Читать книгу Diatom Gliding Motility - Группа авторов - Страница 18

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.

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 viewed from a horizontal perspective. The transapical axis is inclined against the substrate.

In the natural habitat of diatoms there are usually no comparably flat substrates. Trajectories of Craticula cuspidata on a stone from the habitat of diatoms exhibit to a certain extent the typical orbital curvatures, but also disturbances in the direction of motion and frequent reversals, so that they often showed more of the appearance of a random walk. It is to be expected that a substrate of high roughness compared to the size of the diatom will lead to changing and even to simultaneous contact points in all motile diatom species. In the case of large irregularities, the methods presented for determining P fail. The relevance of observing trajectories under laboratory conditions may be considerable, but conclusions about the natural environment are limited. After observing the circular random motion of Nitzschia communis, Gutiérrez-Medina et al. [1.16] came to the conclusion that this motion is not optimized for long distances but for covering a limited 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.