensity decay with resolution.

References

Blundell, T. L. & Johnson, L. N. (1976). Protein crystallography. New York: Academic Press.Google Scholar

Crick, F. H. C. & Magdoff, B. S. (1956). The theory of the method of isomorphous replacement for protein crystals. I. Acta Cryst. 9, 901–908.Google Scholar

Eisenberg, D. (1970). X-ray crystallography and enzyme structure. In The enzymes, edited by P. D. Boyer, Vol. I, 3rd ed., pp. 1–89. New York: Academic Press.Google Scholar

Green, D. W., Ingram, V. M. & Perutz, M. F. (1954). The structure of haemoglobin. IV. Sign determination by the isomorphous replacement method. Proc. R. Soc. London Ser. A, 225, 287–307.Google Scholar

Matthews, B. W. & Czerwinski, E. W. (1975). Local scaling: a method to reduce systematic errors in isomorphous replacement and anomalous scattering measurements. Acta Cryst. A31, 480–487.Google Scholar

Phillips, D. C. (1966). Advances in protein crystallography. In Advances in structure research by diffraction methods, Vol. 2, edited by R. Brill & R. Mason, pp. 75–140. New York: John Wiley.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 245-246