Torsion, Twist, and Writhe: the Elementary Geometry of Axonemal Bending in Three Dimensions

The cilia and flagella of eukaryotic cells function by active bending that produces cell movements and fluid flows in their environment. They contain a common cytoskeleton, known as the axoneme, that is the locus of all the enzymes and mechanical components that are required for production of useful bending. The bending of many flagella, and some cilia, is planar, or at least nearly planar. In these cases, the usual observation is that bends are formed near the basal end and then propagate towards the tip. The flagellum of a sea urchin spermatozoon is a classic example [Gray, 1955; Brokaw, 1990]. Planar bending is easy to record as planar photographic images, and is mathematically simple. The shape of the bent flagellum can be described mathematically by a scalar variable, k, that measures the curvature of the flagellum as a function of length along the flagellum. There are also many cilia and flagella that generate bending in 3 dimensions. These bending patterns are more difficult to record and analyze, and their mathematical description is more complex. However, it is important to understand this complexity in order to understand how a flagellum generates 3-dimensional bending. The pioneering analysis is the paper by Hines and Blum [1983], and to the extent possible, their conventions have been followed here. Mathematical analysis of 3-dimensional bending is simplified by restricting attention to bending of a structure such as a flagellum. Neglecting pathological cases where a flagellum is broken, the elastic bending resistance of the flagellum smooths its bends so that its curvature is a continuous function of length. Restricting the analysis of 3-dimensional bending to cases where the curvature is always greater than 0 (except possibly at the ends of the flagellum), is sufficient to handle most cases of flagellar bending.