Small-angle Scattering Intensity Research Articles

Integrating machine learning with -SAS for enhanced structural analysis in small-angle scattering: applications in biological and artificial macromolecular complexes.

Small-Angle Scattering (SAS), encompassing both X-ray (SAXS) and Neutron (SANS) techniques, is a crucial tool for structural analysis at the nanoscale, particularly in the realm of biological macromolecules. This paper explores the intricacies of SAS, emphasizing its application in studying complex biological systems and the challenges associated with sample preparation and data analysis. We highlight the use of neutron-scattering properties of hydrogen isotopes and isotopic labeling in SANS for probing structures within multi-subunit complexes, employing techniques like contrast variation (CV) for detailed structural analysis. However, traditional SAS analysis methods, such as Guinier and Kratky plots, are limited by their partial use of available data and inability to operate without substantial a priori knowledge of the sample's chemical composition. To overcome these limitations, we introduce a novel approach integrating -SAS, a computational method for simulating SANS with CV, with machine learning (ML). This approach enables the accurate prediction of scattering contrast in multicomponent macromolecular complexes, reducing the need for extensive sample preparation and computational resources. -SAS, utilizing Monte Carlo methods, generates comprehensive datasets from which structural invariants can be extracted, enhancing our understanding of the macromolecular form factor in dilute systems. The paper demonstrates the effectiveness of this integrated approach through its application to two case studies: Janus particles, an artificial structure with a known SAS intensity and contrast, and a biological system involving RNA polymerase II in complex with Rtt103. These examples illustrate the method's capability to provide detailed structural insights, showcasing its potential as a powerful tool for advanced SAS analysis in structural biology.

Desmearing small-angle scattering data by central moment expansions

This article proposes a robust method for desmearing 1D small-angle scattering (SAS) intensity profiles. Using the central-moment-expansion technique, smearing in SAS intensity profiles is achieved in a model-free manner, eliminating the need for presumptive input during the desmearing. This innovative approach enables data analysis of SAS data without the resolution being convoluted with the structural model, thereby facilitating faster implementation and performance of fitting functions directly from the scattering models.

Fractal Analysis of DNA Sequences Using Frequency Chaos Game Representation and Small-Angle Scattering.

The fractal characteristics of DNA sequences are studied using the frequency chaos game representation (FCGR) and small-angle scattering (SAS) technique. The FCGR allows representation of the frequencies of occurrence of k-mers (oligonucleotides of length k) in the form of images. The numerically encoded data are then used in a SAS analysis to enhance hidden features in DNA sequences. It is shown that the simulated SAS intensity allows us to obtain the fractal dimensions and scaling factors at various scales. These structural parameters can be used to distinguish unambiguously between the scaling properties of complex hierarchical DNA sequences. The validity of this approach is illustrated on several sequences from: Escherichia coli, Mouse mitochondrion, Homo sapiens mitochondrion and Human cosmid.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Integral approach to biomacromolecular structure by analytical-ultracentrifugation and small-angle scattering

Currently, a sample for small-angle scattering (SAS) is usually highly purified and looks monodispersed: The Guinier plot of its SAS intensity shows a fine straight line. However, it could include the slight aggregates which make the experimental SAS profile different from the monodispersed one. A concerted method with analytical-ultracentrifugation (AUC) and SAS, named as AUC-SAS, offers the precise scattering intensity of a concerned biomacromolecule in solution even with aggregates as well that of a complex under an association-dissociation equilibrium. AUC-SAS overcomes an aggregation problem which has been an obstacle for SAS analysis and, furthermore, has a potential to lead to a structural analysis for a general multi-component system.

Structural Properties of Molecular Sierpiński Triangle Fractals.

The structure of fractals at nano and micro scales is decisive for their physical properties. Generally, statistically self-similar (random) fractals occur in natural systems, and exactly self-similar (deterministic) fractals are artificially created. However, the existing fabrication methods of deterministic fractals are seldom defect-free. Here, are investigated the effects of deviations from an ideal deterministic structure, including small random displacements and different shapes and sizes of the basic units composing the fractal, on the structural properties of a common molecular fractal—the Sierpiński triangle (ST). To this aim, analytic expressions of small-angle scattering (SAS) intensities are derived, and it is shown that each type of deviation has its own unique imprint on the scattering curve. This allows the extraction of specific structural parameters, and thus the design and fabrication of artificial structures with pre-defined properties and functions. Moreover, the influence on the SAS intensity of various configurations induced in ST, can readily be extended to other 2D or 3D structures, allowing for exploration of structure-property relationships in various well-defined fractal geometries.

Structural Properties of Vicsek-like Deterministic Multifractals

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.

Scattering from surface fractals in terms of composing mass fractals

It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensity I(q) ∝ q^{D_{\rm s}-6}, where 2 < D s < 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝ r −τdr, with D s = τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.

Small-angle scattering from three-phase systems: Investigation of the crossover between mass fractal regimes

In this paper, we construct a three-phase model (that is, a system consisting of three homogeneous regions with various scattering length densities), which illustrate the behavior of small-angle scattering (SAS) scattering curves. Here two phases are a deterministic fractal embedded in another deterministic mass fractal, and they altogether are further embedded in a third phase, which can be a solution or solid matrix. We calculate SAS intensities, derive expressions for the crossover position (that is, the point where the power-law scattering exponent changes) as a function of control parameters, including size, concentration, and volumes of each phase. The corresponding SAS intensities from these models describe a succession of power-law regimes in momentum space where both regimes correspond to mass fractals. The models can be applied to SAS data where the absolute value of the scattering exponent of the first power-law regime is higher than that of the subsequent second power-law regime, that is, the scattering curve of "convex" kind near the crossover position.

A model of small-angle scattering from three-phase fractal systems

In small-angle scattering (SAS) intensities on a double logarithmic scale, one generically observes "knees" arising from changes of the power-law exponents. While the appearance of power-law behavior is associated usually with a fractal structure, the explanation of the knees is a more intricate problem in general. It is shown that the crossover between the exponents can be accounted for by various levels of the structures with different scattering length densities (SLD). Thus, the whole system is a multiphase system, composed of the homogeneous structures embedded into each other. In this way, one can explain the crossover between arbitrary exponents whose values vary from −4 to 0. In order to confirm this conclusion, we calculate the SAS intensity from a set of non-interacting, randomly oriented and uniformly distributed three-phase fractals with controllable Hausdorff dimension. We observe the appearance of crossover (knee), which depends on the values of the SLD of each phase and estimate its position. Random fractals are described as well within the developed model by taking the fractal sizes at random, which leads to polydispersity and, as a consequence, to smearing of the SAS intensity curves.

Small-angle scattering from generalized self-similar Vicsek fractals

An analytical approach for calculating the small-angle X-ray or neutron scattering (SAXS/SANS) from generalized self-similar Vicsek fractals (GSSVF) is presented; each fractal consists of spherical subunits. The system considered is a mass-fractal, generated iteratively from a regular 3D Vicsek fractal structure. Its fractal dimension is controllable and increases with increasing the value of the scaling factor. Small-angle scattering (SAS) intensity is determined from a set of non-interacting, randomly oriented and uniformly distributed GSSVF fractals. It is shown that in the fractal region, the curve I(q)qD is approximately log-periodic with the period equal to the scaling factor of fractal; here D and I(q) are the fractal dimension and the SAS intensity, respectively. In particular, the positions of deepest minima and highest maxima are log-periodic, and their number coincides with the number of fractal iterations. The log-periodicity of the scattering curves is a consequence of the self-similarity of GSSVF.

Relationship between the atomic pair distribution function and small-angle scattering: implications for modeling of nanoparticles

The relationship between the equations used in the atomic pair distribution function (PDF) method and those commonly used in small-angle-scattering (SAS) analyses is explicitly shown. The origin of the sloping baseline, -4pirrho0, in PDFs of bulk materials is identified as originating from the SAS intensity that is neglected in PDF measurements. The nonlinear baseline in nanoparticles has the same origin, and contains information about the shape and size of the nanoparticles.

The puzzle-interlayer model: an approach to the analysis of tightly packed arrangements of hard particles

The small-angle scattering (SAS) structure functions are analyzed for an idealized, random two-phase system: a stationary arrangement of hard puzzle particles, separated by a relatively thin interspace, which can be approximated by an interlayer. The detailed shape of the interlayer is defined by the shape of the particles themselves: The starting point for producing these initial particles is a three-dimensional initial puzzle {P_0} in the state of tessellation. Its pieces, homogeneous particles of random shape, fit together filling the space. In a second step, an expanded puzzle {P_{\tau }} is constructed by translating the initial particles by a certain length \tau (relative to one another). The whole tightly packed particle arrangement depends on {P_0}. The interlayer region between the particles is a connected, homogeneous region. The SAS intensity of {P_{\tau }} depends on the parameter \tau and on the typical shape and size of the pieces of {P_0}. Chord length distributions (CLDs) are used in the description. The random shape of the pieces of {P_0} possesses a CLD 1. The CLD 2 of the intermediate spaces is approximated by that of an idealized layer of constant thickness \tau. The scattering of {P_{\tau}} results in terms of the CLDs of both phases. The approach can be applied to many types of {P_0}. Two initial tessellations of {P_0} are studied, a `dead leaves' tessellation produced by spherical primary grains and the Poisson plane mosaic.

A target function for quaternary structural refinement from small angle scattering and NMR orientational restraints

We present a novel target function based on atomic coordinates that permits quaternary structural refinement of multi-domain protein-protein or protein-RNA complexes. It requires that the high-resolution structures of the individual domains are known and that small angle scattering (SAS) data as well as NMR orientational restraints from residual dipolar couplings (RDCs) of the complex are available. We show that, when used in combination, the translational and rotational restraints contained in SAS intensities and RDCs, respectively, define a target potential function that permits to determine the overall topology of complexes made up of domains with low internal symmetry. We apply the target function on a modestly anisotropic model system, the Barnase/Barstar complex, and discuss factors that influence the structural refinement such as data errors and the geometrical properties of the individual domains.

Tetraalkylammonium bromides in methanol: Small angle neutron scattering and vapor pressure measurements

Vapor pressure measurements and small angle scattering (SANS) experiments are reported and discussed for bispiperidinium (BP) bromide and n-tetrapentylammonium bromide solutions in methanol at 25°C. The BP+ ion which may be considered as a tetraalkylammonium (TAA) ion with pairwise connected alkyl chains, is used for the study of the effects due to flexible TAA alkyl chains. SANS intensities are calibrated with the help of the precise osmotic coefficients from the vapor pressure measurements.

On the determination of the metastable miscibility gap in ternary alloys from small-angle measurements. I. Theoretical background

Metastable precipitation as the formation of GP zones is governed by a metastable miscibility gap. It is shown that the limits of this gap in a ternary system can be determined from combined analysis of the small-angle scattering (SAS) of X-rays and of neutrons. The necessary condition is a significantly different ratio β = (bC −bA )/(bB −bA ) of the atomic scattering amplitudes b of the components A, B and C for both the X-rays and the neutrons. This difference enables one to determine the slope of the tie-line passing through an individual alloy by comparing the integrated SAS intensities of the two types of radiation. Thereafter, the determination of the gap limits follows the same procedure as already known for binary alloys. The change of the mean atomic volume with composition influences the result, but this can be taken into account provided the same Vegard's law is valid for both the depleted matrix and the precipitate.