Analysis of deformation and contraction in cylindrical crystals due to dispirations or dislocations

Abstract

Cylindrical crystal structures are common in biology. The shape changes and movements of cylindrical crystals are basic to the understanding of the contractile mechanisms in biological systems such as tobacco mosaic viruses and the tail sheath of T-even bacteriophages. It has been suggested that the concept of defects in crystal physics can be applied to study these contractile mechanisms. The defect believed to be responsible for the shape changes of cylindrical crystals is known as a dispiration. Dispirations are characterized by the shear displacement on the slip plane through a screw symmetry operation. The elastic field of a dispiration can be decomposed into its translational (dislocation) and rotational (disclination) components. The magnitude of the translational and rotational displacements in a cylindrical crystal has been related to the crystal structural parameters. The passage of a dispiration along a helical plane in a cylindrical crystal can induce one of two types of shape changes. In one type, only the disclination component of the dispiration contributes to contraction, whereas in the other type, both the disclination and dislocation components are responsible for the shape change. Estimates of the magnitude of contraction are made in terms of the dimensional and structural parameters of the cylindrical crystal. The reversal of the direction of helical slip results in extension instead of contraction of the cylindrical crystal. The local elastic deformation of a dispiration dipole situated on the helical plane of a cylindrical crystal is examined. It has been shown that, for the first type of deformation mentioned above, closed form solutions of the stress field can be obtained by superposing the stress fields of two dispiration dipoles with slip planes parallel and normal to the cylinder axis, respectively. The approximations of shallow shell theory are adopted in the analysis. Future problems of biological interests are identified.