Finding heavy atoms with three-dimensional methods

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, p. 243 | 1 | 2 |

Section 2.3.3.3. Finding heavy atoms with three-dimensional methods

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

2.3.3.3. Finding heavy atoms with three-dimensional methods

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A Patterson of a native bio-macromolecular structure (coefficients $[F_{N}^{2}]$ ) can be considered as being, at least approximately, a vector map of all the light atoms (carbons, nitrogens, oxygens, some sulfurs, and also phosphorus for nucleic acids) other than hydrogen atoms. These interactions will be designated as LL. Similarly, a Patterson of the heavy-atom derivative will contain [HH + HL + LL] interactions, where H represents the heavy atoms. Thus, a true difference Patterson, with coefficients $[F_{NH}^{2} - F_{N}^{2}]$ , will contain only the interactions [HH + HL] . In general, the carpet of HL vectors completely dominates the HH vectors except for very small proteins such as insulin (Adams et al., 1969). Therefore, it would be preferable to compute a Patterson containing only HH interactions in order to interpret the map in terms of specific heavy-atom sites.

Blow (1958) and Rossmann (1960) showed that a Patterson with $[(|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2}]$ coefficients approximated to a Patterson containing only HH vectors. If the phase angle between $[{\bf F}_{N}]$ and $[{\bf F}_{NH}]$ is φ (Fig. 2.3.3.2), then $[|{\bf F}_{H}|^{2} = |{\bf F}_{N}|^{2} + |{\bf F}_{NH}|^{2} - 2|{\bf F}_{N}\|{\bf F}_{NH}| \cos \varphi.]$ In general, however, $[|{\bf F}_{H}| \ll |{\bf F}_{N}|]$ . Hence, φ is small and $[|{\bf F}_{H}|^{2} \simeq (|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2},]$ which is the same relation as (2.3.3.1) for centrosymmetric approximations. Since the direction of $[{\bf F}_{H}]$ is random compared to $[{\bf F}_{N}]$ , the root-mean-square projected length of $[{\bf F}_{H}]$ onto $[{\bf F}_{N}]$ will be $[{\bf F}_{H} / \sqrt{2}]$ . Thus it follows that a better approximation is $[|{\bf F}_{H}|^{2} \simeq \sqrt{2} (|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2}, \eqno(2.3.3.2)]$ which accounts for the assumption (Section 2.3.3.2) that $[\varepsilon_{3} = \sqrt{2} \varepsilon_{2}]$ . The almost universal method for the initial determination of major heavy-atom sites in an isomorphous derivative utilizes a Patterson with $[(|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2}]$ coefficients. Approximation (2.3.3.2) is also the basis for the refinement of heavy-atom parameters in a single isomorphous replacement pair (Rossmann, 1960; Cullis et al., 1962; Terwilliger & Eisenberg, 1983).

Figure 2.3.3.2 | top | pdf |

Vector triangle showing the relationship between $[{\bf F}_{N}]$ , $[{\bf F}_{NH}]$ and $[{\bf F}_{H}]$ , where $[{\bf F}_{NH} = {\bf F}_{N} + {\bf F}_{H}]$ .

References

Adams, M. J., Blundell, T. L., Dodson, E. J., Dodson, G. G., Vijayan, M., Baker, E. N., Harding, M. M., Hodgkin, D. C., Rimmer, B. & Sheat, S. (1969). Structure of rhombohedral 2 zinc insulin crystals. Nature (London), 224, 491–495.Google Scholar

Blow, D. M. (1958). The structure of haemoglobin. VII. Determination of phase angles in the noncentrosymmetric [100] zone. Proc. R. Soc. London Ser. A, 247, 302–336.Google Scholar

Cullis, A. F., Muirhead, H., Perutz, M. F., Rossmann, M. G. & North, A. C. T. (1962). The structure of haemoglobin. IX. A three-dimensional Fourier synthesis at 5.5 Å resolution: description of the structure. Proc. R. Soc. London Ser. A, 265, 161–187.Google Scholar

Rossmann, M. G. (1960). The accurate determination of the position and shape of heavy-atom replacement groups in proteins. Acta Cryst. 13, 221–226.Google Scholar

Terwilliger, T. C. & Eisenberg, D. (1983). Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions. Acta Cryst. A39, 813–817.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 243