Describing the structure of a macro molecular complex as a random Signal in noise and a maximum likelihood reconstruction

Instances of biological macromolecular complexes that have identical chemical constituents may not have the same geometry due to, for example, flexibility. Cryo electron microscopy provides one noisy projection image of each of many instances of a complex where the projection directions for the different instances are random. The noise is sufficient severe (SNR ≪ 1) that the projection direction for a particular image cannot be easily estimated from the individual image. The goal is to determine the 3-D geometry of the complex (the 3-D distribution of electron scattering intensity) which requires fusing information from these many images of many complexes. In order to describe the geometric heterogeneity of the complexes, the complex is described as a weighted sum of basis functions where the weights are random. In order to get tractable algorithms, the weights are modeled as Gaussian random variables with unknown statistics and the noise is modeled as additive Gaussian random variables with unknown covariance. The statistics of the weights and the statistics of the noise are jointly estimated by maximum likelihood by a generalized expectation maximization algorithm. The method has been developed to the point where it appears to be able to solve problems of interest to the structural biology community.