research papers
Analysis and characterization of data from twinned crystals
^{a}Department of Biochemistry, University of Leicester, University Road, Leicester LE1 7RH, England, and ^{b}Department of Biology and Biochemistry, University of Bath, Claverton Down, Bath BA2 7AY, England
^{*}Correspondence email: nc42@le.ac.uk
It is difficult but not impossible to determine a macromolecular structure using Xray data obtained from twinned crystals, providing it is noticed and corrected. For perfectly twinned crystals, the structure can probably only be solved by
It is possible to detect and characterize from an analysis of the intensity statistics and crystal packing density. Tables of likely operators and some examples are discussed here.Keywords: twinning.
1. Introduction
Recent advances in the theory and practice of molecular crystallography have increased the speed and scope of ; Wei, 1969; Gao et al., 1994).
enormously, but suitable crystals are still an essential prerequisite. is a possible complication; its symptoms can be recognized at the dataprocessing stage and if suitably treated, the solution of the structure is possible. It is frequently observed in crystals of small molecules and is not regarded as an insoluble problem (Dunitz, 1964Twinning can be described as a crystalgrowth anomaly whereby the orientations of individual crystalline domains within the crystal specimen differ in such a way that their diffraction lattices overlap, either completely or partially (Redinbo & Yeates, 1993; Giacovazzo, 1992; Yeates & Fam, 1999). There is some operator which can be applied to align the crystal axes of each domain (Fig. 1). For instance, a trigonal crystal could contain three blocks, B1, B2 and B3, where the B1 axes (a, b, c) are aligned with the B2 axes (b, a, −c) and the B3 axes (−a, −b, c). The realspace operators [T_{real}] to convert B2 or B3 to B1 would be
respectively. The reciprocalspace T_{recip}] are the inverse of [T_{real}]; so that [h_{1}k_{1}l_{1}] will overlap [h_{2}k_{2}l_{2}] [T_{recip}]. (In both these cases [T_{recip}] = [T_{real}], but this is not necessarily so.)
operators [The different types of ; Giacovazzo, 1992). There are two families: (i) quasitwinlattice symmetry (QTLS) and (ii) twinlattice symmetry (TLS) or (Catti & Ferraris, 1976; Giacovazzo, 1992). The QTLS twins have two or more lattices which do not completely overlap; they give rise to multiple diffraction spots and can often be recognized under a microscope with a polarizing attachment. If only two lattice axes can be aligned, these twins are called nonmerohedral or epitaxial. This is easily recognizable by inspection of the observed threedimensional diffraction pattern, which reveals clearly distinct interpenetrating reciprocal lattices (for details, see Liang et al., 1996; Lietzke et al., 1996). Usually a single lattice can be identified and the unique data integrated.
have been described and classified (Donnay & Donnay, 1974On the other hand, TLS twins are generally indistinguishable even under a powerful optical microscope and the reciprocal lattices of each domain completely overlap, giving rise to single diffraction spots. TLS twins are further divided into class I, where the twin fragments do not have the same diffracting volume (referred to as partial ). (This of course is only possible if the Laue group is capable of supporting a higher order of symmetry, and hence usually occurs in P4, P3, P6 or cubic systems.)
twinning) and the apparent lattice symmetry is the same as the true Laue symmetry, and class II, where the twin fragments have identical volumes (perfect twinning) and the apparent symmetry appears greater than the true Laue symmetry. The independent imposes an extra relationship in the lattice symmetry which is not part of the Laue symmetry of the single crystal (Yeates, 1997The
can either involve two domains (hemihedral), four domains (tetartohedral) or, for some centric forms, eight domains (ogdohedral).To deal with twinned crystals it is essential to (i) identify the nature of the twin operator and (ii) estimate the volume ratio (the . All these operators can be considered as a rotation about some axis defined as the twin axis. Extra complexities in the analysis are introduced when this axis is parallel to some axis of (see, for example, Lea & Stuart, 1995).
fraction) of the domains in the crystals. Likely twin operators for different space groups are tabulated in Table 1

We describe here some general strategies for detecting and characterizing the
problem by analyzing the data on the basis of diffraction pattern, intensity statistics and packing density. Some troubleshooting ideas are also discussed.2. Data collection in the case of twinned crystals
For QTLS, the crystal lattices are not superposable in all three dimensions, so care must be taken to make sure that the data from each lattice can be integrated separately and overlaps excluded, the data can perhaps then be treated for nonmerohedral et al., 1996; Lietzke et al., 1996). No examples of this type are discussed here.
effects (LiangFor TLS, the et al., 1998). In order to estimate the twin accurately, it is essential that a complete data set is collected from a single crystal (for details, see Ito et al., 1995). When the is nearperfect, it is possible to mistake the if only the supposed unique data were collected there would be no chance of deconvolution.
fraction may vary from crystal to crystal (for details, see Valegard3. Diagnostic signals of twinning
It is important to detect
before embarking on the structure solution, and if it is possible to find an untwinned crystal this is by far the best approach!3.1. Packing density
If the supposed
volume of the crystal is too small to hold the molecule, it is likely that there is perfect and the has been wrongly assigned.3.2. Intensity statistics
The intensity statistics from an untwinned crystal are quite different from those of twinned crystals. Wilson showed that for a single crystal the mean and higher moments of centric and acentric intensities and amplitudes follow a predictable pattern (Wilson, 1949). These may be tabulated either as 〈I^{k}〉/〈I〉^{k}, 〈E〉^{k}/〈E^{k}〉 or as functions of Z (defined as I/〈I〉), but they are all related to each other. For measurements I_{tH} from a twinned crystal, each `intensity' is in fact the sum of two or more I_{Hi},
3.3. Useful distributions to inspect
(i) The N(Z) plot (given in TRUNCATE output) shows the number of observed weak reflections plotted against the expected value. For twinned data, there are many fewer weak reflections than expected and hence the acentric distribution is always sigmoidal. This follows from the fact that each observed I_{tH} is a sum of two or more I_{H}, and it is unlikely that all I_{H} contributions will be weak (it is wise to exclude the centric terms, since if the has been wrongly assigned these may be wrongly flagged).
(ii) The kth moments of I or E also provide useful indicators. The expected values are given in various references (Stanley, 1972; Redinbo & Yeates, 1993; Breyer et al., 1999; Yeates, 1997) and the observed values can be extracted from several commonly used programs, for example TRUNCATE and ECALC (Collaborative Computational Project, Number 4, 1994). The ratio of 〈I^{2}〉/〈I〉^{2} for acentric reflections against resolution gives expected values of 2.0 for cases without and 1.5 for perfect hemihedral respectively. Similarly, the acentric Wilson ratio, 〈E〉^{2}/〈E^{2}〉, where E is the normalized is expected to give values of 0.785 for twinned and 0.885 for untwinned data.
(iii) Once the x. If the indices (hkl) of I_{tH1} and I_{tH1}are related so that T[h_{1}k_{1}l_{1}] overlaps T[h_{2}k_{2}l_{2}] [T_{recip}], then (1) and (2) are valid and, providing x is not equal to 0.5, the true I_{H1} and I_{H2} can be determined from (3) and (4). For the correct value of x the detwinned data should satisfy certain criteria. The intensity statistics of the detwinned data should be more `normal'. The number of negative I_{H} should be small (this is the basis of the Britton plot; Britton, 1972). The correlation between I_{H1} and I_{H2} should be minimum. This may not actually fall to zero, since if there is with an axis parallel to the twin axis, there may well be real correlation between the two observations.
operator is known and the twinned partner intensity can be selected, it is often possible to `detwin' the data assuming different values of the ratio3.4. Using twinned data
Twinned data has been used to solve many structures. The techniques fall into two groups, depending on the methods to be used. If the structure solution requires the use of amplitudes, as is the case when using heavyatom derivative or MAD data to give experimental phases, it is essential to detwin the intensities. This can be performed with programs such as DETWIN (Leslie, 1998), providing both the 3 × 3 matrix [T] and the ratio x are known and the value of x is not equal to 0.5 (for details, see Valegard et al., 1998).
Possible Is) rather than structure factors (Fs) is suitable.
operators can usually be deduced from the spacegroup symmetry and the apparent symmetry of the diffraction pattern. The second technique is not to bother to detwin the data but to use the summed intensities for structure solutions. Any technique which exploits intensities (3.4.1. Molecular replacement
MR should be able to find solutions for each crystal block. However, the usual test of whether these solutions overlap cannot be performed since each is partially occupied, but the results are often clear. Such a molecularreplacement search would in fact also indicate the realspace
operator.3.5. Example: native bovine αlactalbumin
Both native recombinant and native bovine αlactalbumin (αLA; Sigma) crystallize as hemihedral twins (N. Chandra, K. Brew & K. R. Acharya, unpublished results). The unitcell parameters are a = 93.5, b = 93.5, c = 67.0 Å; α = 90.0, β = 90, γ = 120°. The pseudoprecession picture (Fig. 2) seems to show that the Laue group is either P6/m or P6/mmm, as the diffraction pattern nicely displays an apparent hexagonal symmetry.
A data set to 2.5 Å resolution was collected from a single crystal on a Siemens area detector using Cu Kα radiation and the data were reduced in Laue symmetry groups 3¯, 3¯1m, 3¯m1, 6/m and 6/mmm, corresponding to the space groups P3, P312, P321, P6 and P622, respectively. The corresponding R_{sym} values for these space groups are 6.9, 7.9, 8.5, 8.4 and 8.9% (99.9% complete in each space group), respectively.
The problem of ), an intensity distribution which did not follow the Wilson statistics (Fig. 4), and the crystal packing density. Calculation of the solvent content for 12 molecules per gives a reasonable value of 57% [the V_{m} (Matthews, 1968) value is 2.99 Å^{3} Da^{−1}]. Therefore, none of the above space groups could be eliminated. The ratio of 〈I^{2}〉/〈I〉^{2} for αLA is found to be 1.47 (Fig. 4) indicating the crystals might be hemihedrally twinned. Similarly, the Wilson ratio 〈E〉^{2}/〈E^{2}〉 calculated for acentric reflections gave a value of 0.886, consistent with the data from αLA crystals being perfectly twinned.
was addressed on the basis of the intensity statistics (Fig. 3We estimated the x from the parameter H (Yeates, 1997), where
fractionH is a function of x [from 0 to (1 − 2x)] and the true crystallographic intensities. The value of x is determined using the fact that
The mean value of x directly estimated from H was found to be 0.49, compared with the average value of 0.50 obtained based on intensity statistics described by Britton (1972), Rees (1982) and Fisher & Sweet (1980).
The selfrotation function also showed more peaks than were expected (Fig. 5). The molecularreplacement translation search fixed the correct as P3. The crystals are twinned along the a, b or −a, −b axes, all of which are equivalent by the symmetry of the That is, the realspace twin matrix in this case is
Detwinning of the observed twinned structure factors was carried out according to the method of Redinbo & Yeates (1993); following this, the structure was refined to R_{free} and R values of 19.8 and 18.8%, respectively. The XPLOR datainput files were modified and used as a means of a convenient way to detwin the twinned data.
4. Partial XAT
When the twin x is less than 0.5, the is called partial the diffraction pattern does not reveal a higher apparent symmetry, but the observed intensities still contain contributions from both the domains. If there is a good estimate of x and it is below a value of about 0.45, the can reliably be corrected using (3) and (4) (Britton, 1972; MurrayRust, 1973; Rees, 1982; Fisher & Sweet, 1980; Redinbo & Yeates, 1993; Yeates, 1997).
A highresolution (1.509 Å) data set (R_{sym} is 0.052 and the completeness of the data is 95.9%) was collected using synchrotron radiation at Daresbury Station 7.2 from crystals of xenobiotic acetyltransferase (XAT). The crystals are in R3, with unitcell parameters a = 123.64, b = 123.64, c = 63.08 Å, α = 90, β = 90, γ = 120°, Z = 1 molecule per V_{m} = 3.96 Å^{3} Da^{−1} (Matthews, 1968), and exhibit partial with an average twin fraction volume x, calculated from (3), (4) and (5), of 0.1982. The only possible twin operator for R3 generates a rotation about the diagonal of the a and b axes, i.e. the realspace twin operator matrix is
When using the measured data, the MR solution of XAT did not refine well, with the R_{free} and R values converging at 39.4 and 36.1%, respectively. There was only one MR solution with a high (after rigidbody refinement) of 71.2% with an R factor of 35.9% and, despite the high solvent content (70%), there was no sign of a second molecule. Furthermore, the selfrotation function showed only one significant peak.
To investigate whether the data could be partially twinned, we reexamined the dataprocessing statistics more carefully. The TRUNCATE program in the CCP4 package (Collaborative Computational Project, Number 4, 1994) produces a table for acentric and centric reflections of the second, third and fourth moments of I. The expected value of the second moment for untwinned acentric data is 2.0 and any gross deviation from this is a probable indication of partial twinning.
The partially twinned data of XAT were detwinned (Fig. 6) using the DETWIN program (Leslie, 1998), which is now released as a part of the CCP4 package. This requires as input the reciprocalspace operator, a range of values of the twin x and the input data as intensities, and outputs a list of supposedly detwinned intensities. It tabulates the between I_{H1} and I_{H2} after detwinning, which should have its minimum value for the best estimate of x.
The detwinned data of XAT using the x = 0.1941 was used in MR and gave the same solution as that obtained from twinned data. The detwinned data were used in of the model to see whether there was an improvement in the values of R_{free} and R. We found that the values of R_{free} and R were reduced by 4.3 and 4.5%, respectively. There was a remarkable improvement in the electron density of the side chains of many of the lysines and arginines and also some of the residues which it was not possible to fit into the electrondensity map previously (before detwinning).
ratioFurther SHELXL (Sheldrick & Schneider, 1997) using the TWIN command option available in the program. We found that the twin x was refined from 0.194 to 0.191, showing that the other methods used are remarkably sensitive.
of the XAT structure was subsequently carried out with5. Correct estimation of x
The exact value of the twin x can be computed if the calculated structurefactor amplitudes for the given model are available. Calculating the [Corr(x)] using the expressions of GomisRüth et al. (1995) gives
where
〈I〉 is the mean intensity and
The x) attains a unique maximum as a function of x, and the optimum value of x may be accurately determined by setting the derivative in (8): dCorr(x)/dx = 0. Therefore, the optimal value of x is calculated from
Corr(where
and represents the calculated structure factors.
However, the accurate value of x, which is quite different from 0.5 and is independent of resolution, can be evaluated from (12) and (13). Then, the intensities of the twinrelated reflections are detwinned using (3) and (4). In the case of XAT, we did not try to obtain the accurate value of the twin x using (13), since the SHELXL program is being used and takes care of the of the twin factor. The structural results and other details of the XAT structure will be published elsewhere (N. Chandra, J. Snidwongse, W. V. Shaw, I. A. Murray & P. C. E. Moody, manuscript in preparation).
6. Troubleshooting
It is always better to avoid Escherichia coli, the data collected from PPAT crystals were reduced in P3, P3_{1}, P6 and P6_{3} space groups (unitcell parameters: a = 65.15, b = 65.15, c = 119.06 Å, α = β = 90, γ = 120°) and exhibit an apparent Laue symmetry of 6/m. The N(z) plots indicated that was likely and thus that the true Laue symmetry is probably 3 (Fig. 7). However, by changing the crystallization conditions, the crystals of the same protein were regrown in an entirely different I23 free from The structure of this protein has been solved using MAD data (Izard & Geerlof, 1999).
altogether so that many difficulties can be eliminated. Growing the crystals under different crystallization conditions may be one way of overcoming the problem. A new crystal form in an entirely different may be obtained. For example, for phosphopantetheine adenylyltransferase (PPAT) fromCephalosporin synthase protein structure was solved by MIR using several data sets collected from different crystals which exhibited different values of twin et al., 1998). This demonstrates the accuracy with which the fraction can be determined and used to deconvolute the data.
(Valegard7. Summary
We have discussed the general strategy regarding the identification, analysis, characterization and correction of the data collected from twinned crystals based on the Xray diffraction pattern, intensity statistics, packing density and
Some suggestions have been given for overcoming twinning.Acknowledgements
We wish to acknowledge the significant contribution of Eleanor J. Dodson to this work and to thank Todd O. Yeates, Matthew R. Redinbo and S. Ramaswamy for helpful discussion and suggestions on the DETWIN program as one of the beta testers, Michael I. Wilson for helping to write a program for detwinning and Tina Izard for providing twinned data. NC thanks the CCP4 organizers for the opportunity to speak at the CCP4 meeting. The αLA project is supported by a Leverhulme Trust grant (F351/U) to Ravi Acharya and the XAT project is supported by a BBSRC grant to Peter C. E. Moody.
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