Information Theory Optimization of Signals From Small-Angle Scattering Measurements

Authors

Robert P Rambo
John A Tainer

Publication Date

8-5-2025

Journal

Biophysical Journal

DOI

10.1016/j.bpj.2025.06.031

PMID

40581817

PMCID

PMC12414664

PubMedCentral® Posted Date

6-27-2025

PubMedCentral® Full Text Version

Author MSS

Abstract

Small-angle X-ray scattering (SAXS) of particles in solution informs on the conformational states and assemblies of biological macromolecules (bioSAXS) outside of cryo- and solid-state conditions. In bioSAXS, the SAXS measurement under dilute conditions is resolution limited, and through an inverse Fourier transform, the measured SAXS intensities directly relate to the physical space occupied by the particles via the P(r)-distribution. Yet, this inverse transform of SAXS data has been historically cast as an ill-posed, ill-conditioned problem requiring an indirect approach. Here, we show that through the applications of matrix and information theories, the inverse transform of SAXS intensity data is a well-conditioned problem. The so-called ill-conditioning of the inverse problem is directly related to the Shannon number. By exploiting the oversampling enabled by modern detectors, a direct inverse Fourier transform of the SAXS data is possible, provided the recovered information does not exceed the Shannon number. The Shannon limit corresponds to the maximum number of significant singular values that can be recovered in a SAXS experiment, suggesting this relationship is a fundamental property of band-limited inverse integral transform problems. This correspondence reduces the complexity of the inverse problem to the Shannon limit and maximum dimension. We propose a hybrid scoring function using an information theory framework that assesses both the quality of the model-data fit as well as the quality of the recovered P(r)-distribution. The hybrid score utilizes the Akaike information criteria and Durbin-Watson statistic that considers parameter-model complexity, i.e., degrees of freedom, and the randomness of the model-data residuals. The described tests and findings extend the boundaries for bioSAXS by completing the information theory formalism initiated by Peter B. Moore to enable a quantitative measure of resolution in SAXS, robustly determine maximum dimension, and more precisely define the best parameter model appropriately representing the observed scattering data.

Keywords

Scattering, Small Angle, X-Ray Diffraction, Information Theory, Fourier Analysis

Published Open-Access

yes

Recommended Citation

Robert P Rambo and John A Tainer, "Information Theory Optimization of Signals From Small-Angle Scattering Measurements" (2025). Faculty, Staff and Student Publications. 5039.
https://digitalcommons.library.tmc.edu/uthgsbs_docs/5039