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

Abstract

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.

Novel Actin-like Filament Structure from Clostridium tetani

“…Note that the handedness was not determined yet at this stage. We initially assumed the selection rule as l ϭ Ϫ1n ϩ 27m (right-handed helix-like F-actin) and produced a three-dimensional structure by using helical reconstruction (25). The layer lines up to the 72nd order (ϳ19.5 Å) were included.…”

Modular Design of Self-Assembling Peptide-Based Nanotubes

“…This absence of layer lines without zero-order Bessel functions is intriguing and indicates that the peptide length and the overall coiled-coil pitch in this case must be related by an integer value, and that this must be 6, the order of the rotation axis (Supporting Information Fig. S10) 29 . Thus, the predicted pitch is 6 x 4.3 nm = 25.8 nm, which is experimentally indistinguishable from that predicted from the Xray crystal structure of CC-Hex (25.88 nm) 22 .…”

Bacteriophage SPP1 Tail Tube Protein Self-assembles into β-Structure-rich Tubes

“…The classes that produced the most symmetrical diffraction patterns were analyzed further. The maxima intensities on a layer line are defined by a distance R from the meridian and radius r of the virus that was described by a Bessel function J of order n: J n (2Rr) (25). The values of Bessel function orders were determined, and the parameters of the tube helix were defined using the selection rule for the helical symmetry as described in Stewart (25).…”

“…The maxima intensities on a layer line are defined by a distance R from the meridian and radius r of the virus that was described by a Bessel function J of order n: J n (2Rr) (25). The values of Bessel function orders were determined, and the parameters of the tube helix were defined using the selection rule for the helical symmetry as described in Stewart (25). The distance between gp17.1 rings was determined from the positions of layer lines in the diffraction patterns, and the angle of rotation around the tube axis between the rings was derived from the helical parameters of the tube.…”