I believe that the first publication to report the use of coefficients 2Fobs-Fc antidates both the Main publication and the use of a downweighting of the Fc and is in Freer, et al., 1975 J. Biol. Chem. 250:46-54, which also gives a heuristic account of the value of such coefficients.
----- Original Message -----
From: Randy Read <
[log in to unmask]>
Date: Friday, January 9, 2009 7:26 am
Subject: Re: [ccp4bb] Published derivation of mFo-DFc formula?
To:
[log in to unmask]> 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
> >
> >
> > Disclaimer
> > This communication is confidential and may contain
> privileged
> > information intended solely for the named addressee(s). It may
> not
> > be used or disclosed except for the purpose for which it has
> been
> > sent. If you are not the intended recipient you must not
> review,
> > use, disclose, copy, distribute or take any action in reliance
> upon
> > it. If you have received this communication in error, please
> notify
> > Astex Therapeutics Ltd by emailing I.Tickle@astex-
> therapeutics.com
> > and destroy all copies of the message and any attached documents.
> > Astex Therapeutics Ltd monitors, controls and protects all
> its
> > messaging traffic in compliance with its corporate email
> policy. The
> > Company accepts no liability or responsibility for any
> onward
> > transmission or use of emails and attachments having left the
> Astex
> > Therapeutics domain. Unless expressly stated, opinions
> in this
> > message are those of the individual sender and not of
> Astex
> > Therapeutics Ltd. The recipient should check this email and
> any
> > attachments for the presence of computer viruses. Astex
> Therapeutics
> > Ltd accepts no liability for damage caused by any virus
> transmitted
> > by this email. E-mail is susceptible to data corruption,
> > interception, unauthorized amendment, and tampering,
> Astex
> > Therapeutics Ltd only send and receive e-mails on the basis
> that the
> > Company is not liable for any such alteration or any
> consequences
> > thereof.
> > Astex Therapeutics Ltd., Registered in England at 436
> Cambridge
> > Science Park, Cambridge CB4 0QA under number 3751674
>
> ------
> Randy J. Read
> Department of Haematology, University of Cambridge
> Cambridge Institute for Medical
> Research Tel: + 44 1223 336500
> Wellcome Trust/MRC
> Building Fax: + 44 1223 336827
> Hills
> Road E-mail:
[log in to unmask]> Cambridge CB2 0XY,
> U.K. www-
> structmed.cimr.cam.ac.uk
>
