research papers
Intensity statistics in the presence of translational noncrystallographic symmetry
^{a}Department of Haematology, Cambridge Institute for Medical Research, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 0XY, England, and ^{b}Lawrence Berkeley National Laboratory, Berkeley, CA 947208235, USA
^{*}Correspondence email: rjr27@cam.ac.uk
In the case of translational
(tNCS), two or more copies of a component in the of the crystal are present in a similar orientation. This causes systematic modulations of the reflection intensities in the diffraction pattern, leading to problems with and methods that assume, either implicitly or explicitly, that the distribution of intensities is a function only of resolution. To characterize the statistical effects of tNCS accurately, it is necessary to determine the translation relating the copies, any small rotational differences in their orientations, and the size of random coordinate differences caused by conformational differences. An algorithm to estimate these parameters and refine their values against a likelihood function is presented, and it is shown that by accounting for the statistical effects of tNCS it is possible to unmask the competing statistical and tNCS and to more robustly assess the crystal for the presence of twinning.Keywords: translational ; intensity statistics; twinning; maximum likelihood.
1. Introduction
There have been great advances in the methods available for macromolecular crystallography, such that a significant fraction of structure determinations are now relatively straightforward. However, there is still the potential for serious complications when the crystals possess features that break the assumptions underlying the routine structuresolution pathways. The presence of translational
(tNCS) is particularly insidious in causing difficulties in all stages of determination, from indexing the diffraction pattern to refining the structure.In tNCS, two or more crystallographically independent copies are in the same (or nearly the same) orientation in the
Their contributions to a have the same (or similar) amplitudes but have relative phases determined by the projection of the translation vector on the diffraction vector. As a result, they interfere constructively for some reflections and destructively for others, so that there is a systematic modulation of the sum of their contributions. The most serious case is when the translation is approximately, but not exactly, equal to a potential translation such as a centring operator or a cell doubling. The exact relationship is often broken by a small rotation (typically less than 10°) in addition to the translation. Such translations are referred to as pseudotranslations or pseudocentrings because of their pseudocrystallographic nature, and they lead to pronounced effects, with large numbers of systematically very weak and very strong reflections. The perturbation of the distribution of intensities leads to difficulties with statistical tests based on intensity statistics, as well as violating the assumptions behind likelihood targets for phasing and which assume that the data follow an isotropic Wilson distribution.Translational NCS is a frequent issue in solved macromolecular crystal structures. The frequency of tNCS has been investigated by Zwart et al. (2005). The existence of tNCS can be detected by the presence of a large nonorigin Patterson peak. Using the criterion that a nonorigin peak greater than 20% of the origin peak was present in a computed using data to 5 Å resolution, it was found that about 8% of structures deposited in the Protein Data Bank (PDB; Berman et al., 2000) probably possess tNCS. Translational NCS can also prevent structure solution, for which there are anecdotal accounts but no statistical records.
In the following, the effect of tNCS on structurefactor intensity statistics is investigated. A method to characterize the parameters describing the tNCS has been developed and tested, and it is shown that corrected intensity statistics can be used to detect the presence of
The implications for experimental phasing and will be explored in subsequent publications.2. Statistical effects of noncrystallographic symmetry
A full
treatment of NCS would cover the very general case of a number of different components that are related by different noncrystallographic symmetries. In practice, the NCSrelated deviations in structurefactor intensities from an isotropic Wilson distribution are most serious when there is exact translational NCS or nearly exact translational NCS (a small rotation is present), particularly if these are translations close to crystallographic centring operators and if only one set of NCS operators is present. For this reason, and for simplicity of notation, we will only deal with the case where there is one set of NCS operators, although the formulae are presented in a way that may be generalized to multiple sets of operators. In order to deal with the very common case that the relationship is not a perfect translation but is rather a translation combined with a small rotation, we start with the case of NCS operations that combine translations with rotations of any size.2.1. Covariance elements sensitive to the effects of noncrystallographic symmetry
The statistical effects of NCS are easiest to evaluate by considering correlations between NCSrelated contributions to the structure factors and then assembling them into a picture of the overall effects of NCS.
As pointed out by Bricogne (1997), the presence of NCS leads to modulations in the intensities, which can be used to characterize the nature of the NCS. The following treatment of intensity statistics is similar in spirit to that of Bricogne, with the addition of an allowance for small random differences among the NCSrelated copies in the positions and scattering factors of the atoms that make them up. As in Bricogne (1997) we will not consider correlations among structure factors, so the structure factors are all implicitly assumed to be for reflection h.
Consider a crystal containing in its F) is made up of contributions from copies related by a combination of N_{ncs} noncrystallographic and N_{sym} crystallographic operations,
two or more copies of components with similar structure. The total (where
In this, there is an allowance for differences in the scattering factors for atoms in different copies (f_{jm} could differ among NCSrelated molecules m, particularly because of differences in the incorporated effects of B factors). The coordinates are represented in terms of those from a canonical copy of the molecule centred on the origin and conformational differences relative to that molecule (_{F}δ_{jm}). For convenience, we can take the canonical copy to be in the same orientation as the copy with k = m = 1, so that x_{j} = x_{j11} − _{F}v_{1} − _{F}δ_{j1} and _{F}V_{1} is an identity matrix. Note that since conformational differences are assigned even to the first copy, the canonical copy can be considered to be an average structure. The number of atoms in one copy of the component is given by N. The NCS rotations could be represented in terms of one matrix, C, in the notation used by Bricogne (1997), but the physical meaning is easier to understand in terms of rotations (_{F}V_{m}) in orthogonal space, so that the transformations from (O) and to (O^{−1}) fractional coordinates must be included explicitly. The operations are represented by a rotation matrix, T_{k}, and a translation vector, t_{k}.
We start by considering the covariances among the contributions to the ) terms between common atoms will dominate,
where (similar to the case of experimental phasing; Read, 2003For covariances involving atoms within the same copy (k = l for and m = n for noncrystallographic symmetry), we can consider the atoms to be independent because we have factored out any relationships leading to correlations,
If the expressions for the transformed coordinates are entered explicitly, the dot product inside the exponential in (2) can be expanded as follows:
With some rearrangement and changes of variable, this can be expressed more succinctly:
where
so that
The first exponential term in (6) accounts for the effect of rotation on interference, with _{FF}h_{klmn} being equal to the difference between two copies of the original index h rotated by different combinations of crystallographic and in the crystal; the closer _{FF}h_{klmn} is to zero, the larger the interference effect. The second exponential accounts for a systematic translationderived phase shift between the contributions of the two copies of the component. The third exponential (along with the scattering factors) accounts for the effects of differences among the NCSrelated copies. Note that if the coordinate differences are considered to be drawn randomly from a spherically symmetric distribution, then rotating these differences (e.g. in the variable _{F}δ_{jm}) will not change the nature of their probability distributions, so that the distribution of _{FF}δ_{jklmn}will be independent of k and l. (The subscripted prefix FF indicates terms relating two contributions to the observed F, to distinguish them from terms relating contributions involving calculated structure factors, G. Such terms will be needed for subsequent work on applications to experimental phasing and refinement.)
For the covariances between copies related purely by m = n but k ≠ l), the presence or absence of tNCS is not relevant. These terms will only differ significantly from zero when the symmetry rotation is parallel to the diffraction vector (T_{k}^{T}h = T_{l}^{T}h, so that _{FF}h_{klmn} = 0). When there is no phase shift between the contributions of these copies, they will contribute to increasing the expected intensity factor; otherwise, they will lead to Such pairs of contributions can be handled in a simple fashion by setting the covariance terms for m = n, k ≠ l to zero and then multiplying the remaining diagonal elements in the covariance matrix by the usual expected intensity factor ∊.
(The interesting covariances are those between copies related by m ≠ n). If we assume that the differences in scattering factors and atomic positions are independent of the positions of the atoms within the components, then the expected value can be treated as a product of expected values, separating the correlation (_{FF}ρ_{mn}) of the structure factors for the components if they were in the same position and orientation from the interference effects,
(where
If there is an atomic model, then at least the approximate locations of the atoms in each component are known, so that the expected value of the rotational interference term can be computed. However, if we are characterizing translational NCS prior to structure solution, the best we will have is some idea of the envelope containing the component. In this case, the expected value of the interference term is an integral over the volume of the envelope (denoted U_{F} for the volume of a unique component contributing to the F), which is equivalent to the Fourier transform of the envelope or a Gfunction (Rossmann & Blow, 1962). Because the envelope is finite in volume and does not possess crystallographic symmetry, it is convenient to index it in terms of a diffraction vector (in units of Å^{−1}),
where
Before the shape of the molecule (or at least its orientation) is known, it may be appropriate to approximate it as a sphere with radius r, so that the Gfunction is the Fourier transform of a sphere (Rossmann & Blow, 1962),
A Gfunction computed from a sphere centred on the origin (Fig. 1) gives insight into the general behaviour of the interference term; the Gfunction differs significantly from zero only for values of _{FF}s_{klmn} with a magnitude substantially less than the reciprocal of the sphere radius. Gfunctions from volumes with finer details in their shapes and lacking symmetry will also lack spherical symmetry and will have features extending to higher resolution, although the largest values will still be close to the origin.
The argument of the Gfunction, _{FF}s_{klmn}, will be near zero either when the two corresponding copies of the structure component (related by combinations of crystallographic and noncrystallographic symmetry) are in nearly the same orientation or when the rotation axis is nearly parallel to the diffraction vector, so that
The former condition will apply for all structure factors, leading to an overall modulation of the diffraction pattern, while the latter condition will lead to spikes in the diffraction pattern with a significant modulation (Bricogne, 1997). The maximum modulation along the direction of the spikes arising from this component of the symmetry would be equal to the number of copies in the However, the maximum would only be reached if the direction of the rotation axis coincided with the diffraction vector and if the disposition of the copies were such that they were equally spaced between the Bragg planes. In principle, knowing the directions of such spikes would contribute to understanding the rotational part of the NCS, and the pattern of intensity modulation along these spikes would give information about the relative positions of copies of components. However, this is a minor contribution to the overall modulation of the structurefactor intensities in the case of translational NCS. Including this term does not significantly alter the corrective factors, but does significantly increase the computation time (results not shown). In the remainder we will neglect the contribution to the covariances of copies in significantly different orientations.
Although a noncrystallographic translation can be generated by a combination of k = l for the pairs we will consider; the covariance elements will be approximated as zero for k ≠ l. (As above, we deal with the case in which the symmetry rotation is parallel to the diffraction vector by multiplying included terms by the expected intensity factor ∊.) This leads to simplification of the expressions in the covariances,
and (for example, a crystallographic twofold and a nearly parallel noncrystallographic twofold), we can choose without loss of generality to consider the copies related by noncrystallographic translations as belonging to the same so thatNote that the phaseshift term containing _{FF}v_{kkmn} now only depends on the translation vector between the NCSrelated copies and not on the translational component of the operators. This has the advantage that an analysis of the effects of tNCS can be carried out when the Laue group is known but not necessarily the particular space group.
2.2. Effect of tNCS on the expected intensity of the observed structure factor
Correlations among the components of the
lead to systematic modulation of the observed intensities.The variance (expected intensity) of the ∊. To allow simply for the possibility of a part of the crystal that does not obey these NCS operators, we can add a term Σ_{Fr} for the rest of the structure. (Note that Σ_{Fr} could include the contribution of another component with a different set of NCS operators, showing how the treatment presented here could easily be generalized.)
that is the sum of the contributions of the different components is the sum of all of the covariances between these contributions. This is simplified by the fact that we are ignoring terms between different operators and collecting their influence in the expected intensity factorIn this expression, terms with m < n have been paired with their complex conjugates, i.e. the terms with m > n, so that the imaginary parts cancel. The unmodulated terms can be collected into a term representing the intensity that would be expected after averaging over the modulations, Σ_{N},
The term in the curly braces can be thought of as an extra ∊ factor accounting for the modulation of the intensities by NCS.
This general expression could be applied when there is an atomic model, which defines the envelope enclosing the parts of the structure that obey tNCS, and the rotations and translations that relate these parts of the structure. Before the structure is solved, there is no way to know the shape of the envelope (or at least how it should be oriented, if there is a molecularreplacement model), so it is simplest to assume a sphere, in which case the Gfunction is real and depends only on the resolution. This approach should capture the most important effects of tNCS even when there is a detailed atomic model,
For the very common special case in which there is only one translational NCS operator, the equation can be simplified further,
where
In this form, the weight _{F}τ_{ncs} applied to the modulation term is effectively the fraction of the scattering of one component in the unit cells that obeys the translational NCS, corrected for the effect of differences among tNCSrelated copies. Note that this automatically allows the presence of a component that does not obey tNCS.
3. Simulations to test the probability distributions
The probability distributions describing the statistical effects of tNCS have been tested by simulations in Mathematica (v.8.0; Wolfram Research, Champaign, Illinois, USA). In these simulations, data have been generated for a crystal in P1 containing two `molecules' related by tNCS. For the first copy of the molecule, atoms were generated randomly within a sphere and copies of these atoms were then generated by applying a small rotation, a translation and a random shift. Since the molecules have a spherical envelope, the Gfunction is the Fourier transform of a sphere, as discussed by Rossmann & Blow (1962). The simulations show that accounting for the effects of orientation and conformation differences between tNCSrelated copies will be essential to gain a good agreement between theory and observation.
3.1. Modulations of observed intensities
As described by (13), tNCS introduces a modulation of the expected intensities depending primarily on the phase shift of the contributions from copies related by tNCS. The modulation drops in strength if there are differences in the conformations or the orientations of the copies. Fig. 2 illustrates the effects of random coordinate differences (assumed to be drawn from a Gaussian distribution) and differences in orientation on the strength of modulation for structure factors obtained from a crystal with two spherical molecules. Note that when the model is complete and the two copies scatter with the same strength then the term _{F}τ_{ncs} in (15) is equal to half of the complex correlation between these copies _{FF}ρ_{12}. When the coordinate differences are drawn from a Gaussian distribution with an r.m.s. coordinate difference of σ_{r}, then this complex correlation can be calculated using the appropriate formula for σ_{A}, which is also a complex correlation (Read, 1990),
As shown in Fig. 2, random conformational differences and rotational differences between the copies can have a similar effect on the strength of the intensity modulation, except that there is a directiondependence of the effect of the rotation difference: a rotation around the diffraction vector has no effect (because it does not change the positions of the atoms relative to the Bragg planes), whereas a rotation around an axis perpendicular to the diffraction vector has a large effect. This figure also shows that the information to distinguish the effects of random conformational differences and rotational differences may be most obvious at higher resolution.
The simulation in Fig. 3 demonstrates that (15) provides an excellent description of the average intensities for different reciprocallattice vectors, even when there is a combination of conformational and orientation differences between the copies.
4. Refining parameters characterizing tNCS
To characterize tNCS from a data set, parameters describing the NCS translation, the difference in orientation of the tNCSrelated copies and the random differences between the structures of the copies must be estimated and refined. This has been implemented with the following algorithm in Phaser (McCoy et al., 2007). The current implementation is optimized for the common case of two copies related by tNCS. Multiple tNCS copies can also be handled, as long as the copies are generated by successive applications of the same translation vector, but a more general treatment has not yet been implemented. The parameters characterizing the tNCS are refined against a likelihood function given by the Wilson (1949) distribution of amplitudes for acentric reflections,
or centric reflections,
In this likelihood function, the expected value of the intensity is computed using (14), so the refined parameters are the parameters from that equation.
An initial estimate of the translation vector between the two copies (or the first two of successive copies), _{F}v_{1} − _{F}v_{2}, is obtained from the largest offorigin peak in a native If the translation is close to a centring operator, symmetryrelated copies of the Patterson peak will merge into a single peak on a special position. would not be able to move this translation vector to one of the equidistant symmetry copies so, if the Patterson peak is on a special position, the translation vector is first perturbed by a small translation of d_{min}/6 in each of the x, y and z directions; we have found this to be sufficient to avoid the being trapped on an exact centring translation.
A x, y and z axes, which behave well in because they are approximately orthogonal. Note that when the exact shape and size of the molecule that obeys tNCS is not known, there is a tradeoff between the assumed radius of the sphere that approximates the molecular envelope and the size of the rotation angles. The rotational difference enters the likelihood target through the Gfunction term, which depends on the amount by which the rotational difference rotates the diffraction vector. For small rotations, the absolute size of the movement of the diffraction vector is, to a good approximation, proportional to the rotation angle, so an error in the assumed sphere radius can be compensated by a reciprocal change in the size of the rotation angle.
of the relative orientation is carried out if there are two copies related by tNCS; for multiple copies, we currently approximate the effect of rotational differences as random differences among copies related by a pure translation. Because the orientation does not always converge uniquely from any starting point, refinements are started from several relative orientations and that giving the best agreement with the data is chosen. The rotational difference between the two copies is parameterized as a combination of small rotations about theFinally, the complex correlation between pairs of tNCSrelated copies (_{FF}ρ_{mn} in 14) is currently assumed to be equivalent for all pairs when there is more than one NCS translation, and we do not currently account for the possibility of different overall B factors among the copies. In this case, we can refine the resolutiondependent parameter _{F}τ_{ncs} assumed to be equivalent for all pairs of tNCSrelated copies. In Phaser this is reported as a Luzzati D factor (Luzzati, 1952). In fact, the refined parameter is given by the corresponding variance term
which has better
properties, as the likelihood function is more nearly quadratic when expressed in terms of this parameter.5. Intensity moments in the presence of tNCS
Intensity moments can be a useful diagnostic for the presence of ; Rees, 1980), but their usefulness can be reduced by other influences on the distribution of intensities, such as overall anisotropy and, in particular, tNCS (Padilla & Yeates, 2003; Lebedev et al., 2006). Corrections for overall anisotropy are now well established (Popov & Bourenkov, 2003; McCoy et al., 2007). We were interested in determining whether a further correction for the statistical effects of tNCS would at least partially unmask the statistical effects of twinning.
(Stanley, 1972Evalues that have been corrected for the statistical effects of tNCS can be computed using the expression for the expected intensity in (14),
and then these Evalues can be used in the standard moment tests.
Several test data sets were selected from the PDB for structures with pairs of molecules or assemblies in the 2fuq (Shaya et al., 2006), 1un7 (Vincent et al., 2004), 1y9r (Fagart et al., 2005), 1eh4 (Mashhoon et al., 2000) and 1upp (Karkehabadi et al., 2003). These cases were chosen to illustrate the effects of anisotropy, and small rotational deviations from a pure translation. One of these cases, 1upp, was also chosen by Lebedev et al. (2006) to illustrate the effect of combining and tNCS.
related by tNCS: PDB entriesTable 1 shows the results that are obtained by computing second intensity moments for centric and acentric reflections before and after correction for overall anisotropy and for the effects of tNCS. Note that if the data obey standard Wilson distributions the expected value for this moment is 3 for centric reflections and 2 for acentric reflections, but in the presence of perfect the moments would be reduced to 2 for centric reflections and 1.5 for acentric reflections (Stanley, 1972). To assess the significance of any deviation from the values expected for untwinned data, a pvalue is also shown; this pvalue is the probability (computed from the observed distribution of intensities) that the true value of the second moment for the acentric reflections is 2 or greater. In Phaser, a pvalue of 0.001 or less triggers a warning that the crystal is likely to be twinned.

As an objective measure of phenix.refine (Afonine et al., 2012) is shown for the structures in cells that support or In addition, Table 2 compares the refined values for the tNCS operators with the values determined from the deposited models to allow an assessment of the simplified model of the crystal used to characterize tNCS.
the twin fraction obtained by twin in
‡PDB translation vector measured as a vector between centres of mass of common mainchain atoms 
These tests demonstrate that the correction for the statistical effects of tNCS can indeed unmask the statistical pvalues for twinned crystals are significantly lower than the threshold of 0.001 even when the twin fraction is as low as about 0.1. However, for the case of nearly perfect in 1upp, the second moment is 1.71, which is significantly larger than the value of 1.5 that would be expected for perfect This may, at least in part, be because the molecular assembly differs significantly from the assumed spherical shape with a radius of about 33 Å; it is a Ushape fitting into a box of approximately 88 × 54 × 42 Å. More importantly, the twinrelated reflections in this case will be affected by different modulations, so that the model of the effects of tNCS will be a compromise. In 1upp the two molecules are related by a translation of approximately 0, 1/2, 1/2 and a rotation of 3.43° about an axis very nearly parallel to the y axis. The largest modulations will therefore be seen for reflections with small h and l indices, for which the rotation has very little effect on scattering. However, the is k, h, −l, so that reflections near the h00 axis, with large values of the h index and thus relatively little modulation, are superimposed on reflections near the 0k0 axis with significant modulation.
TheThe results in Table 2 show that the method is able to detect deviations from exact centring operators, even when the Patterson peaks merge into a single peak consistent with a perfect centring operation. The refined translation vectors agree well with the vectors determined from the refined models. Also, even though the assumption of spherical molecules is not necessarily obeyed well, the refined rotations are correlated to the true rotations. The rotations are determined more accurately when the translations are closer to centring operators. In this situation, more of the reflections are affected by strong modulations, so that there is more signal from which the rotational parameters can be deduced.
To test whether it is important to model the rotational difference between pairs of tNCSrelated molecules, or whether the D parameters can compensate, we repeated the test calculations for two of the crystals that showed a significant rotational difference, 1un7 and 1eh4, but not allowing the modelled rotation to refine away from zero. For 1un7, the mean value of the second moment of the intensity was 2.25, compared with 1.97 when the rotation was modelled. For 1eh4, the second moment without refining the rotational parameters was 1.94, compared with 1.81. Note that a second moment of 1.94 does not differ significantly from the value of 2 expected for an untwinned crystal, with a pvalue of 0.148. These results demonstrate that it is indeed important to model the rotational differences when characterizing tNCS.
of the Luzzati6. Conclusions
This analysis has shown that the effects of tNCS depend on the exact values of the translation, which can be estimated precisely, and on small differences in orientation between the NCSrelated copies, which can be given better than random estimates even under conditions where the simplifying assumptions of spherical molecules are not valid. By taking account of the statistical effects of tNCS, the statistical et al., 2006). If the data have been merged with too high symmetry these tests cannot be applied, but if the data have been merged with too low symmetry then these tests will generate false positives. Note that when the symmetry is correctly assigned, tests such as the Ltest (Padilla & Yeates, 2003) are preferable for their ability to assess the twin fraction reasonably reliably. In the application of the Ltest, reflections with indices differing by even numbers are typically chosen to minimize the statistical effects of tNCS arising from pseudocentring (Padilla & Yeates, 2003); however, when the tNCS differs from a pseudocentring operation it may be helpful to correct for the statistical effects of tNCS before applying the Ltest.
can be unmasked sufficiently to provide a clear diagnostic for This is important in practice because tests that depend on twin laws rely on having the symmetry correctly assigned (LebedevIn future work, we will show how this understanding of the statistical effects of tNCS can be used to improve methods for
phasing by singlewavelength anomalous diffraction and structure refinement.Acknowledgements
This work was supported by the Wellcome Trust (Principal Research Fellowship award 082961 to RJR) and by the NIH/NIGMS (P01GM063210 to PDA and RJR). This work was supported in part by the US Department of Energy under Contract No. DEAC0205CH11231.
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