research papers Intensity statistics in twinned crystals with examples from the PDB 1. Introduction

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1 Acta Crystallograpica Section D Biological Crystallograpy ISSN Intensity statistics in twinned crystals wit examples from te PDB Andrey A. Lebedev,* Alexei A. Vagin and Garib N. Mursudov Structural Biology Laboratory, Department of Cemistry, University of York, Heslington, York YO10 5YW, England Correspondence lebedev@ysbl.york.ac.uk Entries deposited in te Protein Data Bank as of February 2004 for wic bot model and X-ray data were available were analysed to identify cases of twinning using suc simple statistics as te R factor between potential twin-related reflections. Careful consideration of all identified twins sowed tat in many cases twinning was ignored during structure solution and refinement. Manual analysis of te models sowed tat twinning often occurs in association wit rotational pseudosymmetry parallel to te twinning operator. Te coexistence of tese two penomena complicates te detection and diagnostics of twinning using currently available twinning tests. It was concluded tat a twinning-detection step sould be incorporated in every stage of structure analysis from data acquisition to refinement and validation. Received 7 November 2005 Accepted 8 November 2005 # 2006 International Union of Crystallograpy Printed in Denmark all rigts reserved 1. Introduction Te Protein Data Bank (PDB; Bernstein et al., 1977; Berman et al., 2002) is a ric source of biological, biocemical and structural information. It also offers templates for te determination of new structures by molecular replacement. Te uge number of models wit experimental X-ray data provides numerous training cases of varying difficulties useful to bot te practical crystallograper and software developers. Tese cases sould be analysed before approacing real-life difficult cases and, in an ideal world, all new software sould be tested against tem before general release. However, one sould be careful wen extracting information from te PDB because of several problems, some of wic ave been described by Kleywegt (1999, 2000). Currently, a new entry goes troug a careful validation procedure during deposition. Neverteless, at least one potential problem, twinning (Giacovazzo et al., 1992), as not yet been addressed. Twinning tests sould be included in te validation routine and, wen twinning is present, te structure-factor analyses need to be adjusted accordingly. If twinning is not taken into consideration during refinement, te resulting model will inevitably be degraded. Terefore, during deposition, it is important to notify te depositor if tis is te case. Te twinning penomenon in crystals as been recognized for a long time (Friedel, 1926). For small-molecule structures, data collection and processing, structure solution and refinement against data from twinned crystals are routine (Seldrick & Scneider, 1997). However, te situation wit macro- Acta Cryst. (2006). D62, doi: /s

2 molecules is not yet so straigtforward; te software used for data-acquisition and structure-solution procedures ave not addressed tis problem fully. For example, it is particularly difficult to solve a twinned structure using experimental pasing (Dauter, 2003; Rudolp et al., 2003). Te penomenon of twinning sould be considered as a special case of crystal intergrowt. Crystal clusters are often observed, but usually it is possible eiter to optimize crystallization to grow a single crystal or to break off a single-crystal fragment. In some cases tis simple approac does not work and one as to deal wit diffraction data from an intergrown crystal were te diffraction patterns of two or more fragments overlap. If te fragments are orientated in a random manner relative to eac oter te two lattices can be identified from te first images. [A very interesting case of treatment of suc data as been reported by Dauter et al. (2005) in wic te intergrown domains ave different space groups.] However, diffraction patterns from suc intergrown crystals are often deceptive; if te diffraction spots from te two (or more) crystal domains completely overlap, te diffraction pattern will appear normal on initial inspection. In tis case te measured observation at a given reciprocal-lattice point is in fact te sum of te twinned sets of intensities, weigted by te relative volumes ( twinning fractions ) of te different components. Tis is called (pseudo)meroedral twinning and te term twinning in macromolecular crystallograpy usually refers to tis. Te most common case seen in macromolecular crystallograpy is emiedral twinning, in wic tere are only two crystal components related by a twofold operator. However, te situation can be more complicated, as demonstrated by Barends et al. (2005). For two or more lattices to overlap completely, te unit-cell parameters and crystal symmetry must possess some special relationsips. Te unit-cell parameters must allow te possibility of iger symmetry tan te crystal actually sows. Tis is most common in tetragonal, trigonal or cubic crystal classes, were te twinning operator will be one of te symmetry operators of te supergroup. However, it is also possible for triclinic, monoclinic and ortorombic crystals wen te unitcell parameters possess some special properties (Giacovazzo et al., 1992). A tecnique for identifying data sets were te unit-cell parameters and space group can allow (pseudo)- meroedral twinning and finding te possible twinning operators is described by Flack (1987). In tese cases, to detect weter twinning as occurred requires statistical analysis of te wole data set. Wen a problem is detected two options are open: (i) to discard te data set and try to obtain a new untwinned crystal or (ii) try to solve and refine te structure using te twinned data. Wile te first option seems to offer better data, it may turn out to be time-consuming (or even impossible). Moreover, structural genomics imposes strong constraints on te time spent on an individual protein and option (ii), i.e. using wat is at and, is becoming more common, wit new software being developed to meet tis demand. Tis contribution analyzes te PDB to find out ow often suc a problem occurs and to generate ideas for te future automatic treatment of structures using data from twinned crystals. We also describe te major difficulties we faced in te identification of twinning in special cases using te widely available twinning tests. 2. Materials and metods Te PDB February 2004 release containing about structures was screened and only tose entries were bot coordinates and structures factors ad been deposited ( entries) were used in te analysis. Te unit-cell parameters and space group of tese entries were analysed using te tecnique described in Appendix A. If (pseudo)meroedral twinning was possible ten tis data set was selected for furter analysis. 5% deviation from ideal twinning constraints was allowed. Tis tresold is consistent wit Mallard s rule as cited by Grimmer (2003). If observed intensities were present in a CIF file tey were used directly and for oter applications tey were converted to structure factors using TRUNCATE (Frenc & Wilson, 1978). If only observed structure amplitudes were available, estimates of te corresponding intensities were generated, altoug some information must be lost. Tus, in all selected cases tere is at least one potential twinning operator. R twin, defined in (1), was calculated wit respect to eac operator for bot observed intensities and tose calculated from te atomic model. Te matrix for a potential twinning operator, selected from te coset of equivalent operators, and te associated twin were calculated using a program written by one of us (AAL). Te distribution of twin against R calc twin, referred to as an RvR plot and discussed below, can give a clear indication of twinning. Detailed analysis was carried out for all likely twinned structures. Te analysis involved estimation of te likely number of molecules in te asymmetric unit using SFCHECK, self-rotation function (Rossmann & Blow, 1962) as implemented in MOLREP (Vagin & Teplyakov, 1997), twinning tests based on overall reflection statistics, namely cumulative distribution of normalized intensity and moments of acentric reflections (Rees, 1980) as implemented in TRUNCATE, and H-tests (Yeates, 1997) as implemented in SFCHECK (Vaguine et al., 1999). If te interpretation of tese tests was ambiguous, ten molecular replacement using MOLREP and refinement using REFMAC (Mursudov et al., 1997) were carried out using te model from te PDB witout substrates and wit all atomic displacement parameters (ADP) reset to equal values. Te models, Patterson and electron-density maps were visualized using Coot (Emsley & Cowtan, 2004). Furter statistical analyses of te results were performed using te statistical package R (R Development Core Team, 2004). Some figures in tis paper are based on tose generated from CCP4 software (Collaborative Computational Project, Number 4, 1994). 3. RvR plot Detection of twinning sould ideally be performed at te stage of data acquisition before te crystal structure is known. Tis 84 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

3 task is not always trivial; for example, perfect twinning cannot be detected from merging statistics. In some instances, even finer (tan R merge ) statistical properties of te data are too ambiguous for assignment of crystal symmetry and detection of twinning prior to te structure determination. Terefore, we undertook an investigation of all possible twinning cases, known or undetected, deposited in te PDB. Te goal of te work was to understand te symmetry environments most frequently accompanying twinning and to pinpoint problems wit its detection. Since for tese data sets Figure 1 (a) Scematic view of RvR scatter plot. (b) Observed RvR scatter plot: red, (potential) meroedral twins; black, (potential) pseudomeroedral twins. Green ovals sow te area populated by cases wit TNCS (labelled A) and te areas corresponding to mislabelled data (labelled B and C). (c) Observed RvR scatter plot (enlargement of Fig. 1b): black, known to be untwinned and not analysed; blue, found to be untwinned after furter analysis; green, twins witout RPS; red, twins wit some degree of RPS, but te difference between te twinning and RPS operators may be quite large. (d) Middle blue curve, results after refinement of PDB entry 1nq, performed witout taking twinning into consideration, against simulated data sets wit twinning fractions in te range wit standard restraints on ADPs. Left red curve, te same calculations wit relaxed restraints on te ADPs. Rigt green curve, results before refinement. Here R calc twin 0.5. It is expected tat proper twin refinement would preserve tis value. Acta Cryst. (2006). D62, Lebedev et al. Intensity statistics in twinned crystals 85

4 bot te atomic model and te experimental data are available, te analysis is considerably simplified R factor wit respect to twinning operator Let us assume tat in a given crystal te combination of lattice and crystal symmetries allows twinning. Tis means tat tere is at least one potential twinning operator S twin. It can be determined using, for example, te tecnique described in Appendix A. Let R twin be te intensity-based R factor between reflections related by potential twinning operator S twin, P ji I 0j R twin ¼ P ði þ I 0Þ ; ð1þ were summation is over all unique reflections, suc tat intensities for bot and 0 = S twin ave been measured and 6¼ 0. Te definition of R twin (1) is similar to tat of R sym, except tat S twin is not an operator of te crystal point group, but belongs to te point group of te crystal lattice. twin are R twin calculated using observed intensities and (untwinned) intensities derived from te atomic model, respectively. Te relationsip between te two magnitudes is as follows (see also Appendix B) twin R calc twin ðno twinningþ; twin < R calc twin ðpartial twinningþ; twin 0 ðperfect twinningþ: ð2þ Te approximation sign in te equations above is a consequence of model and experimental errors. Our experience sows tat in te majority of te cases tese errors do not affect te qualitative conclusions. If te crystal symmetry as been misspecified 1 ten te analysis of unit-cell parameters and space group identifies missing elements of te point group of te crystal as twinning operators. In tis case it is expected tat bot twin 0 R calc ðmisspecified crystal symmetryþ: ð3þ twin 0 Note tat a small value of twin can be misinterpreted, as tis takes place in two different cases; see (1) and (2). In particular, false positives in detection of twinning can be found in some PDB entries wit misspecified symmetry (see x3.4.2) Twinning interfering wit NCS Let a crystal or an individual crystal of a twin possess noncrystallograpic symmetry (NCS) and let one of te NCS operators be suc tat its rotational component is approximately equal to te (potential) twinning operator. In tis case, te NCS could interfere wit twinning and is furter referred 1 We say tat crystal symmetry is misspecified wen te space group reported in te PDB file is a subgroup of te true space group of te crystal, e.g. P4 instead of P422. Accordingly, in suc cases te PDB file contains more molecules tan sould be in te asymmetric unit of te crystal, but some of tese molecules are actually related by te missing symmetry operator(s). to as rotational pseudosymmetry (RPS). Tere are two reasons wy twins wit RPS are of special interest. Firstly, a correlation between observations related by potential twinning operators could be caused eiter by RPS or by bot RPS and twinning. Te two cases cannot be discriminated by twin alone. Tese cases are particularly difficult for twinning detection prior to te structure determination. Secondly, we expect a relatively ig frequency of twins wit RPS because of ig likeliood of te following two mecanisms of teir formation. Te first mecanism assumes a cange of crystal symmetry (we are interested in symmetry reduction) wic is sometimes observed during crystallization, seeding, soaking, fast cooling and even data collection. It is pysically reasonable to expect tat tis transition starts simultaneously in several areas of te crystal. As a result, several identical domains are formed in two or more different orientations related by te broken symmetry element and tus te crystal becomes twinned and te broken symmetry element becomes a twinning operator. At te same time it becomes an RPS operator, relating molecules in te asymmetric unit wic were equivalent by crystal symmetry before te transition. In te second mecanism an individual crystal is formed by tigtly packed molecular layers wit symmetry tat is iger tan tat of te interfaces between tem. In tese structures te wole layers are (approximately) invariant wit respect to NCS operators. Consequently, any NCS operator in te layer at a twinning interface relates two twinning domains and tus te NCS is RPS. Te ig frequency of twinning interfaces in suc symmetry environments leads to statistical crystals (Bragg & Howells, 1954; Cocran & Howells, 1954). If tere is no NCS, no pronounced anisotropy and no serious experimental errors, te expected value of R calc twin can be estimated to be 0.5 as sown in Appendix B. However, wen RPS is present, te correlation between related reflections causes a decrease in R calc twin. Tus, in addition to (2) te following olds, twin 1=2 ðno RPSÞ; R calc twin < 1=2 ðrpsþ: ð4þ R calc Note tat even if RPS is present, (2) olds. Tus, despite te similar effects of RPS and twinning on twin, te availability of a crystal model in principle allows us to distinguis between twinning, RPS and twinning interfering wit RPS using suc simple statistics as twin. Te relations (2) (4) are illustrated in Fig. 1(a). Te figure sows areas corresponding to different combinations of RPS and twinning Selection of twinning cases Te simplest possible way to select twinning cases from te PDB would be to extract te relevant information from te PDB eaders and/or related papers. However, tis approac is not sufficient because te researcers depositing data and/or writing papers eiter may ave not noticed or not discussed twinning (false negatives) or may ave misinterpreted iger crystal symmetry as twinning (false positive). Terefore, it was 86 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

5 decided to analyse PDB entries directly. Tis direct approac may also lead to a better understanding of te problems wit te detection of twinning. We analysed unit-cell parameters and te reported crystal symmetry of entries present in te PDB at February 2004 containing bot an atomic model and X-ray data. Entries were twinning is impossible or were te data were corrupted and unreadable by our software were rejected from furter consideration. For te remaining 4086 entries, potential twinning operators were determined and twin were computed. If tere were two or more (non-equivalent) potential twinning operators (as, for example, in P3), ten tat wic gave te lowest value of twin was cosen. Tus, eac selected entry was caracterized by only two quantities, twin and R calc twin, and te corresponding point was drawn on te plot of twin versus R calc twin (RvR plot; see Fig. 1b). For eac structure represented in Fig. 1(b), we analysed weter te twinning, if present, is meroedral or pseudomeroedral using te tecnique described in Appendix A.Te points in Fig. 1(b) are coloured according to te results of tis analysis. All cases belonging to twinning areas in te RvR plot (Fig. 1a) were analysed in detail to validate te presence or absence of twinning and to caracterize te NCS if present. Te specific areas and some peculiarities of te RvR plot are discussed below. Table 1 Frequency of twinning in different symmetry environments. No. of twins Crystal symmetry or type of meroedry Total RPS TNCS RPS + TNCS DNA Detwinned P1 1 1 P C2 1 1 P P C Pseudomeroedral total P P P I4 3 2 P P P P P P H P P I Meroedral total Total Observed RvR plot Main cluster. A large cluster around (0.5, 0.5) corresponds to untwinned crystals wit no pronounced pseudosymmetry. However, twinning is not forbidden by te unit-cell parameters and space group and could occur for related crystals. Some of tese points correspond to data sets wic were detwinned before deposition. Tese cases were not included in furter analysis. In particular, all untwinned crystals in space groups P3 x, P3 x 21, P3 x 12, P4 x, I4 x, P6 x, P2 x 3, I2 x 3andF23 belong to tis area. Since te lattices of tese space groups ave iger rotational symmetry tan tat of crystals, no extra constraints on te unit-cell parameters are needed for twinning to occur (Giacovazzo et al., 1992; Sclessman & Litvin, 1995) Misspecified crystal symmetry. Te cluster at te origin corresponds to structures in wic te crystal symmetry is misspecified and is actually iger tan tat used in te refinement and reported in te PDB entry. Tus, bot twin and R calc twin are expected to be close to 0.0. Several randomly cosen cases from tis cluster ave been successfully refined in te iger symmetry space group. It is interesting to note tat for two of tem twinning was reported, presumably on te basis of te low twin; tese are examples of false positives. Te first reliable case of twinning as R calc twin = 0.2. At te same time, in some cases were te space group was misspecified R calc twin goes up to 0.3; tese are mainly low-resolution structures were it is easy to overfit te model and to generate significant differences between independently modelled symmetry-related molecules RPS. Te lower tail of te main cluster corresponds to untwinned crystals wit RPS. Most points in tis area are located on te diagonal, wit R calc twin twin in te range Tis tail extends along te diagonal down to about 0.2. Here we find one of te most extreme examples, te untwinned 1i1j (Lougeed et al., 2001), in wic te rootmean-square deviation of C atoms from te positions corresponding to iger crystal symmetry is about 0.15 Å Translational noncrystallograpic symmetry (TNCS). Te main cluster as an upper diagonal tail around (0.6, 0.6) corresponding to structures wit TNCS, in wic te set of TNCS vectors and consequently te modulation of intensities in te reciprocal space caused by TNCS are not invariant wit respect to S twin. Tus, te intensities related by S twin are modulated differently and te assumptions required for te relation (10) in Appendix B to be valid are violated. Te numerator in (1) for bot twin increases, increasing teir values. Among suc structures we observed no cases of twinning (note te empty area below tis cluster in te RvR plot) Mislabelled and corrupt data. Tere are some extra features on te RvR plot arising from mislabelling of columns in te CIF file. Two small clusters sown by circles in Fig. 1(b), wic are just above and below te main cluster, are wort mentioning. In te first one, at about (0.5, 0.4) on te RvR plot, te structure amplitudes are present in te CIF file but are labelled as intensities. In te second one, at about (0.5, 0.6), te intensities are labelled as structure amplitudes. Acta Cryst. (2006). D62, Lebedev et al. Intensity statistics in twinned crystals 87

6 Tese peculiarities may in teory be identified by simple statistical tecniques. However, if suc factors as twinning, pseudosymmetry or anisotropy affect te data or several deposition inaccuracies (for example, deposition of te Figure 2 Effect of te resolution cutoff on te experimental cumulative distributions of Z. Te plots were produced using X-ray data from PDB entry 1l2 (Rudolp et al., 2003). (a) Experimental cumulative distributions of Z for all te data, resolution range Å,(b) te experimental second moment of Z for acentric reflections versus resolution, (c) experimental cumulative distributions of Z in te resolution range Å,(d) completeness and R standard versus resolution. Te resolution range used in (c) is sown by green boxes in (b) and (d). Te tick blue and red curves correspond to centric and acentric reflections, respectively. Te tin black curves in (a) and (c) are te reference (teoretical) cumulative distributions of Z corresponding to (top curves) centric reflections in te untwinned case, (middle curves) centric reflections in te perfectly emiedrally twinned case (i.e. wit two twinning fractions) and acentric reflections in te untwinned case and (bottom curves) acentric reflections in te perfectly emiedrally twinned case. Te tin black curves in (c) are teoretical second moments of Z for acentric reflections versus resolution for (top line) te untwinned and (bottom line) perfectly emiedrally twinned cases. 88 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

7 detwinned instead of te measured data) are present simultaneously ten suc analysis becomes complicated, if possible Areas of te RvR plot indicating twinning is likely. Te points below te diagonal sould, at least in teory, correspond to twins. In particular, points tat deviate from te diagonal wit R calc twin significantly less tan 0.5 sould correspond to twins wit RPS (see x3.2) and wit an adequate refinement protocol, te deviation from te diagonal sould correlate wit twinning fraction. Te cases wit R calc twin in te range and twin in te range , as well as some randomly cosen cases from oter areas, were furter investigated (coloured circles in Fig. 1c). Te protocol of analysis included validation of te model, various twinning tests performed wit bot observed Figure 3 Te effect of RPS on te perfect twinning tests. Te plots were drawn using X-ray data from PDB entry 1i1j (Lougeed et al., 2001). (a) and (c) Sample cumulative distribution of Z and (b) and (d) second moment of Z for acentric reflections versus resolution for (a) and (b) original data and (c) and (d) data wit simulated emiedral twinning. Te colour legend is te same as for te similar plots in Fig. 2. Acta Cryst. (2006). D62, Lebedev et al. Intensity statistics in twinned crystals 89

8 and calculated intensities wit different resolution cutoffs and caracterization of te NCS. In particular, NCS operators if present were compared wit potential twinning operators to identify RPS. If tere was spatial pseudosymmetry, attempts were made to refine structures in te corresponding iger symmetry space group to ensure tat te reported crystal symmetry was correct. Te rectangular area of te RvR plot under consideration overlaps wit te areas discussed above and terefore includes a number of oter data sets wic turned out to be untwinned but wic ad special features suc as misspecified symmetry, mislabelled structure amplitudes or wic displayed RPS and terefore lay on te diagonal wit bot twin below cases of twinning ave been identified, verified and caracterized. Tey are marked in Fig. 1(c) wit red and green circles corresponding to te presence and absence of RPS interference, respectively. Tese cases are furter discussed in te next subsection Twinning cases All te structures and teir data belonging to te twinning areas of te RvR plot (Figs. 1a and 1c) were analysed in detail to identify actual twinning cases. Table 1 contains symmetry and NCS information for te 78 twinning cases identified wit a ig degree of confidence. NCS for DNA structures was not analysed. Tere are two features of tis table tat are wort mentioning. Firstly, pseudomeroedral twinning is not unusual. Secondly, te cases were twinning interferes wit RPS are more frequent tan simple twinning, especially for te pseudomeroedral twins. One of te important practical conclusions from tese analyses is as follows. If te perfect twinning tests sow te presence of twinning, it does not necessarily mean tat te twinning is meroedral. Tus, even if data from a perfect twin ave point-group symmetry P422, it is not necessary, neiter teoretically nor in practice, tat te point-group symmetry of te individual crystal is P4 and tat te twinning is meroedral, generated by twofold axis ortogonal to crystallograpic fourfold. For example, te crystal symmetry of 1upp (Karkeabadi et al., 2003) is C222 1 and tere are two twinning domains, wit one possible coice of twinning operator being a fourfold axis along one of te crystallograpic twofold axes False negatives It is interesting to note tat only one tird of te cases identified as twins by our analysis were reported as suc in te PDB submission, altoug in some of tese cases analysis of intensities derived from atomic models sows tat te twinning was actually taken into account during refinement. Neverteless, in a significant number of cases tis was not done. Te effect of ignoring twinning on R calc twin is illustrated by te following simulated experiment. Te 3.1 Å data from an untwinned crystal were artificially twinned to produce six data sets wit twinning fractions of 0.0, 0.1,..., 0.5. Te model from te PDB was refined against all data sets following te same protocol, witout model rebuilding and ignoring twinning. twin were computed for tese six data sets and for te intensities calculated from te appropriate refined models. Te result is sown as te central blue curve in Fig. 1(d). If twinning ad been properly taken into account during refinement ten R calc twin would remain constant trougout all tese refinements (vertical green line on te rigt in Fig. 1d). Note tat incorrect refinements ave been carried out starting from te correct model. Even in tese cases te points clearly drift towards te left on te RvR plot. Since real-life crystal structure solution requires many cycles of refinement alternated wit model building, it is anticipated tat tis drift to te left is muc more serious tan in tis simulation. To analyse tis trend furter refinements were carried out wit relaxed restraints on ADPs. Te results are plotted in red in Fig. 1(d) and sow furter reduction of R calc twin. Tis simulated experiment elps to explain wy only some of te twinning cases witout RPS (green points in Fig. 1c) sow R calc twin 0.5. In all of tem proper refinement accounting for twinning as been performed. In some twinning cases witout RPS, R calc twin is significantly less tan 0.5 and, judging by te simulated results sown in Fig. 1(d), we expect tat te refinement protocol was not adequate. Te above simulated experiment is one of te cases were te so-called model bias arises because of an insufficient number of parameters and te addition of only one extra parameter, te twinning fraction, would substantially reduce it. Generally speaking, model bias is not so muc a consequence of a large number of parameters, but of incorrect parameterization; bias is best corrected by re-parameterization of te model rater tan by removing a part of it. 4. Performance of twinning tests During te verification of twinning in te cases selected using te RvR plot a number of problems were encountered. Some of tese problems are of a general nature and are worty of special attention. Tis section discusses te influences of experimental error and pseudosymmetry on perfect twinning tests and te influence of RPS on one particular partial twinning test, te H-test Effect of experimental errors on te perfect twinning tests In te perfect twinning tests, te observed intensities normalized witin resolution sells are assumed to be sampled from te one-dimensional distributions of te random variable Z. Two different distributions are considered, one for centric and one acentric reflections. Derivation of tese distributions (Rees, 1980) is based on te Wilson distribution of structure factors (Wilson, 1949) for untwinned crystals. Two of te major tests are based on comparison of te teoretical and observed curves of (i) te cumulative distribution of Z versus Z and (ii) 90 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

9 te second moment of Z versus resolution, sown in Figs. 2, 3 and 4. It is necessary to use sensible resolution cutoffs to be able to draw any reliable conclusions from tese tests. Te reason for tis is tat ig-resolution reflections as a rule ave larger experimental errors, but te teory does not take tese into account. Our experience sows tat te low-resolution cutoff is not necessary; owever, it is important to remove ig-resolution data, were R standard = (F)i/Fi starts growing and/or were a large variation of te second moment of Z for acentric reflections is observed. Te required plots are available from various software, e.g. TRUNCATE and SFCHECK. An example of ow tis rule of tumb works is sown in Fig. 2. In tis example, te experimental cumulative distribution of Z clearly indicates perfect twinning wit a ig-resolution cutoff at 2.2 Å. In contrast, te same test but wit all data is misleading and te experimental curves are close to te teoretical curves for untwinned crystals. It is important to empasize tat ig-resolution reflections do contain useful information about te structure despite a resolution cutoff being needed for some applications Effect of RPS on te perfect twinning tests Our experience sows tat RPS affects perfect twinning tests only in te presence of twinning, wen it partially compensates for te effect of twinning. Te following numerical experiment illustrates tis effect. Te 1i1j X-ray data set represents an untwinned crystal wit RPS. Te data set wit perfect twinning was simulated from te original untwinned data. All parameters except te twinning fraction are te same in te two data sets. Te experimental second moment of Z versus resolution and te cumulative distribution of Z are sown in Fig. 3. Te experimental curves for te original data set matc teoretical predictions (Figs. 3a and 3b); owever, tis is not so for te simulated data set, were only a marginal deviation from te teoretical curves for untwinned data towards tose for perfect twins is observed (Figs. 3c and 3d). Tis example demonstrates tat te simple teory based on Wilson s distribution assuming uniform distribution of atoms in te asymmetric unit fails for twins wit RPS. Suc beaviour can intuitively be understood by imaginary traversing of te RvR plot (Fig. 1). If we travel from te main cluster at (0.5, 0.5) towards te origin along te diagonal, we start from a point witout anomalies of any kind and finis at te point were crystal symmetry is iger tan reported symmetry but also wit no anomalies. At bot ends we ave untwinned data statistics and te same statistical distributions could be expected all along te diagonal patway (were untwinned data sets wit RPS are located). Anoter limiting pat is from te point (0.5, 0.0), below te main cluster, towards te origin along te abscissa. On tis pat te transition occurs from perfect twin statistics to untwinned statistics. Te above example wit simulated twinning is located on tis pat at a point were tis transition is almost accomplised. Tis beaviour of perfect twinning tests as been observed in a number of real cases were RPS interferes wit twinning (Table 1; see also an example in Dauter et al., 2005). Te closer te NCS operator generating RPS comes to an operator of iger space group, te less contrast tere is between te results of perfect twinning tests for untwinned and twinned data. Tis lack of contrast creates difficulties for diagnostics. Fortunately, te atomic structure in suc circumstances can frequently be solved and refined to a first approximation in a iger symmetry space group and wen te refinement sticks at an unreasonably ig R factor, te structure can be resolved and furter refined in te correct space group. In tis scenario problems wit uncertain diagnosis are avoided, but it is necessary to collect data in te lower symmetry space group and to keep tem unmerged. Reliable diagnosis of twinning terefore becomes an important component of bot data collection and refinement. Te effect of RPS on intensities decreases and terefore effect of twinning becomes more pronounced in iger resolution sells, were te intensities are affected by te small difference between NCS-related molecules. However, as noted above te data in iger resolution sells are less reliable for twinning tests because of experimental errors (see x4.1) TNCS and twinning Table 1 sows tat wen twinning and TNCS coexist te tird ingredient, RPS, is usually also present (see, for example, PDB entry1upp; Karkeabadi et al., 2003). Tis is not surprising because te reduction of symmetry resulting in twinning must involve reduction of crystal point group, i.e. formation of RPS. In tese structures iger point-group symmetry and sorter crystallograpic translations can be accommodated by small, sometimes less tan 1 Å, displacements of atoms. Tus, te modulation of intensities in te reciprocal space caused by TNCS can be considered in terms of sublattices wit different mean intensities (pseudocentering). Note tat in tese structures te sublattices are invariant wit respect to twinning operator (compare wit x3.4.4). Te modulation of intensities owing to TNCS as an effect on te perfect twinning tests wic is opposite to tat of twinning. Tis effect is present in bot twinned and untwinned crystals, e.g. te second moment of Z for acentric reflections becomes greater tan two in te absence of twinning. Te effect becomes stronger wen te deviation from iger crystal symmetry decreases. Demodulation (normalization accounting for TNCS) or examination of te separate sublattices may reduce te effect of TNCS, but te effect of RPS remains. For different sublattices, te effect of te RPS is different and depends differently on te deviation from iger crystal symmetry, but it always partially compensates for te effect of twinning. Acta Cryst. (2006). D62, Lebedev et al. Intensity statistics in twinned crystals 91

10 Te analysis of te data in terms of sublattices can be avoided by using te perfect twinning test suggested by Padilla & Yeates (2003) and implemented in DATAMAN (Kleywegt & Jones, 1996). Wit a proper resolution cutoff tis test indicates perfect twinning, but te contrast is less tan tat teoretically predicted (Padilla & Yeates, 2003). Te presence of RPS in most twins wit TNCS explains tis observation, as te effect of RPS partially compensates te effect of twinning even if te modulation of intensities owing to TNCS is accounted for Partial twinning tests Tere are two partial twinning tests most frequently used in macromolecular crystallograpy, te Britton test (Britton, 1972) and te H-test (Yeates, 1997). In tese tests te data are assumed to be processed in te correct group or its subgroup and not over-merged in a iger group. Tese tests are applied to a given potential twinning operator suggested by te crystal and lattice symmetries. Terefore, all nonequivalent twinning operators (e.g. tere are two of tem in P3) ave to be tested individually. Unfortunately, neiter of tese tests can distinguis between iger symmetry and perfect twinning. Neverteless, in te case of partial twinning tey bot indicate twinning and estimate te twinning fraction. We discuss te H-test in more detail wit an empasis on te effect of pseudosymmetry on its beaviour. In te H-test te joint two-dimensional distribution of te intensities related by potential twinning operator is of interest. Tus, extra information is used compared wit te previously discussed twinning tests, wic are based on one-dimensional distribution of intensities derived from te Wilson distribution. Te idea of tis test is tat te cumulative distribution P(H) of a random variable H is a straigt line over te wole range of possible H. In teory, wic unfortunately does not take account of te effects of RPS, te linearity olds for bot twinned and untwinned data and te slope of te plot P(H) Figure 4 Cumulative distributions of H for tree twins: (a) 1rxf (Morgan et al., 1994), (b) 1ku5 (Li et al., 2002) and (c) 1gwy (Manceno et al., 2003). In 1ku5 and 1gwy, RPS interferes wit twinning. Experimental distributions are represented by red dotted lines. Te intensities derived from atomic models were used to simulate cumulative distribution of H for different twinning fractions (tick blue lines), te values of te twinning fractions being given over te corresponding lines. 92 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

11 versus H depends on te twinning fraction (blue lines in Fig. 4a). In te original version of te H-test te linearity is essential, as te twinning fraction is estimated from mean value of H. However, teoretically impossible difference between intensities related by twinning operator may appear as a result of radiation damage to te crystal if tere was a long time interval between te two measurements. Too large differences could also occur if te X-ray beam was focused at different parts of te crystal during tese two measurements. Te presence of suc outliers distorts te experimental distribution of H at larger H and causes non-linearity as in Fig. 4(a). Tis type of non-linearity is typical of twins witout interfering RPS, altoug te range of H were te plot deviates from te straigt line varies. Suc cases can be treated by a modified H-test in wic te twinning fraction is estimated using te slope of te plot at te origin (Yeates & Fam, 1999). If RPS interferes wit twinning ten all pairs of reflections involved in te H-test are affected and te cumulative distribution of H becomes non-linear over te wole range of te argument (Fig. 4b) and bot versions of H-test fail to give a reasonable estimate of te twinning fraction. In cases similar to tat in Fig. 4(b), owever, te twinning fraction can be estimated from te value of H at te point were te experimental curve approaces te line P(H) = 1. In tis formulation te H-test is equivalent to te Britton test. A disadvantage of suc a formulation is tat te estimate of twinning fraction is based on te rigt tail of te distribution, wic can be seriously corrupted by te effects of experimental errors mentioned above (see Fig. 4c). Tus, furter improvement of te test can only be acieved by accurate modelling of te effect of RPS, TNCS and anisotropy and by accounting for outliers, wile keeping te advantage of te original version of te H-test in wic te wole data set is utilized. 5. Conclusions Tis analysis of te PDB sows tat combinations of crystal and lattice symmetries can allow twinning in more tan 30% of cases, wit bot meroedral and pseudomeroedral cases widespread. For easy identification of twinning, te RvR plot was designed, wic utilizes bot observed intensities and intensities derived from te atomic models. Careful analysis of suspected twins identified from tis plot as flagged 78 cases wit a ig degree of confidence. However, since refinement of te atomic model ignoring twinning causes model bias and tus distorts tis picture, we expect tere may be more actual twins. Moreover, since twinning is one of te factors tat often prevents structure solution, tere are almost certainly many cases of twinning tat ave not been fully analysed and deposited in te PDB. Analysis of all te identified cases sowed tat RPS coexists wit twinning more frequently tan we expected, affecting te intensity distributions and tus increasing te difficulty of detecting twinning and ence te analysis of te structure. We found tat all twinning tests can fail to give convincing results. Te situation becomes even more serious wen TNCS is added to te picture. Ideally, one sould also consider oter crystal-growt anomalies, suc as statistical crystals, nonmeroedral twinning and split crystals. As a result of in-dept analysis of te identified twinning cases, we arrived at te conclusion tat it is important to ceck for tis and oter crystal-growt anomalies at every stage of structure analysis: starting from data acquisition and ending wit refinement and validation. To do tis correctly, it is important to build a model accounting for various abnormalities and utilizing all te information available up to te current stage. For example, at te data-collection stage an awareness of twinning may elp to coose te correct strategy; during refinement proper modelling of tis penomenon can reduce te noise in te electron density and ence elp to reveal finer details of te molecular structure. APPENDIX A A1. Algoritms used in te determination of twinning operators and teir type of meroedry Several autors (Flack, 1987; Le Page, 2002; Grimmer, 2003) ave already described te automatic identification of potential twinning operators using unit-cell parameters and space group. A necessary step in all tese algoritms is reducing te cell to a minimum primitive cell, eiter a Buerger or Niggli cell (see, for example, Migell & Rodgers, 1980, and references terein). Here, we describe an algoritm designed by one of us (AAL) and implemented in a set of routines. Given unit-cell parameters and crystal symmetry, potential twinning operators are determined as follows. Te cell is reduced to te primitive cell wit te sortest unit-cell edges and te point-group operators derived from crystal symmetry are transformed accordingly to give G, a group of 3 3 matrices wit elements from { 1, 0, 1}. Te set of 504 matrices wit elements from { 1, 0, 1} of finite order wit respect to matrix multiplication and wit determinant equal to one is ten generated. Tis set includes all operators of all rotational point groups expressed in fractional coordinates, provided tat te unit cell wit sortest edges is cosen. Tese operators are sorted according to te perturbation tat tey cause to te metric derived from te primitive unit-cell parameters. Te best 24 or less generators satisfying te perturbation tresold of 5% are ten used sequentially according to te above sorting order to expand G to H, te rotational point group of te lattice, H ¼ G [ Ga [ GaGa [...; were a is te tested generator. Expansion (5) is carried until a(ga) n contains no new elements or a new element is inconsistent wit H being finite group. Any new consistent q 2 aðgaþ n generates coset Gq of new elements tat are added to existing subset of H. Tis procedure simultaneously produces H and its coset decomposition wit respect to G. Representatives of te cosets, one from eac excluding G are potential (non-equivalent) twinning operators. ð5þ Acta Cryst. (2006). D62, Lebedev et al. Intensity statistics in twinned crystals 93

12 To draw Fig. 1(b), we also analysed te type of meroedry of potential twinning operators using te following metod. Let G be a rotational point group and M be a metric represented as a set of 6 6 matrices and as a 6-vector, respectively. Let M be invariant wit respect to G, i.e. gm ¼ M; g 2 G: ð6þ Consequently, te projector ¼jGj 1 P g2g g is suc tat M = M. Let R be a 6 6 matrix representing a potential twinning operator. If R ¼ ; ð7þ ten RM ¼ RM ¼ M ¼ M and no constraints are needed for M to be invariant wit respect to R in addition to tose imposed by (6). Terefore, if (7) olds, ten te twinning generated by R is meroedral. Tis type of meroedry ceck requires no tables and can be performed in integers if te 6 6 matrix representation of G is generated from its 3 3 matrix representation in fractional coordinates. APPENDIX B Consider, for example, a treefold twinning operator S twin relating tree twinning domains. Let 0 = S twin and 00 = S twin 0 and, 0 and 00 be different (be in a general position wit respect to S twin ) and corresponding intensities I obs, I obs, 0 Iobs be measured. For twinning fractions 1, 00 1, 1 and neglected errors, I obs I obs 0 I obs 00 1 ¼ 1 þ 2 þ 3 ; ¼ 1 I calc ¼ 3I calc ¼ 2I calc þ 2 I calc 0 þ 1 I calc 0 þ 3 I calc 0 þ 3I calc 00 ; þ 2I calc 00 ; þ 1I calc 00 : ð8þ Relations (2) can be verified as follows. In te absence of twinning, 1 = 1 and 2 = 3 = 0, and terefore for all wit I obs measured, I obs = I calc and ence twin = R calc twin. For perfect twinning 1 = 2 = 3 and I obs twin =0. For partial twinning = I obs 0 =Iobs 00 and ji obs I obs 0 jþjiobs 0 Iobs 00 jþjiobs 00 Iobs j ¼j 1 ði calc I calc Þþ 2ðI calc 0 0 Icalc Þþ 3ðI calc Icalc Þj þ... 1 ji calc I calc jþ 2jI calc 0 0 Icalc jþ 3jI calc Icalc jþ... ¼ð 1 þ 2 þ 3 ÞðjI calc I calc 0 jþjicalc 0 Icalc 00 jþjicalc 00 Icalc jþ ¼jI calc I calc 0 jþjicalc 0 Icalc 00 jþjicalc 00 Icalc j: ð9þ If intensities are not equal to zero, ten te two sides of te above relation are equal only if I calc = I calc = 0 Icalc. Hence, 00 assuming tat te crystal symmetry is correctly specified and tus tere are at least some non-zero nonequal triplets of calculated intensities, we ave twin < R calc twin. Relations (2) can be similarly derived for any kind of twinning, including te usual case of two twinning fractions. Let us estimate te expected value of R calc twin (in te absence of any twinning) defined in (1), assuming tat (i) tere is no RPS and (ii) te overall ADP tensor and te set of TNCS vectors (if TNCS is present) are invariant wit respect to S twin. Formally, tese mean (i) mutual independence of all random variables I and (ii) identical exponential distribution of random variables I and I 0, 0 = S twin. In particular, te random variables I and I 0 possess te following joint probability distribution density pði ; I 0Þ¼pðI ÞpðI 0Þ¼ 2 expð I I 0Þ; I > 0; I 0 > 0; ð10þ were te multipliers ( )ati and I 0 are te same. For new variables r ¼ I þ I 0; s ¼ I I 0; ð11þ te joint probability distribution density is pðs ; r Þ¼ expð 2 r Þ; r < s < r ; ð12þ and, in particular, te conditional probability distribution density of s given r is pðs jr Þ¼ pðs ; r Þ ¼ 1 ; r pðr Þ 2r < s < r : ð13þ Tus, te expected value of s given r is Eðjs j r Þ¼ 1 R r js 2 r j ds ¼ r r 2 : ð14þ Finally, 2P 3 2P js j Eðjs j 3 r Þ EðR twin Þ¼E4 P 5 ¼E4 P 5 ¼E 1 ¼ : r r ð15þ Te last equation means tat R twin averaged over all possible structures obeying te above conditions (i) and (ii) equals one alf exactly. For a particular structure, tis means te approximate equation in (4). Tese calculations can also be applied to two unrelated structures, as was performed by Srinivasan & Partasaraty (1976) for a similar problem but for te conventional R factor. It is important to stress tat tis interpretation does not mean tat in te X-ray experimental data te resolution sells wit R merge iger tan 50% are useless. In te case of experimental data, experimental errors are necessarily present and teir distribution is different from tat used to derive te above relation. Te estimation of te resolution cutoff is a completely different problem and as to be approaced using different notions, suc as te informational content of te data or te informational content of te data per unit of syncrotron time. 94 Lebedev et al. Intensity statistics in twinned crystals Acta Cryst. (2006). D62, 83 95

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