Figure 1.4 Root-mean-square deviation of the angle between the apical axis and the smoothed trajectory of the point x located between the trackers.
Figure 1.5 Histogram of the frequencies of the angular difference between the direction of the diatom (apical axis) and the smoothed curve in P.
Figure 1.6 Craticula cuspidata observed from an almost horizontal view.
Figure 1.6 shows a diatom of the species Craticula cuspidata and its mirror image from an almost horizontal view. On longer sections the diatom moves to the right at this inclination. The inclination of the apical axis to the substrate in this image is about 7.5°. However, a considerable fluctuation of the inclination occurs, so that the contact point P also changes. According to the hypothesis, one finds that the numerically determined value of B is within the range of the variation of P. Due to the fluctuations of P, the position of the minimum of RMSD yields just an average value for P.
The second hypothesis exclusively concerns the fluctuations of the apical axis around the direction of motion. It says that this is a rotation about the point P (Figure 1.7). From the viewing direction perpendicular to the substrate, only the existence of a pivot point T can be concluded, although a match of a pivot point with the contact point seems to be obvious.
Figure 1.7 Hypothesis that there is a point P between apices A1 and A2, so that the diatom performs stochastic rotary motions around P.
Figure 1.8 Root-mean-square deviation of the transverse component of the fluctuations of the hypothetical pivot point.
In order to determine the pivot point, a smoothed trajectory is calculated for a hypothetical point x between the apices and the motion of x is split into a component in the direction of the smoothed curve and a perpendicular, i.e., transversal, direction. The transversal component disappears in the ideal case when x coincides with the pivot point. Due to the random fluctuations of T this is not exactly the case. For the trajectory shown in Figure 1.3, the RMSD of the transverse fluctuations against x is plotted (Figure 1.8). In accordance with the hypothesis, a distinct minimum is apparent, whose position is in good agreement with the obtained value of B.
The two approximation methods for determining the contact point evaluate different information and differ in their applicability. The algorithm described first is based on the orientation of the tangents and cannot be used for straight sections. On the other hand, the method of using the rotation does not make any assumptions about the shape of the paths. However, it requires a sufficiently well observable fluctuation of the orientation of the diatom. None of the methods is suitable for almost straight paths without significant directional fluctuations. In addition, the statistical methods require sufficiently long path sections between the reversal points. In principle, both numerical methods could be generalized to more universal forms of raphes. When determining P, only hypothetical points lying on a numerically modeled raphe are to be used.
In all observed species of the genus Navicula the point P was closer to the leading than to the trailing apex. Within the proposed interpretation, the driving force is acting in the leading half of the diatom. To put it simply, the diatom is pulled. In contrast, in the case of Stauroneis sp. and Craticula cuspidata, point P is near the trailing apex. These diatoms are pushed. In some species there is no clear statement regarding P, which could be due to frequently changing positions of the contact point. From the observations it cannot be concluded in which respect the different positions of the propulsion have significance for a species and which advantages or disadvantages they possess.
When the diatom stops moving and then reverses, the point P in the following section of the path is located again on the same side in relation to the direction of movement. With respect to the cell, it changes to a point on the other branch of the raphe, which lies approximately symmetrically to the center. There are two alternatives for changing the position of P when reversing direction: Either the diatom tilts around the transapical axis at the reversal point so that the opposite raphe branch comes into contact with the substrate. Then the reversal of direction would be the result of an opposite activity of the raphe branches. Or the direction of activity of the driving raphe changes first, so that the direction is reversed at an identical contact point. Then the contact point should shift after the reversal. These alternatives are shown in Figure 1.9 for a diatom of the species Craticula cuspidata, which touches the substrate on longer paths at a point near the trailing apex. The observation of Craticula cuspidata from a horizontal point of view reveals that at reversal points a change of direction occurs first. Within a few valve lengths the apical axis then tilts to the other raphe branch. It obviously has the same direction of activity; otherwise there would be another reversal of direction, this time according to the alternative scheme. It is conceivable that the rapid back and forth jerking observed in some diatom species is caused by such an alternation of driving raphes having opposite activity.
When viewed from a horizontal perspective, it can be seen that the transapical axis is usually inclined against the substrate (Figure 1.10). In this case, the raphe is located at the edge of the contact area. This causes an asymmetric friction with respect to the raphe, with adherent extracellular polymeric substances (EPS) being relevant. The often surprisingly small radii of curvature of the trajectories compared to the curvature of raphe could be a consequence of this asymmetry.
Figure 1.9 The left side (a) illustrates the sequence of steps for reversal of direction, in which the tilting takes place after the direction of motion has been changed. In alternative (b), tilting takes place before reversing the direction.
Figure 1.10 Craticula cuspidata
