Alignment error envelopes for single particle analysis

To determine the structure of a biological particle to high resolution by electron microscopy, image averaging is required to combine information from different views and to increase the signal-to-noise ratio. Starting from the number of noiseless views necessary to resolve features of a given size, four general factors are considered that increase the number of images actually needed: (1) the physics of electron scattering introduces shot noise, (2) thermal motion and particle inhomogeneity cause the scattered electrons to describe a mixture of structures, (3) the microscope system fails to usefully record all the information carried by the scattered electrons, and (4) image misalignment leads to information loss through incoherent averaging. The compound effect of factors 2-4 is approximated by the product of envelope functions. The problem of incoherent image averaging is developed in detail through derivation of five envelope functions that account for small errors in 11 "alignment" parameters describing particle location, orientation, defocus, magnification, and beam tilt. The analysis provides target error tolerances for single particle analysis to near-atomic (3.5 A) resolution, and this prospect is shown to depend critically on image quality, defocus determination, and microscope alignment.