Conversion of MAD data to a pseudo-SIRAS form

Terwilliger, T. C.; Berendzen, J.

doi:10.1107/97809553602060000686

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 14.2, pp. 304-305 | 1 | 2 |

Section 14.2.2.5. Conversion of MAD data to a pseudo-SIRAS form

T. C. Terwilliger^c ^* and J. Berendzen^d

14.2.2.5. Conversion of MAD data to a pseudo-SIRAS form

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The conversion of MAD data to a pseudo-SIRAS form that has almost the same information content requires two important assumptions. The first assumption is that the structure factor corresponding to anomalously scattering atoms in a structure varies in magnitude but not in phase at various X-ray wavelengths. This assumption will hold when there is one dominant type of anomalously scattering atom. The second is that the structure factor corresponding to anomalously scattering atoms is small compared to the structure factor from all other atoms. As long as these two assumptions hold, the information in a MAD experiment is largely contained in just three quantities: a structure factor ( $[F_{o}]$ ) corresponding to the scattering from non-anomalously scattering atoms, a dispersive or isomorphous difference at a standard wavelength $[\lambda_{o}]$ ( $[\Delta^{\rm ISO}_{\lambda_{o}}]$ ), and an anomalous difference ( $[\Delta^{\rm ANO}_{\lambda_{o}}]$ ) at the same standard wavelength (Terwilliger, 1994b). It is easy to see that these three quantities could be treated just like a SIRAS data set with the `native' structure factor $[F_{P}]$ replaced by $[F_{o}]$ , the derivative structure factor $[F_{PH}]$ replaced by $[F_{o} + \Delta^{\rm ISO}_{\lambda_{o}}]$ , and the anomalous difference replaced by $[\Delta^{\rm ANO}_{\lambda_{o}}]$ (Terwilliger, 1994b). This is the approach taken by Solve. In this section, it is briefly shown how these three quantities can be estimated from MAD data.

For a particular reflection and a particular wavelength $[\lambda_{j}]$ , we can write the total normal (i.e., non-anomalous) scattering from a structure ( $[{\bf F}_{{\rm tot},\,\lambda_{j}}]$ ) as the sum of two components. One is the scattering from all non-anomalously scattering atoms ( $[{\bf F}_{o}]$ ). This scattering is wavelength-independent. The second is the normal scattering from anomalously scattering atoms ( $[{\bf F}_{H_{\lambda_{j}}}]$ ) at wavelength $[\lambda_{j}]$ . This term includes wavelength-dependent dispersive shifts in atomic scattering due to the f′ term in the scattering factor, but not the anomalous part due to the f″ term. The magnitude of the total scattering factor can then be written in the form $[F_{{\rm tot},\, \lambda_{j}} = |{\bf F}_{o} + {\bf F}_{H_{\lambda_{j}}}|. \eqno(14.2.2.1)]$ Here $[{\bf F}_{o}]$ and $[{\bf F}_{{\rm tot},\,\lambda_{j}}]$ can be thought of corresponding, respectively, to the native structure factor, $[F_{P}]$ , and the derivative structure factor, $[F_{PH}]$ , as used in the method of isomorphous replacement (Blundell & Johnson, 1976). If the scattering from anomalously scattering atoms is small compared to that from all other atoms, equation (14.2.2.1) can be rewritten in the approximate form $[F_{{\rm tot},\, \lambda_{j}} \simeq F_{o} + F_{H_{\lambda_{j}}} \cos (\alpha), \eqno(14.2.2.2)]$ where α is the phase difference between the structure factors corresponding to non-anomalously and anomalously scattering atoms in the unit cell, $[{\bf F}_{o}]$ and $[{\bf F}_{H_{\lambda_{j}}}]$ , respectively, at this X-ray wavelength.

The data in a MAD experiment consist of observations of structure-factor amplitudes for Bijvoet pairs, $[F^{+}_{\lambda_{j}}]$ and $[F^{-}_{\lambda_{j}}]$ , for several X-ray wavelengths $[\lambda_{j}]$ . These can be rewritten in terms of an average structure-factor amplitude $[\overline{F}_{\lambda_{j}}]$ and an anomalous difference $[\Delta^{\rm ANO}_{\lambda_{j}}]$ (cf. Blundell & Johnson, 1976). We would like to convert these into estimates of the amplitude of the structure factor corresponding to the non-anomalously scattering atoms alone, the amplitude of the structure factor corresponding to the entire structure at a standard wavelength, and the anomalous difference at the standard wavelength.

The normal scattering due to anomalously scattering atoms ( $[{\bf F}_{H_{\lambda_{j}}}]$ ) changes in magnitude but not direction as a function of X-ray wavelength. We can therefore write (Terwilliger, 1994b) $[{\bf F}_{H_{\lambda_{\dot{j}}}} = {\bf F}_{H_{\lambda_{o}}} {f_{o} + f^{\prime}(\lambda_{{j}}) \over f_{o} + f^{\prime}(\lambda_{o})}, \eqno(14.2.2.3)]$ where $[\lambda_{o}]$ is an X-ray wavelength arbitrarily defined as a standard, and the real part of the scattering factor for the anomalously scattering atoms at wavelength $[\lambda_{o}]$ is $[f_{o} + f'(\lambda_{j})]$ . A corresponding approximation for the anomalous differences at various wavelengths can also be written (Terwilliger & Eisenberg, 1987) $[\Delta^{\rm ANO}_{\lambda_{j}} = \Delta^{\rm ANO}_{\lambda_{o}} {f''(\lambda_{j}) \over f''(\lambda_{o})}, \eqno(14.2.2.4)]$ where $[f''(\lambda_{j})]$ is the imaginary part of the scattering factor for the anomalously scattering atoms at wavelength $[\lambda_{j}]$ . Based on equation (14.2.2.4), anomalous differences at any wavelength can be estimated using measurements at the standard wavelength.

An estimate of the structure-factor amplitude ( $[F_{o}]$ ) corresponding to the scattering from non-anomalously scattering atoms and of the dispersive difference at standard wavelength $[\lambda_{o}]$ ( $[\Delta^{\rm ISO}_{\lambda_{o}}]$ ) can be obtained from average structure-factor amplitudes ( $[\overline{F}_{\lambda_{j}}]$ ) at any pair of wavelengths $[\lambda_{i}]$ and $[\lambda_{j}]$ by proceeding in two steps. Using equations (14.2.2.2) and (14.2.2.3), the component of $[{\bf F}_{H_{\lambda_{o}}}]$ along $[{\bf F}_{o}]$ , which we term $[\Delta^{\rm ISO}_{\lambda_{o}}]$ , can be estimated as $[\Delta^{\rm ISO}_{\lambda_{o}} \simeq F_{H_{\lambda_{o}}} \cos(\alpha) \eqno(14.2.2.5)]$ or $[\Delta^{\rm ISO}_{\lambda_{o}} \simeq (\overline{F}_{\lambda_{i}} - \overline{F}_{\lambda_{j}}) {f_{o} + f'(\lambda_{o}) \over f'(\lambda_{i}) - f'(\lambda_{j})}. \eqno(14.2.2.6)]$ Then, in turn, this estimate of $[\Delta^{\rm ISO}_{\lambda_{o}}]$ can be used to obtain $[F_{o}]$ : $[F_{o} \simeq \overline{F}_{\lambda_{j}} - \Delta^{\rm ISO}_{\lambda_{o}} {f_{o} + f'(\lambda_{j}) \over f_{o} + f'(\lambda_{o})}. \eqno(14.2.2.7)]$ This set of $[F_{o}]$ , $[F_{o} + \Delta^{\rm ISO}_{\lambda_{o}}]$ and $[\Delta^{\rm ANO}_{\lambda_{j}}]$ can then be used just as $[F_{P}]$ , $[F_{PH}]$ and $[\Delta^{\rm ANO}]$ are used in the SIRAS (single isomorphous replacement with anomalous scattering) method.

The algorithm described above is implemented in the program segment MADMRG as part of Solve (Terwilliger, 1994b). In most cases, there are more than one pair of X-ray wavelengths corresponding to a particular reflection. The estimates from each pair of wavelengths are averaged, using weighting factors based on the uncertainties in each estimate. Data from various pairs of X-ray wavelengths and from various Bijvoet pairs can have very different weights in their contributions to the total. This can be understood by noting that pairs of wavelengths that yield a large value of the denominator in equation (14.2.2.6) (i.e., those that differ considerably in dispersive contributions) would yield relatively accurate estimates of $[\Delta^{\rm ISO}_{\lambda_{o}}]$ . In the same way, Bijvoet differences measured at the wavelength with the largest value of f″ will contribute the most to estimates of $[\Delta^{\rm ANO}_{\lambda_{j}}]$ .

The standard wavelength choice in this analysis is arbitrary, because values at any wavelength can be converted to values at any other wavelength. The standard wavelength does not even have to be one of the wavelengths in the experiment, though it is convenient to choose one of them.

References

Blundell, T. L. & Johnson, L. N. (1976). Protein crystallography. p. 368. New York: Academic Press.Google Scholar

Terwilliger, T. C. (1994b). MAD phasing: treatment of dispersive differences as isomorphous replacement information. Acta Cryst. D50, 17–23.Google Scholar

Terwilliger, T. C. & Eisenberg, D. (1987). Isomorphous replacement: effects of errors on the phase probability distribution. Acta Cryst. A43, 6–13.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 14.2, pp. 304-305