Hi Ian,

Indeed, I didn't have time to reply during the festive season.

Just to give appropriate credit, the 2mFo-DFc coefficients for  
acentrics in SIGMAA were derived by analogy to arguments made by Peter  
Main, but taking account of the effect of errors in the partial  
model.  According to that argument, one doesn't expect model bias in  
the mFo coefficients for centrics, which is why they have different  
coefficients.

In the original version of SIGMAA, the difference map coefficients  
were mFo-Fc, not mFo-DFc.  At that time, I was thinking that mFo gives  
the best (lowest rms error, though biased) representation of the true  
density, while Fc represents the model so the difference between them  
should show most clearly how you need to change the model.  A year or  
two later, I decided that it was more appropriate to smear out the  
model density according to its uncertainty, hence the addition of the  
D factor.  There were two more advantages to the D factor.  First, if  
the model is complete garbage, both m and D are zero so the difference  
map is flat.  (If you don't know anything about the true phases, you  
can't make a map showing you how to change it!)  Second, it turns out  
that if Fo is not on the same scale as Fc, the D term absorbs the  
necessary scale factor correction.  (Before I added the D term,  
Eleanor Dodson had written a comment in the source code of the CCP4  
version of SIGMAA, saying something along the lines "Randy seems to be  
assuming that the data are
on absolute scale.  What on earth is he thinking?")

These arguments for the difference map coefficients are based simply  
on intuition about what makes sense to show how the model should  
change.  So I was very pleased, when we were working on refinement  
likelihood targets, to see that they can also be justified in terms of  
log-likelihood-gradient maps.  If you take the derivative of the log- 
likelihood functions with respect to Fc, you get

f(|s|)(mFo-DFc)

where f(|s|) is a resolution-dependent function given by

2D/sigma-delta^2 for acentrics, and
  D/sigma-delta^2 for centrics,

where sigma-delta^2 is the variance from the Rice function.

The D/sigma-delta^2 part would give a map that is the convolution of a  
map computed with the coefficients you suggest (i.e. 2mFo-2DFc for  
acentrics, mFo-DFc for centrics) and some shape function, which might  
sharpen the map if D/sigma-delta^2 increases with resolution or smear  
it out, if it decreases with resolution.

Anyway, at the least the factor of two for acentrics should be  
included in the various programs that compute difference map  
coefficients.  Someone should probably look at the effect of the  
convolution with the resolution-dependent part.  You will get  
different results if you consider the effect of coordinate error to be  
part of the model (e.g. D is taken up in the model partly by  
increasing B-factors -- if you take the derivative with respect to  
DFc, the factor D is missing from f(|s|)) or if you work in terms of E- 
values.  It's possible that the LLG map computed when the likelihood  
is expressed in terms of E-values would be optimal in terms of being  
sharpened as much as can be justified by the level of phase error at  
different resolutions.

Thanks for stimulating the discussion about this point!

Regards,

Randy Read

On 8 Jan 2009, at 11:43, Ian Tickle wrote:

>
> All - I didn't get a single response to my posting last week
> (https://www.jiscmail.ac.uk/cgi-bin/webadmin?A2=ind0812&L=CCP4BB&T=0&O=D
> &X=512817322E87355F7F&Y=i.tickle%40astex-therapeutics.com&P=266420)
> concerning the formulae that are widely used for the 'minimally- 
> biased'
> Fourier and difference Fourier coefficients.  It probably didn't help
> that I posted it in the middle of the festive season! - but still
> somewhat surprising since I imagine everyone here is involved with  
> maps
> at one time or another, and has an interest in getting the density  
> that
> shows best what if any further modifications need to be made to the
> current model.  Anyway now that people have hopefully returned to work
> from the rigours of the CCP4 Study Weekend I thought I'd post it again
> and see if I can provoke some discussion this time.  I won't post  
> all my
> calculations again, just a summary of my conclusions.
>
> First, I think I can now prove my conjecture that the optimal  
> difference
> Fourier coefficient dF is given for both acentrics and centrics by:
>
> 	dF = Fm - DFc
>
> where Fm is the 'minimally-biased' Fourier coefficient derived by Read
> (AC 1986,A42,140):
>
> 	Fm(acen) = 2mFo - DFc
> 	Fm(cen)  =  mFo
>
> I'm satisfied now that my alternative conjecture, that dF = Fm - Fc,  
> is
> probably wrong.  Also I can see that there might be an argument to put
> DFc in the FC (FC_ALL) column in place of Fc as appears to be  
> currently
> done by REFMAC, but not by SIGMAA (but I'd still like to see some
> discussion of that).
>
> So here's a summary comparison of theory with what is my understanding
> is actually implemented in software, and with the inconsistencies
> highlighted (>...<):
>
> 	Source       Coefficient   Acentrics        Centrics
> 	======       ===========   =========        ========
>
> 	THEORY(Read)     Fm        2mFo - DFc       mFo
> 	  ..  (me)       dF        2(mFo-DFc)       mFo - DFc
>
> 	SIGMAA           Fm        2mFo - DFc       mFo
> 	                 dF       > mFo - DFc <     mFo - DFc
>                       Fc            Fc              Fc
>
> 	REFMAC           Fm        2mFo - DFc    > 2mFo - DFc <
> 	                 dF       > mFo - DFc <     mFo - DFc
>                       Fc         > DFc <         > DFc <
>
> Even if you don't accept my suggestion for the acentric dF coefficient
> there are clearly some significant inconsistencies between the
> coefficients output by SIGMAA & REFMAC which it would be nice to
> resolve!
>
> Cheers
>
> -- Ian
>
>
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------
Randy J. Read
Department of Haematology, University of Cambridge
Cambridge Institute for Medical Research      Tel: + 44 1223 336500
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