Merging and scaling native and derivative data

Furey, W.

doi:10.1107/97809553602060000724

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. 25.2, pp. 696-697

Section 25.2.1.3. Merging and scaling native and derivative data

W. Furey^a ^*

25.2.1.3. Merging and scaling native and derivative data

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The programs CMBISO and CMBANO (both interactive) are used to combine unique native and derivative data sets into a single file and place the derivative data set on the scale of the native. All common reflections are identified, paired together, scaled and output to a single `scaled' file. With CMBISO, only mean structure-factor amplitudes are used for both native and derivative data, i.e. Bijvoet mates are deemed equivalent and averaged. CMBANO functions similarly, except that for the derivative data the individual Bijvoet mates are not averaged, and both values are output to the scaled file. The overall merging R factor is reported both on F and $[F^{2}]$ , along with tables indicating the R factor as a function of resolution, F magnitude and $[|F/\sigma (F)|]$ . A table is also output indicating the mean value of $[F_{PH} - F_{P}]$ as a function of resolution, where $[F_{PH}]$ and $[F_{P}]$ are the derivative and native structure-factor amplitudes, respectively. By default, scaling is initially carried out by the relative Wilson method (Wilson, 1949), with other optional procedures as outlined below to follow if desired.

25.2.1.3.1. Relative Wilson scaling

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With this method, the derivative scattering, on average, is made equal to the native scattering by plotting $[-\ln \left({\langle F_{PH}^{2}\rangle \over \langle F_{P}^{2}\rangle}\right) \ versus \ \left\langle {\sin^{2} (\theta) \over \lambda^{2}}\right\rangle, \eqno(25.2.1.1)]$ with the averages taken in corresponding resolution shells. A least-squares fit of a straight line to the plot yields a slope equal to $[2(B_{PH}-B_{P})]$ (twice the difference between overall isotropic temperature parameters for derivative and native data sets) and an intercept of ln $[K^{2}]$ . From these values, the derivative data are put on the scale of the native by multiplying each derivative amplitude by $[K \exp \left[(B_{PH} - B_{P}) {\sin^{2} (\theta) \over \lambda^{2}}\right]. \eqno(25.2.1.2)]$

25.2.1.3.2. Global anisotropic scaling

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With this option, applied after relative Wilson scaling, the unique parameters of a symmetric 3 × 3 scaling tensor S are determined by two cycles of least-squares minimization of $[\textstyle\sum\limits_{hkl}\displaystyle W_{hkl} (F_{P} - SF_{PH})^{2} \eqno(25.2.1.3)]$ with respect to S, where $[W_{hkl}]$ is a weighting factor, $[{S = S_{11} O_{x}^{2} + S_{22} O_{y}^{2} + S_{33} O_{z}^{2} + 2(S_{12} O_{x} O_{y} + S_{13} O_{x} O_{z} + S_{23} O_{y} O_{z})} \eqno(25.2.1.4)]$ and $[O_{x}]$ , $[O_{y}]$ , $[O_{z}]$ are direction cosines of the reciprocal-lattice vector expressed in an orthogonal system. The derivative data are then placed on the scale of the native by multiplying each derivative amplitude by the appropriate S.

25.2.1.3.3. Local scaling

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With this option, again applied after relative Wilson scaling, a scale factor for each reflection is also determined by minimizing equation (25.2.1.3) with respect to S, but here S is a scalar and the summation is taken only over neighbouring reflections within a sphere centred on the reflection being scaled. The sphere radius is initially set to include roughly 125 neighbours, and the scale factor is accepted if at least 80 are actually present. If insufficient neighbours are available, then the sphere size is increased incrementally and the process repeated until a preset maximum radius is encountered. If the maximum is reached, the process terminates with the message that the data set is too sparse for local scaling. Scaling is achieved by multiplying each derivative amplitude by the appropriate S.

25.2.1.3.4. Outlier rejection

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Rejection of outliers is often desirable, as erroneously large isomorphous or anomalous differences can lead to streaks in difference-Patterson maps and complicate identification of heavy-atom or anomalous-scatterer sites. The interactive program TOPDEL facilitates identification and rejection of such outliers while selecting reflections for use in difference-Patterson calculations. An input `scaled' file is read in, and user-supplied resolution and $[F/\sigma(F)]$ cutoffs are applied. The data are then sorted in descending order of magnitude of ΔF (either isomorphous or anomalous differences) and the largest differences are listed for examination. The user is then prompted to determine which, if any, of the large differences are to be rejected as outliers and to determine what percentage of the remaining largest differences are to be used in the Patterson-map synthesis. The appropriate Fourier-coefficient file is then created.

References

Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–321.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 25.2, pp. 696-697