Estimation of r.m.s. error

Vijayan, M.; Ramaseshan, S.

doi:10.1107/97809553602060000557

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.4, pp. 273-274 | 1 | 2 |

Section 2.4.4.6. Estimation of r.m.s. error

M. Vijayan^a ^* and S. Ramaseshan^b ^‡

^a Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^bRaman Research Institute, Bangalore 560 080, India
Correspondence e-mail: mv@mbu.iisc.ernet.in

2.4.4.6. Estimation of r.m.s. error

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Perhaps the most important parameters that control the reliability of phase evaluation using the Blow and Crick formulation are the isomorphous r.m.s. error $[E_{i}]$ and the anomalous r.m.s. error $[E'_{i}]$ . For a given derivative, the sharpness of the peak in the phase probability distribution obviously depends upon the value of E and that of E′ when anomalous-scattering data have also been used. When several derivatives are used, an overall underestimation of r.m.s. errors leads to artifically sharper peaks, the movement of $[\alpha_{B}]$ towards $[\alpha_{M}]$ , and deceptively high figures of merit. Opposite effects result when E's are overestimated. Underestimation or overestimation of the r.m.s. error in the data from a particular derivative leads to distortions in the relative contribution of that derivative to the overall phase probability distributions. It is therefore important that the r.m.s. error in each derivative is correctly estimated.

Centric reflections, when present, obviously provide the best means for evaluating E using the expression $[E^{2} = {\textstyle\sum\limits_{n}} (|F_{NH} \pm F_{N}| - F_{N})^{2}/n. \eqno(2.4.4.25)]$ As suggested by Blow & Crick (1959), values of E thus estimated can be used for acentric reflections as well. Once a set of approximate protein phase angles is available, $[E_{i}]$ can be calculated as the r.m.s. lack of closure corresponding to $[\alpha_{B}]$ [i.e. $[\alpha = \alpha_{B}]$ in (2.4.4.20)] (Kartha, 1976). $[E'_{i}]$ can be similarly evaluated as the r.m.s. difference between the observed anomalous difference and the anomalous difference calculated for $[\alpha_{B}]$ [see (2.4.4.24)]. Normally, the value of $[E'_{i}]$ is about a third of that of $[E_{i}]$ (North, 1965).

A different method, outlined below, can also be used to evaluate E and E′ when anomalous scattering is present (Vijayan, 1981; Adams, 1968). From Fig. 2.4.2.2, we have $[\cos \psi = (F_{NH}^{2} + F_{H}^{2} - F_{N}^{2})/2F_{NH}F_{H} \eqno(2.4.4.26)]$ and $[F_{N}^{2} = F_{NH}^{2} + F_{H}^{2} - 2F_{NH}F_{H} \cos \psi, \eqno(2.4.4.27)]$ where $[\psi = \alpha_{NH} - \alpha_{H}]$ . Using arguments similar to those used in deriving (2.4.3.5), we obtain $[\sin \psi = [F_{NH}^{2}(+) - F_{NH}^{2}(-)]/4F_{NH}F''_{H}. \eqno(2.4.4.28)]$ If $[F_{NH}]$ is considered to be equal to $[[F_{NH}(+) + F_{NH}(-)]/2]$ , we obtain from (2.4.4.28) $[F_{NH}(+) - F_{NH}(-) = 2F''_{H} \sin \psi. \eqno(2.4.4.29)]$ We obtain what may be called $[\psi_{\rm iso}]$ if the magnitude of $[\psi]$ is determined from (2.4.4.26) and the quadrant from (2.4.4.28). Similarly, we obtain $[\psi_{\rm ano}]$ if the magnitude of $[\psi]$ is determined from (2.4.4.28) and the quadrant from (2.4.4.26). Ideally, $[\psi_{\rm iso}]$ and $[\psi_{\rm ano}]$ should have the same value and the difference between them is a measure of the errors in the data. $[F_{N}]$ obtained from (2.4.4.27) using $[\psi_{\rm ano}]$ may be considered as its calculated value $[(F_{N{\rm cal}})]$ . Then, assuming all errors to lie in $[F_{N}]$ , we may write $[E^{2} = {\textstyle\sum\limits_{n}} (F_{N} - F_{N{\rm cal}})^{2}/n. \eqno(2.4.4.30)]$ Similarly, the calculated anomalous difference $[(\Delta H_{\rm cal})]$ may be evaluated from (2.4.4.29) using $[\psi_{\rm iso}]$ . Then $[E'^{2} = {\textstyle\sum\limits_{n}} [|F_{NH}(+) - F_{NH}(-)| - \Delta H_{\rm cal}]^{2}/n. \eqno(2.4.4.31)]$ If all errors are assumed to reside in $[F_{H}]$ , E can be evaluated in yet another way using the expression $[E^{2} = {\textstyle\sum\limits_{n}} (F_{HLE} - F_{H})^{2}/n. \eqno(2.4.4.32)]$

References

Adams, M. J. (1968). DPhil thesis, Oxford University, England.Google Scholar

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

Kartha, G. (1976). Protein phase evaluation: multiple isomorphous series and anomalous scattering methods. In Crystallographic computing techniques, edited by F. R. Ahmed, pp. 269–281. Copenhagen: Munksgaard.Google Scholar

North, A. C. T. (1965). The combination of isomorphous replacement and anomalous scattering data in phase determination of non-centrosymmetric reflexions. Acta Cryst. 18, 212–216.Google Scholar

Vijayan, M. (1981). X-ray analysis of 2Zn insulin: some crystallographic problems. In Structural studies on molecules of biological interest, edited by G. Dodson, J. P. Glusker & D. Sayre, pp. 260–273. Oxford: Clarendon Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.4, pp. 273-274