2.2.1 Crystal lattices – unit cells – symmetry

X-ray crystallography requires the generation of crystals, and macromolecular crystals are typically grown under conditions where molecules are reversibly driven out of solution (Weber, Reference Weber1997). Macromolecular samples require that these conditions are gentle and do not cause unfolding or disassociation of complexes. Typically, crystal lattice forces are much weaker than macromolecular folding energies.

Crystals are ordered arrays of atoms related by pure translation (a transformation with only a change in position but not orientation or rotation) in one dimension (fibers), two dimensions (sheets), and three dimensions (lattices). Although fiber diffraction of samples such as DNA can be studied with X-rays, the most common crystals studied by macromolecular crystallography are three-dimensional. The smallest repeating unit of the crystal that is related only by translations is called the unit cell. For three-dimensional crystals, the shape and size of the unit cell is defined by the length of three axes (a, b, and c) and three angles between these axes (α, β, and γ).

Frequently, internal symmetry exists within the unit cell when the unit cell contains multiple molecules. If these symmetries apply to the entire lattice, they are crystallographic symmetries and constrain the parameters of the unit cell. The smallest portion of structural information required to reconstruct the entire lattice through crystallographic symmetries and lattice translations is termed the asymmetric unit. In contrast, symmetries that do not apply to the lattice are non-crystallographic.

In many cases the biologically relevant complex possesses symmetry. These biological symmetries may be observed by some combination of both the crystallographic and non-crystallographic symmetry operators, if the appropriate assembly is present in the crystal structure. For crystallographic analysis, non-crystallographic symmetries can be tremendously useful for map improvement (Section 2.2.6), as well as generation of constraints for the refinement of atomic models, such as have been implemented in the program CNS (Brunger et al. Reference Brunger, Adams, Clore, DeLano, Gros, Grosse-Kuntsleve, Jiang, Kuszewski, Nilges, Pannu, Read, Rice, Simonson and Warren1998). Analogously, particle symmetry provides important constraints on model-based and ab-initio reconstruction of three-dimensional shapes from one-dimensional SAXS data (Sections 4.2 and 4.3). Application of these symmetry constraints, like use of the non-crystallographic symmetries in crystallography, substantially increases the accuracy of the final models.

2.2.2 Diffraction from crystals and Laue conditions

X-rays diffracted from crystals can be mathematically treated as if they were being reflected from a plane of angle θ to the incident X-rays, and hence the diffraction maxima measured during the crystallographic experiment are frequently termed ‘reflections’. By this definition and due to the geometry of the diffraction event, the diffracted X-rays make an angle of 2θ with incident beam, and thus 2θ is the experimentally measured angle between the direct beam position and the diffraction maximum on the X-ray detector (Fig. 1a).

As electromagnetic waves, X-rays possess both a wavelength and phase. In the crystal, X-rays are diffracted from multiple parallel planes simultaneously, and this leads to a path difference through which these X-rays travel. When the path difference corresponds to an integral number of wavelengths of the incident X-ray, then the diffracted X-rays undergo constructive (in-phase) interference and can be detected experimentally as diffraction maxima. If not, then the X-rays interfere destructively (out-of-phase) and are not observed. This requirement can be expressed mathematically using the Laue conditions:

where a, b, and c are vectors corresponding to the orientation of the unit cell edges, S is the vector corresponding to the path difference between incident and scattered X-rays, and h, k, and l must be integers to ensure constructive interference. The h, k, and l values are the Miller indices and are used to identify each reflection. Thus, the planes that scatter X-rays are determined by the wavelength of the incident X-rays (or wavelengths in the case of Laue multi-wavelength diffraction experiments), the unit cell parameters, and the orientation of the crystal. The Laue conditions give rise to a regularized lattice of diffraction spots (Fig. 1c). Importantly, the unit cell size, shape, and orientation, but not the positions of atoms in the unit cell, control which reflections are in the diffraction condition and where these diffracted reflections occur on the detector. This allows crystallographers to collect, index, and process diffraction data prior to knowing the atomic structure.

Bragg's law provides a measure of the distance between the theoretical planes giving rise to X-ray scattering:

By analogy to light scattering through slits, in the case of a theoretical perfect crystal of infinite size the first-order spectrum will occur when n=1, a second-order spectrum will occur at n=2, and so forth. The maximum resolution (smallest spacing between planes) measured from the crystal can provide insights into its suitability for data collection and structure determination. In contrast, evaluating samples in SAXS for suitability for structural reconstruction is more difficult because every macromolecular solution will scatter X-rays. In this sense, determining if a SAXS curve is suitable for further analysis (Section 5) is more reminiscent of determining if a crystal is merohedrally or pseudo-merohedrally twined (Yeats, Reference Yeats1997), rather than whether or not it can diffract X-rays.

2.2.3 Intensities and atomic arrangements

In contrast to the positions of the reflections, the intensities of the diffracted X-rays are dictated by the atomic arrangements in the unit cell. The positions and types of atoms within the unit cell control both the amplitude and phase. Mathematically,

where F(h, k, l) is the structure factor, h, k, l are the Miller indices of the structure factor, f _j is the resolution-dependent atomic scattering factor, and x _j, y _j, and z _j are the fractional positions of the jth atom in the unit cell. Unfortunately, data collection only allows measurement of the intensities, I(h, k, l), which are the square of the amplitude of F(h, k, l), but not the relative phase information necessary to calculate the electronic distribution in the unit cell. Measurement of the relative phase of X-rays striking the detector at any two diffraction spots has not been possible. This problem is the ‘phase problem’ of protein crystallography that must be solved in order for structures to be determined.

In general, measured intensities are on a relative scale, not an absolute one. From a theoretical standpoint, I(0, 0, 0) is the square of the sum of the number of electrons in the unit cell and is directly comparable to the important SAXS result where I(0) is proportional to the square of the number of electrons in the scattering particle (Section 2.3.2). Data in crystallography are generally put on only a quasi-absolute scale using the Wilson plot (Wilson, Reference Wilson1942) using data between 3·0 Å and 1·5 Å resolution. Placing the data on a true absolute scale is, however, quite difficult. In addition to the ordered atoms, the non-ordered bulk solvent must be included in the calculation, which can be demonstrated by the importance of modeling bulk solvent in refining X-ray structures against low-resolution reflections (Urzhumtsev & Podjarny, Reference Urzhumtsev and Podjarny1995). In contrast, the contribution of bulk solvent is explicitly subtracted out during SAXS data processing and typically is involved in the modeling process as a scale factor between calculated and observed intensities. This subtraction is the basis of contrast variation techniques where matching the average electron density of bulk solvent to specific components of a scattering complex causes their contributions to be eliminated from the processed SAXS data (Section 2.3.6).

The crystal lattice has two effects. First, the orientation of the molecules allows the diffraction data to retain information about atomic positions in three-dimensional space, which is lost in SAXS data collected on molecules in solution that are orientationally averaged. Second, the scattering from the atoms in the unit cell is convoluted with the scattering from the lattice so that the crystal diffraction is sampled only at discrete positions defined by unit cell, which also increases signal-to-noise. The measured X-ray intensities in crystallography are the square of the summed amplitudes from the atoms in the unit cell. In contrast, SAXS intensities are the sum of the squared amplitudes from each scattering event. SAXS data are also continuous and vastly over-sampled in comparison to the independent data content as derived from Shannon's theorem. Even with higher signal-to-noise, crystallographic data are substantially under-sampled and cannot take advantage of ‘super-resolution’ techniques that rely upon over-sampling to determine the phases of measured reflections (Koch et al. Reference Koch, Vachette and Svergun2003). Hence, techniques to solve the phase problem use additional information (Section 2.2.5).

2.2.4 The Patterson function

The autocorrelation function for the electron density, which is essentially a three-dimensional map of all of the atom–atom vectors in the crystal, can be written as:

where u, v, w correspond to some difference vector between the position x, y, and z and x+u, y+v, and z+w. Thus N peaks in the electron density map (atoms) will give rise to N ² peaks in the Patterson function. It has been shown that the above formulation is equivalent to the expression:

where F(h, k, l) is the observed amplitude and V is the volume of the unit cell. The importance of the second expression is that this function can be calculated in the absence of any phasing information, and this function is important for several macromolecular phasing techniques (Section 2.2.5).

When the Patterson function is calculated using the measured amplitudes F(h, k, l), the largest peaks in the autocorrelation are the positions of direct translations between molecules in the crystal lattice; this Patterson function is typically called the ‘self-Patterson’ (Fig. 2). The self-Patterson function in crystallography is an autocorrelation function and is related to the autocorrelation function, P(r), calculated from the SAXS intensities (Section 2.3.3). Unlike the SAXS P(r) function, the crystallographic Patterson function is calculated from molecules that are restricted in their rotations and thus retains three-dimensional information about the inter-atomic vectors (Fig. 2). Additionally, the Patterson function includes all vectors between atoms for all molecules within the crystal. The P(r) function, on the other hand, is a histogram of distances that are orientationally averaged and correspond only to the scattering particle.

Fig. 2. Comparison of the Patterson autocorrelation function in X-ray crystallography and the pair-distribution autocorrelation function in SAXS. A theoretical two-dimensional molecule of four atoms is placed in an arbitrary two-dimensional crystal in solution. The Patterson function contains cross peaks for every interatomic distance in the crystal and these cross-peaks in the u,v-plane, are indicated by circles and retain directional information about their positions in the crystal. The cross-peaks between symmetry mates are not shown in the expanded view due to the size of the unit cell. The pair-distribution function, on the other hand, resolves distances but not directions within each scattering unit. Thus, all equivalent distances in the four-atom molecule add together.

2.2.5 Phase determination

To build a map of the electron density in the unit cell by adding together the diffracted X-ray waves, it is necessary to determine phases for each of the reflections whose intensity is measured. The phases of reflections are not constrained mathematically unless they possess special symmetry relationships. The phase for each reflection needs to be determined and one of three techniques is used: experimental methods, direct methods, and molecular replacement.

Experimental phasing techniques systematically perturb the intensities in ways that can be used to extract information about the relative phases of the measured reflections. Isomorphous replacement (IR) depends upon the introduction of ‘heavy atoms’, e.g. atoms with a large number of electrons such as mercury or uranium, into crystals without perturbing the overall lattice (Ke, Reference Ke1997). To take advantage of the heavy atoms for phasing information, the positions of these atoms must first be identified either through analysis of Patterson maps calculated from differences in intensity or through direct methods (as described below). For determining heavy atom positions, the differences in the amplitudes between the heavy atom derivative and the native crystal are used. From the N ² peaks in the u, v, w space of the Patterson function and the symmetry of the crystal, the N peaks in x, y, z space can be calculated. Importantly, as the number of sites where the heavy atom binds increase, so do the number of Patterson peaks. Thus Patterson-based methods for solving heavy atom positions can quickly become challenging to solve. Moreover, as the size of the unit cell increases, the height of each Patterson peak becomes relatively weaker; for particularly large unit cells, heavy atom clusters, such as Ta₆Br₁₂²⁺, are used instead of single heavy atoms (Knablein et al. Reference Knablein, Neuefeind, Schneider, Bergner, Messerschmidt, Lowe, Steipe and Huber1997).

The other major experimental phasing technique relies upon introducing atoms with anomalous scattering or dispersion into the crystal. Anomalous scattering occurs when X-ray energies are near electronic (typically) or nuclear excitations and are typically described as:

where f is the scattering factor of the atom, f ₀ is the wavelength-independent component of the scattering, and f′ and f″ are the real and imaginary parts of the atomic scattering. At any wavelength, f′ is constant, so that f′ differences can only be measured by comparing data at different wavelengths. The imaginary part, the photoelectric absorption f″ (Fig. 3a), however, leads to a breakdown in Friedel's law so that the amplitude of the Friedel pairs [F(h, k, l) and F(−h, −k, −l) are not the same. Phasing experiments that use multiple wavelengths, and can take advantage of both f′ and f″ differences (multi-wavelength anomalous dispersion, MAD) or at a single wavelength that can only use f″ differences (single-wavelength anomalous dispersion, SAD) can be performed. Both SAD and MAD experiments use wavelengths at the atomic transitions to maximize the information. Further, experiments using f″ differences must have carefully measured Friedel pairs. As for MIR, the positions of anomalous scatters can also be determined by Patterson-based techniques (Fig. 3b; Hendrickson & Ogata, Reference Hendrickson and Ogata1997) and used to determine phases (Fig. 3c, d). Because phase information can be readily combined, it is not unusual for the anomalous signal from heavy atom derivatives (MIRAS) to be used and potentially combined with phasing information from other sources such as MAD or SAD experiments or even partial structures (Blow & Crick, Reference Blow and Crick1959; Sim, Reference Sim1959). Anomalous dispersion techniques have become a method of choice for solving crystal structures due to the tunability of synchrotron radiation (Helliwell, Reference Helliwell1997), the ability to cryo-cool and collect datasets from single crystals (Hope, Reference Hope1990; Garman & Schneider, Reference Garman and Schneider1997), and the techniques to introduce anomalous scatters such as selenomethionine in proteins and bromouridine in DNA molecules (Doublie, Reference Doublie1997). Anomalous scattering has also been used in conjunction with SAXS (ASAXS) (Stuhrmann, Reference Stuhrmann1981; Miake-Lye et al. Reference Miake-Lye, Doniach and Hodgson1983); however, the orientational averaging of SAXS data eliminates the f″ component, so that anomalous differences can only be measured between wavelengths. In theory, ASAXS has a number of potential applications such as monitoring the distance between two anomalous scatters; however, the signal is small and most applications have involved simple biological systems such as ion solvation of DNA (Andresen et al. Reference Andresen, Das, Park, Smith, Kwok, Lamb, Kirkland, Herschlag, Finkelstein and Pollack2004), and anomalous scattering is not yet as important in SAXS as it is in macromolecular crystallography.

Fig. 3. The structure of E. coli YgbM determined by selenomethionine MAD. (a) Comparison of theoretical and measured X-ray fluorescence at the selenium edge for the crystal. (b) Anomalous difference Patterson map identifying the selenomethionine Met11-Met11 and Met105-Met105 cross-peaks generated by crystallographic symmetry operators (at the Harker section) calculated using differences between Friedel pairs at the selenium fluorescence inflection point (panel a). (c) Anomalous difference density contoured at 5σ above the mean superimposed with the final refined structure. (d) 2F _o−F _c experimental electron density after MAD phasing and density modification contoured at 1σ (green), 3σ (lime), and 5σ (yellow). (e) Final refined structure and 2F _o−F _c map calculated using final refined phases at 1σ (blue), 3σ (orchid), and 5σ (red). (f) Overall structure of YgbM, a zinc-containing TIM-barrel, is shown as a cartoon.

Direct methods, on the other hand, have been primarily used in macromolecular crystallography as an alternative to Patterson-based methods for solving heavy-atom or anomalous-scattering substructures (Weeks et al. Reference Weeks, Blessing, Miller, Mungee, Potter, Rappleye, Smith, Xu and Furey2002). Direct methods take advantages of relationships between phases between multiple reflections (Giacovazzo et al. Reference Giacovazzo, Monaco, Viterbo, Scordari, Gilli, Zanotti and Catti1992) and require that data are complete, accurate, and are of high enough resolution so that individual scatterers can be resolved (resolutions of better than 1·2 Å). In macromolecular crystallography, direct methods can be readily applied to substructure determination, as atoms in these substructures typically are much more than 1·2 Å apart and fit the ‘atomaticity’ requirements even at moderate resolutions. Direct methods have been able to determine large anomalous substructures (Weeks et al. Reference Weeks, Blessing, Miller, Mungee, Potter, Rappleye, Smith, Xu and Furey2002), despite the fact that the use of the differences between intensities rather than intensities introduces some noise into the substructure determination. The determination of these substructures will likely continue to be one of the major roles for this technique in macromolecular crystallography.

Unlike the other techniques described above, molecular replacement attempts to computationally position an atomic model using experimental intensities. From the positioned molecule or molecules, phases can then be calculated for the calculation of electron density maps. The atomic model must be ‘similar’ to the structure of the crystallized molecule, where the degree of similarity depends on the number of molecules to be found and any potential conformational changes that can occur. Molecular replacement is a six-dimensional search; however, proper rotation of a model will allow a calculated Patterson function containing all interatomic vectors to correlate well with one calculated from the experimental data. This allows the problem to be broken down into a three-dimensional rotational search, followed by a three-dimensional translational search. In the rotation search, typically only vectors within a radius similar to the longest intramolecular distance are considered; however, the close intermolecular vectors between atoms in the atomic packing and non-crystallographically related molecules with other orientations result in ‘noise’ in this search. Increasing the number of molecules in the asymmetric unit typically makes the molecular replacement problem more difficult. Similarly, the translation function can also be calculated by comparing the Patterson function from the experimental data with the Patterson functions calculated from rotated molecules to which different translations have been applied. Although molecular replacement can be performed by calculating and overlaying explicitly calculated Patterson functions, faster algorithms that do the equivalent searches are used in practice (e.g. Navaza, Reference Navaza2001).

Molecular replacement solutions introduce the possibility of model-biased phases, which generate maps that do not show the differences between the atomic model used to solve the structure and the electron density that gives rise to the scattering. Importantly, model bias tends to increase as homology with the atomic model decreases, but can be detected through the use of omit maps, which are calculated with portions of the model omitted in the phase calculation. For true solutions, the electron density for the omitted regions will still be observed. For phase-biased results that are entirely dependent upon the model, no electron density will be observed in these omitted regions. As the number of solved protein structures increases, the ability to use molecular replacement to rapidly screen through all reasonable or all possible molecular replacement targets will increasingly become a reasonable strategy to solve new structures. To this end, the ability to identify overall structural similarities through comparison of experimental and calculated SAXS (see Section 2.4.3) could greatly reduce the number of atomic models to be screened and thereby improve the efficiency and success of molecular replacement methods for crystallography.

2.2.6 Structure determination

Given an initial set of phases, either from experimental or computational sources, electron density maps of the unit cell can be calculated. Frequently initial phase information has substantial errors; however, the goal is to generate a map of sufficiently good quality so that an atomic model can be built. Multiple density modification techniques can be used to improve phase information. The two most important ones are solvent flattening or flipping and non-crystallographic symmetry averaging (Vellieux & Read, Reference Vellieux and Read1997; Zhang et al. Reference Zhang, Cowtan and Main1997). These density modification techniques mainly operate by directly modifying the electron density maps and back calculating new phases. Solvent flattening and solvent flipping operate on the assumption that the bulk solvent regions in crystals should have uniform density and that both positive and negative deviations should either be flattened to this average density or flipped in magnitude. Non-crystallographic symmetry averaging, on the other hand, averages the density between non-crystallographically related molecules. Use of these phase modification techniques not only allows for the improvement of initial phases, but also allows for phase extension in cases where experimental phasing information is at lower resolution than the native dataset (Fig. 3d).

From the initial interpretable maps, an atomic structure is usually fit through rounds of atom placement followed by automated refinement (Kleywegt & Jones, Reference Kleywegt and Jones1997). Although experimental maps after phase modification techniques can be of excellent quality, it is possible to place atoms into maps that retain substantial errors in the phases. Thus it is quite common, although not required, to use maps calculated from phases from the updated model itself (Fig. 3e). In the case of atomic partial models or excellent experimental phases, the phases calculated from the model itself can also be combined with experimental phasing information. The crystallographers normally evaluate the experimental electron density map, calculated with the Fourier coefficients 2F _o−F _c, and the difference electron density map, calculated with the coefficients F _o−F _c, where F _o(h, k, l) are observed amplitudes and F _c(h, k, l) are calculated amplitudes. Placement of residues into maps can now be automated, such as by the program suite ARP/wARP (Perrakis et al. Reference Perrakis, Morris and Lamzin1999) and RESOLVE (Terwilliger, Reference Terwilliger2003), which can allow for rapid building of macromolecular structures by cycling between an automated density modeling and model refinement programs, given initial phases with sufficient quality. In these cases, the crystallographer supervises the process and corrects regions that are wrong or are trapped by refinement into local minima.

The overall agreement of the model of the asymmetric unit with the experimental data is measured by the ‘R-factor’:

where F _o(h, k, l) are observed amplitudes and F _c(h, k, l) are amplitudes calculated from the model. This number allows the crystallographer to monitor the effects of making manual modifications to the structure as well as providing a numerical target for minimization by automated refinement packages.

One important advance in helping detect the problem of overfitting is the R _free parameter, which calculates an R-factor for the current model using a set of reflections, typically several thousand, that are withheld from the refinement calculation (Brunger, Reference Brunger1992). However, R _free is a global parameter and while it can help determine overfitting, it is not sensitive enough to evaluate the validity of small changes to the crystallographic model. Moreover, choosing the number of reflections and which reflections to include in an R _free set can be difficult, particularly when substantial non-crystallographic symmetry exists. Although R _free is imperfect in some ways, it is a universally accepted measure of quality which is useful for both crystallographers and external reviewers. In contrast to X-ray crystallography, an appropriate analog to the R-factor is under debate (Section 4.1), and there currently is no suitable SAXS analog to R _free (Section 6).

2.2.7 Structure refinement

Crystallographers are aware that molecules in crystals are not completely rigid and are composed of atoms held together by electrons in specific orbitals. However, crystallographers are constrained in their ability to fit these features by the amount of unique data observed in the experiment. Thus, crystallographers are rarely able to fit all of the features of the molecules that are present. Since macromolecular crystals have fairly consistent density ranges (Matthews, Reference Matthews1968), ‘rules of thumb’ of how the molecules can be modeled based on the data-to-parameter ratio can be given as a function of the highest resolution data measured from the crystal.

Traditionally, a crystallographic model is constructed of primarily one conformation. Additional alternate conformations are typically added at high resolutions (<2·0 Å) when clear evidence for their existence can be observed in difference electron density maps. Atomic positions in the model are most frequently described using three positional parameters, x, y, and z and some number of parameters to describe the displacement of the atom from an equilibrium position. An additional parameter describing the ‘occupancy’ of a particular atom is normally only refined for structures with alternate conformations or partially bound ligands. For most structures at moderate to high resolutions (3·0 Å to 1·3 Å), a single parameter is used to describe the Gaussian motion of each atom about their equilibrium positions. This isotropic atomic displacement factor (ADF), alternately called the B-factor, the temperature factor, or the Debye–Waller factor, assumes that all atoms can be treated as an isotropic Gaussian distribution of atomic positions centered at an equilibrium position. However, this model fails to capture atomic displacement directed along a single direction (anisotropic displacements) or if the disorder is due to the superposition of multiple static conformations. Thus, in this resolution range, each atom is typically described by four parameters. Introducing geometric restraints, such as bond lengths, bond angles, torsion angles, chiral volumes, and planar restraints, plays an important role in constraining atomic positions to chemically reasonable positions (Engh & Huber, Reference Engh and Huber1991). These constraints are required to maintain a ratio of data and constraints that prevents over-refinement and is applied to both molecular dynamics (MD)-based refinement, such as implemented in CNS (Brunger et al. Reference Brunger, Adams, Clore, DeLano, Gros, Grosse-Kuntsleve, Jiang, Kuszewski, Nilges, Pannu, Read, Rice, Simonson and Warren1998), as well as generalized least-squares-based refinement, such as implemented in SHELX (Sheldrick & Schneider, Reference Sheldrick and Schneider1997).

At lower resolutions (below 3 Å), individual isotropic ADFs for all non-hydrogen atoms can introduce too many fittable parameters, and typically a single isotropic ADF is then refined for groups of atoms (such as side-chains, residues, or even whole domains). In contrast, very high-resolution structures (1·3 Å or better), use anisotropic ADFs that describe probability ellipsoids with six parameters (Willis & Pryor, Reference Willis and Pryor1975). And at subatomic resolutions (0·7 Å or better), the treatment of atoms as spheres of electrons begins to become inappropriate as valence electrons in the protein backbone atoms and unpaired electrons on oxygen atoms become visible in difference electron density maps (Jelsch et al. Reference Jelsch, Teeter, Lamzin, Pichon-Pesme, Blessing and Lecomte2000; Ko et al. Reference Ko, Robinson, Gao, Cheng, DeVries and Wang2003). At these resolutions, additional modeling can be used to fit the experimental data using ‘multipolar models’ for fitting the non-spherical valence shell electrons, ‘dummy atoms’ to account for valence bond electrons with additionally Gaussian scatterers, and quantum mechanics modeling methods (reviewed in Petrova & Podjarny, Reference Petrova and Podjarny2004). In each of these cases, the non-spherical treatment of electrons corresponding to the atoms introduces additional parameters that require these extraordinarily high resolutions to be fit.

Importantly, the decision on how to model the ADFs is not dictated by whether or not anisotropic motions (or valence electrons) are present in the crystal, but whether or not any particular model introduces too many parameters. For example, more economical parameterizations of non-isotropic motion have been recently applied, recognizing that much of the anisotropic motion of atoms can be correlated to domain motions within crystals. These schemes simultaneously model the motion of groups of atoms by translation-libration-screw (TLS) models (Schomaker & Trueblood, Reference Schomaker and Trueblood1968) or normal mode models (Kidera & Go, Reference Kidera and Go1990) and have been used to explain motion and help refine crystal structures at moderate resolutions (Howlin et al. Reference Howlin, Butler, Moss, Harris and Driessen1993; Winn et al. Reference Winn, Isupov and Murshudov2001, Reference Winn, Murshudov and Papiz2003). The decisions on how to properly model the structure given the information content of the data is as important in the generation of SAXS models as it is in X-ray crystallography. For SAXS, the application of use of external constraints, such as symmetry or atomic structures of individual domains, can be very important to ensure the reproducible reconstruction of solution structures (Section 4), and these constraints are analogous to the use of geometric constraints during crystallographic refinement derived from chemistry or non-crystallographic symmetry.

2.2.8 Flexibility and disorder in crystals

The crystallographic Debye–Waller or B-factor has been used as a surrogate for flexibility and local disorder in crystals. A number of alternative ways to model these features in crystal structures have emerged more recently. These schemes seek to better fit the disorder to improve R-factors as well as to better understand disorder in the crystallized molecules. For example, the use of multiple models provides a possible means to analyze disorder within crystal structures (Furnham et al. Reference Furnham, Blundell, DePristo and Terwilliger2006). Two types of ‘crystallographic ensembles’ can be envisioned. In the first, the different structures represent independent refinements against the raw data and do not ‘see’ each other. These are most equivalent to the independently calculated models generated during NMR refinement. From a number of test cases, it has been suggested that multiple distinct isotropic models can fit the experimental data equally well and thereby suggests that classic measures of model accuracy fail to capture inaccuracies and ambiguities in single model refinements (dePristo et al. Reference dePristo, de Bakker and Blundell2004). In the second sort of crystallographic ensemble, multiple structures can be simultaneously calculated against the raw data. This refinement would be most appropriate for structures that have the types of disorder that tend to limit the resolution of diffraction. Unfortunately, these ensembles also introduce the real possibility of introducing far more parameters than can be justified by the raw data. Attempts to ensure that individual refinements are restrained have been performed by refining only single models with isotropic ADFs at a time while monitoring R _free to monitor overfitting (Rejto & Freer, Reference Rejto and Freer1996). However, in at least one case, TLS refinement performed better than multiple model refinement (Wilson & Brunger, Reference Wilson and Brunger2000).

In addition to alternative modeling techniques to fit potential information about disorder in the X-ray diffraction data, theories for the interpretation of diffuse scatter, which is normally ignored in X-ray diffraction experiments, have emerged (Faure et al. Reference Faure, Micu, Perahia, Doucet, Smith and Benoit1994; Mizuguchi et al. Reference Mizuguchi, Kidera and Go1994). This diffuse scatter arises from transient and static imperfections in the crystal lattice and causes scattered X-ray intensities to be observed at positions other than the Bragg peaks. Since these motions occur in crystals trapped in the lattice, the diffuse scatter is not radially averaged as it is in SAXS. Fitting of this diffuse scatter by techniques like normal mode analysis (NMA; Section 3.3) has suggested that they involve correlated motions of domains of the proteins. Importantly, this diffuse scatter does not include distortions affecting distances between unit cells that give rise to streaks due to lattice distortions. Although these experiments have been largely restricted to model systems (Wall et al. Reference Wall, Clarage and Phillips1997a,Reference Wall, Ealick and Grunerb; Meinhold & Smith, Reference Meinhold and Smith2007), they hold potentially valuable information regarding biologically relevant motions.

These methods for understanding flexibility of molecules in the context of a crystal lattice can directly complement more direct measurements of flexibility derived from solution experiments including NMR, SAXS, and fluorescence studies. However, care must be taken as the crystal lattice can directly influence what conformations can be observed and what range of motions are possible.

2.3.1 Measuring SAXS data

Unlike X-ray crystallography, SAXS is inherently a contrast method where the scattering signal is derived from the difference in the average electron density, Δρ(r), of solute molecules of interest, ρ(r), and bulk solvent ρ_S (~0·33 e⁻/ų for pure water):

Proteins, for example, have an average electron density of ~0·44 e⁻/ų. Larger Δρ(r) values give rise to larger signals (Table 1), which is important to maximize scattering from dilute solutions as well as for contrast variation techniques (Section 2.3.6). This result also makes SAXS particularly attractive for determining RNA and DNA structures, which have higher contrasts than proteins. In practice, data is collected on a buffer blank and on a sample. Subtraction of observed scattering yields the signal from the scattering due to the macromolecule. Subtracting scattering of the blank from the sample must be done as precisely as possible to accurately measure differences of over three orders of magnitude (Section 5.1).

Table 1. Common parameters defined by SAXS for monodisperese and homogeneous scatterers

The scattering curve resulting from the subtraction of the buffer from the sample, I(q), is radially symmetric (isotropic) due to the randomly oriented distribution of particles in solution (Fig. 4). I(q) is a function of the momentum transfer q=(4π sin θ)/λ, where 2θ is the scattering angle, as in X-ray crystallography, and λ is the wavelength of the incident X-ray beam. In various treatments, the symbols s and h can be used for q. Confusingly other treatments define S=(2 sin θ)/λ, so that q=2πS, and others define θ, rather than 2θ, as the scattering angle. Each of these definitions is equivalent; the convention being followed must be defined. Here we will consistently use q as defined above with 2θ as the scattering angle. The units of q are the inverse of units used in the wavelength, typically Å⁻¹ or nm⁻¹, and the value is a measure of the directional momentum change that the photons undergo. By comparison with Bragg's law in X-ray crystallography, q=2π/d, where 1/d is the reciprocal resolution. Regardless of the incident wavelength, a plot of I(q) vs. q should be identical for the same sample, except at wavelengths where anomalous scattering of atoms within the sample occurs.

Fig. 4. Experimental SAXS curves and parameters measured for the Pyrococcus furiosis PF1282 rubredoxin (magenta), the ‘designed’ scaffoldin protein S4 (red) (Hammel et al. Reference Hammel, Fierobe, Czjzek, Kurkal, Smith, Bayer, Finet and Receveur-Brechot2005), the ‘designed’ minicellulosome containing three catalytic subunits (green), and the DNA-dependent protein kinase (blue). (a) D _max of the scattering particle is a simple function of molecular weight for perfect spheres (spheres), but not for proteins that adopt different shapes (diamonds). Envelopes correspond to ab-initio models calculated from experimental curves using GASBOR. (b) The experimental scattering curves for each protein show that the intensity of scattering falls more slowly for rubredoxin (R _G 11 Å; magenta) than the minicellulosome (R _G 82 Å; green). (c) The linear region of the Guinier plot, from which R _G and I(0) can be derived, is function of the R _G. (d) Each protein has both a substantially different D _max as well as pair-distribution function, reflecting the different atomic arrangements.

Unlike X-ray crystallography, where diffraction provides a clear measure of quality, it can be more difficult to confirm that a measured scattering curve is appropriate for further analysis. In general this is an unsolved problem; however, some empirical guidelines do exist for assessing data quality (Sections 5.2–5.5). Many issues are primarily understood anecdotally, and a directed effort on the best methods to assess sample quality will benefit from a growing group of researchers adopting SAXS methodologies. We encourage researchers to describe problems as well as solutions in the literature.

2.3.2 Scattering from macromolecules

The theoretical basis for solution scattering has been the subject of an excellent review (Koch et al. Reference Koch, Vachette and Svergun2003). Here we briefly consider the most common situation for structure reconstruction (Section 4), in which samples are homogeneous, monodisperse, and lacking long-range interactions in solution. Many of the most commonly used relationships relevant to this case are tabulated in Table 1. More complicated or recalcitrant samples require additional experimental and theoretical treatment (Section 5).

The scattering curve of a homogeneous sample can be derived from the electron distribution of the particle [the pair-distribution function, P(r), Section 2.3.3]:

where D _max is the maximum distance present in the scattering particle.

From a practical standpoint, the lowest resolution portion of the SAXS curve is dictated by a single size parameter (Fig. 4). This size parameter, the radius of gyration (R _G), is the square root of the average squared distance of each scatterer from the particle center (Table 1). For example, a sphere of radius r with uniform electron density, for example, has a R _G=(3/5)^½r. R _G, like the hydrodynamic or Stokes' radius (R _S), is shape-dependent and a poor measure of the actual molecular weight (volume) of the molecule of interest. R _G and R _S are different, however, in that R _S is the radius of an equivalent sphere that diffuses identically to the molecule of interest, hence R _S=r for a perfect sphere.

At low resolution, the scattering can be described by the Guinier approximation:

The Guinier plot of log(I(q)) against q ² will give a straight line from which R _G and I(0) can be extracted (Fig. 4c; Guinier & Fournet, Reference Guinier and Fournet1955). The q-range over which the Guinier approximation is valid (qR _G<1·3 for globular proteins) is much larger for particles with small R _G than larger particles (Fig. 4c). In practice, this estimation of R _G must be performed iteratively or interactively (Konarev et al. Reference Konarev, Volkov, Sokolova, Koch and Svergun2003), since new estimates of R _G can alter the q-range for which the estimate can be made. Lack of linearity in the Guinier plot is a sign that more care needs to be taken to evaluate the sample (Section 5), or that samples are elongated. For these samples, other methods for estimating R _G may be more appropriate (Table 1). Similarly, R _G should not vary with concentration for well-behaved samples with no interparticle interference or aggregation. R _G shows some dependence on the contrast difference between bulk solvent and the sample comparisons of different samples should be performed in the same buffer.

The second important parameter that can be evaluated from the lowest q values is I(0), the intensity measured at zero angle (q=0), which must be determined by extrapolation, as it is coincident with the direct beam. On an absolute scale, I(0) is the square of the number of electrons in the scatterer and is unaffected by particle shape and is useful for molecular weight determination (Section 2.3.4). I(0) is equivalent to the value of I(0, 0, 0) in X-ray crystallography (Section 2.2.3). For well-behaved samples, a plot of I(0) vs. concentration gives a straight line. Additionally, since I(0) depends on the square of the number of electrons (molecular weight), SAXS is particularly sensitive to the assembly state of the scatterers.

Higher q values contain details regarding molecular shape. For folded macromolecules, the intensity of the scattering falls off by Porod's law (Porod, Reference Porod1951):

This relationship, however, assumes a uniform density for the scatterer, which breaks down at high q values when atomic resolution information begins to contribute significantly. Hence, Porod's law, like the Guinier approximation, holds only in a portion of the scattering curve, and we have observed some samples that possess little or no scattering following Porod's law. For arbitrary polymers, this region of scattering is typically termed the ‘power law regime’, where the resolution-dependence of the scattering can be expressed as:

where d _f is the fractal degrees of freedom. For example, scattering comprised of spheres has a d _f=4, flat (oblate) ellipsoids has a d _f=2 in the high q-range, whereas scattering from needle-like (prolate) ellipsoids has a d _f=1 in the high q-range. Random coils in ‘good solvent’ have d _f=5/3.

Thus, SAXS is an ideal method for identifying and characterizing polymers without folded domains. The Kratky plot [q ²I(q) as a function of q], which can be calculated directly from the scattering curve, provides an excellent tool for evaluating the folding of samples. For folded domains, the Kratky plot yields a peak roughly shaped like a parabola. The position of the peak provides some information about its overall size; however, our experience has shown that the position is shape-dependent like R _G and thus cannot directly provide information regarding molecular weight. In contrast, extended semi-stiff polymers, such as random coil peptides, follow the Porod–Kratky worm-like chain model (Kratky & Porod, Reference Kratky and Porod1949). Random coil or unstructured peptides lack the characteristic folded peak and are linear with respect to q in the large q-region. At low resolutions the scattering can be described by (Brulet et al. Reference Brulet, Boue and Cotton1996):

where y=q ²Lb/6 and R _c is the radius of gyration of the cross section, L is the total length of the polymer, and b is twice the persistence length, the maximum length that the polymer chain persists in any one direction (Table 1). This relationship holds in the resolution range q<3/b. For peptides, b varies between 19 Å and 25 Å, yielding an average persistence length of 9·5–12·5 Å or roughly 3–4 amino acids (Perez et al. Reference Perez, Vachette, Russo, Desmadril and Durand2001). The expected R _G for the unfolded polypeptide can be calculated with the equation:

where x=L/b. This equation is useful as the R _G value for unfolded or chemically denatured samples is so large that the scattering region following Guinier's approximation is typically not recorded in normal beamline geometries (Calmettes et al. Reference Calmettes, Durand, Desmadril, Minard, Receveur and Smith1994).

2.3.3 Pair-distribution function

The pair-distribution function P(r), also called the pair-density distribution function (PDDF; Fig. 4d) is the SAXS function corresponding to the Patterson function (Section 2.2.4). This autocorrelation function can be directly calculated through a Fourier transform of the scattering curve (Table 1), and the result provides direct information about the distances between electrons in the scattering particles in the sample, in a manner similar to the Patterson function. The P(r) function can also be calculated directly from the electron density:

The important differences between P(r) and the Patterson function are that the P(r) is radially averaged and lacks vectors corresponding to vectors between scattering particles, which gives rise to large ‘origin peaks’ at (0, 0, 0) and other positions corresponding to pure crystallographic translations in the Patterson function. Typically, the P(r) function is calculated by an indirect Fourier transformation to avoid problems due to discrete sampling of the I(q) curve over a finite range (Glatter, Reference Glatter1977). The indirect Fourier transform essentially constructs trial P(r) functions that are Fourier transformed and evaluated in comparison with the experimental scattering. In the GNOM program (Semenyuk & Svergun, Reference Semenyuk and Svergun1991), a regularizing multiplier is used to balance the smoothness of the trial P(r) functions with the goodness of fit to the data. In the GIFT program (Bergmann et al. Reference Bergmann, Fritz and Glatter2000), the inverse transformation is solved using Boltzmann simplex simulated annealing to solve the nonlinear dependencies of the scattering curve with the P(r) structure factor parameters and iteratively fits the parameters. Additionally, GIFT also simultaneously fits contributions from the scattering due to interparticle interactions (Brunner-Popela & Glatter, Reference Brunner-Popela and Glatter1997).

Theoretically, the P(r) function is zero at r=0 and at r⩾D _max, where D _max corresponds to the maximum linear dimension in the scattering particle. For the processing of real data, the P(r) function is typically constrained in the calculation to be zero at these values. This constraint is often not necessary for well-behaved (globular) samples and can be an indicator of good quality data (Section 5.3). On the other hand unfolded proteins are often not zero at r=0 in unconstrained P(r) functions, and non-zero values at r=D _max may indicate aggregation or improper background subtraction. D _max is useful for characterizing the sample; however, accurately determining D _max for samples can be difficult. The scattering data should be measured at q⩾2π/D _max. More problematically, the indirect Fourier transformation methods to calculate P(r) rely upon the value of D _max, giving the value more importance than if the P(r) function could be calculated from a direct Fourier transformation. Moreover, the P(r) curves are typically small in the vicinity of D _max, and hence contribute little to the overall scattering. Thus, errors in estimates of D _max can be difficult to identify, including extended structures and globular structures with disordered extensions, such as unstructured N- and C-termini in proteins. In practice, estimation of D _max by the inverse Fourier transformation involves choosing multiple D _max values and evaluation of the resulting P(r) functions for their fit to the experimental scattering.

The P(r) function has many important usages. First, a value for R _G and I(0) can be calculated from the P(r) function that takes into account all of the collected data and is not limited to the small region about the direct beam that is used in the Guinier approximation. Thus, this real space approximation is likely to be a better estimate for samples complicated by small amounts of aggregation that most strongly affect the lowest resolution information. Second, P(r) functions can be readily calculated from atomic models (Section 4.2.1). This has important implications for many different methods of using atomic models in conjunction with SAXS data (Section 4.2). The P(r) function can also give some initial indication of the overall shape from its overall shape, for example spherical objects with bell-shaped P(r) functions can be readily distinguished from rod-like shapes (Fig. 5) and are particularly useful in conjunction with Kratky plots (Section 2.3.2).

Fig. 5. Theoretical and experimental SAXS from a 76mer double-stranded DNA fragment. (a) Theoretical scattering from a 76mer B-DNA fragment without counter ions was calculated using CRYSOL (green; Svergun et al. Reference Svergun, Baraberato and Koch1995) and compared to the experimental scattering (black) or ab-initio reconstructions generated by GASBOR (red; Svergun et al. Reference Svergun, Petoukhov and Koch2001a). The experimental data has a R _G of 69 Å as compared to the 67 Å for the theoretical B-DNA structure. (b) Comparison of the P(r) functions calculated from the theoretical and experimental scattering by GNOM. The early peak and roughly linear fall off of the P(r) are characteristic of a linear or extended molecule and can be seen with protein samples as well (Fig. 10). The observed D _max is 260 Å, whereas the D _max calculated for the theoretical DNA fragment is 255 Å, and the linear length expected for 76 bases of B-DNA is 250 Å. C. Ab-initio reconstructions of the DNA generated by GASBOR are quite similar in thickness and length to the size of B-DNA.

2.3.4 Information content in scattering curves

One of the central strengths of SAXS is that measurements are done in solution with little preparation relative to other techniques. The downside is that the measured data are orientationally averaged and cannot, for example, be used to distinguish between enantiomorphs. Furthermore, the scattering curve only has a small number of independent data points, typically estimated by the number of Shannon channels (Shannon & Moore, Reference Shannon and Moore1949):

The number of independent values that can be extracted from the scattering (reciprocal space) has been shown to be equivalent to the number of independent data-points in real space (Moore, Reference Moore1980). For most SAXS curves, N _s usually does not exceed 10–15. As the lowest-resolution value describes the overall size of the scatterer (R _G), the first data point q _min ought to be measured at q _min⩽π/D _max. An alternative measure for the number of experimentally determined parameters has been suggested that uses a maximum entropy method (Mueller et al. Reference Mueller, Hansen and Puerschel1996). This method accounts for the problem of determining a uniquely defined q _max in the presence of experimental noise; however, this measure does not substantially change the number of parameters available (Vestergaard & Hansen, Reference Vestergaard and Hansen2006). SAXS data are dramatically over-sampled, in that Δq between two adjacent points measured in the scattering curve are, however, much less than π/D _max. This fact has been used to argue that the effective information content is higher than predicted from the number of Shannon channels (Koch et al. Reference Koch, Vachette and Svergun2003).

Given this estimate of information content in a solution scattering curve it is remarkable that accurate shapes can be derived. This intuitively daunting feature may explain why the development of SAXS has not been pursued at the same rate as crystallography. Studies have demonstrated situations in which more and more detailed structural information can be extracted utilizing SAXS data when additional constraints are imposed on the reconstructions. Modern ab-initio algorithms include constraints that attempt to force final solutions to have protein-like properties. For example GASBOR enforces penalties on its shape reconstructions for compactness (Svergun et al. Reference Svergun, Petoukhov and Koch2001b). The number of restraints added by this type of external information is not easily estimated. Regardless of the precise number of fitted parameters that can be justified, we would suggest that there is a clear analogy with the difficulties of refining X-ray crystal structures at different resolutions (Section 2.2.7) in which the best use of the available experimental data includes external information. Thus, using known crystal structures as a basis for fitting low-resolution SAXS data mirrors the use of chemical bonding parameters in the case of moderate or ‘low’-resolution X-ray diffraction data, and we detail theoretical and practical methods to do this in Sections 4 and 5.

2.3.5 Molecular weight and multimerization state in solution

In a monodisperse, ideal solution of identical particles, the observed scattering is linearly related to the number of particles, N, in the sample. The measured I(0) obtained after scaling for concentration corresponds to the scattering of the single particle and it is proportional to the square of the total excess scattering length in the particle. If the measurements are made on an absolute scale (cm⁻¹), I(0) can be directly related to the molecular weight of the particle:

where m is the number of electrons of the particle, ρ₀ is the average electron density of the solvent, and ψ is the ratio of the volume of the particle to its number of the electron. If the scattering curve is scaled by concentration in units of the molarity of the particle, then I(0) is proportional to the mass squared. However typically only the molarity of the monomeric unit is known and the concentration of the target particle, c, is reported as mass per volume (mg/ml) and is c=Nμm/N _A, where N _A is Avogadro's number, and μ is the ratio M/m of the molecular weight to the number of electrons, which depends on the chemical composition of the particle (for proteins a good approximation is M/m=1·87). Therefore,

If ψ, ρ₀, and c are known, and the intensity of the incident beam is known on an absolute scale, then the intensity at the origin provides a determination of the molecular weight (Vachette & Svergun, Reference Vachette, Svergun, Fanchon, Geissler, Hodeau, Regnard and Timmins2000; Koch et al. Reference Koch, Vachette and Svergun2003).

An experimentally more tractable measurement of mass can be obtained using relative I(0) values after proper calibration with reference samples, such as lysozyme (14·3 kDa), bovine serum albumin (BSA, 66·2 kDa), and glucose isomerase (172 kDa) (Kozak, Reference Kozak2005; Mylonas & Svergun, Reference Mylonas and Svergun2007). Samples composed of multiple components with different average electron densities, such as protein–DNA complexes, can be more problematic. Nevertheless even with mixed electron density systems, bounding the mass between values may be sufficient to establish the multimeric state as has been demonstrated for membrane protein systems collected above the critical micelle concentration of the solubilizing detergent (Columbus et al. Reference Columbus, Lipfert, Klock, Millett, Doniach and Lesley2006).

Each of these techniques requires accurate determinations of I(0) values from Guinier or Debye approximations (Table 1), or from estimates using the P(r) function:

The P(r) is calculated from the entire scattering curve. Thus, extracting I(0) from P(r) has several advantages over the I(0) measured from Guinier plots, particularly for data where only a few points have been measured in the Guinier region or where the Guinier region is affected by interparticle interactions. The P(r)-based I(0) value is typically reported by programs performing P(r) calculations (Svergun, Reference Svergun1992; Bergmann et al. Reference Bergmann, Fritz and Glatter2000).

Since most macromolecules have fairly uniform densities, the molecular weight is also directly related to volume. Volume information can be derived from SAXS curves in experiments where neither absolute I(0) nor reference samples have been measured. In this approach the theoretical excluded volume calculated from sequence or from an atomic model, such as reported by the program CRYSOL (Svergun et al. Reference Svergun, Baraberato and Koch1995), can be compared to volumes generated by ab-initio shape-determination algorithms (Section 4.3) (Hammel et al. Reference Hammel, Kriechbaum, Gries, Kostner, Laggner and Prassl2002; Krebs et al. Reference Krebs, Durchschlag and Zipper2004) and/or volumes derived from the scattering according the Porod law (Porod, Reference Porod, Glatter and Kratky1982).

The volume of the macromolecule undergoing scattering can be calculated from I(0) and the Porod invariant Q (Porod, Reference Porod, Glatter and Kratky1982):

where the invariant is calculated by:

This calculation does not require data normalization. For globular proteins, Porod volumes in nm³ are typically twice the molecular masses in kDa and is a valuable conformation of mass estimates using I(0) (Petoukhov et al. Reference Petoukhov, Svergun, Konarev, Ravasio, van den Heuvel, Curti and Vanoni2003; Gherardi et al. Reference Gherardi, Sandin, Petoukhov, Finch, Youles, Ofverstedt, Miguel, Blundell, Vande Woude, Skoglund and Svergun2006). These volume determinations are, however, subject to error as they rely on the accurate data over the entire q-range (due to extrapolation of high q using the fall off of intensity with q ⁻⁴). The contribution of internal particle structure to scattering at larger angles becomes significant at q values above 0·2 Å⁻¹, and this contributes error to the calculation. Thus, the large angle portions of the curves should be discarded in the computation (Glatter, Reference Glatter, Glatter and Kratky1982). Additionally, this technique for extracting mass is very inaccurate for asymmetric particles. Excluded volumes can be easily calculated using programs such as PRIMUS (Konarev et al. Reference Konarev, Volkov, Sokolova, Koch and Svergun2003).

Importantly, the I(0) method for molecular weight determination and volumes derived from ab-initio shape determination (Hammel et al. Reference Hammel, Kriechbaum, Gries, Kostner, Laggner and Prassl2002; Krebs et al. Reference Krebs, Durchschlag and Zipper2004) should yield consistent results (Petoukhov et al. Reference Petoukhov, Svergun, Konarev, Ravasio, van den Heuvel, Curti and Vanoni2003, Reference Petoukhov, Monie, Allain, Matthews, Curry and Svergun2006; Gherardi et al. Reference Gherardi, Sandin, Petoukhov, Finch, Youles, Ofverstedt, Miguel, Blundell, Vande Woude, Skoglund and Svergun2006; Nemeth-Pongracz et al. Reference Nemeth-Pongracz, Barabas, Fuxreiter, Simon, Pichova, Rumlova, Zabranska, Svergun, Petoukhov, Harmat, Klement, Hunyadi-Gulyas, Medzihradszky, Konya and Vertessy2007; Qazi et al. Reference Qazi, Bolgiano, Crane, Svergun, Konarev, Yao, Robinson, Brown and Fairweather2007), and thus each can be used to independently confirm the results from a single sample (Table 1). In practice, SAXS provides a powerful approach to determining the molecular weight and assembly state in solution that can be extremely useful for crystallization efforts (Section 2.4), for modeling solution assemblies (Section 4), and for interpreting biochemical and mutational results.

2.3.6 Contrast variation

Measurable scattering from a solute is contingent on the contrast in scattering density between the solute and solvent. For length scales larger then 15 Å the scattering density for biomolecules is approximately homogeneous. Thus, most internal structural features can be successfully ignored at resolutions lower than q=0·2 Å⁻¹. SAXS can be used to extract internal structural features of systems comprised of two or more components with distinct average electron densities. Since SAXS is a contrast method, variation of the average solvent electron density can extract information about the inner structure of multicomponent systems (Stuhrmann, Reference Stuhrmann1973, Reference Stuhrmann, Glatter and Kratky1982). Choosing appropriate solvent electron densities with high concentrations of sugars, glycerol, or salt can mask out the scattering of one of the components (Pilz, Reference Pilz, Glatter and Kratky1982), and contrast variation studies have been successfully performed with SAXS (Muller et al. Reference Muller, Laggner, Glatter and Kostner1978). In practice, however, the dramatic difference in the interaction of neutrons with hydrogen atoms (¹H) and deuterons (²H), makes neutron scattering (SANS) with specifically deuterated components, and not SAXS, the technique of choice for contrast variation studies.

SANS contrast variation studies on two component systems have been performed for samples such as the ribosome (Stuhrmann et al. Reference Stuhrmann, Haas, Ibel, Wolf, Koch, Parfait and Crichton1976; Svergun et al. Reference Svergun, Burkhardt, Pedersen, Koch, Volkov, Kozin, Meerwink, Stuhrmann, Diedrich and Nierhaus1997a, Reference Svergun, Burkhardt, Pedersen, Koch, Volkov, Kozin, Meerwink, Stuhrmann, Diedrich and Nierhausb), human plasma lipoprotein (Stuhrmann et al. Reference Stuhrmann, Tardieu, Mateu, Sardet, Luzzati, Aggerbeck and Scanu1975), and DNA–protein complexes (Chamberlain et al. Reference Chamberlain, Receveur, Spencer, Redfield and Dobson2001). One recent SANS study used contrast variation to eliminate the contribution of the detergent molecules required to solubilize the hydrophobic membrane-binding protein apolipoprotein B-100 (apoB-100) (Johs et al. Reference Johs, Hammel, Waldner, May, Laggner and Prassl2006). ApoB-100 is the protein component of human low-density lipoproteins (LDL), which triggers the receptor-mediated cellular uptake of LDL. Due to its size (550 kDa and 4536 residues) and hydrophobicity, the apoB-100 structure had not been well characterized. The authors have calculated low-resolution models (Fig. 6a, b) that reveal a pronounced cavity in the center of the molecule with alternating wide and narrow sections, which have been interpreted as folded domains connected by linkers. These linkers may confer flexibility, allowing apoB-100 to rearrange when forming the lipoprotein particle (Fig. 6b–d).

Fig. 6. Reconstructed three-dimensional ab-initio model of the detergent solubilized apoB-100 protein. (a) Ten independent single models with similar goodness of fit were restored by DAMMIN (Svergun, Reference Svergun1999). (b) The average envelope of apoB-100 was calculated for 10 independent DAMMIN models using the program DAMAVER (Volkov & Svergun, Reference Volkov and Svergun2003), and the secondary structure information was mapped onto the low-resolutionmodel. (c) Hypothetical model of the spatial arrangement of apo-100B in LDL. (d) Model of the LDL particle with apoB-100 and a superimposed 250 Å sphere, representing the lipid components (after Johs et al. Reference Johs, Hammel, Waldner, May, Laggner and Prassl2006).

2.4.1 Validating sample quality and assessing crystallographic targets

A frequently underappreciated feature of a SAXS experiment is that even in the absence of an atomic resolution structure, SAXS experiments are useful in directly assessing sample quality. In a structural genomics project involving protein complexes isolated from Pyrococcus furiosis, features of the SAXS were excellent indicators of sample folding, assembly, and aggregation. In turn each of these features was a useful predictor for identifying samples that were readily crystallized (G. L. Hura, J. A. Tainer, M. W. Adams unpublished observations).

Experimental evidence has shown that homogenous solutions of macromolecules with little or no aggregation have a tendency to readily crystallize both when multiple proteins (D'Arcy, Reference D'Arcy1994; Ferre-D'Amare & Burley, Reference Ferre-D'Amare and Burley1997) or multiple preparations of the same protein are compared (Habel et al. Reference Habel, Ohren and Borgstahl2001). Dynamic light scattering (DLS) has emerged as an important technique to characterize macromolecules (Zulauf & D'Arcy, Reference Zulauf and D'Arcy1992) and screen for buffer conditions for crystallization (Jancarik et al. Reference Jancarik, Pufan, Hong, Kim and Kim2004) as it can evaluate the overall aggregation of the sample. As SAXS uses wavelengths that probe atomic resolution information, the presence of structure in the SAXS curves as compared to featureless decay has been used to distinguish between samples where aggregation can be ameliorated by varying conditions, centrifugation, or size-exclusion chromatography from those that are hopelessly aggregated.

Similarly, for some molecules that form complexes, control of the molecular assembly state has been important for crystallization, such as for the requirement of multimerization by the Schizosaccharomyces pombe cell cycle-regulatory protein suc1 for crystallization (Parge et al. Reference Parge, Arvai, Murtari, Reed and Tainer1993; Bourne et al. Reference Bourne, Arvai, Bernstein, Watson, Reed, Endicott, Noble, Johnson and Tainer1995), although many others examples also exist based on anecdotal evidence. As has been described in Section 2.3.4, SAXS is exquisitely sensitive to the assembly state and thus can be used to identify buffer conditions that help stabilize particular assemblies for crystallization.

Finally, SAXS is also sensitive to the overall shape of the macromolecule, and samples that are unfolded are clearly visible in the Kratky plot (Section 2.3.2). This evaluation is particular important for engineering and characterizing truncation mutations for crystallographic studies, such as the α-subunit of DNA polymerase III (Lamers et al. Reference Lamers, Georgescu, Lee, O'Donnell and Kuriyan2006), as well as identification of natively unfolded proteins (Shell et al. Reference Shell, Putnam and Kolodner2007).

Taken together, these uses of SAXS data can be tremendously helpful for characterizing proteins being expressed and purified for biochemical, biophysical and crystallographic studies. In general, the typical concentrations and total amounts of sample required for SAXS are readily accessible when crystallography or other biophysical experiments are being pursued. In addition to evaluating constructs and buffer conditions for aggregation or unfolding (Section 5), SAXS can be used to provide initial information regarding the shape and the assembly of complexes from ab-initio structure reconstruction as discussed below (Section 4.2) and direct information for establishing the biologically relevant solution assemblies as discussed below (Section 4.2.5).

2.4.2 Using low-resolution SAXS envelopes for phasing

From a one-dimensional experimental SAXS curve, it is possible to reconstruct a three-dimensional envelope (Section 4.3.1). Even in the absence of a high-resolution structure, these envelopes may be useful for low-resolution phasing of crystallographic data. For example, specialized programs such as FSEARCH can place envelopes derived from EM or SAXS into the crystal by molecular replacement (Hao et al. Reference Hao, Dodd, Grossmann and Hasnain1999; Ockwell et al. Reference Ockwell, Hough, Grossmann, Hasnain and Hao2000; Hao, Reference Hao2001, Reference Hao2006). This has been successful for low-resolution phasing of phytase (Liu et al. Reference Liu, Weaver, Xiang, Thiel and Hao2003) and lobster clottable protein (Kollman & Quispe, Reference Kollman and Quispe2005). Many standard molecular replacement programs, such as AMoRe (Navaza, Reference Navaza2001), can also do molecular replacements using structure factors calculated from a low-resolution envelope in an appropriate cell with P1 symmetry (Urzhumtsev & Podjarny, Reference Urzhumtsev and Podjarny1995). Ab-initio phasing from fairly simple envelope models has been used for phasing viral particles. For these systems, success is due to the ability to perform rounds of phase extension with non-crystallographic symmetry averaging (reviewed in Rossmann, Reference Rossmann1995).

For average macromolecular crystals lacking the extensive non-crystallographic symmetry of viruses, the major challenge for the use of low-resolution phasing techniques is to extend the phase information to a resolution high enough that atomic models can be built. One intriguing possibility that may be more accessible for general crystallographic problems might be to leverage the power of multicrystal averaging, if molecular replacement solutions can be found for each non-isomorphous crystal (Cowtan & Main, Reference Cowtan and Main1996, Reference Cowtan and Main1998). However, even in the absence of appropriate density modification techniques for phase extension, low-resolution phasing by envelopes has the potential to help locate heavy atoms, particularly heavy-atom clusters used for phasing crystals with large asymmetric units, and thereby allow the first heavy-atom positions to be identified from difference Fourier maps rather than the more difficult Patterson maps.

2.4.3 Structural database of SAXS

With the increase in speed of modern computers, one potential crystallographic phasing strategy is a molecular replacement search using all solved domains from the Protein Data Bank (PDB) (Bernstein et al. Reference Bernstein, Koetzle, Williams, Meyer, Brice, Rodgers, Kennard, Shimanouchi and Tasumi1977). These searches still take substantial amounts of computer time and one useful strategy would be to focus the molecular replacement search on models that are most like the protein of question. Measurement of a solution scattering curve might be able to provide sufficient structural information to focus such a search.

Recently, a database of calculated scattering curves, DARA, was created for a large portion of the structures deposited in the PDB (Sokolova et al. Reference Sokolova, Volkov and Svergun2003a, Reference Sokolova, Volkov and Svergunb). DARA ranks experimental scattering curves relative to their fit to the precalculated DARA scattering curves. Theoretical scattering curves that were scored as similar tended to fit both overall shape and secondary structural features. For a brute-force molecular replacement search, these hits are likely to be much more useful for phasing.

There are a number of difficulties in performing these comparisons, even in the absence of experimental noise. First, determining appropriate criteria by which to compare scattering curves can be difficult (Section 4.1). Second, many PDB entries lack appropriate or correct information for the biologically relevant assemblies in the file meta-data, which are problematic for DARA searches. Moreover, multiple conformations of residues and heavy atoms in the crystal structures may cause problems for calculating the theoretical scattering.

These difficulties may be responsible for the mixed success in studies using the current version of DARA (Hamada et al. Reference Hamada, Higurashi, Mayangi, Miyata, Fukui, Iida, Honda and Yanagihara2007). Nevertheless we have found it useful in several cases (Fig. 7), and it is currently unclear if success is dependent upon specific features of the macromolecules being investigated. However, the possibility of identifying similarly shaped molecules using SAXS curves has the potential to be particularly powerful for high-throughput applications. Fine tuning parameters for analysis of SAXS data may also be aided by applying the analysis on structures identified by DARA. For example, the database search may help identify hollow protein shells like ferritin (Trikha et al. Reference Trikha, Theil and Allewell1995), which tend to be biased against in ab-initio shape restoration (Section 4.3).

Fig. 7. An example of a successful DARA (Sokolova et al. Reference Sokolova, Volkov and Svergun2003a, Reference Sokolova, Volkov and Svergunb) hit from a protein of unknown structure and function. Using SAXS data, an ab-initio GASBOR envelope was calculated that superimposes well with the crystal structure of the best DARA hit.

4.2.1 Calculation of scattering curves from atomic models

Theoretical SAXS curves can be directly calculated from atomic models. The observed scattering profile is largely the difference between the scattering of the target molecule with its ordered solvation layer and the excluded volume that takes into account the missing scattering of bulk solvent due to the existence of the solute. The excluded volume term can be determined by defining the shape of the molecule and calculating the scattering from it as if it were filled by an electron density equivalent to bulk solvent (Fraser et al. Reference Fraser, MacRae and Suzuki1978; Lattman, Reference Lattman1989; Svergun et al. Reference Svergun, Baraberato and Koch1995).

In principle, the P(r) function and hence the scattering from the solute can be calculated by evaluating all of the interatomic distances in the structure: a procedure that scales by the square of the number of atoms. This algorithm is inappropriate for fitting purposes for which calculated scattering must be calculated thousands of times. Faster alternatives include the calculation of the P(r) function using Monte Carlo integration routines (Zhao et al. Reference Zhao, Hoye, Boylan, Walsh and Trewhella1998) and is implemented in ORNL_SAS (Tjioe & Heller, Reference Tjioe and Heller2007) or from spherical harmonics (multipole expansion) envelopes that cover the entire model as implemented in CRYSOL (Svergun et al. Reference Svergun, Baraberato and Koch1995). The spherical harmonics procedures scale as N(L _max+1)², where the default values of L _max is typically in a range from 15 to 17 with different programs.

Although solute atoms dominate the scattering signal at small angles, the scattering from ordered solvent atoms must also be considered (reviewed in Perkins, Reference Perkins2001). For proteins, this ordered solvation layer corresponds to 0·3 g water per g protein and is 15% denser than bulk water (Merzel & Smith, Reference Merzel and Smith2002a) due to both geometric effects and changes in the water structure such as shorter oxygen–oxygen distances and increased water coordination numbers. Strongly bound water molecules are known to fill surface grooves and channels stabilizing their structures and smoothing the excluded volume (Kuhn et al. Reference Kuhn, Siani, Pique, Fisher, Getzoff and Tainer1992). Computationally, the hydration shell has been modeled by explicitly placing water molecules on the surface (Hubbard et al. Reference Hubbard, Hodgson and Doniach1988; Grossmann et al. Reference Grossmann, Abraham, Adman, Neu, Eady, Smith and Hasnain1993; Fujisawa et al. Reference Fujisawa, Uruga, Yamaizumi, Inoko, Nishimura and Ueki1994) or by surrounding the particle by a continuous envelope representing the solvation shell of 3 Å with a density that can differ both from bulk density and the solute (Svergun et al. Reference Svergun, Baraberato and Koch1995). Including the hydration shell improves the accuracy of the calculated scattering profiles; however, the contribution from the ordered solvent layer is several orders of magnitude lower than the scattering from the solute and the excluded volume.

Fitting experimental scattering at higher resolutions (q>0·4 Å⁻¹) is more problematic for spherical harmonic reconstructions, as they do not account for the internal structure of the scattering particles. Even globular proteins, such as glucose isomerase, typically require extreme values for adjustable parameters in CRYSOL to fit the high-resolution portion of the experimental curve (Fig. 8a). Calculation of scattering profiles by algorithms that explicitly include all atoms are more accurate but require more intensive computation (Merzel & Smith, Reference Merzel and Smith2002b). For example the program solX (Tiede & Zuo, Reference Tiede, Zuo, Aartsma and Matysik2007) shows good agreement throughout the q-range with measured scattering profiles using default settings and values even without a solvation layer. Fitting proteins with flexibility at high q values is more problematic, as exemplified by a truncated form of Cel5A (Fig. 8b), for which multiple conformations must be explicitly included (Section 4.4). Moreover, the theoretical calculation of higher resolution scattering (q>0·25 Å⁻¹) is influenced by atomic displacement factors (Section 2.2.8) in the atomic model (ADFs; Zhang et al. Reference Zhang, Thiyagarajan and Tiede2000a). The crystallographically refined ADFs may not be appropriate for molecules in solution; hence, proper calculation of higher resolution scattering curves remains an unsolved problem.

Fig. 8. A. Comparison of experimental scattering curves of glucose isomerase (black) with the scattering curve calculated from the atomic model using the program CRYSOL (Svergun et al. Reference Svergun, Baraberato and Koch1995) with the default parameters (red line; solvation shell contrast 0·03 e/ų, average atomic radius 1·61 Å, excluded volume 2·13×10⁵ Å³) or adjusted parameters (blue line; solvation shell contrast 0·005 e/ų, average atomic radius 1·4 Å, exclude volume 2·29×10⁵ Å³). Theoretical scattering calculated with program solX using all-atom methods is shown in green line (Tiede & Zuo, Reference Tiede, Zuo, Aartsma and Matysik2007). B. Comparison of the experimental scattering curves of truncated Cel5A (black) with the scattering curves calculated using CRYSOL with default parameters (red line; solvation shell contrast 0·03 e/ų, average atomic radius 1·61 Å, excluded volume 5·35×10⁴ Å³), CRYSOL with adjusted parameters (blue line; solvation shell contrast 0·01 e/ų, average atomic radius 1·8 Å, excluded volume 5·48×10⁴ Å³), or solX using all atom-methods (green line).

In many of the modeling schemes described in the following sections, the methods for manipulating the structures of interest are directly tied to the calculation engines for generating the theoretical scattering curves for the models. In some sense, this tight coupling and limited scripting abilities have introduced pratical limitations for altering modeling schemes. In X-ray crystallography, X-PLOR and its successor CNS (Brunger et al. Reference Brunger, Adams, Clore, DeLano, Gros, Grosse-Kuntsleve, Jiang, Kuszewski, Nilges, Pannu, Read, Rice, Simonson and Warren1998) were designed to be highly modifiable, which we believe is part of its success as becoming an important crystallographic refinement package. Thus, we are excited about the stated intent of the authors of ORNL_SAS (Tjioe & Heller, Reference Tjioe and Heller2007) to ensure that the scattering-curve calculation engine is easily scriptable for integration so that other programs can be readily integrated into SAXS modeling even if these programs were never written with X-ray scattering in mind.

4.2.2 Rigid-body modeling with SAXS data

When atomic resolution structures of individual domains are known, SAXS can be used to determine the relative orientation and placement of these domains in a complex by maximizing the agreement between the theoretical and experimentally observed scattering (reviewed in Wall et al. Reference Wall, Gallagher and Trewhella2000). In rigid-body modeling, the domains are considered static objects and only their relative orientations are changed. In general, building an assembly with M subunits has 6(M-1) fittable parameters, as one subunit can be fixed and M-1 subunits are mobile with three translations and three rotations each. The number of parameters can be reduced if symmetry is present, such as with homo-oligomers. Depending on the problem, rigid-body modeling with atomic models can reduce the dimensionality of the fitting problem and be a more economical and powerful way to use the information content in the SAXS curve.

Evaluation of each trial rotation and translation in the rigid-body search can be computationally expensive. A brute-force search scales as n ^6(M-1), where n is the number of steps searched for each parameter. Different simplifications, however, can be performed for rapid screening. An important simplification is the representation of rigid bodies in forms from which scattering profiles are easily computed. For example, individual components have been modeled using triaxial ellipsoids (Gallagher et al. Reference Gallagher, Callaghan, Zhao, Dalton and Trewhella1999), such as in the case of a heterodimeric cAMP-dependent protein kinase which was used as a basis for construction of an atomic model by energy minimization (Tung et al. Reference Tung, Walsh and Trewhella2002). Spherical harmonic representations calculated from atomic models have also been applied (Svergun et al. Reference Svergun, Baraberato and Koch1995) using the algorithm of Svergun and Stuhrmann (Svergun & Stuhrmann, Reference Svergun and Stuhrmann1991).

Several programs are freely available for rigid-body docking using these spherical harmonic envelopes of atomic domains (Table 2). Interactive docking can be performed using ASSA (Kozin et al. Reference Kozin, Volkov and Svergun1997) or MASSHA (Konarev et al. Reference Konarev, Petoukhov and Svergun2001); however, in many cases, being able to generate automated rigid-body modeling is preferable (Koch et al. Reference Koch, Vachette and Svergun2003; Petoukhov & Svergun, Reference Petoukhov and Svergun2005; Svergun, Reference Svergun2007). For two-component complexes, DIMFOM exhaustively searches rotation space by rolling one monomer over the other. For larger symmetric complexes that reduce the dimensionality of the search space, GLOBSYMM is appropriate. For complexes containing several subunits that may or may not be symmetrically related, a heuristic search algorithm has been implemented in the program SASREF. An advantage of SASREF is the ability to use additional constraints to incorporate other information about the system, such as known subunit interfaces (Mattinen et al. Reference Mattinen, Paakkonen, Ikonen, Craven, Drakenberg, Serimaa, Waltho and Annila2002; Grishaev et al. Reference Grishaev, Wu, Trewhella and Bax2005). Recently SASREF was used to simultaneously fit X-ray and neutron-scattering curves that included contrast variation data from selectively deuterated complexes (Gherardi et al. Reference Gherardi, Sandin, Petoukhov, Finch, Youles, Ofverstedt, Miguel, Blundell, Vande Woude, Skoglund and Svergun2006).

Table 2. Rigid-body modeling SAXS programs

With symmetry constraints, rigid-body modeling has been successful at constructing the dimeric complexes of hepatocyte growth factor and tyrosine kinase with p2 point-group symmetry (Gherardi et al. Reference Gherardi, Sandin, Petoukhov, Finch, Youles, Ofverstedt, Miguel, Blundell, Vande Woude, Skoglund and Svergun2006) or glyceraldehyde-3-phosphate dehydrogenase (Ferreira da Silva et al. Reference Ferreira da Silva, Pereira, Gales, Roessle, Svergun, Moradas-Ferreira and Damas2006) and glutamate synthase (Petoukhov et al. Reference Petoukhov, Svergun, Konarev, Ravasio, van den Heuvel, Curti and Vanoni2003) with p222 point-group symmetry. Even with the small number of parameters of rigid-body modeling, the small number of independent data values can lead to overfitting (Section 2.3.4). From a practical standpoint, many models should be generated independently and examined for uniqueness and tested by other biophysical techniques. For example, automated rigid-body modeling of bacterial release factor 1 (RF1) generated multiple solutions with similar fits to the experimental data (Vestergaard et al. Reference Vestergaard, Sanyal, Roessle, Mora, Buckingham, Kastrup, Gajhede, Svergun and Ehrenberg2005). These models were evaluated by SAXS analysis of additional deletion constructs and by cryo-EM studies.

Rigid-body modeling can also be combined with ab-initio modeling (Table 3). The unknown portions of the macromolecule may be modeled as in ab-initio methods (Section 4.3.1) where each amino acid is represented by ‘dummy residues’ (also called ‘beads’ or ‘big atoms’). The program BUNCH models full-length proteins as rigid-body domains linked by their termini to flexible chains or domains (Petoukhov & Svergun, Reference Petoukhov and Svergun2005) and determines a best-fit conformation through a simulated annealing protocol (e.g. Fig. 9). Flexible regions have also been modeled using a Monte Carlo dihedral angle sampling of hinge regions (Akiyama et al. Reference Akiyama, Fujisawa, Ishimori, Morishima and Aono2004) and an ‘automated constrained fit’ procedure generates thousands of possible models by applying MD on the linker region in exhaustive search of the best-fit conformation (Boehm et al. Reference Boehm, Woof, Kerr and Perkins1999) applied to a number of human complement regulating proteins (Aslam & Perkins, Reference Aslam and Perkins2001; Aslam et al. Reference Aslam, Guthridge, Hack, Quigg, Holers and Perkins2003; Sun et al. Reference Sun, Reid and Perkins2004; Gilbert et al. Reference Gilbert, Eaton, Hannan, Holers and Perkins2005, Reference Gilbert, Asokan, Holers and Perkins2006) and the cellulosome (Hammel et al. Reference Hammel, Fierobe, Czjzek, Kurkal, Smith, Bayer, Finet and Receveur-Brechot2005). Frequently, the simulation is performed on the linker regions at very high temperature (~1500 K) to prevent the molecule from becoming trapped in a local minimum (Leach, Reference Leach and Hall2001). Different conformations of the protein were produced at regular intervals along the trajectory of subsequent calculations of the theoretical SAXS profiles. The comparison of the MD-generated configurations with the experimental data enabled discrimination of a finite number of structures with the best fit and with R _G closest to the experimental values (Hammel et al. Reference Hammel, Fierobe, Czjzek, Kurkal, Smith, Bayer, Finet and Receveur-Brechot2005). This modeling, however, is likely to be inappropriate for macromolecules with substantial flexibility that must be treated as conformational ensembles lacking a ‘best-fit’ structure (Section 4.4).

Fig. 9. Experimental SAXS curves (black circles) and scattering profiles computed from the models of a multidomain protein reconstructed by the rigid-body modeling using BUNCH (red line; Petoukhov & Svergun, Reference Petoukhov and Svergun2005) and ab-initio model using the program GASBOR (green line; Svergun et al. Reference Svergun, Petoukhov and Koch2001a). The secondary structural elements of the known atomic structures of the single modules are shown by a multicolor ball representation (yellow, gold, gray, red). Restored linker conformations between the modules are displayed by a blue beads representation and varied from run to run. Single best-fit models did not fit the smallest q region (q<0·03 Å⁻¹) well, but fits involving multiple models did. The position of the unknown structure of the small domain has been modeled as the globular region at the position correlated with its position in the sequence (marked with the arrow). One GASBOR model is shown as gold beads superposed on the average shape shown as light blue spheres.

Table 3. Reconstruction of missing components

4.2.3 Computational docking combined with SAXS data

Atomic structures have long held the promise of allowing the structures of unknown complexes to be determined computationally. Substantial effort has been directed towards using only chemical and geometric parameters to determine docking sites (Mandell et al. Reference Mandell, Roberts, Pique, Kotlovyi, Mitchell, Nelson, Tsigelny and Ten Eyck2001; Bonvin, Reference Bonvin2006); however, this problem has not proven to be simple. Computational docking has two major challenges: (1) dealing with flexibility in macromolecules in a computationally practical manner and (2) determining an appropriate scoring function that successfully distinguishes correct solutions. A number of different strategies have been developed to introduce flexibility. Protocols allowing some degree of interpenetration of the molecules (‘soft docking’), side-chain flexibility, and truncating surface side-chains have been developed (Schnecke et al. Reference Schnecke, Swanson, Getzoff, Tainer and Kuhn1998; Palma et al. Reference Palma, Krippahl, Wampler and Moura2000; Chen et al. Reference Chen, Li and Weng2003; Heifetz & Eisenstein, Reference Heifetz and Eisenstein2003; Li et al. Reference Li, Chen and Weng2003). Additionally, docking has also employed ensembles of conformation (Smith et al. Reference Smith, Sternberg and Bates2005) from NMR structure determination (Dominguez et al. Reference Dominguez, Bonvin, Winkler, van Schaik, Timmers and Boelens2004), MD simulations (Rajamani et al. Reference Rajamani, Thiel, Vajda and Camacho2004), NMA (Mustard & Ritchie, Reference Mustard and Ritchie2005), or residual dipolar coupling (Tai, Reference Tai2004). A number of web servers generate conformational ensembles specifically for docking (Lei et al. Reference Lei, Zavodszky, Kuhn and Thorpe2004; Suhre & Sanejouand, Reference Suhre and Sanejouand2004; Barrett & Noble, Reference Barrett and Noble2005). Regardless of the protocol involved, an energy minimization is frequently run for top solutions to resolve steric clashes at the interface (Li et al. Reference Li, Chen and Weng2003).

SAXS can provide an experimental target function to score hits from the docking search and can substantially aid in the problem of development of scoring functions that successfully distinguish between good and bad solutions. For example, this combination has been used to show that the histone-like domain of the Ras activator son of sevenless folds onto a helical linker between two other son-of-sevenless domains (SOS; Sondermann et al. Reference Sondermann, Nagar, Bar-Sagi and Kuriyan2005). In this work, the docking of the SOS^DH-PH-cat and SOS^Histone crystal structures was performed separately from comparisons to SAXS results. Of the top 40 docking solutions, the top-scoring solution was also the best fit to the SAXS. Computation docking was also used to generate 100 dimeric structures of purine nucleoside phosphorylase and to verify that the crystallographic trimer was the assembly state in solution (Filgueira de Azevedo et al. Reference Filgueira de Azevedo, dos Santos, dos Santos, Olivieri, Canduri, Silva, Basso, Renard, da Fonseca, Mendes, Palma and Santos2003). We anticipate that computational docking of the supramolecular complexes and validation of models by SAXS will become a particularly powerful combination.

4.2.4 Differences in crystallographic and solution structures

The ability to directly compare atomic models to SAXS curves is a remarkably useful tool for deciphering the influences that the crystal lattice has on the observed structure. The crystal structure is likely the lowest energy state within the lattice and under crystallization conditions; however, it not necessarily the lowest energy state in solution. Many studies have suggested that the effects of the crystal lattice do not alter the folding of domains, but rather influence the conformations adopted by flexible termini or linkers between domains. Moreover, proteins that undergo allosteric rearrangements and have multiple stable conformations separated by small energy differences can be affected by lattice forces. The classic example for incompatibility between crystal lattices and conformational states is the cracking of reduced deoxyhemoglobin crystals upon the addition of oxygen (Perutz et al. Reference Perutz, Bolton, Diamond, Muirhead and Watson1964). By their very nature, crystal structures tend not to reveal the flexibility of the crystallized macromolecules. Crystal structures, such as for the c-Abl protein kinase (PDB id 1opl; Nagar et al. Reference Nagar, Hantschel, Young, Scheffzek, Veach, Bornmann, Clarkson, Superti-Furga and Kuriyan2003) that has two molecules in the asymmetric unit with substantially different domain arrangements tend to be the exceptions, so we favor continued research to optimize the combination of computational searches and SAXS data to characterize solution conformations.

A naive assumption is that crystallization will mostly force macromolecules to adopt a more compact structure. SAXS has revealed that crystallographically induced compaction is likely to be frequent, such as with the ligand-bound ‘relaxed’ state (but not the unliganded ‘tense’ state) of aspartate transcarbamoylase. These differences, however, could be modeled by small rigid-body subunit rotations on the order of 10 degrees (reviewed in Koch et al. Reference Koch, Vachette and Svergun2003). Despite these sorts of results, compaction by the lattice is not the only possible effect. For example, in a study of four thiamine diphosphate-dependent enzymes, one was found to be identical in the crystal and in solution, two appeared to be less compact in solution, and one complex with weaker subunit interactions was clearly more compact in solution (Svergun et al. Reference Svergun, Petoukhov, Koch and Koenig2000).

The differences between solution and crystal structures can be important for deciphering biological mechanisms. For example, the single conformation of the dimeric bacterial DNA mispair recognition protein MutS is likely constrained by lattice forces, despite the fact that multiple states are expected from biochemical results. MutS recognizes mispairs in DNA when in a nucleotide-free or ADP-bound state. Binding of ATP by bacterial or eukaryotic versions of the DNA–MutS complex induces a state in which the protein acts like a rigid ring that can freely diffuse along the length of the DNA and is no longer restricted to the mispair (Mendillo et al. Reference Mendillo, Mazur and Kolodner2005). Crystal structures of MutS–DNA complexes with ATP or ATP analogs have yet to reveal the conformational changes anticipated from biochemical studies (Lamers et al. Reference Lamers, Georgijevic, Lebbink, Winterwerp, Agianian, de Wind and Sixma2004). These predicted domain motions that would allow the two halves of the composite ATP-binding pockets to engage with their bound nucleotide have been anticipated by the effects of dominant mutations in the eukaryotic homologs (Hess et al. Reference Hess, Gupta and Kolodner2002) and studies in related systems for which both states are known (Hopfner et al. Reference Hopfner, Karcher, Shin, Craig, Arthur, Carney and Tainer2000). Regardless of whether or not these conformational changes are incompatible with the crystal lattice or due to the concentration effects of the trapped DNA, attempts to decipher these conformational changes have not yet been successful through crystallography alone. Properly assessing these conformational changes in the case of the efficient DNA repair machinery is important as toxins that interfere with the repair machinery are expected to be more dangerous to public health than toxins that directly damage DNA (McMurray & Tainer, Reference McMurray and Tainer2003).

4.2.5 Determining biological assemblies in crystal structures

Crystallization typically requires the formation of symmetric contacts that exist solely in the crystal and may not be important in biology. However biologically relevant multimers can also be symmetric, and biologically relevant symmetry operators can also be part of the underlying symmetry of the crystal structure so the contents of the asymmetric unit cannot be used as a guide for the biological assembly. Thus, crystallographers must decipher which macromolecular contacts mediate multimerization in solution and which are only crystal contacts.

In many cases identifying the biological multimer can be a difficult problem. Systematic investigation of atomic resolution structures has shown that authentic interfaces tend to be large and involve hydrophobic interactions on the surfaces of the protein (Jones & Thornton, Reference Jones and Thornton1996) and tend to be better packed than crystal contacts (Li et al. Reference Li, Keskin, Ma, Nussinov and Liang1985; Getzoff et al. Reference Getzoff, Tainer and Olson1986; Bahadur et al. Reference Bahadur, Chakrabarti, Rodier and Janin2004; Keskin et al. Reference Keskin, Ma and Nussinov2005). Additionally, these interfaces are less hydrated (Rodier et al. Reference Rodier, Bahadur, Chakrabarti and Janin2005) and tend to have greater sequence conservation (Valdar & Thornton, Reference Valdar and Thornton2001a, Reference Valdar and Thorntonb) with frequent tryptophans, phenylalanines, and methionines (Ma et al. Reference Ma, Elkayam, Wolfson and Nussinov2003). These analyses, although useful in the absence of other information, are theoretical in nature. SAXS data collected on the biologically relevant assemblies in solution has the potential to provide experimental information to help test and identify the biologically relevant interfaces in the crystal.

Identifying biological dimers is particularly difficult, as crystallographic symmetries tend to generate many dimeric interactions. For example, the bacterial mismatch repair protein MutL is known to form functional dimers. MutL is comprised of an N-terminal ATPase domain, a flexible internal linker, and a C-terminal domain that mediates constitutive dimerization (Drotschmann et al. Reference Drotschmann, Aronshtam, Fritz and Marinus1998). The dimeric C-terminal domain crystallized in the space group P4₃22 with four different dimer interfaces (PDB id 1x9z; Guarne et al. Reference Guarne, Ramon-Maiques, Wolff, Ghirlando, Hu, Miller and Yang2004). Two interfaces have remarkably small buried surface areas, but two others were more sizable. The dimer interface proposed by Guarne et al. later proved to be incorrect, whereas the other large dimer interface was consistent with crosslinking experiments and identifiable by computations analysis of packing (Kosinski et al. Reference Kosinski, Steindorf, Bujnicki, Giron-Monzon and Friedhoff2005). The biologically relevant dimer is more extended than the dimers generated by crystal contacts, and theoretical calculations suggest that SAXS could have easily distinguished it by size (R _G=34·1 Å, D _max=120 Å vs. R _G=28·1 Å, D _max=80 Å) as well as by the P(r) function that is characteristic of an elongated shape, having a peak at short distances and a long extended tail (R _G=28·1 Å, D _max=80 Å) (Fig. 10).

Fig. 10. SAXS could have readily distinguished between alternative dimer structures of the C-terminus of MutL observed in the crystal structure (PDB id 1x9z; Guarne et al. Reference Guarne, Ramon-Maiques, Wolff, Ghirlando, Hu, Miller and Yang2004). (a) Each of the four different dimers has remarkably different overall shapes, giving rise to measurable differences in SAXS and parameters such as R _G and D _max. Dimer 1, with a buried surface area of the monomer of 755 Å², is the asymmetric unit of the crystal, whereas dimer 2, with a buried surface area of 923 Å², is the solution dimer assembly (Kosinski et al. Reference Kosinski, Steindorf, Bujnicki, Giron-Monzon and Friedhoff2005). (b) Theoretical P(r) functions calculated for dimer 1 (black) and dimer 2 (red) are readily distinguished. Dimer 1 has a characteristic globular P(r) that is bell-shaped, whereas dimer 2 has a characteristic extended P(r) with an early peak and a long tail.

A more insidious challenge is when the authentic multimer does not exist in the crystal at all. Generating a solution assembly model given only the high resolution structure of a subunit is possible with SAXS data, and modeling assemblies with SAXS data will be discussed below. But before modeling can be performed, it is critical to identify that the biologically relevant assembly is not present in the crystal and to recognize that modeling must be done in the first place. In these cases, the lack of agreement between SAXS data collected on multimers in solution and theoretical scattering curves calculated from assembles observed in the crystals could be an important clue. For example, the hexameric assembly for the Holliday junction ATPase motor RuvB could be characterized by SAXS results in combination with the crystal structure of the subunit from a non-biologically relevant screw assembly (Putnam et al. Reference Putnam, Clancy, Tsuruta, Gonzalez, Wetmur and Tainer2001).

One particularly difficult problem was identifying the biological dimer of ATP-binding cassette (ABC) transporters (Fig. 11). ABC transporters couple ATP hydrolysis with the transport of ligands across membranes and include the cystic fibrosis transmembrane regulator. The nucleotide-binding domains of the ABC ATPase transporters were known to dimerize. The first of these structures, the Salmonella typhimurium histidine permease HisP was crystallized in the space group P4₃2₁2 with two symmetry-related dimers in the crystal packing (PDB id 1b0u; Hung et al. Reference Hung, Wang, Nikaido, Liu, Ames and Kim1998); the largest dimer interface was proposed to be the biologically relevant one. Other transporter ABC ATPases were solved later, including the Thermococcus litoralis maltose transporter MalK (PDB id 1g29; Diederichs et al. Reference Diederichs, Diez, Greller, Mueller, Breed, Schnell, Vornrhein, Boos and Welte2000), and the Methanococcus jannaschii ABC ATPases MJ0796 (PDB id 1f3o; Yuan et al. Reference Yuan, Blecker, Martsinkevich, Millen, Thomas and Hunt2001) and MJ1267 (PDB id 1g6h and 1gaj, Karpowich et al. Reference Karpowich, Martsinkevich, Millen, Yuan, Dia, MacVey, Thomas and Hunt2001). Remarkably, each of these ABC ATPases had dimers generated by crystallographic symmetry, yet none shared common dimer interfaces. The biologically relevant dimer was first proposed using sequence analysis and the structure of the HisP monomer (Jones & George, Reference Jones and George1999). Dimerization generates an ‘ATP sandwich’ with the active site being made up of a Walker A ATP-binding and hydrolysis site on one molecule and an ABC ATPase ‘signature’ motif from the other subunit. These interfaces were first observed crystallographically in the ABC ATPase domains from the DNA repair proteins Rad50 (PDB id 1f2u; Hopfner et al. Reference Hopfner, Karcher, Shin, Craig, Arthur, Carney and Tainer2000) and MutS (PDB id 1e3m; Lamers et al. Reference Lamers, Perrakis, Enzlin, Winterwerp, de Wind and Sixma2000; PDB id 1ewq; Obmolova et al. Reference Obmolova, Ban, Hsieh and Yang2000), and the Rad50 structure in particular was subsequently used for modeling of the ABC transporters. The inference that the membrane transporters shared the Rad50/MutS dimer assembly was later established by the crystal structure of the E. coli BtuCD heterotetramer (PDB id 1l7v; Locher et al. Reference Locher, Lee and Rees2002).

Fig. 11. SAXS has the potential to indicate mismatches between the dimers in the ABC ATPase assemblies observed in crystal and with their solution conformation, if dimeric states could have been stabilized for solution scattering. (a) The biological dimeric assemblies of the ABC ATPases were first observed with the DNA repair proteins Rad50 (PDB id 1f2u; Hopfner et al. Reference Hopfner, Karcher, Shin, Craig, Arthur, Carney and Tainer2000) and MutS (PDB id 1e3m, Lamers et al. Reference Lamers, Perrakis, Enzlin, Winterwerp, de Wind and Sixma2000; PDB id 1ewq, Obmolova et al. Reference Obmolova, Ban, Hsieh and Yang2000) that were normally stable as dimers in solution. Subunits are displayed as yellow and red ribbons; bound nucleotides are orange ball-and-stick models, and blue ribbons indicate the position of the signature motif that forms the second half of the ATP-binding surface. (b) Different crystallographic dimers were observed for HisP, MalK, MJ0796, and MJ1267 [PDB ids 1b0u, 1g29, 1f3o, 1g6h (Hung et al. Reference Hung, Wang, Nikaido, Liu, Ames and Kim1998; Diederichs et al. Reference Diederichs, Diez, Greller, Mueller, Breed, Schnell, Vornrhein, Boos and Welte2000; Karpowich et al. Reference Karpowich, Martsinkevich, Millen, Yuan, Dia, MacVey, Thomas and Hunt2001; Yuan et al. Reference Yuan, Blecker, Martsinkevich, Millen, Thomas and Hunt2001)] that lack special positioning of the signature motifs. Theoretical P(r) scattering curves clearly distinguish between the crystallographically observed dimers and those modeled using the Rad50 assembly.

SAXS can be used to distinguish between alternative assemblies in different crystal structures. The bacterial HslUV chaperone/protease complex had been observed in two distinct crystallographic assemblies (Fig. 12; Bochtler et al. Reference Bochtler, Hartmann, Song, Bourenkov, Bartunik and Huber2000; Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000). In the center of both assemblies was an HslV hexamer, but in the two different crystal forms the globular N- and C-terminal domains of HslU were either packed against HslV (PDB id 1g3i; Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000) or held away from HslV by an extended internal domain (PDB id 1doo; Bochtler et al. Reference Bochtler, Hartmann, Song, Bourenkov, Bartunik and Huber2000). The P(r) functions for these different assemblies and hence their X-ray scattering were easily distinguishable and SAXS revealed that in solution the globular N- and C-terminal domains of HslU pack against HslV and that the extended HslU domain extends into solvent (Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000).

Fig. 12. The two different HslUV protease/chaperone assemblies observed crystallographically have different orientations of the HslU hexamers (green) that interact on both sides of the double-ringed HslV dodecamer (gold). In the first, the globular domains of HslU are separated from HslV, giving a bimodal P(r) function with a R _G of 90 Å and a D _max of 265 Å (PDB id 1doo; Bochtler et al. Reference Bochtler, Hartmann, Song, Bourenkov, Bartunik and Huber2000). In the second, the globular domains of HslU pack against HslV, giving a monomodal P(r) function with a R _G of 65 Å and a D _max of 220 Å (PDB id 1g3i; Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000). The experimental P(r) function (not shown; Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000) strongly argues for a monomodal distribution and the more compact 1g3i-like assembly (after Sousa et al. Reference Sousa, Trame, Tsuruta, Wilbanks, Reddy and McKay2000).

Finally, SAXS can establish the validity of weak assemblies observed in crystal packing that might be ignored as artifacts of the crystallization process. The E. coli mismatch repair protein MutS forms a dimer that recognizes mispairs in DNA. At high protein concentrations, MutS forms tetramers that depend upon the presence of the C-terminal 53 amino acids (Bjornson et al. Reference Bjornson, Blackwell, Sage, Baitinger, Allen and Modrich2003). Fusions of this C-terminal domain onto maltose binding protein (MBP) mediated its tetramerization, and the crystal structure revealed a helix–loop–helix domain that made a symmetric interaction to form a tight dimer (PDB id 2ok2; Mendillo et al. Reference Mendillo, Putnam and Kolodner2007). A potential tetramer contact was observed in the crystal form; however, the interface was small and weakly packed. Despite this, SAXS data collected on the MBP fusion protein and subsequent mutagenesis of salt bridges that stabilized it revealed that the tetramer in solution closely resembled the assembly observed crystallographically. Similarly, a combination of SAXS, ultracentrifugation, and EM was used to validate the biologically relevant Mre11₂/Rad50₂ heterotetrameric DNA processing head used in double-strand break repair, whose structure was inferred from the atomic resolution structures of Mre11, Rad50, and biochemical results (Hopfner et al. Reference Hopfner, Karcher, Craig, Woo, Carney and Tainer2001).

4.3.1 Calculation of ab-initio SAXS envelopes

Several programs have been created for the purpose of calculating so called ab-initio shapes from scattering profiles (Table 4). In practice many of these programs use information external to the scattering profiles and the term ab initio solely refers to lack of a pre-defined input structure. These programs implicitly or explicitly assume the shape is a continuous object, which substantially reduces the search space. We do not attempt to provide detailed descriptions of these programs here and the citations listed in Table 4 should be referenced. The general approach taken by these programs is to propose shapes, calculate scattering curves or P(r) functions and optimize the agreement to the experimental data (Fig. 13).

Table 4. Ab-initio SAXS envelope reconstruction programs

Many of the early programs restricted searches to shapes defined by a small number of parameters. For example, R _G can be thought of as describing an ab-initio model with a single parameter: a spherical envelope of radius r=(5/3)^½R _G. Whole-body methods approach the problem by attempting to fit X-ray scattering using spheres, oblate, prolate, and triaxial ellipsoids and are also used for modeling hydrodynamic properties (Rallison & Harding, Reference Rallison and Harding1985). A modern version of this algorithm that can also combine several simple shapes is available with the program ELLSTAT (Heller, Reference Heller2006). Spherical harmonics also have been used (Svergun & Stuhrmann, Reference Svergun and Stuhrmann1991) as implemented in SASHA (Fig. 13a; Svergun et al. Reference Svergun, Volkov, Kozin and Stuhrmann1996). However, spherical harmonics descriptions cannot properly represent all shapes, such as structures with cavities or holes. In the specialized case of icosohedrally symmetric virus particles, icosohedral harmonics have also been employed to describe low-resolution structures (Zheng et al. Reference Zheng, Doerschuk and Johnson1995).

Recently, a number of over-parameterized methods have been used for ab-initio shape determination (Volkov & Svergun, Reference Volkov and Svergun2003). These methods attempt to generate a bead model (or dummy atom model) to fill a volume consistent with the experimental scattering. External constraints, such as smoothness, connectivity, and particle symmetry have been used to reduce the search space (Koch et al. Reference Koch, Vachette and Svergun2003). The number of possible arrangements increases combinatorially, so all programs use computational tricks to search through the solution space including Monte Carlo approaches, genetic algorithms, and simulated annealing (Table 4). Several programs fit directly to the scattering data involving a Fourier transform for each shape proposed. This becomes computationally intensive and cheap methods of calculating scattering curves from proposed shapes have been developed (Section 4.1).

Shape-restoration programs using bead models do not uniquely define the position of each bead within a volume, nor do the beads represent positions of specific residues. Rather, the bead positions are non-unique and define a volume for the scattering particle, as illustrated by six independent GASBOR reconstructions of the human OGG1 DNA glycosylase/lyase (Fig. 14). Each of the reconstructions fits the experimental scattering equally well and describes similar shapes. GASBOR is designed specifically for proteins (though modeling of other macromolecules is possible); dummy atoms have radii approximating amino acids with penalties imposing chain connectivity. In the case of OGG1, the envelope generated by averaging the different GASBOR runs fits the volume and shapes the truncated crystal structure (Fig. 14).

Fig. 14. GASBOR reconstructions of OGG1, a 39 kDa protein critical to the recognition and repair of oxidized guanine in double-stranded DNA by the DNA base-excision repair pathway. (Protein courtesy of Tapas Hazra and Sankar Mitra University of Texas, Galveston, reconstructions for functional analyses in collaboration with Cynthia McMurray.) (a) Six independent runs of GASBOR showing the variation among models all of which produce scattering curves in agreement with the experimental data. Each model is composed of dummy atoms whose radii approximate those of amino acids. Data were collected at a concentration of 5 mg/ml with a 30-s exposure. (b) The crystal structure of a truncated version of OGG1 fits well within the ab-initio SAXS envelope defined by the average of the six GASBOR runs. Each run takes around two hours on computers typically found in most laboratories.

The final resolution attainable from ab-initio envelopes varies with data quality, the program used, particle size, shape and flexibility (Section 5.6). Several studies have compared available programs (Takahashi et al. Reference Takahashi, Nishikawa and Fujisawa2003; Zipper & Durschschlag, Reference Zipper and Durschschlag2003); however, a systematic study on the accuracy of ab-initio envelopes has not been conducted. Figure 6 shows 10 ab-initio reconstructions and the average shape from the highly flexible detergent solubilized apoB-100 generated by DAMMIN. A larger variation exists between the models in comparison to OGG1 (Fig. 14); although, the overall dimensions are in agreement and suggest relative domain motions. Very flexible structures typically have fewer features in their scattering profiles, which can be fit by a greater variety of shapes. A comparison of the output of multiple independent modeling runs (at least 6–10) provides some measure of the uniqueness of the models.

One parameter used to characterize the agreement among models is the normalized spatial discrepancy (NSD) (Kozin & Svergun, Reference Kozin and Svergun2001). Briefly, if models 1 and 2 are expressed in two sets of points P ₁={p _1i, i=1, …, N ₁} and P ₂={p _2i, i=1, …, N ₂}, then, NSD between P ₁ and P ₂ is defined as

where N _i is the nuber of points in P _i, d is the average distance between neighboring points in P _i, and ρ(p _1i, P ₂) is the distance from arbitrary points, p _1i, in P ₁ to the nearest point in P ₂. NSD is 0 for identical models. NSD enables quantitative comparison of similarities of models if the modes have the same resolutions, but it is not straightforward to compare similarities of models with different resolutions because N _i and d are very different. In these cases, using the program SITUS, which was originally developed for EM, is more appropriate (Wriggers et al. Reference Wriggers, Milligan and McCammon1999; Wriggers & Chacon, Reference Wriggers and Chacon2001).

The combination of multiple runs and external constraints has been remarkably successful for the low-resolution envelope reconstruction of many systems (Fig. 15). Imposing correct symmetry greatly enhances the resolution of final results. In general, symmetry information will need to be introduced from external information. However, in the case of the cytosolic portion of a voltage-gated potassium channel (Pioletti et al. Reference Pioletti, Findeisen, Hura and Minor2006), the fourfold symmetry could be derived from the scattering using GASBOR, as shown by the result of imposing various symmetries (Fig. 16). Each of the five symmetries shown was run eight times resulting in a total of 40 GASBOR runs. Each run required 12 h of computer time for the 1288 amino-acid complex. In this case, the overall shape could be reconstructed even when fourfold symmetry was not explicitly enforced. Imposed symmetries below and including fourfold-derived shapes with similar χ² agreement to the experimental data. Imposing higher symmetries resulted in noticeably higher χ² values and marginally poorer fits to the data. On the other hand symmetries of fourfold and higher had low NSD values, implying a much smaller variation in shapes. An imposed fourfold symmetry optimized both the χ² agreement and NSD values (Fig. 16f). We note that the mass was a critical parameter required as input for this study, and the ability to determine overall symmetry by SAXS will likely depend greatly on the overall shape of the target. In this case, the dramatic X-shape of the potassium channel, which is apparent even at extremely low resolutions, was likely critical for determining the proper symmetry.

Fig. 15. Crystallographic and SAXS based models of DNA ligase-PCNA complex. (a) A model of human DNA-ligase-PCNA postulated from two independent crystal structures: human PCNA (purple surface PDB id 1w69; Kontopidis et al. Reference Kontopidis, Wu, Zheleva, Taylor, McInnes, Lane, Fischer and Walkinshaw2005) and human ligase I (green) complexed with nicked DNA (orange, PDB id 1x9n; Pascal et al. Reference Pascal, O’Brien, Tomkinson and Ellenberger2004). The model assumes the two ring-like structures are co-axial around DNA. B. The averaged ab-initio SAXS model derived from experimental SAXS data of the Sulfolobus solfataricus ligase-PCNA complex without DNA (purple transparent surface) superposed on the complex reconstructed from the ligase (PDB id 2hiv) and PCNA (PDB id 2hii) models using SASREF (Petoukhov & Svergun, Reference Petoukhov and Svergun2005). Ab-initio shapes were calculated by GASBOR (Svergun et al. Reference Svergun, Petoukhov and Koch2001a) and average using DAMAVER (Volkov & Svergun, Reference Volkov and Svergun2003) (after Pascal et al. Reference Pascal, Tsodikov, Hura, Song, Cotner, Classen, Tomkinson, Tainer and Ellenberger2006).

Fig. 16. SAXS envelopes calculated with different point-group symmetries determined from experimental data from the cytosolic portion of a voltage-gated potassium channel (a tetramer generated from two heterodimers) and overlaid on the crystal structure (Pioletti et al. Reference Pioletti, Findeisen, Hura and Minor2006). Each of the envelopes is the result of averaging eight GASBOR (Svergun et al. Reference Svergun, Petoukhov and Koch2001a) runs and is shown in two orientations, 90° apart. Models were calculated with the point group symmetries of (a) p1 (no symmetry), (b) p2 (one twofold), (c) p222 (three perpendicular twofolds), (d) p4 (one fourfold), and (e) p8 (one eightfold). For each applied symmetry indicated by the value in the circle, the discrepancy between the individual models (NSD; Section 4.3.1) is plotted against the average agreement with the experimental scattering (χ²; Section 4.1). The model generated with fourfold symmetry has nearly equivalent χ² with lower symmetry models but also has low NSD values. All models suggest a fourfold symmetric structure, excepting the model forced to have p8 point-group symmetry.

One of the temptations of ab-initio programs is that they require much less information and are easier to run. However, atomic resolution information provides substantially more restraints on possible solutions and is in our opinion the future of SAXS computational development. Ab-initio derived models can support models built with or proposed by other methods. Moreover, ab-initio structures can be used directly to build atomic models through rigid-body (Section 4.3.2) or flexible (Section 4.3.3) docking into the envelopes.

4.3.2 Rigid-body docking into low-resolution envelopes

A candidate ab-initio SAXS structure is essentially a low-resolution envelope to which an atomic structure may be docked. Finding a manual best fit using specifically developed SAXS software (Kozin et al. Reference Kozin, Volkov and Svergun1997; Konarev et al. Reference Konarev, Petoukhov and Svergun2001) or with standard molecular graphics programs using envelopes expressed as atomic positions can be a surprisingly challenging process. The program SUPCOMB (Kozin & Svergun, Reference Kozin and Svergun2001) automatically superimposes atomic structures with dummy atom models; however, this problem is essentially the same as that faced when fitting high-resolution structures into EM maps. Substantial progress has been made in the EM community in this area (Volkmann & Hanein, Reference Volkmann and Hanein1999; Roseman, Reference Roseman2000; Rossmann, Reference Rossmann2000; Wriggers & Chacon, Reference Wriggers and Chacon2001; Chacon & Wriggers, Reference Chacon and Wriggers2002; Navaza et al. Reference Navaza, Lepault, Rey, Alvarez-Rua and Borge2002; Craig et al. Reference Craig, Volkmann, Arvai, Pique, Yeager, Egelman and Tainer2006), and the EM program SITUS (Wriggers et al. Reference Wriggers, Milligan and McCammon1999) has also been used for SAXS (Rosenberg et al. Reference Rosenberg, Deindl, Sung, Nairn and Kuriyan2005). In SITUS the distribution of atoms within the high-resolution structure as well as the low-resolution reconstructions are approximated by a small number of vectors that are calculated by vector quantization. Vector quantization-based fitting is limited to cases in which all density in the SAXS envelope is accounted for by the atomic model. In practice missing or disordered regions of the atomic model need to be modeled before vector quantization-based fitting can be applied (Volkmann & Hanein, Reference Volkmann and Hanein2003). Additionally, more accurate correlation-based approaches have been proposed. In these methods the solution sets lead to the possibility of defining confidence intervals and error margins for the fitting parameters, which is particularly important in the context of docking atomic structures into low-resolution density maps as no independent information is available on how a correct fit should look (Volkmann & Hanein, Reference Volkmann and Hanein2003).

The ability to align crystal structures into SAXS envelopes is particularly informative when there are functionally related conformational changes. For example, both symmetry and the existence of atomic resolution structures combined with SAXS analysis have provided substantial information regarding the mechanism by which the AAA+-ATPase p97 undergoes ATP-coupled conformational changes. Ab-initio reconstruction of SAXS envelopes for this hexameric ATPase with p3 or p6 point-group symmetry provided substantial information on conformational changes induced upon nucleotide binding (Davies et al. Reference Davies, Tsuruta, May and Weis2005). States induced by the non-hydrolysable analog AMP-PNP, the transition-state like mimic ADP-AlF_x, ADP, and no nucleotide were readily distinguishable in GASBOR-generated envelopes and consistent with previous EM studies (Zhang et al. Reference Zhang, Shaw, Bates, Newman, Bowen, Orlova, Gorman, Kondo, Dokurno and Lally2000b; Rouiller et al. Reference Rouiller, DeLaBarre, May, Weis, Brunger, Milligan and Wilson-Kubalek2002; Beuron et al. Reference Beuron, Flynn, Ma, Kondo, Zhang and Freemont2003). Using the crystal structure of the ADP-AlF_x bound state, different conformations were proposed, revealing a coordinated mechanism for transferring the energy of nucleotide hydrolysis through two parallel rings of domains.

Modeling atomic resolution structures into SAXS envelopes has been used to study the conformational states of the ATPase domain of the Aquifex aeolicus enhancer-binding protein NtrC1 in various nucleotide-bound states. NtrC1, which is also an AAA+-ATPase, acts upon a quiescent σ⁵⁴–RNA polymerase complex to activate transcription initiation. Superposition of the ADP-bound crystal structure of the heptameric ATPase domain with low-resolution envelopes derived from SAXS with enforced sevenfold symmetry could be used to model the nucleotide control of different conformational states. In conjunction with EM, the authors demonstrated an ATP-bound state that stabilized the EBP–⁵⁴–RNA polymerase complex, while subsequent hydrolysis and phosphate release drove the conformational changes necessary to generate an open polymerase/promoter complex (Chen et al. Reference Chen, Doucleff, Wemmer, De Carlo, Huang, Nogales, Hoover, Kondrashkina, Guo and Nixon2007).

4.3.3 Flexible docking into low-resolution envelopes

In addition to rigid-body fitting of domains into low-resolution envelopes, ab-initio SAXS envelopes, like experimental EM maps, may correspond to states that are different from the crystallized form of the macromolecule. Hence, computational techniques that can take single conformational states and account for potential flexibility during the docking protocols become quite useful. This problem has been faced frequently in EM studies and a variety of computational methods are currently being applied (Suhre et al. Reference Suhre, Navaza and Sanejouand2006). Large conformational changes have been shown to frequently correspond to highly collective movements that can be described by a small number of low-frequency normal modes of protein (Section 3.3; Tama & Sanejouand, Reference Tama and Sanejouand2001), and structures modified by NMA (Section 3.3) have been used for fitting EM density maps (Tama et al. Reference Tama, Miyashita and Brooks2004; Hinsen et al. Reference Hinsen, Reuter, Navaza, Stokes and Lacapere2005; Mitra et al. Reference Mitra, Schaffitzel, Shaikh, Tama, Jenni, Brooks, Ban and Frank2005). For example, the NORMA software package (Suhre et al. Reference Suhre, Navaza and Sanejouand2006) allows flexible fitting to EM maps and is well suited for SAXS envelopes; however, the SAXS envelopes generally need to be converted into synthetic EM envelopes with SITUS first (Wriggers et al. Reference Wriggers, Milligan and McCammon1999).

The elastic normal modes computed based on the low-resolution envelopes compare well with the normal modes obtained at atomic resolution (Tama et al. Reference Tama, Wriggers and Brooks2002; Chacon et al. Reference Chacon, Tama and Wriggers2003). Thus the motions of large macromolecular assemblies can be directly extracted from low-resolution envelopes derived from SAXS or EM, as has been shown for the DNA-dependent protein kinase and pyruvate dehydrogenase (Boskovic et al. Reference Boskovic, Rivera-Calzada, Maman, Chacon, Willison, Pearl and Llorca2003; Kong et al. Reference Kong, Ming, Wu, Stoops, Zhou and Ma2003). NMA has also been used to study the motion in free and complexed cellulase built from SAXS models (Hammel et al. Reference Hammel, Fierobe, Czjzek, Finet and Receveur-Brechot2004a) when submitted to the ElNémo server (Suhre & Sanejouand, Reference Suhre and Sanejouand2004). Computational methods other than NMA could be used to deform molecules to fit into these low-resolution envelopes; however, NMA has proven to be quite useful due to the efficiency for which altered conformations can be calculated.

4.4.1 Multiple well-defined conformations

Assuming that different conformational states do not interact, the resulting scattering from a mixed sample is a population-weighted average of scattering from individual states. This scattering poses a significant challenge for all shape reconstruction techniques. Nevertheless, data determined from the consensus population in solution, under a variety of conditions can be advantageous. A concern with cryo-EM is that an observed minor population overly biases the determined shape. This was dramatically demonstrated by cryo-EM and SAXS DNA bound and free structures of the important DNA damage response protein p53, which is mutated in 50% of all human cancer (Tidow et al. Reference Tidow, Melero, Mylonas, Freund, Grossmann, Carazo, Svergun, Valle and Fersht2007).

In general, the characterization of these mixed states by SAXS is most straightforward if the different states can be isolated independently and treated as homogeneous samples. Driving the population to a single conformation might be induced by buffer conditions or through binding specific ligands or substrates. For example, the dodecameric Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) converts from a compact autoinhibited state to a loosely tethered state with independent kinase domains upon Ca²⁺ binding (Rosenberg et al. Reference Rosenberg, Deindl, Sung, Nairn and Kuriyan2005).

Alternatively, direct comparisons of different conformational states with theoretical scattering calculated from atomic-resolution structures has been quite successful in identifying and deconvoluting the relative fractions in the sample, such as with the archaeal secretion ATPase GspE (Fig. 17). The structure was solved as a mixed hexamer of open and closed conformational states of the component monomers (Yamagata & Tainer, Reference Yamagata and Tainer2007). Fitting of the solution X-ray scattering curves of the Mg²⁺ and AMP-PNP bound enzyme fit the experimental crystal structure less well than a computational hexamer generated from the closed state of the monomers alone. Moreover, the scattering of the ADP-bound state was well fit by a mixture of scattering from the crystal structure, the all-open model, and the all-closed model, demonstrating considerable flexibility in the system only using experimentally determined states of the monomer for modeling.

Fig. 17. Determination of the solution conformation of the hexameric archaeal secretion ATPase GspE from a combination of SAXS and crystallography. (a) Side view of the 2 alternating configurations of the ATPase GspE monomer bound to AMPPNP found in the hexameric structure determined via crystallography. (b) Top views of the crystal structure and two models proposed by modifying subunits to adopt either all open (brown) or all closed (green) conformations. (c) In solution the ATPase in excess AMPPNP adopts a conformation most similar to the all closed model while a solution with excess ADP is best described as a mixture of all the models (after Yamagata & Tainer, Reference Yamagata and Tainer2007).

Identifying that a solution contains a mixed population of macromolecules can be challenging and often requires additional information from other techniques such as native gel electrophoresis. Since R _G² corresponds to the average square distance of each scatterer from the center of the particle contributing to the scattering, mixed samples containing components with different R _G values will yield observed R _G's that are the square root of the weighted sum of the R _G² values of the different components. Thus, mixed populations continue to have linear Guinier regions (Heller, Reference Heller2005). Moreover, the signal indicating heterogeneity is highly dependent upon how much the conformational changes alter SAXS (Heller, Reference Heller2005). In the calculation of the collapse of calmodulin, for example, the 2·6 Å difference in R _G and 20 Å differences in D _max were readily apparent, even when simulated noise was added to the scattering curves. Ab-initio reconstructions of mixed populations show substantial conformational components of both extended and collapsed states. In contrast, the conformational changes involving protein kinase A, with a change in R _G of 0·13 Å, were lost when noise was added to the theoretical scattering (Heller, Reference Heller2005). Therefore, SAXS studies of mixed populations that give rise to large changes in SAXS will be the most straightforward. For example, scattering power in the small-angle region goes by the square of the mass; hence, mixed populations of different multimeric states will be more readily identified. Similarly, unfolding of macromolecules involves large-scale changes in the overall structure and will dramatically change the observed scattering; thus, SAXS has been a method of choice for studying protein folding (Provencher & Glockner, Reference Provencher and Glockner1983; Doniach, Reference Doniach2001; Perez et al. Reference Perez, Vachette, Russo, Desmadril and Durand2001).

The best way to identify heterogeneity is by following it through titration of the states from one form to another or through situations in which atomic models are available so that they can be directly compared to the observed scattering. For example, SAXS has been used to follow large-scale changes due the pH-induced maturation of viral capsids of the HK97 bacteriophage and Nudaurelia capensis ω virus, which give rise to substantial changes in the SAXS curve (Canady et al. Reference Canady, Tsuruta and Johnson2001; Lee et al. Reference Lee, Gan, Tsuruta, Hendrix, Duda and Johnson2004). Moreover, several schemes have been used to prove the existence of transient, substrate-induced conformational changes and characterize them (Akiyama et al. Reference Akiyama, Fujisawa, Ishimori, Morishima and Aono2004; Goettig et al. Reference Goettig, Brandstetter, Groll, Gohring, Konarev, Svergun, Huber and Kim2005; Graille et al. Reference Graille, Zhou, Receveur-Brechot, Collinet, Declerck and van Tileurgh2005; Vestergaard et al. Reference Vestergaard, Sanyal, Roessle, Mora, Buckingham, Kastrup, Gajhede, Svergun and Ehrenberg2005; Nowak et al. Reference Nowak, Panjikar, Konarev, Svergun and Tucker2006a, Reference Nowak, Panjikar, Morth, Jordanova, Svergun and Tuckerb).

Using experimentally determined SAXS and theoretical scattering from individual components (form factors), volume fractions in each conformation can be determined by solving a systems of linear equations with the program OLIGOMER (Konarev et al. Reference Konarev, Volkov, Sokolova, Koch and Svergun2003). The bacterial class I release factors, for example, adopt a compact structure in the crystal, but unlike the crystal, the solution scattering is consistent with a population containing 92·5% in an open conformation and only 7·5% in the compact form (Vestergaard et al. Reference Vestergaard, Sanyal, Roessle, Mora, Buckingham, Kastrup, Gajhede, Svergun and Ehrenberg2005). Similarly, the transcriptional antiterminator LicT exhibits a heterogeneous population in solution with 61% being open and 39% being compact and active (Graille et al. Reference Graille, Zhou, Receveur-Brechot, Collinet, Declerck and van Tileurgh2005).

In addition to systems of linear equations, single-value decomposition (SVD; Press et al. Reference Press, Teukolsky, Vetterling and Flannery1992) can deconvolute SAXS data from mixtures. SVD was introduced into SAXS in the early 1980s (Fowler et al. Reference Fowler, Foote, Moody, Vachette, Provencher, Gabriel, Bordas and Koch1983) and has been applied to the problem of protein folding (Chen et al. Reference Chen, Hodgson and Doniach1996; Perez et al. Reference Perez, Vachette, Russo, Desmadril and Durand2001) as well as deconvoluting mixtures of protein–RNA complexes (Bilgin et al. Reference Bilgin, Ehrenberg, Ebel, Zaccai, Sayers, Koch, Svergun, Barberato, Volkov, Nissen and Nyborg1998) and transient protein conformations (Fetler et al. Reference Fetler, Tauc, Herve, Moody and Vachette1995). SVD requires multiple scattering curves collected from samples containing different populations of states. Unlike solving systems of linear equations, SVD does not require scattering curves from individual components, but in order to extract real scattering curves external information may be required, such as thermodynamic models for transitions.

In SVD, all the collected scattering profiles I _n(q) are reduced to a common minimal basis set as described in the following equation:

where w _jb _jⁿ is the weighting contribution of u_j basis vector to I _n(q), the nth experimentally determined scattering profile. Determining the values of basis vectors and the relative weights is accomplished through creating an M×N matrix A(M×N) where the N columns are the intensity values of the scattering profiles determined at M values of q. Such a matrix may be represented as A(M×N)=U(M×N) W(N×N) B(N×N) where U is also an M×N matrix containing the u_j basis vectors, W is a diagonal N×N matrix composed of the so-called singular values w _j and B is an N×N matrix containing b _jⁿ. Although U contains N vectors, only important basis vectors will have associated significant w _j values. Many vectors in U will fit noise in the data rather than the significant parts of scattering profiles. Several commercially available mathematical packages have built-in SVD routines.

SVD will identify the minimum number of curves required to describe all the scattering profiles and can readily distinguish a system with a single transition from one with multiple transitions. Only two basis vectors were necessary to describe the scattering datasets collected from the allosteric enzyme aspartate transcarbamoylase measured in the presence of various substrate analogs, activators, and inhibitors (Fetler et al. Reference Fetler, Tauc, Herve, Moody and Vachette1995).

In an illustrative use of SVD, the temperature denaturation of neocarzinostatin was characterized by SAXS (Perez et al. Reference Perez, Vachette, Russo, Desmadril and Durand2001). At least three SVD basis vectors were required for a complete description of the unfolding, suggesting the presence of at least one intermediate state. With the scattering from the starting protein and the final unfolded state, they attempted to derive the scattering from the intermediate state by reprojecting the basis set. However, the derived scattering curve was degenerate, and the authors concluded that the folding pathway of neocarzinostatin involves an ensemble of related flexible intermediate states.

Another promising application of SVD is toward characterizing detergent solubilized membrane proteins (Lipfert & Doniach, Reference Lipfert and Doniach2007). The experimental matrix for SVD used SAXS data collected from different concentration ratios of membrane protein to detergent. All concentrations of detergents were above the critical micelle concentration. They reprojected the SVD vectors as scattering due to membrane protein–detergent complex, micelles, a micelle–micelle interaction component and a membrane protein–detergent to micelle interaction component. Of course the component of interest is the membrane protein–detergent complex. Although their experimental application using membrane protein TM0026 did not provide details on structural parameters other than R _G, the potential to isolate the scattering profile of a membrane protein–detergent complex is very exciting and well worth further research effort.

SVD is potentially very powerful; however, some care must be taken in employing the technique. Robust analysis requires many scattering curves and relatively error-free data. Systematic errors from background subtraction among the data sets may cause particularly insidious problems for SVD, as some of the basis sets may be required to fit error rather than be truly representative of solutions. If background-subtracted data is used in the SVD analysis, great care must be taken in titration experiments to guarantee the buffer is properly matched to the solution for background subtraction. SVD is also particularly powerful for time-resolved experiments of induced conformational changes. Unfortunately, time-resolved experiments with short exposures also suffer from weaker signals. Finally, SVD identifies the minimum number of scattering curves and not necessarily all states that may exist (Koch et al. Reference Koch, Vachette and Svergun2003), which might be particularly problematic when a continuum of states exist, such as with partially unfolded peptides.

4.4.2 Conformational disorder explored by SAXS

In contrast to samples that can be described as containing multiple well-defined conformations, many multidomain proteins and protein complexes contain flexible linkers that allow them to adopt large numbers of conformations. This situation is substantially more complex to model; however, it is likely to be quite common for large numbers of eukaryotic proteins. For these types of samples, attempting shape reconstructions to derive a single model with a ‘best-fit’ conformation can be misleading and at best provides a model representing an average of the conformations. At times an averaged model can be informative. For example, SAXS studies of the c-Abl tyrosine kinase that is dysregulated by gene fusions in chronic myelogenous leukemia (Nagar et al. Reference Nagar, Hantschel, Seeliger, Davies, Weis, Superti-Furga and Kuriyan2006) have provided additional insight into the autoinhibition of the kinase. The N-terminal half of Abl is comprised of three domains connected by flexible linkers. Binding of the myristoylated N-terminus within a binding pocket on the protein generates a compact autoinhibitated state. SAXS data revealed that mutants predicted to disrupt the autoinhibited state were much more elongated than wild-type Abl that closely matched the more compacted crystal structure. Fully extended models, created from the crystal structure (PDB id 1opl; Nagar et al. Reference Nagar, Hantschel, Young, Scheffzek, Veach, Bornmann, Clarkson, Superti-Furga and Kuriyan2003), fit the averaged SAXS envelope well, despite the fact that the molecule was proposed to be in a conformational ensemble with a wide range of heterogeneous states. Further interpretation about the extent of flexibility would require analysis beyond ab-initio shape restoration.

In some cases, the lack of convergence of ab-initio models has been correlated with flexibility. A number of models calculated by CREDO from scattering data collected from the cellulase Cel48F did not generate a single conformation (Fig. 18; Hammel et al. Reference Hammel, Fierobe, Czjzek, Finet and Receveur-Brechot2004a). Combining and weighting the scattering of individual models with OLIGOMER improved the overall fit (Fig. 18). In this case, more parameters were added which makes an improvement in the fit unsurprising; however, it is noteworthy that the individual models were generated independently of each other and from the weighting and merging steps. What is truly remarkable, however, is the fact that CREDO, a program that generates an over-parameterized ab-initio model (Section 4.3.1), was unable to come up with a better single model fit to the raw data. In the case of scattering from a heterogeneous population, the measured scattering is derived from the population-weighted thermodynamic ensemble and describes some population-weighted distribution of electrons. The inability of CREDO to generate and converge to a ‘best fit’ model likely derives from incompatibilities between the constraints that CREDO has in generating models and those model features, such as partial occupancy, necessary to completely describe this population-weighted electron density.

Fig. 18. Partial ab-initio models of free and complexed cellulase Cel48F. (a) Five typical CREDO (Petoukhov et al. Reference Petoukhov, Eady, Brown and Svergun2002) models of linker–dockerin region of free Cel48F are displayed in different colors together superposed on the crystal structure of Cel48F catalytic domain. (b) Two restored models (green and blue) of the dockerin/cohesin complex of the complexed Cel48F using the program CREDO superposed on the crystal structure of Cel48F (PDB id 1fbw; Parsiegla et al. Reference Parsiegla, Reverbel-Leroy, Tardif, Belaich, Driguez and Haser2000). The corresponding subdomains are schematically represented below each construct. The observed displacements between individual runs of partial ab-initio restoration are highlighted with the orange arrows. (c) Experimental SAXS profiles of free (bottom curve) and complexed (upper curve) fitted by the averaged form factors of the CREDO models obtained by the program OLIGOMER (red line; Konarev et al. Reference Konarev, Volkov, Sokolova, Koch and Svergun2003), and SAXS profiles calculated from the single CREDO models (green line) (after Hammel et al. Reference Hammel, Fierobe, Czjzek, Finet and Receveur-Brechot2004a).

The biggest challenge in trying to model conformationally very flexible systems using SAXS data is to avoid overfitting the raw data. One strategy to avoid overfitting the raw data with multiple models is to leverage existing atomic structures to reduce the parameter space of the model by describing the ensemble as a set of most probable structures. To minimize potential problems with overfitting, individual conformations to be tested as probably components of the population ought to be generated independently of the SAXS data. A number of creative techniques have begun to address the problem of quantitatively modeling flexible macromolecules observed in SAXS experiments (Akiyama et al. Reference Akiyama, Fujisawa, Ishimori, Morishima and Aono2004). Various modeling approaches can be used to generate atomic models that sample conformational space for use in fitting experimental SAXS curves. Monte Carlo techniques (Buey et al. Reference Buey, Monterroso, Menendez, Diakun, Chacon, Hermoso and Diaz2007; Shell et al. Reference Shell, Putnam and Kolodner2007), exploring the dihedral space of linkers (Tai, Reference Tai2004), CONCOORD, a non-dynamical method of generating conformation sets (Schlick, Reference Schlick2001) and MD (Levy & Becker, Reference Levy and Becker2002) have all been employed.

Conventional MD methods are computationally intensive; however, several advances have increased the size of tractable conformational changes (Yuzawa et al. Reference Yuzawa, Yokochi, Hatanaka, Ogura, Kataoka, Miura, Mandiyan, Schlessinger and Inagaki2001). These new techniques include multiple time-step MD and high temperature MD (Boehm et al. Reference Boehm, Woof, Kerr and Perkins1999; Aslam & Perkins, Reference Aslam and Perkins2001; Yuzawa et al. Reference Yuzawa, Yokochi, Hatanaka, Ogura, Kataoka, Miura, Mandiyan, Schlessinger and Inagaki2001; Aslam et al. Reference Aslam, Guthridge, Hack, Quigg, Holers and Perkins2003; Hammel et al. Reference Hammel, Fierobe, Czjzek, Kurkal, Smith, Bayer, Finet and Receveur-Brechot2005; von Ossowski et al. Reference von Ossowski, Eaton, Czjzek, Perkins, Frandsen, Schulein, Panine, Henrissat and Receveur-Brechot2005; Gilbert et al. Reference Gilbert, Asokan, Holers and Perkins2006) in which simulations are run at very high temperatures (~1000 K) to prevent molecules from becoming trapped in local minima (Leach, Reference Leach and Hall2001). Similarly, many of the simulations are sped up by including only van der Waals terms and distance restraints, but not electrostatic terms (Losonczi et al. Reference Losonczi, Andrec, Fischer and Prestegard1999). Comparison of rigid-body modeling, ab-initio shape reconstruction, and reconstruction of missing domains using CREDO have given similar results to MD sampling (Marino et al. Reference Marino, Zou, Svergun, Garcia, Edlich, Simon, Wilmanns, Muhle-Goll and Mayans2006). SAXS profiles calculated using MD trajectories are particularly useful for determining the average overall shape of the minicellulosomes and for proposing plausible atomic conformations they may adopt (Fig. 19). Given the restricted amount of independent data values from SAXS experiments, the best these techniques can provide is a set of conformations that are consistent with experimental data. In the absence of additional experimental evidence, the combination of SAXS and atomic models generated by MD trajectories cannot prove the existence of any particular conformation.

Fig. 19. The solution structure of the cellulosome. (a) The average envelope shape of different cellulosome constructs were calculated with GASBOR (Svergun et al. Reference Svergun, Petoukhov and Koch2001a). The corresponding modules are schematically represented on the top of each shape. (b) Restored partial ab-initio models of the different cellulosomes constructs calculated with CREDO (Petoukhov et al. Reference Petoukhov, Eady, Brown and Svergun2002). The CREDO models are displayed in surface representation (yellow). The fixed known atomic structures are in C_α tube representation. The secondary structural elements of superimposed atomic structures are shown as C_α tubes. (c) Best-fit models were calculated by MD of different constructs and superimposed on cohesin from S4 or complexed Cel48F (highlighted with the black circle). Three models colored by blue, brown, green for each model, except for FtS4Fc complex where only two models (green and blue) for better visualization, are presented (after Hammel et al. Reference Hammel, Fierobe, Czjzek, Kurkal, Smith, Bayer, Finet and Receveur-Brechot2005).

Once generated from atomic structures, these models can then be evaluated for their relative contribution to the scattering using SVD or the algorithm underlying OLIGOMER (Section 4.4.1). In the case of β₂-glycoprotein I (β₂GPI), SAXS fits poorly with the crystallized conformation or individual models with systematic modified conformations (Hammel et al. Reference Hammel, Kriechbaum, Gries, Kostner, Laggner and Prassl2002) but does fit well using a multiple model approach (Fig. 20). Not only is the inherent flexibility in β₂GPI likely to be important to its function, but it also illustrates the common case in which external information must be introduced in order to generate the correct types of models to fit the observed scattering.

Fig. 20. Solution structure of β₂-glycoprotein I (β₂GPI). (a) The crystal structure of β₂GPI (left) is shown with attached sugars (red). (b) The best single-conformation model derived from experimental SAXS data is superposed with the averaged ab-initio model calculated by program DAMMIN (Svergun, Reference Svergun, Baraberato and Koch1999) displayed as gray cages. (c) The theoretical scattering profile calculated for the mixture of different conformations improved the fit to the experimental data (root mean squared deviation for single and multiconformation fit are 3·0×10⁻³ and 8·5×10⁻³, respectively). Three different atomic conformations of β₂GPI with the indicated fractional occupancies in solution are shown. Different β₂GPI conformations were constructed by simple rotations between domains CCP2 and CCP3 of the best fit single structure (after Hammel et al. Reference Hammel, Kriechbaum, Gries, Kostner, Laggner and Prassl2002).

We are enthusiastic about a new approach to analyze the presence of multiple conformations of proteins contributing to the experimental scattering profile (Bernado et al. Reference Bernado, Mylonas, Petoukhov, Blackledge and Svergun2007). Bernado and co-workers define an ensemble optimization method (EOM) in which a pool of possible conformations (N>1000) is randomly generated to cover the possible conformational space. A genetic algorithm is then applied to select subsets (N=50) of configurations that fit the experimental scattering. The advantage of this method is the use of quantitative criteria for analyzing the EOM-selected models and for determining the optimal number of conformers in the subset. The best subsets are then selected for further evolution. Using both theoretical and experimental data for unstructured and multidomain proteins, EOM was able to distinguish between rigid and flexible proteins and assess interdomain contacts.

We also are enthusiastic about the potential offered by the recently developed residual dipolar coupling (RDC) NMR, which has been used to identify relative orientations in multidomain proteins relative to an external coordinate system (the ‘alignment tensor’). For multidomain proteins, the orientation of each domain can be determined separately within this coordinate frame, and the relative interdomain orientation can be deduced (Bernado et al. Reference Bernado, Blanchard, Timmins, Marion, Ruigrok and Blackledge2005). RDC data allow for unbiased determination of interdomain orientations in solution, albeit with a fourfold degeneracy, but require structural models for interpretation. In the study of the first two Ig domains of titin, Z1Z2, SAXS data was used to resolve the RDC degeneracy (Marino et al. Reference Marino, Zou, Svergun, Garcia, Edlich, Simon, Wilmanns, Muhle-Goll and Mayans2006). Calculations of the 200 RDC conformers showed that the 50 models with the lowest-discrepancy values were a well-defined cluster of conformations that superimposed with the ab-initio SAXS model, whereas the more compact crystal structure failed to fit the solution structures. Furthermore, conformational sampling using simulated averaged RDCs was applied in the analysis of SAXS profiles of partially unfolded protein. The close agreement of experimental and simulated RDC to SAXS data validated the conformational sampling results and provided a description of local structure, dynamics and average dimensions of the ensemble of unfolded protein. Thus, the synergy of SAXS with molecular modeling, crystallography, and NMR promises to provide unique insights into the structural characterization of proteins with intrinsic flexibility (Mattinen et al. Reference Mattinen, Paakkonen, Ikonen, Craven, Drakenberg, Serimaa, Waltho and Annila2002; Bax & Grishaev, Reference Bax and Grishaev2005).

4.4.3 Flexibility, oligomerization, or aggregation?

In general multidomain proteins with long linkers or that adopt extended conformations have smooth scattering profiles with few prominent features at high resolution and extended tails in P(r) functions. These proteins also typically have heterogeneous conformations with a variety of R _G and D _max values. Guinier plots of these macromolecules are linear over a smaller region: qR _G<0·8 instead of qR _G<1·3 which is more typical for globular samples (Table 1). Unfortunately, many of these features are also observed in the presence of small amounts of aggregation, which primarily affects the same low-resolutionregion of the scattering curve.

Studies on the soybean lipoxygenase-1 and rabbit 15-lipoxygenase-1 illustrate the need for careful interpretation of SAXS data in cases where flexibility is proposed (Fig. 21). The major discrepancies between the experimental scattering curve of rabbit lipoxygenase in solution and the curve calculated from the atomic coordinates were interpreted in terms of a large movement of the N-terminal domain with respect to the C-terminal domain (Hammel et al. Reference Hammel, Walther, Prassl and Kuhn2004b). Rigid-body modeling was applied, and the improvement of the fit in the entire scattering profile was observed using a mixture of different swing out conformations (Fig. 21, red, two conformations are shown; Hammel et al. Reference Hammel, Walther, Prassl and Kuhn2004b). The modeling of rabbit lipoxygenase was entirely dependent on establishing that the solution studied was monodisperse. In a study of soybean lipoxygenase, Dainese and co-workers found that the modeled structure was quite similar to the crystal structure and that slightly aggregated samples would give rise to elongated signals, as reported for rabbit lipoxygenase (Dainese et al. Reference Dainese, Sabatucci, van Zadelhoff, Angelucci, Vachette, Veldink, Agro and Maccarrone2005). Re-evaluation of rabbit lipoxygenase indicated that partial (20%) aggregation could also explain the scattering of rabbit lipoxygenase (Fig. 21).

Fig. 21. The experimental P(r) of the rabbit 15-lipoxyganase-1 (black). Theoretical P(r) calculated for a mixture of two protein conformations adopting different extensions in the N-terminal regions with the indicated occupancies (red) and for a mixture of 80% monomeric and 20% oligomeric assemblies (blue) are shown. The crystal structures used in the P(r) calculation (PDB id 1lox; Gillmor et al. Reference Gillmor, Villasenor, Fletterick, Sigal and Browner1997) using in the P(r) calculation are shown as surfaces using the same coloring (after Hammel et al. Reference Hammel, Walther, Prassl and Kuhn2004b).

Therefore, samples that are suspected of possessing intrinsic flexibility must be carefully characterized to ensure monodispersity prior to any SAXS modeling. Native gels can be very useful as an indicator of the homogeneity of samples. We have also found dynamic light scattering (DLS) useful for providing overall size and sample polydispersity measurements under the same concentrations and buffer conditions as for SAXS experiments. We have used DLS as an important pre-screening tool to optimize samples and buffers for maximum monodispersity. Our empirically based approaches to dealing with intrinsic flexibility, heterogeneous conformations, aggregation, and multidomain proteins are presented in Section 5 below.