Swimming eukaryotic microorganisms exhibit a universal speed distribution
Abstract
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 lognormal 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.
https://doi.org/10.7554/eLife.44907.001Introduction
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 worldwide (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, singlecelled 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 taillike 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 crosslinking 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 denselypacked and collectivelymoving cilia, which are short hairlike 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 lognormal 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 (${N}_{\text{fl}}=112$ flagellates and ${N}_{\text{cil}}=77$ 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.
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 FreedmanDiaconis 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 submm/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 lowsize 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 $\mathrm{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), shows that ciliates and flagellates operate in different ranges of $\mathrm{Re}$, although for both groups $\mathrm{Re}<1$, imposing on them the same limitations of inertialess Stokes flow (Purcell, 1977; Lauga and Powers, 2009).
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 ($L}_{\text{fl}}=31\text{}\mu \text{m$, $L}_{\text{cil}}=132\mu \text{m$) and the median swimming speeds ($U}_{\text{fl}}=127\mathrm{m}/\mathrm{s$, $U}_{\mathrm{c}\mathrm{i}\mathrm{l}}=784\mathrm{m}/\mathrm{s$), and taking $D={10}^{3}(\mu \mathrm{m}{)}^{2}/\mathrm{s}$, we find $P{e}_{\mathrm{fl}}\sim 3.9$ and $P{e}_{\mathrm{cil}}\sim 103$, 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 welldescribed by lognormal distributions,
normalized as ${\int}_{0}^{\mathrm{\infty}}\mathit{d}UP(U)=1$, where $\mu $ and $\sigma $ are the mean and the standard deviation of $\mathrm{ln}U$. The median $M$ of the distribution is ${\mathrm{e}}^{\mu}$, with units of speed. Lognormal 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 $M}_{\text{fl}}=127\mathrm{m}/\mathrm{s$ and ${\sigma}_{\text{fl}}=0.978$ while for ciliates, we obtain $M}_{\text{cil}}=784\mathrm{m}/\mathrm{s$ and ${\sigma}_{\text{cil}}=0.936$. Lognormal 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.
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 $P(U,M)={M}^{1}F(U/M)$ for some function $F$. For the lognormal distribution, with $M$ the median, we find
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, 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 suggest a very similar range of variability of the fitted distributions. Furthermore, both the integrated absolute difference between the distributions (0.028) and the KullbackLeibler 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.
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 lognormal 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 lognormal 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
Request a detailed protocolData 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 lognormal distribution
Request a detailed protocolBin widths in histograms in Figure 2 and Figure 3 have been chosen separately for ciliates and flagellated eukaryotes according to the FreedmanDiaconis 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 $\mathrm{IQR}$ as
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 ${N}_{j}$ 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 lognormal distribution Equation (1). In general, from from data comprising $N$ measurements, labelled ${x}_{i}$ ($i=1,\mathrm{\dots},N$), the $n$th arithmetic moment ${\mathcal{M}}_{n}$ is the expectation $\mathbb{E}({X}^{n})$, or
Medians of the data were found by sorting the list of values and picking the middlemost value. For a lognormal distribution, the arithmetic moments are given solely by $\mu $ and $\sigma $ of the associated normal distribution as
where we have defined $M=\mathrm{exp}(\mu )$ and $\mathrm{\Sigma}=\mathrm{exp}({\sigma}^{2}/2)$, and note that $M$ is the median of the distribution. Thus, the mean is $M\mathrm{\Sigma}$ and the variance is ${M}^{2}{\mathrm{\Sigma}}^{2}\left({\mathrm{\Sigma}}^{2}1\right)$. From the first and second moments, we estimate
Having estimated $\mu $ and $\sigma $, 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 ${U}^{\ast}=\frac{1}{{N}_{i}}{\sum}_{i=1}^{{N}_{i}}{U}_{i}$, 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
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 lognormal distribution, no analogous explicit analytic result is available. Despite that, there is general agreement that a sum of independent lognormal random variables can be well approximated by another lognormal random variable. It has been proven by Szyszkowicz and Yanikome (2009) that the sum of identically distributed equally and positively correlated joint lognormal distributions converges to a lognormal 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
Request a detailed protocolIn order to quantify the differences between the fitted distributions, we define the integrated absolute difference $\mathrm{\Delta}$ between two probability distributions $p(x)$ and $q(x)$ ($x>0$) as
As the probability distributions are normalized, this is a measure of their relative ’distance’. As a second measure, we use the KullbackLeibler divergence (Kullback and Leibler, 1951),
Note that $D(p,q)\ne D(q,p)$ 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
Species  Phylum  Class  $L$[$\mu \mathrm{m}$]  $U$[$\mu \mathrm{m}/s$]  References 

Alexandrium minutum  dflg.  dph.  21.7  222.5  (Lewis et al., 2006) 
Alexandrium ostenfeldii  dflg.  dph.  41.1  110.5  (Lewis et al., 2006) 
Alexandrium tamarense  dflg.  dph.  26.7  200  (Lewis et al., 2006) 
Amphidinium britannicum  dflg.  dph.  51.2  68.7  (Bauerfeind et al., 1986) 
Amphidinium carterae  dflg.  dph.  16  81.55  (Gittleson et al., 1974; Bauerfeind et al., 1986) 
Amphidinium klebsi  dflg.  dph.  35  73.9  (Gittleson et al., 1974) 
Apedinella spinifera  ochph. (het.)  dict.  8.25  132.5  (Throndsen, 1973) 
Bodo designis  euglenozoa  kinetoplastea  5.5  39  (Visser and Kiørboe, 2006) 
Brachiomonas submarina  chlph.  chlorophyceae  27.5  96  (Bauerfeind et al., 1986) 
Cachonina (Heterocapsa) niei  dflg.  dph.  21.4  302.8  (Levandowsky and Kaneta, 1987; Kamykowski and Zentara, 1977) 
Cafeteria roenbergensis  bygira (heterokont)  bicosoecida  2  94.9  (Fenchel and Blackburn, 1999) 
Ceratium cornutum  dflg.  dph.  122.3  177.75  (Levandowsky and Kaneta, 1987; Metzner, 1929) 
Ceratium furca  dflg.  dph.  122.5  194  (Peters, 1929) 
Ceratium fusus  dflg.  dph.  307.5  156.25  (Peters, 1929) 
Ceratium hirundinella  dflg.  dph.  397.5  236.1  (Levandowsky and Kaneta, 1987) 
Ceratium horridum  dflg.  dph.  225  20.8  (Peters, 1929) 
Ceratium lineatus  dflg.  dph.  82.1  36  (Fenchel, 2001) 
Ceratium longipes  dflg.  dph.  210  166  (Peters, 1929) 
Ceratium macroceros  dflg.  dph.  50  15.4  (Peters, 1929) 
Ceratium tripos  dflg.  dph.  152.3  121.7  (Peters, 1929; Bauerfeind et al., 1986) 
Chilomonas paramecium  cryptophyta  crypt.  30  111.25  (Lee, 1954; Jahn and Bovee, 1967; Gittleson et al., 1974) 
Chlamydomonas reinhardtii  chlph.  chlorophyceae  10  130  (Gittleson et al., 1974; Roberts, 1981; Guasto et al., 2010) 
Chlamydomonas moewusii  chlph.  chlorophyceae  12.5  128  (Gittleson et al., 1974) 
Chlamydomonas sp.  chlph.  chlorophyceae  13  63.2  (Lowndes, 1944; Lowndes, 1941; Bauerfeind et al., 1986) 
Crithidia deanei  euglenozoa  kinetoplastea  7.4  45.6  (Gadelha et al., 2007) 
Crithidia fasciculata  euglenozoa  kinetoplastea  11.1  54.3  (Gadelha et al., 2007) 
Crithidia (Strigomonas) oncopelti  euglenozoa  kinetoplastea  8 .1  18.5  (Roberts, 1981; Gittleson et al., 1974) 
Crypthecodinium cohnii  dflg.  dph.  n/a  122.8  (Fenchel, 2001) 
Dinophysis acuta  dflg.  dph.  65  500  (Peters, 1929) 
Dinophysis ovum  dflg.  dph.  45  160  (Buskey et al., 1993) 
Dunaliella sp.  chlph.  chlorophyceae  10.8  173.5  (Gittleson et al., 1974; Bauerfeind et al., 1986) 
Euglena gracilis  euglenozoa  euglenida (eugl.)  47.5  111.25  (Lee, 1954; Jahn and Bovee, 1967; Gittleson et al., 1974) 
Euglena viridis  euglenozoa  euglenida (eugl.)  58  80  (Holwill, 1975; Roberts, 1981; Lowndes, 1941) 
Eutreptiella gymnastica  euglenozoa  euglenida (aphagea)  23.5  237.5  (Throndsen, 1973) 
Eutreptiella sp. R  euglenozoa  euglenida  50  135  (Throndsen, 1973) 
Exuviaella baltica (Prorocentrum balticum)  dflg.  dph.  15.5  138.9  (Wheeler, 1966) 
Giardia lamblia  srcm.  zoomastigophora  11.25  26  (Lenaghan et al., 2011; Campanati et al., 2002; Chen et al., 2012) 
Gonyaulax polyedra  dflg.  dph.  39.2  254.05  (Hand et al., 1965; Gittleson et al., 1974; Kamykowski et al., 1992) 
Gonyaulax polygramma  dflg.  dph.  46.2  500  (Levandowsky and Kaneta, 1987) 
Gymnodinium aureolum  dflg.  dph.  n/a  394  (Meunier et al., 2013) 
Gymnodinium sanguineum (splendens)  dflg.  dph.  47.6  220.5  (Kamykowski et al., 1992; Levandowsky and Kaneta, 1987) 
Gymnodinium simplex  dflg.  dph.  10.6  559  (Jakobsen et al., 2006) 
Gyrodinium aureolum  dflg.  dph.  30.5  139  (Bauerfeind et al., 1986; Throndsen, 1973) 
Gyrodinium dorsum (biflagellated)  dflg.  dph.  37.5  324  (Hand et al., 1965; Gittleson et al., 1974; Kamykowski et al., 1992; Levandowsky and Kaneta, 1987; Brennen and Winet, 1977) 
Gyrodinium dorsum (uniflagellated)  dflg.  dph.  34.5  148.35  (Hand and Schmidt, 1975) 
Hemidinium nasutum  dflg.  dph.  27.2  105.6  (Levandowsky and Kaneta, 1987; Metzner, 1929) 
Hemiselmis simplex  cryptophyta  crypt.  5.25  325  (Throndsen, 1973) 
Heterocapsa pygmea  dflg.  dph.  13.5  102.35  (Bauerfeind et al., 1986) 
Heterocapsa rotundata  dflg.  dph.  12.5  323  (Jakobsen et al., 2006) 
Heterocapsa triquetra  dflg.  dph.  17  97  (Visser and Kiørboe, 2006) 
Heteromastix pyriformis  chlph.  nephrophyseae  6  87.5  (Throndsen, 1973) 
Hymenomonas carterae  haptophyta  prym.  12.5  87  (Bauerfeind et al., 1986) 
Katodinium rotundatum (Heterocapsa rotundata)  dflg.  dph.  10.8  425  (Levandowsky and Kaneta, 1987; Throndsen, 1973) 
Leishmania major  euglenozoa  kinetoplastea  12.5  36.4  (Gadelha et al., 2007) 
Menoidium cultellus  euglenozoa  euglenida (eugl.)  45  136.75  (Holwill, 1975; Votta et al., 1971) 
Menoidium incurvum  euglenozoa  euglenida (eugl.)  25  50  (Lowndes, 1941; Gittleson et al., 1974) 
Micromonas pusilla  chlph.  mamiellophyceae  2  58.5  (Bauerfeind et al., 1986; Throndsen, 1973) 
Monas stigmata  ochph. (het.)  chrys.  6  269  (Gittleson et al., 1974) 
Monostroma angicava  chlph.  ulvophyceae  6.7  170.55  (Togashi et al., 1997) 
Nephroselmis pyriformis  chlph.  nephrophyseae  4.8  163.5  (Bauerfeind et al., 1986) 
Oblea rotunda  dflg.  dph.  20  420  (Buskey et al., 1993) 
Ochromonas danica  ochph. (het.)  chrys.  8.7  77  (Holwill and Peters, 1974) 
Ochromonas malhamensis  ochph. (het.)  chrys.  3  57.5  (Holwill, 1974) 
Ochromonas minima  ochph. (het.)  chrys.  5  75  (Throndsen, 1973) 
Olisthodiscus luteus  ochph. (het.)  raphidophyceae  22.5  90  (Bauerfeind et al., 1986; Throndsen, 1973) 
Oxyrrhis marina  dflg.  oxyrrhea  39.5  300  (Boakes et al., 2011; Fenchel, 2001) 
Paragymnodinium shiwhaense  dflg.  dph.  10.9  571  (Meunier et al., 2013) 
Paraphysomonas vestita  ochph. (het.)  chrys.  14.7  116.85  (ChristensenDalsgaard and Fenchel, 2004) 
Pavlova lutheri  haptophyta  pavlovophyceae  6.5  126  (Bauerfeind et al., 1986) 
Peranema trichophorum  euglenozoa  euglenida (heteronematales)  45  20  (Lowndes, 1941; Gittleson et al., 1974; Brennen and Winet, 1977) 
Peridinium bipes  dflg.  dph.  42.9  291  (Fenchel, 2001) 
Peridinium cf. quinquecorne  dflg.  dph.  19  1500  (Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987; Horstmann, 1980) 
Peridinium cinctum  dflg.  dph.  47.5  120  (Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987; Metzner, 1929) 
Peridinium (Protoperidinium) claudicans  dflg.  dph.  77.5  229  (Peters, 1929) 
Peridinium (Protoperidinium) crassipes  dflg.  dph.  102  100  (Peters, 1929) 
Peridinium foliaceum  dflg.  dph.  30.6  185.2  (Kamykowski et al., 1992) 
Peridinium (Bysmatrum) gregarium  dflg.  dph.  32.5  1291.7  (Levandowsky and Kaneta, 1987) 
Peridinium (Protoperidinium) ovatum  dflg.  dph.  61  187.5  (Peters, 1929) 
Peridinium (Peridiniopsis) penardii  dflg.  dph.  28.8  417  (Sibley et al., 1974) 
Peridinium (Protoperidinium) pentagonum  dflg.  dph.  92.5  266.5  (Peters, 1929) 
Peridinium (Protoperidinium) subinerme  dflg.  dph.  50  285  (Peters, 1929) 
Peridinium trochoideum  dflg.  dph.  25  53  (Levandowsky and Kaneta, 1987) 
Peridinium umbonatum  dflg.  dph.  30  250  (Levandowsky and Kaneta, 1987; Metzner, 1929) 
Phaeocystis pouchetii  haptophyta  prym.  6.3  88  (Bauerfeind et al., 1986) 
Polytoma uvella  chlph.  chlorophyceae  22.5  100.9  (Lowndes, 1944; Gittleson et al., 1974; Lowndes, 1941) 
Polytomella agilis  chlph.  chlorophyceae  12.4  150  (Gittleson and Jahn, 1968; Gittleson and Noble, 1973; Gittleson et al., 1974; Roberts, 1981) 
Prorocentrum mariaelebouriae  dflg.  dph.  14.8  141.05  (Kamykowski et al., 1992; Bauerfeind et al., 1986; Miyasaka et al., 1998) 
Prorocentrum micans  dflg.  dph.  45  329.1  (Bauerfeind et al., 1986; Levandowsky and Kaneta, 1987) 
Prorocentrum minimum  dflg.  dph.  15.1  107.7  (Bauerfeind et al., 1986; Miyasaka et al., 1998) 
Prorocentrum redfieldii Bursa (P.triestinum)  dflg.  dph.  33.2  333.3  (Sournia, 1982) 
Protoperidinium depressum  dflg.  dph.  132  450  (Buskey et al., 1993) 
Protoperidinium granii (Ostf.) Balech  dflg.  dph.  57.5  86.1  (Sournia, 1982) 
Protoperidinium pacificum  dflg.  dph.  54  410  (Buskey et al., 1993) 
Prymnesium polylepis  haptophyta  prym.  9.1  45  (Dölger et al., 2017) 
Prymnesium parvum  haptophyta  prym.  7.2  30  (Dölger et al., 2017) 
Pseudopedinella pyriformis  ochph. (het.)  dict.  6.5  100  (Throndsen, 1973) 
Pseudoscourfieldia marina  chlph.  pyr.  4.1  42  (Bauerfeind et al., 1986) 
Pteridomonas danica  ochph. (het.)  dict.  5.5  179.45  (ChristensenDalsgaard and Fenchel, 2004) 
Pyramimonas amylifera  chlph.  pyr.  24.5  22.5  (Bauerfeind et al., 1986) 
Pyramimonas cf. disomata  chlph.  pyr.  9  355  (Throndsen, 1973) 
Rhabdomonas spiralis  euglenozoa  euglenida (aphagea)  27  120  (Holwill, 1975) 
Rhodomonas salina  cryptophyta  crypt.  14.5  588.5  (Jakobsen et al., 2006; Meunier et al., 2013) 
Scrippsiella trochoidea  dflg.  dph.  25.3  87.6  (Kamykowski et al., 1992; Bauerfeind et al., 1986; Sournia, 1982) 
Spumella sp.  ochph. (het.)  chrys.  10  25  (Visser and Kiørboe, 2006) 
Teleaulax sp.  cryptophyta  crypt.  13.5  98  (Meunier et al., 2013) 
Trypanosoma brucei  euglenozoa  kinetoplastea  18.8  20.5  (Rodríguez et al., 2009) 
Trypanosoma cruzi  euglenozoa  kinetoplastea  20  172  (Jahn and Fonseca, 1963; Brennen and Winet, 1977) 
Trypanosoma vivax  euglenozoa  kinetoplastea  23.5  29.5  (Bargul et al., 2016) 
Trypanosoma evansi  euglenozoa  kinetoplastea  21.5  16.1  (Bargul et al., 2016) 
Trypanosoma congolense  euglenozoa  kinetoplastea  18  9.7  (Bargul et al., 2016) 
Tetraflagellochloris mauritanica  chlph.  chlorophyceae  4  300  (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
Species  Phylum  Class  $L$[$\mu \mathrm{m}$]  $U$[$\mu \mathrm{m}/s$]  References 

Amphileptus gigas  imnc.  lit.  808  608  (Bullington, 1925) 
Amphorides quadrilineata  imnc.  spir.  138  490  (Buskey et al., 1993) 
Balanion comatum  imnc.  prostomatea  16  220  (Visser and Kiørboe, 2006) 
Blepharisma  pcdph.  hettr.  350  600  (Sleigh and Blake, 1977; Roberts, 1981) 
Coleps hirtus  imnc.  prostomatea  94.5  686  (Bullington, 1925) 
Coleps sp.  imnc.  prostomatea  78  523  (Bullington, 1925) 
Colpidium striatum  imnc.  olig.  77  570  (Beveridge et al., 2010) 
Condylostoma patens  pcdph.  hettr.  371  1061  (Bullington, 1925; Machemer, 1974) 
Didinium nasutum  imnc.  lit.  140  1732  (Bullington, 1925; Machemer, 1974; Roberts, 1981; Sleigh and Blake, 1977) 
Euplotes charon  imnc.  spir.  66  1053  (Bullington, 1925) 
Euplotes patella  imnc.  spir.  202  1250  (Bullington, 1925) 
Euplotes vannus  imnc.  spir.  82  446  (Wang et al., 2008; Ricci et al., 1997) 
Eutintinnus cf. pinguis  imnc.  spir.  147  410  (Buskey et al., 1993) 
Fabrea salina  pcdph.  hettr.  184.1  216  (Marangoni et al., 1995) 
Favella panamensis  imnc.  spir.  238  600  (Buskey et al., 1993) 
Favella sp.  imnc.  spir.  150  1080  (Buskey et al., 1993) 
Frontonia sp.  imnc.  olig.  378.5  1632  (Bullington, 1925) 
Halteria grandinella  imnc.  spir.  50  533  (Bullington, 1925; Gilbert, 1994) 
Kerona polyporum  imnc.  spir.  107  476.5  (Bullington, 1925) 
Laboea strobila  imnc.  spir.  100  810  (Buskey et al., 1993) 
Lacrymaria lagenula  imnc.  lit.  45  909  (Bullington, 1925) 
Lembadion bullinum  imnc.  olig.  43  415  (Bullington, 1925) 
Lembus velifer  imnc.  olig.  87  200  (Bullington, 1925) 
Mesodinium rubrum  imnc.  lit.  38  7350  (Jonsson and Tiselius, 1990; Riisgård and Larsen, 2009; Crawford and Lindholm, 1997) 
Metopides contorta  imnc.  armophorea  115  359  (Bullington, 1925) 
Nassula ambigua  imnc.  nassophorea  143  2004  (Bullington, 1925) 
Nassula ornata  imnc.  nassophorea  282  750  (Bullington, 1925) 
Opalina ranarum  placidozoa (heterokont)  opalinea  350  50  (Blake, 1975; Sleigh and Blake, 1977) 
Ophryoglena sp.  imnc.  olig.  325  4000  (Machemer, 1974) 
Opisthonecta henneg  imnc.  olig.  126  1197  (Machemer, 1974; Jahn and Hendrix, 1969) 
Oxytricha bifara  imnc.  spir.  282  1210  (Bullington, 1925) 
Oxytricha ferruginea  imnc.  spir.  150  400  (Bullington, 1925) 
Oxytricha platystoma  imnc.  spir.  130  520  (Bullington, 1925) 
Paramecium aurelia  imnc.  olig.  244  1650  (Bullington, 1925; Bullington, 1930) 
Paramecium bursaria  imnc.  olig.  130  1541.5  (Bullington, 1925; Bullington, 1930) 
Paramecium calkinsii  imnc.  olig.  124  1392  (Bullington, 1930; Bullington, 1925) 
Paramecium caudatum  imnc.  olig.  225.5  2489.35  (Bullington, 1930; Jung et al., 2014) 
Paramecium marinum  imnc.  olig.  115  930  (Bullington, 1925) 
Paramecium multimicronucleatum  imnc.  olig.  251  3169.5  (Bullington, 1930) 
Paramecium polycaryum  imnc.  olig.  91  1500  (Bullington, 1930) 
Paramecium spp.  imnc.  olig.  200  975  (Jahn and Bovee, 1967; Sleigh and Blake, 1977; Roberts, 1981) 
Paramecium tetraurelia  imnc.  olig.  124  784  (Funfak et al., 2015) 
Paramecium woodruffi  imnc.  olig.  160  2013.5  (Bullington, 1930) 
Porpostoma notatum  imnc.  olig.  107.7  1842.2  (Fenchel and Blackburn, 1999) 
Prorodon teres  imnc.  prostomatea  175  1066  (Bullington, 1925) 
Spathidium spathula  imnc.  lit.  204.5  526  (Bullington, 1925) 
Spirostomum ambiguum  pcdph.  hettr.  1045  810  (Bullington, 1925) 
Spirostomum sp.  pcdph.  hettr.  1000  1000  (Sleigh and Blake, 1977) 
Spirostomum teres  pcdph.  hettr.  450  640  (Bullington, 1925) 
Stenosemella steinii  imnc.  spir.  83  190  (Buskey et al., 1993) 
Stentor caeruleus  pcdph.  hettr.  528.5  1500  (Bullington, 1925) 
Stentor polymorphus  pcdph.  hettr.  208  887  (Bullington, 1925; Sleigh and Aiello, 1972; Sleigh, 1968) 
Strobilidium spiralis  imnc.  spir.  60  330  (Buskey et al., 1993) 
Strobilidium velox  imnc.  spir.  43  150  (Gilbert, 1994) 
Strombidinopsis acuminatum  imnc.  spir.  80  390  (Buskey et al., 1993) 
Strombidium claparedi  imnc.  spir.  69.5  3740  (Bullington, 1925) 
Strombidium conicum  imnc.  spir.  75  570  (Buskey et al., 1993) 
Strombidium sp.  imnc.  spir.  33  360  (Buskey et al., 1993) 
Strombidium sulcatum  imnc.  spir.  32.5  995  (Fenchel and Jonsson, 1988; Fenchel and Blackburn, 1999 Fenchel and Blackburn, 1999) 
Stylonichia sp.  imnc.  spir.  167  737.5  (Bullington, 1925; Machemer, 1974) 
Tetrahymena pyriformis  imnc.  olig.  72.8  475.6  (Sleigh and Blake, 1977; Roberts, 1981; Brennen and Winet, 1977) 
Tetrahymena thermophila  imnc.  olig.  46.7  204.5  (Wood et al., 2007) 
Tillina magna  imnc.  colpodea  162.5  2000  (Bullington, 1925) 
Tintinnopsis kofoidi  imnc.  spir.  100  400  (Buskey et al., 1993) 
Tintinnopsis minuta  imnc.  spir.  40  60  (Buskey et al., 1993) 
Tintinnopsis tubulosa  imnc.  spir.  95  160  (Buskey et al., 1993) 
Tintinnopsis vasculum  imnc.  spir.  82  250  (Buskey et al., 1993) 
Trachelocerca olor  pcdph.  karyorelictea  267.5  900  (Bullington, 1925) 
Trachelocerca tenuicollis  pcdph.  karyorelictea  432  1111  (Bullington, 1925) 
Uroleptus piscis  imnc.  spir.  203  487  (Bullington, 1925) 
Uroleptus rattulus  imnc.  spir.  400  385  (Bullington, 1925) 
Urocentrum turbo  imnc.  olig.  90  700  (Bullington, 1925) 
Uronema filificum  imnc.  olig.  25.7  1372.7  (Fenchel and Blackburn, 1999) 
Uronema marinum  imnc.  olig.  56.9  1010  (Fenchel and Blackburn, 1999) 
Uronema sp.  imnc.  olig.  25  1175  (Sleigh and Blake, 1977; Roberts, 1981) 
Uronychia transfuga  imnc.  spir.  118  6406  (Leonildi et al., 1998) 
Uronychia setigera  imnc.  spir.  64  7347  (Leonildi et al., 1998) 
Uronemella spp.  imnc.  olig.  28  250  (Petroff et al., 2015) 
Data availability
All data generated or analysed during this study are included in the manuscript.
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Decision letter

Arup K ChakrabortySenior and Reviewing Editor; Massachusetts Institute of Technology, United States

Matthew HerronReviewer; 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 wellwritten, wellsourced, and has sufficient detail provided by the authors to replicate the findings. This methodology has the potential to enrich parallel geneticsbased 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 wellillustrate the authors' point, but comprise fitting and rescaling of just these two sets of data. The demonstration in Figure 4 that two lognormal 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)?
https://doi.org/10.7554/eLife.44907.016Author 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 wellillustrate the authors' point, but comprise fitting and rescaling of just these two sets of data. The demonstration in Figure 4 that two lognormal 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.
https://doi.org/10.7554/eLife.44907.017Article and author information
Author details
Funding
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.
Acknowledgements
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).
Senior and Reviewing Editor
 Arup K Chakraborty, Massachusetts Institute of Technology, United States
Reviewer
 Matthew Herron, Georgia Tech, United States
Publication history
 Received: January 5, 2019
 Accepted: June 29, 2019
 Version of Record published: July 16, 2019 (version 1)
Copyright
© 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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Further reading

 Physics of Living Systems
Little is known about how muscle length affects residual force enhancement (rFE) in humans. We therefore investigated rFE at short, long, and very long muscle lengths within the human quadriceps and patellar tendon (PT) using conventional dynamometry with motion capture (rFE_{TQ}) and a new, noninvasive shearwave tensiometry technique (rFE_{WS}). Eleven healthy male participants performed submaximal (50% max.) EMGmatched fixedend reference and stretchhold contractions across these muscle lengths while muscle fascicle length changes of the vastus lateralis (VL) were captured using Bmode ultrasound. We found significant rFE_{TQ} at long (7±5%) and very long (12±8%), but not short (2±5%) muscle lengths, whereas rFE_{WS} was only significant at the very long (38±27%), but not short (8±12%) or long (6±10%) muscle lengths. We also found significant relationships between VL fascicle length and rFE_{TQ} (r=0.63, p=.001) and rFE_{WS }(r=0.52, p=.017), but relationships were not significant between VL fascicle stretch amplitude and rFE_{TQ} (r=0.33, p=.126) or rFE_{WS }(r=0.29, p=.201). PT shearwave speedangle relationships did not agree with estimated PT forceangle relationships, which indicates that estimating PT loads from shearwave tensiometry might be inaccurate. We conclude that increasing muscle length rather than stretch amplitude contributes more to rFE during submaximal voluntary contractions of the human quadriceps.

 Cell Biology
 Physics of Living Systems
In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. Here, we demonstrate that the shapes of airinemes in zebrafish are consistent with a finite persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from livecell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding that there is a theoretical tradeoff between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of noncanonical cellular protrusions in general.