Information Theory Optimization of Signals from Small Angle Scattering Measurements

Abstract

Solution-state small angle scattering (SAS) using X-rays (SAXS) or neutrons (SANS), informs on the conformational states and assemblies of biological macromolecules (bioSAS) outside of cryo- and solid-state conditions. BioSAS measurements are resolution-limited and through an inverse Fourier transform, the measured SAS intensities directly relate to the physical space occupied by the particles via the P(r)-distribution. Yet, this inverse transform of SAS data is canonicaly cast as an ill-posed, ill-conditioned problem requiring an indirect approach. Here, we show that with modern instrumentation and the application of information theory, the inverse transform of SAXS intensity data can be cast as a well-conditioned problem and that the 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 limit. The limit corresponds to the maximum number of significant singular values that can be recovered in a SAXS experiment suggesting the relationship between the Shannon limit and significant singular values is a fundamental property of band-limited inverse integral transform problems. An intrinsic challenge to the type of problem found in SAS, is identifying the best model while avoiding over-fitting. We propose a hybrid scoring function employing 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 (DW) statistic that considers model complexity, i.e., degrees-of-freedom and randomness of the model-data residuals. Importantly, 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 SAS, robustly determine maximum dimension, and more precisely define the best model to appropriately represent the observed scattering data. SIGNIFICANCEThe inverse transform problem in small angle scattering (SAS) of dispersed particles in solution is canonically considered an ill-posed, ill-conditioned problem. We apply information theory to show that the so-called Shannon number, Ns, of band-width limited signals defines the boundary at which a problem transitions from well-conditioned to ill-conditioned. In this framework, Ns represents the maximum number of orthogonal elements that are available to the SAS measurement and defines a fundamental relationship that determines the effective resolution of the recovered P(r)-distribution. We find that the inverse problem can be solved directly using a Shannon-limited approach. The Shannon-limit establishes model complexity and when incorporated into the Akaike Information Criteria aids in selecting the most likely P(r)-distribution.