International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 18.12, pp. 548-551
https://doi.org/10.1107/97809553602060000866

Chapter 18.12. Structure determination in the presence of twinning by merohedry

T. O. Yeatesa* and M. R. Sawayab

aDepartment of Chemistry and Biochemistry, University of California, Los Angeles, PO Box 951569, Los Angeles, CA 90095–1569, USA, and bInstitute for Genomics and Proteomics, UCLA-DOE, 611 Young Drive East, Los Angeles, CA 90095, USA
Correspondence e-mail:  yeates@mbi.ucla.edu

This chapter details strategies for determining structures based on intensity data suffering from twinning. Progress can often be made in the early stages of structure determination without taking the twinning into account, whereas proper refinement of structures must account for the effects of twinning. Complications that can occur when handling anomalous-scattering information are discussed. Opportunities are identified for more formal treatments of the problem of phasing twinned data. The facility with which twinning can be analysed and discriminated from local noncrystallographic symmetry during the final stages of analysis is emphasized.

18.12.1. Introduction

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Twinning by merohedry (see Chapter 11.7[link] ) presents challenges for structure determination because the correct intensities are not observed directly. This is a key problem, since the ultimate goal of structure determination is to arrive at agreement between the observed crystallographic quantities and the corresponding values calculated from a model. Common strategies for handling this problem at different stages of the structure-determination process are considered here. In the following discussion, it is helpful to note that, in any given strategy, one of three approaches is generally being taken. Either (i) the effects of twinning are ignored and treated essentially as a source of noise, (ii) the effect twinning has on the observed intensities is reversed – a process called `detwinning' – in an attempt to recover the true underlying crystallographic intensities, or (iii) the effects of twinning are applied to the model in order to match the twinning present in the observed data. The following discussions apply primarily to twinning by hemihedry, in which just two domain orientations are present. For the treatment of higher forms of twinning (Yu et al., 2009[link]; Barends et al., 2005[link]) or other types of disorder (Wang et al., 2005[link]; Tsai et al., 2009[link]; Hare et al., 2009[link]), the reader is referred to the recent literature.

18.12.2. Detwinning based on observed intensities

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Equation (11.7.4.1)[link] in Chapter 11.7[link] describes how the observed twinned intensities arise from linear combinations of true crystallographic intensities. By inverting that system of equations, one can obtain the crystallographic intensities from the observed intensities, assuming the twin fraction α is known: [\eqalign{I({\bf h}_1) &= \left [ (1 - \alpha) I_{\rm obs}({\bf h}_1) - \alpha I_{\rm obs}({\bf h}_2) \right ] / (1 - 2\alpha),\cr I({\bf h}_2) &= \left [ - \alpha I_{\rm obs}({\bf h}_1) + (1 - \alpha) I_{\rm obs}({\bf h}_2) \right ] / (1 - 2\alpha),}]where I are the true crystallographic intensities for two reflections h1 and h2 related by the twin operation, and Iobs are the corresponding observed intensities. The form of this equation makes it clear that the observed data cannot be detwinned when α = 1/2 (i.e. perfectly twinned). When α ≠ 1/2, the crystallographic intensities can be calculated but measurement errors in the observed intensities are magnified according to the term 1/(1 − 2α) (Fisher & Sweet, 1980[link]; Grainger, 1969[link]). Therefore, detwinning based on observed intensities alone tends to be problematic when the twin fraction is high. If the twin fraction is low then detwinning is robust, although in those cases the effects of twinning are the least important to consider. As a result, detwinning based only on observed intensities is not practised often. The situation is different once calculated intensities from a model become available, as discussed below.

18.12.3. Molecular replacement with twinning

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Even in the case of perfect twinning, it is often possible to arrive at a correct molecular-replacement solution without taking twin­ning into account, as demonstrated by Redinbo & Yeates (1993[link]). At the rotation-function stage, the effects of twinning are much like the effects of high crystal symmetry; one is attempting to detect the correct orientation of a single search model in the context of noise due to other molecules in distinct orientations within the crystal. In favourable cases such a search can be successful. At the translation-function stage, recognizing the true space-group symmetry is critical. As described in Chapter 11.7[link] , even the correct point-group symmetry can be misclassified in cases of high or perfect twinning. Therefore, it is generally advisable to evaluate potential molecular-replacement solutions in multiple space groups consistent with the data. In practice, twinning is usually ignored throughout the molecular-replacement process (Breyer et al., 1999[link]; Contreras-Martel et al., 2001[link]; Luecke et al., 1998[link]; Redinbo & Yeates, 1993[link]; Larsen et al., 2002[link]; Heffron et al., 2006[link]; Yuan et al., 2003[link]; Lee et al., 2003[link]; Anand et al., 2007[link]), although some workers have detwinned the data and then proceeded with molecular replacement (Rabijns et al., 2001[link]; Taylor et al., 2000[link]). In evaluating potential molecular-replacement solutions, it could be useful to consider the contributions of both domain orientations to the observed intensities; this has generally not been done.

18.12.4. Multiple isomorphous replacement and anomalous phasing with twinning

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Strategies for phasing twinned data by multiple isomorphous replacement (MIR) and anomalous-scattering methods have been reviewed (Terwisscha van Scheltinga et al., 2003[link]; Dauter, 2003[link]). To date, several successful solutions of twinned structures by MIR and anomalous phasing approaches have ignored the effects of twinning, at least during the steps of heavy-atom substructure determination and phasing (Rees & Lipscomb, 1980[link]; Mueller, Muller et al., 1999[link]; Mueller, Schübel et al., 1999[link]; MacRae & Doudna, 2007[link]). As noted above, such an approach effectively treats the contributions from the alternate (minor) twin domain as noise. If a correct heavy-atom substructure can be obtained, from Patterson analysis for example, that solution reflects only the heavy-atom contents of one twin domain, presumably the dominant one. Additional peaks not directly accounted for by the identified heavy-atom positions will be present in the native Patterson map in twin-related positions. Adding to the potential confusion, cross-peaks between the heavy-atom positions identified and those related by the twin operation are not present in a Patterson map. Likewise, contributions of the minor twin domain to the measured differences between native and derivative structure-factor amplitudes are ignored when estimating the best phase for the native structure factor. In favourable cases, useful heavy-atom substructures and phases can be obtained despite these serious oversimplifications. In less favourable cases, some workers have overcome twinning in MIR experiments by detwinning the native and derivative data sets, then executing isomorphous replacement with detwinned structure factors (Declercq & Evrard, 2001[link]; Terwisscha van Sheltinga et al., 2001[link]; Hillig & Renault, 2006[link]; Ban et al., 1999[link]). In a few cases, results indicate that detwinning the data produces maps of a quality similar to or worse than the observed twinned data (Rudolph et al., 2003[link]; Yang et al., 2000[link]; Dong et al., 2001[link]).

A proper treatment of the problem of phasing highly twinned data by MIR has been developed (Yeates & Rees, 1987[link]) which takes into account the simultaneous contributions from both twin domains. The phasing of ordinary MIR data is often presented in terms of a phase circle for the native structure factor. The circle describes the requirement for the sum of the squared real and imaginary components of a structure factor to equal the observed intensity. Additional equations derived from heavy-atom information then lead to the identification of the correct point (or native phase) on this circle. In the case of twinning, taking the case of perfect twinning as an example, what is known is not an individual crystallographic intensity but the sum of two twin-related crystallographic intensities. As a result, each observed intensity leads to an equation specifying the sum of four terms: the squared real and imaginary structure-factor components of two distinct twin-related reflections. This equation is a hypersphere, and equations derived from heavy-atom information can specify the correct point on that hypersphere. This effectively identifies the correct or most probable phases for both twin-related reflections simultaneously. The utility of a proper treatment of twinned MIR data has been demonstrated in synthetic test cases (Yeates & Rees, 1987[link]), but such treatments have not been implemented in standard crystallographic programs. It has been unnecessary in a number of successful MIR cases (Rees & Lipscomb, 1980[link]; Declercq & Evrard, 2001[link]; Terwisscha van Sheltinga et al., 2001[link]; Hillig & Renault, 2006[link]; Ban et al., 1999[link]; Rudolph et al., 2003[link]; Yang et al., 2000[link]; Dong et al., 2001[link]), though there are likely to be situations where such treatment is important for success. The handling of anomalous scattering in the presence of twinning parallels that for MIR. Successful phasing has been achieved using standard programs and approaches while ignoring the presence of twinning, while more exact treatments are possible along lines similar to those advocated for MIR.

When dealing with anomalous data from twinned crystals, sometimes special care is required. It may be counterintuitive, but the twinning operation can mix together reflections carrying an F+ and an F designation in a data set reduced in the correct space-group symmetry. For example, in space group P4, one possible choice of asymmetric region in reciprocal space would be the octant with h, k and l all non-negative (ignoring, for simplicity, the redundancy in the principal zones). Reflections recorded in this region (and those related by P4 rotational symmetry) would receive F+ designations, while their Bijvoet pairs would be assigned as F. Reflections 123 and 213 are distinct in P4, and both would typically be labelled F+ since they fall in the chosen asymmetric region. These two reflections have Bijvoet pairs 12[{\overline 3}] and 21[{\overline 3}], respectively, and these would be designated F. The twin operation (a dyad about the xy diagonal, for example) would mix the 123 and 21[{\overline 3}] reflections together and the 213 and 12[{\overline 3}] reflections together (Fig. 18.12.4.1[link]). It is therefore important to bear in mind, particularly when contemplating detwinning data having an anomalous signal, that reflections designated F+ and F in the true lower symmetry may be mixed together by the twin operation. This presents no real obstacle as long as the Bijvoet pairs are handled correctly.

[Figure 18.12.4.1]

Figure 18.12.4.1 | top | pdf |

An illustration of how twinning operations can mix together reflections denoted F+ and F. The case of space group P4 is illustrated. Filled squares represent a reflection [F_{(h1)}^+] and its symmetry mates related by crystallographic operations (i.e. 90° rotations), while open squares represent their Friedel pairs [F_{(h1)}^-]. Filled circles represent distinct but twin-related reflections, [F_{(h2)}^+], while open circles represent their Friedel pairs, [F_{(h2)}^-]. On the basis of the asymmetric unit shown (shaded), assuming reflections falling in the designated asymmetric unit are assigned as F+, the twin operators (hatched lines in the ab plane) relate reflection [F_{(h1)}^+] to reflection [F_{(h2)}^-] and reflection [F_{(h2)}^+] to reflection [F_{(h1)}^-]. Detwinning anomalous data – if such a step is deemed appropriate – therefore requires that the symmetry relationships be considered properly rather than relying on F+ and F designations.

Finally, although direct methods of phasing are not discussed in detail here, in cases where the resolution is high enough those methods can also be employed successfully in cases of twinning, as exemplified by the SHELXD program (Sheldrick, 2008[link]).

18.12.5. Atomic refinement with twinning

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If twinning is present, it must be taken into account during atomic refinement, otherwise the R values remain high. If twinning is present in combination with pseudo-crystallographic noncrystallographic symmetry (NCS), it may be possible to achieve acceptable R values at low resolution but not generally at high resolution. To account for twinning during refinement, the best approach is to use the observed intensity measurements as the target values and incorporate the effects of twinning in the calculated crystallographic quantities. Current refinement programs provide for this (Sheldrick, 2008[link]; Adams et al., 2010[link]; Murshudov et al., 1997[link]; Brünger et al., 1998[link]). Typically, an accurate value for the twin fraction can be determined during the refinement process. Note that it is straightforward to solve for α based on a system of equations of the form given in Chapter 11.7[link] [equation (11.7.4.1[link] )], where I are calculated from the model and Iobs are the observed intensities.

As noted in Chapter 11.7[link] , the presence of twinning is sometimes difficult to detect early in structure determination based on observed intensity data alone, so atomic refinement presents an important opportunity to re-examine the possibility of twinning. As described above, the presence of twinning and an accurate value for the twin fraction can usually be obtained during refinement by comparing observed and calculated intensities. Another method of analysis, which is particularly useful for discriminating between twinning and pseudo-crystallographic NCS, is to examine Rtwin (i.e. the R value between potentially twin-related reflections) and compare the values obtained for Rtwin when it is evaluated over either observed intensity data or intensities calculated from the model (Fig. 18.12.5.1[link]). This has been referred to as an RvR calculation (Lebedev et al., 2006[link]). The result can be especially instructive when examined as a function of resolution. For cases where twinning occurs together with NCS, the calculated and observed Rtwin values may be low at low resolution, with Rtwin over the calculated intensities climbing steeply at higher resolution and Rtwin over the observed intensities remaining relatively low. When twinning is present in the absence of interfering NCS, the Rtwin value should be nearly random for the calculated intensities but significantly lower for the observed intensities. An analysis of RvR in a survey of macromolecular structures and diffraction data in the Protein Data Bank (PDB; Berman et al., 2000[link]) has illuminated a number of cases of twinned structures (both merohedral and pseudo-merohedral) where the phenomenon was apparently overlooked during structure determination (Lebedev et al., 2006[link]). This survey emphasizes the importance of checking for twinning during or after atomic refinement.

[Figure 18.12.5.1]

Figure 18.12.5.1 | top | pdf |

A comparison of calculated versus observed Rtwin values indicates the presence of twinning in a crystal of alanyl-tRNA synthetase (PDB entry 1YFS; Swairjo & Schimmel, 2005[link]). The Iobs values continue to obey the NCS/twin-law at high resolution, while the pseudo-symmetry in the Icalc values breaks down much more quickly. This particular case depicts pseudo-merohedral twinning in space group P212121, allowed by nearly equal values of the unit-cell b and c dimensions.

One cautionary point regarding refinement is that the final R values tend to be artificially low for a twinned refinement as a purely statistical effect. Twinning produces an averaging effect on the intensities as a whole. Therefore, two data sets of twinned intensities (e.g. observed and calculated) will tend to differ less from each other than untwinned data sets. It has been suggested that the R values for refining a structure against perfectly twinned data are artificially lower by a factor of approximately 21/2/2 (Redinbo & Yeates, 1993[link]). This is in line with the related observation that the expected value of |L| is 1/2 for untwinned data but only 3/8 for perfectly twinned data [see equation (11.7.6.1)[link] in Chapter 11.7 for the definition of |L|]. R values must therefore be judged carefully. In particular, the statistical effect means that a drop in the R value when a problem is treated as if it is twinned is generally not good evidence for twinning. Alternative measures of discrepancy could be useful in evaluating the validity of twinned refinement in individual cases. The standard Pearson correlation (e.g. between Fcalc and Fobs) is likely to be less adversely affected by twinning than the crystallographic R value, but its utility has not been examined in practice.

An important concern when refining twinned structures is how to select test reflections when using the Rfree value for cross-validation (Brünger, 1992[link]). Selection must be made so that both members of a pair of reflections related by the twin operation are included in the test set. This is relatively simple in cases of twinning by true merohedry, but can be more complicated when dealing with pseudomerohedry.

18.12.6. Detwinning on the basis of model Fcalc values

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As noted above, detwinning based only on observed intensities is usually not advisable. However, once the true crystallographic intensities can be estimated based on model calculations, the situation is improved. Detwinning can be important in two situations. The primary one is to enable electron-density-map calculations. Interpretable electron-density maps require proper estimates for the true crystallographic structure-factor amplitudes (Fobs). These values can be obtained effectively by detwinning, even in the case of perfect twinning, based on knowledge of the Fcalc values. At least two different types of equations have been employed, which are different in detail but similar in spirit. We illustrate them in the case of perfect twinning, where the measured Iobs = 1/2[I(h1) + I(h2)] [i.e. I(h1) = I(h2)], and the calculated quantities are I(h1)calc and I(h2)calc. In one approach (Wei, 1969[link]), the observed intensity is simply partitioned into estimates for I(h1) and I(h2) according to the proportion of I(h1)calc to I(h2)calc. This leads to [\eqalign{I({\bf h}_1) &= {{I({\bf h}_1)_{\rm calc}} \over {\left [ I({\bf h}_1)_{\rm calc} + I({\bf h}_2)_{\rm calc}\right ]} } \, 2I_{\rm obs},\cr I({\bf h}_2) &= {{I({\bf h}_2)_{\rm calc}} \over {\left [ I({\bf h}_1)_{\rm calc} + I({\bf h}_2)_{\rm calc}\right ]} } \, 2I_{\rm obs} . }]

Alternatively, one can take a least-squares approach and say that there are two equations for I(h1), namely I(h1) = I(h1) calc and I(h1) = 2Iobs − I(h2)calc. Assuming equal estimated errors, the best choice is the average, [I({\bf h}_1) = I_{\rm obs} + {{\left [I({\bf h}_1)_{\rm calc} - I({\bf h}_2)_{\rm calc} \right ]} \over {2}},]and likewise for I(h2), [I({\bf h}_2) = I_{\rm obs} + {{\left [I({\bf h}_2)_{\rm calc} - I({\bf h}_1)_{\rm calc} \right ]} \over {2}}. ]Somewhat more complicated expressions arise when α ≠ 1/2. In either case, one obtains estimates for the true crystallographic intensities. Square roots of these quantities can then be used to calculate electron-density maps. It is critical to note that these values are useful estimates of Fobs but they are biased by the current atomic model. Therefore, when calculating electron-density maps using model-based detwinning procedures, it is important to remove model bias locally. As in other kinds of `omit' maps, atoms in and around a particular region of interest can be removed from the model calculations before applying the detwinning procedure.

Model-based detwinning is also useful in certain atomic refinement situations. If a twinning scenario arises whose treatment is not possible using available refinement programs, model-based detwinning can be used in an iterative fashion to produce structure factors for use as target values in an otherwise ordinary refinement protocol. This strategy was used in the refinement of early cases of protein crystals twinned by hemihedry (Redinbo & Yeates, 1993[link]) and in recent cases of protein crystals twinned by tetartohedry (Yu et al., 2009[link]; Barends et al., 2005[link]; Gayathri et al., 2007[link]).

18.12.7. Summary

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Structure determination is often possible even when the only available data are from a perfectly twinned specimen. This is especially true for molecular replacement, but also in favourable cases for MIR and anomalous phasing methods. In general, at the stage where phase determination is being attempted, the available programs attempt to obtain phases (by molecular replacement or heavy-atom methods) while ignoring the presence of twinning. While this has been successful in several cases, there are opportunities for more sophisticated treatments of phase determination in the presence of twinning. Such improvements may make it possible to phase data successfully from a larger number of twinned crystals. At the stage of atomic refinement, the problem of handling twinned structures has been treated rigorously in a number of programs. The proper use of these programs makes it relatively straightforward to refine structures successfully in the presence of twinning.

Acknowledgements

The authors thank Yingssu Tsai, Regine Herbst-Irmer and George Sheldrick for helpful comments on the manuscript.

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