The coexistence of multiple distinct structural states often obstructs the application of three-dimensional cryo-electron microscopy to large macromolecular complexes. Maximum likelihood approaches are emerging as robust tools for solving the image classification problems that are posed by such samples. Here, we propose a statistical data model that allows for a description of the experimental image formation within the formulation of 2D and 3D maximum likelihood refinement. The proposed approach comprises a formulation of the probability calculations in Fourier space, including a spatial frequency-dependent noise model and a description of defocus-dependent imaging effects. The Expectation-Maximization like algorithms presented are generally applicable to the alignment and classification of structurally heterogeneous projection data. Their effectiveness is demonstrated with various examples, including 2D classification of top views of the archaeal helicase MCM, and 3D classification of 70S E.coli ribosome and Simian Virus 40 large T-antigen projections.
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