Constrained Maximum Likelihood Estimation of the Abundances of Protein Conformation in a Heterogeneous Structural Ensemble from Small Angle X-ray Scattering Intensity Measurements

Abstract

A method is described for maximum likelihood estimation (MLE) of the relative abundances of different conformations of a protein in a heterogeneous mixture based on small angle X-ray scattering (SAXS) intensities.This approach is of particular interest in situations where there are unknown, intermediate conformations, for instance, during catalytic cycling of a protein. First, an ensemble of structures is generated using molecular dynamics, crystallography or other technique. This ensemble is then clustered into sub-sets based on k-means clustering and the Cramer-Rao bound on the mixture coefficient estimation error. A sparse basis set that represents the space spanned by the measured SAXS intensities of the conformations of a protein is then generated from representative members of each cluster. Based on a statistical model for the intensity measurements, we show that the MLE approach can be expressed as a constrained convex optimization problem. Starting with a basis set generated from known conformations of the enzyme, adenylate kinase (ADK), we carried out Monte Carlo simulations to assess the performance of the proposed estimation scheme. We demonstrate the utility of the approach by identification of dominant conformations under different solution conditions and provide estimates of the abundance of minor species as a function of concentration of different ligands.