focus here. Influences by inhomogeneous illumination, chemical gradients, thermal gradients, flow, unevenness or other anisotropy of the substrate, such as inclination to the horizontal, cannot be completely excluded in the experiment, but were avoided where possible. Such controlled conditions can also be achieved over a longer observation period by the use of diatoms cultures. All observations described below were performed either directly using batch cultures in Petri dishes or after transfer of the diatoms from the culture into a suitable test vessel.
Figure 1.1 Drawing of a pennate diatom with two raphe branches on its valve.
The analysis of the trajectories of diatoms has long been of qualitative nature. This has changed with the use of video technology, manual or automatic tracking of motion and the use of computers to analyze movement data. Edgar [1.10] determined speeds and accelerations of diatoms and the speed of particles transported along the dorsal raphe of Nitzschia sigmoidea. A statistical analysis of the trajectory of Navicula sp. was published by Murase et al. [1.23] whereby the movement of the diatoms was confined by a micro chamber. Murguía et al. [1.24] performed a time series analysis of the Hurst exponent. The above-mentioned experiments on chemotaxis [1.5] [1.6] are based on the statistical evaluation of frames of video recordings.
1.2 Kinematics and Analysis of Trajectories in Pennate Diatoms with Almost Straight Raphe along the Apical Axis
In the following, pennate diatoms will be considered whose raphe system is almost straight and is located centrally between the apices. Furthermore, it is assumed that the valve has a convex surface, so that the diatom contacts a flat smooth substrate with only one point P of its valve. When the diatom moves on such a substrate, it can therefore be assumed that the driving force acts at the contact point P or in its immediate vicinity.
Observations on diatoms of the genera Navicula, Craticula or Stauroneis show at least slightly curved and often circular or spiral paths in valve view. The radii of the trajectories in these genera are large compared to the size of the diatoms. Particularly in smaller species, the orientation of the apical axis shows random fluctuations around the direction of movement, which are visually perceived as “wagging.” The question arises whether and in what way the position of P has an influence on this movement and what consequences this has for the analytical description of the movement. If there is such an influence, then changes in the position of P are also relevant.
Two hypotheses are formulated which connect the position of point P with the movement. The first hypothesis states that the alignment of the diatom is tangential to the trajectory of the diatom at the point of contact P, except for random fluctuations. It is based on the assumption that at the contact point the direction of the propulsion force lies in the direction of the raphe with only minor deviations. In Figure 1.2 the hypothesis is visualized with a point P near the leading apex. Since P is not located in the center of the cell, the radii of the trajectories of the apices A1 and A2 differ. An analysis of the movement of the diatoms from a view perpendicular to the valves allows answering the question whether there is a point B on the connecting line of the apices, with the property that the apical axis lies tangentially to its trajectory on statistical average. A match between the kinematic point B and the contact point cannot be proven in this way.
Figure 1.2 Hypothesis that there is a point P between apices A1 and A2, so that the apical axis is tangential to the trajectory of P.
Figure 1.3 Traces of two trackers attached close to the apices of a diatom of Navicula sp.
To investigate the hypothesis, individual diatoms from a culture were transferred to a Petri dish of polystyrene and their movement was recorded under an inverted microscope. Using the Video Spot Tracker1 software tool, movement data can be obtained from the video files, which include the orientation of the diatom. For this purpose, two circular trackers can be attached to the apices of the diatom in the first analyzed frame. Sometimes the use of a rectangular tracker proves to be more suitable. The tracker coordinates determined for each frame are imported into a spreadsheet program for analysis, using scripts for complex evaluations.
Figure 1.3 shows a diatom of the genus Navicula with two trackers and their trajectories. As in Figure 1.2, the curve of the tracker trailing behind has the larger radius. Fluctuations are clearly visible, mainly due to random changes in the orientation of the apical axis. Because of these fluctuations, long paths between reversal points were preferred for analysis. The numerically determined trajectory of a hypothetical point x between the trackers is smoothed by means of a finite impulse response (FIR) filter (low pass). Then the angle between the tangent in x with the apical axis can be determined. The root-mean-square deviation (RMSD) of these angles over all frames of the included video sequence serves as a measure for the deviation from meeting the tangential condition. When x is varied, the curve shown in Figure 1.4 is obtained in this example. The positions of the trackers are used as a reference for the position of x. The value x = 0 on the abscissa corresponds to a match of P with the tracker trailing behind, the value x = 1 corresponds to the leading tracker and the value x = 0.5 represents the center of the diatom. In accordance with the hypothesis, the variance has a minimum, which is about 0.65 in this case. As the trackers do not sit exactly on the apices, but are slightly shifted inwards, a correction must be made to determine the position of the minimum with respect to the apices. On the normalized line segment A₁A₂, the minimum and thus the position of B is at x = 0.62. B is located on the side of the leading apex, as shown in Figure 1.2. To illustrate the correctness of the evaluation, the frequencies of the angular deviation between the smoothed trajectory in B and the apical axis were calculated (Figure 1.5). According to the hypothesis, the density function is in good approximation symmetrical to the origin. The standard deviation amounts to 4.15 degrees. The minima of the RMSD for 10 analyses of the same Navicula sp. were between 0.58 and 0.77, thus yielding similar values. In contrast, in Craticula cuspidata the minimum RMSD is typically at around 0.2, so that B is close to the trailing apex.
To clarify the question of whether the determined point B actually corresponds to the contact point P, the diatoms were viewed from an angle that is almost parallel to the substrate. For observation, a coverslip can be placed almost vertically in a Petri dish with diatoms and examined with an inverse microscope. If there are enough diatoms in the Petri dish, after some time diatoms will have migrated onto the coverslip. Otherwise, the coverslip is first laid flat on the bottom of the Petri dish, diatoms are placed on its surface and then carefully tilted vertically.
