Table 1.
Steady (Re) and oscillatory (Reω) Reynolds numbers for five representative organisms from the database.
The values of the mass density (ρ) and dynamic viscosity (η) used correspond to water at 25°C.
Fig 1.
Top: Geometrical and kinematic parameters of flagellated swimmers, illustrated here for a bacterium; we use the same symbols for cells employing planar or helical waves for simplicity. Bottom: Geometrical and kinematic parameters of ciliated swimmers. Drawings by Marcos F. Velho Rodrigues.
Fig 2.
Number of appendages, i.e. cilia or flagella, of each organism (whenever available) plotted against the cell body length.
Both characteristics span orders of magnitude but the data cluster within taxonomic groups.
Fig 3.
The taxonomy was obtained from the Open Tree of Life [65]. Ciliates are indicated by an asterisk *, and spermatozoa by a dagger † beside their species’ names. The drawings are not to scale and were inspired by real microscopy images or by illustrations. All drawings by Marcos F. Velho Rodrigues.
Fig 4.
Histograms of body lengths, B (μm, left), and swimming speeds, U (μm s−1, right), for rod-shaped bacteria (excluding spirochaetes and Spiroplasma) (〈B〉 = 5.79 ± 9.33 μm (n = 66), 〈U〉 = 48.33 ± 98.47 μm s−1 (n = 77)), spirochaetes (〈B〉 = 18.59 ± 13.02 μm (n = 17), 〈U〉 = 17.94 ± 18.84 μm s−1 (n = 15)), Spiroplasma (〈B〉 = 5.72 ± 0.28 μm (n = 2), 〈U〉 = 1.69 ± 0.81 μm s−1 (n = 2)) and archaea (〈B〉 = 2.71 ± 2.12 μm (n = 10), 〈U〉 = 89.18 ± 126.57 μm s−1 (n = 10)) from our database.
Most organisms have sizes below 10 μm (〈B〉 = 7.75 ± 10.85 μm (n = 95)) and swimming speeds below 100 μm s−1 (〈U〉 = 46.98 ± 95.42 μm s−1 (n = 104)).
Fig 5.
Histograms of aspect ratios W/B (left) and body-to-flagellum length B/L (right) for rod-shaped bacteria (excluding spirochaetes and Spiroplasma) (〈W/B〉 = 0.33 ± 0.20 (n = 63), 〈B/L〉 = 0.93 ± 1.19 (n = 28)), spirochaetes (〈W/B〉 = 0.02 ± 0.01 (n = 17)),Spiroplasma (〈W/B〉 = 0.03 ± 0.00 (n = 2)) and archaea (〈W/B〉 = 0.11 ± 0.06 (n = 2), 〈B/L〉 = 0.63 ± 0.24 (n = 9)).
All bacteria in our study are prolate, with an average aspect ratio 〈W/B〉 = 0.25 ± 0.22 (n = 84), with a notable slenderness of spirochaetes and Spiroplasma. If the prokaryotes possess freely rotating flagella, their length often exceeds the body size 〈B/L〉 = 0.86 ± 1.05 (n = 37) (both spirochaetes and Spiroplasma are not included in the B/L graph).
Fig 6.
Swimming speed, U (μm s−1), as function of the cell body length, B (μm, top), and body width, W (μm, bottom), for all our registered prokaryotes.
Error bars represent standard deviations, whenever available, or the span between the recorded maximum and minimum values.
Fig 7.
Propulsion speed of rod-shaped prokaryotes vs morphological factor 1/ξ2.
Bacteria are plotted in squares and archaea in circles with colours used to distinguish between the different taxonomic classes. The plot, along with Eq (14), allows to estimate the range of bacterial motor torques 27.48 − 1907 pN nm, represented by the shaded area.
Fig 8.
Histograms of body lengths, B (μm, left), and swimming speeds, U (μm s−1, right), for flagellated eukaryotic swimmers (excluding spermatozoa and ciliates) in our dataset.
The average cell length is 〈B〉 = 38.87±56.64 μm (n = 113) and the average swimming speed 〈U〉 = 186.70 ± 208.77 μm s−1 (n = 116).
Fig 9.
Histograms of aspect ratios W/B (left) and body-to-flagellum length ratios B/L (right) for flagellated eukaryotic swimmers.
For all organisms in this category, the aspect ratios do not exceed ≈ 1.1, and the shape distribution indicates a slightly prolate shape on average, with 〈W/B〉 = 0.60 ± 0.27 (n = 73). The distribution of body-to-flagellum length ratios shows that flagella tend to be of length comparable to the cell body, with a few exceptions 〈B/L〉 = 1.03 ± 0.79 (n = 49).
Fig 10.
Swimming speed, U (μm s−1), plotted versus the frequency of flagellar beat, f (s−1), for flagellated eukaryotes in our dataset (excluding spermatozoa and ciliates).
Colours mark different classes and sub-classes. Wave-producing organisms are plotted in squares and the remaining flagellated eukaryotes are plotted in circles.
Fig 11.
Swimming speed, U (μm s−1), vs length of flagella, L (μm), for flagellated eukaryotes in our database (excluding spermatozoa and ciliates).
Taxonomic classes are marked by colours. Wave-producers are again plotted in squares, while other flagellates are plotted in circles.
Fig 12.
Sketch of a swimming eukaryote (spermatozoon of Chaetopterus, Annelida) propelled by a single flagellum.
We distinguish a section of length δs inclined at an angle θ to the direction of motion ex, which we use to determine the local hydrodynamic forces exerting on the flagellum. Drawing by Marcos F. Velho Rodrigues.
Fig 13.
Swimming speeds of flagellated eukaryotes (excluding spermatozoa and ciliates) reported in the database plotted against the theoretical prediction of Eq (28).
Colours mark different classes. Square symbols mark organisms for which the prediction was directly calculated from the available data, while circles represent organisms for which either the body width or one of the flagellar characteristics has been estimated (see text for details).
Fig 14.
Histograms of body lengths, B (μm, left), and swimming speeds, U (μm s−1, right), for the spermatozoa in the database.
The average cell length is 〈B〉 = 12.21 ± 17.25 μm (n = 39), while the the average swimming speed is 〈U〉 = 127.23 ± 78.49 μms−1 (n = 52) over a wide distribution. We use colours to distinguish between the different taxonomic classes.
Fig 15.
Histograms of aspect ratios W/B (left) and body-to-flagellum length B/L (right) for spermatozoa (colours mark the different taxonomic classes).
The distribution of cell aspect ratios is rather wide, and yields an average value of 〈W/B〉 = 0.47±0.30 (n = 31). The size-to-flagellum length ratios are mostly close to the average 〈B/L〉 = 0.17 ± 0.18 (n = 38), showing that in spermatozoa the flagellum length is typically much larger than the cell body.
Fig 16.
Swimming speeds, U (μm s−1), as function of flagellar beat frequency f (s−1), for spermatozoa.
A strong correlation between U and f is apparent on the figure.
Fig 17.
Swimming speeds, U (μm s−1), as function of flagellar lengths, L (μm), for spermatozoa.
In contrast with the result in Fig 16, no clear correlation between U and L is observed here.
Fig 18.
Reported propulsion speed of spermatozoa compared with the values predicted by the theoretical model in Eq (30).
Colour scheme distinguishes between the different taxonomic classes. Squares represent spermatozoa that had all parameters available in the literature, while the circles mark cases where at least one parameter had to be estimated (via 〈W/B〉 = 0.47 from Fig 15 or through Eq (21)).
Fig 19.
Histograms of body lengths, B (μm, left), and swimming speeds, U (μm s−1, right), for the 93 ciliates in the database.
Ciliates are by far the largest organisms in our database, with the average cell length of 〈B〉 = 194.87 ± 207.45 μm (n = 91), and an average swimming speed 〈U〉 = 1147.57 ± 1375.64 μm s−1 (n = 81).
Fig 20.
Histograms of aspect ratios W/B (left) and body-to-cilium length B/ℓ (right) for ciliates.
Most of the cells are prolate, with the mean aspect ratio 〈W/B〉 = 0.49 ± 0.22 (n = 86). The size-to-flagellum length ratios have average values 〈B/ℓ〉 = 23.13 ± 27.03 (n = 26).
Fig 21.
The swimming speed U for ciliates plotted versus the numerical factor accompanying the constant tangential stress assumed in model (A) and Eq (39).
The shaded area encloses all organisms and serves as an estimate of the average effective tangential stress for all organisms, with the lower bound of τmin = 0.55 mPa, and the upper bound of τmax = 580 mPa. Colours distinguish between classes of ciliated organisms. The scatter of data suggests that only a large range of values for the stress of individual organisms can be inferred.
Fig 22.
Reported swimming speed U plotted against the numerical prefactor of Eq (42), assuming a constant effective force per cilium in the propulsion model (B).
Square symbols mark organisms for which the prediction was directly calculated from the available data, while circles represent those for which we estimated the number N of cilia. Colours distinguish the different taxonomic classes. The visible large scatter of data sets the bounds for the effective force per cilium to be in the range of 1.10 10−3 to 3.19 pN, represented by the shaded area in grey.
Table 2.
Estimated values of the effective tangential force F exerted by each cilium for the species in Fig 22.
Fig 23.
Reported swimming speeds from our database U plotted against the geometric factor from Eq (45) for the constant-flow model (C).
The data can be used to estimate the range of effective surface velocities to be in the range from 63.0 μm s−1 to 1.10 104 μm s−1, in the grey shaded area. Colours allow to distinguish different taxonomic classes.
Table 3.
List of symbols used in the database, together with their explanation and units.
Table 4.
Data for swimming bacteria (Spirochaetes and Spiroplasma excluded).
Table 5.
Data for swimming Spirochaetes and Spiroplasma.
Table 6.
Data for swimming Archaea.
Table 7.
Data for swimming flagellated eukaryotes.
Table 8.
Data for spermatozoa.
Table 9.
Data for ciliates.