Generating motion with cilia or flagella

Microorganisms operate at Reynolds numbers (Re) << 1, and so in a world where all fluid motions are reversible, they are excluded from any form of propulsion that makes use of the inertia of the water. Octopuses or jellyfish shrunk to the size of protozoa and trying to move would simply be moving back and forth on one spot. In order to move, microorganisms instead make use of the difference in drag of a cylinder moving perpendicular compared to parallel to the flow; the resistance to normal motions of a cylinder is somewhat higher than the resistance to tangential motions. This is the principle behind motion by smooth and hispid flagella as well as cilia. Because of the insignificance of inertial effects at low Re, the motion of an object is only possible as long as a force acts upon it. If one attempted at low Re to throw a ball, it would never leave the hand (see, e.g., Ref. [19]).

13.2.1 Smooth Flagella

Because of the differences in drag, a moving cylinder tilted toward the direction of motion will exert a force on the fluid normal to the length of the cylinder (Figure 13.3). This is the principle behind flagellar propulsion first noted by Taylor [20,21]. Thus, contrary to appearance, the mechanics of flagellar motion is more closely related to that of a snake moving through sand than that of eels or water snakes swimming. Motion is generated by the propagation of planar or three-dimensional helical waves along the length of the flagellum. This generates a force normal to the segments of the flagellum that are tilted to the direction of the wave propagation (Figure 13.3). The propulsive effect depends on this force exceeding the retarding components of tangential forces acting along the body [22-24].

FIGURE 13.3 The generation of motion with smooth (upper) or hispid (lower) flagella. The black arrows represent propulsive force or thrust, the gray arrow shows the direction of flagellar wave propagation, and the white arrows indicate the overall direction of the resulting fluid motion with respect to the cell.

FIGURE 13.3 The generation of motion with smooth (upper) or hispid (lower) flagella. The black arrows represent propulsive force or thrust, the gray arrow shows the direction of flagellar wave propagation, and the white arrows indicate the overall direction of the resulting fluid motion with respect to the cell.

The fluid dynamics of bacterial and eukaryotic flagella are similar, but they differ in all other respects. Eukaryote cilia and flagella are around 0.2 ^m in diameter and composed of the well-known 9 + 2 structure of inflexible microtubules that slide relative to each other. The bacterial flagellum is about 0.02 ^m in diameter and is in itself completely immobile. It is composed of molecules of the protein flagellin that form a hollow tube. Perhaps unique in the biological world, the bacterial flagellum rotates continuously around its own axis [25] because of two rings that rotate relative to each other [26-28]. In this way, helical waves are propagated along the flagellum. Bacteria often have many flagella, which tend to form a bundle or bundles because beating filaments in the vicinity of each other tend to be synchronized through viscous coupling [20,29]. The flagella of this bundle are only separated during tumbles (see Section 13.5).

Both helical and planer waveforms have energetic disadvantages. In planar waves, the segments of the flagellum nearly parallel to the wave direction produce only drag and no thrust [30]. All segments of the helical waves produce thrust but also generate a torque on the organism that must be counterbalanced by the counter-rotation of the cell body. This reduces the swimming speed proportionally to the effective rotation rate [31]. It has been proposed that the body movements of microorganisms rotated around their axis could contribute to the thrust in the manner of a rotating inclined plane [32,33], but this contribution would in most cases be negligible [34]. One interesting exception is the bacterium Spirillum, which has a spiral-shaped body. It uses the flagella to generate a rotation of the cell body, which then moves through the fluid much in the manner of a corkscrew through a cork [35-37].

13.2.2 Hispid Flagella

Hispid flagella have rigid hairs, or mastigonemes, protruding from the flagellum (Figures 13.1B and 13.3). They are curious in that they pull the cell body in the same direction as that of the wave propagation, opposite to that of smooth flagella. This is because the movement of the individual mastigonemes produces thrust in the direction of the cell body (Figure 13.3), and given sufficiently large numbers of mastigonemes, it is predominantly these, and not the flagellum itself, that moves the fluid [38,39]. The number and characteristics of the mastigonemes required may depend on the relative amplitude and wavelength of the flagellum [38,40,41]. Theoretically, only relatively inflexible mastigomenes should be capable of moving the fluid [39,42]. Dinoflagellates, however, have mastigonemes that appear flexible, and yet they always rotate counterclockwise in the direction of the flagellar beat of the transverse flagellum [43]. Thus, it seems that our understanding of the functioning of mastigonemes is not yet complete.

Flagellates with hispid flagella are common in all aquatic habitats, and this type of flagellum is present in a number of unrelated families. Because the presence of mastigonemes is not a primary character, it must provide important competitive advantages in microbial ecosystems. The importance of mastigonemes in evolution and ecology is, however, as yet poorly understood. They could improve the swimming efficiency of the cells, because the previously described energetic disadvantages of smooth flagella are not present in hispid flagella. The flagellum itself, however, will work against the motion of the fluid, so the picture is not clear. Another possibility is that the mastigonemes can function as mechanoreceptors or in feeding because the anterior position of the hispid flagellum would make it well placed to work also as a sensory or food-intercepting organelle. The mastigonemes also make it possible for hispid flagella to move fluid across or even perpendicular to the flagellar axis, something that is not possible for smooth flagella [44]. It has already been shown that Paraphysomonas uses its flagellum for intercepting prey and increases its effective feeding area by utilizing this possibility [44]. This, however, may be a special case.

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