Treatment of errors

Stubbs, M. T.; Huber, R.

doi:10.1107/97809553602060000680

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. 12.2, pp. 258-259 | 1 | 2 |

Section 12.2.4.1. Treatment of errors

M. T. Stubbs^a ^* and R. Huber^b

^a Institut für Pharmazeutische Chemie der Philipps-Universität Marburg, Marbacher Weg 6, D-35032 Marburg, Germany, and ^bMax-Planck-Institut für Biochemie, 82152 Martinsried, Germany
Correspondence e-mail: stubbs@mailer.uni-marburg.de

12.2.4.1. Treatment of errors

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Until now, we have dealt with cases involving perfect data. Although this ideal may now be attainable using MAD techniques, this is not necessarily the usual laboratory situation. In the first place, it is necessary to scale the derivative data $[F_{PH}]$ to the native $[F_{P}]$ . One of the most common scaling procedures is based on the expected statistical dependence of intensity on resolution (Wilson, 1949). This may not be particularly accurate when only low-resolution data are available, in which case a scaling through equating the Patterson origin peaks of native and derivative sets may provide better results (Rogers, 1965).

A model to account for errors in the data, determination of heavy-atom positions etc. was proposed by Blow & Crick (1959), in which all errors are associated with $[|F_{PH}|_{\rm obs}]$ (Fig. 12.2.4.1); a more detailed treatment has been provided by Terwilliger & Eisenberg (1987). Owing to errors, the triangle formed by $[F_{P}]$ , $[F_{PH}]$ and $[F_{H}]$ fails to close. The lack of closure error ɛ is a function of the calculated phase angle $[\varphi_{P}]$ : $[\varepsilon (\varphi_{P}) = |F_{PH}|_{\rm obs} - |F_{PH}|_{\rm calc}.]$ Once an initial set of heavy-atom positions has been found, it is necessary to refine their parameters (x, y, z, occupancy and thermal parameters). This can be achieved through the minimization of $[{\textstyle\sum\limits_{\bf S}} \varepsilon^{2} / E,]$ where E is the estimated error $[(\simeq \langle (|F_{PH}|_{\rm obs} - |F_{PH}|_{\rm calc})^{2}\rangle)]$ (Rossmann, 1960; Terwilliger & Eisenberg, 1983). This procedure is safest for noncentrosymmetric reflections (φ restricted to 0 or π) if enough are present. Phase refinement is generally monitored by three factors: $[R_{\rm Cullis} = {\textstyle\sum} \| F_{PH} + F_{P}| - |F_{H}|_{\rm calc}|\big/ {\textstyle\sum} |F_{PH} - F_{P}|]$ for noncentrosymmetric reflections only; acceptable values are between 0.4 and 0.6; $[R_{\rm Kraut} = {\textstyle\sum} \|F_{PH}|_{\rm obs} - |F_{PH}|_{\rm calc}| \big/ {\textstyle\sum} |F_{PH}|_{\rm obs},]$ which is useful for monitoring convergence; and the $[\hbox{phasing power} = {\textstyle\sum} |F_{H}|_{\rm calc} /{\textstyle\sum} \|F_{PH}|_{\rm obs} - |F_{PH}|_{\rm calc}|,]$ which should be greater than 1 (if less than 1, then the phase triangle cannot be closed via $[F_{H}]$ ).

Figure 12.2.4.1| top | pdf |

The treatment of phase errors. The calculated heavy-atom structure results in a calculated value for both the phase and magnitude of $[F_{H}]$ (red). According to the value of $[\varphi_{P}]$ , the triangle $[F_{P}]$ – $[F_{H}]$ – $[F_{PH}]$ will fail to close by an amount ɛ, the lack of closure (green). This gives rise to a phase distribution which is bimodal for a single derivative. The combined probability from a series of derivatives has a most probable phase (the maximum) and a best phase (the centroid of the distribution), for which the overall phase error is minimum.

The resulting phase probability is given by $[P (\varphi_{P}) = \exp \{- \varepsilon^{2} (\varphi_{P}) / 2E^{2}\}.]$ The phases have a minimum error when the best phase $[\varphi_{\rm best}]$ , i.e. the centroid of the phase distribution, $[\varphi_{\rm best} = {\textstyle\int} \varphi_{P} P(\varphi_{P})\ \hbox{d}\varphi_{P},]$ is used instead of the most probable phase. The quality of the phases is indicated by the figure of merit m, where $[m = {\textstyle\int} P(\varphi_{P}) \exp (i\varphi_{P})\ \hbox{d}\varphi_{P} \big/ {\textstyle\int} P(\varphi_{P})\ \hbox{d}\varphi_{P}.]$ A value of 1 for m indicates no phase error, a value of 0.5 represents a phase error of about 60°, while a value of 0 means that all phases are equally probable.

The best Fourier is calculated from $[\rho_{\rm best}({\bf r}) = (1/V) {\textstyle\sum\limits_{\bf S}} m|F_{P}({\bf S})| \exp \{i\varphi_{P{\rm best}}({\bf S})\},]$ where the electron density should have minimal errors.

References

Blow, D. M. & Crick, F. H. C. (1959). The treatment of errors in the isomorphous replacement method. Acta Cryst. 12, 794–802.Google Scholar

Rogers, D. (1965). In Computing methods in crystallography, edited by J. S. Rollett, pp. 133–148. Oxford University Press.Google Scholar

Rossmann, M. G. (1960). The accurate determination of the position and shape of heavy-atom replacement groups in proteins. Acta Cryst. 13, 221–226.Google Scholar

Terwilliger, T. C. & Eisenberg, D. (1983). Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions. Acta Cryst. A39, 813–817.Google Scholar

Terwilliger, T. C. & Eisenberg, D. (1987). Isomorphous replacement: effects of errors on the phase probability distribution. Acta Cryst. A43, 6–13.Google Scholar

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. 12.2, pp. 258-259