 1. Introduction
 2. Tetartohedral twinning is (pseudo)merohedral twinning with Ntwins = 4
 3. Structural refinement against tetartohedrally twinned data
 4. Tetartohedrally twinned structures in the literature
 5. Tetartohedrally twinned crystals of human complement factor I
 6. Conclusions
 A1. Twinning operators coinciding with the rotational part of pseudosymmetry
 References
 1. Introduction
 2. Tetartohedral twinning is (pseudo)merohedral twinning with Ntwins = 4
 3. Structural refinement against tetartohedrally twinned data
 4. Tetartohedrally twinned structures in the literature
 5. Tetartohedrally twinned crystals of human complement factor I
 6. Conclusions
 A1. Twinning operators coinciding with the rotational part of pseudosymmetry
 References
research papers
Tetartohedral
could happen to you too^{a}Sir William Dunn School of Pathology, Oxford University, South Parks Road, Oxford OX1 3RE, England,^{b}King's College Centre for Bioinformatics (KCBI), School of Physical Sciences and Engineering, King's College London, Strand, London WC2R 2LS, England, and ^{c}MRC Centre for Developmental Neurobiology, New Hunt's House, King's College London, Guy's Campus, London SE1 1UL, England
^{*}Correspondence email: pietro.roversi@path.ox.ac.uk
Tetartohedral crystal
is discussed as a particular case of (pseudo)merohedral when the number of twinned domains is four. Tetartohedrally twinned crystals often possess with the rotational part of the pseudosymmetry operators coinciding with the operators. Tetartohedrally twinned structures from the literature are reviewed and the recent of tetartohedrally twinned triclinic crystals of human complement factor I is discussed.Keywords: tetartohedral twinning.
1. Introduction
When crystal (pseudo)merohedral ; Rosendal et al., 2004; Barends et al., 2005; Gayathri et al., 2007; FernándezMillán et al., 2008; Yu et al., 2009; Anand et al., 2007; Leung et al., 2011) and find that this type of is almost always accompanied by with the operators coinciding with the rotational parts of the To discuss a few of the issues that arise when working with tetartohedrally twinned crystals, we also illustrate the determination of the structure of tetartohedrally twinned triclinic crystals of human complement factor I (Roversi et al., 2011).
arises from four twinned crystal domains, the is called tetartohedral. In this manuscript, we review published tetartohedrally twinned macromolecular structures (see Table 1
‡Rosendal et al. (2004). §Barends et al. (2005). ¶Gayathri et al. (2007). ††Yu et al. (2009). ‡‡Leung et al. (2011). §§Joint Center for Structural Genomics, unpublished work. ¶¶Anand et al. (2007). †††Roversi et al. (2011). 
2. Tetartohedral is (pseudo)merohedral with N_{twins} = 4
If the crystalline sample exposed to Xrays is made of N_{twins} single crystals, the is described in terms of the relative sizes of the N_{twins} domains (the twin fractions α_{k}) and the set of matrices T_{k} that represent the operators. When the operators leave the crystal lattice (almost) unchanged, the is called (pseudomerohedral). The Xray diffraction spots from all of the twinned domains (almost) overlap and the diffracted intensity can be written
As already mentioned, when N_{twins} = 4 the structure is said to be tetartohedrally twinned. As for any type of (pseudo)merohedral detection of tetartohedral is possible at an early stage, after a set of diffraction intensities has been collected, by performing a number of tests that analyse the crystal intensity statistics (see Yeates, 1997)^{1}. The formulae needed to estimate twin fractions from tetartohedrally twinned data have been described in Yeates & Yu (2008).
If the extent of et al. (2006) and Zwart et al. (2008).
is small and/or is obscured by the presence of and especially when the NCS axes coincide with the directions of (the latter introducing deviations from the intensity statistics used to derive the tests), it can also be the case that can only be confirmed at a stage as late as that of of the model. Fortunately, in this case the availability of the model allows statistical tests on the calculated intensities, which can help in estimation of the twin fractions: intensity statistics in the presence of NCS and have been discussed and illustrated in LebedevFor a discussion of experimental phasing see Dauter (2003) and for a discussion of in the presence of crystal see Redinbo & Yeates (1993), Breyer et al. (1999) and Jameson et al. (2002). Generally speaking, whenever data sets from several different twinned samples have been measured and estimates of the fractions have been obtained for each sample, if possible one should avoid working with data sets from crystals for which all the twinned fractions are close to 1/N_{twins} (`perfect twinning'). Of course, the closer the sample is to perfect the greater the need for the accurate estimation of fractions based on I_{h}^{calc} from each twin domain, which is only possible if the structure is available. This in turn means that only towards the end of the structuredetermination process will the details of each twinned sample be properly understood and the optimal choice of sample/data set be possible.
3. Structural against tetartohedrally twinned data
Various strategies are possible when refining against twinned data and tetartohedral I_{h}^{calc} from each twin domain. Structural can then be carried out against these intensities, leading to a new model and a new round of estimation of twin ratios and so on, hopefully to convergence (see, for example, the of PDB entry 3eop ; Yu et al., 2009). This strategy may suffer from instability and its convergence may be slow.
is not an exception. The simplest approach would involve detwinning the experimental intensities on the basis of the current estimates for the twin ratios by using the current model andIn a second approach, the B factors, occupancies etc.), possibly including joint second derivatives of the target function with respect to twin fractions and other parameters. The leastsquares program SHELXL97 has long allowed joint structural against tetartohedrally twinned diffraction intensities (HerbstIrmer & Sheldrick, 1998). It refines all parameters in the same conjugategradient or matrixinversion run. If the matrix of the second derivatives of the target function with respect to the parameters is inverted, it is possible to obtain the correlations between the twin fractions and the other parameters of the model and error estimates of the twin fractions.
target function can be defined taking into account and carried out against the twinned intensities. Ideally, of the ratios should be carried out at the same time as the of the structural parameters (scale factors, atomic coordinates andTo make vice versa, alternating cycles of of twin fractions and structural parameters. A protocol to perform of the model against tetartohedrally twinned intensities was included in the supplementary information of Barends et al. (2005). This protocol makes use of the program CNS and it relies on initial estimation of the twin fractions, which are subsequently kept fixed during the leastsquares structural More recently, the program REFMAC5 enabled the initial detection of tetartohedral twin operators, initial estimation of the twin fractions and their optimization in between cycles of of structural parameters (Murshudov et al., 2011).
computationally simpler, the twin fractions can be optimized while holding the other parameters fixed andOf course, as is the case with all refinements against intensities from merohedrally twinned crystals and/or crystals that possess NCS, special care should be taken in assigning free R flags so that NCSrelated and/or twinrelated reflections either belong to the free or to the working set, i.e. NCS/twinrelated reflections should not be distributed across the two sets (Kleywegt & Brünger, 1996). REFMAC5 internally changes free R flags so that twinrelated reflections belong to the either the free or the working set.
4. Tetartohedrally twinned structures in the literature
Keyword searches in the Protein Data Bank and the literature (via the PubMed server) returned a number of published crystal structures from tetartohedrally twinned crystals.^{2} We summarize them in Table 1.
In many of these structures the et al., 2008; Appendix A) and the group of the NCS rotations coincides with that of the operators. Interestingly, most of these structures are trigonal, with the operators and the NCS belonging to 222; the twofold axes are aligned along a, a* and c so as to create apparent 622 One structure (PDB entry 2xrc ; see below) is pseudomerohedrally twinned, triclinic P1, but with a pseudoorthorhombic cell and the NCS and the twofolds also aligned with crystal axes. The only published tetartohedrally twinned structure for which the group of the NCS rotations and one of the operators do not coincide is PDB entry 3eop , where the twofold NCS operator and the crystal symmetry together have 321 symmetry, while the has 222 symmetry (the two groups sharing only the twofold along a).
operators are close to true (pseudosymmetry; Zwart5. Tetartohedrally twinned crystals of human complement factor I
The crystal structure of human complement factor I (fI) was described in Roversi et al. (2011). The crystals were triclinic and tetartohedrally twinned. In this manuscript, we examine the analysis of the crystal symmetry, the detection of the tetartohedral and the protocol followed for initial phasing, model building and of the structure against the tetartohedrally twinned diffraction data.
The fI crystals appeared to be frayed at the ends, which may indicate several crystalline layers stacking to form each sample, but otherwise had sharp edges, could be grown reproducibly and gave diffraction patterns that could be successfully indexed by invoking a single lattice (Roversi et al., 2011).
Several samples were exposed to Xrays and diffraction data sets were measured, the best diffracting of which (2.4 Å resolution) was collected in September 2009 at 100 K using Xrays of wavelength 0.97630 Å on beamline I03 at the Diamond Light Source, Harwell, England. The data were originally indexed and scaled in a primitive orthorhombic 222 lattice with the unitcell parameters reported in Table 2. Analysis with POINTLESS and phenix.xtriage (Zwart et al., 2008) suggested a primitive 222 lattice and P2_{1}2_{1}2.

No problems were initially noticed, apart from the fact that the cumulative intensity distribution (not shown), other overall intensity statistics and the results of the Ltest (see Table 3) departed from what would be expected from good to reasonable untwinned data. As there are no (pseudo)merohedral twin laws possible for these orthorhombic crystals, phenix.xtriage concluded that
there could be a number of reasons for the departure of the intensity statistics from normality. Overmerging pseudosymmetric or twinned data, intensitytoamplitude conversion problems as well as bad data quality might be possible reasons. It could be worthwhile considering reprocessing the data.
Symmetry (Z) P1 (4) P2_{1} (4) P2_{1}2_{1}2 (4) No twin Perfect twin Resolution range (Å) 10–2.43 10–2.87 10–2.77 〈I^{2}〉/〈I〉^{2}, acentric (centric) 1.748 1.745 (2.282) 1.666 (2.436) 2.0 (3.0) 1.5 (2.0) 〈F^{2}〉/〈F〉^{2}, acentric (centric) 0.842 0.843 (0.740) 0.856 (0.719) 0.785 (0.637) 0.885 (0.785) 〈E^{2} − 1〉, centric (acentric) 0.642 0.631 (0.841) 0.602 (0.896) 0.736 (0.968) 0.541 (0.736) L (acentric) 0.424 0.418 0.403 0.500 0.375 〈L^{2}〉 (acentric) 0.250 0.244 0.228 0.333 0.200 Multivariate Z score Ltest 5.8 6.8 8.1 <3.5 >3.5
Had one attempted scaling in a lower symmetry ). In agreement with the these scaling statistics, the κ = 180° section of the selfrotation function for this crystal shows almost perfect 222 symmetry, with three peaks at 94, 93 and 92% of the origin and along the directions (ω = 89.9°, φ = 0.0°), (ω = 89.9°, φ = 89.9°) and (ω = 0.0°, φ = 0.0°), respectively. Retrospectively, once the structure was solved in P1 and the tetartohedral twin fractions were calculated with REFMAC5 it appeared that this crystal (like all other fI triclinic crystals measured but one) was almost perfectly tetartohedrally twinned, i.e. the four twin fractions were all close to 1/4 (see Table 6, last column), a special case of the condition α_{k} + α_{k′} = 1/2 that makes twinned crystals most problematic (`perfect twin'; Yeates, 1997).
the scaling statistics would have shown only a marginal improvement upon lowering of the symmetry (see Table 2Further clues to the fact that the crystals were not orthorhombic came from molecularreplacement efforts in P2_{1}2_{1}2 using Phaser and searching with several models of domains homologous to the serine protease domain (43% of the structure). The searches consistently yielded a pair of placements (with very equivalent scores) which shared the rotationfunction maximum but differed by a shift of almost 6 Å along c in the translationfunction maximum. This could be interpreted as an indication of lower symmetry, but the observation was originally ignored and model building attempted starting from the topscoring placement in P2_{1}2_{1}2, without much success.
In November 2009, an fI crystal gave a 2.70 Å resolution diffraction data set on beamline I02 at the Diamond Light Source, on analysis of which phenix.xtriage (Zwart et al., 2008) indicated the need to lower the symmetry to P2_{1} with the monoclinic axis along the longest dimension and a β angle of 90.2°. The scaling statistics also agreed with the data merging better as monoclinic (see Table 4). The 222 symmetry in the κ = 180° section of the selfrotation function computed from these data in P1 is still apparent in Fig. 1, but the selfrotation function maxima are only 80% of the origin and along directions ω = 90.7°, φ = 90.6°, ω = 89.7°, φ = 0.0° and ω = 0.0°, φ = 0.0° and thus are neither as intense nor as orthogonal to each other as would be expected for orthorhombic crystals.

Indeed, reprocessing the data in P2_{1} improved the value of R_{meas} a.k.a. R_{r.i.m.} (see Table 4). This assumed two molecules per and with operators h, k, l and −h, −k, l, with the two twin fractions estimated at around 0.49–0.5. However, the intensity statistics still indicated problems with the data (see Table 5). After a few more unsuccessful attempts at refining the structure in P2_{1}, the symmetry was eventually lowered to P1, invoking four molecules in the and tetartohedral along the crystal axes (operators h, k, l; −h, −k, l; h, −k, −l; −h, k, −l).

In keeping with triclinic symmetry and pointing to the fact that the crystals are not monoclinic P2_{1}, the reflection 050 had nonzero intensity in more than one data set (Fig. 2 shows one such measurement). Although violations of the of higher symmetry space groups can be explained for example by multiple scattering (Renninger, 1937) and/or anisotropy of (Templeton & Templeton, 1980), in the context provided by the merging and intensity statistics, selfrotation function and molecularreplacement hits, the repeated measurements of such a reflection from more than one crystal sample were taken as additional evidence that the fI crystals were indeed triclinic.
The structure of the triclinic human complement fI crystals was eventually determined by sequential P1, searching against the tetartohedrally twinned intensities with Phaser and search models from homologous individual domains. The initial solution was followed by iterative model building in Coot (Emsley et al., 2010) and in REFMAC5 (Roversi et al., 2011). The four copies of the molecule in the cell are arranged in a pseudoorthorhombic packing which almost follows P2_{1}2_{1}2 symmetry except that the two molecules related by the twofold axis along c are also shifted with respect to each other by about 6 Å along the same direction.
inTight NCS restraints were initially used and gradually released during the course of model building and
whenever the current electron density showed surface loops and crystal contacts that differed in the four copies of the molecule these regions were omitted from the part of the structure that was NCSrestrained. In the final model, approximately 35% of the structure had to be excluded from the NCS restraints.Refinement statistics are reported in Roversi et al. (2011). The REFMAC5 estimates of the fractions at the end of the and building process are reported in Table 6. Table 6 also reports the R^{obs}_{twin} and R^{calc}_{twin} values (Lebedev et al., 2006; for the use of statistical agreement indicators on observed and calculated intensities in order to investigate and NCS, see also Lee et al., 2003). As expected, R^{obs}_{twin} < R^{calc}_{twin} for all operators, placing the factor I crystals in the regions of the RvR plot that is characteristic of twinned crystals with rotational (RPS; Lebedev et al., 2006).

6. Conclusions
The relatively recent occurrence in the literature of several cases of tetartohedrally twinned structures suggests that this form of
is not as infrequent as one might wish it to be (with the additional possibility that further tetartohedrally twinned structures may be lurking in the PDB, having been determined and deposited with the going undetected). Tetartohedral could happen to you too!Fortunately, careful analysis of intensity statistics and rotational symmetry can help to overcome the difficulties associated with this type of e.g. phenix.xtriage, to detect potential laws and guide the crystallographer towards the correct symmetry, laws and fraction. Once detected and characterized, tetartohedral is also relatively simple to handle thanks to a number of good macromolecular programs, notably CNS, the leastsquares program SHELXL97 and the latest version of the program REFMAC5. Tetartohedral is not a fatal disease. Only, to quote Petrus Zwart
even in the presence of the potentially confusing shared rotational NCS and symmetry. In addition, excellent statistical tools are now available in a number of dataprocessing/analysis programs,By now you should be a crystallographic hypochondriac
APPENDIX A
A1. operators coinciding with the rotational part of pseudosymmetry
A survey of the tetartohedrally twinned structures that have appeared to date in the literature suggested that in most cases the tetartohedral et al., 2008). In this Appendix, we derive a formula that illustrates the contributions to the diffracted intensity from the part of the structure that follows the and the part that does not and their interplay with the fractions.
operators are aligned with the rotational parts of the operators and that the latter in turn are close to true crystallographic symmetry, a situation known as (ZwartLet us write the electron density in the
of the crystal aswhere ρ^{NCS}(x) is the part of density in the that follows (NCS), i.e. electron density from a reference set of atoms and its NCSrelated copies, and ρ^{noNCS}(x) is the electron density for the remaining part of the which cannot be described using a reference copy and NCS operators. The portion of the electron density that obeys the NCS can be written using the J NCS operators starting from the electron density for the reference copy, labelled ρ_{1}(x),
where R_{j} is the rotation matrix and t_{j} is the translation vector of the jth operator.
The ρ_{cell}(x); following the above notation,
is the Fourier transform of the unitcell electron densitywhere the ith crystallographic operator G_{i} in the crystal G acts as
In other words, S_{i} is the rotation matrix and t_{i} is the translation vector of the ith spacegroup symmetry operator. Λ is the set of crystal lattice translations.
As we saw earlier, when the crystals are (pseudo)merohedrally twinned evaluation of the diffraction intensity (1) involves the calculation of the structure factors evaluated at reciprocallattice vectors rotated by the
operatorsLet us now assume that
Under these hypotheses, we have
In formulae (6), the notation {Tt} symbolizes the action of the operator defined by the rotation matrix T and the translational vector t, while {Ss} {Tt} means the result of acting sequentially first with the operator defined by T and t and then with the operator defined by S and s. These equalities show that under the hypotheses stated above and for all choices of spacegroup symmetry operator i, operator j and operator k, there exist operators labelled i′ and j′ that allow a simplification of the effect of a chosen operator k on the symmetry copy i of the NCS copy j.
Replacing this expression in (1) gives
where I^{noNCS}(h) = F^{noNCS}*(h)·F^{noNCS}(h) and I_{1}(h) = F_{1}*(h)·F_{1}(h).
The terms within the summation over the k in (8) describe the joint dependency of the observed intensity on the twin fractions and on the structural parameters. The part of the structure that does not follow the [ρ^{noNCS}(x) in (2)] contributes to both terms within the summation. The part of the structure that does follow it, besides mixing with the noNCS part within the summation, also makes a contribution to the intensity that does not depend on the twin fractions (the first term on the righthand side of equation 8). The larger the proportion of the structure following the the smaller the dependency of the measured intensity on the ratios. Notably, in the limit of no violations of the [i.e. F^{noNCS}(h) = 0] the diffracted intensity tends to the value computed for an untwinned crystal with pseudosymmetry.
Footnotes
^{1}Of course, can be also thought of as a very special case of powder diffraction, intensity statistics for which are discussed in Bricogne (1991).
^{2}The structure of MJ0729, a CBSdomain protein from Methanococcus jannaschii, was reported to suffer from tetartohedral (FernándezMillán et al., 2008), but it has not been fully refined nor deposited at the time of this writing and was therefore not included in Table 1.
Acknowledgements
PR and SJ were funded by the Wellcome Trust (083599) and MRC (G0400775) Project Grants to SML. We thank Garib Murshudov, Dale Tronrud and Jade Li for discussions on aspects of the work and the referees for helpful suggestions on the manuscript.
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