Map coefficients to reduce model bias

Read, R. J.

doi:10.1107/97809553602060000688

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. 15.2, pp. 327-328 | 1 | 2 |

Section 15.2.5. Map coefficients to reduce model bias

R. J. Read^a ^*

^a Department of Haematology, University of Cambridge, Wellcome Trust Centre for Molecular Mechanisms in Disease, CIMR, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 2XY, England
Correspondence e-mail: rjr27@cam.ac.uk

15.2.5. Map coefficients to reduce model bias

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15.2.5.1. Model bias in figure-of-merit weighted maps

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A figure-of-merit weighted map, calculated with coefficients $[m|{\bf F}_{O}|\exp(i\alpha_{C})]$ , has the least r.m.s. error from the true map. According to the normal statistical (minimum variance) criteria, then, it is the best map. However, such a map will suffer from model bias; if its purpose is to allow the detection and repair of errors in the model, this is a serious qualitative defect. Fortunately, it is possible to predict the systematic errors leading to model bias and to make some correction for them.

Main (1979) dealt with this problem in the case of a perfect partial structure. Since the relationships among structure factors are the same in the general case of a partial structure with various errors, once $[D{\bf F}_{C}]$ is substituted for $[{\bf F}_{C}]$ , all that is required to apply Main's results more generally is a change of variables (Read, 1986, 1990).

In Main's approach, the cosine law is used to introduce the cosine of the phase error, which is converted into a figure of merit by taking expected values. Some manipulations allow us to solve for the figure-of-merit weighted map coefficient, which is approximated as a linear combination of the true structure factor and the model structure factor (Main, 1979; Read, 1986). Finally, we can solve for an approximation to the true structure factor, giving map coefficients from which the systematic model bias component has been removed. $[\eqalignno{&m|{\bf F}_{O}|\exp(i\alpha_{C}) = F/2 + D{\bf F}_{C}/2 + \hbox{ noise terms},\cr &F \simeq (2m|{\bf F}_{O}| - D|{\bf F}_{C}|)\exp(i\alpha_{C}).\cr}]$

A similar analysis for centric structure factors shows that there is no systematic model bias in figure-of-merit weighted map coefficients, so no bias correction is needed in the centric case.

15.2.5.2. Model bias in combined phase maps

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When model phase information is combined with, for instance, multiple isomorphous replacement (MIR) phase information, there will still be model bias in the acentric map coefficients, to the extent that the model influences the final phases. However, it is inappropriate to continue using the same map coefficients to reduce model bias, because some phases could be determined almost completely by the MIR phase information. It makes much more sense to have map coefficients that reduce to the coefficients appropriate for either model or MIR phases, in extreme cases where there is only one source of phase information, and that vary smoothly between those extremes.

Map coefficients that satisfy these criteria (even if they are not rigorously derived) are implemented in the program SIGMAA. The resulting maps are reasonably successful in reducing model bias. Two assumptions are made: (1) the model bias component in the figure-of-merit weighted map coefficient, $[m_{\rm com}|{\bf F}_{O}|\exp(i\alpha_{\rm com})]$ , is proportional to the influence that the model phase has had on the combined phase; and (2) the relative influence of a source of phase information can be measured by the information content, H (Guiasu, 1977), of the phase probability distribution. The first assumption corresponds to the idea that the figure-of-merit weighted map coefficient is a linear combination of the MIR and model phase cases. $[\!\matrix{\hbox{MIR:} \hfill& m_{\rm MIR} | {\bf F}_{O} | \exp (i\alpha_{\rm MIR}) \hfill& \simeq {\bf F} \hfill\cr \hbox{Model:} \hfill& m_{C} | {\bf F}_{O} | \exp (i\alpha_{C}) \hfill& \simeq {\bf F}/2 + D{\bf F}_{C}/2 \hfill\cr \hbox{Combined:} \hfill& m_{\rm com} | {\bf F}_{O} | \exp (i\alpha_{\rm com}) \hfill& \simeq [1 - (w/2)] {\bf F} + (w/2) D{\bf F}_{C}, \hfill\cr}]$ where $[w = H_{C} / (H_{C} + H_{\rm MIR})]$ and $[H = \int\limits_{0}^{2\pi} p(\alpha) \ln {p(\alpha) \over p_{0} (\alpha)} \kern2pt\hbox{d} \alpha\hbox{;} \quad\ p_{0} (\alpha) = {1 \over 2\pi}.]$

Solving for an approximation to the true F gives the following expression, which can be seen to reduce appropriately when w is 0 (no model influence) or 1 (no MIR influence): $[{\bf F} \simeq {2m|{\bf F}_{O}| \exp(i\alpha_{\rm com}) - wD{\bf F}_{C} \over 2 - w}.]$

References

Guiasu, S. (1977). Information theory with applications. London: McGraw-Hill.Google Scholar

Main, P. (1979). A theoretical comparison of the β, γ′ and 2F_o − F_c syntheses. Acta Cryst. A35, 779–785.Google Scholar

Read, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.Google Scholar

Read, R. J. (1990). Structure-factor probabilities for related structures. Acta Cryst. A46, 900–912.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.2, pp. 327-328