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Re: [ccp4bb]: 3Fo-2Fc maps

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On Tue, 28 Nov 2000 17:00:14 -0200 (EDT), Stefano Trapani <stefano@if.sc.usp.br> wrote:
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> A question about electron density maps.
> Does it make any sense to use a weighted (3mFo-2DFc) difference map
> (with phases from a MR solution) during model building and refinement,
> where m and D are the Sigma-A-style weights of Read (Acta Cryst
> A42,1986)?

Fred Vellieux first suggested combining my SIGMAA map coefficients with
different relative weights.  I'm afraid I can't track down a reference at
the moment.  The unweighted case (nFo-(n-1)Fc) was considered
systematically by Vijayan, who showed that, as the amount of missing
structure increases, the relative height of peaks for missing atoms drops,
so a higher n is needed to equalise peaks from modeled and missing atoms.

In principle, this shouldn't apply to the m and D weighted coefficients, 
where the weighting should put the two kinds of peaks on the same
relative scale.  Nonetheless, Fred found by examining maps that he had
a better indication of what was wrong when looking at maps with higher n,
such as 3mFo-2DFc.

I think the reason is that the derivation of the SIGMAA map coefficients
assumes a certain probability distribution relating Fo and Fc, where
the expected value of Fo is given by DFc and the components of the error
parallel and perpendicular to Fc are equally large, on average.  But if
you think about what is happening when we overfit the data (which is
inevitable given typical observation to parameter ratios), the overfitting
must cause the error in the parallel direction (amplitude) to be reduced
systematically compared to the error in the perpendicular direction
(essentially the phase component of the error).  Overweighting the 
difference component of the map will correct to some extent for such

Another way to look at this is that our model is represented by DFc,
which corresponds to model density smeared out by overall coordinate
uncertainty.  The difference map coefficient, mFo-DFc, turns out to
be proportional to the gradient of the likelihood function with respect
to Fc, the Fourier transform of which tells us where the likelihood
function would like to see changes in the density.  Varying the value
of n in Fred's map coefficients corresponds to adding different amounts
of this gradient map to the DFc map.

I've been planning for some time to look into this more systematically, by
devising a measure of overfitting (e.g. by comparing the relationship
between Fo and Fc in the working data with that seen in the
cross-validation data) and using that to choose an appropriate map 
coefficient.  The more overfitting (generally corresponding to lower
resolution), the bigger the value of n would have to be.  But there's 
always too much to do.

In the meantime, it's doesn't hurt to look at 2mFo-DFc and 3mFo-2DFc
maps to see which shows regions of error more clearly.  The other
thing to note is that the difference map (mFo-DFc) is much less affected
by considerations of relative heights of peaks for included and missing

Randy J. Read
Department of Haematology, University of Cambridge
Wellcome Trust Centre for Molecular Mechanisms in Disease
Cambridge Institute for Medical Research
Wellcome Trust/MRC Building     Tel: + 44 1223 336500
Hills Road                      Fax: + 44 1223 336827
Cambridge CB2 2XY, U.K.         E-mail: rjr27@cam.ac.uk