Non-sinusoidal bending waves of sperm flagella.

The analyses of flagellar movement stimulated by Sir James Gray's (1955) photographic study of the movement of sea-urchin spermatozoa have used sine waves as a convenient mathematical model for the active bending waves of flagella. Machin (1958), following a suggestion of Pringle (1957), outlined an attractively simple model for the mechanical co-ordination of bending elements distributed along a flagellum to give propagated sine waves. However, in a later paper, Machin (1963) showed that these bending elements must behave non-linearly, in which case the bending waves will probably not be sinusoidal. Brokaw & Wright (1963) presented photographs showing that in at least one case—the large posterior flagellum of the dinoflageUate Ceratium—the bending waves are not sine waves, but instead contain regions of constant bending, forming circular arcs, separated by shorter unbent regions. In this paper photographs of flagellar bending waves of spermatozoa from marine invertebrate representatives of three animal phyla will be presented and discussed with reference to the description of bending waves proposed by Brokaw and Wright. Differences have been observed between the spermatozoa of these three species in respect of their movements under certain experimental conditions, but the common features of their movements will be emphasized in this paper.