Refinement

Kabsch, W.

doi:10.1107/97809553602060000676

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. 11.3, pp. 220-221 | 1 | 2 |

Section 11.3.2.8. Refinement

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

11.3.2.8. Refinement

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For a fixed detector, the diffraction pattern depends on the parameters $[{\bf S}_{0}, {\bf m}_{2}, {\bf b}_{1}, {\bf b}_{2}, {\bf b}_{3}, X_{0}, Y_{0}]$ and F. Starting values for the parameters can be obtained by the procedures described above that do not rely on prior knowledge of the crystal orientation, space-group symmetry or unit-cell metric. Better estimates of the parameter values, as required for the subsequent integration step, can be obtained by the method of least squares from the list of n observed indexed reflection centroids $[h_{i}]$ , $[k_{i}]$ , $[l_{i}]$ , $[X_{i}']$ , $[Y_{i}']$ , $[Z_{i}']$ $[(i = 1, \ldots, n)]$ . In this method, the parameters are chosen to minimize a weighted sum of squares of the residuals $[E = w_{X} {\textstyle\sum\limits_{i=1}^{n}} (\Delta_{X}^{i})^{2} + w_{Y} {\textstyle\sum\limits_{i=1}^{n}} (\Delta_{Y}^{i})^{2} + w_{Z} {\textstyle\sum\limits_{i=1}^{n}} (\Delta_{Z}^{i})^{2}.]$ The residuals between the calculated $[(X_{i}, Y_{i}, Z_{i})]$ and observed spot centroids are $[\eqalign{\Delta_{X}^{i} &=X_{i} - X_{i}' =X_{0} + F{\bf S}_{i} \cdot {\bf d}_{1}/{\bf S}_{i} \cdot {\bf d}_{3} - X_{i}'\cr \Delta_{Y}^{i} &=Y_{i} - Y_{i}' =Y_{0} + F{\bf S}_{i} \cdot {\bf d}_{2}/{\bf S}_{i} \cdot {\bf d}_{3} - Y_{i}'\cr \Delta_{Z}^{i} &=Z_{i} - Z_{i}' =\varphi_{0} + \Delta_{\varphi} {\textstyle\sum\limits_{j = -\infty}^{\infty}} (\;j - 1/2) R_{j}^{i} - Z_{i}'.}]$

Let $[s_{\mu}\ (\mu = 1, \ldots, k)]$ denote the k independent parameters for which initial estimates are available. Expanding the residuals to first order in the parameter changes $[\delta s_{\mu}]$ gives $[\Delta (s_{\mu} + \delta s_{\mu}) \approx \Delta (s_{\mu}) + \sum\limits_{\mu = 1}^{k} {\partial\Delta\over \partial s_{\mu}}\delta s_{\mu}.]$ The parameters should be changed in such a way as to minimize $[E(\delta s_{\mu})]$ , which implies $[\partial E/\partial \delta s_{\mu} = 0]$ for $[\mu = 1, \ldots, k]$ . The $[\delta s_{\mu}]$ are found as the solution of the k normal equations $[\eqalign{&\sum\limits_{\mu'= 1}^{k} \left(w_{X} \sum\limits_{i=1}^{n} {\partial\Delta_{X}^{i} \over \partial s_{\mu}} {\partial\Delta_{X}^{i} \over \partial s_{\mu'}} + w_{Y}\sum\limits_{i=1}^{n} {\partial\Delta_{Y}^{i} \over \partial s_{\mu}} {\partial\Delta_{Y}^{i} \over \partial s_{\mu'}} + w_{Z} \sum\limits_{i=1}^{n} {\partial\Delta_{Z}^{i} \over \partial s_{\mu}} {\partial\Delta_{Z}^{i} \over \partial s_{\mu'}}\right) \delta s_{\mu'}\cr &\quad = - \left(w_{X}\sum\limits\limits_{i=1}^{n} \Delta_{X}^{i} {\partial \Delta_{X}^{i} \over \partial s_{\mu}} + w_{Y}\sum\limits\limits_{i=1}^{n}\Delta_{Y}^{i} {\partial \Delta_{Y}^{i} \over \partial s_{\mu}} + w_{Z}\sum\limits\limits_{i=1}^{n} \Delta_{Z}^{i} {\partial \Delta_{Z}^{i} \over \partial s_{\mu}}\right).}]$ The parameters are corrected by $[\delta s_{\mu}]$ and a new cycle of refinement is started until a minimum of E is reached. The weights $[w_{X} = 1/{\textstyle\sum\limits_{i=1}^{n}}(\Delta_{X}^{i})^{2}, \quad w_{Y} = 1/{\textstyle\sum\limits_{i=1}^{n}} (\Delta_{Y}^{i})^{2}, \quad w_{Z} = 1/{\textstyle\sum\limits_{i=1}^{n}} (\Delta_{Z}^{i})^{2}]$ are calculated with the current guess for $[s_{\mu}]$ at the beginning of each cycle.

The derivatives appearing in the normal equations can be worked out from the definitions given in Sections 11.3.2.2 and 11.3.2.4, and only the form of the gradient of the Z residuals is shown. Assuming $[\sigma_{i} = \sigma_{M}/|\zeta_{i}|\ (i = 1,\ldots, n)]$ is constant for each reflection, the gradients of the Z residuals are obtained from the chain rule and the relation $[\hbox{d erf}(z)/\hbox{d}z = [2/(\pi)^{1/2}] \exp (-z^{2})]$ . $[\eqalign{{\partial \Delta_{Z}^{i} \over \partial s_{\mu}} &= {\partial \Delta_{Z}^{i} \over \partial \varphi_{i}} {\partial \varphi_{i} \over \partial s_{\mu}}\cr {\partial \Delta_{Z}^{i} \over \partial \varphi_{i}} &= {\Delta_{\varphi} \over (2\pi)^{1/2} \sigma_{i}} \sum\limits_{j=-\infty}^{\infty} \exp [-(\varphi_{0} + j\Delta_{\varphi} -\varphi_{i})^{2}/2\sigma_{i}^{2}]\cr {\partial \varphi_{i} \over \partial s_{\mu}} &= \cos \varphi_{i} {\partial \sin \varphi_{i} \over \partial s_{\mu}} - \sin \varphi_{i} {\partial \cos \varphi_{i} \over \partial s_{\mu}}.}]$ Obviously, $[\partial \Delta_{Z}^{i}/\partial s_{\mu}]$ is small for a fully recorded reflection because of the small values of all exponentials appearing in $[\partial \Delta_{Z}^{i}/\partial \varphi_{i}]$ . In contrast, the gradient for a partial reflection, equally recorded on two adjacent images, is most sensitive to parameter variations because one of the exponentials assumes its maximum value. In the limiting case of infinitely fine-sliced data, it can be shown that $[\lim_{\Delta_{\varphi \rightarrow 0}} \partial \Delta_{Z}^{i}/\partial \varphi_{i} = 1]$ . Thus, the refinement scheme based on observed Z centroids, as described here and implemented in XDS, is applicable to fine-sliced data – and to data recorded with a large oscillation range as well.

References

International Tables for Crystallography (2006). Vol. F. ch. 11.3, pp. 220-221