Received 25 October 2001
Detection and characterization of merohedral twinning in two protein crystals: bacteriorhodopsin and p67phox
Antoine Royant,a,b+++ Sylvestre Grizot,a+++++++ Richard Kahn,a Hassan Belrhali,b+++++ Franck Fieschi,a Ehud M. Landauc and Eva Pebay-Peyroulaa*
aInstitut de Biologie Structurale, CEA-CNRS-Université Joseph Fourier UMR5075, 41 Rue Jules Horowitz, F-38027 Grenoble CEDEX 1, France,bEuropean Synchrotron Radiation Facility, 6 Rue Jules Horowitz, BP 220, F-38043 Grenoble CEDEX, France, and cMembrane Protein Laboratory, Department of Physiology and Biophysics and Sealy Centers for Structural Biology and Molecular Science, The University of Texas Medical Branch, 301 University Boulevard, Galveston, TX 77555-0641, USA
The estimation of the twinning ratio in crystals of two proteins, bacteriorhodopsin and a truncated form of p67phox, is described. Various approaches are combined to detect and determine accurately the twinning ratio. For each protein, three data sets from crystals exhibiting twinning ratios ranging from 0 to almost 50% are analysed. Self-rotation functions and Rsym values considering space groups of higher symmetry are indicative of twinning. Precise values of the twinning ratios are derived from Yeates' statistical approaches and Britton plots. The twinning ratios are also obtained by analysing the second-, third- and fourth-order moments of the intensity distribution. The twinning ratios obtained from the various approaches are compared. Second-, third- and fourth-order moments of the intensity distribution are useful indicators for evaluating the twinning precisely in order to correct the intensities or to screen rapidly for non-twinned crystals.
Twinning has recently been observed in numerous protein crystal systems (Luecke et al., 1998; Valegard et al., 1998; Royant et al., 2000; Frazão et al., 1999; Ban et al., 1999; Taylor et al., 2000). Twinning is a crystal-growth defect that arises when two or more domains within a crystal have different orientations and may even vary among crystals grown in the same batch. If the lattices of the various domains overlay perfectly in three dimensions, the twinning is merohedral. As a consequence, the reciprocal lattices of these domains coincide, summing the intensities of the Bragg spots corresponding to different Miller indices. Hemihedral twinning is a special case of merohedral twinning that occurs when two crystal domains exist. In the case of a 50% hemihedral twinning, the crystal is composed of two domains of equal volume and it is impossible to retrieve the diffracting intensities for each of the individual domains. For twinning ratios smaller than 50% diffraction intensities can be corrected. It is known from small-molecule studies that some space groups favour twinning. This is particularly the case for hexagonal, trigonal or tetragonal space groups (for reviews, see Yeates, 1997; Chandra et al., 1999). Because small-molecule crystals usually diffract to very high resolution, the ratio between the number of refined parameters and the number of experimental observations is large enough to allow a global refinement of both crystal domains at the same time.
The mixing of diffraction spots with different Miller indices affects the global intensity distribution and adds additional symmetry relationships, relative to crystallographic symmetries, among the Bragg reflections. The characterization of twinning (existence, type and ratio) is a crucial step when diffraction data from a new crystal form are analysed. Various twinning analyses for protein crystals have been developed in the last 20 y (Fisher & Sweet, 1980; Rees, 1980, 1982; Yeates, 1988, 1997; Gomis-Rüth et al., 1995; Chandra et al., 1999). Presently, although twinning poses an additional difficulty, it is not necessarily an obstacle to solving structures. The possibility of molecular replacement from twinned crystals has been discussed (Breyer et al., 1999), various refinement programs such as SHELX (Sheldrick & Schneider, 1997) or CNS (Brünger et al., 1998) consider twinning, and even multiple-wavelength anomalous diffraction methods have been used to solve a structure from twinned crystals (Yang et al., 2000). Many analyses used to detect twinning exploit the additional symmetry by examining the Rsym values in higher space groups, self-rotation functions or deviations from Wilson statistics (Rees, 1980, 1982; Yeates, 1988, 1997). These criteria are usually used to detect non-crystallographic symmetry. Packing considerations might possibly be sufficient to exclude the presence of two or more molecules in the asymmetric unit related by non-crystallographic symmetry. Alternatively, one can follow the number of negative intensities obtained during the correction of the diffraction data from the twinning effect as a function of the twinning ratio (Fisher & Sweet, 1980). Moreover, when a first model of the structure is available, it is also possible to compare the observed Bragg intensities with calculated intensities (Redinbo & Yeates, 1993; Gomis-Rüth et al., 1995).
Here, we report analyses of two protein crystals: bacteriorhodopsin (BR) and a truncated form of p67phox, a cytosolic factor of the NADPH oxidase from neutrophils. Highly ordered three-dimensional microcrystals of bacteriorhodopsin, a light-driven proton pump located in the membrane of Halobacterium salinarum, were obtained for the first time by a novel crystallization approach using lipidic cubic phases (Landau & Rosenbusch, 1996). The space group is P63 and the structure was initially solved by molecular replacement to 2.5 Å resolution (Pebay-Peyroula et al., 1997). The refinement was further extended to 2.35 Å, taking a twinning ratio of 18% into account (PDB code 1ap9 ). The resolution was subsequently improved owing to the availability of larger crystals and the structure was refined to 2.2 Å resolution using diffraction data from 35% twinned crystals (Pebay-Peyroula, unpublished results). Finally, a structure at 1.9 Å resolution was obtained from non-twinned crystals (Belrhali et al., 1999). Using the same lipidic cubic phase crystallization method, the BR structure was refined by a different group using almost perfectly twinned crystals (Luecke et al., 1998) to 2.3 Å resolution and to 1.55 Å resolution (Luecke et al., 1999) from 24% twinned crystals. The two highest resolution structures refined with CNS (Belrhali et al., 1999) and refined with SHELX (Luecke et al., 1999) are highly similar. In the case of p67phox, a truncated form of the protein crystallizes in space group P31. Since no similar structure was known, experimental phases were determined by the SAD (single-wavelength anomalous diffraction) method from a protein containing selenomethionines instead of methionines (Grizot et al., 2001). Merohedral twinning was observed for both protein crystals, with twinning ratios ranging from 0 to 50%. Knowing the twinning ratio for each data set, we were able to correct the experimental data and extract the diffraction intensities produced by a single crystal domain. In the following, we report how twinning was detected qualitatively from self-rotation functions and Rsym values and how it was assessed quantitatively from Yeates statistics and Britton plots. In addition, we derive the second-, third- and fourth-order moments of the intensity distribution as a function of twinning ratio and discuss the possibility of quantitatively determining the twinning ratios from these values. Analysis of twinning ratios allowed efficient screening for non-twinned crystals, which was necessary to collect SAD data of high quality in the case of p67phox (Grizot et al., 2001). It also enabled screening for non-twinned crystals for the ground state and the K-state structure of BR (Belrhali et al., 1999; Edman et al., 1999) and to correct the data from twinning for the L-state of BR (Royant et al., 2000).
For merohedral twinning, the symmetry operation relating domain 1 to domain 2 within a crystal is such that the Bragg spots with different Miller indices of both domains coincide exactly, giving rise to mixed intensities. Thus, Bragg spots associated with h1 from one domain coincide perfectly with those associated with h2 from the second domain, where h2 is related to h1 by a symmetry operation S. The intensities measured from the twinned crystal can be expressed as
where x is the volume fraction of the smallest domain, i.e. the twinning ratio, h2 = S(h1) and IU is the intensity diffracted by a non-twinned crystal. These equations are reduced into a single equation in the case of perfect (50%) twinning,
introducing a perfect additional symmetry on intensities. As a consequence, there is no way to retrieve the intensities of the individual domains in the case of perfect twinning and the only possibility is to refine both structures at the same time as performed by the program SHELX (Sheldrick & Schneider, 1997). The number of parameters to refine is not increased because the symmetry between the two crystals is imposed during the refinement, but (3) divides the number of independent reflections by two. The ratio of the number of observations to the number of parameters is therefore reduced by a factor of two compared with the non-twinned case. For small-molecule crystals diffracting to very high resolution, this ratio still remains acceptable for refinement. For macromolecules, the ratio is usually limiting. This is especially critical when the solvent content is low and therefore the number of atoms in a unit cell is relatively high. If the twinning is not perfect, i.e. the twinning ratio is smaller than 50%, (1) and (2) can be inverted and the intensities for reflections h1 and h2 diffracted by one domain are directly extractable,
However, the errors in the intensities are summed during this calculation and increase dramatically when x approaches 0.5,
These equations can be applied provided that the twinning ratio is known and the structure can be refined in a similar way as with non-twinned data. The ratio of observed reflections to refined parameters is the same as for a non-twinned crystal. Additionally, the refinement can be performed with the programs commonly used in protein crystallography and therefore the results (R factors, electron-density maps etc.) are easy to compare with those obtained with other structures.
Twinning can be detected from a self-rotation function (SRF), similar to the determination of non-crystallographic symmetries. The tight packing observed in space group P63 for BR or P31 for p67phox does not allow an additional internal crystallographic or non-crystallographic symmetry and additional peaks in the SRF can only be explained by the presence of two (or more) symmetry-related domains within the crystal. The relative height of the extra peak compared with the peak related to crystallographic symmetry is indicative of the importance of the twinning. Three data sets are analysed for BR (sets 1-3) and for p67phox (sets 4-6) and the data-collection statistics are summarized in Table 1. Experimental details are reported in Belrhali et al. (1999) for BR and in Grizot et al. (2001) for p67phox. The SRFs were calculated with the program GLRF (Tong & Rossmann, 1990).
The self-rotation functions are shown in Fig. 1. They indicate an additional twofold symmetry axis parallel to the a axis for data sets 2 and 3 (Figs. 1b and 1c). This corresponds to the existence of two domains for which the Bragg spots [hkl] of domain 1 superimpose with spots [kh] of domain 2. The relative ratio of the crystallographic twofold peak to the peak induced by the twinning varies from 35 to 97% for the three data sets, indicating a broad range of twinning (Figs. 1a, 1b and 1c). Surprisingly, the SRF on simulated data, based on the electron-microscopy model (Grigorieff et al., 1996) positioned in the three-dimensional crystals, shows a twofold symmetry axis in addition to the crystallographic c axis (Fig. 1d). However, this peak is of low intensity and located in the ab plane, a few degrees off the a axis. In the absence of any twinning, this additional peak probably results from an inherent symmetry of BR induced by the helices which are perpendicular to the ab plane. From simulated data with increasing twinning ratios, the extra peak increases in intensity and shifts progressively toward the a axis (data not shown). Initially, we used the intensity ratio between the crystallographic and this extra peak from the calculated SRF as a calibration in order to determine the twinning ratio of experimental data. However, the results obtained using the SRF were shown to be biased by strong diffraction spots and/or by poor completeness.
| || Figure 1 |
Self-rotation functions. (a), (b) and (c) represent the SRFs for data sets 1, 2 and 3 of BR, respectively. (d) is obtained from calculated data using the BR model determined from electron microscopy.
In addition to the crystallographic threefold axis, the SRF indicates a twofold axis parallel to the c axis (data not shown). The Bragg spots [hkl] of domain 1 superimpose with spots [l] of domain 2. The additional twofold symmetry peak displayed on experimental SRFs ranges from 0 to 100% of the height of the crystallographic threefold peak. In the absence of any model, the twinning ratio cannot be estimated from the SRFs, but it is clear that the twinning ranges from almost 0% for data set 4 to almost 50% for data set 6, while data set 5 displays an intermediate twinning ratio.
The calculation of Rsym in a higher symmetry space group is also indicative of twinning. For BR, the values of Rsym in space group P622 are 17, 12 and 5.3% for data sets 1, 2 and 3 respectively, while for p67phox the Rsym values calculated in space group P6 are 26, 24 and 8.1% for data sets 4, 5 and 6, respectively. These values are comparable to the Rsym values in the correct space group (Table 1) for a twinning ratio higher than 40%. However, for an intermediate twinning value the Rsym in the higher space group does not necessarily indicate twinning, as seen for data set 5.
After scaling and merging the intensities (Collaborative Computational Project, Number 4, 1994), a small number of negative intensities resulting from the background correction are still present. If the twinning is overestimated, correction of the diffraction intensities according to (4) and (5) leads to an increased number of negative intensities, as shown in the Britton plots in Fig. 2 (Fisher & Sweet, 1980). The value for which the number of negative intensities increases drastically gives an estimation of the twinning ratio. The values estimated for data sets 1 to 6 are given in Table 2. Yeates (1988, 1997) developed a statistical approach based on the cumulative distribution of H, where H is defined as
For acentric reflections, the theoretical cumulative distribution is
Graphic representation of the experimental S(H) compared with theoretical distributions obtained for different twinning ratios is shown in Fig. 3. For BR it shows 5, 25 and 43% for data sets 1, 2 and 3, respectively, and for p67phox it shows 9, 21 and 44% for data sets 4, 5 and 6, respectively. The twinning ratio can also be derived from the mean value or from the mean-square value of H which, for acentric reflections, are given theoretically as
The twinning ratios derived from the Britton plots, from the graphic representation, as well as from the mean values and mean-square values of H of the Yeates statistics are summarized in Table 2. They are consistent for the three data sets for both proteins within a few percent. The determination of the twinning ratio using different approaches prevents the obtaining of values that could be biased by artefacts such as incomplete data.
| || Figure 2 |
Britton plots. The number of negative intensities as a function of twinning ratio for (a) data sets 1, 2 and 3 of BR and (b) data sets 4, 5 and 6 of p67phox.
| || Figure 3 |
Cumulative distributions S(H). The dashed lines represent the theoretical cumulative distribution of H for twinning ratios of 0, 0.1, 0.2, 0.3 and 0.4, respectively. (a) shows the distributions for data sets 1, 2 and 3 of BR. (b) shows the distributions for data sets 4, 5 and 6 of p67phox.
Diffraction data obey the Wilson statistics and these were shown to be affected by twinning. Stanley (1972) derived the intensity distributions for x = 0 (no twinning) and x = 1/2 (perfect twinning). The moments of the distribution are usually calculated from the second to the fourth order with the usual analysis programs (TRUNCATE; Collaborative Computational Project, Number 4, 1994), and the experimental values can be compared with the expected theoretical values. As described earlier, only the second, third and fourth moments are of interest, as higher order moments are too sensitive to experimental errors (Giacovazzo, 1980a). Twinning will result in shrinking the distribution. The distributions of centric and acentric reflections are examined separately. In our case, because of the small number of centric reflections, only the acentric reflections were examined. The theoretical moment of order n of the intensity distribution as a function of the twinning ratio is (Appendix A)
The second-, third- and fourth-order moments are shown in Fig. 4. For p67phox, M2, M3 and M4 (Table 3) indicate twinning ratios of 2-4, 18-19% and 38-40% for data sets 4, 5 and 6, respectively. These values are consistent with those obtained from the Britton plot or the Yeates statistics (Table 2). After correcting from twinning, the values agree with a normal distribution for data sets 4 and 5. The deviations observed for data set 6 denote the difficulty of correcting data that are highly twinned. In the case of BR, the moments are larger than the theoretical ones. In order to exclude the influence of data treatment, particularly the intensity distribution, which might be affected by the way that weak reflections are integrated, BR data were integrated both with DENZO (Otwinowski & Minor, 1997) and XDS (Kabsch, 1993). Although these programs deal with the negative reflections in a different manner, the same distributions were obtained from both programs. In certain cases, pseudosymmetries or non-crystallographic symmetries were also reported to affect intensity distributions of some reciprocal zones and to induce centrosymmetric type distributions for non-centrosymmetric reflections (Giacovazzo, 1980b). Indeed, self-rotation functions on BR data from non-twinned crystals, in addition to the crystallographic symmetry-induced peaks, exhibit peaks that represent a twofold axis in the ab plane a few degrees off the a axis (Fig. 1a). This pseudo-symmetry can be explained by an intrinsic symmetry of the molecule at low and medium resolution. Another contribution to the deviation of Mn versus the normal values may arise from anisotropic diffraction. Although BR diffraction is not strongly anisotropic, this hypothesis can be tested by simulating anisotropy on calculated data. We were not able to ascertain the values of the two contributions, but we used the relative Mn values prior and after twinning correction to quantify the twinning ratio. The moments obtained after correcting the data from the twinning are still higher than the values for the theoretical distribution, but the corrected M2 and M3 are comparable for the three data sets. Knowing the deviations of M2 and M3 from the theoretical values, we were able to determine almost on-line the twinning ratio and to select crystals with low twinning ratios.
| || Figure 4 |
Normalized theoretical second-, third-, fourth- and fifth-order moment of intensity distribution as a function of twinning ratio. (a) Acentric, (b) centric. The values of Table 3 can be compared with the curves by multiplying Mn by n!.
We combined several methods in order to determine the presence and quantify the ratio of twinning in crystals of BR and p67phox. In addition to known methods such as Yeates statistics or Wilson plots, we derived the Mn as a function of twinning ratios and show how the values deviate from the theoretical values. We analysed the experimental Mn before and after twinning corrections and found that in the case of BR crystals the absolute values of the moments depend not only on twinning but also on pseudo-symmetries, non-crystallographic symmetries or anisotropic diffraction. BR data from almost perfectly twinned crystals have a M2 value larger than 1.5. As twinning and pseudo-symmetry induce deviations in opposite directions, M2 alone, without comparison, is not sufficient to detect twinning. In conclusion, the various methods are consistent and deviations for one of them could highlight pseudo- or non-crystallographic symmetry and should be analysed carefully. As the twinning ratios may vary from 0 to 0.5 among crystals grown under the same conditions, twinning should be analysed for various data sets. Inspection of second-, third- and fourth-order moments of the intensity distributions is a very rapid and efficient way of screening for non-twinned (or slightly twinned) crystals almost on-line during data collection.
Let us consider a twinned crystal composed of a fraction x1 of one domain and a fraction x2 of a second domain (x1 + x2 = 1). For a given reflection h, the intensity from the twinned crystal, IT(h), is the linear combination x1IU(h1) + x2IU(h2), where IU(hj) denotes the intensity of reflection hj from the non-twinned crystal and the relationship between h, h1 and h2 is known from the twin law.
For simplicity, the relationship IT(h) = x1IU(h1) + x2IU(h2) will be noted I = x1I1 + x2I2.
The moment of order n, <In> = Inp(I) dI, of the probability distribution p(I) can be derived from the joint probability distribution p(I1, I2) of I1 and I2, since
where (x) is the Dirac distribution of x and thus
If the probability distributions p1(I1) and p2(I2) of I1 and I2 are independent, then p(I1, I2) = P1(I1)P2(I2) and
The Wilson distribution of the intensity, Ij, for an acentric reflection is
where H(x) is the Heaviside distribution of x defined as
and <I> is the mean intensity in a given resolution shell.
From this probability distribution, it can readily be shown that
Therefore, if we consider the probability distributions of I1 and I2 as independent,
Using x to denote the twin fraction of one of the crystals and taking into account the relationship x1 + x2 = 1 one derives
For x = 0 or x = 1 (non-twinned crystal), the standard relationship for acentric reflections Mn(0) = Mn(1) = n! is obtained.
For x ½ (perfectly twinned crystal), one obtains
The variation versus x of Mn(x)/Mn(0) for the first values of n is shown in Fig. 4, from which it can be seen that the normalized moments Mn(x) of the intensity distribution of acentric reflections from a twinned crystal are always smaller than the corresponding normalized moments for an non-twinned crystal.
For a centric reflection, the Wilson distribution of the intensity Ij is
where is the multiplicity of the reflection and <I/> is the mean value of I/ in a given resolution shell.
From this probability distribution, it can readily be shown that
Therefore, if we consider the probability distributions of I1 and I2 as independent,
Using x to denote the twin fraction for one crystal and taking into account the relationship x1 + x2 = 1, one derives
For x = 0 or x = 1, one obtains the standard relationship for centric reflections from a non-twinned crystal,
For x = ½ one obtains
which shows that the intensity distribution of centric reflections for a perfectly twinned crystal is similar to the intensity distribution of acentric reflections for an non-twinned crystal.
We thank R. Henderson, K. Edman, J. Hajdu, R. Neutze and J. Navaza for helpful discussions and L. Nauton for technical assistance. Diffraction experiments were performed at ESRF (Grenoble) on beamlines ID14, ID29 and BM30A.
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