Wilson Statistics: Derivation, Generalization, and Applications to Electron Cryomicroscopy

The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in electron cryomicroscopy (cryo-EM). Specifically, modern computational methods for three-dimensional map sharpening and atomic modelling of macromolecules by single particle cryo-EM are based on Wilson statistics. Here we provide the first rigorous mathematical derivation of Wilson statistics. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher order spectra. These in turn provide theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation based methods in cryo-EM.