Three-dimensional reconstruction of single particles from random and nonrandom tilt series

To overcome the radiation damage-induced limitations to the resolution of three-dimensional reconstructions from electron microscopic tilt series, novel reconstruction schemes have been developed that require only a single exposure of the specimen. The tilt series collected with these methods have random projection directions. First, three-dimensional reconstruction techniques are described that are applicable to data obtained from tilt series with regular tilt geometry, followed by the extensions of these techniques to permit analysis of projection series with randomly spaced tilts. The main emphasis is placed on the weighted back-projection methods, which have recently been extended so as to be applicable to random tilt series. Besides a description of the algorithms, the complete procedure for a three-dimensional reconstruction from a single-exposure, random conical tilt series is explained, including the determination of the azimuthal angles, the alignment scheme for conical tilt series, the dependence of the achievable resolution on the number of projections for regular conical and single-axis geometries, and the method to calculate the actual resolution of two-dimensional image averages and of three-dimensional reconstructions using the phase residual and Fourier ring correlation criteria. Examples are given of biological specimens to which these three-dimensional reconstruction methods have been applied.