A constrained optimization approach to fit protein structure to X-ray solution scattering measurements

The study of protein structure is of great interest in fields like biology, pharmacology and medicine due to the great impact that structural configuration has on protein functioning and behaviour. However, fully atomistic models are inaccurate or directly unavailable for many protein conformations because of the constraints that protein crystallography, the default method to obtain atomic configurations, imposes on protein structures. In that regard, Small Angle X-Ray Scattering (SAXS) has become a widely used tool to study proteins in solution, as it can retrieve structural information from proteins in their native environment without the need for crystallography, but at the cost of significant loss of information, and the need to interpret or reconstruct structure from the measurements. SAXS measurements consist of a band-pass filtered spherically averaged Fourier transform of the protein's atomic pair-distribution function, resulting in a 1D signal with limited support and with a restricted resolution range between 50 to 10 angstroms. Determining protein configurations from SAXS measurements is, thus, a very ill-posed challenge as only 1D information is available to reconstruct the protein's 3D structure and requiring regularized and constrained optimization frameworks to successfully solv it.;In this work we propose a modular optimization framework to tackle this inverse problem. One key challenge in SAXS modelling is to account for the water that surrounds and physically interacts with the protein, affecting the system's scattering; here we use a convex hull-based approach to geometrically model the hydration shell volume of the protein. A second key problem is to constrain the inverse solution in a physically meaningful way; we propose a subdomain-based elastic network to model the global motions of the protein present in conformation transitions. Finally, to solve the resulting non-convex optimization problem, we adapt the standard formulation of Alternate Directions Method of Multipliers to allow for (local) convergence in non-convex scenarios. Results for the proposed approach show comparable accuracy to similar methods without the need for supervised choice of parameters while its modularity holds potential for incorporating additional prior information.