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

International Tables for Crystallography (2006). Vol. F, ch. 11.4, p. 233   | 1 | 2 |

Section 11.4.8. Multiplicative corrections (scaling)

Z. Otwinowskia* and W. Minorb

aUT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390-9038, USA, and bDepartment of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA
Correspondence e-mail:  zbyszek@mix.swmed.edu

11.4.8. Multiplicative corrections (scaling)

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Proper error estimation requires the use of Bayesian reasoning and a multi-component error model (Schwarzenbach et al., 1989[link]; Evans, 1993[link]). In SCALEPACK, the estimated error of the measurement is enlarged by a fraction of the expected, rather than the observed, intensity. This algorithm reduces the bias towards reflections with an integrated intensity below the average.

The scaling model allows for a large number of diverse components to contribute to the multiplicative correction factor s for each observation, [s = \exp \left[{\textstyle\sum\limits_{i}}\ p_{i}\ f_{i} (hkl)\right], \eqno(11.4.8.1)] where [p_{i}] are a priori unknown coefficients of the scaling components and [f_{i}] represent different functional dependence of the scale factor for each observation. The simplest scaling model has a separate scale factor for each group (batch) of data, for example, one scale factor per image. In such a case, [f_{i} = \delta_{ij}, \eqno(11.4.8.2)] where j is the batch index for a particular reflection. For resolution-dependent decay, represented by one temperature factor per batch of data, [f_{i + n} = [({\bf S} \cdot {\bf S}) \big/ 2]\delta_{ij}, \eqno(11.4.8.3)] where n is the number of batches needed to make [p_{i}] represent the logarithm of the overall ith batch scale factor and [p_{i+n}] represents the temperature factor of batch i. S is the scattering vector for each reflection.

Coefficients of crystal absorption are much more complex. Scaling coefficients are associated with spherical harmonics (Katayama, 1986[link]; Blessing, 1995[link]) as a function of the direction of vector S, expressed in the coordinate system of the rotating crystal. Each spherical harmonic index lm has two (or, in the case of [m = 0], one) [p_{g}] coefficients. One of these spherical harmonic functions is given by [f_{g} = \left[{(2l + 1) \over 4\pi} {(l - m)! \over (l + m)!}\right]^{1/2} P_{lm} (\cos \Psi )\cos (m\Phi ), \eqno(11.4.8.4)] where Ψ and Φ are polar-coordinate angles of vector S in the crystal coordinate system, and [P_{lm}] is a Legendre polynomial (Press et al., 1989[link]). The other spherical harmonic of index lm has a sine instead of a cosine as the last term.

The multiplicative factor is applied to each observation and its σ to obtain the corrected intensity [I_{\rm corr}] and associated σ. The averaged intensity over all symmetry-related reflections [\langle I_{\rm corr}\rangle] is obtained by solving the two following equations: [W = 1 \Big/ [(\sigma E_{1})^{2} + (\langle I_{\rm corr}\rangle E_{2})^{2}], \eqno(11.4.8.5)] where [E_{1}] and [E_{2}] are the user-specified error scale factor and estimated error, respectively, and [\langle I_{\rm corr}\rangle = {\textstyle\sum} I_{\rm corr} W \Big/ {\textstyle\sum} W . \eqno(11.4.8.6)] Thus, [\sigma (I) = I \Big/[{\textstyle\sum} W]^{1/2}. \eqno(11.4.8.7)] During parameter refinement, the scale (and B, if requested) for all scaled batches are refined against the difference between the [\langle I_{\rm corr}\rangle]'s and [I_{\rm corr}]'s for individual measurements, summed over all reflections (Fox & Holmes, 1966[link]; Arndt & Wonacott, 1977[link]; Stuart & Walker, 1979[link]; Leslie & Tsukihara, 1980[link]; Rossmann & Erickson, 1983[link]; Walker & Stuart, 1983[link]; Rossmann, 1984[link]; Schutt & Evans, 1985[link]; Stuart, 1987[link]; Takusagawa, 1987[link]; Tanaka et al., 1990[link]). [\langle I_{\rm corr}\rangle]'s are recalculated in each cycle of refinement. There is full flexibility in the treatment of anomalous pairs. They can be assumed to be equivalent (or not) and they may be merged (or not). This approach allows the crystallographer to choose the best scaling and merging strategy.

11.4.8.1. Polarization

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A polarization correction may be applied during DENZO calculations. Sometimes the exact value of polarization is not known. This error may be corrected during the scaling procedure. This feature can be used to refine the polarization at synchrotron beamlines. Very high resolution data should be used for this purpose.

References

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Evans, P. (1993). Data reduction: data collection and processing. In Proceedings of the CCP4 study weekend. Data collection and processing, 29–30 January, edited by L. Sawyer, N. Isaac & S. Bailey, pp. 114–123. Warrington: Daresbury Laboratory.
Fox, G. C. & Holmes, K. C. (1966). An alternative method of solving the layer scaling equations of Hamilton, Rollett and Sparks. Acta Cryst. 20, 886–891.
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