Analysis and characterization of data from twinned crystals

Chandra, N.; Acharya, K.R.; Moody, P.C.E.

doi:10.1107/S0907444999009968

research papers

BIOLOGICAL
CRYSTALLOGRAPHY

ISSN: 1399-0047

Volume 55| Part 10| October 1999| Pages 1750-1758

doi:10.1107/S0907444999009968

Analysis and characterization of data from twinned crystals

N. Chandra,^a,^b ^* K. Ravi Acharya ^b and P. C. E. Moody ^a

^aDepartment of Biochemistry, University of Leicester, University Road, Leicester LE1 7RH, England, and ^bDepartment of Biology and Biochemistry, University of Bath, Claverton Down, Bath BA2 7AY, England
^*Correspondence e-mail: nc42@le.ac.uk

(Received 11 March 1999; accepted 21 July 1999)

It is difficult but not impossible to determine a macromolecular structure using X-ray data obtained from twinned crystals, providing it is noticed and corrected. For perfectly twinned crystals, the structure can probably only be solved by molecular replacement. It is possible to detect and characterize twinning from an analysis of the intensity statistics and crystal packing density. Tables of likely twinning 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 structure determination enormously, but suitable crystals are still an essential pre-requisite. Twinning is a possible complication; its symptoms can be recognized at the data-processing 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, 1964 ; Wei, 1969 ; Gao et al., 1994 ).

Twinning can be described as a crystal-growth 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 twinning 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 real-space twinning operators [T_real] to convert B2 or B3 to B1 would be

$[\left (\matrix { 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & -1} \right) \left (\matrix {a \cr b \cr c} \right)\,\, {\rm and} \,\, \left (\matrix {-1 & 0 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & 1} \right) \left (\matrix {a \cr b \cr c} \right),]$

respectively. The reciprocal-space twinning operators [T_recip] are the inverse of [T_real]; so that [h₁k₁l₁] will overlap [h₂k₂l₂] [T_recip]. (In both these cases [T_recip] = [T_real], but this is not necessarily so.)

Figure 1
The alignment of crystal axes by twinning operator. All the four blocks (a)–(d) align axes related by crystal symmetry in the case of trigonal P3_i (see Table 1

The different types of twinning have been described and classified (Donnay & Donnay, 1974 ; Giacovazzo, 1992). There are two families: (i) quasi-twin-lattice symmetry (QTLS) and (ii) twin-lattice symmetry (TLS) or merohedral twinning (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 non-merohedral or epitaxial. This is easily recognizable by inspection of the observed three-dimensional 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.

On 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 merohedral 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 merohedral twinning) and the apparent twin lattice symmetry appears greater than the true Laue symmetry. The independent twin operation imposes an extra relationship in the lattice symmetry which is not part of the Laue symmetry of the single crystal (Yeates, 1997 ). (This of course is only possible if the Laue group is capable of supporting a higher order of symmetry, and hence twinning usually occurs in P4, P3, P6 or cubic systems.)

The twinning 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 twinning fraction) of the domains in the crystals. Likely merohedral twin operators for different space groups are tabulated in Table 1. 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 non-crystallographic symmetry (see, for example, Lea & Stuart, 1995 ).

Table 1
Lookup tables

Possible merohedral twin operators for tetragonal, trigonal, hexagonal and cubic space groups. [Source: likely twinning operators (CCP4: General) under CCP4 program suite.]

It is a requirement for merohedral twinning that two independent lattices overlap. Since for tetragonal, trigonal, hexagonal or cubic systems the symmetry requires that two cell axes are equal, twinning often occurs in these systems. For these space groups axes can be aligned in the following ways: (a,b,c) or (-a,-b,c) or (b,a,-c) or (-b,-a,-c), with corresponding reciprocal axes: (a^*,b^*,c^*) or (-a^*,-b^*,c^*) or (b^*,a^*,-c^*) or (-b^*,-a^*,-c^*). The corresponding indexing systems are: (h,k,l) or (-h,-k,l) or (k,h,-l) or (-k,-h,-l^*). All P4_i and related 4_i space groups: (hkl) is equivalent to ( $[\overline h \overline k]$ l), so we only need to check real axes (a,b,c) and (b,a,-c) and reciprocal axes (b^*,a^*,c^*) and (b^*,a^*,-c^*), i.e. check if reindexing (hkl) to (kh $[\overline l]$ ) gives a better match to previous data sets. Twinning possible with this operator: apparent Laue symmetry for perfect twin would be P422.

Space-group number	Space group	Point group	Possible twin operator
75	P4	PG4	k, h, −l
76	P4₁	PG4	k, h, −l
77	P4₂	PG4	k, h, −l
78	P4₃	PG4	k, h, −l
79	I4	PG4	k, h, −l
80	I4₁	PG4	k, h, −l

For all P4_i2_i2 and related 4_i2_i2 space groups, (hkl) is equivalent to ( $[\overline h]$ $[\overline k]$ l), (kh $[\overline l]$ ) and ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ), so all axial pairs are already equivalent as a result of the crystal symmetry. No twinning possible, but a perfect twin for the Laue group P4 might appear to have this symmetry.

Space-group number	Space group	Point group	Twin operators
89	P422	PG422	None
90	P42₁2	PG422	None
91	P4₁22	PG422	None
92	P4₁2₁2	PG422	None
93	P4₂22	PG422	None
94	P4₂2₁2	PG422	None
95	P4₃22	PG422	None
96	P4₃2₁2	PG422	None
97	I422	PG422	None
98	I4₁22	PG422	None

All P3_i and R3: (hkl) neither equivalent to ( $[\overline h]$ $[\overline k]$ l) nor (kh $[\overline l]$ ) nor ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ), so we need to check all four possibilities. These are the only cases where tetartohedral twinning can occur: real axes (a,b,c) (-a,-b,c), (b,a,-c) and (-b,-a,c), reciprocal axes (a^*,b^*,c^*), (-a^*,-b^*,c^*), (b^*,a^*,-c^*) and (-b^*,-a^*,c^*), i.e. for P3, consider reindexing (hkl) to ( $[\overline h]$ $[\overline k]$ l) or (kh $[\overline l]$ ) or ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ). For R3, the indices must satisfy the relationship −h + k + l = 3n, so it is only possible to reindex as (kh $[\overline l]$ ). For trigonal space groups, symmetry equivalents do not seem as `natural' as in other systems. Replacing the four basic sets with other symmetry equivalents gives a bewildering range of apparent possibilities, but all are equivalent to one of the above. Twofold twinning possible with this operator: apparent Laue symmetry or twofold perfect twin could be P312 (operator k, h, −l), P321 (operator −k, −h, −l) or P6 (operator −h, −k, l). Fourfold twinning with these operators could generate apparent Laue symmetry P622.

Space-group number	Space group	Point group	Possible twin operator
143	P3	PG3	−h, −k, l; k, h, −l; −k, −h, −l
144	P3₁	PG3	−h, −k, l; k, h, −l; −k, −h, −l
145	P3₂	PG3	−h, −k, l; k, h, −l; −k, −h, −l
146	R3	PG3	k, h, −l

All P3_i12: (hkl) already equivalent to ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ), so we only need to check real axes (a,b,c) and (b,a,-c) and reciprocal axes (a*, b*, c*) and (b*, a*, −c*), i.e. reindex (hkl) to (kh $[\overline l]$ ) [or its equivalent operator (−h, −k, l)]. Twinning possible with this operator: apparent symmetry for twofold perfect twin would be P622 (operator −h, −k, l).

Space-group number	Space group	Point group	Possible twin operator
149	P312	PG312	−h, −k, l or k, h, −l
151	P3₁12	PG312	−h, −k, l or k, h, −l
153	P3₂12	PG312	−h, −k, l or k, h, −l

All P3_i21: (hkl) already equivalent to (kh $[\overline l]$ ), so we only need to check real axes (a,b,c) and (-a,-b,-c) and reciprocal axes (a^*,b^*,c^*) and (-a^*,-b^*,-c^*), i.e. reindex (hkl) to ( $[\overline h]$ $[\overline k]$ l) [or its equivalent operator (−k, −h, −l)]. Twinning possible with this operator: apparent symmetry for twofold perfect twin would be P622 (operator −h, −k, l).

Space-group number	Space group	Point group	Possible twin operator
150	P321	PG321	−h, −k, l or −k, −h, −l
152	P3₁21	PG321	−h, −k, l or −k, −h, −l
154	P3₂21	PG321	−h, −k, l or −k, −h, −l

All P6_i: (hkl) already equivalent to ( $[\overline h]$ $[\overline k]$ l), so we only need to check real axes (a,b,c) and (b,a,-c) and reciprocal axes (a^*,b^*,c^*) and (b^*,a^*,-c^*), i.e. reindex (hkl) to (kh $[\overline l]$ ). Twinning possible with this operator: apparent symmetry for twofold perfect twin would be P622 (operator k, k, −l).

Space-group number	Space group	Point group	Possible twin operator
168	P6	PG6	k, h, −l
169	P6₁	PG6	k, h, −l
170	P6₅	PG6	k, h, −l
171	P6₂	PG6	k, h, −l
172	P6₄	PG6	k, h, −l
173	P6₃	PG6	k, h, −l

All P6_i22: (hkl) already equivalent to ( $[\overline h]$ $[\overline k]$ l) and (kh $[\overline l]$ ) and ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ), so no twinning is possible. However, a perfect twin for the Laue group P312, P321 or P6 might appear to have this symmetry.

Space-group number	Space group	Point group	Possible twin operator
177	P622	PG622	k, h, −l
178	P6₁22	PG622	None
179	P6₅22	PG622	None
180	P6₂22	PG622	None
181	P6₄22	PG622	None
182	P6₃22	PG622	None

All P2_i3 and related 2_i3 space groups: (hkl) already equivalent to ( $[\overline h]$ $[\overline k]$ l), so we only need to check real axes (a, b, c) and (b, a, −c) and reciprocal axes (a*, b*, c*) and (b*, a*, −c*), i.e. reindex (hkl) to (kh $[\overline l]$ ). Twinning possible with this operator: apparent symmetry for twofold perfect twin would be P43 (operator k, h, −l).

Space-group number	Space group	Point group	Possible twin operator
195	P23	PG23	k, h, −l
196	F23	PG23	k, h, −l
197	I23	PG23	k, h, −l
198	P2₁3	PG23	k, h, −l
199	I2₁3	PG23	k, h, −l

All P4_i32 and related 4_i32 space groups: (hkl) already equivalent to ( $[\overline h]$ $[\overline k]$ l) and (kh $[\overline l]$ ) and ( $[\overline k]$ $[\overline h]$ $[\overline l]$ ), so we do not need to check.

Space-group number	Space group	Point group	Possible twin operator
207	P432	PG432	None
208	P4₂32	PG432	None
209	F432	PG432	None
210	F4₁32	PG432	None
211	I432	PG432	None
212	P4₃32	PG432	None
213	P4₁32	PG432	None
214	I4₁32	PG432	None

We describe here some general strategies for detecting and characterizing the merohedral twinning 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 non-merohedral twinning effects (Liang et al., 1996; Lietzke et al., 1996). No examples of this type are discussed here.

For TLS, the twinning fraction may vary from crystal to crystal (for details, see Valegard et al., 1998 ). In order to estimate the twin volume fraction accurately, it is essential that a complete data set is collected from a single crystal (for details, see Ito et al., 1995 ). When the merohedral twinning is near-perfect, it is possible to mistake the space group; if only the supposed unique data were collected there would be no chance of deconvolution.

3. Diagnostic signals of twinning

It is important to detect twinning 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 asymmetric unit volume of the crystal is too small to hold the molecule, it is likely that there is perfect merohedral twinning and the space group 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_{H_i},

$[\eqalignno {I_{tH_{1}} & = [(1-x)I_{H_{1}} + xI_{H_{2}}], & (1)\cr I_{tH_{2}} & = [xI_{H_{1}} + (1-x)I_{H_{2}}], & (2) \cr I_{H_{1}} & = [(1-x)I_{tH_{1}} - xI_{tH_{2}}]/(1-2x), & (3)\cr I_{H_{2}} & = [(1-x)I_{tH_{2}} - xI_{tH_{1}}]/(1-2x). & (4)}]$

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 space group 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 〈I²〉/〈I〉² for acentric reflections against resolution gives expected values of 2.0 for cases without twinning and 1.5 for perfect hemihedral twinning, respectively. Similarly, the acentric Wilson ratio, 〈|E|〉²/〈|E|²〉, where E is the normalized structure factor, is expected to give values of 0.785 for twinned and 0.885 for untwinned data.

(iii) Once the twinning operator is known and the twinned partner intensity can be selected, it is often possible to `detwin' the data assuming different values of the twinning ratio x. If the indices (hkl) of I_tH₁ and I_tH₁are related so that T[h₁k₁l₁] overlaps T[h₂k₂l₂] [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 non-crystallographic symmetry with an axis parallel to the twin axis, there may well be real correlation between the two observations.

3.4. 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 heavy-atom 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 twinning matrix [T] and the twinning ratio x are known and the value of x is not equal to 0.5 (for details, see Valegard et al., 1998).

Possible twinning operators can usually be deduced from the space-group 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 (Is) rather than structure factors (Fs) is suitable.

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 molecular-replacement search would in fact also indicate the real-space twinning operator.

3.4.2. Refinement against Is

Once the solution is obtained, SHELXL can be used with the option to refine the twinning factor x as well as the coordinates. SHELXL is a versatile program which can handle refinement with twinned data of any kind, be it non-merohedral, merohedral or racemic twinning.

3.5. Example: native bovine α-lactalbumin

Both native recombinant and native bovine α-lactalbumin (α-LA; Sigma) crystallize as hemihedral twins (N. Chandra, K. Brew & K. R. Acharya, unpublished results). The unit-cell parameters are a = 93.5, b = 93.5, c = 67.0 Å; α = 90.0, β = 90, γ = 120°. The pseudo-precession 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.

Figure 2
A HKLVIEW (Collaborative Computational Project, Number 4, 1994

) of the pseudo-precession photograph of the diffraction pattern (for l = 3) of the crystals of α-LA. The diffraction pattern indicates an apparent (hexagonal) symmetry. The observed diffraction appears to be normal, but each of the observed intensities contains equal contributions from the two domains. This is an indication of the symptoms of merohedral twinning.

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 twinning was addressed on the basis of the intensity statistics (Fig. 3), 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 unit cell gives a reasonable value of 57% [the V_m (Matthews, 1968 ) value is 2.99 Å³ Da⁻¹]. Therefore, none of the above space groups could be eliminated. The ratio of 〈I²〉/〈I〉² for α-LA 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 α-LA crystals being perfectly twinned.

Figure 3
A graphical representation of N(Z) curves against Z for acentric data from α-LA. The term N(Z) represents the cumulative distribution function, where Z represents the intensity relative to the mean intensity (i.e. Z = I/〈I〉, where I is the intensity), gives the fraction of reflections having an intensity less than Z. As indicated, a comparison with the theoretical distribution shows that a sigmoid habitus follows the distribution for perfect merohedral twinning (x = 1/2) (for details, see Gomis-Rüth et al., 1995

Figure 4
A test for perfect merohedral (hemihedral) twinning for acentric data from α-LA. The ratios are computed in thin resolution bins of 300–400 reflections.

We estimated the twinning fraction x from the parameter H (Yeates, 1997), where

$[H = |I_{tH{_1}} - I_{tH_{2}}| / (I_{tH_{1}} + I_{tH_{2}}). \eqno (5)]$

H 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

$[\eqalignno {\langle H \rangle & = 0.5 - x, & (6) \cr \langle H^{2} \rangle & = (0.5 - x)^{2}/3. & (7)}]$

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 self-rotation function also showed more peaks than were expected (Fig. 5). The molecular-replacement translation search fixed the correct space group as P3. The crystals are twinned along the a, b or −a, −b axes, all of which are equivalent by the symmetry of the space group. That is, the real-space twin matrix in this case is

$[\left (\matrix {0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & -1} \right).]$

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 X-PLOR data-input files were modified and used as a means of a convenient way to detwin the twinned data.

Figure 5
Representation of the Patterson self-rotation function of the set of structure factors in space group P3 for α-LA (using a spherical integration radius of 21 Å and data between 8 and 4.5 Å resolution) was produced using GLRF (Tong & Rossmann, 1990

). The diagram indicates the presence of a strong twofold along the crystallographic c axis (ψ = 90, φ = 90, κ = 180°), as is usually expected for a crystallographic sixfold axis. Additionally, twofold peaks occur every 30° in the ab plane. The total number of peaks is 12 and this would suggest an erroneously high space group of the P6/mmm Laue family with 12 copies of the asymmetric unit in the unit cell (for a nice discussion on the interpretation of peaks in the Patterson self-rotation function in the case of human lactoferrin, see Breyer et al., 1999

4. Partial twinning: XAT

When the twin volume fraction x is less than 0.5, the twinning is called partial merohedral twinning; 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 twinning can reliably be corrected using (3) and (4) (Britton, 1972; Murray-Rust, 1973 ; Rees, 1982; Fisher & Sweet, 1980; Redinbo & Yeates, 1993; Yeates, 1997).

A high-resolution (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 space group R3, with unit-cell parameters a = 123.64, b = 123.64, c = 63.08 Å, α = 90, β = 90, γ = 120°, Z = 1 molecule per asymmetric unit, V_m = 3.96 Å³ Da⁻¹ (Matthews, 1968), and exhibit partial merohedral twinning 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 real-space twin operator matrix is

$[\left (\matrix {0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & -1} \right).]$

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 correlation coefficient (after rigid-body 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 self-rotation function showed only one significant peak.

To investigate whether the data could be partially twinned, we re-examined the data-processing 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 reciprocal-space twinning operator, a range of values of the twin volume fraction x and the input data as intensities, and outputs a list of supposedly detwinned intensities. It tabulates the correlation coefficient between I_H₁ and I_H₂ after detwinning, which should have its minimum value for the best estimate of x.

Figure 6
Differences observed between partial merohedral and perfect merohedral twinning. A HKLVIEW (Collaborative Computational Project, Number 4, 1994

) of the pseudo-precession photograph of the diffraction pattern of the crystals of XAT. The diffraction pattern indicates no apparent symmetry. The observed diffraction appears to be normal with a threefold symmetry. Because the observed intensities of the twin-related reflections are not equal, the diffraction pattern does not acquire an additional symmetry unlike in the case of perfect twinning. Therefore, it is very difficult to recognize partial merohedral twinning from the observed diffraction pattern.

The detwinned data of XAT using the twinning ratio x = 0.1941 was used in MR and gave the same solution as that obtained from twinned data. The detwinned data were used in refinement 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 electron-density map previously (before detwinning).

Further refinement of the XAT structure was subsequently carried out with SHELXL (Sheldrick & Schneider, 1997 ) using the TWIN command option available in the program. We found that the twin volume fraction x was refined from 0.194 to 0.191, showing that the other methods used are remarkably sensitive.

5. Correct estimation of x

The exact value of the twin volume fraction x can be computed if the calculated structure-factor amplitudes for the given model are available. Calculating the correlation coefficient [Corr(x)] using the expressions of Gomis-Rüth et al. (1995 ) gives

$[{\rm Corr}(x) = \textstyle \sum \limits_{H} \Big[(\Delta_{\rm calc})(\Delta_{\rm obs})\Big]\Big/ \Big[\sum \limits_{H}(\Delta_{\rm calc})^{2}\sum \limits_{H}(\Delta_{\rm obs})^{2}\Big]^{1/2}, \eqno (8)]$

where

$[\eqalignno {\Delta_{\rm cal} & = \Big|I_{tH_{1}}^{\rm calc{\hbox {-}}twin} (x) - \langle I_{tH_{1}}^{\rm calc{\hbox {-}}twin} (x) \rangle \Big|, & (9) \cr \Delta_{\rm obs} & = \Big|I_{tH_{1}}^{\rm obs} (x) - \langle I_{tH_{1}}^{\rm obs} (x) \rangle \Big|, & (10)}]$

〈I〉 is the mean intensity and

$[I_{tH_{1}}^{\rm calc{\hbox {-}}twin} (x) = (1-x) I_{H_{1}}^{\rm calc} + x I_{H_{2}}^{\rm calc}. \eqno (11)]$

The correlation coefficient Corr(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

$[\eqalignno {\alpha &= \{ [\langle C_{tH_{1}}^{\rm calc} C_{tH_{1}}^{\rm calc} \rangle - (C_{tH_{1}}^{\rm calc} S_{tH_{12}}^{\rm calc} \rangle \langle C_{tH_{1}}^{\rm obs}C_{tH_{1}}^{\rm calc})\rangle] \cr &\ \quad{\times}\ [C_{tH_{1}}^{\rm calc} (C_{tH_{1}}^{\rm calc} - C_{tH_{2}}^{\rm calc})]\}, &(12) \cr x &= \alpha/(\langle C_{tH_{1}}^{\rm obs} S_{tH_{12}}^{\rm calc} \rangle), & (13)}]$

where

$[S_{tH_{12}}^{\rm calc} = C_{tH_{1}}^{\rm calc} + C_{tH_{2}}^{\rm calc} \eqno (14)]$

and $[C_{tH_{i}}^{\rm calc}]$ 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 twin-related reflections are detwinned using (3) and (4). In the case of XAT, we did not try to obtain the accurate value of the twin volume fraction x using (13), since the SHELXL program is being used and takes care of the refinement 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 twinning 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 space group may be obtained. For example, for phosphopantetheine adenylyltransferase (PPAT) from Escherichia coli, the data collected from PPAT crystals were reduced in P3, P3₁, P6 and P6₃ space groups (unit-cell 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 twinning 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 space group I23 free from twinning. The structure of this protein has been solved using MAD data (Izard & Geerlof, 1999 ).

Figure 7
Representation of the Patterson self-rotation function of the set of structure factors in space group P6₃ for PPAT (using a spherical integration radius of 30 Å and diffraction data between 10 and 3.0 Å resolution) was produced using GLRF (Tong & Rossmann, 1990

). The diagram shows the presence of a strong twofold along the crystallographic c axis (ψ = 90, φ = 90, κ = 180°) as is usually expected for a crystallographic sixfold axis. Additionally, there are twofold peaks occurring every 30° in the ab plane. The total number of peaks is 12 and this suggests an erroneously high space group of the P6/mmm Laue family.

Cephalosporin synthase protein structure was solved by MIR using several data sets collected from different crystals which exhibited different values of twin volume fraction (Valegard et al., 1998). This demonstrates the accuracy with which the twinning fraction can be determined and used to deconvolute the data.

7. Summary

We have discussed the general strategy regarding the identification, analysis, characterization and correction of the data collected from twinned crystals based on the X-ray diffraction pattern, intensity statistics, packing density and refinement statistics. Some suggestions have been given for overcoming merohedral 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 merohedral twinning problem, George Sheldrick and Geoffrey B. Jameson on the refinement strategy using twinned data, Andrew G. W. Leslie for allowing us to use 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 α-LA 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.

References

Breyer, W. A., Kingston, R. L., Anderson, B. F. & Baker, E. N. (1999). Acta Cryst. D55, 129–138. Web of Science CrossRef CAS IUCr Journals Google Scholar
Britton, D. (1972). Acta Cryst. A28, 296–297. CrossRef IUCr Journals Web of Science Google Scholar
Catti, M. & Ferraris, G. (1976). Acta Cryst. A32, 163–165. CrossRef IUCr Journals Web of Science Google Scholar
Collaborative Computational Project, Number 4 (1994). Acta Cryst. D50, 760–763. CrossRef IUCr Journals Google Scholar
Donnay, G. & Donnay, J. D. H. (1974). Can. Miner. 12, 422–425. Google Scholar
Dunitz, D. (1964). Acta Cryst. 17, 1299–1304. CrossRef IUCr Journals Web of Science Google Scholar
Fisher, R. G. & Sweet, R. M. (1980). Acta Cryst. A36, 755–760. CrossRef CAS IUCr Journals Web of Science Google Scholar
Gao, Q., Weber, H.-P. & Craven, B. M. (1994). Acta Cryst. B50, 695–703. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Giacovazzo, C. (1992). Editor. Fundamentals of Crystallography, pp. 61–140. IUCr/Oxford University Press. Google Scholar
Gomis-Rüth, F. X., Fita, I., Kiefersauer, R., Huber, R., Avilés, F. X. & Navaza, J. (1995). Acta Cryst. D51, 819–823. CrossRef Web of Science IUCr Journals Google Scholar
Ito, N., Komiyama, N. H. & Fermi, G. (1995). J. Mol. Biol. 250, 648–658. CrossRef CAS PubMed Web of Science Google Scholar
Izard, T. & Geerlof, A. (1999). EMBO J. 18, 2021–2030. Web of Science CrossRef PubMed CAS Google Scholar
Lea, S. & Stuart, D. (1995). Acta Cryst. D51, 160–167. CrossRef CAS Web of Science IUCr Journals Google Scholar
Leslie, A. G. W. (1998). DETWIN. A Program for Detwinning of Merohedrally Twinned Data. Google Scholar
Liang, J., Ealick, S., Nielsen, S., Schreiber, S. L. & Clardy, J. (1996). Acta Cryst. D52, 207–210. CrossRef CAS Web of Science IUCr Journals Google Scholar
Lietzke, S. E., Carperos, V. E. & Kundrot, C. E. (1996). Acta Cryst. D52, 687–692. CrossRef CAS Web of Science IUCr Journals Google Scholar
Matthews, B. W. (1968). J. Mol. Biol. 33, 491–497. CrossRef CAS PubMed Web of Science Google Scholar
Murray-Rust, P. (1973). Acta Cryst. B29, 2559–2566. CrossRef CAS IUCr Journals Web of Science Google Scholar
Redinbo, M. R. & Yeates, T. O. (1993). Acta Cryst. D49, 375–380. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rees, D. C. (1982). Acta Cryst. A38, 201–207. CrossRef CAS Web of Science IUCr Journals Google Scholar
Sheldrick, G. M. & Schneider, T. R. (1997). Methods Enzymol. 277, 319–343. CrossRef PubMed CAS Web of Science Google Scholar
Stanley, E. (1972). J. Appl. Cryst. 5, 191–194. CrossRef CAS IUCr Journals Web of Science Google Scholar
Tong, L. & Rossmann, M. G. (1990). Acta Cryst. A46, 783–792. CrossRef CAS Web of Science IUCr Journals Google Scholar
Valegard, K., Scheltinga, A. C. T. V., Lloyd, M. D., Hara, T., Ramaswamy, S., Perrakis, A., Thompson, A., Lee, H., Baldwin, J. E., Schofield, C. J., Hajdu, J. & Andersson, I. (1998). Nature (London), 394, 805–809. Web of Science CAS PubMed Google Scholar
Yeates, T. O. (1997). Methods Enzymol. 276, 344–358. CrossRef CAS PubMed Web of Science Google Scholar
Yeates, T. O. & Fam, B. C. (1999). Structure, 7, R25–R29. Web of Science CrossRef PubMed CAS Google Scholar
Wei, C. H. (1969). Inorg. Chem. 8, 2384–2397. CSD CrossRef CAS Web of Science Google Scholar
Wilson, A. J. C. (1949). Acta Cryst. 2, 318–321. CrossRef IUCr Journals Web of Science Google Scholar

© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.

BIOLOGICAL
CRYSTALLOGRAPHY

ISSN: 1399-0047

Volume 55| Part 10| October 1999| Pages 1750-1758

doi:10.1107/S0907444999009968