 1. Introduction
 2. Treatment of AS in macromolecular crystallography: a brief survey
 3. Anisotropy of anomalous scattering in a nutshell: monoclinic crystals of pbromobenzamide
 4. AASinduced symmetry breaking in crystals of a brominated DNA molecule
 5. A formal description of AAS and its connection to macromolecular phasing
 6. Modelling and parameter refinement for AAS
 7. AAS and symmetrybreaking effects in crystals of the selenated protein PPAT
 8. AASinduced phasing on `standard' data: improved SAD phasing of the selenated protein IMPDH
 9. Discussion
 10. Conclusion
 References
 1. Introduction
 2. Treatment of AS in macromolecular crystallography: a brief survey
 3. Anisotropy of anomalous scattering in a nutshell: monoclinic crystals of pbromobenzamide
 4. AASinduced symmetry breaking in crystals of a brominated DNA molecule
 5. A formal description of AAS and its connection to macromolecular phasing
 6. Modelling and parameter refinement for AAS
 7. AAS and symmetrybreaking effects in crystals of the selenated protein PPAT
 8. AASinduced phasing on `standard' data: improved SAD phasing of the selenated protein IMPDH
 9. Discussion
 10. Conclusion
 References
research papers
Exploiting the anisotropy of
boosts the phasing power of SAD and MAD experiments^{a}École Polytechnique Fédérale de Lausanne (EPFL), Laboratoire de Cristallographie, CH1015 Lausanne, Switzerland, and ^{b}Global Phasing Ltd, Sheraton House, Castle Park, Cambridge CB3 0AX, England
^{*}Correspondence email: marc.schiltz@epfl.ch
We would like to dedicate this article to Lieselotte and David Templeton, whose pioneering studies of AAS provided much inspiration through the course of this work.
The Xray polarization anisotropy of SHARP. Refinements using actual data show that exploitation of the anisotropy of can deliver substantial extra phasing power compared with conventional approaches using the same raw data. Examples are given that show improvements in the phases which are typically of the same order of magnitude as those obtained in a conventional approach by adding a secondwavelength data set to a SAD experiment. It is argued that such gains, which come essentially for free, i.e. without the collection of new data, are highly significant, since radiation damage can frequently preclude the collection of a secondwavelength data set. Finally, further developments in synchrotron instrumentation and in the design of datacollection strategies that could help to maximize these gains are outlined.
in crystals of brominated and selenated proteins is shown to have significant effects on the diffraction data collected at an For conventionally collected single or multiwavelength anomalous diffraction data, the main manifestation of the anisotropy of is the breakage of the equivalence between symmetryrelated reflections, inducing intensity differences between them that can be exploited to yield extra phase information in the structuresolution process. A new formalism for describing the anisotropy of which allows these effects to be incorporated into the general scheme of experimental phasing methods using an extended Harker construction is introduced. This requires a paradigm shift in the dataprocessing strategy, since the usual separation of the datamerging and phasing steps is abandoned. The data are kept unmerged down to the Harker construction, where the symmetrybreaking is explicitly modelled and refined and becomes a source of supplementary phase information. These ideas have been implemented in the phasing programKeywords: anisotropy of ; phasing; SAD; MAD; polarized resonant diffraction.
1. Introduction
Experimental phasing methods based on f′ and f′′ for certain heavier atoms. These quantities are scalarvalued and hence describe an isotropic AS behaviour without reference to any notion of directionality. It is known, however, that the phenomenon of AS at or very near absorption edges involves an extra degree of complexity, namely anisotropic behaviour through its dependence on the polarization of the Xray beam, that has not so far been exploited for phase determination.
(AS), as currently implemented, describe the phenomenon of AS in terms of the wavelengthdependent factorsIt is the purpose of this paper to correct this omission and thus complete the task of fully exploiting one further aspect of AS, its anisotropy, for macromolecular phasing. Our rationale is that the move towards ever smaller crystals and more intense Xray beams has led to radiation damage, rather than beamtime availability, becoming the main limiting factor in the amount of data that can be collected from a single sample for the purpose of experimental phasing. In these circumstances, it becomes of the utmost importance to be able to derive the maximum amount of phase information from data collected at a smaller number of wavelengths (often one), as one cannot count on being able to collect data at further wavelengths from that sample, even if beam time is not the limiting factor.
We show here that the anisotropy of
(AAS), which has so far been overlooked for phasing purposes, is a significant and ubiquitous effect that can deliver substantially enhanced phasing power from conventionally collected single or multiwavelength anomalous diffraction (SAD or MAD) data sets after a suitable treatment for it has been introduced into standard phasing software. We also discuss how this effect could be further exploited by suitable adaptations of current instrumentation and experimental protocols.To situate the present work in its proper context, we first survey certain basic aspects of the use of 2). As AAS may be unfamiliar to many readers, we give two simple easily visualisable concrete examples of it: firstly in a smallmolecule crystal, where it manifests itself through violations of certain (§3), and secondly through the related effect of breaking the symmetry equivalence of peaks in an anomalous difference Fourier map for a BrDNA crystal (§4). Both examples illustrate the simple conceptual basis of AAS and its ability to give rise to new phase information by breaking the pointgroup symmetry of the diffraction pattern of a crystal, in much the same way as isotropic AS breaks the Friedel symmetry. We then present a formal treatment of AAS (§5) and show that a small approximation to the full expression of an AASmodulated intensity allows it to be cast in such a form that these intensity modulations, together with a suitable parametrization of the anomalousscatterer substructure, can be used as extra phasing contributors in an extended Harker construction. These extra contributions arise from unmerged intensity measurements between which the usual symmetry equivalence has been broken by AAS. Technical aspects of the modelling of AAS effects and of the of the corresponding parameters are dealt with in §6 and are illustrated on the BrDNA example of §4. We then present two other instances of AS phasing supplemented by AAS effects arising from Se atoms: one from a data set specifically collected to further investigate the symmetrybreaking picture of AAS (§7) and another from a totally standard data set collected almost a decade ago, without any special concerns regarding methodology, towards a MAD (§8). In both cases we demonstrate substantial gains in phasing power compared with the conventional treatment of AS from the same data, gains that occur, so to speak, `for free'.
for experimental macromolecular phasing (§We conclude (§9) by emphasizing the ubiquitousness of AAS effects in data sets measured near the absorption edges of Br or Se and hence the possibility of revisiting such data sets and deriving further phase information from them. We discuss the reasons why AAS had so far remained inconspicuous, how future experimental strategies can be designed with AAS effects in mind so as to maximize the phaseinformation gain they can afford and how some judicious instrumental choices at synchrotron beamlines could further increase the usefulness of the AAS phasing signal in such experiments.
2. Treatment of AS in macromolecular crystallography: a brief survey
; Hendrickson, 1999). These methods exploit the resonance behaviour of certain heavier atoms in the macromolecule when the photon energy of the incident Xrays is in the vicinity of one of their absorption edges (Hendrickson, 1991). Resonance effects lead to the breakdown of Friedel's law and the resulting intensity differences provide bimodal indications for the structurefactor phases of acentric reflections in the singlewavelength anomalous diffraction (SAD) method. The wavelengthdependent modulation of the factors f′ and f′′ in the vicinity of an can be further exploited in multiwavelength anomalous diffraction (MAD) experiments to resolve the bimodality of singlewavelength phase indications of acentric reflections and complement them with phase indications for centric reflections. In all these procedures, the f′ and f′′ factors are treated as scalars.
methods have achieved tremendous success in the of proteins and using Xray crystallography (Ogata, 1998It has been known since the pioneering work of Templeton & Templeton (1980) that in the vicinity of an AS can display anisotropic behaviour with respect to the polarization of the Xray beam and is then best described by representing f′ and f′′ as tensor rather than scalar quantities. This was subsequently confirmed by numerous studies on salt and smallmolecule crystals (Templeton & Templeton, 1982, 1985a,b, 1986, 1987, 1989, 1991, 1992, 1995; Dmitrienko, 1983, 1984; Petcov et al., 1990; Kirfel et al., 1991; Kirfel & Petcov, 1992; Lippmann et al., 1996; Dmitrienko et al., 2005). The effects of AAS are most pronounced with (although not restricted to) linearly polarized synchrotron radiation. AAS originates from resonant transitions between the core electrons and antibonding valence molecular orbitals that are rendered nonspherically symmetric by the chemical bonding of the absorbing atom. The thus depends on the relative orientation of the incident and diffracted electric fields with respect to these molecular orbitals. This is illustrated in Figs. 1 and 2 for Se in selenomethionine and Br in brominated respectively, which are the two most important anomalous scatterers used for phasing purposes in macromolecular crystallography.
The relevance of AAS to macromolecular phasing was first pointed out by Templeton & Templeton (1988) in their study of selenolanthionine, the same smallmolecule compound used by Hendrickson (1985) to demonstrate the possibility of at the Se K edge. AAS effects were visible in the dependence of the intensity of Xray reflections on the orientation of the C—Se—C moieties in the crystal with respect to the polarization direction of the Xray beam. The variations in factors produced by changes in crystal orientation were found to be of comparable magnitude to those associated with the changes of wavelength used in the MAD method. Templeton & Templeton (1988) commented that
the polarization effects reported here may be useful for phasing, but can cause error if not taken into account.
This led Fanchon & Hendrickson (1990) to investigate the practical impact of AAS on the MAD method. Concerns about the possible detrimental effects of AAS on the accuracy of were alleviated by estimating typical phase errors through numerical simulations and by actual measurements on crystals of selenobiotinyl streptavidin. The authors concluded that
the results show that the AAS does not cripple the MAD method, and that phases uncorrupted by these effects can be recovered.
Our own interest in exploring the potential of AAS for more accurate phase determination, evoked by Templeton & Templeton (1988), has been longstanding. In an earlier paper (de La Fortelle & Bricogne, 1997) describing the phasing program SHARP, plans to implement a tensorial description for the f′ and f′′ factors were alluded to. Also, the implementation of refineable `anomalous nonisomorphism' parameters in SHARP was in part justified by the fact that AAS effects could be present in SAD or MAD data sets and would give rise to nonisomorphism if the factors were treated as scalar quantities only. The work described here was preceded by extensive developments of computer code for simulating data affected by AAS and analysing them for the purpose of phase determination (Schiltz & Bricogne, unpublished results). Experimental investigations of AAS effects in macromolecules at the level of polarized absorption spectra were reported by Bricogne et al. (2005) and an example of their impact on the success or failure of a SAD phasing experiment was given by Sanishvili et al. (2007). Here, we finally tackle the central topic of incorporating AAS effects into the general process of macromolecular phasing.
3. Anisotropy of in a nutshell: monoclinic crystals of pbromobenzamide
To illustrate what is involved, we briefly present the example of AASinduced symmetrybreaking effects in crystals of pbromobenzamide. In this smallmolecule compound, a Br atom is attached to a benzene ring. Thus, the AAS of Br in this compound can be expected to be very similar to that of brominated (Sanishvili et al., 2007). This was confirmed by a series of polarized absorption spectra recorded on triclinic crystals of pbromobenzamide, in which all the C—Br bonds in the are parallel (Schiltz et al., unpublished results). As is the case with brominated the white line was found to be most pronounced along the direction parallel to the C—Br bond, whereas it was completely absent if the polarization vector of the Xray beam was perpendicular to the C—Br bonds. The same molecule can also crystallize in the monoclinic P2_{1}/c (unitcell parameters a = 4.491, b = 5.415, c = 28.013 Å, β = 93.58°) with four symmetryrelated molecules of pbromobenzamide in the The crystal packing is such that two molecules related by the glide plane have their C—Br bonds in directions that are almost perpendicular to each other (Fig. 3). Xray diffraction experiments were carried out at station BM01A of the Swiss–Norwegian Beamlines (SNBL) at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. The Xray beam displays a high degree of linear polarization and the photon energy was tuned to 13.474 keV, which corresponds to the position of the white line at the Br K edge.
In an initial experiment, the crystal was oriented so that the polarization direction of the Xray beam was aligned with the C—Br bond of one of the molecules in the ). In this configuration, the C—Br bond of the glideplane related molecule is almost perpendicular to the polarization direction. Thus, the AAS will cause the two symmetryrelated Br atoms to display different factors at the Br K edge: the Br atom that has the C—Br bond aligned with the polarization direction will exhibit a strong white line (large f′′), but this will not be the case for its symmetryrelated mate, which has its C—Br bond perpendicular to the polarization direction. These Br atoms are therefore no longer equivalent as far as their properties are concerned and indeed the glideplane forbidden reflections in the (h0l) layer are observed to have weak but nonzero intensities (Fig. 3). It should be mentioned that the diffraction data were collected using a single scan axis which was oriented parallel to the direction of polarization of the Xray beam. Thus, even though the sample was rotated during data collection, the direction of polarization of the incident beam did not change with respect to the crystal. It was also checked that at energies away from the Br K edge the glideplane forbidden reflections returned to being systematically absent (data not shown).
(Fig. 3In a second experiment, the polarization direction of the incident beam was aligned parallel to the glide plane (i.e. perpendicular to the crystal b axis) and Xray diffraction data were again collected at the whiteline energy position (13.474 keV). This time, the glideplane forbidden reflections in the (h0l) layer were observed to have zero intensities (Fig. 3). The reason for this is easily understood: if the polarization direction of the incident beam is parallel to the glideplane the C—Br bonds of glideplane related molecules are oriented at identical angles with respect to the direction of polarization. The equivalence of the symmetryrelated Br atoms is thus restored and the forbidden reflections disappear.
This simple example illustrates some of the important aspects of AAS. The electric field vector of the incident Xray beam can break the symmetry equivalence between anomalously scattering atoms. As a consequence, the symmetry in 7), we will give an example of AAS in a macromolecular crystal where certain symmetries are broken whilst others are preserved owing to the particular orientation of the crystal symmetry elements with respect to the Xray beam polarization.
is also broken and forbidden reflections can appear. Furthermore, the example shows that the symmetrybreaking effects depend on the relative orientation of the Xray polarization direction with respect to the symmetry elements in the crystal. If the polarization direction remains invariant under the action of a of the crystal the corresponding symmetry is preserved, even in the presence of AAS. Below (in §4. AASinduced symmetry breaking in crystals of a brominated DNA molecule
The intensity of forbidden reflections stems exclusively from anisotropy in the behaviour of the anomalously scattering atoms. In macromolecules, the proportion of these atoms in the
is usually rather small. As a consequence, forbidden reflections are difficult to observe even if there is significant AAS. On the other hand, the intensity of nonforbidden reflections is a consequence of the superposition of the scattering from both the anomalously scattering atoms and the `normally' scattering atoms in the Thus, AASinduced intensity modulations in nonforbidden reflections can become significant in macromolecular crystals. In the example presented here, we show that AAS by a few Br atoms can give rise to measurable intensity modulations of the nonforbidden reflections in crystals of a brominated DNA molecule.The brominated ZDNA duplex d(CGCG[BrU]G) crystallizes in P2_{1}2_{1}2_{1}, with unitcell parameters a = 17.34, b = 32.07, c = 44.34 Å. There are two brominated residues per DNA duplex. The packing of the molecules in the crystal is such that all C—Br bonds in the are located in planes almost perpendicular to [001], but they are not parallel to each other within these planes (see Fig. 4). We have previously shown (Sanishvili et al., 2007) that there is a pronounced directional dependence of the anomalous signal strength in Xray diffraction data collected at the Br K from these crystals and that choosing the correct orientation for crystals of such molecules can be a crucial determinant of success or failure when using SAD or MAD methods to solve their structure. Here, we show that beyond the directional dependence of the overall anomalous signal, the strength of each Br site in the crystal is modulated by the relative orientation of the corresponding C—Br bond with respect to the Xray polarization direction.
Xray diffraction data were recorded from a single cryofrozen crystal on the SBCCAT beamline 19ID at the Advanced Photon Source (APS) in Argonne, Illinois, USA (Rosenbaum et al., 2006) at a wavelength of 0.9199 Å, corresponding to the energy position of the Br K edge white line (13.477 keV). The crystal was oriented with the aid of a computercontrolled kappa goniostat and data sets were collected for three different crystal orientations (see Table 1). The data were recorded using a single scan axis oriented parallel to the direction of polarization of the Xray beam. Hence, during each data collection the direction of polarization remained constant with respect to the crystal and thus also with respect to the C—Br bonds. Each data set was separately integrated with MOSFLM (Leslie, 1993). Further data processing was carried out with programs from the CCP4 software suite (Collaborative Computational Project, Number 4, 1994). Details of the data collection and reduction are given in Table 1. Each data set was internally scaled by minimizing the disagreement between symmetryequivalent reflections in Laue group mmm. These scaled intensities were used to produce two reflection files: one with the data merged in the crystal 222 and a second with the data merged in 1, i.e. not imposing any symmetry.
‡Multiplicityweighted merging R factor, keeping Bijvoet pairs separate (i.e. computed in the crystal point group). §Multiplicityweighted merging R factor, not keeping Bijvoet pairs separate (i.e. computed in the crystal Laue group). 
The eight Br sites in the crystal (a) for data set (II)]. However, in anomalous difference Fourier maps computed with the data merged in 1, symmetryrelated Br sites display widely differing peak heights. Furthermore, for each Br site the peak height also varies considerably among the three data sets, i.e. it is a function of crystal orientation. A clear correlation can be established between the height of each peak in the anomalous difference Fourier maps and the angle between the C—Br bond direction and the direction of Xray polarization for the corresponding Br sites (see Fig. 5). The strength is largest for the Br atoms which have their C—Br bonds closely aligned with the Xray polarization direction. We will analyse this correlation quantitatively in §6.2 (see also Fig. 6) and conclude that these large variations in strengths between symmetryrelated Br sites are indeed a manifestation of AAS.
feature prominently in an anomalous difference Fourier map computed with data merged in the true crystal 222. As expected, symmetryrelated sites display identical peak heights [see Fig. 5When reflection data are merged in a certain point or Laue group one actually imposes a symmetry on the , each of the three data sets recorded at different crystal orientations on the brominated DNA gives rise to a different configuration of anomalous scatterers in the (provided that the data are kept unmerged). The situation is analogous to a series of in which the `anomalous occupancies' of each individual heavyatom site would vary from crystal to crystal.
so that any genuine intensity differences between symmetryrelated reflections are averaged out and are instead viewed as contributions to variance estimates. In macromolecular crystallography, data are usually merged in the crystal before starting the phase calculation. For this reason, AAS has not up to now been a major obstacle in SAD or in the vast majority of cases. However, if the data are kept unmerged the intensity differences in symmetryrelated reflections can be exploited to model the AAS of anomalously scattering atoms. Furthermore, as will be demonstrated below, the AASinduced intensity differences can yield phase information, which essentially comes `for free'. This can be of particular interest for breaking the phase ambiguity in SAD experiments. Another possibility would be to record data at several crystal orientations and to extract phase information from the observed intensity variations. As can be seen from Fig. 55. A formal description of AAS and its connection to macromolecular phasing
Formal derivations that describe the effects of AAS on the intensities of diffracted beams have been reported in the literature in terms of optical models, which were first introduced by Templeton & Templeton (1982) and further refined by Dmitrienko (1983), Fanchon & Hendrickson (1990) and Kirfel et al. (1991). The emphasis in these studies was to describe the intensity modulations of reflections as a function of crystal orientation. Indeed, most experiments that exploit AAS effects in inorganic or smallmolecule crystals are carried out by performing azimuthal scans about certain (mainly forbidden) reflections. Thus, the derivations presented, for example, in Kirfel et al. (1991) aim at expressing the diffracted intensity of a reflection as a function of goniometer angles. The experimental measurement of azimuthal scans then allows the extraction of the tensorial properties of factors.
In macromolecular crystallography the situation is very different, as the vast majority of diffraction experiments use the screenless rotation method with an area detector and with a single scan axis. On most synchrotron beamlines, the scan axis is parallel to the direction of linear polarization of the incident beam (i.e. in the plane defined by the orbit of the electron beam and perpendicular to the incident Xray beam direction). Thus, as the crystal is rotated during data collection, the direction of polarization of the incident beam does not change with respect to the crystal. However, the effects of AAS become apparent through the nonequivalence of symmetryrelated reflections. Fanchon & Hendrickson (1990) have presented a general theory of AAS effects in protein crystallography. However, the general expression for the dependence of diffracted intensities on AAS which they derived (equation 9 in their paper) does not clearly unveil how this phenomenon can be exploited to generate phase information in the standard framework of SAD or Thus, in their approach phase determination is essentially carried out along conventional lines in the isotropic approximation and the AAS tensors are only refined at the end against unmerged intensities. Here, we show that a considerable simplification in the formalism of AAS can be achieved by considering that those atoms that exhibit AAS properties form only a small of all the atoms in the of a macromolecular crystal. This allows us to develop a simple model for the AASinduced intensity modulations of nonforbidden reflections in macromolecular crystals which can be neatly integrated into the general framework of experimental phasing methods. Using an extended Harker construction, phase information can then be generated through the intensity differences of symmetryrelated reflections.
5.1. An optical model of AAS in macromolecular crystals
Our model is based on the dipole approximation of et al., 1992; Templeton & Templeton, 1994; Templeton, 1998; Ovchinnikova & Dmitrienko, 2000). The AAS properties of an atom can then be expressed by a secondrank tensor. In macromolecular crystallography, the anomalously scattering atoms are usually in an environment of low symmetry. This is certainly the case for Se in selenomethionine residues and for Br in brominated Thus, the use of the dipole approximation is plainly justified, since higher order effects in AAS are usually substantially weaker and only become visible when the anomalously scattering atoms are in an environment of spherical or cubic symmetry (Finkelstein et al., 1992; Templeton & Templeton, 1994). We further assume that the incident Xray beam is completely linearly polarized. This is a fairly reasonable assumption in the context of macromolecular crystallography since most SAD/MAD experiments are nowadays carried out on undulator beamlines at thirdgeneration synchrotrons, where the degree of linear polarization is usually very high. It is easy to generalize the model presented here to the cases of elliptically and/or partially polarized Xrays, but this will be presented elsewhere.
(FinkelsteinBy convention, all vectors involved in the forthcoming derivations are expressed as column matrices of three components in a crystal Cartesian basis (e_{x}, e_{y}, e_{z}). Thus, our frame of reference is always the crystal, not the laboratory. In the dipole approximation, the Xray scattering factor of an atom which exhibits AAS is given by a secondrank tensor f, which is then represented by a symmetric 3 × 3 matrix with complexvalued entries (Templeton & Templeton, 1982; Dmitrienko, 1983),
For a given reflection h, recorded in a certain geometry, we define a pair of unit vectors, u and v, that are mutually perpendicular and perpendicular to the incidentbeam direction. Similarly, we define a pair of unit vectors, u′ and v′, that are mutually perpendicular and perpendicular to the scatteredbeam direction (Fanchon & Hendrickson, 1990). The scattering of Xrays from an atom which exhibits AAS is described by a matrix of four elements corresponding to polarization transfers from the incidentbeam polarization components u, v to the scatteredbeam polarization components u′, v′ (Templeton & Templeton, 1982; Fanchon & Hendrickson, 1990; Kirfel et al., 1991),^{1}
Here, the left superscript t denotes matrix transposition.
We now assume that the incident beam is completely linearly polarized along a direction given by the unit vector p and we choose the vector u to coincide with this direction. Since there is then no polarization component in the incident beam along the direction v, only the matrix elements Φ_{u′u} and Φ_{v′u} need to be considered. For an atom that displays no AAS, a scattered Xray beam will also be completely linearly polarized along a direction which we denote by the unit vector p′. The direction of p′ is obtained by projecting p onto a plane perpendicular to the scatteredbeam direction. We choose u′ = p′ and denote v′ = p′_{⊥}. Thus, the two matrix elements that need to be considered are
and
However, for atoms that do not display AAS, only the matrix element Φ_{p′p} is relevant, since there is no polarization component in the scattered beam along the direction p′_{⊥}.
The scatteringfactor tensor for a given atom can be expressed as a sum of a purely isotropic scattering factor (including isotropic anomalous scattering) and an AAS tensor (Kirfel et al., 1991),
where I is the 3 × 3 identity matrix and f′ and f′′ are the real and imaginary components, respectively, of the scatteringfactor tensor that describes the AAS behaviour of that atom. The scatteringmatrix elements can then be expressed as
and
where ψ = ^{t}p′p. It should be noted that ψ^{2} is the standard polarization factor of a reflection when the incident Xray beam is completely linearly polarized.
Corresponding structure factors are obtained by summing up the scattering contributions of all N_{uc} atoms in the crystal
and
where O_{j} denotes the occupancy of atom j, T_{j}(h) its Debye–Waller factor (temperature factor) and other symbols have their usual meaning.
Using (6) and (7), these summations can be grouped into contributions that are a consequence of purely isotropic scattering on the one hand and of AAS on the other,
and
where N_{uc}^{aas} indicates the number of atoms in the that exhibit AAS. F(h) corresponds to the normal isotropic including isotropic anomalous scattering.
Since there is no interference between complex contributions with orthogonal polarizations, the intensity of the scattered beam can be expressed as a sum of intensities scattered along the polarization components p′ and p′_{⊥},
These intensities are related to the squared moduli of the corresponding complex structure factors
and
The proportionality factor k includes the usual geometric and experimental quantities which relate the integrated intensity of a reflection h to the squared modulus of its e.g. incidentbeam intensity, irradiated sample volume, Lorentz correction etc., but excluding the polarization correction, which is explicitly given by the factor ψ^{2}.
To summarize, the total intensity can be written as
where
and
and F(h) is the normal isotropic scattering factor. In macromolecular crystallography, the proportion of anomalously scattering atoms in the is usually rather small. The summation in (12) is only over the relatively few atoms which exhibit AAS, whereas the expressed by (10) is obtained by summing the scattering contributions over all atoms in the We can therefore reasonably assume that the second term in (19) is negligible for all but the very weakest reflections and we will use the following approximation for the total intensity of a diffracted beam:
This approximation was also used by Templeton & Templeton (1991) for the special case of diffraction in the vertical plane to determine phases by polarized dispersion in vanadyl sulfate pentahydrate (i.e. using their notation, the sp′ scattering factor is ignored).
The physical interpretation of this result is as follows. When the incident Xray beam is linearly polarized along a direction p, the scattering from all the atoms which do not exhibit AAS will give rise to a diffracted beam which is linearly polarized along a direction p′. This scattering is expressed by the normal F(h). Those atoms which do exhibit AAS will scatter the Xrays in a more complicated way, with resulting polarization components along both the directions p′ and p′_{⊥}. The scattering in each of these polarization components is expressed by the structure factors G_{p′p}(h) and , respectively. However, since the scattering in the polarization component p′_{⊥} only arises from the relatively few atoms that exhibit AAS, the diffracted intensity in that component, , will be a great deal weaker than I_{p′}(h). Furthermore, since p′ and p′_{⊥} are mutually orthogonal, there can be no interference effects between and F(h). In other words, G_{p′p}(h) represents that part of AAS which can actually interfere with the scattering from all the other atoms in the Experimental phasing methods in macromolecular crystallography are based precisely on interference effects of this kind between the scattering from a socalled consisting of a relatively small number of atoms (heavy atoms and/or anomalously scattering atoms) that can be modelled and refined, and the scattering from the unknown part of the structure, which consists of the vast majority of atoms. Thus, if the effects of AAS are to be exploited for phase determination, only the contribution G_{p′p}(h) is of relevance.^{2}
5.2. Extracting phase information from AASinduced symmetrybreaking effects
(22) reveals how structurefactor phases for F(h) can be derived if G_{p′p}(h) is varied. This equation is formally identical to the equation that defines a circle on the socalled Harker construction (Harker, 1956),
where the total complex F^{Tot}(h) is split up into a part F^{P}(h) that is constant and which arises from a common structure and a variable part F^{H}(h) which arises from a subset of atoms, often called the In order to generate phase information, intensity modulations are induced by some physical or chemical changes in the only [e.g. a modulation of F^{H}(h) by wavelength changes in MAD experiments]. It is further required that F^{H}(h) can be modelled in real space in terms of refineable atomic parameters.
In the presence of AAS, the variable part F^{H}(h) is given by G_{p′p}(h). The modulations in G_{p′p}(h) can be induced by changes in the crystal orientation, i.e. by changing p. However, even if p is fixed, there can be differences in G_{p′p}(h) and G_{p′p}(h′), where h and h′ are symmetryequivalent reflections.
Even in the absence of AAS, the complex structure factors of symmetryequivalent reflections are not identical: only their moduli are. Let denote the g of will be written as
of the crystal. The operation of an elementReflections h and h_{g} are symmetryrelated^{3} if
The normal structure factors (i.e. the parts which are not affected by AAS) of symmetryrelated reflections are related by (Waser, 1955)
If the symmetryrelated reflections F(h_{g}) and F(h) are to be used on the same Harker construction, it is necessary to rotate F(h_{g}) back to the phase angle of F(h), i.e. undo the phase shift^{4} exp(−2π^{t}ht_{g}).
We therefore define
and
Then, for all ,
but
as will be demonstrated in §5.4.
It can now be seen how AASinduced phasing can be naturally incorporated into the general framework of de novo phasing methods. The measured data are kept unmerged and used as individual contributions (circles) on the Harker construction. Phasing power is generated through the intensity differences between symmetryrelated reflections and the associated complex offsets which, according to (30), differ between unmerged observations. In this procedure, the operation that is equivalent to data merging is deferred to the phasing stage. Data merging is effectively replaced by data comparison carried out in the complex plane, i.e. through the Harker construction: from all the symmetryrelated reflections, a single quantity F^{P}(h) is estimated, but as a complex value! Evidently, if measured data are available from different crystal orientations, they can also be very naturally incorporated into this scheme.
5.3. AASinduced symmetry breaking in direct space
Using (20), we can write the AAS part of a as
where the AAS contribution of a given atom j is noted as
Here, we have made the dependence of f_{j} on p′ and p explicit. Using the spacegroup symmetry, (31) can be rewritten as a summation over N_{au}^{aas} atoms in an (assuming that the atoms are not on special positions),
where
Under the action of the spacegroup symmetry, the AAS tensors are transformed by a similarity transformation involving the point operators R_{s} (Dmitrienko, 1983). Thus,
where
and
We can now see that the AAS contributions from two atoms that are related by the s are identical only if p_{s} = ±p and p_{s}′ = ±p′, i.e. if both p and p′ are invariant, up to sign changes, under the operation ^{t}R_{s}.
As was mentioned above, the direction of p′ is obtained by projecting p onto the plane that is perpendicular to the scatteredbeam direction. At zero scattering angle, p′ = p and this also holds approximately for small scattering angles.^{5} In macromolecular crystallography, the resolution of the data is usually rather limited and the measurable diffraction is confined to relatively small scattering angles. This is even more true in SAD and MAD experiments that are performed at the K edges of Se or Br, which are both located at relatively short wavelengths. Therefore, in many cases, a valid firstorder approximation is given by
The AAS contribution to a
may then be written asThis shows that the dominating effects in AASinduced intensity modulations in macromolecular crystals arise from the orientation of the incidentbeam polarization direction p with respect to the arrangements of bonds around anomalously scattering atoms. Thus, in a firstorder approximation, the AAS contributions from two atoms that are related by the s are identical if p is invariant, up to a sign change, under the operation ^{t}R_{s}. In brominated the AAS tensors are almost uniaxial at the whiteline energy (see Fig. 2). Therefore, the AAS of a Br atom is essentially governed by the orientation of p with respect to the principal direction of the tensor, i.e. the direction of the C—Br bond, as is apparent from the experimental data on brominated DNA presented above.
On synchrotron beamlines, diffraction data for macromolecular crystals are usually recorded with a single scan axis that is oriented parallel to the direction of linear polarization of the incident beam. Thus, as the crystal is rotated during a data collection, the direction of polarization of the incident beam does not change with respect to the crystal so that p is the same for all reflections. The factors f_{j}(p_{s}) are then identical for all reflections and we can drop the dependence on p and write
(40) is very similar to the standard expression of the as a summation over all atoms in the The only difference is that for each symmetryrelated atom an individual factor f_{j,s} is defined which depends on s ∈ and not only on the label j of the symmetryunique atom. This is the simplest way to model the effects of AAS in macromolecular crystallography. Although this model is strictly speaking only valid at zero scattering angle, we have found it to be a sufficiently good approximation for modelling the major features of AAS in many SAD/MAD data sets from macromolecules. Some examples will be given below (in §§6.2 and 7).
5.4. AASinduced symmetry breaking in reciprocal space
Using (35), we can rewrite (33) as
and for a symmetryrelated reflection h_{g}
where
and
Because of the group properties of , the product gs is just another element of , which we denote by k. Then, s = , where is the inverse of g. Thus,
where
By comparing (41) and (45), it can now be seen that G_{p′p}(h) and are identical only if = ±p_{k} and = ±p′_{k}, i.e. if both p and p′ remain invariant, up to sign changes, under the action of R_{g}.
In the forwardscattering approximation (38), it is enough to state that p must be invariant under the action of R_{g} for two reflections h and h_{g} to remain equivalent.
6. Modelling and parameter for AAS
The theory presented in the previous section has been implemented as new code in the heavyatom SHARP (de La Fortelle & Bricogne, 1997; Bricogne et al., 2003). The program had already been extended for the use of unmerged data in the context of radiationinduced phasing (Schiltz et al., 2004; Schiltz & Bricogne, 2007). In this new enhanced version, SHARP now reads and processes goniometric information in various forms and computes the vectors p and p′ for each reflection record. By applying the transformations (27) and (28), symmetryrelated and/or identical reflections measured at different crystal orientations can be used in the Harker construction, together with data recorded at any other wavelength and/or from any other heavyatom derivative or native.
and phasing programThe parameters of the heavy atoms are usually first refined in the approximation of isotropic i.e. with one set of (possibly refineable) f′ and f′′ parameters per atom or per chemically distinct type of atoms. Once the of the positional and thermal parameters of the anomalously scattering atoms has converged in the isotropic approximation, it is possible to switch on the of AAS parameters. We have implemented two different parametrizations for AAS.
6.1. of AAS tensors
This parametrization implements (33) and (35). For each anomalously scattering atom j in the the AAS is described by a tensor , represented by a symmetric 3 × 3 matrix with refineable elements. The contributions of symmetryrelated sites are obtained from the transformations (35). The elements of either f_{j}′ or f_{j}′′ or both can be refined. Thus, for each atom in the either six or 12 tensorial elements are refined.
This parametrization was tested on the data recorded on the brominated ZDNA duplex d(CGCG[BrU]G) presented in §4. The three data sets corresponding to different crystal orientations were used. For each of the two Br atoms, six individual tensor elements of f_{j}′′ were refined. Occupancies were fixed to 1 and f′ factors were held fixed. No AAS tensors were refined for f′, since at the peak wavelength (corresponding to the white line) there is very little anisotropy in f′ of Br (see Fig. 2). The results of the are presented in Table 2. An eigenvalue/eigenvector analysis yielded the principal directions of the refined tensors. For each Br atom, the principal direction that corresponds to the largest eigenvalue of f_{j}′′ is in very good agreement with the direction of the C—Br bond (agreement is better than 5.5°), thus validating the In principle (though probably not in practice) the determination of the tensors could thus be helpful for map interpretation as they give information about the direction of the Se or Br bonding.
The p (i.e. if the crystal is rotated around p during data collection, as is almost universally the case in macromolecular synchrotron crystallography) and if the crystal is of low symmetry or if p is aligned along a symmetry axis, the of some combinations of tensorial elements may be ill conditioned. This can be seen from (35): if there are only a small number of different p_{s}, only some linear combinations of tensor elements are well defined. In such cases, certain tensor elements or linear combinations of tensor elements can be kept fixed during the refinement.
of AAS tensors is very general since it can accommodate data from different crystal orientations. However, if all reflections have been recorded with the same polarization directionWe have also implemented an alternative description of f_{j}′ and f_{j}′′ tensors in terms of principal values and orientational parameters (Euler angles), as was previously suggested by Fanchon & Hendrickson (1990). This allows a reduction of the number of parameters since the principal values are identical for all atoms of the same chemical type (at a given wavelength), whereas for a given atom the orientational parameters are identical for different data sets (e.g. recorded at different wavelengths).
6.2. of `symmetryunrolled' factors
This parametrization implements the forwardscattering approximation (40), where for each anomalously scattering atom in the (labelled by j and s) individual factors f_{j,s}′ and f_{j,s}′′ are refined. This is only valid if all reflections have been recorded with the same polarization direction p. It should be noted that this is not the same as refining all the heavy atoms in P1. Only the factors of symmetryrelated atoms are refined individually; the positional and thermal parameters are constrained to obey the spacegroup symmetry, i.e. the symmetrybreaking effects are assumed to only originate from the properties. Also, atoms which are related by latticecentring translations are constrained to have identical factors, since the latticecentring symmetries are not broken by AAS. Either the factors f_{j,s}′ or f_{j,s}′′ or both can be refined. Thus, for each anomalously scattering atom in the either N_{PG} or 2 × N_{PG} factors are refined, where N_{PG} is the order of the crystal For lowsymmetry space groups, this parametrization can therefore be more economical, as far as the number of refined parameters is concerned, than the of AAS tensors. Another advantage of this parametrization is that no goniometric or other information about the crystal orientation is needed. It is therefore the simplest way to implement AAS capabilities in an existing heavyatom and phasing program. On the other hand, if several data sets have been recorded at different crystal orientations, an individual set of f_{j,s}′ and f_{j,s}′′ factors must be refined for each data set.
This parametrization was also tested on the data from the brominated DNA. The three data sets were declared as individual batches within the hierarchical organization of data implemented in SHARP (de La Fortelle & Bricogne, 1997). Thus, for each of the two Br atoms, four individual f_{j,s}′′ factors (N_{PG} = 4 in P2_{1}2_{1}2_{1}) were refined in each data set. The positional parameters and atomic displacement parameters were constrained to remain identical across all data sets. Occupancies were fixed to 1 and all f_{j,s}′ factors were held fixed. The results of the are presented in Table 3. The refined f_{j,s}′′ values correlate very well with the the relative orientation of the C—Br bond with respect to p (see Fig. 6), thus validating the refinement.

6.3. Phasing with AAS parameters and unmerged data
For both of the refinements described above (using AAS parameters and unmerged data), phases were computed and compared with the phases obtained from a standard SAD f′′ factors and merged data. The quality of the phases was assessed through map correlation coefficients (before density modification; presented in Fig. 7). As can be seen, a substantial improvement of the quality of the phases can be achieved with either of the two alternative AAS parametrizations.
using isotropicIn practical applications, a decision will have to be made at some stage of the
on whether the signal is strong enough to support the extra AAS parameters. We are developing a method to compute directional residual maps which can be computed at the end of a standard with isotropic parameters (Schiltz & Bricogne, unpublished work). These maps will show residual features whenever the data are clearly affected by AAS and will allow an initial estimation of the AAS tensors prior to their refinement.The modelling and parametrization of nonisomorphism in the case of data affected by AAS is significantly more complex than for standard cases. The error model that is currently implemented in SHARP assumes that the effects of all sources of nonisomorphism are uncorrelated between different observations of a given reflection (de La Fortelle & Bricogne, 1997; Bricogne et al., 2003). In essence, a diagonal approximation is used for the nonisomorphism covariance matrix. Such an approximation may not always be entirely justified since nonisomorphism can be correlated across observations that have been recorded under similar geometric conditions (Bricogne et al., 2003). For a more general treatment it will be necessary to resort to multivariate likelihood functions which are capable of accommodating adequate patterns of covariances between the various observations (Bricogne, 2000; Pannu et al., 2003). The implementation of these functions in SHARP is currently under way.
7. AAS and symmetrybreaking effects in crystals of the selenated protein PPAT
The symmetrybreaking effects of AAS were further investigated on crystals of selenomethionine phosphopantetheine adenylyltransferase (PPAT; Izard & Geerlof, 1999). The motivation behind this study was severalfold. First of all, we aimed at studying the AAS in representative proteins, which are typically much larger molecules than the brominated DNA molecule on which we previously explored AAS. Secondly, the anomalous scatterer investigated here is Se in selenomethionine, which plays a major role in the application of the SAD/MAD methods in protein crystallography as it can be used to replace methionine residues by (Hendrickson et al., 1990). Polarized dispersion at the Se K edge has been observed in crystals of selenomethioninecontaining proteins (Hendrickson et al., 1990; Bricogne et al., 2005) and related compounds (Templeton & Templeton, 1988; Fanchon & Hendrickson, 1990) and revealed that the AAS properties are represented by biaxial tensors (see Fig. 1). The geometric interpretation of AAS in selenated protein crystals is therefore somewhat more complicated than in brominated DNA or RNA molecules, where the AAS of Br is described to a good approximation by a uniaxial tensor. As a final incentive, the particular protein that was chosen here crystallizes in a cubic It is sometimes thought that the effects of AAS are less pronounced in protein crystals of high symmetry and/or that contain a large number of anomalously scattering atoms in the because the polarized from the various atoms would `average out' to isotropy. This is indeed the case for linear dichroism and birefringence, which are global (macroscopic) consequences of AAS and which follow the pointgroup symmetry of the crystal (Bricogne et al., 2005; Sanishvili et al., 2007). Thus, within the validity of the dipole approximation, there can be no dichroism in cubic crystals. However, this is not the case for AAS effects in diffraction, which are microscopic (local) effects to which each individual atom contributes with its own phase shift.^{6}
Crystals of PPAT belong to I23, with unitcell parameter a = 136.23 Å (Izard et al., 1999). In solution and in the the enzyme forms a homohexamer with 32. The threefold axis of the hexamer coincides with the crystallographic ternary axis and its twofold axis gives rise to (Izard & Geerlof, 1999). There are thus two polypeptide chains in the crystal each consisting of 159 amino acids and having a molecular weight of 17.8 kg mol^{−1} (kDa). In the selenated form, there are 2 × 9 selenomethionines in the but two of them, located in the Nterminal residues, are disordered. The structure of PPAT was solved by threewavelength at the Se K edge (Izard & Geerlof, 1999).
For the present study, a single crystal of selenomethionine PPAT of approximate dimensions 200 × 200 × 200 µm was mounted on a nylonfibre loop and flashfrozen in liquid nitrogen. All measurements were carried out on station X25 of the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory, Upton, New York, USA. This beamline features a threecircle kappa goniometer and is equipped with an ADSC Q315 CCD detector. An K edge was recorded in mode in order to select the wavelength corresponding to the white line. A series of 20 diffraction images were collected in order to compute a crystal orientation matrix. The crystal was then oriented with the aid of the motorized and computercontrolled kappa goniostat so that the [010] direction was aligned with the direction of linear polarization of the incident Xray beam (p). A 240° data set was collected by rotating the crystal about [010]. Thus, the vector p remained constantly aligned with the crystal b axis. The diffraction images were integrated with MOSFLM (Leslie, 1993). Further data processing was carried out with programs from the CCP4 software suite (Collaborative Computational Project, Number 4, 1994). Details of the datacollection and reduction parameters are given in Table 4. The data set was internally scaled by minimizing the disagreement between symmetryrelated reflections in Laue group m3 and two final reflection files were computed, one with the data merged in crystal 23 and a second with the data merged in crystal 1.
around the Se
‡Redundancyindependent (multiplicityweighted) merging R factor, not keeping Bijvoet pairs separate (i.e. computed in the crystal Laue group). 
The Se sites feature prominently in an anomalous difference Fourier map computed with the data merged in the true crystal ). On the other hand, Se sites which are related by a binary axis show similar peak heights (see Fig. 9).
23. As expected, symmetryrelated sites display identical peak heights. However, in an anomalous difference Fourier map computed with the data merged in 1, Se sites which are related by a ternary axis display widely differing peak heights (see Fig. 8The parameters of the Se atoms were refined in SHARP. Firstly, a standard with isotropic factors was carried out on the merged data. Coordinates and occupancy parameters were refined for all sites. Based on careful inspection of residual maps, anisotropic displacement parameters were refined for eight Se sites; five additional minor sites, corresponding to alternate positions of main sites, were included in the Starting from these refined parameters, the unmerged data were used to refine AAS parameters in the form of individual factors f_{j,s}′′ for each Se atom in the (i.e. the parametrization described in §6.2). No AAS parameters were refined for the minor sites. The final values of the refined AAS parameters are presented in Table 5. It can be seen that for most symmetryunique sites, the refined values for the f_{j,s}′′ factors fall into three groups. Sites which are related by twofold symmetry operations refine to similar f_{j,s}′′ values, whereas this is not the case for sites related by a threefold symmetry operation.

This example nicely illustrates the discussion given in §5.3. Here, the incidentbeam polarization vector p was aligned with the [010] direction and thus remains invariant (up to sign changes) under the action of any of the twofold symmetry operations, but not under the action of the threefold symmetry operations along 〈111〉. Thus, to a firstorder approximation, the I23 symmetry is reduced to I222, as is confirmed by the anomalous difference Fourier maps and by the refined values of the AAS parameters.
Phases were computed after the f′′ factors and merged data (see Fig. 10). A very significant improvement of the quality of the phases, typically corresponding to a 10–15% increase in map correlation coefficients (before density modification), is achieved by using unmerged data and AAS parameters.
of AAS parameters against unmerged data and were compared with the phases obtained from a standard SAD using isotropic8. AASinduced phasing on `standard' data: improved SAD phasing of the selenated protein IMPDH
As a final example, we present the K edge on crystals of selenated inosine5′monophosphate dehydrogenase (IMPDH; Zhang et al., 1999). No new data were recorded for this study. Instead, we reused the data from which the structure was initially solved. Our aim was to assess the improvement in phases that can be achieved on `standard' data (i.e. data collected according to conventional strategies), by keeping the data unmerged and by exploiting the AAS effects.
of AAS parameters and phasing from unmerged data collected at the SeThe structure of IMPDH had been solved by threewavelength et al., 1999). IMPDH crystals belong to I422, with unitcell parameters a = b = 151.49, c = 101.67 Å. IMPDH is a homotetramer, with its four subunits related by the crystallographic fourfold axis. The molecular weight of the IMPDH monomer is 52.7 kg mol^{−1} (kDa). It comprises 491 residues, of which 13 are selenomethionines in the selenated form. The MAD data were collected at the SBCCAT beamline 19ID at the Advanced Photon Source (APS), Argonne, Illinois, USA (Rosenbaum et al., 2006). At each wavelength, a 90° data set had been recorded to 2.5 Å resolution, using a single scan axis oriented parallel to the direction of polarization of the Xray beam. The crystal had not been oriented in any special way.
from a single crystal of selenated IMPDH (ZhangIn the following, we only used the data set that was recorded at the peak wavelength (λ = 0.9791 Å). Firstly, a standard with isotropic factors was carried out in SHARP on the merged data. Coordinates and occupancy parameters were refined for all sites. Based on the inspection of residual maps, two Se sites were found to have alternate positions and were modelled accordingly. Another Se site showed signs of disorder and was refined with anisotropic displacement parameters. A single f′′ parameter was refined and converged to a value of 6.2. The quality of the resulting SAD phases, gauged through map correlation coefficients, is indicated in Fig. 11.
The raw data were then reprocessed in order to keep the reflections unmerged and goniometric information was extracted for each reflection record. This data set consists of a total of 272 576 observations (for 20 627 unique reflections). Starting from the refined parameters of the Se atoms, AAS tensors ( + ) were refined for all sites, along with positional and thermal parameters. The occupancies of all Se atoms were held fixed at their previously refined values. This was necessary in order to avoid instabilities in the f′′ proceeded smoothly and the final refined tensors all had eigenvalues of the order of 3, 6, 8.5 (±1.5). The corresponding eigenvectors concur with the principal molecular directions in the various C—Se—C moieties. The refined f′ tensors all have eigenvalues which are not widely spread, in the range of −5, −6, −7 (±1.5), thus reflecting the fact that there is relatively little anisotropy in the f′ factors at the peak energy of the Se K edge (se Fig. 1).
which arise from the fact that the occupancy parameters strongly correlate with the diagonal elements of the AAS tensors. The of the elements ofPhases were computed and their quality in terms of map correlation coefficients is indicated in Fig. 11. For reference, the quality of the phases obtained by conventional SAD, as well as two and threewavelength MAD (on merged data, with isotropic factors), is shown in Fig. 11. By using unmerged data and a parametrization for AAS, a very substantial improvement in the quality of the SAD phases is achieved over the standard procedure. This is quite remarkable since no new data were used to obtain this improvement: only the processing, heavyatom and phasing differed. In fact, the quality of the AASimproved SAD phases is comparable with that of conventional twowavelength MAD phases. Thus, the gains in terms of phasing power that can be achieved by fully exploiting the AAS effects present in the data (i.e. without collecting new data) are comparable to those which in a conventional approach would require the collection of a complete second data set at a different wavelength. Although the conventional threewavelength MAD phases are still superior to the AASimproved SAD phases, it must be stressed that three times more data needed to be collected for the former. Also, it can be seen that even though very good data were obtained (three wavelengths, very high redundancy, no radiation damage), there is still a small improvement in the quality of phases that can be achieved by exploiting AAS (compare the phasing statistics for the threewavelength with and without AAS).
9. Discussion
The previous examples show that AAS is likely to have an impact on most standard SAD or MAD experiments performed at an e.g. those involving azimuthal scans). The question then arises as to why the effects of AAS have until now largely gone unnoticed in protein crystallography. This state of affairs is the consequence of a combination of several circumstances.
Indeed, AAS is intrinsic to the phenomenon of Xray scattering and its occurrence is therefore not restricted to special types of experiments (

The reasons listed above explain why AAS has so far remained a relatively invisible phenomenon in protein crystallography and also why it does not generally have a deleterious effect on the success of the SAD and MAD methods. On the other hand, we have shown here that if AAS is fully exploited, substantial gains in phase information can be obtained.
The prospect of fully exploiting the phase information generated through AAS brings into consideration entirely new strategies of data collection. Crystals could be intentionally misaligned in order to maximize the AASinduced inequivalence between symmetryrelated reflections. Furthermore, if the rotation axis is chosen along a direction that does not coincide with the direction of Xray polarization (e.g. using a vertical scan axis) the effects of AAS in the data set will be significantly boosted since the f′ and f′′ tensors will then be sampled over a much wider range of polarization directions than is currently the case (Schiltz & Bricogne, work to be published). Indeed, on undulatorbased beamlines at thirdgeneration synchrotrons there are no compelling reasons to limit oneself to a horizontal rotation axis.
These new approaches will be greatly facilitated by the systematic deployment of standard or minikappa goniometers on synchrotron beamlines. Moreover, the future use of goniometers with a vertical spin axis, designed for the purposes of gaining mechanical stability in the handling of microcrystals, will result in the most general AAS effects being ubiquitous in data sets recorded on such instruments. The magnitude of the AAS effects themselves could be enhanced by reducing the bandwidth of the Xray beam through the use of higher order reflections (e.g. 311) from Si crystals in monochromators. This would come at the expense of a reduced Xray beam but it would provide a more productive alternative for attenuating the incident beam at thirdgeneration synchrotron beamlines than the widely used practice of using absorbers.
10. Conclusion
In retrospect, it appears that AAS, which is a well known physical complication of
at an was considered at some stage as a potential threat to the simplicity of the MAD method but was then ignored to all intents and purposes. We have shown in this work that if appropriate steps are taken to preserve the original measurements in which these effects appear, a suitable generalization of current phasing methods can accommodate them and deliver extra phasing power compared with conventional approaches using the same data. Examples were given that show improvements in the phases which are typically of the same order of magnitude as those obtained in a conventional approach by adding a secondwavelength data set to a SAD experiment. Thus, the exploitation of AAS can give access to a twowavelength map quality with singlewavelength measurements. Such a gain is particularly significant, since radiation damage can frequently preclude the collection of a secondwavelength data set. It may thus also be worthwhile revisiting SAD or MAD data sets where the quality of the phases was too marginal to provide an interpretable map.Now that we have established how to handle and exploit AAS effects, new possibilities are opened to use them deliberately by incorporating them in the design of experiments. We conclude by citing Templeton & Templeton (1982) who wrote that AAS
adds a new dimension of complexity to the theory of Xray scattering. By introducing an error into the conventional methods of computation, it offers a handicap […] to exploit the maximum effects at the absorption edges for solving the
Thus from a pessimistic point of view it is a setback. We adopt the opposite view: where there is a complication there is the opportunity of sharper, more penetrating methods for extracting information from diffraction experiments.
Footnotes
^{1}These relations are equivalent to equation (3) in Templeton & Templeton (1982). In Fanchon & Hendrickson (1990) they also appear in equation (3) and in Kirfel et al. (1991) they appear in equation (8). Both Templeton & Templeton (1982) and Kirfel et al. (1991) explicitly chose for the vectors u, v, u′ and v′ directions that are perpendicular and parallel, respectively, to the plane of diffraction. These directions are denoted s, s′, p and p′ by Templeton & Templeton (1982) or σ, σ′, π and π′ in Kirfel et al. (1991). However, the choice of these directions is completely free, as was noticed by Fanchon & Hendrickson (1990). Below, we show that a particular choice of vectors u, v, u′ and v′ can considerably simplify the expressions in the case of linearly polarized Xrays.
^{2}This also explains why in macromolecular crystallography the effects of AAS are much more pronounced in nonforbidden reflections than in forbidden ones. In forbidden reflections, the normal part of the F(h), is always zero, so that the AASinduced intensity modulations only arise from the terms G_{p′p}(h) and , which are always small. However, for nonforbidden reflections (22) can be explicitly written as I(h) ≅ . As a consequence of the quadrature relationship between intensities and structure factors, the modulations arising from G_{p′p}(h) in the third term of the righthand side of this equation are `boosted' by the multiplication by the usually much larger F(h).
^{3}We will henceforth use the term symmetryrelated reflections rather than symmetryequivalent reflections, since in the presence of AAS these reflections are no longer equivalent.
^{4}This operation is similar to the reflection of F(−h) through the real axis on the complex plane when building the Harker construction with Friedel pairs in the case of (North, 1965). In that case, the Harker construction is actually set up with F(h) and , where is the complex conjugate of F(−h).
^{5}It also holds exactly, at any scattering angle, for reflections h ⊥ p, i.e. for diffraction in the vertical plane. This was, for example, the case for all measurements that were carried out on fourcircle diffractometers with a point detector by Templeton and Templeton.
^{6}To put it more succinctly: is similar to Xray absorption, but with phase shifts (Bricogne & Schiltz, 2000).
Acknowledgements
We thank P. Pattison, S. Capelli, P. Roversi, G. Evans and G. Chapuis for their many contributions at various stages of this project. We are grateful to R. Sanishvili for collecting the Xray data on the brominated DNA molecule, to T. Izard for providing selenated crystals of PPAT and to R. M. Sweet for providing inhouse beamtime on NSLSX25 and for his support and advice during the measurements. We acknowledge financial support from the European Commission Grant No. LSHGCT2003503420 within the BIOXHIT project and from the Swiss National Science Foundation Grant No. 200021101519/1. The provision of PPAT crystals was supported by NIH grant AI55894 to T. Izard. We acknowledge the Advanced Photon Source (APS), the Structural Biology Center CAT, the European Synchrotron Radiation Facility (ESRF), the Swiss–Norwegian Beamlines (SNBL) and the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory for provision of synchrotronradiation facilities and user beamtime. The APS and NSLS are supported by the US Department of Energy, Basic Energy Sciences, Office of Science.
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