Dirichlet Priors for MAP Inference of Protein Conformation Abundances from SAXS

Estimation of mixture coefficients of protein conformations in solution find applications in understanding protein behavior. We describe a method for maximum a posteriori (MAP) estimation of the mixture coefficients of ensemble of conformations in a protein mixture solution using measured small angle X-ray scattering (SAXS) intensities. The proposed method builds upon a model for the measurements of crystallographically determined conformations. Assuming that a priori information on the protein mixture is available, and that priori information follows a Dirichlet distribution, we develop a method to estimate the relative abundances with MAP estimator. The Dirichlet distribution depends on concentration parameters which may not be known in practice and thus need to be estimated. To estimate these unknown concentration parameters we developed an expectation-maximization (EM) method. Adenylate kinase (ADK) protein was selected as the test bed due to its known conformations Beckstein et al. (Journal of Molecular Biology, 394(1), 160 1). Known conformations are assumed to form the full vector bases that span the measurement space. In Monte Carlo simulations, mixture coefficient estimation performances of MAP and maximum likelihood (ML) (which assumes a uniform prior on the mixture coefficients) estimators are compared. MAP estimators using known and unknown concentration parameters are also compared in terms of estimation performances. The results show that prior knowledge improves estimation accuracy, but performance is sensitive to perturbations in the Dirichlet distribution’s concentration parameters. Moreover, the estimation method based on EM algorithm shows comparable results to approximately known prior parameters.