How to approach the analysis of such a problem:
For any sample, crystalline or not, a generally valid description of
diffraction intensity is it being a Fourier transform of electron density
autocorrelation function. There are obvious normalizations involved. For
crystals, this autocorrelation function is periodic and is called a
Patterson function when it is derived from diffraction data.
In the case of statistical disorder, an important factor characterizing it
is autocorrelation of alternative conformations when they are displaced by
unit cell periodicities. If such autocorrelation is zero, we have a pure
statistical disorder; in such a case, we should add structure factors of
alternative conformations to create a calculated F. There will be also
diffused scattering from the disorder, but it will not be aligned with
Bragg diffraction. More often, the presence of a particular alternative
conformation will affect the probability of alternative conformation a
unit cell away, and this needs to be considered separately for every unit
cell translation. If this correlation is very strong - close to 1 - we
have a situation similar or identical to merohedral twinning, and one
should add F^2 from alternative models. In an intermediate case, when
autocorrelation in a particular direction is between zero and one, the
Fourier transform will produce streaks in diffraction pattern and the
alignment of these streaks will be related to the properties of the
autocorrelation function. Unfortunately, this creates problems when
dealing with reduced data sets.
Mosaicity is a very different phenomenon. It describes a range of angular
alignments of microcrystals with the same unit cell within the sample. It
broadens diffraction peaks by the same angle irrespective of the data
resolution, but it cannot change the length of diffraction vector for each
Bragg reflection. For this reason, the elongation of the spot on the
detector resulting from mosaicity will be always perpendicular to the
diffraction vector. This is distinct from the statistical disorder, where
spot elongation will be aligned with the crystal lattice and not the
detector plane.
Obviously, no phase information can be derived from the spot shapes
resulting from mosaicity. Interestingly, there is a potential for
extracting phase information from spot shapes induced by statistical
disorder. However, it is far from simple and can be used only to improve
phases. It is not promising as an ab initio phasing method.
This discussion assumed only one unit cell periodicity in the sample,
which is the desired state in all cases. In cryo-cooled crystals, the rate
of cooling is different for different parts of the sample, resulting quite
often in different unit cell periodicities across the sample. Now there
are multiple possibilities to consider; quite typically, the crystal
symmetry is the same and the range of unit cell variability is small. This
results in variable spot shape elongation, with angular range being
resolution-dependent and elongation not necessarily perpendicular to the
diffraction vector. By just looking at diffraction pattern, it is easy to
distinguish this case from mosaicity. In such samples, a problem arises
when rotation exposes distinctly different phases at different
orientations. The resulting diffraction data will merge with poor
statistics, as distinct structure factors will be merged together. Such
condition is quite typical when large crystals are exposed with
microbeams.
Presence of different crystal forms also provides phasing opportunities
known as averaging between crystals. However, this requires separate data
set collection rather than mixing such crystals during one rotation sweep.
Presence of multiple, similar unit cells in the sample is completely
different and unrelated condition to statistical disorder.
Zbyszek Otwinowski
> Not sure I understand why having statistical disorder makes for
> streaks--does the crystal then have a whole range of unit cell constants,
> with the spot at the most prevalent value, and the streaks are the "tails"
> of the distribution? If so, doesn't having the streak imply a really wide
> range of constants? And how would this be different from mosaicity? My
> guess is that this is not the right picture, and this is indeed roughly
> what mosaicity is.
>
> Alternatively, perhaps the streaks are interpreted as the result of a
> duality between the "unit cell," which yields spots, and a "super cell"
> which is so large that it yields extremely close "spots" which are
> indistinguishable from lines/streaks. Usually this potential super cell is
> squelched by destructive interference due to each component unit cell
> being very nearly identical, but here the destructive interference doesn't
> happen because each component unit cell differs quite a bit from its
> fellows.
>
> And I guess in the latter case the "supercell" would have its cell
> constant (in the direction of the streaks) equal to (or a function of) the
> coherence length of the incident radiation?
>
> I know some attempts have been (successfully) made to use diffuse
> scattering, but has anyone used the streak intensities to determine
> interesting features of the crystallized protein?
>
> JPK
>
>
>
> -----Original Message-----
> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of
> Andrew Leslie
> Sent: Wednesday, March 12, 2014 12:25 PM
> To: [log in to unmask]
> Subject: Re: [ccp4bb] twinning problem ?
>
> Dear Stephen,
>
> I have seen a similar effect in the structure of
> F1-ATPase complexed with the full length inhibitor
> protein. The inhibitor is a dimer, and it actually
> couples 2 copies of the ATPase, but it
> crystallised with only one copy of the ATPase per
> asymmetric unit. When I solved the structure by
> MR, I saw additional density that could not be
> accounted for. The extra density was, in fact, a
> second ATPase molecule that was related to the
> first by a 120 degree rotation about the pseudo
> 3-fold axis of the enzyme. The "dimers" were
> packing with statistical disorder in the crystal
> lattice. This gave rise to clear streaking between
> Bragg spots in the diffraction images in a
> direction that was consistent with that expected
> from the statistical packing of the inhibitor
> linked dimers.
>
> Two copies of F1 were included in the refinement, each with occupancy 0.5.
> the final Rfree was 27.7% (2.8A data). Prior to introduction of the second
> copy of F1, the Rfree was 37%.
>
> More details are in Cabezon et al., NSMB 10, 744-750, 2003
>
> Best wishes,
>
> Andrew
>
>
>
> On 11 Mar 2014, at 14:04, Stephen Cusack <[log in to unmask]> wrote:
>
>> Dear All,
>> I have 2.6 A data and unambiguous molecular replacement solution
>> for two copies/asymmetric unit of a 80 K protein for a crystal
>> integrated in P212121 (R-merge around 9%) with a=101.8, b=132.2,
>> c=138.9.
>> Refinement allowed rebuilding/completion of the model in the noraml
>> way but the R-free does not go below 30%. The map in the model regions
>> looks generally fine but there is a lot of extra positive density in
>> the solvent regions (some of it looking like weak density for helices
>> and strands) and unexpected positive peaks within the model region.
>> Careful inspection allowed manual positioning of a completely different,
>> overlapping solution for the dimer which fits the extra density
>> perfectly.
>> The two incompatible solutions are related by a 2-fold axis parallel to
>> a.
>> This clearly suggests some kind of twinning. However twinning analysis
>> programmes (e.g. Phenix-Xtriage), while suggesting the potentiality of
>> pseudo-merohedral twinning (-h, l, k) do not reveal any significant
>> twinning fraction and proclaim the data likely to be untwinned. (NB.
>> The programmes do however highlight a non-crystallographic translation
>> and there are systematic intensity differences in the data). Refinement,
>> including this twinning law made no difference since the estimated
>> twinning fraction was 0.02. Yet the extra density is clearly there and I
>> know exactly the real-space transformation between the two packing
>> solutions.
>> How can I best take into account this alternative solution (occupancy
>> seems to be around 20-30%) in the refinement ?
>> thanks for your suggestions
>> Stephen
>>
>> --
>>
>> **********************************************************************
>> Dr. Stephen Cusack,
>> Head of Grenoble Outstation of EMBL
>> Group leader in structural biology of protein-RNA complexes and viral
>> proteins Joint appointment in EMBL Genome Biology Programme Director
>> of CNRS-UJF-EMBL International Unit (UMI 3265) for Virus Host Cell
>> Interactions (UVHCI)
>> **********************************************************************
>>
>> Email: [log in to unmask]
>> Website: http://www.embl.fr
>> Tel: (33) 4 76 20 7238 Secretary (33) 4 76 20 7123
>> Fax: (33) 4 76 20 7199
>> Postal address: EMBL Grenoble Outstation, 6 Rue Jules Horowitz, BP181,
>> 38042 Grenoble Cedex 9, France
>> Delivery address: EMBL Grenoble Outstation, Polygone Scientifique,
>> 6 Rue Jules Horowitz, 38042 Grenoble, France
>> **********************************************************************
>
Zbyszek Otwinowski
UT Southwestern Medical Center at Dallas
5323 Harry Hines Blvd.
Dallas, TX 75390-8816
Tel. 214-645-6385
Fax. 214-645-6353
|
