Swimming eukaryotic microorganisms exhibit a universal speed distribution

  1. Maciej Lisicki  Is a corresponding author
  2. Marcos F Velho Rodrigues
  3. Raymond E Goldstein
  4. Eric Lauga  Is a corresponding author
  1. University of Cambridge, United Kingdom
  2. University of Warsaw, Poland


One approach to quantifying biological diversity consists of characterizing the statistical distribution of specific properties of a taxonomic group or habitat. Microorganisms living in fluid environments, and for whom motility is key, exploit propulsion resulting from a rich variety of shapes, forms, and swimming strategies. Here, we explore the variability of swimming speed for unicellular eukaryotes based on published data. The data naturally partitions into that from flagellates (with a small number of flagella) and from ciliates (with tens or more). Despite the morphological and size differences between these groups, each of the two probability distributions of swimming speed are accurately represented by log-normal distributions, with good agreement holding even to fourth moments. Scaling of the distributions by a characteristic speed for each data set leads to a collapse onto an apparently universal distribution. These results suggest a universal way for ecological niches to be populated by abundant microorganisms.



Unicellular eukaryotes comprise a vast, diverse group of organisms that covers virtually all environments and habitats, displaying a menagerie of shapes and forms. Hundreds of species of the ciliate genus Paramecium (Wichterman, 1986) or flagellated Euglena (Buetow, 2011) are found in marine, brackish, and freshwater reservoirs; the green algae Chlamydomonas is distributed in soil and fresh water world-wide (Harris et al., 2009); parasites from the genus Giardia colonize intestines of several vertebrates (Adam, 2001). One of the shared features of these organisms is their motility, crucial for nutrient acquisition and avoidance of danger (Bray, 2001). In the process of evolution, single-celled organisms have developed in a variety of directions, and thus their rich morphology results in a large spectrum of swimming modes (Cappuccinelli, 1980).

Many swimming eukaryotes actuate tail-like appendages called flagella or cilia in order to generate the required thrust (Sleigh, 1975). This is achieved by actively generating deformations along the flagellum, giving rise to a complex waveform. The flagellar axoneme itself is a bundle of nine pairs of microtubule doublets surrounding two central microtubules, termed the '9 + 2' structure (Nicastro et al., 2005), and cross-linking dynein motors, powered by ATP hydrolysis, perform mechanical work by promoting the relative sliding of filaments, resulting in bending deformations.

Although eukaryotic flagella exhibit a diversity of forms and functions (Moran et al., 2014), two large families, ‘flagellates’ and ‘ciliates’, can be distinguished by the shape and beating pattern of their flagella. Flagellates typically have a small number of long flagella distributed along the bodies, and they actuate them to generate thrust. The set of observed movement sequences includes planar undulatory waves and traveling helical waves, either from the base to the tip, or in the opposite direction (Jahn and Votta, 1972; Brennen and Winet, 1977). Flagella attached to the same body might follow different beating patterns, leading to a complex locomotion strategy that often relies also on the resistance the cell body poses to the fluid. In contrast, propulsion of ciliates derives from the motion of a layer of densely-packed and collectively-moving cilia, which are short hair-like flagella covering their bodies. The seminal review paper of Brennen and Winet (1977) lists a few examples from both groups, highlighting their shape, beat form, geometric characteristics and swimming properties. Cilia may also be used for transport of the surrounding fluid, and their cooperativity can lead to directed flow generation. In higher organisms this can be crucial for internal transport processes, as in cytoplasmic streaming within plant cells (Allen and Allen, 1978), or the transport of ova from the ovary to the uterus in female mammals (Lyons et al., 2006).

Here, we turn our attention to these two morphologically different groups of swimmers to explore the variability of their propulsion dynamics within broad taxonomic groups. To this end, we have collected swimming speed data from literature for flagellated eukaryotes and ciliates and analyze them separately (we do not include spermatozoa since they lack (ironically) the capability to reproduce and are thus not living organisms; their swimming characteristics have been studied by Tam and Hosoi, 2011). A careful examination of the statistical properties of the speed distributions for flagellates and ciliates shows that they are not only both captured by log-normal distributions but that, upon rescaling the data by a characteristic swimming speed for each data set, the speed distributions in both types of organisms are essentially identical.

Results and discussion

We have collected swimming data on 189 unicellular eukaryotic microorganisms (Nfl=112 flagellates and Ncil=77 ciliates) (see Appendix 1 and Source data 1). Figure 1 shows a tree encompassing the phyla of organisms studied and sketches of a representative organism from each phylum. A large morphological variation is clearly visible. In addition, we delineate the branches involving aquatic organisms and parasitic species living within hosts. Both groups include ciliates and flagellates.

The tree of life (cladogram) for unicellular eukaryotes encompassing the phyla of organisms analyzed in the present study.

Aquatic organisms (living in marine, brackish, or freshwater environments) have their branches drawn in blue while parasitic organisms have their branches drawn in red. Ciliates are indicated by an asterisk after their names. For each phylum marked in bold font, a representative organism has been sketched next to its name. Phylogenetic data from Hinchliff et al. (2015).


Due to the morphological and size differences between ciliates and flagellates, we investigate separately the statistical properties of each. Figure 2 shows the two swimming speed histograms superimposed, based on the raw distributions shown in Figure 2—figure supplement 1, where bin widths have been adjusted to their respective samples using the Freedman-Diaconis rule (see Materials and methods). Ciliates span a much larger range of speeds, up to 7 mm/s, whereas generally smaller flagellates remain in the sub-mm/s range. The inset shows that the number of flagella in both groups leads to a clear division. To compare the two groups further, we have also collected information on the characteristic sizes of swimmers from the available literature, which we list in Appendix 1. The average cell size differs fourfold between the populations (31 µm for flagellates and 132 µm for ciliates) and the distributions, plotted in Figure 2—figure supplement 2, are biased towards the low-size end but they are quantitatively different. In order to explore the physical conditions, we used the data on sizes and speeds to compute the Reynolds number Re=UL/ν for each organism, where ν=η/ρ is the kinematic viscosity of water, with η the viscosity and ρ the density. Since almost no data was available for the viscosity of the fluid in swimming speed measurements, we assumed the standard value ν=106m2/s for water for all organisms. The distribution of Reynolds numbers (Figure 2—figure supplement 3), shows that ciliates and flagellates operate in different ranges of Re, although for both groups Re<1, imposing on them the same limitations of inertia-less Stokes flow (Purcell, 1977; Lauga and Powers, 2009).

Figure 2 with 3 supplements see all
Histograms of swimming speed for ciliates and flagellates demonstrate a similar character but different scales of velocities.

Data points represent the mean and standard deviation of the data in each bin; horizontal error bars represent variability within each bin, vertical error bars show the standard deviation of the count. Inset: number of flagella displayed, where available, for each organism exhibits a clear morphological division between ciliates and flagellates.


Furthermore, studies of green algae (Short et al., 2006; Goldstein, 2015) show that an important distinction between the smaller, flagellated species and the largest multicellular ones involves the relative importance of advection and diffusion, as captured by the Péclet number Pe=UL/D, where L is a typical organism size and D is the diffusion constant of a relevant molecular species. Using the average size L of the cell body in each group of the present study (Lfl=31 μm, Lcil=132μm) and the median swimming speeds (Ufl=127m/s, Ucil=784m/s), and taking D=103(μm)2/s, we find Pefl3.9 and Pecil103, which further justifies analyzing the groups separately; they live in different physical regimes.

Examination of the mean, variance, kurtosis, and higher moments of the data sets suggest that the probabilities P(U) of the swimming speed are well-described by log-normal distributions,

(1) P(U)=1Uσ2πexp(-(lnU-μ)22σ2),

normalized as 0𝑑UP(U)=1, where μ and σ are the mean and the standard deviation of lnU. The median M of the distribution is eμ, with units of speed. Log-normal distributions are widely observed across nature in areas such as ecology, physiology, geology and climate science, serving as an empirical model for complex processes shaping a system with many potentially interacting elements (Limpert et al., 2001), particularly when the underlying processes involve proportionate fluctuations or multiplicative noise (Koch, 1966).

The results of fitting (see Materials and methods) are plotted in Figure 3, where the best fits are presented as solid curves, with the shaded areas representing 95% confidence intervals. For flagellates, we find the Mfl=127m/s and σfl=0.978 while for ciliates, we obtain Mcil=784m/s and σcil=0.936. Log-normal distributions are known to emerge from an (imperfect) analogy to the Gaussian central limit theorem (see Materials and methods). Since the data are accurately described by this distribution, we conclude that the published literature includes a sufficiently large amount of unbiased data to be able to see the whole distribution.

Figure 3 with 1 supplement see all
Probability distribution functions of swimming speeds for flagellates (a) and ciliates (b) with the fitted log-normal distributions.

Data points represent uncertainties as in Figure 2. Despite the markedly different scales of the distributions, they have similar shapes.


We next compare the statistical variability within groups by examining rescaled distributions (Goldstein, 2018). As each has a characteristic speed M, we align the peaks by plotting the distributions versus the variable U/M for each group. Since P has units of 1/speed, we are thus led to the form P(U,M)=M-1F(U/M) for some function F. For the log-normal distribution, with M the median, we find

(2) F(ξ)=1ξσ2πexp(ln2ξ2σ2),

which now depends on the single parameter σ and has a median of unity by construction. To study the similarity of the two distributions we plot the functions F=MP(U/M) for each. As seen in Figure 4, the rescaled distributions are essentially indistinguishable, and this can be traced back to the near identical values of the variances σ, which are within 5% of each other. The fitting uncertainties shown shaded in Figure 4 suggest a very similar range of variability of the fitted distributions. Furthermore, both the integrated absolute difference between the distributions (0.028) and the Kullback-Leibler divergence (0.0016) are very small (see Materials and methods), demonstrating the close similarity of the two distributions. This similarity is robust to the choice of characteristic speed, as shown in Figure 4—figure supplement 1, where the arithmetic mean U* is used in place of the median.

Figure 4 with 1 supplement see all
Test of rescaling hypothesis.

Shown are the two fitted log-normal curves for flagellates and ciliates, each multiplied by the distribution median M, plotted versus speed normalized by M. The distributions for show remarkable similarity and uncertainty of estimation.


In living cells, the sources for intrinsic variability within organisms are well characterized on the molecular and cellular level (Kirkwood et al., 2005) but less is known about variability within taxonomic groups. By dividing unicellular eukaryotes into two major groups on the basis of their difference in morphology, size and swimming strategy, we were able to capture in this paper the log-normal variability within each subset. Using a statistical analysis of the distributions as functions of the median swimming speed for each population we further found an almost identical distribution of swimming speeds for both types of organisms. Our results suggest that the observed log-normal randomness captures a universal way for ecological niches to be populated by abundant microorganisms with similar propulsion characteristics. We note, however, that the distributions of swimming speeds among species do not necessarily reflect the distributions of swimming speeds among individuals, for which we have no available data.

Materials and methods

Data collection

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Data for ciliates were sourced from 26 research articles, while that for flagellates were extracted from 48 papers (see Appendix 1). Notably, swimming speeds reported in the various studies have been measured under different physiological and environmental conditions, including temperature, viscosity, salinity, oxygenation, pH and light. Therefore we consider the data not as representative of a uniform environment, but instead as arising from a random sampling of a wide range of environmental conditions. In cases where no explicit figure was given for U in a paper, estimates were made using other available data where possible. Size of swimmers has also been included as a characteristic length for each organism. This, however, does not reflect the spread and diversity of sizes within populations of individual but is rather an indication of a typical size, as in the considered studies these data were not available. Information on anisotropy (different width/length) is also not included.

No explicit criteria were imposed for the inclusion in the analyses, apart from the biological classification (i.e. whether the organisms were unicellular eukaryotic ciliates/flagellates). We have used all the data found in literature for these organisms over the course of an extensive search. Since no selection was made, we believe that the observed statistical properties are representative for these groups.

Data processing and fitting the log-normal distribution

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Bin widths in histograms in Figure 2 and Figure 3 have been chosen separately for ciliates and flagellated eukaryotes according to the Freedman-Diaconis rule (Freedman and Diaconis, 1981) taking into account the respective sample sizes and the spread of distributions. The bin width b is then given by the number of observations N and the interquartile range of the data IQR as

(3) b=2IQRN1/3.

Within each bin in Figure 3, we calculate the mean and the standard deviation for the binned data, which constitute the horizontal error bars. The vertical error bars reflect the uncertainty in the number of counts Nj in bin j. This is estimated to be Poissonian, and thus the absolute error amounts to Nj. Notably, the relative error decays with the number of counts as 1/Nj.

In fitting the data, we employ the log-normal distribution Equation (1). In general, from from data comprising N measurements, labelled xi (i=1,,N), the n-th arithmetic moment n is the expectation 𝔼(Xn), or

(4) n=1Ni=1Nxin

Medians of the data were found by sorting the list of values and picking the middlemost value. For a log-normal distribution, the arithmetic moments are given solely by μ and σ of the associated normal distribution as

(5) n=MnΣn2,

where we have defined M=exp(μ) and Σ=exp(σ2/2), and note that M is the median of the distribution. Thus, the mean is MΣ and the variance is M2Σ2(Σ2-1). From the first and second moments, we estimate

(6) μ=ln(122) and σ2=ln(212).

Having estimated μ and σ, we can compute the higher order moments from Equation (5) and compare to those calculated directly from the data, as shown in Figure 3—figure supplement 1.

To fit the data, we have used both the MATLAB fitting routines and the Python scipy.stats module. From these fits we estimated the shape and scale parameters and the 95% confidence intervals in Figure 3 and Figure 4. We emphasize that the fitting procedures use the raw data via the maximum likelihood estimation method, and not the processed histograms, hence the estimated parameters are insensitive to the binning procedure.

For rescaled distributions, the average velocity for each group of organisms was calculated as U=1Nii=1NiUi, with i{cil,fl}. Then, data in each subset have been rescaled by the area under the fitted curve to ensure that the resulting probability density functions pi are normalized as

(7) 0pi(x)dx=1.

In characterizations of biological or ecological diversity, it is often assumed that the examined variables are Gaussian, and thus the distribution of many uncorrelated variables attains the normal distribution by virtue of the Central Limit Theorem (CLT). In the case when random variables in question are positive and have a log-normal distribution, no analogous explicit analytic result is available. Despite that, there is general agreement that a sum of independent log-normal random variables can be well approximated by another log-normal random variable. It has been proven by Szyszkowicz and Yanikome (2009) that the sum of identically distributed equally and positively correlated joint log-normal distributions converges to a log-normal distribution of known characteristics but for uncorrelated variables only estimations are available (Beaulieu et al., 1995). We use these results to conclude that our distributions contain enough data to be unbiased and seen in full.

Comparisons of distributions

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In order to quantify the differences between the fitted distributions, we define the integrated absolute difference Δ between two probability distributions p(x) and q(x) (x>0) as

(8) Δ=0|p(x)-q(x)|dx.

As the probability distributions are normalized, this is a measure of their relative ’distance’. As a second measure, we use the Kullback-Leibler divergence (Kullback and Leibler, 1951),

(9) D(p,q)=0p(x)ln(p(x)q(x))dx.

Note that D(p,q)D(q,p) and therefore D is not a distance metric in the space of probability distributions.

Appendix 1

The Appendix contains the data which form the basis of our study. The tables contain data on the sizes and swimming speed of ciliates organisms and flagellated eukaryotes from the existing literature. Data for ciliates were sourced from 26 research articles, while data for the flagellates were extracted from 48 papers. In the cases where two or more sources reported contrasting figures for the swimming speed, the average value is reported in our tables. The data itself is available in Source data 1.

Data for swimming flagellates

Abbreviations: dflg. – dinoflagellata; dph – dinophyceae; chlph. – chlorophyta; ochph. (het.) –ochrophyta (heterokont); srcm. – sarcomastigophora, pyr. – pyramimonadophyceae; prym. – prymnesiophyceae; dict. – dictyochophyceae; crypt. – cryptophyceae; chrys. – chrysophyceae

Alexandrium minutumdflg.dph.21.7222.5(Lewis et al., 2006)
Alexandrium ostenfeldiidflg.dph.41.1110.5(Lewis et al., 2006)
Alexandrium tamarensedflg.dph.26.7200(Lewis et al., 2006)
Amphidinium britannicumdflg.dph.51.268.7(Bauerfeind et al., 1986)
Amphidinium carteraedflg.dph.1681.55(Gittleson et al., 1974; Bauerfeind et al., 1986)
Amphidinium klebsidflg.dph.3573.9(Gittleson et al., 1974)
Apedinella spiniferaochph. (het.)dict.8.25132.5(Throndsen, 1973)
Bodo designiseuglenozoakinetoplastea5.539(Visser and Kiørboe, 2006)
Brachiomonas submarinachlph.chlorophyceae27.596(Bauerfeind et al., 1986)
Cachonina (Heterocapsa) nieidflg.dph.21.4302.8(Levandowsky and Kaneta, 1987; Kamykowski and Zentara, 1977)
Cafeteria roenbergensisbygira (heterokont)bicosoecida294.9(Fenchel and Blackburn, 1999)
Ceratium cornutumdflg.dph.122.3177.75(Levandowsky and Kaneta, 1987; Metzner, 1929)
Ceratium furcadflg.dph.122.5194(Peters, 1929)
Ceratium fususdflg.dph.307.5156.25(Peters, 1929)
Ceratium hirundinelladflg.dph.397.5236.1(Levandowsky and Kaneta, 1987)
Ceratium horridumdflg.dph.22520.8(Peters, 1929)
Ceratium lineatusdflg.dph.82.136(Fenchel, 2001)
Ceratium longipesdflg.dph.210166(Peters, 1929)
Ceratium macrocerosdflg.dph.5015.4(Peters, 1929)
Ceratium triposdflg.dph.152.3121.7(Peters, 1929; Bauerfeind et al., 1986)
Chilomonas parameciumcryptophytacrypt.30111.25(Lee, 1954; Jahn and Bovee, 1967; Gittleson et al., 1974)
Chlamydomonas reinhardtiichlph.chlorophyceae10130(Gittleson et al., 1974; Roberts, 1981; Guasto et al., 2010)
Chlamydomonas moewusiichlph.chlorophyceae12.5128(Gittleson et al., 1974)
Chlamydomonas sp.chlph.chlorophyceae1363.2(Lowndes, 1944; Lowndes, 1941; Bauerfeind et al., 1986)
Crithidia deaneieuglenozoakinetoplastea7.445.6(Gadelha et al., 2007)
Crithidia fasciculataeuglenozoakinetoplastea11.154.3(Gadelha et al., 2007)
Crithidia (Strigomonas) oncopeltieuglenozoakinetoplastea8 .118.5(Roberts, 1981; Gittleson et al., 1974)
Crypthecodinium cohniidflg.dph.n/a122.8(Fenchel, 2001)
Dinophysis acutadflg.dph.65500(Peters, 1929)
Dinophysis ovumdflg.dph.45160(Buskey et al., 1993)
Dunaliella sp.chlph.chlorophyceae10.8173.5(Gittleson et al., 1974; Bauerfeind et al., 1986)
Euglena graciliseuglenozoaeuglenida (eugl.)47.5111.25(Lee, 1954; Jahn and Bovee, 1967; Gittleson et al., 1974)
Euglena viridiseuglenozoaeuglenida (eugl.)5880(Holwill, 1975; Roberts, 1981; Lowndes, 1941)
Eutreptiella gymnasticaeuglenozoaeuglenida (aphagea)23.5237.5(Throndsen, 1973)
Eutreptiella sp. Reuglenozoaeuglenida50135(Throndsen, 1973)
Exuviaella baltica (Prorocentrum balticum)dflg.dph.15.5138.9(Wheeler, 1966)
Giardia lambliasrcm.zoomastigophora11.2526(Lenaghan et al., 2011; Campanati et al., 2002; Chen et al., 2012)
Gonyaulax polyedradflg.dph.39.2254.05(Hand et al., 1965; Gittleson et al., 1974; Kamykowski et al., 1992)
Gonyaulax polygrammadflg.dph.46.2500(Levandowsky and Kaneta, 1987)
Gymnodinium aureolumdflg.dph.n/a394(Meunier et al., 2013)
Gymnodinium sanguineum (splendens)dflg.dph.47.6220.5(Kamykowski et al., 1992; Levandowsky and Kaneta, 1987)
Gymnodinium simplexdflg.dph.10.6559(Jakobsen et al., 2006)
Gyrodinium aureolumdflg.dph.30.5139(Bauerfeind et al., 1986; Throndsen, 1973)
Gyrodinium dorsum (bi-flagellated)dflg.dph.37.5324(Hand et al., 1965; Gittleson et al., 1974; Kamykowski et al., 1992; Levandowsky and Kaneta, 1987Brennen and Winet, 1977)
Gyrodinium dorsum (uni-flagellated)dflg.dph.34.5148.35(Hand and Schmidt, 1975)
Hemidinium nasutumdflg.dph.27.2105.6(Levandowsky and Kaneta, 1987; Metzner, 1929)
Hemiselmis simplexcryptophytacrypt.5.25325(Throndsen, 1973)
Heterocapsa pygmeadflg.dph.13.5102.35(Bauerfeind et al., 1986)
Heterocapsa rotundatadflg.dph.12.5323(Jakobsen et al., 2006)
Heterocapsa triquetradflg.dph.1797(Visser and Kiørboe, 2006)
Heteromastix pyriformischlph.nephrophyseae687.5(Throndsen, 1973)
Hymenomonas carteraehaptophytaprym.12.587(Bauerfeind et al., 1986)
Katodinium rotundatum (Heterocapsa rotundata)dflg.dph.10.8425(Levandowsky and Kaneta, 1987; Throndsen, 1973)
Leishmania majoreuglenozoakinetoplastea12.536.4(Gadelha et al., 2007)
Menoidium cultelluseuglenozoaeuglenida (eugl.)45136.75(Holwill, 1975; Votta et al., 1971)
Menoidium incurvumeuglenozoaeuglenida (eugl.)2550(Lowndes, 1941; Gittleson et al., 1974)
Micromonas pusillachlph.mamiellophyceae258.5(Bauerfeind et al., 1986; Throndsen, 1973)
Monas stigmataochph. (het.)chrys.6269(Gittleson et al., 1974)
Monostroma angicavachlph.ulvophyceae6.7170.55(Togashi et al., 1997)
Nephroselmis pyriformischlph.nephrophyseae4.8163.5(Bauerfeind et al., 1986)
Oblea rotundadflg.dph.20420(Buskey et al., 1993)
Ochromonas danicaochph. (het.)chrys.8.777(Holwill and Peters, 1974)
Ochromonas malhamensisochph. (het.)chrys.357.5(Holwill, 1974)
Ochromonas minimaochph. (het.)chrys.575(Throndsen, 1973)
Olisthodiscus luteusochph. (het.)raphidophyceae22.590(Bauerfeind et al., 1986; Throndsen, 1973)
Oxyrrhis marinadflg.oxyrrhea39.5300(Boakes et al., 2011; Fenchel, 2001)
Paragymnodinium shiwhaensedflg.dph.10.9571(Meunier et al., 2013)
Paraphysomonas vestitaochph. (het.)chrys.14.7116.85(Christensen-Dalsgaard and Fenchel, 2004)
Pavlova lutherihaptophytapavlovophyceae6.5126(Bauerfeind et al., 1986)
Peranema trichophorumeuglenozoaeuglenida (heteronematales)4520(Lowndes, 1941; Gittleson et al., 1974; Brennen and Winet, 1977)
Peridinium bipesdflg.dph.42.9291(Fenchel, 2001)
Peridinium cf. quinquecornedflg.dph.191500(Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987; Horstmann, 1980)
Peridinium cinctumdflg.dph.47.5120(Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987; Metzner, 1929)
Peridinium (Protoperidinium) claudicansdflg.dph.77.5229(Peters, 1929)
Peridinium (Protoperidinium) crassipesdflg.dph.102100(Peters, 1929)
Peridinium foliaceumdflg.dph.30.6185.2(Kamykowski et al., 1992)
Peridinium (Bysmatrum) gregariumdflg.dph.32.51291.7(Levandowsky and Kaneta, 1987)
Peridinium (Protoperidinium) ovatumdflg.dph.61187.5(Peters, 1929)
Peridinium (Peridiniopsis) penardiidflg.dph.28.8417(Sibley et al., 1974)
Peridinium (Protoperidinium) pentagonumdflg.dph.92.5266.5(Peters, 1929)
Peridinium (Protoperidinium) subinermedflg.dph.50285(Peters, 1929)
Peridinium trochoideumdflg.dph.2553(Levandowsky and Kaneta, 1987)
Peridinium umbonatumdflg.dph.30250(Levandowsky and Kaneta, 1987; Metzner, 1929)
Phaeocystis pouchetiihaptophytaprym.6.388(Bauerfeind et al., 1986)
Polytoma uvellachlph.chlorophyceae22.5100.9(Lowndes, 1944; Gittleson et al., 1974; Lowndes, 1941)
Polytomella agilischlph.chlorophyceae12.4150(Gittleson and Jahn, 1968; Gittleson and Noble, 1973; Gittleson et al., 1974; Roberts, 1981)
Prorocentrum mariae-lebouriaedflg.dph.14.8141.05(Kamykowski et al., 1992; Bauerfeind et al., 1986; Miyasaka et al., 1998)
Prorocentrum micansdflg.dph.45329.1(Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987)
Prorocentrum minimumdflg.dph.15.1107.7(Bauerfeind et al., 1986; Miyasaka et al., 1998)
Prorocentrum redfieldii Bursa (P.triestinum)dflg.dph.33.2333.3(Sournia, 1982)
Protoperidinium depressumdflg.dph.132450(Buskey et al., 1993)
Protoperidinium granii (Ostf.) Balechdflg.dph.57.586.1(Sournia, 1982)
Protoperidinium pacificumdflg.dph.54410(Buskey et al., 1993)
Prymnesium polylepishaptophytaprym.9.145(Dölger et al., 2017)
Prymnesium parvumhaptophytaprym.7.230(Dölger et al., 2017)
Pseudopedinella pyriformisochph. (het.)dict.6.5100(Throndsen, 1973)
Pseudoscourfieldia marinachlph.pyr.4.142(Bauerfeind et al., 1986)
Pteridomonas danicaochph. (het.)dict.5.5179.45(Christensen-Dalsgaard and Fenchel, 2004)
Pyramimonas amyliferachlph.pyr.24.522.5(Bauerfeind et al., 1986)
Pyramimonas cf. disomatachlph.pyr.9355(Throndsen, 1973)
Rhabdomonas spiraliseuglenozoaeuglenida (aphagea)27120(Holwill, 1975)
Rhodomonas salinacryptophytacrypt.14.5588.5(Jakobsen et al., 2006; Meunier et al., 2013)
Scrippsiella trochoideadflg.dph.25.387.6(Kamykowski et al., 1992; Bauerfeind et al., 1986; Sournia, 1982)
Spumella sp.ochph. (het.)chrys.1025(Visser and Kiørboe, 2006)
Teleaulax sp.cryptophytacrypt.13.598(Meunier et al., 2013)
Trypanosoma bruceieuglenozoakinetoplastea18.820.5(Rodríguez et al., 2009)
Trypanosoma cruzieuglenozoakinetoplastea20172(Jahn and Fonseca, 1963; Brennen and Winet, 1977)
Trypanosoma vivaxeuglenozoakinetoplastea23.529.5(Bargul et al., 2016)
Trypanosoma evansieuglenozoakinetoplastea21.516.1(Bargul et al., 2016)
Trypanosoma congolenseeuglenozoakinetoplastea189.7(Bargul et al., 2016)
Tetraflagellochloris mauritanicachlph.chlorophyceae4300(Barsanti et al., 2016)

Appendix 2

Data for swimming ciliates

Abbreviations: imnc. = intramacronucleata; pcdph. = postciliodesmatophora; olig. – oligohymenophorea; spir. – spirotrichea; hettr. – heterotrichea; lit. – litostomatea; eugl. – euglenophyceae

Amphileptus gigasimnc.lit.808608(Bullington, 1925)
Amphorides quadrilineataimnc.spir.138490(Buskey et al., 1993)
Balanion comatumimnc.prostomatea16220(Visser and Kiørboe, 2006)
Blepharismapcdph.hettr.350600(Sleigh and Blake, 1977; Roberts, 1981)
Coleps hirtusimnc.prostomatea94.5686(Bullington, 1925)
Coleps sp.imnc.prostomatea78523(Bullington, 1925)
Colpidium striatumimnc.olig.77570(Beveridge et al., 2010)
Condylostoma patenspcdph.hettr.3711061(Bullington, 1925; Machemer, 1974)
Didinium nasutumimnc.lit.1401732(Bullington, 1925; Machemer, 1974; Roberts, 1981Sleigh and Blake, 1977)
Euplotes charonimnc.spir.661053(Bullington, 1925)
Euplotes patellaimnc.spir.2021250(Bullington, 1925)
Euplotes vannusimnc.spir.82446(Wang et al., 2008Ricci et al., 1997)
Eutintinnus cf. pinguisimnc.spir.147410(Buskey et al., 1993)
Fabrea salinapcdph.hettr.184.1216(Marangoni et al., 1995)
Favella panamensisimnc.spir.238600(Buskey et al., 1993)
Favella sp.imnc.spir.1501080(Buskey et al., 1993)
Frontonia sp.imnc.olig.378.51632(Bullington, 1925)
Halteria grandinellaimnc.spir.50533(Bullington, 1925; Gilbert, 1994)
Kerona polyporumimnc.spir.107476.5(Bullington, 1925)
Laboea strobilaimnc.spir.100810(Buskey et al., 1993)
Lacrymaria lagenulaimnc.lit.45909(Bullington, 1925)
Lembadion bullinumimnc.olig.43415(Bullington, 1925)
Lembus veliferimnc.olig.87200(Bullington, 1925)
Mesodinium rubrumimnc.lit.387350(Jonsson and Tiselius, 1990Riisgård and Larsen, 2009; Crawford and Lindholm, 1997)
Metopides contortaimnc.armophorea115359(Bullington, 1925)
Nassula ambiguaimnc.nassophorea1432004(Bullington, 1925)
Nassula ornataimnc.nassophorea282750(Bullington, 1925)
Opalina ranarumplacidozoa (heterokont)opalinea35050(Blake, 1975; Sleigh and Blake, 1977)
Ophryoglena sp.imnc.olig.3254000(Machemer, 1974)
Opisthonecta hennegimnc.olig.1261197(Machemer, 1974; Jahn and Hendrix, 1969)
Oxytricha bifaraimnc.spir.2821210(Bullington, 1925)
Oxytricha ferrugineaimnc.spir.150400(Bullington, 1925)
Oxytricha platystomaimnc.spir.130520(Bullington, 1925)
Paramecium aureliaimnc.olig.2441650(Bullington, 1925; Bullington, 1930)
Paramecium bursariaimnc.olig.1301541.5(Bullington, 1925; Bullington, 1930)
Paramecium calkinsiiimnc.olig.1241392(Bullington, 1930; Bullington, 1925)
Paramecium caudatumimnc.olig.225.52489.35(Bullington, 1930; Jung et al., 2014)
Paramecium marinumimnc.olig.115930(Bullington, 1925)
Paramecium multimicronucleatumimnc.olig.2513169.5(Bullington, 1930)
Paramecium polycaryumimnc.olig.911500(Bullington, 1930)
Paramecium spp.imnc.olig.200975(Jahn and Bovee, 1967; Sleigh and Blake, 1977; Roberts, 1981)
Paramecium tetraureliaimnc.olig.124784(Funfak et al., 2015)
Paramecium woodruffiimnc.olig.1602013.5(Bullington, 1930)
Porpostoma notatumimnc.olig.107.71842.2(Fenchel and Blackburn, 1999)
Prorodon teresimnc.prostomatea1751066(Bullington, 1925)
Spathidium spathulaimnc.lit.204.5526(Bullington, 1925)
Spirostomum ambiguumpcdph.hettr.1045810(Bullington, 1925)
Spirostomum sp.pcdph.hettr.10001000(Sleigh and Blake, 1977)
Spirostomum terespcdph.hettr.450640(Bullington, 1925)
Stenosemella steiniiimnc.spir.83190(Buskey et al., 1993)
Stentor caeruleuspcdph.hettr.528.51500(Bullington, 1925)
Stentor polymorphuspcdph.hettr.208887(Bullington, 1925; Sleigh and Aiello, 1972;
Sleigh, 1968)
Strobilidium spiralisimnc.spir.60330(Buskey et al., 1993)
Strobilidium veloximnc.spir.43150(Gilbert, 1994)
Strombidinopsis acuminatumimnc.spir.80390(Buskey et al., 1993)
Strombidium claparediimnc.spir.69.53740(Bullington, 1925)
Strombidium conicumimnc.spir.75570(Buskey et al., 1993)
Strombidium sp.imnc.spir.33360(Buskey et al., 1993)
Strombidium sulcatumimnc.spir.32.5995(Fenchel and Jonsson, 1988; Fenchel and Blackburn, 1999
Fenchel and Blackburn, 1999)
Stylonichia sp.imnc.spir.167737.5(Bullington, 1925; Machemer, 1974)
Tetrahymena pyriformisimnc.olig.72.8475.6(Sleigh and Blake, 1977; Roberts, 1981; Brennen and Winet, 1977)
Tetrahymena thermophilaimnc.olig.46.7204.5(Wood et al., 2007)
Tillina magnaimnc.colpodea162.52000(Bullington, 1925)
Tintinnopsis kofoidiimnc.spir.100400(Buskey et al., 1993)
Tintinnopsis minutaimnc.spir.4060(Buskey et al., 1993)
Tintinnopsis tubulosaimnc.spir.95160(Buskey et al., 1993)
Tintinnopsis vasculumimnc.spir.82250(Buskey et al., 1993)
Trachelocerca olorpcdph.karyorelictea267.5900(Bullington, 1925)
Trachelocerca tenuicollispcdph.karyorelictea4321111(Bullington, 1925)
Uroleptus piscisimnc.spir.203487(Bullington, 1925)
Uroleptus rattulusimnc.spir.400385(Bullington, 1925)
Urocentrum turboimnc.olig.90700(Bullington, 1925)
Uronema filificumimnc.olig.25.71372.7(Fenchel and Blackburn, 1999)
Uronema marinumimnc.olig.56.91010(Fenchel and Blackburn, 1999)
Uronema sp.imnc.olig.251175(Sleigh and Blake, 1977; Roberts, 1981)
Uronychia transfugaimnc.spir.1186406(Leonildi et al., 1998)
Uronychia setigeraimnc.spir.647347(Leonildi et al., 1998)
Uronemella spp.imnc.olig.28250(Petroff et al., 2015)

Data availability

All data generated or analysed during this study are included in the manuscript.


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Article and author information

Author details

  1. Maciej Lisicki

    1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
    2. Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland
    Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Methodology, Writing—original draft, Writing—review and editing
    Contributed equally with
    Marcos F Velho Rodrigues
    For correspondence
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6976-0281
  2. Marcos F Velho Rodrigues

    Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
    Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing—original draft, Writing—review and editing
    Contributed equally with
    Maciej Lisicki
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-8744-6966
  3. Raymond E Goldstein

    Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
    Investigation, Methodology, Writing—review and editing
    Competing interests
    Reviewing editor, eLife
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-2645-0598
  4. Eric Lauga

    Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
    Conceptualization, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Methodology, Writing—original draft, Project administration, Writing—review and editing
    For correspondence
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-8916-2545


H2020 European Research Council (682754)

  • Eric Lauga

Engineering and Physical Sciences Research Council (EP/M017982/)

  • Raymond E Goldstein

Gordon and Betty Moore Foundation (7523)

  • Raymond E Goldstein

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.


This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 682754 to EL), and from Established Career Fellowship EP/M017982/1 from the Engineering and Physical Sciences Research Council and Grant 7523 from the Gordon and Betty Moore Foundation (REG).

Version history

  1. Received: January 5, 2019
  2. Accepted: June 29, 2019
  3. Version of Record published: July 16, 2019 (version 1)


© 2019, Lisicki et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.


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  1. Maciej Lisicki
  2. Marcos F Velho Rodrigues
  3. Raymond E Goldstein
  4. Eric Lauga
Swimming eukaryotic microorganisms exhibit a universal speed distribution
eLife 8:e44907.

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