bi-flagellate swimmers in a shear flow
5.3 Estimating the effective cell eccentricity
5.3.1 Averaging technique for estimating the effective eccentricity
Requiring that the number of time-steps equals the number of recorded aspects of the sampled data reduces the computational time and storage requirements when estimating α0. However,
0 20 40 60 80 100
Figure 5.7: The eccentricity against the number of individual aspects of the flagellar beat (time-steps), T .
even with only 11 time-steps solving the mobility problem is still computationally expensive and time-consuming. Furthermore, the current approach requires high-speed cameras to capture the complete ordered sequence of flagellar positions. Ideally it would be desirable to be able to apply a technique to find the effective eccentricity from a disordered collection of images. Therefore, we consider a cell with a beat pattern comprised of a single stroke. The stroke represents the average position of the flagella over the course of a single normal flagellar beat. We consider two averages, the mean and median over time of individual node positions. Estimates obtained using the mean flagellar beat will be denoted ¯α0 and estimates using the median technique are denoted ˜α0.
The average beat patterns are displayed in Figure 5.8, solid lines, and the estimates for the effective eccentricity are displayed in the third column of Table 5.2. The results shown are for the mean of a fifty time-step beat pattern. Comparisons between the eccentricity calculated using the full beat pattern, α0, and using the average beat pattern, ¯α0, are shown in Figure 5.9. For the I and R beats ¯α0, over estimates α0 by 14% and 3%, respectively, whereas ¯α0 for the F and RNL beats under estimates α0 by 11% and 24%, respectively. However, the estimate for the RNR beat is a quarter of the full beat estimate. The reason for such a large discrepancy appears to be due to a consistent error across all beat patterns; the estimated values, ¯α0, are all within ±0.05 of the converged values of α0, and because ¯α0 for the RNR beat is so close to zero the error is
(a) I beat (b) F beat
(c) R beat (d) RN beats
Figure 5.8: The average beat patterns for the five distinct beats. The solid flagella represents the mean value over time of the individual node positions. The dashed line represents the median over time of the individual nodes.
more noticable. Since the estimated values can be obtained within ±0.05 of the full beat values it implies that the mean technique provides a computationally and experimentally cheap means of establishing the effective cell eccentricity.
Table 5.2: Estimates for the effective cell eccentricity for the five distinct beat patterns. The estimates are calculated using three techniques; α0 estimate using full beat pattern, ¯α0 employing a mean flagellar beat, and ˜α0 using the median of the beat pattern
Beat Pattern α0 α¯0 α˜0 I 0.3381 0.3838 0.3935 F 0.2337 0.2086 0.1740
R 0.3530 0.3618 0.4950
RNL 0.1503 0.1130 0.0728 RNR 0.0648 0.0153 -0.1065
The results for the second technique using the median of the individual nodes in a single flagellar beat are shown in the last column in Table 5.2, the beat patterns are displayed in Figure 5.8 as dashed lines. While this technique preserves the length of the flagella to a greater degree than the mean approximation the results are not as good as those achieved using the mean technique.
There also seems to be no consistent error across the beat patterns and consequently, adopting the median approach is not beneficial.
10 20 30 40 50 0
0.2 0.4
I F R RNL RNR
10 20 30 40 50
0 0.2 0.4
T T α0¯α0
Figure 5.9: A comparison of the numerical estimates for the effective eccentricity computed using two different techniques. The top figure shows the results using a full beat, whereas the bottom plot shows the results using an average beat. The eccentricity is plotted as a function of the number of time-steps T.
5.4 Discussion
In this chapter we have examined how the shear flow affects various properties of the cell’s swimming dynamics. For small values of e we observed increases in the translational and angular velocity compared to a free-swimmer in an ambient fluid. However, for large e we observe that the cell tumbles due to the vorticity of the fluid and obtaining estimates for swimming speeds etc is not meaningful. For the realistic RN beats we observe that the shear has a greater effect on the recovery stroke, when the flagella lie at the anterior end of the cell, than the effective stroke, when the flagella lie perpendicular to the cell body. Limiting the effects of shear may be a possible reason as to why Chlamydomonas have an average flagellar extension, during the effective stroke, along the cell’s minor axis.
By considering the mean swimming behaviour of a cell within a shear flow we are able to conclude that the intricacies of the flagellar geometry have an enormous impact on the effective behaviour of the cell. Previous studies had assumed that a swimming bi-flagellate could be well represented as a self-propelled spheroid. Here we have adopted a similar approach and established that while idealisations of bi-flagellate locomotion are consistent with previous estimates of the effective eccentricity. The estimates for the realistic beat patterns suggest that cells are best described as self-propelled spheres. Providing such a simplification may be pertinent in the studies
of the collective dynamics of cells.
Additionally, we have outlined a procedure by which the effective cell eccentricity can be esti-mated, within a known error bound, by considering a time-averaged flagellar beat. Exploiting such a procedure offers benefits to both reductions in computational time and the ability to estimate the eccentricity to a wider range of organisms and circumstances. In the latter case we can compute the eccentricity given unordered data as well as situations where the beginning and end of the beat are vague.