Assessment of structure factors for analysis of small-angle scattering data from desired or undesired aggregates

Aggregation processes are central features of many systems ranging from colloids and polymers to inorganic nanoparticles and biological systems. Some aggregated structures are controlled and desirable, e.g. in the design of size-controlled clustered nanoparticles or some protein-based drugs. In other cases, the aggregates are undesirable, e.g. protein aggregation involved in neurodegenerative diseases or in vitro studies of single protein structures. In either case, experimental and analytical tools are needed to cast light on the aggregation processes. Aggregation processes can be studied with small-angle scattering, but analytical descriptions of the aggregates are needed for detailed structural analysis. This paper presents a list of useful small-angle scattering structure factors, including a novel structure factor for a spherical cluster with local correlations between the constituent particles. Several of the structure factors were renormalized to get correct limit values in both the high-q and low-q limit, where q is the modulus of the scattering vector. The structure factors were critically evaluated against simulated data. Structure factors describing fractal aggregates provided approximate descriptions of the simulated data for all tested structures, from linear to globular aggregates. The addition of a correlation hole for the constituent particles in the fractal structure factors significantly improved the fits in all cases. Linear aggregates were best described by a linear structure factor and globular aggregates by the newly derived spherical cluster structure factor. As a central point, it is shown that the structure factors could be used to take aggregation contributions into account for samples of monomeric protein containing a minor fraction of aggregated protein. After applying structure factors in the analysis, the correct structure and oligomeric state of the protein were determined. Thus, by careful use of the presented structure factors, important structural information can be retrieved from small-angle scattering data, both when aggregates are desired and when they are undesired.

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Small-angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems

Journal of Applied Crystallography ◽

10.1107/s0021889898011790 ◽

1999 ◽

Vol 32 (2) ◽

pp. 197-209 ◽

Cited By ~ 152

Author(s):

B. Weyerich ◽

J. Brunner-Popela ◽

O. Glatter

Keyword(s):

Form Factor ◽

Small Angle ◽

Structure Factor ◽

Fourier Transformation ◽

Hard Spheres ◽

Small Angle Scattering ◽

Scattering Data ◽

Model Free ◽

Angle Scattering ◽

The Gift

The indirect Fourier transformation (IFT) is the method of choice for the model-free evaluation of small-angle scattering data. Unfortunately, this technique is only useful for dilute solutions because, for higher concentrations, particle interactions can no longer be neglected. Thus an advanced technique was developed as a generalized version, the so-called generalized indirect Fourier transformation (GIFT). It is based on the simultaneous determination of the form factor, representing the intraparticle contributions, and the structure factor, describing the interparticle contributions. The former can be determined absolutely free from model assumptions, whereas the latter has to be calculated according to an adequate model. In this paper, various models for the structure factor are compared,e.g.the effective structure factor for polydisperse hard spheres, the averaged structure factor, the local monodisperse approximation and the decoupling approximation. Furthermore, the structure factor for polydisperse rod-like particles is presented. As the model-free evaluation of small-angle scattering data is an essential point of the GIFT technique, the use of a structure factor without any influence of the form amplitude is advisable, at least during the first evaluation procedure. Therefore, a series of simulations are performed to check the possibility of the representation of various structure factors (such as the effective structure factor for hard spheres or the structure factor for rod-like particles) by the less exact but much simpler averaged structure factor. In all the observed cases, it was possible to recover the exact form factor with a free determined parameter set for the structure factor. The resulting parameters of the averaged structure factor have to be understood as apparent model parameters and therefore have only limited physical relevance. Thus the GIFT represents a technique for the model independent evaluation of scattering data with a minimum ofa prioriinformation.

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Resolution ofab initioshapes determined from small-angle scattering

IUCrJ ◽

10.1107/s2052252516016018 ◽

2016 ◽

Vol 3 (6) ◽

pp. 440-447 ◽

Cited By ~ 65

Author(s):

Anne T. Tuukkanen ◽

Gerard J. Kleywegt ◽

Dmitri I. Svergun

Keyword(s):

Ab Initio ◽

Small Angle ◽

Three Dimensional ◽

Simulated Data ◽

Shape Reconstruction ◽

Small Angle Scattering ◽

Scattering Data ◽

Structural Method ◽

X Rays ◽

Angle Scattering

Spatial resolution is an important characteristic of structural models, and the authors of structures determined by X-ray crystallography or electron cryo-microscopy always provide the resolution upon publication and deposition. Small-angle scattering of X-rays or neutrons (SAS) has recently become a mainstream structural method providing the overall three-dimensional structures of proteins, nucleic acids and complexes in solution. However, no quantitative resolution measure is available for SAS-derived models, which significantly hampers their validation and further use. Here, a method is derived for resolution assessment forab initioshape reconstruction from scattering data. The inherent variability of theab initioshapes is utilized and it is demonstrated how their average Fourier shell correlation function is related to the model resolution. The method is validated against simulated data for proteins with known high-resolution structures and its efficiency is demonstrated in applications to experimental data. It is proposed that henceforth the resolution be reported in publications and depositions ofab initioSAS models.

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Structure and order in lamellar phases determined by small-angle scattering

Journal of Applied Crystallography ◽

10.1107/s0021889804012956 ◽

2004 ◽

Vol 37 (5) ◽

pp. 703-710 ◽

Cited By ~ 53

Author(s):

Thomas Frühwirth ◽

Gerhard Fritz ◽

Norbert Freiberger ◽

Otto Glatter

Keyword(s):

Form Factor ◽

Small Angle ◽

Structure Factor ◽

Scattering Length ◽

Small Angle Scattering ◽

Scattering Data ◽

X Rays ◽

Model Free ◽

Angle Scattering

Multilamellar phases can be identified and characterized by small-angle scattering of X-rays (SAXS) or neutrons (SANS). Equidistant peaks are the typical signature and their spacing allows the fast determination of the repeat distance,i.e.the mean distance between the midplane of neighbouring bilayers. The scattering function can be described as the product of a structure factor and a form factor. The structure factor is related to the ordering of the bilayers and is responsible for the typical equidistant peaks, but it also contains information about the bilayer flexibility and the number of coherently scattering bilayers. The form factor depends on the thickness and the internal structure (scattering length density distribution) of a single bilayer. The recently developed generalized indirect Fourier transformation (GIFT) method is extended to such systems. This method allows the simultaneous determination of the structure factor and the form factor of the system, including the correction of instrumental broadening effects. A few-parameter model is used for the structure factor, while the determination of the form factor is completely model-free. The method has been tested successfully with simulated scattering data and by application to experimental data sets.

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Update for BayesApp: a web site for analysis of small-angle scattering data

Journal of Applied Crystallography ◽

10.1107/s1600576714013156 ◽

2014 ◽

Vol 47 (4) ◽

pp. 1469-1471 ◽

Cited By ~ 10

Author(s):

Steen Hansen

Keyword(s):

Maximum Entropy ◽

Bayesian Methods ◽

Small Angle ◽

Web Site ◽

Small Angle Scattering ◽

Scattering Data ◽

Structure Factors ◽

Angle Scattering ◽

Ellipsoid Of Revolution ◽

The Web

An update for BayesApp, a web site for analysis of small-angle scattering data, is presented. The indirect transformation of the scattering data now includes an option for a maximum-entropy constraint in addition to the conventional smoothness constraint. The maximum-entropy constraint uses an ellipsoid of revolution as a prior, and the dimensions of the ellipsoid as well as the overall noise level of the experimental data are estimated using Bayesian methods. Furthermore, a correction for slit smearing has been added. The web site also includes options for calculation of the scattering intensity from simple models as well as the estimation of structure factors for polydisperse spheres and nonspherical objects of axial ratios between 0.4 and 2.5.

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Scatter: software for the analysis of nano- and mesoscale small-angle scattering

Journal of Applied Crystallography ◽

10.1107/s0021889810008289 ◽

2010 ◽

Vol 43 (3) ◽

pp. 639-646 ◽

Cited By ~ 121

Author(s):

S. Förster ◽

L. Apostol ◽

W. Bras

Keyword(s):

Small Angle ◽

Calculated Data ◽

Form Factors ◽

Real Space ◽

Small Angle Scattering ◽

Scattering Data ◽

Mathematical Treatment ◽

Mesoscale Structures ◽

Structure Factors ◽

Angle Scattering

Scatteris a new software for analysis, modeling and fitting of one- and two-dimensional small-angle scattering data of non-ordered, partially ordered or fully ordered nano- and mesoscale structures. The calculations are based on closed analytical expressions for the scattering intensity, enabling efficient evaluation of form factors and structure factors. The software allows one to sequentially fit large series of scattering curves and scattering patterns automatically. It provides further tools for data loading, beam centering, calibration, zooming, binning, lattice identification, calculation of density profiles and size distributions, and visualization of real-space structures. Presentations of experimental and calculated data can be saved as is for presentations or exported for further graphical or mathematical treatment.

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2017 publication guidelines for structural modelling of small-angle scattering data from biomolecules in solution: an update

Acta Crystallographica Section D Structural Biology ◽

10.1107/s2059798317011597 ◽

2017 ◽

Vol 73 (9) ◽

pp. 710-728 ◽

Cited By ~ 107

Author(s):

Jill Trewhella ◽

Anthony P. Duff ◽

Dominique Durand ◽

Frank Gabel ◽

J. Mitchell Guss ◽

...

Keyword(s):

Small Angle ◽

Task Force ◽

Data Bank ◽

Small Angle Scattering ◽

Scattering Data ◽

Quality Data ◽

Angle Scattering ◽

Publication Guidelines ◽

Guidelines And Recommendations

In 2012, preliminary guidelines were published addressing sample quality, data acquisition and reduction, presentation of scattering data and validation, and modelling for biomolecular small-angle scattering (SAS) experiments. Biomolecular SAS has since continued to grow and authors have increasingly adopted the preliminary guidelines. In parallel, integrative/hybrid determination of biomolecular structures is a rapidly growing field that is expanding the scope of structural biology. For SAS to contribute maximally to this field, it is essential to ensure open access to the information required for evaluation of the quality of SAS samples and data, as well as the validity of SAS-based structural models. To this end, the preliminary guidelines for data presentation in a publication are reviewed and updated, and the deposition of data and associated models in a public archive is recommended. These guidelines and recommendations have been prepared in consultation with the members of the International Union of Crystallography (IUCr) Small-Angle Scattering and Journals Commissions, the Worldwide Protein Data Bank (wwPDB) Small-Angle Scattering Validation Task Force and additional experts in the field.

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