Features of three-dimensional reconstruction of spirals based on small-angle x-ray scattering data

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Abstract

The interest in spiral particles lies in their resemblance to authentic nanostructures that emerge through the self-organisation of biopolymers (such as carrageenans, DNA, and so forth). Conversely, the determination of the structural parameters of such particles based on small-angle scattering data is challenging due to the lack of conditioning in the inverse problem. This is demonstrated by the utilisation of established bead structure modelling software. This paper considers a modification of the search algorithm in a limited area of space and the behaviour of solutions depending on the values of the parameters of the objective function responsible for the connectivity and looseness of the structure, the type of weighing of the scattering intensity curve, and the width of the angular range of data. In order to statistically assess the stability of the solutions, a sequential model search mode was applied, with varying amounts of contributions of penalty terms. The empirical dependences of the optimal values of the search parameters with respect to the parameters of the distribution curve of paired distances were determined.

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About the authors

V. A. Grigorev

Shubnikov Institute of Crystallography of Kurchatov Complex of Crystallography and Photonics of NRC “Kurchatov Institute”

Author for correspondence.
Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

P. V. Konarev

Shubnikov Institute of Crystallography of Kurchatov Complex of Crystallography and Photonics of NRC “Kurchatov Institute”

Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

V. V. Volkov

Shubnikov Institute of Crystallography of Kurchatov Complex of Crystallography and Photonics of NRC “Kurchatov Institute”

Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

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Supplementary files

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2. Fig. 1. Theoretical models of helices: top – side view, horizontal – top view. Helical pitch (from left to right): 45, 50, 55 and 60 Å for a diameter of 100 and 62 Å for a diameter of 120 Å.

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3. Fig. 2. Averaged sorted NSD values ​​for calculation groups consisting of five and seven models, and their mean values ​​(horizontal lines).

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4. Fig. 3. Examples of found structures with a defect of the “gap” type (1) and the “connection” type (2).

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5. Fig. 4. Estimation of f1-measure depending on the threshold value of the 1st quartile of NSD.

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6. Fig. 5. Calculation evaluation values ​​depending on different values ​​of the algorithm parameters for models I, II, III and IV (from top to bottom). The logarithms of the penalty weights for the structure rupture are plotted along the abscissa axis, and for its looseness along the horizontal axis. The black color marks the areas of successful search.

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7. Fig. 6. Approximations of the calculation estimates for models I, II, III and IV (from top to bottom).

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8. Fig. 7. Graphs of pairwise distance functions for helices with a pitch h = 45, 50, 55 and 60 Å (models I, II, III, IV Fig. 1).

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9. Fig. 8. Optimal values ​​of the fines for the looseness (wL) and discontinuity (wD) of a particle depending on the resolution of the peaks R on the curve p(r) for different pairs of values ​​of the degree of the weighting function (3) n and the number of Shannon channels Nsh.

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10. Fig. 9. Average values ​​of the calculation evaluation for the test model V. On the left is the real result, on the right is the ideal.

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11. Fig. 10. Typical reconstructed test models V for the central (columns 1, 3) and upper right (columns 2, 4) cells from Fig. 9.

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