Bending Moments in Free-Swimming Flagella*

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Abstract
Starting with the simple hydrodynamic assumptions of earlier work on flagellar movement, an exact analysis of the bending moments in a flagellum moving through a viscous solution has been developed. The results illustrate the variation in the resistance to bending and unbending resulting from viscous forces as a function of time as bends move along a flagellum. These bending moments are o at free ends of the flagellum, and rise to peak values of 2–3 times their average value near the middle of the flagellum, for movement patterns similar to those normally found on sea-urchin spermatozoa. The approximate expressions for bending moments and energy expenditures developed in earlier work are reasonable estimates of the average values of these quantities. Bending and unbending points propagate at reasonably constant velocities in the distal portion of free-swimming flagella, in spite of the large variations in the resisting moment resulting from viscous forces. The propagation velocity of individual bending and unbending points, or an individual bent region, is therefore probably not controlled by the resistance to bending and unbending at these points. Some mechanism for internal coordination of the activity of different regions of the flagellum appears to be required. In flagella, as in cilia, the external work done against viscous resistances is not uniformly distributed along the length of the flagellum. These conclusions provide additional support for ‘sliding filament’ models of flagellar bending.

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