Bending moments in free-swimming flagella.

The bending of a flagellum, such as the flagellum which forms the tail of a seaurchin spermatozoon, is the result of active mechanochemical processes occurring within the flagellum itself, which cause it to bend actively throughout its length (Gray, 1955; Machin, 1958). To understand how flagella operate, we must understand not only the mechanisms which generate bending, but also the control mechanisms which initiate and coordinate the active bending of different parts of the flagellum in order to generate smoothly propagated bending waves which will efficiently propel a cell. Spermatozoa, such as sea-urchin spermatozoa, which propagate nearly planar waves of bending along their flagella, can be readily photographed to reveal the shape of the flagellum during active movement. Such photographs indicate that the bending waves are ordinarily composed of bent regions, in which the flagellum is bent into a circular arc, separated by shorter regions in which the flagellum is approximately straight (Brokaw & Wright, 1963; Brokaw, 1965). As bent regions move along a flagellum, bending and unbending of the flagellum are therefore occurring only in rather short transition zones between the bent and straight regions. These transition zones, which I will refer to as bending or unbending points, move relatively uniformly along the flagellum to produce propagated bending waves. After sudden breakage of the flagellum by a laser microbeam, bent regions already established in the distal fragment of the flagellum continue to propagate to the end of the flagellum, although no new bent regions are formed (Goldstein, 1969). Therefore, either the bent region as a whole, or the bending and unbending points at its ends, appear to be autonomous, self-propagating, events. Other aspects of this picture of flagellar bending have been discussed elsewhere (Brokaw, 1966a, 1968). The present paper has been motivated by the following question: Is the propagation velocity of a bent region, or of its bending and unbending points, determined by the resistance which these events encounter as they move from point to point along the flagellum? Since the propagation velocity can vary, for example, when the viscosity of the medium is altered (cf. Brokaw, 19666), some mechanism must exist to control this potentially variable velocity, in order to produce a particular bending wave pattern. The resistance to bending (or to unbending) can be expressed as a bending moment which will be a function of time and position along the length of the flagellum. This