I'm afraid I have to disagree with summary point (i): that
crystallographic and noncrystallographic symmetry are incomparable.
Crystallographic symmetry is a special case of ncs where the symmetry
happens to synchronize with the lattice symmetry. There are plenty
of cases where this synchronization is not perfect and the ncs is
"nearly" crystallographic.
For some reason this situation seems to be particularly popular
with P21 space group crystals with a dimer in the asymmetric unit.
Quite often the two-fold of the dimer is nearly parallel to the
screw axis resulting in a nearly C2 space group crystal. These
crystals form a bridging case in the continuum between ncs, where
the symmetry is unrelated to the lattice symmetry, and those cases
where the unit cell symmetry is perfectly compatible with the
lattice.
The only saving grace of the "nearly centered" ncs crystals is
that the combination of the crystal and noncrystallographic symmetry
brings the potential "contamination" of a reflection in the working
set back to itself. Unless you have a very high copy number, and
a corresponding large G function, you can't have any feedback from
a working set reflection to a test reflection.
Crystallographic symmetry is just a special case of noncrystallographic
symmetry, but our computational methods treat them in very different
ways. This choice of ours creates a discontinuity in the treatment
of symmetry that is quite artificial, and I believe, is the root
cause of many of the problems we have with ncs in refinement and
structure solution.
Dale Tronrud
Dirk Kostrewa wrote:
> Dear Dean and others,
>
> Peter Zwart gave me a similar reply. This is very interesting
> discussion, and I would like to have a somewhat closer look to this to
> maybe make things a little bit clearer (please, excuse the general
> explanations - this might be interesting for beginners as well):
>
> 1). Ccrystallographic symmetry can be applied to the whole crystal and
> results in symmetry-equivalent intensities in reciprocal space. If you
> refine your model in a lower space group, there will be reflections in
> the test-set that are symmetry-equivalent in the higher space group
> to reflections in the working set. If you refine the
> (symmetry-equivalent) copies in your crystal independently, they will
> diverge due to resolution and data quality, and R-work and R-free will
> diverge to some extend due to this. If you force the copies to be
> identical, the R-work & R-free will still be different due to
> observational errors. In both cases, however, the R-free will be very
> close to the R-work.
>
> 2). In case of NCS, the continuous molecular transform will reflect this
> internal symmetry, but because it is only a local symmetry, the observed
> reflections sample the continuous transform at different points and
> their corresponding intensities are generally different. It might,
> however, happen that a test-set reflection comes _very_ close in
> reciprocal space to a "NCS-related" working-set reflection, and in such
> a case their intensities will be very similar and this will make the
> R-free closer to the R-work. If you do not apply NCS-averaging in form
> of restraints/constraints, these accidentally close reflections will be
> the only cases where R-free might be too close to R-work. If you apply
> NCS-averaging, then in real space you multiply the electron density with
> a mask and average the NCS-related copies within this mask at all
> NCS-related positions. In reciprocal space, you then convolute the
> Fourier-transform of that mask with your observed intensities in all
> NCS-related positions. This will force to make test-set reflections more
> similar to NCS-related working-set reflections and thus the R-free will
> be heavily based towards R-work. The range of this influence in
> reciprocal space can be approximated by replacing the mask with a sphere
> and calculate the Fourier-transform of this sphere. This will give the
> so-called G-function, whose radius of the first zero-value determines
> its radius of influence in reciprocal space.
>
> To summarize:
> (i) One can't directly compare crystallographic and non-crystallographic
> symmetry
> (ii) In case of NCS, I have to admit, that even if you do not apply
> NCS-restraints/constraints, there will be some effect on the R-free by
> chance. So, my original statement was too strict in this respect. But
> only if you really apply NCS-restraints/constraints, you force to bias
> the R-free towards the R-work with an approximte radius of the
> G-function in reciprocal space.
>
> What an interesting discussion!
>
> Best regards,
>
> Dirk.
>
> Am 07.02.2008 um 18:57 schrieb Dean Madden:
>
>> Hi Dirk,
>>
>> I disagree with your final sentence. Even if you don't apply NCS
>> restraints/constraints during refinement, there is a serious risk of
>> NCS "contaminating" your Rfree. Consider the limiting case in which
>> the "NCS" is produced simply by working in an artificially low
>> symmetry space-group (e.g. P1, when the true symmetry is P2): in this
>> case, putting one symmetry mate in the Rfree set, and one in the Rwork
>> set will guarantee that Rfree tracks Rwork. The same effect applies to
>> a large extent even if the NCS is not crystallographic.
>>
>> Bottom line: thin shells are not a perfect solution, but if NCS is
>> present, choosing the free set randomly is *never* a better choice,
>> and almost always significantly worse. Together with multicopy
>> refinement, randomly chosen test sets were almost certainly a major
>> contributor to the spuriously good Rfree values associated with the
>> retracted MsbA and EmrE structures.
>>
>> Best wishes,
>> Dean
>>
>> Dirk Kostrewa wrote:
>>> Dear CCP4ers,
>>> I'm not convinced, that thin shells are sufficient: I think, in
>>> principle, one should omit thick shells (greater than the diameter of
>>> the G-function of the molecule/assembly that is used to describe
>>> NCS-interactions in reciprocal space), and use the inner thin layer
>>> of these thick shells, because only those should be completely
>>> independent of any working set reflections. But this would be too
>>> "expensive" given the low number of observed reflections that one
>>> usually has ...
>>> However, if you don't apply NCS restraints/constraints, there is no
>>> need for any such precautions.
>>> Best regards,
>>> Dirk.
>>> Am 07.02.2008 um 16:35 schrieb Doug Ohlendorf:
>>>> It is important when using NCS that the Rfree reflections be selected is
>>>> distributed thin resolution shells. That way application of NCS
>>>> should not
>>>> mix Rwork and Rfree sets. Normal random selection or Rfree + NCS
>>>> (especially 4x or higher) will drive Rfree down unfairly.
>>>>
>>>> Doug Ohlendorf
>>>>
>>>> -----Original Message-----
>>>> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of
>>>> Eleanor Dodson
>>>> Sent: Tuesday, February 05, 2008 3:38 AM
>>>> To: [log in to unmask] <mailto:[log in to unmask]>
>>>> Subject: Re: [ccp4bb] an over refined structure
>>>>
>>>> I agree that the difference in Rwork to Rfree is quite acceptable at
>>>> your resolution. You cannot/ should not use Rfactors as a criteria
>>>> for structure correctness.
>>>> As Ian points out - choosing a different Rfree set of reflections
>>>> can change Rfree a good deal.
>>>> certain NCS operators can relate reflections exactly making it hard
>>>> to get a truly independent Free R set, and there are other reasons
>>>> to make it a blunt edged tool.
>>>>
>>>> The map is the best validator - are there blobs still not fitted?
>>>> (maybe side chains you have placed wrongly..) Are there many
>>>> positive or negative peaks in the difference map? How well does the
>>>> NCS match the 2 molecules?
>>>>
>>>> etc etc.
>>>> Eleanor
>>>>
>>>> George M. Sheldrick wrote:
>>>>> Dear Sun,
>>>>>
>>>>> If we take Ian's formula for the ratio of R(free) to R(work) from
>>>>> his paper Acta D56 (2000) 442-450 and make some reasonable
>>>>> approximations,
>>>>> we can reformulate it as:
>>>>>
>>>>> R(free)/R(work) = sqrt[(1+Q)/(1-Q)] with Q = 0.025pd^3(1-s)
>>>>>
>>>>> where s is the fractional solvent content, d is the resolution, p is
>>>>> the effective number of parameters refined per atom after allowing for
>>>>> the restraints applied, d^3 means d cubed and sqrt means square root.
>>>>>
>>>>> The difficult number to estimate is p. It would be 4 for an
>>>>> isotropic refinement without any restraints. I guess that p=1.5
>>>>> might be an appropriate value for a typical protein refinement
>>>>> (giving an R-factor
>>>>> ratio of about 1.4 for s=0.6 and d=2.8). In that case, your
>>>>> R-factor ratio of 0.277/0.215 = 1.29 is well within the allowed range!
>>>>>
>>>>> However it should be added that this formula is almost a
>>>>> self-fulfilling prophesy. If we relax the geometric restraints we
>>>>> increase p, which then leads to a larger 'allowed' R-factor ratio!
>>>>>
>>>>> Best wishes, George
>>>>>
>>>>>
>>>>> Prof. George M. Sheldrick FRS
>>>>> Dept. Structural Chemistry,
>>>>> University of Goettingen,
>>>>> Tammannstr. 4,
>>>>> D37077 Goettingen, Germany
>>>>> Tel. +49-551-39-3021 or -3068
>>>>> Fax. +49-551-39-2582
>>>>>
>>>>>
>>>>>
>>> *******************************************************
>>> Dirk Kostrewa
>>> Gene Center, A 5.07
>>> Ludwig-Maximilians-University
>>> Feodor-Lynen-Str. 25
>>> 81377 Munich
>>> Germany
>>> Phone: +49-89-2180-76845
>>> Fax: +49-89-2180-76999
>>> E-mail: [log in to unmask]
>>> <mailto:[log in to unmask]>
>>> *******************************************************
>>
>> --
>> Dean R. Madden, Ph.D.
>> Department of Biochemistry
>> Dartmouth Medical School
>> 7200 Vail Building
>> Hanover, NH 03755-3844 USA
>>
>> tel: +1 (603) 650-1164
>> fax: +1 (603) 650-1128
>> e-mail: [log in to unmask] <mailto:[log in to unmask]>
>
>
> *******************************************************
> Dirk Kostrewa
> Gene Center, A 5.07
> Ludwig-Maximilians-University
> Feodor-Lynen-Str. 25
> 81377 Munich
> Germany
> Phone: +49-89-2180-76845
> Fax: +49-89-2180-76999
> E-mail: [log in to unmask] <mailto:[log in to unmask]>
> *******************************************************
>
>
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