research papers
Autoindexing the diffraction patterns from crystals with a pseudotranslation
^{a}Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
^{*}Correspondence email: nksauter@lbl.gov
Rotation photographs can be readily indexed if enough candidate Bragg spots are identified to properly sample the a priori by considering all possible pseudotranslations. Care must be exercised to distinguish between true lattice diffraction and spurious signals contributed by neighboring overlapping Bragg spots, nonBragg diffraction and noise. Such procedures have been implemented within the autoindexing program LABELIT and applied to known cases from publicly available data sets. Routine use of this type of signal search adds only a few seconds to the typical run time for autoindexing. The program can be downloaded from https://cci.lbl.gov/labelit .
However, while automatic indexing algorithms are widely used for macromolecular data processing, they can produce incorrect results in special situations where a subset of Bragg spots is systematically overlooked. This is a potential outcome in cases where a noncrystallographic translational symmetry operator closely mimics an exact crystallographic translation. In these cases, a visual inspection of the diffraction image will reveal alternating strong and weak reflections. However, reliable detection of the weakintensity reflections by software requires a systematic search for a diffraction signal targeted at specific reciprocalspace locations calculatedKeywords: subgroups; sublattices; cosets; noncrystallographic symmetry.
1. Introduction
The ability to process a large number of rotation data sets sequentially is a prerequisite for many largescale projects, including the screening of crystalgrowth conditions for optimal diffraction (Page et al., 2005), the discovery of protein–ligand complexes and the acquisition of multicrystal data sets involving radiationsensitive samples. Synchrotron beamlines can facilitate highthroughput work by deploying software packages such as DNA (Leslie et al., 2002) or WebIce (González et al., 2008), which present the initial diffraction results in summary form. Under these systems, the underlying computations are automatically delegated to established crystallography programs. This represents an efficiency gain for the end user, who is freed from the burden of managing the dataprocessing steps separately for each new sample. However, it requires that routine calculations such as autoindexing (the determination of the basis vectors that span the crystal lattice) work flawlessly despite the diversity present in real experimental samples.
To deduce the e.g. Kabsch, 1988, 1993; Steller et al., 1997; Sauter et al., 2004) take the brightest candidate Bragg spots as a starting point. An implicit assumption is that no matter which bright spots are chosen, the subset is representative of the lattice as a whole. This is valid for most macromolecular crystals, as bright spots in each resolution shell are generally distributed randomly; in particular, given some simple prior assumptions about the placement of atoms in the (Read, 2001), the probability density of observing an acentric reflection with intensity I is
many autoindexing algorithms (where Σ is the mean reflection intensity of the appropriate resolution shell (Wilson, 1949; French & Wilson, 1978). Higher values of Σ are more common in shells of lower resolution.
One exception to this general probability distribution occurs if the structure contains pseudotranslational symmetry (i.e. noncrystallographic translational symmetry such that the translational symmetry operator is close to a rational fraction of the cell length; Hauptman & Karle, 1953, 1959; Gramlich, 1984). This causes the reflections to divide into a of strong intensities and coset(s) of weak intensities. Data of this nature are by no means unusual. Chook et al. (1998), for example, report two crystal structures where the average weak intensity is about 10% of the average strong intensity when considering the lowest resolution shells, in which the disparity between alternating strong and weak intensities is most pronounced. Unfortunately, the normal autoindexing strategy is not robust for these cases, as it is not possible to guarantee that a randomly chosen subset of bright reflections will include members of the weak If the chosen reflections are concentrated at low resolution, are few in number and/or if the is intrinsically weak, the strongintensity set dominates. Autoindexing will then produce an (incorrect) model lattice missing the altogether, in which the noncrystallographic translation is taken to be an exact crystallographic operation.
This problem would disappear if one could just lower the intensity threshold used to include Bragg spot observations for indexing. Regrettably this does not work consistently in practice, as it is necessary to maintain a high enough cutoff to remove artifacts that would otherwise confuse the indexing algorithm. Instead, we introduce an automated procedure that is meant to emulate the empirical process reported by various groups (e.g. Warkentin et al., 2005). Firstly, the data set is autoindexed normally to produce a presumptive basis set that may or may not span the of weak reflections, if any is present. The raw data are then reexamined to ascertain whether there is additional Bragg diffraction in between positions on the modelled lattice. If so, the presumptive basis vectors are transformed accordingly, producing a new lattice model spanning both the strong and the weak reflections. When implemented within the autoindexing program LABELIT (Sauter et al., 2004), this approach takes only a few extra seconds of computational time and identifies cases of pseudotranslation with high fidelity. The procedure has the additional benefit of being able to identify the presence of pseudotranslational symmetry at the stage of autoindexing, in contrast to which rely on the availability of reasonably complete data (Zwart et al., 2005).
2. Mathematical background
Autoindexing gives a complete description of the presumptive L and its relation to the laboratory coordinate system in the form of an orientation matrix
where the matrix components are the orthonormal projections of the unitcell basis vectors a, b, c (and reciprocalcell basis vectors a*, b*, c*) that have been converted to reduced form (GrosseKunstleve et al., 2004). The presence of pseudotranslation, associated with alternating weak Bragg spots that are not on the lattice L, leads to the identification of the true L′ given by the orientation matrix
where M is a transformation matrix whose integer determinant n is the ratio of unitcell volumes, n = A′/A. Using the terminology of Rutherford (2006), we call L′ a of L and n the index of the Although there are an infinite number of indexn sublattices of L, a key result from group theory (Billiet & Rolley Le Coz, 1980) is that the number of distinct sublattices is finite and small. Unitcell doubling, for example, leads to only seven unique sublattices: those with doubled a, b or c basis vectors, those with pseudo A, B or Cface centering and one with pseudo bodycentering. Table 1 shows the uppertriangular matrices M and transformed basis vectors associated with each of these cases.

Borrowing nomenclature from group theory, this paper uses the term L. Reciprocallattice vectors form an under the operation of vector addition, with L being a of L′. The decomposition of L′ with respect to L,
to refer to the weak reflections on the that are not part of the main latticeidentifies cosets (or subsets) g_{2}L, …, g_{n}L obtained by adding the vectors g_{2}, …, g_{n} to each vector of L. For example, the doubling of unitcell vector a leads to a single with g_{2} = ½a* and its tripling leads to cosets with g_{2} = 1/3a* and g_{3} = 2/3a*.
3. Computational approach
It is straightforward to enumerate all distinct transformations M that give sublattices of index n (Billiet & Rolley Le Coz, 1980; Zwart et al., 2006). Having performed this, the following algorithm is used to detect sublattices in the raw data. After autoindexing to determine A, perform a loop over all matrices M to give A′. For each A′ and for each rotation photograph used in autoindexing (LABELIT normally uses two 1° rotations positioned 90° apart in φ), predict the positions of all reflections on the detector out to a certain resolution limit. For each reflection with h′, backtransform the into the original (L) reciprocal basis,
Miller indices are then divided into two sets. Those with allinteger h components ({h_{integer}}) are spanned by the main lattice L, while those containing a fractional h component ({h_{fractional, M}}) are associated only with the L′. Focusing exclusively on this latter the raw data are investigated to see if there is (weak) Bragg scattering at these predicted spot positions. If so, it is concluded that the correct lattice is L′. After the loop over all matrices M is finished, the final orientation matrix (A′ if a has been discovered, otherwise A) is analyzed to determine the metric symmetry as previously described (Sauter et al., 2004, 2006).
This approach completely avoids the original dilemma of lowering the spotpicking threshold sufficiently to sample the shows the detection of a with n = 2.
which carries the risk of introducing artifacts. Instead, we target the search at specific detector positions and thus can detect weak signals down to very low signaltonoise levels. Fig. 1Since the signal of interest is inherently of low intensity, it is necessary to carefully eliminate phenomena that could be falsely interpreted as Bragg scattering from a –3.3.
Such decoy signals are treated in §§3.13.1. Rejection of intensity outliers
One potential pitfall arises from the undesirable presence of outlying pixel intensities in the raw data caused by ice crystals, zingers or other processes (Bourgeois, 1999). An example is seen in Fig. 2(a), in which a small group of saturated CCD pixels occurs by chance at the Bragg spot position predicted by pseudobody centering. A naïve method for confirming pseudotranslation would be to calculate the average centerpixel intensity 〈I〉 over the of predicted spots {h_{fractional, M}}. Unfortunately, this simple metric produces a false result in the case of Fig. 2(a) as the single outlying spot biases 〈I〉 enough that the incorrect pseudo bodycentered lattice scores higher than the correct lattice shown in Fig. 2(c). In order to reliably analyze the data, one must consider the intensity distribution over the entire population. Disregarding the single outlier, intensities measured on the incorrect are distributed on a Gaussian profile (Fig. 2b) as expected from background noise. In contrast, intensities from the correct form an approximate exponential distribution (Fig. 2d) consistent with Bragg diffraction [equation (1); Gramlich, 1984]^{1}.
To implement population modeling in software, the raw data are initially conditioned by removing the background signal of the image as described previously (Zhang et al., 2006). The background is modelled on a 50pixel grid, and we improve upon the previous work by treating the background within each grid area as an inclined (rather than a flat) plane, deriving the local plane constants by the method of Rossmann (1979). After subtraction of the background, pixel values are reexpressed in terms of the background variance (specifically, in units of the rootmeansquared deviation of local background pixels away from the bestfit background plane). We then take the set of centerpixel intensity measurements at predicted Bragg spot positions (within a suitably thin resolution shell) and attempt to fit the population to both a Gaussian distribution
with mean μ and standard deviation σ, and to an exponential distribution as in (1). Outliers are rejected by randomsample consensus (Fischler & Bolles, 1981). Briefly, model parameters (μ and σ for a Gaussian distribution; Σ for an exponential distribution) are calculated from a very small randomly chosen subset of the population. This process is repeated a large number of times, allowing the selection of a final model and a final distribution type (Gaussian or exponential) that fits the largest number of data from the whole set. The criteria for evaluating model fit are explained in Appendix A. The method is useful for distinguishing Bragg diffraction from noise, even though the analysis is performed before the dataintegration step and before and partiality corrections are applied to improve the accuracy of the potential Bragg intensities.
3.2. Rejection of nonconforming spot profiles
In addition to testing whether the intensity distribution is consistent with Bragg diffraction, it is also necessary to confirm that the observed spot positions match the candidate . Here, the Bragg spots on the main lattice {h_{integer}} are round in shape and are perfectly centered at their predicted positions. However, while the spot intensities on the candidate {h_{fractional, M}} form an acceptable exponential distribution, the spot shapes appear to be broken and are not well centered on the lattice. Rather than being an indication of pseudotranslation, this diffraction pattern is likely to arise from some other phenomenon such as fragmentation of the crystal sample.
to high precision. This guards against unwarranted conclusions from diffraction patterns such as that shown in Fig. 3The automatic rejection of this candidate (b) shows a 16 × 16 pixel mask with greyscale shading to indicate the normalized intensity of each profile pixel. The mask is now positioned on the image at every h′ (Fig. 3a) and the location of the strongest observed pixel within the mask is noted. This is performed separately for the main lattice {h_{integer}} and candidate {h_{fractional, M}}. In order for the candidate to be accepted as valid, the pixel maxima must be clustered normally around the predicted spot positions, just as they are for the main lattice. Bivariate Gaussian statistics are used to model the population of pixel maxima from the main lattice, and the candidate is rejected if too many of the maxima (e.g. > 50%) fall outside this distribution, as in Fig. 3(b).
is accomplished by a statistical analysis of spot positions. The brightest spots used for autoindexing are grouped together to form an average spot profile, after which a rectangular mask is constructed that accommodates the profile plus a strip of background pixels on each side. Fig. 33.3. Avoidance of overlapping Bragg reflections
The above examination of Bragg spots relies on the assumption that lattice positions are well separated across the detector face. Yet it is understood (Dauter, 1999) that factors such as large and high mosaicity will inevitably produce spot overlap. Therefore, before any signal analysis is performed, pairs of h_{integer} and h_{fractional,M} Bragg spots are removed if there is mutual overlap of the masks described in §3.2 (and depicted as boxes in Figs. 1–4). This is performed separately for each L′ and the is rejected as a candidate if the remaining nonoverlapped spots are too few in number for meaningful statistics (e.g. < 200). The efficient identification of overlapping masks is facilitated by the use of the Approximate Nearest Neighbor software library (Arya et al., 1998).
Discovery of spot overlaps relies on the accurate ability to predict whether particular A and the effective mosaicity m) are only initial estimates derived from autoindexing. The parameters are postrefined in a subsequent dataprocessing step (Winkler et al., 1979; Rossmann et al., 1979) after integration, but the postrefined values are not yet available at the step utilized here for considering pseudotranslation. Furthermore, the use of a single effective mosaicity parameter m is a simplification that does not account for separate contributions from different physical sources of crystal disorder, anisotropic disorder or the distinct effects of beam divergence. We do not attempt to create a highly accurate or detailed model, and consequently must allow for the possibility that the diffraction intensity at the position of a candidate spot might actually arise from the rockingcurve tail of a nearby mainlattice spot that is not predicted to diffract based on the available model.
will be in diffracting position for a given rotational setting of the crystal. In this context it is important to consider two limitations. Firstly, the lattice parameters used here (including the orientation matrixThis safeguard is implemented by adding a second overlapdetection step. h_{fractional,M} are individually considered, enumerating all on the main lattice h_{integer} that are immediately adjacent in Detector positions are calculated for each of these h_{integer} spots, even if the crystal rotational setting needed to satisfy the reflecting conditions (given the available model parameters A and m) is outside the rotational range used to acquire the image. In this way, we can reject spots that could potentially be overlapped if the model value m is unrealistically small (Fig. 4). Enumeration of all neighboring spots is computationally intensive, so the calculation is limited to a small set of representative spots distributed across the face of the detector. Candidate spots are rejected if the nearest representative spot is potentially overlapped.
spots3.4. validation using integrated intensities
The strategy outlined above examines individual pixel intensities in the raw data to detect any e.g. Chook et al., 1998). To implement this, the raw data (typically the one or two rotation images that have been used for autoindexing) are integrated with MOSFLM (Leslie, 1999) based on the triclinic basis set A′. For each potential transformation matrix M, the h′ of the integrated data are then backtransformed by (5), dividing the intensities into a main set with indices {h_{integer,M}} and a with indices {h_{fractional,M}}. A particular pseudotranslation is inferred if both (i) the has significant data, i.e. the coset's average intensitytoerror ratio 〈I/σ(I)〉 is greater than 1.0, and (ii) the average intensities on the main lattice 〈I_{M}〉 are significantly greater than those on the 〈I_{C}〉, at least in the lowest resolution bins. These calculations can be performed on single images (see Table 2), offering the potential for pseudotranslation to be validated prior to acquiring the complete data set.
that may have been ignored during the autoindexing step. To verify the presence of a it is useful to reexamine the Bragg intensities after the data have been integrated, as other authors have done (

4. Application to experimental data
The public availability of an archive of complete diffraction data sets from the Joint Center for Structural Genomics (Burley et al., 2008; https://www.jcsg.org ) provides an excellent opportunity to test new methods on real data (Baker et al., 2008). Table 2 illustrates two cases where pseudotranslation is clearly detected by examining individual rotation images.
JCSG's published structure of Haemophilus somnus XaaHis dipeptidase (PDB entry 2qyv ) is based on 90° wedges of rotation data taken at four Xray wavelengths from a single crystal with P2_{1}2_{1}2 and unitcell parameters a = 174, b = 84, c = 123 Å. The contains two protomers related by a pseudocrystallographic translation, as evidenced (for example) by the presence of a strong peak at the fractional coordinates (½, ½, ½) of a native It is possible to detect a pattern of alternating strong and weak Bragg spots on each image, but the ability to reliably choose the correct lattice during autoindexing is completely dependent on the details of the spotpicking procedure. In this particular data set, the spotpicking program DISTL (Zhang et al., 2006) typically detects over 1000 candidate Bragg spots per image. If this entire set is used, separate autoindexing of each image always produces the correct P2_{1}2_{1}2 lattice because the weak is always adequately represented. However, using all the candidate spots generally carries the risk that the lattice will be obscured by spurious signals that only appear to be Bragg spots; therefore, autoindexing success is often improved by fitting the lattice to a smaller subset consisting of the brightest spots. For the 2qyv data set, a safer algorithm (using the 300 brightest spots on each image for autoindexing) completely misses the alternating weak spots, invariably producing basis vectors consistent with an Icentered orthorhombic lattice with unitcell parameters a = 84, b = 123, c = 174 Å, where the contains only one protomer. The situation is reconciled by applying the methodology of §3 (see Figs. 1 and 2, and Table 2), allowing us to validate the presence of alternating weak spots after using the robust autoindexing method with the brightest spots.
JCSG's structure of Thermotoga maritima aspartate aminotransferase (PDB entry 2gb3 ) provides a contrasting example where the normal autoindexing approach is sufficient to detect pseudotranslation. Here, the is P2_{1} with unitcell parameters a = 75, b = 214, c = 77 Å, β = 112°. The contains three α_{2} dimers related by threefold pseudocrystallographic translation, as shown by the presence of a strong peak at the fractional coordinates (0, ⅓, 0) of a native In this case, the safe method (sorting the candidate Bragg spots and indexing on the brightest ones) does not limit the ability the choose the correct lattice. In all the images archived (86° wedges composed of 0.25° rotation images taken at two Xray wavelengths and 130° wedges composed of 1° rotation images taken at three Xray wavelengths), as long as enough candidate Bragg spots are used to produce any lattice solution, the correct is found without the additional computational steps taken in §§3.1–3.3 (see Table 2).
5. Software availability
The procedures described here are included in LABELIT v.1.1 (and above), which is available for download by noncommercial users at https://cci.lbl.gov/labelit . A benefit of the present treatment is that pseudotranslation can be detected from a single image, automatically and without visual inspection, immediately after the autoindexing step. It is therefore possible for the results to play a role in decisionmaking during data aquisition, if for example the identification of the correct permits an advantageous choice of datacollection strategy (Dauter, 1999).
Adjustments to the exact algorithm used can be made by setting runtime parameters, potentially assisting with the analysis of difficult cases where the default settings do not produce the desired result. Sample parameters are listed in Table 3 and a complete listing is given in the online documentation.

APPENDIX A
Parameter fitting
Given a Gaussian distribution with mean μ and standard deviation σ as in (6), the cumulative distribution function, or probability that an observation will have a value between −∞ and I, is
where erf is the error function. As a result, if there are N observations sorted in order of increasing value and indexed by the symbol k (k = 0, 1, …, N − 1), the expected intensity of the kth value can be modeled as
In §3.1 we reject the kth observation I_{obs}(k) as an outlier if the difference between observation and expectation is too large, e.g.
A computationally tractable approximation for the erf^{−1} function is given by Winitzki (2008), which while low in precision is sufficient for the present application.
For exponential distributions with mean value Σ as in (1), the cumulative distribution function is
The expected intensity of the kth value in a set of N observations is
where b is an ad hoc parameter allowing the in the distribution of intensities to begin at a nonzero value. Here, a useful criterion for rejecting outliers is
Footnotes
^{1}The intensity distribution of (1) applies only to acentric reflections, not to centric reflections (Wilson, 1949; French & Wilson, 1978). However, since the Bravais symmetry has not yet been identified at this stage of the analysis, we do not strictly know which measurements arise from acentric or centric Bragg spots. Here, we take the simple expedient of treating all observations as if they arise from acentric reflections, relying on the prevalence of acentric spots to allow the population to be modelled according to (1).
Acknowledgements
We thank Ashley Deacon (Joint Center for Structural Genomics) for creating the unique archive of full data sets from published JCSG structures. Ralf GrosseKunstleve and Paul Adams (Lawrence Berkeley National Laboratory) provided invaluable collaboration, aiding our software development. This work was supported in part by DOE contract No. DEAC0205CH11231 and was principally funded by NIH/NIGMS grant No. 1R01GM77071. PHZ was supported in part by NIH/NIGMS grant No. Y1GM906411.
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