area. The authors assume that the biological role is the efficient formation of a biofilm. This conclusion is based on comparable isotropic surfaces in nature. In the case in which the trajectories of the diatoms are severely disturbed in their habitat, a movement in the form of a typical random walk would allow bridging larger distances.
1.3 Curvature of the Trajectory at the Reversal Points
Additional information on the relationship between the shape of the raphe and the propulsion can be obtained from the behavior of the curvature of the trajectory during reversal of motion. For certain shapes of the raphe, it is possible to derive plausible statements about the location of propulsion from the curvature of the path. In many cases at least the assignment to the driving raphe branch is feasible. The absolute value of the curvature will not be the focus of the discussion, but the direction or sign of the curvature. The curvature is defined as positive if the diatom moves counterclockwise from the observer’s point of view. If it moves clockwise, the curvature is negative. An important prerequisite for the following considerations therefore is that the trajectories show clearly recognizable curved and smooth path sections, which implies sufficiently long paths. Furthermore, only movements in which the raphe is in direct contact with the substrate are regarded. We limit ourselves to uncomplicated changes of direction in which the diatoms do not rotate around the apical axis or pervalvarous axis and do not straighten up [1.2]. Many motile diatoms meet these requirements over a sufficiently long observation time. Diatoms whose valves are symmetrical with respect to the transapical plane shall be considered, such as diatoms of the genera Navicula or Rhopalodia, but not sigmoid forms. Figure 1.11 shows the movement of such a diatom with reversal points. The curvature of the raphe is intentionally exaggerated compared to the curvature of the trajectory. The division into two raphe branches shown in the picture and the possibly changing position of point P does not play a significant role here. Furthermore, it is irrelevant whether P is on the side of the leading or trailing apex. But it is essential that the position of P changes only between points whose centers of curvature are on the same side of the diatom. If one assumes that the curvature of the path follows the curvature of the raphe or that a constant lateral oblique position determines the path curvature due to friction effects, then the direction of curvature always changes at the reversal points. This behavior can be well observed in Navicula sp. (Figure 1.12).
The behavior becomes more complex when regions of the raphe with different directions of curvature can act as the center of the propulsion. A Cymbella (Figure 1.13) does not meet the requirement of a simple numerical analysis for the determination of P. According to the predominant curvature of the raphe, it is reasonable to assume that the diatom would move on curved paths in such a way that the center of curvature is located on the ventral side. This can be observed in many Cymbella spp. and in these cases the sign of curvature alternates at the reversal points. In Cymbella cistula and probably other Cymbella spp. the center of curvature is surprisingly often on the dorsal side. A change of the center of curvature between dorsal and ventral often occurs at the reversal points, but also during an uninterrupted movement (point of inflection). Here, both alternating and identical directions of curvature exist. When looking at the path of the leading apex at the dorsal center of curvature, the alignment of the diatom is approximately tangential to it (Figure 1.14). There seems to be a point near the apex from which the diatom is drawn. A dorsal center of curvature corresponds to the direction of curvature of the helictoglossa, so that the propulsion should originate from it. Often the movement at the dorsal center of curvature changes into a rotation around the apex in the same direction of rotation (Figure 1.15). A movement in which the diatom is pushed from the distal end of the raphe could not be observed, but occasionally a reversal of the direction of rotation.
Figure 1.11 Trajectory of a diatom with reversal points. The point P never changes to a place of the raphe with opposite curvature.
Figure 1.12 Path of a diatom of the genus Navicula with reversal points. The direction of curvature changes at each reversal point.
Figure 1.13 Outline and raphe of a Cymbella.
As a last example, the behavior at direction reversal of Surirella biseriata will be presented. Figure 1.16 shows that two raphe sections symmetrical to the apical axis can cause the movement when the diatom lies on its valve. Depending on the lateral inclination, one or the other raphe couples to a flat substrate. Accordingly, paths with a center of curvature occur on one side or the other of the diatom. Occasionally, the coupling can change between the sides, leading to sigmoid trajectories. At the reversal points, a conservation of the direction of curvature is most frequently observed. In this case the driving side of the diatom changes. Figure 1.17 shows the principle and Figure 1.18 an image that was created by superimposing frames from a video. The direction of movement only rarely reverses with the driving side being retained, which leads to a changing direction of curvature.
Figure 1.14 Overlay of four images from a video showing a trajectory of a diatom of the species Cymbella cistula. The yellow line shows the trajectory of the leading apex. The line segment between the apices is marked in white.
Figure 1.15 On the left (a) the superimposition of images of a Cymbella rotating around a point near the helictoglossa is shown, on the right (b) a sketch of the diatom with raphe.
Figure 1.16 Surirella biseriata in valve view.
Figure 1.17 Trajectory of a Surirella biseriata. The driving side changes at the reversal points.