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Dave, thanks for all that research! I had seen the S & J and Bricogne references, but not your Ladd & Palmer 3rd Ed. one: I have only the 1st Ed. of L & P on my bookshelf which apparently only uses monoclinic & orthorhombic examples. In the apparent absence of unambiguous definitions, I was really looking for specific examples of non-axial 2-fold rotation/screw or mirror/glide planes in trigonal, tetragonal, hexagonal and/or cubic space groups which left no leeway for any ambiguity in the definition of Harker vectors. It's good to see that out of all the publications on this subject, at least David Harker has got it right for sure, since clearly Harker lines & planes arising from NCS symmetry will generally have completely arbitrary orientations. The information in the Rossmann & Arnold table is actually not only incomplete with respect to non-axial 2-fold axes & screws, mirrors & glides, it's also in part superfluous and in part just wrong! In part (c)(i) of Table 2.3.2.3. their equation for the 3-fold plane can be simplified by dividing throughout by 0.57735 to get x+y+z=0. It then becomes clear that it's the u+v+w=0 section that they're talking about. Similarly, applying the same simplification to their equation for the 3_1 plane in part (c)(ii) it becomes clear that it's the plane x+y+z=1 (since p = sqrt(3)/3 = 0.57735). But obviously the Patterson has lattice translation symmetry, so this plane is identical to the first one, hence there is only one unique Harker plane in this case! At the end of section (c), the statement that rhombohedral 3-folds produce Harker planes 'whose description depends on the interaxial angle' is wrong: such axes produce Harker planes with the equation u+v+w=0 (exactly as for the equivalent 3-folds in cubic), and this equation plainly does not depend on the interaxial angle. I'm still baffled as to why they consider non-axial 3-folds as 'special', but not non-axial 2-folds! Cheers -- Ian > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On > Behalf Of David W Borhani > Sent: Tuesday, August 08, 2006 8:10 PM > To: [email protected] > Subject: Re: [ccp4bb]: Definition of Harker vector & section? > > > I think that it is clear that self-vector = Harker vector. > > In David Harker's original paper (Harker, D. "The Application > of the Three-Dimensional Patterson Method and the Crystal > Structures of Proustite, Ag3AsS3, and Pyrargyrite, Ag3SbS3" > J. Chem. Physics (1936) v. 4 pp. 381-390), he explicitly > derives the formulae for Harker planes (two-fold [screw] axis > parallel to b) and Harker lines (mirror plane perpendicular > to b, & glide plane along c) in the Patterson function, and > shows the attendant dramatic speed-up in calculation of these > particular sections (lines) of the Patterson. [For a Harker > line, "This is a one-dimensional Fourier series and can be > completely evaluated in a few hours."!!] > > Harker states but does not derive that similar formulae hold > for 3-, 4-, and 6-fold [screw] axes as well, and that these > can be used to advantage. Also, it is obvious that the > symmetry axis need not lie along b. Lastly, he states "There > are other cases in which the values of some of the atomic > coordinates in a crystal can be determined by evaluation of > P(x,y,z) throughout a plane or along a line only." He gives a > few examples, including (pseudo-)non-crystallographic > symmetry, concluding with "Such examples could be multiplied > almost indefinitely." Although Harker did not explicitly > state the formulae for the "diagonal" symmetry arrangement > discussed below, I believe that he gave enough examples to > render this unnecessary: simply determine the Harker plane > (line) as the locus of points satisfied by the vectors: > (SymOp-of-choice)*(x,y,z) - (x,y,z). > > The text in Rossmann & Arnold (Int. Tables B, 2.3.2.2) does > state that all self-vectors give rise to "a concentration of > vectors at characteristic locations in the Patterson." Yet, > their table is indeed incomplete, as is, apparently, the > output from FFT. Perhaps the code-meisters at CCP4 could fix > the latter? > > ( > By the way: > > Stout & Jensen got it right (2nd Ed., p. 285; their table > 12.1 lists "some" of the Harker lines & planes; the text > states "Those for any other symmetry element can be derived > along the lines given above.") > > Ladd & Palmer also got it right, with extensive worked > examples (3rd Ed., pp. 286-305). > > Bricogne also (Harker peaks only, but no mention of these > peaks forming a locus [Harker lines planes]; his restriction > that t(1)g is non-zero would appear to eliminate Harker > planes/peaks due to simple two-fold axes, but his notation > confuses me and perhaps I misinterpret; (Int. Tables B, end > of section 1.3.3.1.2.10)). > ) > > Dave > ~~~~~~~~~~~~~~~~~~~~~~~~ > David W. Borhani, Ph.D. > Structural Biology Group Leader > Chemistry Department > Abbott Bioresearch Center > Vox: 508-688-3944 > Fax: 508-754-7784 > Email: [EMAIL PROTECTED] > Smail: Abbott Bioresearch Center, Inc. > 100 Research Drive > Worcester, MA 01605 U.S.A. > http://abbott.com > ~~~~~~~~~~~~~~~~~~~~~~~~ > > > > > Bart Hazes <[EMAIL PROTECTED]> > Sent by: [EMAIL PROTECTED] > > 08/04/2006 11:30 AM > Please respond to > [EMAIL PROTECTED] > > To > Ian Tickle <[EMAIL PROTECTED]> > cc > [email protected], Tom Davies <[EMAIL PROTECTED]> > Subject > Re: [ccp4bb]: Definition of Harker vector & section? > > > > > > > *** For details on how to be removed from this list visit the *** > *** CCP4 home page http://www.ccp4.ac.uk *** > > > Hi Ian, > > Most of the time the International Tables is the ultimate > authority on > matters crystallographic because most of the time it is > right. However, > I agree with you that this does not make sense. The difficulty of > plotting a section should have nothing to do with the definition of > Harker sections/vectors. > > Bart > > Ian Tickle wrote: > > *** For details on how to be removed from this list visit the *** > > *** CCP4 home page http://www.ccp4.ac.uk *** > > > > > > > > All - > > > > I was just having a discussion here concerning the precise > definition of > > the terms 'Harker vector' & 'Harker section'. This discussion was > > prompted by this webpage: > > > > http://www.doe-mbi.ucla.edu/~sawaya/m230d/Patterson/patterson.html > > > > - specifically the 'Table of Patterson Difference Vectors' for space > > group P43212 (right at the end). The legend states "Table of all > > possible self-vectors in space group P43212. Colored > blocks appear on > > Harker sections.". The implication of this is that some > self-vectors do > > not appear on Harker sections and thus are not Harker vectors. This > > conclusion is backed up by this table at the same site: > > > > http://www.doe-mbi.ucla.edu/~sawaya/tutorials/Phasing/crystal.lib > > > > - scroll down a bit to the second section with the header: > > > > #CCP4I DATA PATTERSON > > # > > # Header lines are: > > # spgp_# spgp_name patterson_spgp_# patterson_spgp_name > > #_harker_sections > > # > > # Harker sections are listed on the subsequent lines > > # > > # Info generated using EJDs harker.f jiffy > > > > with this entry for P43212: > > > > 96 P43212 123 PG422 5 > > X = 0.5000 1/2 2Y+1/2 2Z+3/4 > > Y = 0.5000 2X+1/2 1/2 2Z+1/4 > > Z = 0.2500 X+Y+1/2 -X+Y+1/2 1/4 > > Z = 0.7500 X-Y+1/2 X+Y+1/2 3/4 > > Z = 0.5000 2X 2Y 1/2 > > > > There will always be N-1 self-vectors where N is the number > of primitive > > general equivalent positions, so there are 7 in P43212, > hence 2 of the > > self-vectors are missing from the above entry, and > similarly for most of > > the high-symmetry space-group entries (trigonal, > tetragonal, hexagonal, > > cubic). > > > > I would contend that _all_ self-vectors are also Harker > vectors without > > exception (in fact surely the terms are totally synonymous?). The 2 > > Harker vectors missing from the above entry are: X-Y, X-Y, > 2Z and X+Y, > > X+Y, 1/2+2Z, both of which lie on the U-V = 0 Harker section. Just > > because this isn't a section perpendicular to a principal > axis, hence > > FFT is unable print it on one page (though you can of > course do it via > > MAPROT) seems a totally inadequate reason for excluding it from the > > definition! The locus of the vector between 2 points related by any > > rotation/screw axis is necessarily a plane perpendicular to > that axis, > > and the U-V=0 section is merely the plane perpendicular to > the 2-fold in > > the Laue group 422 along (1,1,0). > > > > Then, further seeking to justify my view, I turned to International > > Tables, which I've always regarded as the ultimate > authority on matters > > crystallographic. The only explicit tabulation of Harker sections I > > could find there is in Vol. B, Chap. 2.3 by Rossmann & Arnold, Table > > 2.3.2.3 on p. 240. Worryingly for my point of view, this > concurs with > > the definition in the webpage, and also omits the 2-fold U-V=0 and > > similar sections in trigonal, tetragonal, hexagonal and cubic space > > groups, though it does include the (1,1,1) 3-fold section > in cubic (it > > calls it a 'special Harker plane' though it's not clear > what's special > > about it!). Bizarrely, in the same table, space groups > with 3, 4 and > > 6-folds are defined with the unique axis parallel to the _b_ axis (I > > wonder which version of the space-group tables they were using?). > > > > Finally I recalled that FFT helpfully prints out the Harker sections > > whenever you do a Patterson, so using some data in space > group P321 I > > get: > > > > Symmetry operators: 1: X, Y, Z: 2:-Y, X-Y, Z > > Harker Vector: 2 X+Y, -X+2Y, 0 > > Harker Section is: Z = 0.00 > > > > Harker vector Matrix: > > 1.00 1.00 0.00 0.00 > > -1.00 2.00 0.00 0.00 > > 0.00 0.00 0.00 0.00 > > 0.00 0.00 0.00 0.00 > > > > Symmetry operators: 1: X, Y, Z: 3:-X+Y, -X, Z > > Harker Vector: 3 2X-Y, X+Y, 0 > > Harker Section is: Z = 0.00 > > > > Harker vector Matrix: > > 2.00 -1.00 0.00 0.00 > > 1.00 1.00 0.00 0.00 > > 0.00 0.00 0.00 0.00 > > 0.00 0.00 0.00 0.00 > > > > That's it! Looking at the code, FFT specifically excludes the > > additional 3 Harker vectors that I think should be listed (on the 3 > > sections U+V=0, 2U-V=0 & U-2V=0), so am I alone in the > belief that all > > self-vectors are Harker vectors, or do we accept the apparent > > alternative definition that only sections that can be printed on one > > page by the FFT program can be Harker sections (with the apparent > > special exception of U+V+W=0 in cubic)? > > > > It would be interesting to know how David Harker in his > original paper > > viewed his eponymous section, unfortunately I don't have > easy access to > > it. > > > > -- Ian > > > > ******************************************** > > Ian J. Tickle, DPhil. > > Director of X-ray Technology > > Astex Therapeutics Ltd > > 436 Cambridge Science Park > > Milton Road, Cambridge > > CB4 0QA, UK > > Tel: +44(0)1223 226214 > > Fax: +44(0)1223 226201 > > www.astex-therapeutics.com > > > > 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 > [EMAIL PROTECTED] 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. > > > > > > > > > > > -- > > ============================================================== > ================ > > Bart Hazes (Assistant Professor) > Dept. of Medical Microbiology & Immunology > University of Alberta > 1-15 Medical Sciences Building > Edmonton, Alberta > Canada, T6G 2H7 > phone: 1-780-492-0042 > fax: 1-780-492-7521 > > ============================================================== > ================ > > > 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. 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