Computer image processing of electron micrographs of biological structures with helical symmetry

Methods are described for the analysis of electron micrographs of biological objects with helical symmetry and for the production of three-dimensional models of these structures using computer image reconstruction methods. Fourier-based processing of one- and two-dimensionally ordered planar arrays is described by way of introduction, before analysing the special properties of helices and their transforms. Conceiving helical objects as a sum of helical waves (analogous to the sum of planar waves used to describe a planar crystal) is shown to facilitate analysis and enable three-dimensional models to be produced, often from a single view of the object. The corresponding Fourier transform of such a sum of helical waves consists of a sum of Bessel function terms along layer lines. Special problems deriving from the overlapping along layer lines of terms of different Bessel order are discussed, and methods to separate these terms, based on analysing a number of different azimuthal views of the object by least squares, are described. Corrections to alleviate many technical and specimen-related problems are discussed in conjunction with a consideration of the computer methods used to actually process an image. A range of examples of helical objects, including viruses, microtubules, flagella, actin, and myosin filaments, are discussed to illustrate the range of problems that can be addressed by computer reconstruction methods.