Introduction {#s1}
============

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* ([@bib94]) or flagellated *Euglena* ([@bib12]) are found in marine, brackish, and freshwater reservoirs; the green algae *Chlamydomonas* is distributed in soil and fresh water world-wide ([@bib37]); parasites from the genus *Giardia* colonize intestines of several vertebrates ([@bib1]). One of the shared features of these organisms is their motility, crucial for nutrient acquisition and avoidance of danger ([@bib10]). 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 ([@bib17]).

Many swimming eukaryotes actuate tail-like appendages called flagella or cilia in order to generate the required thrust ([@bib82]). 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 ([@bib71]), 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 ([@bib70]), 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 ([@bib46]; [@bib11]). 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 [@bib11] 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 ([@bib2]), or the transport of ova from the ovary to the uterus in female mammals ([@bib64]).

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 [@bib87]). 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 {#s2}
======================

We have collected swimming data on 189 unicellular eukaryotic microorganisms ($N_{\text{fl}} = 112$ flagellates and $N_{\text{cil}} = 77$ ciliates) (see Appendix 1 and [Source data 1](#sdata1){ref-type="supplementary-material"}). [Figure 1](#fig1){ref-type="fig"} 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 [@bib38].](elife-44907-fig1){#fig1}

Due to the morphological and size differences between ciliates and flagellates, we investigate separately the statistical properties of each. [Figure 2](#fig2){ref-type="fig"} shows the two swimming speed histograms superimposed, based on the raw distributions shown in [Figure 2---figure supplement 1](#fig2s1){ref-type="fig"}, 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](#fig2s2){ref-type="fig"}, 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 ${Re} = {{UL}/\nu}$ for each organism, where $\nu = {\eta/\rho}$ is the kinematic viscosity of water, with $\eta$ the viscosity and $\rho$ the density. Since almost no data was available for the viscosity of the fluid in swimming speed measurements, we assumed the standard value $\nu = 10^{- 6}m^{2}/s$ for water for all organisms. The distribution of Reynolds numbers ([Figure 2---figure supplement 3](#fig2s3){ref-type="fig"}), 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 ([@bib74]; [@bib55]).

![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.](elife-44907-fig2){#fig2}

Furthermore, studies of green algae ([@bib79]; [@bib32]) 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 ($L_{\text{fl}} = 31\ \mu\text{m}$, $L_{\text{cil}} = 132\mu\text{m}$) and the median swimming speeds ($U_{\text{fl}} = 127m/s$, $U_{cil} = 784m/s$), and taking $D = 10^{3}(\mu m)^{2}/s$, we find ${Pe_{fl}} \sim 3.9$ and ${Pe_{cil}} \sim 103$, 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,$${{P{(U)}} = {\frac{1}{U\sigma\sqrt{2\pi}}{\exp\left( {- \frac{{({{\ln U} - \mu})}^{2}}{2\sigma^{2}}} \right)}}},$$normalized as ${\int_{0}^{\infty}{{dU}P{(U)}}} = 1$, where $\mu$ and $\sigma$ are the mean and the standard deviation of $\ln U$. The median $M$ of the distribution is $e^{\mu}$, 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 ([@bib61]), particularly when the underlying processes involve proportionate fluctuations or multiplicative noise ([@bib53]).

The results of fitting (see Materials and methods) are plotted in [Figure 3](#fig3){ref-type="fig"}, where the best fits are presented as solid curves, with the shaded areas representing 95% confidence intervals. For flagellates, we find the $M_{\text{fl}} = 127m/s$ and $\sigma_{\text{fl}} = 0.978$ while for ciliates, we obtain $M_{\text{cil}} = 784m/s$ and $\sigma_{\text{cil}} = 0.936$. 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.

![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](#fig2){ref-type="fig"}. Despite the markedly different scales of the distributions, they have similar shapes.](elife-44907-fig3){#fig3}

We next compare the statistical variability within groups by examining rescaled distributions ([@bib33]). 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 ${P{(U,M)}} = {M^{- 1}F{({U/M})}}$ for some function $F$. For the log-normal distribution, with $M$ the median, we find$$F(\xi) = \frac{1}{\xi\sigma\sqrt{2\pi}}\exp\left( {- \frac{\ln^{2}\xi}{2\sigma^{2}}} \right),$$which now depends on the single parameter $\sigma$ 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](#fig4){ref-type="fig"}, the rescaled distributions are essentially indistinguishable, and this can be traced back to the near identical values of the variances $\sigma$, which are within 5% of each other. The fitting uncertainties shown shaded in [Figure 4](#fig4){ref-type="fig"} 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](#fig4s1){ref-type="fig"}, where the arithmetic mean $U^{*}$ is used in place of the median.

![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.](elife-44907-fig4){#fig4}

In living cells, the sources for intrinsic variability within organisms are well characterized on the molecular and cellular level ([@bib52]) 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 {#s3}
=====================

Data collection {#s3-1}
---------------

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 {#s3-2}
-------------------------------------------------------

Bin widths in histograms in [Figure 2](#fig2){ref-type="fig"} and [Figure 3](#fig3){ref-type="fig"} have been chosen separately for ciliates and flagellated eukaryotes according to the Freedman-Diaconis rule ([@bib25]) 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$${b = {2\frac{IQR}{N^{1/3}}}}.$$

Within each bin in [Figure 3](#fig3){ref-type="fig"}, 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 $N_{j}$ in bin $j$. This is estimated to be Poissonian, and thus the absolute error amounts to $\sqrt{N_{j}}$. Notably, the relative error decays with the number of counts as $1/\sqrt{N_{j}}$.

In fitting the data, we employ the log-normal distribution [Equation (1)](#equ1){ref-type="disp-formula"}. In general, from from data comprising $N$ measurements, labelled $x_{i}$ ($i = {1,\ldots,N}$), the $n$-th arithmetic moment $\mathcal{M}_{n}$ is the expectation ${\mathbb{E}}{(X^{n})}$, or$$\mathcal{M}_{n} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}x_{i}^{n}}}$$

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 $\mu$ and $\sigma$ of the associated normal distribution as$${\mathcal{M}_{n} = {M^{n}\Sigma^{n^{2}}}},$$where we have defined $M = {\exp{(\mu)}}$ and $\Sigma = {\exp{({\sigma^{2}/2})}}$, and note that $M$ is the median of the distribution. Thus, the mean is $M\Sigma$ and the variance is $M^{2}\Sigma^{2}\left( {\Sigma^{2} - 1} \right)$. From the first and second moments, we estimate$${{\mu = {{\ln\left( \frac{\mathcal{M}_{1}^{2}}{\sqrt{\mathcal{M}_{2}}} \right)}\operatorname{}{and}}}\operatorname{}{\sigma^{2} = {\ln\left( \frac{\mathcal{M}_{2}}{\mathcal{M}_{1}^{2}} \right)}}}.$$

Having estimated $\mu$ and $\sigma$, we can compute the higher order moments from [Equation (5)](#equ5){ref-type="disp-formula"} and compare to those calculated directly from the data, as shown in [Figure 3---figure supplement 1](#fig3s1){ref-type="fig"}.

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](#fig3){ref-type="fig"} and [Figure 4](#fig4){ref-type="fig"}. 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 $U^{\ast} = {\frac{1}{N_{i}}{\sum_{i = 1}^{N_{i}}U_{i}}}$, with $i \in \{\text{cil},\text{fl}\}$. Then, data in each subset have been rescaled by the area under the fitted curve to ensure that the resulting probability density functions $p_{i}$ are normalized as$${{\int_{0}^{\infty}{p_{i}{(x)}{dx}}} = 1}.$$

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 [@bib86] 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 ([@bib6]). We use these results to conclude that our distributions contain enough data to be unbiased and seen in full.

Comparisons of distributions {#s3-3}
----------------------------

In order to quantify the differences between the fitted distributions, we define the integrated absolute difference $\Delta$ between two probability distributions $p{(x)}$ and $q{(x)}$ ($x > 0$) as$${\Delta = {\int_{0}^{\infty}{{|{{p{(x)}} - {q{(x)}}}|}{dx}}}}.$$

As the probability distributions are normalized, this is a measure of their relative 'distance'. As a second measure, we use the Kullback-Leibler divergence ([@bib54]),$${{D{(p,q)}} = {\int_{0}^{\infty}{p{(x)}{\ln\left( \frac{p{(x)}}{q{(x)}} \right)}{dx}}}}.$$

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

Funding Information
===================

This paper was supported by the following grants:

-   http://dx.doi.org/10.13039/100010663H2020 European Research Council 682754 to Eric Lauga.

-   http://dx.doi.org/10.13039/501100000266Engineering and Physical Sciences Research Council EP/M017982/ to Raymond E Goldstein.

-   http://dx.doi.org/10.13039/100000936Gordon and Betty Moore Foundation 7523 to Raymond E Goldstein.

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).

Additional information {#s4}
======================

Reviewing editor, *eLife*.

No competing interests declared.

Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Methodology, Writing---original draft, Writing---review and editing.

Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing---original draft, Writing---review and editing.

Investigation, Methodology, Writing---review and editing.

Conceptualization, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Methodology, Writing---original draft, Project administration, Writing---review and editing.

Additional files {#s5}
================

10.7554/eLife.44907.011

###### Spreadsheet data for swimming eukaryotes listed in Appendix 1 and Appendix 2.

10.7554/eLife.44907.012

Data availability {#s6}
=================

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

10.7554/eLife.44907.013

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](#sdata1){ref-type="supplementary-material"}.

Data for swimming flagellates {#s7}
=============================

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

SpeciesPhylumClass$L$\[$\mu m$\]$U$\[${\mu m}/s$\]References*Alexandrium minutum*dflg.dph.21.7222.5([@bib60])*Alexandrium ostenfeldii*dflg.dph.41.1110.5([@bib60])*Alexandrium tamarense*dflg.dph.26.7200([@bib60])*Amphidinium britannicum*dflg.dph.51.268.7([@bib5])*Amphidinium carterae*dflg.dph.1681.55([@bib29]; [@bib5])*Amphidinium klebsi*dflg.dph.3573.9([@bib29])*Apedinella spinifera*ochph. (het.)dict.8.25132.5([@bib88])*Bodo designis*euglenozoakinetoplastea5.539([@bib90])*Brachiomonas submarina*chlph.chlorophyceae27.596([@bib5])*Cachonina (Heterocapsa) niei*dflg.dph.21.4302.8([@bib59]; [@bib51])*Cafeteria roenbergensis*bygira (heterokont)bicosoecida294.9([@bib23])*Ceratium cornutum*dflg.dph.122.3177.75([@bib59]; [@bib67])*Ceratium furca*dflg.dph.122.5194([@bib72])*Ceratium fusus*dflg.dph.307.5156.25([@bib72])*Ceratium hirundinella*dflg.dph.397.5236.1([@bib59])*Ceratium horridum*dflg.dph.22520.8([@bib72])*Ceratium lineatus*dflg.dph.82.136([@bib22])*Ceratium longipes*dflg.dph.210166([@bib72])*Ceratium macroceros*dflg.dph.5015.4([@bib72])*Ceratium tripos*dflg.dph.152.3121.7([@bib72]; [@bib5])*Chilomonas paramecium*cryptophytacrypt.30111.25([@bib56]; [@bib43]; [@bib29])*Chlamydomonas reinhardtii*chlph.chlorophyceae10130([@bib29]; [@bib77]; [@bib34])*Chlamydomonas moewusii*chlph.chlorophyceae12.5128([@bib29])*Chlamydomonas sp.*chlph.chlorophyceae1363.2([@bib63]; [@bib62]; [@bib5])*Crithidia deanei*euglenozoakinetoplastea7.445.6([@bib27])*Crithidia fasciculata*euglenozoakinetoplastea11.154.3([@bib27])*Crithidia (Strigomonas) oncopelti*euglenozoakinetoplastea8 .118.5([@bib77]; [@bib29])*Crypthecodinium cohnii*dflg.dph.n/a122.8([@bib22])*Dinophysis acuta*dflg.dph.65500([@bib72])*Dinophysis ovum*dflg.dph.45160([@bib15])*Dunaliella* sp.chlph.chlorophyceae10.8173.5([@bib29]; [@bib5])*Euglena gracilis*euglenozoaeuglenida (eugl.)47.5111.25([@bib56]; [@bib43]; [@bib29])*Euglena viridis*euglenozoaeuglenida (eugl.)5880([@bib40]; [@bib77]; [@bib62])*Eutreptiella gymnastica*euglenozoaeuglenida (aphagea)23.5237.5([@bib88])*Eutreptiella* sp. Reuglenozoaeuglenida50135([@bib88])*Exuviaella baltica (Prorocentrum balticum)*dflg.dph.15.5138.9([@bib93])*Giardia lamblia*srcm.zoomastigophora11.2526([@bib57]; [@bib16]; [@bib18])*Gonyaulax polyedra*dflg.dph.39.2254.05([@bib35]; [@bib29]; [@bib50])*Gonyaulax polygramma*dflg.dph.46.2500([@bib59])*Gymnodinium aureolum*dflg.dph.n/a394([@bib68])*Gymnodinium sanguineum (splendens)*dflg.dph.47.6220.5([@bib50]; [@bib59])*Gymnodinium simplex*dflg.dph.10.6559([@bib47])*Gyrodinium aureolum*dflg.dph.30.5139([@bib5]; [@bib88])*Gyrodinium dorsum* (bi-flagellated)dflg.dph.37.5324([@bib35]; [@bib29]; [@bib50]; [@bib59]; [@bib11])*Gyrodinium dorsum* (uni-flagellated)dflg.dph.34.5148.35([@bib36])*Hemidinium nasutum*dflg.dph.27.2105.6([@bib59]; [@bib67])*Hemiselmis simplex*cryptophytacrypt.5.25325([@bib88])*Heterocapsa pygmea*dflg.dph.13.5102.35([@bib5])*Heterocapsa rotundata*dflg.dph.12.5323([@bib47])*Heterocapsa triquetra*dflg.dph.1797([@bib90])*Heteromastix pyriformis*chlph.nephrophyseae687.5([@bib88])*Hymenomonas carterae*haptophytaprym.12.587([@bib5])*Katodinium rotundatum (Heterocapsa rotundata)*dflg.dph.10.8425([@bib59]; [@bib88])*Leishmania major*euglenozoakinetoplastea12.536.4([@bib27])*Menoidium cultellus*euglenozoaeuglenida (eugl.)45136.75([@bib40]; [@bib91])*Menoidium incurvum*euglenozoaeuglenida (eugl.)2550([@bib62]; [@bib29])*Micromonas pusilla*chlph.mamiellophyceae258.5([@bib5]; [@bib88])*Monas stigmata*ochph. (het.)chrys.6269([@bib29])*Monostroma angicava*chlph.ulvophyceae6.7170.55([@bib89])*Nephroselmis pyriformis*chlph.nephrophyseae4.8163.5([@bib5])*Oblea rotunda*dflg.dph.20420([@bib15])*Ochromonas danica*ochph. (het.)chrys.8.777([@bib41])*Ochromonas malhamensis*ochph. (het.)chrys.357.5([@bib39])*Ochromonas minima*ochph. (het.)chrys.575([@bib88])*Olisthodiscus luteus*ochph. (het.)raphidophyceae22.590([@bib5]; [@bib88])*Oxyrrhis marina*dflg.oxyrrhea39.5300([@bib9]; [@bib22])*Paragymnodinium shiwhaense*dflg.dph.10.9571([@bib68])*Paraphysomonas vestita*ochph. (het.)chrys.14.7116.85([@bib19])*Pavlova lutheri*haptophytapavlovophyceae6.5126([@bib5])*Peranema trichophorum*euglenozoaeuglenida (heteronematales)4520([@bib62]; [@bib29]; [@bib11])*Peridinium bipes*dflg.dph.42.9291([@bib22])*Peridinium cf. quinquecorne*dflg.dph.191500([@bib5]; [@bib59]; [@bib42])*Peridinium cinctum*dflg.dph.47.5120([@bib5]; [@bib59]; [@bib67])*Peridinium (Protoperidinium) claudicans*dflg.dph.77.5229([@bib72])*Peridinium (Protoperidinium) crassipes*dflg.dph.102100([@bib72])*Peridinium foliaceum*dflg.dph.30.6185.2([@bib50])*Peridinium (Bysmatrum) gregarium*dflg.dph.32.51291.7([@bib59])*Peridinium (Protoperidinium) ovatum*dflg.dph.61187.5([@bib72])*Peridinium (Peridiniopsis) penardii*dflg.dph.28.8417([@bib80])*Peridinium (Protoperidinium) pentagonum*dflg.dph.92.5266.5([@bib72])*Peridinium (Protoperidinium) subinerme*dflg.dph.50285([@bib72])*Peridinium trochoideum*dflg.dph.2553([@bib59])*Peridinium umbonatum*dflg.dph.30250([@bib59]; [@bib67])*Phaeocystis pouchetii*haptophytaprym.6.388([@bib5])*Polytoma uvella*chlph.chlorophyceae22.5100.9([@bib63]; [@bib29]; [@bib62])*Polytomella agilis*chlph.chlorophyceae12.4150([@bib30]; [@bib31]; [@bib29]; [@bib77])*Prorocentrum mariae-lebouriae*dflg.dph.14.8141.05([@bib50]; [@bib5]; [@bib69])*Prorocentrum micans*dflg.dph.45329.1([@bib5]; [@bib59])*Prorocentrum minimum*dflg.dph.15.1107.7([@bib5]; [@bib69])*Prorocentrum redfieldii Bursa (P.triestinum)*dflg.dph.33.2333.3([@bib85])*Protoperidinium depressum*dflg.dph.132450([@bib15])*Protoperidinium granii (Ostf.) Balech*dflg.dph.57.586.1([@bib85])*Protoperidinium pacificum*dflg.dph.54410([@bib15])*Prymnesium polylepis*haptophytaprym.9.145([@bib21])*Prymnesium parvum*haptophytaprym.7.230([@bib21])*Pseudopedinella pyriformis*ochph. (het.)dict.6.5100([@bib88])*Pseudoscourfieldia marina*chlph.pyr.4.142([@bib5])*Pteridomonas danica*ochph. (het.)dict.5.5179.45([@bib19])*Pyramimonas amylifera*chlph.pyr.24.522.5([@bib5])*Pyramimonas cf. disomata*chlph.pyr.9355([@bib88])*Rhabdomonas spiralis*euglenozoaeuglenida (aphagea)27120([@bib40])*Rhodomonas salina*cryptophytacrypt.14.5588.5([@bib47]; [@bib68])*Scrippsiella trochoidea*dflg.dph.25.387.6([@bib50]; [@bib5]; [@bib85])*Spumella* sp.ochph. (het.)chrys.1025([@bib90])*Teleaulax* sp.cryptophytacrypt.13.598([@bib68])*Trypanosoma brucei*euglenozoakinetoplastea18.820.5([@bib78])*Trypanosoma cruzi*euglenozoakinetoplastea20172([@bib44]; [@bib11])*Trypanosoma vivax*euglenozoakinetoplastea23.529.5([@bib3])*Trypanosoma evansi*euglenozoakinetoplastea21.516.1([@bib3])*Trypanosoma congolense*euglenozoakinetoplastea189.7([@bib3])*Tetraflagellochloris mauritanica*chlph.chlorophyceae4300([@bib4])

10.7554/eLife.44907.014

Data for swimming ciliates {#s8}
==========================

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

SpeciesPhylumClass$L$\[$\mu m$\]$U$\[${\mu m}/s$\]References*Amphileptus gigas*imnc.lit.808608([@bib13])*Amphorides quadrilineata*imnc.spir.138490([@bib15])*Balanion comatum*imnc.prostomatea16220([@bib90])*Blepharisma*pcdph.hettr.350600([@bib84]; [@bib77])*Coleps hirtus*imnc.prostomatea94.5686([@bib13])*Coleps* sp.imnc.prostomatea78523([@bib13])*Colpidium striatum*imnc.olig.77570([@bib7])*Condylostoma patens*pcdph.hettr.3711061([@bib13]; [@bib65])*Didinium nasutum*imnc.lit.1401732([@bib13]; [@bib65]; [@bib77]; [@bib84])*Euplotes charon*imnc.spir.661053([@bib13])*Euplotes patella*imnc.spir.2021250([@bib13])*Euplotes vannus*imnc.spir.82446([@bib92]; [@bib75])*Eutintinnus cf. pinguis*imnc.spir.147410([@bib15])*Fabrea salina*pcdph.hettr.184.1216([@bib66])*Favella panamensis*imnc.spir.238600([@bib15])*Favella* sp.imnc.spir.1501080([@bib15])*Frontonia* sp.imnc.olig.378.51632([@bib13])*Halteria grandinella*imnc.spir.50533([@bib13]; [@bib28])*Kerona polyporum*imnc.spir.107476.5([@bib13])*Laboea strobila*imnc.spir.100810([@bib15])*Lacrymaria lagenula*imnc.lit.45909([@bib13])*Lembadion bullinum*imnc.olig.43415([@bib13])*Lembus velifer*imnc.olig.87200([@bib13])*Mesodinium rubrum*imnc.lit.387350([@bib48]; [@bib76]; [@bib20])*Metopides contorta*imnc.armophorea115359([@bib13])*Nassula ambigua*imnc.nassophorea1432004([@bib13])*Nassula ornata*imnc.nassophorea282750([@bib13])*Opalina ranarum*placidozoa (heterokont)opalinea35050([@bib8]; [@bib84])*Ophryoglena* sp.imnc.olig.3254000([@bib65])*Opisthonecta henneg*imnc.olig.1261197([@bib65]; [@bib45])*Oxytricha bifara*imnc.spir.2821210([@bib13])*Oxytricha ferruginea*imnc.spir.150400([@bib13])*Oxytricha platystoma*imnc.spir.130520([@bib13])*Paramecium aurelia*imnc.olig.2441650([@bib13]; [@bib14])*Paramecium bursaria*imnc.olig.1301541.5([@bib13]; [@bib14])*Paramecium calkinsii*imnc.olig.1241392([@bib14]; [@bib13])*Paramecium caudatum*imnc.olig.225.52489.35([@bib14]; [@bib49])*Paramecium marinum*imnc.olig.115930([@bib13])*Paramecium multimicronucleatum*imnc.olig.2513169.5([@bib14])*Paramecium polycaryum*imnc.olig.911500([@bib14])*Paramecium* spp.imnc.olig.200975([@bib43]; [@bib84]; [@bib77])*Paramecium tetraurelia*imnc.olig.124784([@bib26])*Paramecium woodruffi*imnc.olig.1602013.5([@bib14])*Porpostoma notatum*imnc.olig.107.71842.2([@bib23])*Prorodon teres*imnc.prostomatea1751066([@bib13])*Spathidium spathula*imnc.lit.204.5526([@bib13])*Spirostomum ambiguum*pcdph.hettr.1045810([@bib13])*Spirostomum* sp.pcdph.hettr.10001000([@bib84])*Spirostomum teres*pcdph.hettr.450640([@bib13])*Stenosemella steinii*imnc.spir.83190([@bib15])*Stentor caeruleus*pcdph.hettr.528.51500([@bib13])*Stentor polymorphus*pcdph.hettr.208887([@bib13]; [@bib83];\
[@bib81])*Strobilidium spiralis*imnc.spir.60330([@bib15])*Strobilidium velox*imnc.spir.43150([@bib28])*Strombidinopsis acuminatum*imnc.spir.80390([@bib15])*Strombidium claparedi*imnc.spir.69.53740([@bib13])*Strombidium conicum*imnc.spir.75570([@bib15])*Strombidium* sp.imnc.spir.33360([@bib15])*Strombidium sulcatum*imnc.spir.32.5995([@bib24]; [@bib23]\
[@bib23])*Stylonichia* sp.imnc.spir.167737.5([@bib13]; [@bib65])*Tetrahymena pyriformis*imnc.olig.72.8475.6([@bib84]; [@bib77]; [@bib11])*Tetrahymena thermophila*imnc.olig.46.7204.5([@bib95])*Tillina magna*imnc.colpodea162.52000([@bib13])*Tintinnopsis kofoidi*imnc.spir.100400([@bib15])*Tintinnopsis minuta*imnc.spir.4060([@bib15])*Tintinnopsis tubulosa*imnc.spir.95160([@bib15])*Tintinnopsis vasculum*imnc.spir.82250([@bib15])*Trachelocerca olor*pcdph.karyorelictea267.5900([@bib13])*Trachelocerca tenuicollis*pcdph.karyorelictea4321111([@bib13])*Uroleptus piscis*imnc.spir.203487([@bib13])*Uroleptus rattulus*imnc.spir.400385([@bib13])*Urocentrum turbo*imnc.olig.90700([@bib13])*Uronema filificum*imnc.olig.25.71372.7([@bib23])*Uronema marinum*imnc.olig.56.91010([@bib23])*Uronema sp.*imnc.olig.251175([@bib84]; [@bib77])*Uronychia transfuga*imnc.spir.1186406([@bib58])*Uronychia setigera*imnc.spir.647347([@bib58])*Uronemella* spp.imnc.olig.28250([@bib73])

10.7554/eLife.44907.016

Decision letter

Chakraborty

Arup K

Reviewing Editor

Massachusetts Institute of Technology

United States

Herron

Matthew

Reviewer

Georgia Tech

United States

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article \"Swimming eukaryotic microorganisms exhibit a universal speed distribution\" for consideration by *eLife*. Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by Arup Chakraborty as the Senior and Reviewing Editor. The following individual involved in review of your submission has agreed to reveal his identity: Matthew Herron (Reviewer \#1).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

This study analyzes published data on swimming speed among flagellated and ciliated microorganisms. Data from both groups fit similar, but separate, log normal distributions, leading the authors to infer a \"universal way for ecological niches to be populated by abundant microorganisms.\" This short report is well-written, well-sourced, and has sufficient detail provided by the authors to replicate the findings. This methodology has the potential to enrich parallel genetics-based studies and provide deeper insights into the connections between ecology and evolution. However, we feel that the main claims of the paper need to be toned down or further substantiated in a revised submission.

Essential revisions:

1\) The ecological implication of the results, expressed in the Abstract and at the end of the Discussion, concern how ecological niches are partitioned by microorganisms. The limitation of the analyzed data is that it is per species, and all species are treated equally, with no information about abundance. It is undoubtedly an important result that aquatic ecosystems contain many species that swim slowly and far fewer that swim quickly, at least within each category. However, without information about abundances, this result says very little about the distribution of swimming speeds within a given ecosystem; that is, the data are equally compatible with slow swimmers being rare (but diverse) and fast swimmers being common (but homogenous) or the opposite. So, the distribution of swimming speeds among all individuals is unclear. Similarly, flagellates could be common and ciliates rare or vice versa. The point that the distributions of swimming speeds among species do not necessarily reflect the distributions of swimming speeds among individuals needs to be addressed.

2\) Can universality really be claimed with a set of just two distributions? Figures 2, 3, and 4 well-illustrate the authors\' point, but comprise fitting and rescaling of just these two sets of data. The demonstration in Figure 4 that two log-normal distributions may be collapsed onto one another is not too surprising. It seems with such a data set on hand, there is much potential to explore a variety of questions that can enrich or explain the authors conclusions. In particular, addressing a few of the following points will enhance the paper:

• Did the authors examine any other distributions, such as cell size?

• Did the authors attempt a physical rescaling of the distributions, e.g. in terms of the Reynolds or Peclet numbers?

• Is there sufficient resolution in the data set (i.e. number of samples) to explore/compare subsets of the data such as uniflagellates versus multiflagellates and ciliates?

• Similar to the above comment, examining the swimming speed distributions of other taxonomic groups from the data set -- i.e. corresponding to an early bifurcation in Figure 1 -- may provide additional insights and strengthen the authors\' argument for universality in the swimming speed distribution.

• While perhaps outside of the scope of the present short report, the potential implications of the authors\' observation abound for other organismal systems. Did they consider examining other dataset, e.g. for higher organisms as in Gazzola et al., Nature Physics 10 (2014)?

10.7554/eLife.44907.017

Author response

> Essential revisions:
>
> 1\] The ecological implication of the results, expressed in the Abstract and at the end of the Discussion, concern how ecological niches are partitioned by microorganisms. The limitation of the analyzed data is that it is per species, and all species are treated equally, with no information about abundance. It is undoubtedly an important result that aquatic ecosystems contain many species that swim slowly and far fewer that swim quickly, at least within each category. However, without information about abundances, this result says very little about the distribution of swimming speeds within a given ecosystem; that is, the data are equally compatible with slow swimmers being rare (but diverse) and fast swimmers being common (but homogenous) or the opposite. So, the distribution of swimming speeds among all individuals is unclear. Similarly, flagellates could be common and ciliates rare or vice versa. The point that the distributions of swimming speeds among species do not necessarily reflect the distributions of swimming speeds among individuals needs to be addressed.

This is a very interesting point and we agree with the reviewers. Alas, since there is little or no available data for abundance, it is difficult to make claim concerning particular ecosystems. We are implicitly assuming in our paper that the sampling of the underlying distributions was random. Of course, in real ecosystems there are interactions (symbiotic, mutualistic, etc.) among species so they are not necessarily "independent". To state our point clearly, we have added a sentence in the last paragraph of the main text.

> 2\] Can universality really be claimed with a set of just two distributions? Figures 2, 3, and 4 well-illustrate the authors\' point, but comprise fitting and rescaling of just these two sets of data. The demonstration in Figure 4 that two log-normal distributions may be collapsed onto one another is not too surprising. It seems with such a data set on hand, there is much potential to explore a variety of questions that can enrich or explain the authors conclusions. In particular, addressing a few of the following points will enhance the paper:
>
> • Did the authors examine any other distributions, such as cell size?

Yes we have. We have now added to the paper the available data on cell sizes that we believe helps present a broader picture. Figure 2---figure supplement 2 contains histogram of cell sizes, produced using values given in the revised tables in Appendix 1. The cell sizes represent the \'characteristic\' size for each cell (largest of the available sizes if different width/length were given in literature). The size distributions are distinct and no apparent similarity between them is visible. We have then used them to calculate the Reynolds and Péclet numbers, as suggested below.

> • Did the authors attempt a physical rescaling of the distributions, e.g. in terms of the Reynolds or Peclet numbers?

Rescaled distributions have been added as Figure 2---figure supplement 3. We have plotted there the Reynolds number for each organism. Due to the paucity of reported viscosities in the analysed works, we assumed the viscosity to be that of water at standard conditions in each case. The distributions are different, since ciliates are generally larger and faster swimmers compared to flagellates. The Péclet number is proportional to the Reynolds number, since both contain the product of cell size and swimming velocity. Therefore, we choose one (Re) as a measure of the character of the fluid transport. We have modified the paper to indicate these points.

> • Is there sufficient resolution in the data set (i.e. number of samples) to explore/compare subsets of the data such as uniflagellates versus multiflagellates and ciliates?

Unfortunately, the resolution of the subsets is not sufficient for the proposed comparison. The listed data was the information available in literature on the swimming problem. Our focus was to collect it and analyse it together. Moreover, some of the values given represent averages over samples analyzed in the papers, given with no further information on the uncertainty. In the plots, we have included the fitting errors and estimated the errors associated to the binning procedures. Within the available data, this statistical uncertainty was the only accessible measure.

> • Similar to the above comment, examining the swimming speed distributions of other taxonomic groups from the data set -- i.e. corresponding to an early bifurcation in Figure 1 -- may provide additional insights and strengthen the authors\' argument for universality in the swimming speed distribution.

Pointing back to the previous remark, we think the statistics for individual groups would not be sufficient to justify potential conclusions and thus we refrained from this analysis in the paper.

> • While perhaps outside of the scope of the present short report, the potential implications of the authors\' observation abound for other organismal systems. Did they consider examining other dataset, e.g. for higher organisms as in Gazzola et al., Nature Physics 10 (2014)?

The paper by Gazzola et al. concerns the relation that can be established between the Reynolds number for the flow created by a swimming organism and its swimming number, which involves the temporal details of the actuation (frequency of the periodic body motion which gives rise to the flow). In our study, we focus on unicellular microscale swimmers, and thus the Reynolds numbers rarely exceed 1 (see Figure 2---figure supplement 3), so the realm of higher Re is outside of the scope of the report.

[^1]: These authors contributed equally to this work.
