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:CCP4BB@JISCMAIL.AC.UK] On Behalf Of > Andrew Leslie > Sent: Wednesday, March 12, 2014 12:25 PM > To: CCP4BB@JISCMAIL.AC.UK > 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 <cus...@embl.fr> 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: cus...@embl.fr >> 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
