Evaluation of the profile-fitted intensity

Leslie, A. G. W.

doi:10.1107/97809553602060000675

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. 11.2, p. 215 | 1 | 2 |

Section 11.2.6.2. Evaluation of the profile-fitted intensity

A. G. W. Leslie^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: andrew@mrc-lmb.cam.ac.uk

11.2.6.2. Evaluation of the profile-fitted intensity

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Given an appropriate standard profile, the reflection intensity for fully recorded reflections is evaluated by determining the scale factor K and background plane constants a, b, c which minimize $[R_{3} = {\textstyle\sum} \ w_{i} \left(KP_{i} + ap_{i} + bq_{i} + c - \rho_{i} \right)^{2}, \eqno(11.2.6.5)]$ where the summation is over all valid pixels in the measurement box. As before, $[w_{i} = 1 / \sigma_{i}^{2} \eqno(11.2.6.6)]$ and $[\eqalignno{\sigma_{i}^{2} &= \hbox{expectation value of the counts at pixel } i\cr &= ap_{i} + bq_{i} + c + JP_{i}. &(11.2.6.7)}]$ In order to calculate the weights, the background plane constants and summation integration intensity $[I_{s}]$ are evaluated as described in Section 11.2.5, at the same time identifying any outliers in the background. The summation integration intensity is used to evaluate the scale factor J in equation (11.2.6.7) using $[I_{s} = J {\textstyle\sum\limits_{i}} P_{i}. \eqno(11.2.6.8)]$ In equation (11.2.6.5), the summation is over all valid pixels within the measurement box. This excludes pixels that are overlapped by neighbouring spots (if any) and any outliers identified in the background region.

Minimizing $[R_{3}]$ with respect to K, a, b and c leads to four linear equations from which K, a, b and c can be determined: $[{\left(\matrix{{\textstyle\sum}wP^{2} &{\textstyle\sum}wpP &{\textstyle\sum}wqP &{\textstyle\sum}wP\hfill\cr {\textstyle\sum}wpP &{\textstyle\sum}wp^{2} &{\textstyle\sum}wpq &{\textstyle\sum}wp\hfill\cr {\textstyle\sum}wqP &{\textstyle\sum}wpq &{\textstyle\sum}wq^{2} &{\textstyle\sum}wq\hfill\cr {\textstyle\sum}wP\hfill &{\textstyle\sum}wp\hfill &{\textstyle\sum}wq\hfill&{\textstyle\sum}w\hfill \cr}\right)\left(\matrix{K\hfill\cr a\hfill\cr b\hfill\cr c\hfill\cr}\right) = \left(\matrix{{\textstyle\sum}wP\rho\hfill\cr {\textstyle\sum}wp\rho\hfill\cr {\textstyle\sum}wq\rho\hfill\cr {\textstyle\sum}w\rho\hfill\cr}\right).}\eqno(11.2.6.9)]$ The profile-fitted intensity $[I_{p}]$ is then given by $[I_{p} = K {\textstyle\sum\limits_{i}} P_{i}. \eqno(11.2.6.10)]$ The standard deviation in the profile-fitted intensity is given by $[\eqalignno{\sigma_{I_{p}}^{2} &= \sigma_{K}^{2} \left({\textstyle\sum\limits_{i}} P_{i}\right)^{2} &(11.2.6.11)\cr &= \left({\textstyle\sum\limits_{i}^{N}} w_{i} \Delta_{i}^{2} \big/ (N - 4)\right)A_{KK}^{-1} \left({\textstyle\sum\limits_{i}} P_{i}\right)^{2}, &(11.2.6.12)}]$ where $[\Delta_{i} = (KP_{i} + ap_{i} + bq_{i} + c - \rho_{i}), \eqno(11.2.6.13)]$ N is the number of pixels in the summation and $[A_{KK}^{-1}]$ is the diagonal element for the scale factor K of the inverse normal matrix (used to minimize $[R_{3}]$ ).

In the case of partially recorded reflections, it is no longer valid to fit the sum of the scaled standard profile and a background plane to all pixels in the measurement box. Partially recorded reflections can have a profile that differs significantly from the standard profile, with the result that the background plane constants take on physically unreasonable values in an attempt to compensate for this difference. Therefore, for partially recorded reflections, the summation in equation (11.2.6.5) is restricted to pixels in the peak region of the measurement box. Minimizing $[R_{3}]$ with respect to the scale factor K then gives $[\openup 6pt\eqalignno{I_{p} &= K {\textstyle\sum\limits} P_{i} &(11.2.6.14)\cr &=({\textstyle\sum} w_{i}P_{i}\rho_{i} - a {\textstyle\sum} w_{i}P_{i}p_{i} - b{\textstyle\sum} w_{i}P_{i}q_{i} - c {\textstyle\sum} w_{i}P_{i}) {\textstyle\sum} P_{i} \big/ {\textstyle\sum} w_{i} P_{i}^{2}, &\cr &&(11.2.6.15)}]$ where all summations are over the peak region only.

It is not possible to derive a standard deviation for partially recorded reflections based on the fit of the scaled standard profile (because partially recorded reflections have a different spot profile). For these reflections, the standard deviation can be calculated using equation (11.2.5.17).

References

International Tables for Crystallography (2006). Vol. F. ch. 11.2, p. 215