Biological small-angle X-ray scattering

# Overview
Small-angle scattering (SAS) of X-rays (SAXS) and neutrons (SANS) is a fundamental method for structure analysis of materials. SAXS allows one to study the structure of a variety of objects such as solutions of biological macromolecules, nanocomposites, alloys, synthetic polymers, etc. SAXS is an analogous method to X-ray diffraction and wide angle X-ray scattering, but in addition SAXS yields information on the sizes and shapes of non-crystalline particles.
SAXS patterns are collected at very small angles (less than a degree). SAXS is capable of delivering structural information of macromolecules between 1 and 25 nm, of repeat distances in partially ordered systems of up to 150 nm. Ultra small angle X-ray scattering (USAXS) can resolve even larger dimensions. The grazing incidence small angle X-ray scattering (GISAXS) is a very powerful technique for studying the surface and interface problems.
Usually SAXS is used to determine the structure of particle systems in terms of average particle sizes and shapes. The materials can be solid or liquid and they can contain solid, liquid or gaseous domains (so-called particles) of the same or another material in combination. The method is accurate, non-destructive and usually requires only a minimum of sample preparation. Applications are very broad and include colloids of all type, metals, cement, oil, polymers, plastics, proteins, foods and pharmaceuticals and can be found in research as well as in quality control. Furthermore the method allows to determine parameters, e.g., inner structure and surface-to-volume ratio.
Conceptually, SAXS experiment is simple: the sample is exposed to X-rays and the scattering radiation is registered by a detector. As the SAXS measurements are done very close to the primary beam ("small angles"), the technique profits immensely from the brilliance of X-ray photon beams provided by synchrotron storage rings. The X-ray scattering curve (intensity versus scattering angle) is used to create a low-resolution model of a protein, shown on the right picture. One can further use the X-ray scattering data and fit separate domains (X-ray or NMR structures)  into SAXS envelope.
In comparison to other structure determination methods, like NMR or X-ray crystallography, SAXS allows one to overcome some restraints. For example, NMR is limited to protein size, whereas SAXS can be used for small molecules as well as for large multimolecular assemblies. Structure determination by X-ray may take several weeks or even years, whereas SAXS measurements take days.

# Definition
In a scattering experiment, a solution of macromolecules is exposed to X-rays (with wavelength 0.15 nm) or thermal neutrons (0.5 nm). The scattered intensity I(s) is recorded as a function of momentum transfer s (s=4sin/, where 2 is the angle between the incident and scattered radiation) and the solvent scattering is subtracted. The random positions and orientations of particles result in an isotropic intensity distribution which, for monodisperse solutions of non-interacting particles, is proportional to the scattering from a single particle averaged over all orientations. The net particle scattering is proportional to the squared difference in scattering length density (electron density for X-rays and nuclear/spin density for neutrons) between particle and solvent  the so-called contrast. The latter can be effectively varied in neutron scattering using H2O/D2O mixtures or selective deuteration to yield additional information. 
The information content of SAS data is illustrated in figure on the left, which represents X-ray patterns from proteins with different folds and molecular masses. At low angles (2-3 nm resolution) the curves are rapidly decaying functions of s essentially determined by the particle shape, which clearly differ. At medium resolution (2 to 0.5 nm) the differences are already less pronounced and above 0.5 nm resolution all curves are very similar.  SAS thus contains information about the gross structural features  shape, quaternary and tertiary structure  but is not suitable for the analysis of the atomic structure.

# History
First x-ray applications date back to the late 1930s when the main principles of SAXS were developed in the fundamental work of Guinier following his studies of metallic alloys. In the first monograph on SAXS by Guinier and Fournet it was already demonstrated that the method yields not just information on the sizes and shapes of particles but also on the internal structure of disordered and partially ordered systems.
In the 1960s, the method became increasingly important in the study of biological macromolecules in solution as it allowed one to get low-resolution structural information on the overall shape and internal structure in the absence of crystals. A breakthrough in SAXS and SANS experiments came in the 1970s, thanks to the availability of synchrotron radiation (SR) and neutron sources, the latter paving the way for contrast variation by solvent exchange of H2O for D2O and specific deuteration methods. It was realised that scattering studies on solution provide, for a minimal investment in time and effort, useful insights into the structure of non-crystalline biochemical systems. Moreover, SAXS/SANS also make it possible to investigate intermolecular interactions including assembly and large-scale conformational changes in real time.
The main challenge of SAS as a structural method is to extract information about the three-dimensional structure of the object from the one-dimensional experimental data. In the past, only overall particle parameters (e.g. volume, radius of gyration) of the macromolecules were directly determined from the experimental data, whereas the analysis in terms of three-dimensional models was limited to simple geometrical bodies (e.g. ellipsoids, cylinders, etc) or was performed on an ad hoc trial-and-error basis. Electron microscopy was often used as constraint in building consensus models. In the 1980s, progress in other structural methods led to a decline of the interest of biochemists in SAS studies drawing structural conclusions from a couple of overall parameters or trial-and-error models.
The 1990s brought a breakthrough in SAXS/SANS data analysis methods, allowing reliable ab initio shape and domain structure determination and detailed modelling of macromolecular complexes using rigid body refinement. This progress was accompanied by further advances in instrumentation, and time resolutions down to the sub-ms were achieved on third generation SR sources in studies of protein and nucleic acid folding.
In 2005 a four-year project was started. Small-Angle X-Ray scattering Initiative for EuRope (SAXIER) goal is to combine SAXS method with other analytical techniques and create automated software to rapidly analyse the huge quantities of data. The project will create a unified European SAXS infrastructure using the most advanced methods.

# SAXS data analysis

## Low-resolution models
Basically, the problem in SAXS data analysis is to get a three-dimensional structure from an one-dimensional scattering pattern. The SAXS data  does not imply a single solution. Many different proteins, for example, may have same scattering curve. Reconstruction of 3D structure might result in big number of different models. To avoid this problem a number of simplifications is considered.
In the first general ab initio approach an angular envelope function of the particle r=F(), where (r,) are spherical coordinates, is described by a series of spherical harmonics. The low resolution shape is thus defined by a few parameters  the coefficients of this series  which fit the scattering data. The approach was further developed and implemented in computer program SASHA. It was demonstrated that under certain circumstances a unique envelope can be extracted from the scattering data. This method is only applicable to globular particles with relatively simple shapes without significant internal cavities. To overcome these limitations, there was another approach developed, which uses different types of Monte-Carlo searches. DALAI_GA is an elegant program, that takes sphere with diameter equal to the maximum particle size Dmax, which is determined from the scattering data, and fills it with beads. Each bead belongs either to the particle (index=1) or to the solvent (index=0). The shape is thus described by the binary string of length M. Starting from a random string, a genetic algorithm searches for a model that fits the data. Compactness and connectivity constrains are imposed in the search, implemented in the program DAMMIN. If the particle symmetry is known, SASHA and DAMMIN, can utilise it as useful constraints. The 'give-n-take' procedure SAXS3D and the program SASMODEL based on interconnected ellipsoids are ab initio Monte Carlo approaches without limitation of the search space.
An approach that uses ensemble of Dummy Residues (DRs) and simulated annealing to build a locally chain-compatible DR-model inside a sphere of diameter Dmax lets one extract more details from SAXS data. This method is implemented in the program GASBOR.
Solution scattering patterns of multi-domain proteins and macromolecular complexes can also be fitted using models built from high resolution (NMR or X-ray) structures  of individual domains or subunits assuming that their tertiary structure is preserved. Depending on the complexity of the object, different approaches are employed for the global search of the optimum configuration of subunits fitting the experimental data.

## Consensus model
The Monte-Carlo based models contain hundreds or thousand parameters, and caution is required to avoid overinterpretation. A common approach is to align a set of models resulting from independent shape reconstruction runs to obtain an average model retaining the most persistent- and conceivably also most reliable-features (e.g. using the program SUPCOMB).

## Adding missing loops
Missing loops is a known problem for NMR or crystallographic studies. Such missing fragments still contribute to the SAS intensity and their probable configurations can be found by fixing the known part of the structure and adding the missing parts to fit the SAS pattern from the entire particle. The Dummy Residue approach was extended and algorithm for adding missing loops or domains were implemented in the program suite CREDO.

## Hybrid methods
Recently there were a few methods proposed, which use SAXS data as constrains. The authors aimed to improve results of fold recognition and de novo protein structure prediction methods. SAXS data provide the Fourier transform of the histogram of atomic pair distances (pair distribution function) for a given protein. This can serve as a structural constraint on methods used to determine the native conformational fold of the protein. Threading or fold recognition assumes that 3D structure is more conserved than sequence.  Thus, very divergent sequences may have similar structure. Ab initio methods, on the other hand, challenge one of the biggest problem in molecular biology, namely, to predict the folding of protein from the scratch, using no homologous sequences or structures. Using "SAXS filter", the authors were able to purify the set of de novo protein models significantly. This was further proved by structure homology searches. It was also shown, that combination of SAXS scores with scores, used in threading methods, significantly improves the performance of fold recognition. On one example it was demonstrated how approximate tertiary structure of modular protein can be assembled from high resolution NMR structures of domains, using SAXS data, confining the translational degrees of freedom. Another example shows how SAXS data can be combined together with NMR, X-ray crystallography and electron microscopy to reconstruct the quarternary structure of multidomain protein.

## Simulations (GISAXS)
IsGISAXS is a software dedicated to the simulation and analysis of Grazing Incidence Small Angle X-Ray Scattering (GISAXS) from nanostructures. IsGISAXS only encompasses the scattering by nanometric sized particles which are buried in a matrix subsurface or supported on a substrate or buried in a thin layer on a substrate. The case of holes is also handled. The geometry is restricted to a plane of particles. The scattering cross section is decomposed in terms of interference function and particle form factor. The emphasis is put on the grazing incidence geometry which induces a "beam refraction effect". The particle form factor is calculated within the Distorted Wave Born Approximation (DWBA), starting as an unperturbed state with sharp interfaces or with the actual perpendicular profile of refraction index. Various kinds of simple geometrical shapes are available with a full account of size and shape distributions in the Decoupling Approximation (DA), in the Local Monodisperse Approximation (LMA) and also in the Size-Spacing Correlation Approximation (SSCA). Both, disordered systems of particles defined by their particle-particle pair correlation function and bidimensional crystal or paracrystal are considered.

# SAXS Projects
SAXIER  Small-Angle X-Ray scattering Initiative for EuRope. This project is coordinated by the European Molecular Biology Laboratory (EMBL) in Hamburg, Germany. The project includes Europe's main SAXS laboratories at synchrotrons in France, Germany, Italy and the United Kingdom. 
	SAXIER is a four-year project. It started in December 1, 2005. The coordinator is Dmitri Svergun. 
	Basically, SAXIER's goal is to make SAXS suitable for multidisciplinary studies by combining it with different other methods, such as Raman spectroscopy, gel filtration chromatography, cryogenic technology, mass spectrometry, etc. The new software pipeline will allow to rapidly analyse the huge quantities of data produced by the novel techniques. 
In the era of systems biology and structural genomics SAXIER goals are of particular importance,
as they will make SAXS suitable for detailed study of different proteins and macromolecular machines, as well as
biological and advance nanomaterials in their native conditions.