International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.4, pp. 273-274
Section 2.4.4.6. Estimation of r.m.s. error^{a}Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^{b}Raman Research Institute, Bangalore 560 080, India |
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 and the anomalous r.m.s. error . 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 towards , 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 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, can be calculated as the r.m.s. lack of closure corresponding to [i.e. in (2.4.4.20)] (Kartha, 1976). can be similarly evaluated as the r.m.s. difference between the observed anomalous difference and the anomalous difference calculated for [see (2.4.4.24)]. Normally, the value of is about a third of that of (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 and where . Using arguments similar to those used in deriving (2.4.3.5), we obtain If is considered to be equal to , we obtain from (2.4.4.28) We obtain what may be called if the magnitude of is determined from (2.4.4.26) and the quadrant from (2.4.4.28). Similarly, we obtain if the magnitude of is determined from (2.4.4.28) and the quadrant from (2.4.4.26). Ideally, and should have the same value and the difference between them is a measure of the errors in the data. obtained from (2.4.4.27) using may be considered as its calculated value . Then, assuming all errors to lie in , we may write Similarly, the calculated anomalous difference may be evaluated from (2.4.4.29) using . Then If all errors are assumed to reside in , E can be evaluated in yet another way using the expression
References
Adams, M. J. (1968). DPhil thesis, Oxford University, England.Google ScholarBlow, 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