International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 
International Tables for Crystallography (2012). Vol. F, ch. 11.7, pp. 311316
https://doi.org/10.1107/97809553602060000836 Chapter 11.7. Detecting twinning by merohedry^{a}Department of Chemistry and Biochemistry, University of California, Los Angeles, PO Box 951569, Los Angeles, CA 90095–1569, USA The chapter defines various kinds of twinning and focuses on the detection and analysis of twinning by hemihedry. The perfect overlap of lattices causes distinct intensities to be added together, thereby affecting the intensity distribution as well as the apparent symmetry properties of the crystal, to a degree determined by the twin fraction. Equations are presented for identifying the presence of twinning and estimating the twin fraction from an analysis of diffraction intensities. Complicating factors, including anisotropy, pseudocentring and local noncrystallographic symmetry, are discussed. 
Twinning is a growth disorder in which different regions or domains of a crystal specimen are oriented differently with respect to each other (reviewed in Yeates, 1997; Dauter, 2003; Parsons, 2003; Helliwell, 2008). Numerous types of twinning are possible and a hierarchy among the types has been articulated (Yeates & Fam, 1999) (Fig. 11.7.1.1). In the most general type, differently oriented twin domains in the crystal have lattices that are distinguishable from each other. As a result, the different domains are associated with distinct reciprocal lattices and therefore with distinguishable interpenetrating diffraction patterns. This type of twinning is therefore generally recognized in Xray diffraction images. The situation is referred to as twinning by nonmerohedry and has also been referred to occasionally in the literature as `epitaxial twinning'. However, the latter term is usually used to describe the growth of a crystal of heterogeneous molecules. Treatment of twinning by nonmerohedry usually amounts to a datacollection problem, as only a fraction of the diffraction spots in a given image correspond to a single lattice. A further complication can arise from the presence of a subset of reflections (e.g. those lying on an axis of rotation that relates the twin domains) whose multiple distinct contributions from the different domains overlap in the diffraction pattern. Strategies for unitcell determination and reflection integration have been developed and implemented in various Xray data processing programs (Sparks, 1999; Sheldrick, 1997, 2002, 2004, 2008; Duisenberg, 1992; HerbstIrmer & Sheldrick, 1998; Duisenberg et al., 2003; Dix et al., 2007). Those programs have been developed mainly in the context of smallmolecule applications, but are often suitable for problems involving macromolecular crystals as well. We do not consider twinning by nonmerohedry further in this chapter, but focus instead on a special category (Fig. 11.7.1.1) known as twinning by merohedry. The synonymous term `merohedral twinning' is also widely used in the macromolecular literature, although it has been noted that the latter usage introduces potential confusion in that the adjective `merohedral', or `mérohèdre' (Friedel, 1926), is used in mathematical crystallography in a somewhat different context as a specific property of certain symmetries.
In twinning by merohedry, the lattices of the differently oriented domains are superimposable, although the operations that relate the domains are not operations of the spacegroup symmetry (Fig. 11.7.2.1). This is possible only in special symmetries, namely those where the rotational symmetry of the lattice exceeds the underlying rotational symmetry of the crystal space group; we consider only rotational operations here, in order to focus on situations relevant to biological macromolecules. Symmetries where this occurs are listed in Table 11.7.2.1; they include crystal point groups 4, 3, 312, 321, 6 and 23. Space group P4 provides a simple example. It requires a tetragonal lattice, and that lattice contains twofold axes of rotational symmetry (perpendicular to the fourfold axis) which are not utilized by the P4 symmetry of the crystal. Therefore, two separate crystal domains, each obeying P4 symmetry, can grow together within the same specimen in a way such that they are rotated with respect to each other by one of these twofold symmetry axes of the tetragonal lattice. The operation used to describe the rotation is referred to as the twin operation. Because the lattices of the two domains are indistinguishable, the separate domains give rise to diffraction patterns that exactly superimpose in reciprocal space. Therefore, multiple distinct reflections (e.g. one from each of the two twin domains) fall precisely on top of each other (Fig. 11.7.2.1). Because warning signs are usually not present during inspection of diffraction patterns, twinning by merohedry must be detected after data collection, and special steps must be taken during structure determination if unaffected crystals cannot be grown.

The case of space group P4 described above is an example of the simplest type of twinning by merohedry, namely where only two distinct domain orientations are possible. The case of two domain orientations (n = 2) is referred to as twinning by hemihedry. We focus mainly on hemihedry in this chapter but provide brief comments on higherorder cases, such as tetartohedry (n = 4), at the end. Another distinct category of twinning arises in crystals where the lattice very nearly obeys some higher rotational symmetry, not by virtue of a strict requirement, as in P4 and its tetragonal lattice, but because of a fortuitous geometry of the unit cell. These cases are referred to as twinning by pseudomerohedry. A simple example is provided by a monoclinic crystal with the unique β angle very nearly equal to 90°. The lattice is nearly orthorhombic, in which it is possible to have two domain orientations related by a twofold rotation about one of the principal axes perpendicular to the crystallographic axis of twofold symmetry. Situations of twinning by pseudomerohedry are relatively common (Rudolph et al., 2004; Hamdane et al., 2009). The method of treatment is similar to those of typical merohedry, but detection requires special attention since the phenomenon depends on specific unitcell geometries. Modern software programs are able to detect cases where twinning by pseudomerohedry is possible by identifying potentially higher metric symmetry within an apparently lowersymmetry unit cell (Adams et al., 2002, 2004; Zwart et al., 2005b). Yet another special case is reticular twinning, wherein only a subset of reflections are superimposed on others by the twinning operation while the remaining reflections are not overlapped (HerbstIrmer & Sheldrick, 2002). The simplest case is space group R3, twinned about a twofold axis along c (for example); reflections for which l = 3n overlap with other reflections in the same layer under the twin operation, while reflections in the other layers do not overlap. To date, reticular twinning is not well documented in macromolecular crystals and is not covered further here.
Twinning is distinguished from other types of disorder by the length scale over which the disorder occurs. In cases of twinning, individual twin domains behave as ideal crystals whose sizes are presumed to be large compared with the coherence length of the Xray beam. Therefore, the scattering from twinned specimens behaves as a sum of intensities from the separate twin domains, rather than as a sum of complexvalued structure factors; the Xray scattering from one domain is presumed not to interfere with the scattering from the other domain(s). The organization of differently oriented domains in twinned crystal specimens has been studied in select cases (Efremov et al., 2004; Ko et al., 2001), but is generally not well understood for macromolecular crystals. Based on the behaviour of twinned crystals of small molecules, various types of physical arrangements are likely.
The twin fraction describes the fractional volumes of the different domain orientations within a crystal specimen. In the case of twinning by hemihedry, just one twin fraction, α, needs to be specified. Its value is taken to be the minor component, meaning a value less than or equal to 1/2. The twin fraction dictates the relative weights with which two twinrelated reflections contribute to each observed intensity. The basic system of equations is where h_{1} and h_{2} are twinrelated reflections, I represents the true crystallographic intensity of a reflection (i.e. the square of the structurefactor magnitude) and I_{obs} represents an observed intensity (Fig. 11.7.2.1).
The value of the twin fraction has a critical effect on how twinning manifests itself in practice. Two essentially distinct scenarios arise. First, if the twin fraction is significantly less than 1/2, a situation typically referred to as `partial twinning' occurs, and partial higher symmetry may be evident during data processing and analysis. For example, if the true space group is P4 and the crystal specimen is partially twinned, one might find that the merging R value is lower than random when attempts are made to reduce the data under the higher tetragonal symmetry P422. Similarly, when the data set reduced in P4 is analysed using a selfrotation function, partial higher symmetry may be evident along what would be symmetry axes in P422. However, such symptoms can also be consistent with an alternative explanation, namely local or noncrystallographic symmetry (NCS) nearly coincident with potentially higher crystallographic symmetry (P422 in the present example). Discriminating partial twinning from pseudocrystallographic NCS is therefore a key problem in analysing data in space groups that support twinning.
In the second scenario, the twin fraction α is very nearly 1/2, giving rise to a situation typically referred to as `perfect twinning'. In that case, pairs of twinrelated observations are necessarily equal. The recorded data set therefore obeys higher symmetry. For example, if a crystal in P4 is perfectly twinned by merohedry, the data set will obey 422 symmetry. The challenge in a case of perfect twinning is to recognize that the correct spacegroup symmetry is lower than that implied by the observed intensity data.
Unusual crystal morphology sometimes provides an early suggestion of twinning. In particular, concave crystals or crystals having `reentrant' faces are suspect. However, most cases of twinning in macromolecules are not identified until structure determination is in progress. It is relatively rare, but in some cases twinning can be identified decisively when an impossibly dense molecular packing would be required to generate a crystal with symmetry as high as that implied by the observed data (Redinbo & Yeates, 1993). Sometimes, the possibility of twinning is raised during atomic refinement when, for unexplained reasons, the refinement R values cannot be brought down to acceptable limits. However, it should be emphasized that problematic refinement is never sufficient evidence by itself to conclude that a crystal is twinned; instead, a careful analysis of the observed data is required. Modern datareduction programs now incorporate various statistical analyses for this purpose, making it relatively straightforward to identify the presence of twinning before proceeding further with structure determination.
Twinning has two related but distinct effects on intensity data. First, the observed intensities of twinrelated reflections tend to be more similar to each other than expected for independent reflections; in the limit of perfect twinning they become equal. Second, the overall statistical distribution of intensities is affected, so that classic Wilson statistics are not obeyed. Therefore, two essentially different kinds of tests exist to analyse intensity statistics for the presence of twinning.
Twinning causes a shift in the typical intensity distribution from an exponential to a sigmoidal shape (Fig. 11.7.6.1a). Various measures of the intensity distribution have been used to diagnose twinning. The Wilson ratio, <F>^{2}/<F^{2}> (where the angle brackets denote the average), is 0.785 for ordinary acentric data and 0.885 for perfectly twinned data (Stanley, 1972; Wilson, 1949). Similarly, the value of <E^{2} − 1> (where E^{2} is defined as F^{2}/<F^{2}>) is 0.736 for ordinary acentric data and 0.541 for perfectly twinned data (Bruker, 2000). The moments of the intensity distribution have simple, rational values. For example, the ratio <I^{2}>/<I>^{2}, sometimes referred to as the Stanley factor, is 2.0 for ordinary acentric data and 1.5 for perfectly twinned data (Stanley, 1972). For partial twinning, intermediate behaviour is observed and complex expressions for the resulting intensity distributions have been worked out (Rees, 1980). A key difficulty with these statistical tests is that they assume that the members of the set of reflections being evaluated all have the same expected value. If the diffraction is nearly isotropic, this condition can be satisfied relatively easily by normalizing the reflection intensities in thin resolution shells. In contrast, anisotropy presents a problem that must be corrected if these traditional tests are to be useful. If the anisotropy has a relatively simple form (e.g. if it follows an elliptical Gaussian function), then correction may be straightforward, but otherwise correction can remain problematic. Crystals in which distinct molecules in the asymmetric unit are related by translational shifts create similar problems: classes of strong and weak reflections are present in a single resolution shell. Cases of anisotropy and translational pseudosymmetry both give intensity distributions that are shifted counter to the shift caused by twinning, and can therefore mask its presence.
One method for examining overall intensity statistics in a way that seeks to overcome the challenges described above has been described (Padilla & Yeates, 2003). Reflections near each other in reciprocal space are considered in pairs. The difference between the two intensities divided by their sum is taken as the quantity to be analysed, where L is the socalled local intensity difference, and I_{obs,A} and I_{obs,B} are the observed intensities of two reflections that are not related by any crystal symmetry or potential twin operation. Choosing the reflection pairs to be near in reciprocal space ensures that anisotropy effects will not affect their expected values differently. Countering translational pseudosymmetry is a more difficult problem, but common cases of pseudocentring or unitcell doubling can be treated effectively by requiring that reflection pairs have the same parity (i.e. differ by an even value for all three indices). In principle, more complex types of pseudosymmetry can be handled with additional care (Zwart et al., 2005a).
For two ordinary (untwinned) acentric intensities following Wilson statistics, L has a uniform probability distribution from 0 to 1. For perfectly twinned intensities, the distribution of L has a quadratic form. An observed data set can therefore be analysed and the cumulative distribution of L compared with simple theoretical curves (Fig. 11.7.6.1b). Likewise, the expected values and moments of L are simple fractions for both untwinned and perfectly twinned cases (Table 11.7.6.1). Values for <L> are 1/2 and 3/8 for untwinned and perfectly twinned data, respectively.

When applying tests of overall intensity statistics, situations of partial twinning produce values intermediate between those expected for untwinned and perfectly twinned data. The twinning fraction can therefore be estimated by such tests. However, owing to systematic deviations of the types noted above, this is generally not the best way to estimate α. In particular, if α is small the presence of partial twinning can be missed. The strength of these tests is in identifying cases of high or perfect twinning. Consequently, an examination of overall intensity statistics is sometimes referred to as a test for perfect twinning. Such tests typically provide decisive support for a claim of high or perfect twinning.
Once modelderived intensities become available during structure determination, the presence of partial twinning can be detected reliably and α can be estimated with high precision (see Chapter 18.12 ). However, at the outset of structure determination, identifying partial twinning unambiguously and estimating α are more challenging problems, and numerous methods have been developed. One method, referred to as a Britton plot and popularized by Fisher & Sweet (1980), is based on the behaviour of equation (11.7.4.1). Given a pair of twinrelated observed intensities and a presumed value for α, the two underlying crystallographic intensities can be calculated by inverting the linear equations. If α is chosen to be higher than its correct value, this process leads to some of the crystallographic intensities having negative calculated values. The correct value of α can therefore be estimated by plotting the number of negative calculated intensities obtained as a function of α. For errorfree data, the correct value for α is the highest value that gives no negative calculated intensities.
Other methods of estimating α evaluate the degree of similarity between twinrelated intensities. Rees (1982) analysed the behaviour of the R value when calculated as a comparison over potentially twinrelated intensities. If twinning is absent, the comparison is effectively between independent intensities, resulting in a random R value, which is 1 for acentric data. In the presence of twinning, the expected R value is 1 − 2α, providing an equation for extracting α from the R value over twinrelated pairs. A related method takes twinrelated reflections one pair at a time and evaluates the difference divided by their sum, to give a variable designated H (Yeates, 1988),where I_{obs}(h_{1}) and I_{obs}(h_{2}) are the observed intensities of two twinrelated reflections, h_{1} and h_{2}. The value of H ranges from 0 to 1 for untwinned (acentric data) and has a uniform probability distribution. A plot of the cumulative distribution of H is therefore linear. In the presence of twinning, H takes on a restricted range from 0 to 1 − 2α but remains uniformly distributed. A plot of H therefore provides a relatively simple means of extracting α (Fig. 11.7.7.1). In addition, the expected value and moments for H relate simply to α. The value of <H> is (1 − 2α)/2, while the value of <H^{2}> is (1 − 2α)^{2}/3, providing equations for determining α from measured values of <H> or <H^{2}>.
Measurement errors complicate methods for estimating α. In a Britton plot, the ideally sharp transition that indicates the correct value for α becomes more gradual, leading to uncertainty. In methods that use measures of the average difference between twinrelated reflections, ignoring the effects of errors causes α to be underestimated. Equations for incorporating the effects of errors have been discussed by Rees (1982), Fisher & Sweet (1980), Dumas et al. (1999) and Lunin et al. (2007). In general, measurement errors have a relatively small effect on estimates of α when the twin fraction is low, but have a much larger effect when the intensity measurement errors (on a fractional scale) are comparable with the deviation of α from 1/2.
Local or noncrystallographic symmetry (NCS) often presents challenging complications in relation to twinning. If a local symmetry operator nearly coincides with a crystallographic operator of some higher crystal symmetry, this gives rise to higher pseudocrystallographic symmetry. Especially at lower resolution, reflections related by this operation will have similar intensities, and this can be mistaken for partial twinning. A further complication is that bona fide twinning often occurs in conjunction with NCS. One approach for resolving such complications is to analyse intensity statistics as a function of resolution. In cases where NCS is present but twinning is not, the similarity between potentially twinrelated reflections typically decreases at higher resolution. In contrast, similarity between twinrelated reflections persists out to the maximum resolution if twinning is present.
NCS can also occur in conjunction with perfect twinning. As before, resolutiondependent behaviour can be informative. Tests of overall intensity statistics (i.e. tests of perfect twinning) described earlier can appear relatively normal at low resolution where, owing to NCS, the true crystallographic intensities of twinrelated reflections may be very similar to each other. If the true crystallographic intensities of twinrelated reflections are nearly equal, twinning amounts to an averaging of values that are already nearly equal, which has little effect on intensity statistics. The effects of twinning on intensity statistics may be prominent only at higher resolution, where the true crystallographic intensities of twinrelated reflections have diverged.
Types of twinning beyond hemihedry are possible. When four distinct twin domain orientations are involved, the situation is referred to as tetartohedry. For macromolecular crystals, twinning by tetartohedry is possible when the true symmetry of the crystal is primitive trigonal, e.g. P3. The crystal lattice has 622 rotational symmetry (order 12), which exceeds the rotational symmetry of P3 by a factor of four. Multiple cases have been reported (Rosendal et al., 2004; Barends et al., 2005; Gayathri et al., 2007; Yu et al., 2009). Twinning by pseudotetartohedry is possible in symmetries other than P3 if the unitcell geometry obeys fortuitously high symmetry (Anand et al., 2007). In twinning by tetartohedry, four crystallographic intensities contribute to each observed intensity. The effects on intensity statistics can therefore be even more pronounced compared with twinning by hemihedry. This is especially true when all four domain orientations are equally represented (i.e. the four underlying twin fractions are all 1/4), a situation referred to as perfect tetartohedry. In that case, the ratio <I^{2}>/<I>^{2} is expected to be 5/4 (Stanley, 1972) and the value of <L> is expected to be 35/128 ≃ 0.273 (Table 11.7.6.1). Less extreme values are expected for partial tetartohedry.
Twinning by partial tetartohedry can manifest itself as a crystal that appears to be partially twinned according to multiple distinct twin laws. However, cases of tetartohedry cannot be described properly in terms of a set of estimated hemihedral twin fractions (e.g. one for each twin law). Whereas the twin fraction α (and its complement 1 − α) can be estimated in the case of hemihedry by comparing pairs of twinrelated reflections, the four twin fractions (which sum to unity) in the case of tetartohedry must be estimated by performing statistical tests on quadruplets of twinrelated reflections. Equations and methods for that purpose are provided by Yeates & Yu (2008).
Macromolecular crystals can suffer from disorders other than twinning. These are not covered in depth here, but similarities and differences between twinning and a class of problems known as order–disorder (OD) phenomena are worth noting. In OD phenomena, alternative molecular positions or orientations occur over a short range within a crystal, from one layer to another for instance, in a stochastic fashion (DornbergerSchiff, 1956; DornbergerSchiff & GrellNiemann, 1961). When the stochastic variation between layers or between rows is a translation, this is referred to as latticetranslocation disorder (LTD). Several cases of OD have been described and most have been of the LTD type (Wang, Kamtekar et al., 2005; Wang, Rho et al., 2005; Hwang et al., 2006; Tanaka et al., 2008; Zhu et al., 2008; Hare et al., 2009; Tsai et al., 2009), while cases involving alternately oriented layers are known, but less common (Pletnev et al., 2009). Similar to twinning, OD phenomena can complicate structure determination and prevent proper atomic refinement. However, the symptoms and treatments differ from those for twinning. A distinguishing feature is the length scale of the disorder. In OD cases, differently oriented or translated molecules are nearby in the crystal and therefore scatter coherently. The intensity statistics are therefore not shifted in the same way as for twinning. Furthermore, the interference between alternatively positioned molecules in the case of LTD is typically evident by major nonorigin peaks in a native Patterson map, often appearing to be impossibly close to the origin. In addition, LTD diffraction patterns often show prominent streaking in all or a subset of reflections. Another situation that often arises in OD cases is the appearance of sterically impossible (e.g. interpenetrating) electron density during model building. This can arise when the model accounts for some of the molecules in the unit cell, but others (typically in alternative, mutually exclusive, orientations or positions) have not been accounted for. In contrast, unexplained or interpenetrating electron density is generally not observed in twinning, owing to the absence of coherent scattering between differently oriented twin domains.
In situations where symmetry permits it, twinning is fairly common in macromolecular crystals. If twinning is present but goes undetected, problems arise in later stages of structure determination. Partial and perfect twinning lead to distinct scenarios, but statistical tests make it possible to uncover both kinds of problem. Local or noncrystallographic symmetry, which very often occurs together with twinning, can complicate the interpretation. Resolutiondependent analysis can help to clarify these cases. Additional challenges occur with more complex types of twinning, such as tetartohedry, which is being recognized with increasing frequency.
Acknowledgements
The authors thank Michael Sawaya, Regine HerbstIrmer and George Sheldrick for helpful comments on the manuscript.
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