Randy, 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 <[email protected]> Date: Friday, January 9, 2009 7:26 am Subject: Re: [ccp4bb] Published derivation of mFo-DFc formula? To: [email protected] > 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.tic...@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: [email protected] > Cambridge CB2 0XY, > U.K. www- > structmed.cimr.cam.ac.uk >
