Standard profiles

Kabsch, W.

doi:10.1107/97809553602060000676

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 222 | 1 | 2 |

Section 11.3.3.3. Standard profiles

W. Kabsch^a ^*

^a Max-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail: kabsch@mpimf-heidelberg.mpg.de

11.3.3.3. Standard profiles

| top | pdf |

Reflection profiles are represented on the Ewald sphere within a domain $[D_{0}]$ comprising $[2n_{1} + 1, 2n_{2} + 1, 2n_{3} + 1]$ equidistant gridpoints along $[{\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}]$ , respectively. The sampling distances between adjacent grid points are then $[\Delta_{1} = \delta_{D}/(2n_{1} + 1)]$ , $[\Delta_{2} = \delta_{D}/(2n_{2} + 1), \Delta_{3} = \delta_{M}/(2n_{3} + 1)]$ . Thus, grid coordinate $[\nu_{3}\ (\nu_{3} = -n_{3},\ldots, n_{3})]$ covers the set of rotation angles $[\Gamma_{\nu_{3}} = \{\varphi' | (\nu_{3} - 1/2) \Delta_{3} \leq (\varphi' - \varphi)\cdot \zeta \leq (\nu_{3} + 1/2)\Delta_{3}\}.]$ Contributions to the spot intensity come from one or several adjacent data images $[(\;j = j_{1},\ldots,\ j_{2})]$ , each covering the set of rotation angles $[\Gamma_{j} = \{\varphi' | \varphi_{0} + (j - 1) \Delta_{\varphi} \leq \varphi' \leq \varphi_{0} + j\Delta_{\varphi}\}.]$ Assuming Gaussian profiles along $[{\bf e}_{3}]$ for all reflections (see Section 11.3.2.3), the fraction of counts (after subtraction of the background) contributed by data frame j to grid coordinate $[\nu_{3}]$ is $[\eqalign{f_{\nu_{3}j}&\approx {\textstyle\int\limits_{\Gamma_{j} \cap \Gamma_{\nu_{3}}}} \exp [-(\varphi' - \varphi)^{2}/2\sigma^{2}]\ \hbox{d} \varphi' \cr &\quad\times\bigg\{{\textstyle\int\limits_{\Gamma_{j}}} \exp [-(\varphi' - \varphi)^{2}/2\sigma^{2}]\ \hbox{d} \varphi'\bigg\}^{-1},}]$ where $[\sigma = \sigma_{M}/|\zeta|]$ . The integrals can be expressed in terms of the error function, for which efficient numerical approximations are available (Abramowitz & Stegun, 1972). Finally, each pixel on data image j belonging to the reflection is subdivided into $[5 \times 5]$ areas of equal size, and $[f_{\nu_{3}j}/25]$ of the pixel signal is added to the profile value at grid coordinates $[\nu_{1},\nu_{2},\nu_{3}]$ corresponding to each subdivision.

This complicated procedure leads to more uniform intensity profiles for all reflections than using their untransformed shape. This simplifies the task of modelling the expected intensity distribution needed for integration by profile fitting. As implemented in XDS, reference profiles are learnt every 5° of crystal rotation at nine positions on the detector, each covering an equal area of the detector face. In the learning phase, profile boxes of the strong reflections are normalized and added to their nearest reference profile boxes. The contributions are weighted according to the distance from the location of the reference profile. Each grid point within the average profile boxes is classified as signal if it is above 2% of the peak maximum. Finally, each profile is scaled such that the sum of its signal pixels normalizes to one. The analytic expression $[\omega(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})]$ defined in Section 11.3.2.3 for the expected intensity distribution is only a rough initial approximation which is now replaced by the empirical reference profiles.

References

Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover Publications.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 222