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
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