Dynamics of ellipsoidal tracers in swimming algal suspensions

Abstract

Enhanced diffusion of passive tracers immersed in active fluids is a universal feature of active fluids and has been extensively studied in recent years. Similar to microrheology for equilibrium complex fluids, the unusual enhanced particle dynamics reveal intrinsic properties of active fluids. Nevertheless, previous studies have shown that the translational dynamics of spherical tracers are qualitatively similar, independent of whether active particles are pushers or pullers---the two fundamental classes of active fluids. Is it possible to distinguish pushers from pullers by simply imaging the dynamics of passive tracers? Here, we investigated the diffusion of isolated ellipsoids in algal C. reinhardtii suspensions---a model for puller-type active fluids. In combination with our previous results on pusher-type E. coli suspensions [Peng et al., Phys. Rev. Lett. 116, 068303 (2016)], we showed that the dynamics of asymmetric tracers show a profound difference in pushers and pullers due to their rotational degree of freedom. Although the laboratory-frame translation and rotation of ellipsoids are enhanced in both pushers and pullers, similar to spherical tracers, the anisotropic diffusion in the body frame of ellipsoids shows opposite trends in the two classes of active fluids. An ellipsoid diffuses fastest along its major axis when immersed in pullers, whereas it diffuses slowest along the major axis in pushers. This striking difference can be qualitatively explained using a simple hydrodynamic model. In addition, our study on algal suspensions reveals that the influence of the near-field advection of algal swimming flows on the translation and rotation of ellipsoids shows different ranges and strengths. Our work provides not only new insights into universal organizing principles of active fluids, but also a convenient tool for detecting the class of active particles.

Full text

arXiv:1610.01641v1 [cond-mat.soft] 5 Oct 2016
Dynamics of ellipsoidal tracers in swimming algal suspensions
Ou Yang,1Yi Peng,1Zhengyang Liu,1Chao Tang,1Xinliang Xu,2and Xiang Cheng1, ∗
1Department of Chemical Engineering and Materials Science,
University of Minnesota, Minneapolis, MN 55455
2Beijing Computational Science Research Center, Beijing 100193, China
(Dated: October 3, 2018)
Enhanced diffusion of passive tracers immersed in active fluids is a universal feature of active
fluids and has been extensively studied in recent years. Similar to microrheology for equilibrium
complex fluids, the unusual enhanced particle dynamics reveal intrinsic properties of active fluids.
Nevertheless, previous studies have shown that the translational dynamics of spherical tracers are
qualitatively similar, independent of whether active particles are pushers or pullers—the two funda-
mental classes of active fluids. Is it possible to distinguish pushers from pullers by simply imaging
the dynamics of passive tracers? Here, we investigated the diffusion of isolated ellipsoids in algal
C. reinhardtii suspensions—a model for puller-type active fluids. In combination with our previous
results on pusher-type E. coli suspensions [Peng et al., Phys. Rev. Lett. 116, 068303 (2016)], we
showed that the dynamics of asymmetric tracers show a profound difference in pushers and pullers
due to their rotational degree of freedom. Although the laboratory-frame translation and rotation
of ellipsoids are enhanced in both pushers and pullers, similar to spherical tracers, the anisotropic
diffusion in the body frame of ellipsoids shows opposite trends in the two classes of active fluids. An
ellipsoid diffuses fastest along its major axis when immersed in pullers, whereas it diffuses slowest
along the major axis in pushers. This striking difference can be qualitatively explained using a sim-
ple hydrodynamic model. In addition, our study on algal suspensions reveals that the influence of
the near-field advection of algal swimming flows on the translation and rotation of ellipsoids shows
different ranges and strengths. Our work provides not only new insights into universal organizing
principles of active fluids, but also a convenient tool for detecting the class of active particles.
PACS numbers: 87.17Jj, 05.40-a, 47.63.Gd
I. INTRODUCTION
Active fluids are a novel class of nonequilibrium soft
materials, which are composed of a large number of self-
propelled particles suspended in simple fluids [1–5]. The
self-propelled particles can convert ambient or internal
free energy into persistent motions with a direction de-
pending on the local configuration of particles and inter-
particle interactions. Suspensions of swimming microor-
ganisms and synthetic colloidal swimmers are the most
widely studied active fluids in experiments and frequently
serve as models for theoretical investigations [4, 6]. Joint
efforts of experiments, simulations, and theories have
shown that active fluids exhibit surprising behaviors such
as giant number fluctuations [7–9], ordered phases with
collective particle motions [3, 10, 11], and abnormal rhe-
ology [12–15], unknown to conventional equilibrium com-
plex fluids. Among these features, the enhanced diffusion
of passive tracers in active fluids has attracted probably
the most extensive and sustained interests in recent years.
Wu and Libchaber first showed that a spherical tracer
immersed in suspensions of swimming Escherichia coli
exhibits a superdiffusive motion at short times and an
enhanced diffusion at long times [16]. Such an enhanced
diffusion can be orders of magnitude stronger than the
tracer’s intrinsic Brownian motion at high bacterial con-
centrations. Following their pioneering work, the en-
hanced diffusion of passive spherical tracers has been re-
ported and systematically studied in different active flu-
ids including suspensions of swimming microorganisms
such as prokaryotic cells E. coli [17–22, 24], Bacillus sub-
tilis [25] and Pseudomonas sp. [26] and eukaryotic cell
Chlamydomonas reinhardtii [27, 28] as well as synthetic
colloidal microswimmers [19]. The motion of passive
tracers in active fluids is induced by the fluid flow of
microswimmers in the far field and the direct steric in-
teraction between tracers and microswimmers in the near
field [21, 29–36]. Thus, similar to microrheology in equi-
librium systems [37], the study of the dynamics of passive
tracers can reveal the intrinsic properties of active fluids.
In addition, understanding the motion of passive trac-
ers in suspensions of swimming microorganisms is also of
biological relevance [16, 34]. The enhanced diffusion of
passive particles such as nutrient granules, dead bacte-
rial bodies, and liquid droplets of macromolecules boosts
fluid mixing at microscopic scales [28, 38], which helps
to maintain an active ecological balance and promotes
intercellular signaling and metabolite transport.
Although the dynamics of spherical passive tracers
have been extensively studied [16–35], the motion of
asymmetric tracers that possess degrees of freedom be-
yond simple translation has not been investigated un-
til very recently. Peng et al. studied the dynamics
of isolated ellipsoids in E. coli suspensions and showed
that both the translational and rotational diffusion of
ellipsoids are enhanced with increasing bacterial con-

2
centrations [39]. More importantly, they found an ab-
normal anisotropic diffusion in the body frame of ellip-
soids, where an ellipsoid diffuses fastest along its minor
axis at high bacterial concentrations. Such an abnor-
mal anisotropic diffusion leads to a negative translation-
rotation coupling in the laboratory frame that is strictly
forbidden for Brownian particles. Based on a simple
mean-field hydrodynamic calculation, they argued that
the unusual anisotropic diffusion is a result of the uni-
versal straining flow created by pusher-type active parti-
cles with E. coli as one specific example. Moreover, the
calculation predicts that asymmetric tracers immersed in
puller-type active fluids should show an opposite trend,
where the diffusion of ellipsoids along the major axis
should be more strongly enhanced. However, such a pre-
diction has not been experimentally verified.
Depending on the forces exerted on the surround-
ing fluid, isolated self-propelled microswimmers can be
generally categorized in two classes, i.e., “pushers” and
“pullers” [2, 3]. In the first order of a multipole expan-
sion, a pusher creates a tensile dipole flow in the far field,
where the fluid is pushed out parallel to the swimming
direction of the swimmer and is pulled in at its midpoint
perpendicular to the swimming direction (Fig. 1(a)). In
contrast, a puller creates a contractile dipole flow, where
the fluid is pulled in parallel to the swimming direction
and is pushed out at its midpoint (Fig. 1(b)). Most bac-
teria including E. coli and B. subtilis propel by rotating
long thin flagella, which push fluid backward and, there-
fore, are pushers [40]. Algae such as C. reinhardtii propel
by beating two anterior flagella that pull in fluid in the
front and are well-known examples of pullers [41, 42].
Although the two classes of active particles lead to pro-
found differences in the behaviors of active fluids, such as
their rheological response under shear [43–45] and their
stability in ordered swarming phases [1, 29, 46, 47], the
influences of pushers and pullers on the enhanced diffu-
sion of spherical tracers are identical in the dilute limit
[26, 28, 31, 32]. In both cases, the enhanced diffusion of
passive tracers increases linearly with the active flux, de-
fined as the product of the number density and the speed
of microswimmers [19, 21, 22, 27, 29–32].
In contrast, the calculation by Peng et al. showed
that when the rotational degree of freedom is considered,
the swimming mechanism of microswimmers directly af-
fects the coupling between the translation and rotation
of asymmetric passive tracers. Therefore, it is possible to
distinguish the swimming mechanism of active particles
by simply imaging the diffusion of asymmetric tracers
immersed in active fluids. The method should be easier
and faster to implement when compared with other com-
monly used procedures for probing the swimming mech-
anism of microswimmers such as imaging the periodic
time-irreversible stroke of flagella or cilia of microorgan-
isms [48, 49] or mapping the far-field flow field through
particle imaging velocimetry (PIV) [40–42]. For instance,
FIG. 1: Schematics showing the flow patterns induced by a
single pusher (a) and a single puller (b). The color indicates
the magnitude of the flow, which decays from the origin fol-
lowing a r−2scaling. The arrows indicates the direction of
the flow. E. coli is a typical pusher-type microswimmer [39],
whereas C. reinhardtii we study here is a typical puller-type
microswimmer.
the width of a single flagellum of E. coli is ∼20 nm,
which cannot be observed using optical microscopy with-
out special treatments [48]. It is hard, if not impossible,
to image the concentration gradient of various chemical
species around the body of self-propelled catalytic Janus
particles, which can be either pushers or pullers depend-
ing on the area fraction of the active catalytic region
on the surface of the particles [50]. Although PIV can
resolve the strength of multipole flow fields to higher or-
ders, it requires a large number of measurements to re-
duce statistical noise and demands manipulation of the
position and orientation of individual microswimmers for
image analysis [40–42]. Hence, the study of the dynam-
ics of asymmetric particles in active fluids will provide
not only new insights into the universal features of ac-
tive fluids, but also a convenient tool for characterizing
the fundamental properties of active particles.
Although the dynamics of spherical tracers have been
studied in both pushers and pullers [16–35], the dynam-
ics of asymmetric tracers have only been investigated in
pusher-type active fluids so far [39]. Here, we reported
our study on the dynamics of isolated ellipsoids in a
premier puller-type active fluid—suspensions of algae C.
reinhardtii. We studied the rich dynamics of ellipsoids
in algal suspensions and drew a detailed comparison be-
tween the dynamics of ellipsoids and spherical tracers in
both puller-type and pusher-type active fluids. Partic-
ularly, we investigated the anisotropic diffusion of ellip-
soids in the body frame and verified the prediction of
the simple hydrodynamic model. Moreover, thanks to
the large size of C. reinhardtii, the algal system allows
us to probe the detail of the near-field translational and
rotational advection of ellipsoidal tracers under the influ-
ence of single alga that cannot be easily achieved with E.
coli [27]. Through this study, we show that the influence
of an algal swimming flow on the rotation of ellipsoids

3
FIG. 2: Experimental setup and method. (a) The left panels
show a schematic of the adjustable wire-frame device. The
right panel shows a zoomed view of a region of 130×130 µm2
in the center of the free-standing film. The solid red line indi-
cates the trajectory of the ellipsoid over a time interval of 10 s.
The dashed lines indicate the trajectories of algae. Scale bar:
20 µm. (b) Reference frame transformation. The displace-
ment vector, δxn, of an ellipsoid in a small time interval can
be decomposed into two orthogonal components (δxn,δyn) in
the laboratory frame based on the x-yaxis or (δ˜xn,δ˜yn) in
the body frame based on the ˜x-˜yaxis. The two coordinates
are connected through a 2D rotational matrix specified by the
orientation of the ellipsoid in the laboratory frame, θn.
is weaker and has a shorter range, when compared with
the influence of the same flow on the translation of ellip-
soids. We provide a quantitative estimate of the range of
influences on different degrees of freedom.
Our paper is organized as follows. We introduce ma-
terials and experimental methods in Section II and show
results and discussions in Section III. Specifically, the
laboratory-frame dynamics and the body-frame dynam-
ics of ellipsoids are presented in two separate subsections
of Section III. The anisotropic diffusion of ellipsoids—the
key feature that distinguishes between pusher-type and
puller-type active fluids—is then discussed in subsection
III C. Conclusions are summarized in Section IV.
II. EXPERIMENT
A. Materials
We used wild-type C. reinhardtii (CC125+) as our ac-
tive particles. C. reinhardtii is a unicellular green alga
with a spherical body of diameter d≈7–10 µm. It has
two anterior flagella about 10–12 µm in length. The flag-
ella beat at ∼50 Hz and propel the cell at a mean speed
of 100–200 µm/s [51]. We cultured algae in minimal me-
dia (M1) on a light cycle of 14 h bright and 10 h dark.
The procedure helps to increase the uniformity of the size
and the speed of algae [28]. To vary the concentration
of algal suspensions, we centrifuged suspensions at 1800
rpm for 1.5 min and then resuspended the concentrated
suspensions in M1 media to desired concentrations.
We used polystyrene (PS) ellipsoids as our passive
asymmetric tracers. The ellipsoids were obtained by
stretching monodispersed micron-sized PS spheres at 150
◦C above the glass transition temperature of PS [52]. The
lengths of the semi-principal axes of the ellipsoids were
fixed at a=b= 2.8±0.2µm and c= 14.2±0.5µm,
the same as the ellipsoids used in the previous study on
pusher-type E. coli suspensions [39]. The aspect ratio of
ellipsoids is p≡c/a = 5.1. A small number of PS ellip-
soids were mixed into algal suspensions with the volume
fraction of ellipsoids below 0.05%. The concentration is
so low that the hydrodynamic coupling between ellipsoids
can be safely neglected in our experiments.
B. Methods
A drop of an algal suspension containing PS ellipsoids
was first deposited onto a small gap suspended by four
thin wires made of human hairs. By enlarging the dis-
tance between the four wires, we stretched the suspen-
sion into a free-standing 4 ×4 mm2liquid film with a
thickness ∼20 µm (Fig. 2(a)). The construction of the
adjustable wire-frame device is similar to those used in
previous experiments on enhanced diffusions of passive
tracers [10, 28, 39, 42]. To stabilize the thin liquid film,
a trace amount of surfactant (Tween 20, 0.03 vol%) was
added into the algal suspensions. We quantify the con-
centration of algae in the thin film by measuring the area
fraction of algae φ=N πhdi2/4A, where Nis the num-
ber of algae in the field of view, hdi= 8 µm is the mean
diameter of algae, and Ais the total area of the field
of view. Note that, differently from E.coli suspensions
where high bacterial concentrations can be achieved [39],
the maximal φof algal suspensions is limited in the dilute
regime [27]. When φis above 4%, C. reinhardtii start to
shed flagella and stop swimming in our experiments.
We recorded the motions of algae and ellipsoids at a
frame rate of 10 frames/second using a Nikon Ti-E in-
verted microscope with ×20 0.5 NA (numerical aperture)

4
objective (Fig. 2(a) and Supplemental Movies 1 and 2
[53]). To eliminate the phototaxis of algae, a long-pass
filter with a cut-on wavelength of 620 nm was used for
illumination. We extracted the position and orientation
of ellipsoids using a custom particle tracking algorithm.
The laboratory-frame trajectories of the center of mass
of ellipsoids, x(t), and the orientation of ellipsoids, θ(t),
can then be obtained. Here, θis defined as the angle of
the major axis of ellipsoids with respect to the xaxis, ar-
bitrary selected and fixed in the laboratory frame. Due
to the centrosymmetry of ellipsoids, θis limited between
−π/2 and π/2.
In order to probe the anisotropic diffusion of ellipsoids,
we transformed the motion of ellipsoids from the labora-
tory frame into the body fame (Fig. 2(b)). The body
frame is a special frame of reference that rotates along
with ellipsoids. Particle displacements in the body frame
were obtained through rotation of particle displacements
in the laboratory frame (Fig. 2(b)) [39, 54, 55]. Specif-
ically, the displacement of an ellipsoid in the laboratory
frame, δx(tn) = x(tn+1)−x(tn), in a small time inter-
val δt =tn+1 −tnwas transformed into its displacement
in the body frame, δ˜x(tn), via δ˜x(tn) = R(tn)δx(tn),
where R(tn) is a 2D rotation matrix with R(tn) =
cos θnsin θn
−sin θncos θn. Here, θn= [θ(tn) + θ(tn+1)]/2 is
the average orientation of the ellipsoid in δt. Since
the rotation of ellipsoids in δt is small, choosing either
θn=θ(tn) or θn=θ(tn+1) does not change the final re-
sult [39, 54]. Last, we constructed the total body-frame
displacement by summing displacements in each small
time step, ∆˜x(t) = Pk
n=0 δ˜x(tn), where tk=t0+t.
∆˜x(t) has two orthogonal components, ∆˜xand ∆˜y, in-
dicating the displacements of ellipsoids along the major
and minor axes, respectively.
III. RESULTS AND DISCUSSIONS
A. Dynamics in the laboratory frame
We first investigated the dynamics of ellipsoids in the
laboratory frame. Figure 3(a) and (b) show translational
mean-squared displacements (MSDT), h[∆x(t)]2i=
h[x(t+t0)−x(t0)]2i, and rotational mean-squared dis-
placements (MSDR), h[θ(t)]2i=h[θ(t+t0)−θ(t0)]2i, re-
spectively. In both cases, the average is taken over the
initial time t0. Similar to the dynamics of ellipsoids in
pusher-type active fluids [39], ellipsoids in algal suspen-
sions show superdiffusive motions at short times and dif-
fusive motions at long times. The effective translational
and rotational diffusivity, Dtand Dr, can be extracted by
fitting the long-time diffusions with h[∆x(t)]2i= 4DTt
and h[θ(t)]2i= 2DRt. We found that both DTand DR
increase linearly with algal concentrations (Fig. 3(c)). It
is worth noting that the intrinsic Brownian translation
and rotation of the large ellipsoids used in our experi-
ments are orders of magnitude smaller, even when com-
pared with the diffusivity of ellipsoids in algal suspensions
of the lowest concentration we studied. Indeed, we barely
observed any diffusive motions of ellipsoids without algae.
The weak diffusion led to large experimental uncertain-
ties when we attempted to measure the bare diffusivity
of ellipsoids in the liquid films. Instead, we estimated
the Brownian diffusivity from the Stokes-Einstein rela-
tion and the anisotropic drag coefficients of ellipsoids in
three dimensions [56], which gives DT0= 0.038 µm2/s
and DR0= 3.1×10−4rad2/s.
The linear relationship between diffusivity and algal
concentrations has also been found for enhanced diffu-
sions of spherical tracers in both pushers and pullers in-
cluding E. coli and algal suspensions [19, 21, 22, 27, 29–
32]. However, the result is different from a measurement
on the enhanced diffusion of spherical tracers in quasi-
two-dimensional algal suspensions, where the transla-
tional diffusivity increases nonlinearly with φfollowing
DT∼φ3/2[28]. The difference is unlikely due to the
additional rotational degree freedom and probably arises
from the three-dimensional (3D) nature of our experi-
ments, where we used thicker liquid films and smaller
algae. It is known that DT∼φfor spherical tracers in
3D algal suspensions [27]. Moreover, DTis also ∼φfor
spherical tracers in dilute E. coli suspensions confined in
liquid films with a thickness of a couple of bacterial body
lengths [16]. Last, for ellipsoids in E. coli suspensions,
both DTand DRincrease linearly with bacterial concen-
trations in the dilute limit [39], qualitatively similar to
our results in algal suspensions.
We also studied the probability distribution of transla-
tional and rotational displacements within a small time
interval ∆t= 0.1 s. Figures 4(a) and (b) show re-
spectively the translational probability distribution func-
tion (PDFT) and the rotational probability distribution
function (PDFR) at different algal concentrations. Both
PDFTand PDFRshow Gaussian cores at small displace-
ments and long exponential tails at large displacements.
The width of the distributions broadens significantly with
increasing algal concentrations. The exponential tails of
the large displacements are induced by the advection of
the fluid flow of a single algae swimming close to ellip-
soids, whereas the Gaussian cores indicate an effective
diffusion induced by the average fluid flow of numerous
algae further away from ellipsoids [27]. It should be em-
phasized that although it is unambiguous that the large
displacements of ellipsoids result from close encounters
between individual algae and ellipsoids, the effective dif-
fusion of the small displacements is also a direct conse-
quence of algal flows instead of the thermal fluctuation of
the surrounding media. In fact, the PDFs derived from
the bare diffusivity DT0and DR0are significantly nar-
rower than the PDFs of the lowest-concentration algal
suspensions (Fig. 4(a) and (b)).

5
FIG. 3: Dynamics of ellipsoids in the laboratory frame. (a) Mean-square translational displacements of ellipsoids (MSDT) at
different algal concentrations φ. (b) Mean-square rotational displacements of ellipsoids (MSDR) at the same φ. The dashed
lines indicate the slope of the data. (c) Effective translational diffusivity, DT(the left axis), and effective rotational diffusivity,
DR(the right axis), as a function of φ.DTand DRwere extracted from the long-time diffusions of the corresponding MSDs.
The dashed line indicates a linear relation between DT,R and φ.
The relative contributions of the near-field advection
and the far-field diffusion can be further quantified using
a formula originally proposed for spherical tracers [27],
PDF(∆λ) = (1 −f)
2πδ2
g
e−(∆λ)2/2δ2
g+f
2δe
e−|∆λ|/δe,(1)
where ∆λ≡∆xfor translation and ∆θfor rotation,
δgindicates the standard deviation of the Gaussian core
and δeindicates the decay length of the exponential tails.
The weighting factor, f(φ)∈[0,1], quantifies the rela-
tive contributions of the Gaussian and the exponential
terms. f= 0 gives a pure Gaussian distribution and
f= 1 gives an exponential (Laplace) distribution. f(φ)
extracted from both PDFTand PDFRincreases with
the algal concentration φ(Fig. 4(c)), indicating an in-
creasing influence of the near-field advection. Although
ffrom PDFTand PDFRfollow a qualitatively similar
trend, the rotational f,fr, is consistently smaller than
the translational f,fT. At high φ,fTgradually grows to
a value around 0.67, whereas fRapproaches 0.5. Based
on the study of spherical tracers, Leptos et al. pro-
posed a heuristic physical picture, where they suggested
that there exists a sphere of influence surrounding each
swimming alga [27]. Tracers within the sphere are dom-
inantly influenced by the advection induced by the fluid
flow of the alga, whereas outside the sphere tracer dy-
namics are affected by the fluid flows of multiple algae,
which on average lead to diffusive motions. The radius
of the sphere is estimated as r∼ hdi(f /φ)1/3based on
dimensional analysis, where hdi ≈ 8µm is the average
diameter of algae. Our results suggest that the sphere
of influence for translational advection is systematically
larger than that for rotational advection. The size ra-
tio between the two spheres of influence at high φis
rT/rR= (fT/fR)1/3= 1.10 ±0.06.
The difference between the sizes of the sphere of in-
fluence for the two degrees of freedom can be under-
stood from the nature of the swimmer’s velocity field.
Without external forces, the fluid velocity around a sin-
gle swimming microorganism, u(r), follows a dipole form
[57], which decays as u∼r−2in 3D [1, 2]. The ve-
locity field of the dipole flow determines the transla-
tional motion of ellipsoids. On the other hand, the ro-
tational motion of ellipsoids is dictated by the rate of
strain and the vorticity of the dipole flow, which de-
cays faster following Ω ∼r−3, where Ω is the vortic-
ity [58]. Hence, the influence of the rotational advection
imparted by swimming algae has a shorter range, lead-
ing to a smaller sphere of influence. Quantitatively, the
oscillatory flow field around a single alga can be approx-
imated as u(r) exp(iωt), where ω≈2π×50 rad/s is the
beating frequency of the flagella of algae [27, 42]. The
radius of the sphere of influence in translation can be es-
timated as the location where the translational displace-
ment due to the advection in a half cycle, 2u(rT)/ω, is
comparable with the diffusive displacement in the same
time interval, (2DTt)1/2= (2πDT/ω)1/2[27]. Note that
t=π/ω. The argument leads to u(rT)∼(πDTω/2)1/2.
Similarly, the radius of the sphere of influence in rota-
tion can be estimated as Ω(rR)∼(πDRω/2)1/2. From
the dipole flow of swimming microorganisms, we have
u(r)∼k/r2and Ω(r)∼1
2|∇ × u| ∼ k/r3, where k
is the dipole strength depending on the specific swim-
ming mechanism of microorganisms. By using the rela-
tion u(rT)/Ω(rR) = (DT/DR)1/2, we reach
rT
rR
=r2
TDR
DT1/6
.(2)
The radius of the sphere of influence in translation, rT,
depends on the dipole strength of algae, which can be
estimated as k=U0l2≈4.9×104µm3/s [60]. Here,
U0≈150 µm/s is the characteristic speed of algae from
our direct measurements. A similar value has also been
reported in Ref. [59]. lis the length of the force dipole.

6
FIG. 4: Probability distribution functions of laboratory-frame
translational displacements (PDFT) (a) and rotational dis-
placements (PDFR) (b) in a time interval ∆t= 0.1 s at dif-
ferent algal concentrations φ. The dashed lines are the PDFs
of Brownian ellipsoids with bare translational and rotational
diffusivity given in the text. The solid lines are fits using
Eq. 1. (c) The weighting factor, f, as a function of φfor both
PDFTand PDFR.findicates the relative contributions of
diffusion and advection. The solid lines are visual guides.
l=hdi+lf≈18 µm, where hdiis the average length
of algae and lf≈10 µm is the length of algal flagella.
Using the relation k/r2
T= (πDTω/2)1/2and DT≈27
µm2/s at high φfrom our measurements (Fig. 3(c)), we
have rT≈20.5µm [61]. Inserting rT,DTand DR≈0.14
rad2/s from our measurements at high φinto Eq. 2, we
have rT/rR≈1.14, consistent with our estimate from
f. Finally, as an interesting comparison, we can also
calculate the ratio of length scales, r′
T/r′
R.r′
Tand r′
R
are the lengths where the advection in a half cycle is
comparable with the size of tracers. Previous work has
shown that r′
Tis the same order of magnitude as rT[27].
For translation, r′
Tcan be obtained from 2u(r′
T)/ω ≈
hai, where we use the average length of the three semi-
axes of ellipsoids, hai= (a+b+c)/3 = 6.6µm, as the
characteristic size of ellipsoids. Similarly, we also have
2Ω(rR)/ω ≈π/2 for rotation, where we set π/2 as a
typical angle of rotation. Combining these two relations,
we have r′
T/r′
R= [πr′
T/(2hai)]1/3= 1.17, quantitatively
similar to rT/rRwe obtained above. Note that we use
r′
T≈6.8µm here, which is obtained from the relation
2k/(r′
Tω) = hai.
The laboratory-frame dynamics of ellipsoids show that
the translational and rotational motions of ellipsoids arise
from the same origin, i.e. the dipole flow of swimming
algae. In comparison with previous studies on passive
tracers in active fluids, we found that the short-time su-
perdiffusion and the long-time diffusion of ellipsoids in al-
gal suspensions are qualitatively similar to the dynamics
of ellipsoids in pusher-type bacterial suspensions. More-
over, the linear relation between the enhanced diffusivity
and algal concentrations is also quantitatively the same
as the enhanced translational diffusion of spherical trac-
ers in both pushers and pullers. Thus, if treated inde-
pendently in the laboratory frame, the translational and
rotational degrees of freedom of ellipsoids show identi-
cal behaviors regardless of the swimming mechanism of
active particles. However, the nature of dipole flow dic-
tates a coupling between the two degrees of freedom as
we shall show next.
B. Dynamics in the body frame
The body-frame dynamics of ellipsoids show profound
differences in pusher-type and puller-type active fluids.
Figure 5(a) shows the mean-squared displacements of
ellipsoids along the major axis (MSDa) and along the
minor axis (MSDb). Similar to the laboratory-frame
MSDs, we found a superdiffusive regime at short times
and a diffusive regime at long times in both MSDaand
MSDb. The diffusivity along the major and minor axis
can be obtained by fitting the long-time diffusions with
MSDa≡ h∆˜x2i= 2Datand MSDb≡ h∆˜y2i= 2Dbt,
respectively. Both Daand Dbincrease with algal con-
centrations (Fig. 5(b)). The bare diffusivity of Brownian

7
FIG. 5: Dynamics of ellipsoids in the body frame. (a) Mean-square translational displacements of ellipsoids along the major
axis (MSDa) and along the minor axis (MSDb) at different algal concentrations φ. (b) Effective diffusivity along the major
axis, Da, and along the minor axis, Db, as a function of φ.Daand Dbwere extracted from the long-time diffusions of MSDa
and MSDb. The dashed line is a linear fit of Dbfor guiding the eye. (c) Anisotropic diffusion of ellipsoids, Da/Db, as a function
of φ.
ellipsoids in the body frame without algae is again or-
ders of magnitude smaller, with Da0= 0.044 µm2/s and
Db0= 0.033 µm2/s.
More interestingly, ellipsoids show an anisotropic dif-
fusion with Da/Db6= 1 (Fig. 5(c)) [62]. The anisotropic
diffusion in the body frame gives rise to a non-zero cross-
correlation between the translation and rotation of par-
ticles in the laboratory frame [39, 54]. Thus, the trans-
lational and rotational degrees of freedom of asymmetric
tracers are coupled in algal suspensions. Da/Dbvaries
with algal concentrations. At low φ, the diffusion along
the major axis is faster than that along the minor axis
with Da/Db≈1.15. As φincreases, the diffusion along
the major axis is more strongly enhanced with Da/Db
increasing to ∼1.35 at high φ. As a comparison, for
the same size Brownian ellipsoids, Da0/Db0= 1.33. The
upper limit of the anisotropic diffusion of Brownian el-
lipsoids in 3D is 2, which is reached when the aspect
ratio of ellipsoids p→ ∞. More importantly, the in-
creasing trend of Da/Dbwith φis opposite to the concen-
tration dependence of the anisotropic diffusion of ellip-
soids in pusher-type active fluids, where Da/Dbdecreases
monotonically with increasing φ, to such an extent that
Da/Dbbecomes smaller than 1 at high enough concen-
trations (Figs. 6(a)(c)) [39]. This difference manifests no-
ticeably in the motion of ellipsoids: an ellipsoid diffuses
fastest along the minor axis in pushers when Da/Db<1,
whereas it diffuses fastest along the major axis in pullers
with Da/Db>1. The origin of this striking difference
will be discussed in Sec. III C below. As a final com-
ment, it should be noted that since the dynamics of el-
lipsoids are dominated by the fluid flows of swimming
algae, the contribution of the thermal fluctuation to the
anisotropic diffusion is negligible in our experiments. Be-
cause Da0/Db0= 1.33 for Brownian ellipsoids, Da/Db
should eventually increase with diminishing algal con-
tributions when the thermal fluctuation starts to play
the dominant role. Hence, Da/Dbshould show a non-
monotonic trend as φ→0. However, due to the small
bare diffusivity of ellipsoids, any small drift currents in
liquid films will induce large experimental uncertainties
in our measurements. We were not able to resolve this
increasing trend at small φin our experiments.
Last, for completeness, we also show the PDFs of el-
lipsoids in the body frame with a time interval ∆t= 0.1
s. Figures 7(a) and (b) show PDFs along the major axis
(PDFa) and along the minor axis (PDFb), respectively.
The results are qualitatively similar to PDFTin the lab-
oratory frame. The body-frame PDFs can also be fitted
with Eq. 1. The resulting ffor PDFa,fa, and for PDFb,
fb, are shown in Fig. 7(c). faand fbfollow a qualitatively
similar trend as fTand fR, which increase with φand
saturate toward a constant at high φ.fais larger than
fbat high φ, indicating a stronger influence of advection
on particles’ motion along the major axis.
C. Discussions on the body-frame anisotropic
diffusion
We shall now discuss the origin of the anisotropic dif-
fusion of ellipsoids in the body frame. From the Green-
Kubo formula [63], the diffusivity of a random motion
equals the integral of the velocity autocorrelation of the
motion,
D=Z∞
0
dthv(t0+t)v(t0)i.(3)
Assume the velocity autocorrelation follows a simple ex-
ponential decay, hv(t0+t)v(t0)i=hv2
0iexp(−t/τ), where
τis the correlation time and hv2
0iis the mean-square
velocity [39]. We have D=hv2
0iτ, a relation that can

8
FIG. 6: Comparison of anisotropic diffusions in pullers and pushers. (a) Anisotropic diffusion of ellipsoids, Da/Db, in puller-type
C. reinhardtii, and in pusher-type E. coli. The concentrations of active particles are normalized by the maximal concentration
studied in experiments. φmax = 3.9% for C. reinhardtii.φmax = 40n0for E. coli, where n0= 8 ×108cells/ml. (b) The ratio of
the correlation times along the ma jor and minor axis, τa/τb, in C. reinhardtii and in E. coli. Horizontal dashed lines indicate
the ratio of 1. Since the maximal algal concentration we can achieve is much smaller than the maximal bacterial concentration
due to different microbial physiology of C. reinhardtii and E. coli (see Sec IIB), it may seem arbitrary to normalize φby φmax.
Thus, we also plot Da/Db(c) and τa/τb(d) as a function of the normalized laboratory-frame translational diffusivity DT/DT,0,
which show similar increasing and decreasing trends for pullers and pushers, respectively. The data for E. coli are extracted
from Ref. [39].
also be obtained based on dimensional analysis. hv2
0i1/2
indicates the step size of the random motion per unit
time, whereas τindicates the persistence of the motion.
Naturally, a motion that has larger steps and is more
persistent in its moving direction shows a large diffu-
sivity. For Brownian ellipsoids, hv2
0i1/2=hv0iis the
speed of Brownian particles in the ballistic regime [65].
In this superdiffusive regime, we can further approximate
hv2
0i ≈ h∆x2i/∆t2, where h∆x2iis the mean-square dis-
placement in a small time interval ∆t.
Applying the above general consideration in the
anisotropic diffusion of ellipsoids in active fluids, we have
Da
Db
=h∆˜x2i
h∆˜y2i·τa
τb
,(4)
which leads to
τa
τb
=Da/Db
h∆˜x2i/h∆˜y2i.(5)
Here, τaand τbare the correlation times of ellipsoids’ mo-
tions along the ma jor and minor axis, respectively. We
measured the average ratio of the mean-square displace-
ments in the superdiffusive regime, hh∆˜x2i/h∆˜y2ii∆t, at
different φ, where the average is taken for all ∆t≤0.8 s,
the upper limit of the superdiffusive regime (Fig. 5(a)).
We found that hh∆˜x2i/h∆˜y2ii∆t≈1.13 ±0.02 indepen-
dent of φat low φand increases slightly at the highest
φ(Fig. 8). With hh∆˜x2i/h∆˜y2ii∆tand Da/Db, we ob-
tained τa/τbfrom Eq. 5 (Fig. 8). Although τa/τbthus
obtained shows large errors due to drift flows in the thin
film and the variation of film thickness and algal activity
in different experimental runs, the increasing trend of the
mean value of τa/τbis clear.

9
FIG. 7: Probability distribution functions of body-frame
translational displacements along the major axis (PDFa) (a)
and along the minor axis (PDFb) (b) in a time interval
∆t= 0.1 s at different algal concentrations φ. The dashed
lines are the PDFs of Brownian ellipsoids with bare diffusiv-
ity along the major and minor axis given in the text. The
solid lines are fits using Eq. 1. (c) The weighting factor, f, as
a function of φfor both PDFaand PDFb. The solid lines are
visual guides.
FIG. 8: Origin of the anisotropic diffusion. Black squares
are the ratio of the correlation times along the major and
minor axes, τa/τb, as a function of algal concentrations φ.
Red circles are the average ratio of the mean-square displace-
ments along the major and minor axis in the superdiffusive
regime, hh∆˜x2i/h∆˜y2ii∆t, as a function of φ. The horizontal
dashed lines indicate hh∆˜x2i/h∆˜y2ii∆t= 1.13 (upper, red)
and τa/τb= 1 (lower, black), respectively.
The decomposition of Da/Dbinto hh∆˜x2i/h∆ ˜y2ii∆t
and τa/τbin Eq. 4 helps to illustrate the origin of
the anisotropic diffusion of ellipsoids. First, the step
size along the ma jor axis is larger than that along
the minor axis. However, the ratio between the two,
hh∆˜x2i/h∆˜y2ii∆t, keeps roughly constant except at the
highest φ. The result is consistent with the previous ob-
servation of fa/fb, where advection exerts a stronger in-
fluence on the motion of particles along the major axis at
small time intervals (Fig. 7(c)). The ratio of the correla-
tion times, τa/τb, shows a more interesting trend. At low
φ,τa≈τb, indicating a similar persistence for the mo-
tions along the major and minor axes. However, τa/τb
increases with φat high φ, which leads to the increase of
Da/Dbwith φ. Hence, the anisotropic diffusion of ellip-
soids in algal suspensions arises from the increase of the
persistence of the motion along the major axis relative to
that along the minor axis. This observation is in sharp
contrast with the dynamics of ellipsoids in E. coli suspen-
sions, where τa/τbdecreases monotonically with bacterial
concentrations (Figs. 6(b)(d)) [39]. As a comparison, the
correlation time of thermal Brownian motions is given by
the inertial time of Brownian particles, τ=m/ζ, where
mis the mass of the particles and ζis the drag coeffi-
cient [63]. Hence, τa0/τb0=ζb/ζa=Da0/Db0= 1.33.
ζa,b are the drag coefficients along the major and minor
axes of ellipsoids, which are related to the anisotropic dif-
fusion through the Stokes-Einstein relation. τis on the
order of micro-seconds beyond the time resolution of our
experiments [64]. We should again emphasize that the

10
origins of the correlation time of Brownian diffusion and
enhanced diffusion are completely different. Thus, the
limit of zero algal concentrations (φ→0) is not equiva-
lent to Brownian diffusion with φ= 0.
Therefore, to understand the effect of pushers and
pullers on the dynamics of asymmetric tracers, it is im-
portant to reveal how the swimming of microswimmers
affects the correlation times of ellipsoids’ motions. The
simple hydrodynamic calculation by Peng et al. provides
a qualitative guideline [39]. First, the motion of an ellip-
soid in the dipole flow of a single microswimmer can be
calculated in Stokes flow as [58]
vp=u,(6)
ωp=1
2∇ × u+p2−1
p2+ 1 ˆa×(ǫ·ˆa).(7)
Here, vpand ωpare the translational and angular veloc-
ity of the ellipsoid with aspect ratio p. The ellipsoid is
treated as a force-free and torque-free point particle. The
assumption is valid in the far field when the concentra-
tion of microswimmers is low. u=−k(r/r3−3x2r/r5) is
the dipole flow at r= (x, y) induced by a microswimmer
at origin. The dipole strength, k, is positive for push-
ers and negative for pullers. ǫis the rate-of-strain tensor
of the dipole flow. ˆais the unit vector along the ma-
jor axis of the ellipsoid. In the dilute limit, the motion
of the ellipsoid under the influence of many microswim-
mers can be obtained by averaging all the possible ori-
entations and positions of microswimmers relative to the
ellipsoid—a mean-field approximation that ignores the
correlation between microswimmers. The method has
been successfully used for interpreting the linear relation
between the enhanced diffusivity of spherical tracers and
the concentration of active particles [31, 34]. A coupling
scaler, S≡ωp·ˆa′×vp
|ˆa′×vp|can then be defined, which
quantifies the intrinsic coupling between the translation
and rotation of an ellipsoid in the dipole flow [39]. ˆa′is
a unit vector that satisfies ˆa′= ˆawhen ˆa′·v≥0 and
ˆa′=−ˆawhen ˆa·v<0. In other words, ˆa′is along
the major axis that always forms an acute angle with
v. From the definition, the amplitude of Scharacter-
izes the speed of the particle rotation, |S|=|ωp|, and
the sign of Sindicates the direction of the rotation. A
negative Scorresponds to a rotation that tends to align
the minor axis of the ellipsoid along with its translational
direction vp/|vp|, whereas a positive Scorresponds to a
rotation that aligns the major axis of the ellipsoid along
with vp/|vp|. By taking an average over all possible ori-
entations and positions of microswimmers, Peng et al.
showed that the average of Sfollows
hSi=−p2−1
p2+ 1
3k
V0
φ, (8)
where V0indicates the volume of microswimmers. hSi=
0 for spherical tracers with p= 1. More importantly,
hSi>0 for pullers and hSi<0 for pushers.
The effect of a non-zero hSiis equivalent to a straining
flow, which rotates an asymmetric tracer in a direction
depending on the swimming mechanism of microswim-
mers and the orientation of the tracer with respect to
its translational direction. The dynamics of asymmet-
ric tracers can thus be modeled as the diffusion of the
tracers in a thermal bath with an effective temperature
Teff under the influence of an external rotational poten-
tial characterized by hSi[39]. The correlation time of
the diffusion can be obtained by solving an over-damped
Langevin equation with a rotational potential, which
leads to τa=tc/(1−2tchSi/π) and τb=tc/(1+2tchSi/π).
Here, tcindicates the correlation time of the random fluc-
tuation of the effective thermal bath, which is related to
the transition time between the superdiffusive regime to
the diffusion regime (Fig. 5(a)) [16]. Thus, we finally
have
τa
τb
=1 + sgn (hSi)tc/t0
1−sgn (hSi)tc/t0
,(9)
where t0≡π/2
|hSi| is a characteristic time for the rota-
tion of tracers over an angle of π/2 due to a non-zero
|hSi| =|ωp|. Note that sgn(x) = 1 when x > 0,
sgn(x) = −1 when x < 0 and sgn(x) = 0 when x= 0.
From Eq. 8, t0∼1/|hSi| should decrease with increas-
ing φ. Thus, for pushers, since hSi<0, τa/τb=
(1 −tc/t0)/(1 + tc/t0)≤1 and decreases with increasing
φ, a prediction that has been confirmed experimentally
from the study of the anisotropic diffusion of ellipsoids
in E. coli suspensions (Figs. 6(b)(c)) [39]. For pullers,
hSi>0. Thus, τa/τb= (1 + tc/t0)/(1 −tc/t0)≥1, which
should increase with increasing φ. Our experiments with
C. reinhardtii suspensions qualitatively agree with this
prediction (Fig. 8). Finally, for spherical tracers, hSi= 0.
τa=τb, independent of whether active fluids are pushers
or pullers.
IV. CONCLUSIONS
We have studied the dynamics of ellipsoids immersed
in C. reinhardtii suspensions, a premier model of puller-
type active fluids. Different from spherical tracers that
have been extensively studied, ellipsoidal tracers possess
an additional rotational degree of freedom and, therefore,
show much richer dynamics.
In the laboratory frame, both translation and rotation
of ellipsoids show superdiffusive motions at short times
and enhanced diffusions at long times, similar to the dy-
namics of spherical tracers. By analyzing the probability
distribution functions of the displacements of ellipsoids,
we showed that the translational and rotational motions

11
of ellipsoids can be quantitatively understood from the
balance of advection and diffusion induced by the swim-
ming of algae. Due to the nature of the dipole flow of mi-
croswimmers, the translational advection shows a longer
range of influence than the rotational advection.
The body-frame dynamics of ellipsoids reveal the dis-
tinct difference between pushers and pullers. Although
when viewed independently the motions of ellipsoids
along the major and minor axes show qualitatively simi-
lar trends, ellipsoids in the body frame show an unusual
anisotropic diffusion, where the long-time diffusion along
the major axis is more strongly enhanced with increasing
algal concentrations than that along the minor axis. Such
an anisotropic diffusion dictates a coupling between the
translation and rotation of ellipsoids in the laboratory
frame. By decomposing diffusivity into different com-
ponents, we demonstrated that the anisotropic diffusion
arises from the differential variation of the correlation
times of ellipsoids’ motions along the ma jor and minor
axes. The motion of an ellipsoid along its ma jor axis
becomes more persistent when algal concentration in-
creases. This trend is in sharp contrast to the dynamics
of ellipsoids in pusher-type E. coli suspensions, where the
persistence of the motion along the ma jor axis decreases
with increasing bacterial concentrations. This sharp dif-
ference can be qualitatively explained at the mean-field
level by considering the translation-rotation coupling in-
duced by the straining component of the dipole flow of
microswimmers. As such, our study showed that the
anisotropic diffusion of asymmetric tracers is a universal
feature of active fluids, which can be used as a convenient
experimental indicator for distinguishing pushers versus
pullers.
Finally, our work also provided experimental results on
the dynamics of asymmetric particles in suspensions of
eukaryotic microorganisms. Since asymmetric particles
(e.g. macromolecules and dead bodies of microorgan-
isms) are naturally more abundant than spherical par-
ticles, our results should be more relevant to realistic
biological systems and be useful for understanding the
physiology of swimming eukaryotes.
ACKNOWLEDGMENTS
We acknowledge P. Lefebvre for providing us with
the C.reinhardtii strain and helping us with algae cul-
turing. We also thank B. Zhang, L. Gordillo and D.
Samanta for help with experiments and L. Lai for fruit-
ful discussions on theory. The research was supported by
ACS Petroleum Research Fund (54168-DNI9) and by the
David & Lucile Packard Foundation. C. T. thanks sup-
port from the Coating Process Fundamentals Program
(CPFP) at University of Minnesota. X. X. acknowledges
support by the National Natural Science Foundation of
China No. 11575020.
∗Electronic address: xcheng@umn.edu
[1] S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1,
323 (2010).
[2] D. L. Koch and G. Subramanian, Annu. Rev. Fluid Mech.
43, 637 (2011).
[3] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B.
Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod.
Phys. 85, 1143 (2013).
[4] J. Schwarz-Linek, J. Arlt, A. Jepson, A. Dawson, T. Vis-
sers, D. Miroli, T. Pilizota, V. A. Martinez, and W. C.
K. Poon, Colloid. Surface. B 137, 2 (2016).
[5] C. Bechinger, R. Di Leonardo, H. Lowen, C. Reichhardt,
G. Volpe, and G. Volpe, arXiv:1602.00081 (2016).
[6] S. J. Ebbens and J. R. Howse, Soft Matter 6, 726 (2010).
[7] V. Narayan, S. Ramaswamy, and N. Menon, Science 317,
105 (2007).
[8] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and
P. M. Chaikin, Science 339, 936 (2013).
[9] H. P. Zhang, A. Beer, E.-L. Florin, and H. L. Swinney,
Proc. Natl. Acad. Sci. U.S.A. 107, 13626 (2010).
[10] A. Sokolov and I. S. Aranson, Phys. Rev. Lett. 109,
248109 (2012).
[11] H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher,
R. E. Goldstein, H. Lowen, and J. M. Yeomans, Proc.
Natl. Acad. Sci. U.S.A. 109, 14308 (2012).
[12] A. Sokolov and I. S. Aranson, Phys. Rev. Lett. 103,
148101 (2009).
[13] S. Rafai, L. Jibuti, and P. Peyla, Phys. Rev. Lett. 104,
098102 (2010).
[14] J. Gachelin, G. Mino, H. Berthet, A. Lindner, A. Rousse-
let, and E. Clement, Phys. Rev. Lett. 110, 268103 (2013).
[15] H. M. Lopez, J. Gachelin, C. Douarche, H. Auradou, and
E. Clement, Phys. Rev. Lett. 115, 028301 (2015).
[16] X.-L. Wu and A. Libchaber, Phys. Rev. Lett. 84, 3017
(2000).
[17] M. J. Kim and K. S. Breuer, Phys. Fluid 16, L78 (2004).
[18] D. T. N. Chen, A. W. C. Lau, L. A. Hough, M. F. Islam,
M. Goulian, T. C. Lubensky, and A. G. Yodh, Phys. Rev.
Lett. 99, 148302 (2007).
[19] G. Mino, T. E. Mallouk, T. Darnige, M. Hoyos, J.
Dauchet, J. Dunstan, R. Soto, Y. Wang, A. Rousselet,
and E. Clement, Phys. Rev. Lett. 106, 048102 (2011).
[20] L. G. Wilson, V. A. Martinez, J. Schwarz-Linek, J.
Tailleur, G. Bryant, P. N. Pusey, and W. C. K. Poon,
Phys. Rev. Lett. 106, 018101 (2011).
[21] G. L. Mino, J. Dunstan, A. Rousselet, E. Clement, and
R. Soto, J. Fluid Mech. 729, 423 (2013).
[22] A. Jepson, V. A. Martinez, J. Schwarz-Linek, A. Moro-
zov, and W. C. K. Poon, Phys. Rev. E 88, 041002(R)
(2013).
[23] T. V. Kasyap, D. L. Koch, and M. Wu, Phys. Fluid 26,
081901 (2014).
[24] A. E. Patteson, A. Gopinath, P. K. Purohit, and P. E.
Arratia, Soft Matter 12 2365 (2016).
[25] C. Valeriani, M. Li, J. Novosel, J. Arlt, and D. Maren-
duzzo, Soft Matter 7, 5228 (2011).
[26] L. Vaccari, D. B. Allan, N. Sharifi-Mood, A. R. Singh, R.
L. Leheny, and K. J. Stebe, Soft Matter 11, 6062 (2015).
[27] K. C. Leptos, J. S. Guasto, J. P. Gollub, A. I. Pesci, R.
E. Goldstein, Phys. Rev. Lett. 103, 198103 (2009).
[28] H. Kurtuldu, J. S. Guasto, K. A. Johnson, and J. P.

12
Gollub, Proc. Natl. Acad. Sci. U.S.A. 108, 10391 (2011).
[29] P. T. Underhill, J. P. Hernandez-Ortiz, and Michael D.
Graham, Phys. Rev. Lett. 100, 248101 (2008).
[30] T. Ishikawa, J. T. Locsei, and T. J. Pedley, Phys. Rev.
E82, 021408 (2010).
[31] Z. Lin, J. L. Thiffeault, and S. Childress, J. Fluid Mech.
669, 167 (2011).
[32] D. O. Pushkin and J. M. Yeomans, Phys. Rev. Lett. 111,
188101 (2013).
[33] D. O. Pushkin, H. Shum, and J. M. Yeomans, J. Fluid
Mech. 726, 5 (2013).
[34] A. Morozov and D. Marenduzzo, Soft Matter 10, 2748
(2014).
[35] A. J. T. M. Mathijssen, D. O. Pushkin, and J. M. Yeo-
mans, J. Fluid Mech. 773, 498 (2015).
[36] K. Yeo, E. Lushi, and P. M. Vlahovska, Soft Matter 12,
5645 (2016).
[37] T. M. Squires and T. G. Mason, Annu. Rev. Fluid Mech.
42, 413 (2010).
[38] K. Katija, J. Exp. Biol. 215, 1040 (2012).
[39] Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X.
Cheng, Phys. Rev. Lett. 116, 068303 (2016).
[40] K. Drescher, J. Dunkel, L. H. Cisneros, S. Ganguly, and
R. E. Goldstein, Proc. Natl. Acad. Sci. U.S.A. 108, 10940
(2011).
[41] K. Drescher, R. E. Goldstein, N. Michel, M. Polin, and
I. Tuval, Phys. Rev. Lett. 105, 168101 (2010).
[42] J. S. Guasto, K. A. Johnson, and J. P. Gollub, Phys.
Rev. Lett. 105, 168102 (2010).
[43] Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha,
Phys. Rev. Lett. 92, 118101 (2004).
[44] M. E. Cates, S. M. Fielding, D. Marenduzzo, E. Or-
landini, J. M. Yeomans, Phys. Rev. Lett. 101, 068102
(2008).
[45] L. Giomi, T. B. Liverpool, and M. C. Marchetti, Phys.
Rev. E 81, 051908 (2010).
[46] D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 100,
178103 (2008).
[47] A. Baskaran and M. C. Marchetti, Proc. Natl. Acad. Sci.
U.S.A. 106, 15567 (2009).
[48] L. Turner, W. S. Ryu, and Howard C. Berg, J. Bacteriol.
182, 2793 (2000).
[49] M. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E.
Goldstein, Science 325, 487 (2009).
[50] N. Sharifi-Mood, A. Mozaffari, and U. Cordova-Figueroa,
arXiv:1510.03000 (2015).
[51] E. H. Harris, The Chlamydomonas Sourcebook (Academic
Press, San Diego, CA, 1989)
[52] C. C. Ho, A. Keller, J. A. Odell, and R. H. Ottewill,
Colloid. Poly. Sci. 271, 469 (1993).
[53] See Supplemental Material at [URL will be inserted by
publisher] for videos showing the enhanced diffusion of
ellipsoids in algal suspensions of low and high concentra-
tions.
[54] Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Luben-
sky, and A. G. Yodh, Science 314, 626 (2006).
[55] D. Mukhija and M. J. Solomon, J. Colloid Interface Sci.
314, 98 (2007).
[56] Y. Han, A. Alsayed, M. Nobili, and A. G. Yodh, Phys.
Rev. E 80, 011403 (2009).
[57] Close to the body of swimming microorganisms, flow field
of higher orders may become important. Nevertheless,
measurements on the flow field around C. reinhardtii have
shown that there is only a small deviation from the dipole
form within 2 −3 body lengths of algae [41].
[58] S. Kim and S. J. Karrila, Microhydrodynamics: Princi-
ples and Selected Applications (Butterworth-Heinemann,
Boston, MA, 1991).
[59] J. S. Guasto, K. A. Johnson, and J. P. Gollub, Phys.
Fluids 23, 091112 (2011).
[60] Note that kshould be negative for pullers such as C.
reinhardtii. Here, we only consider the magnitude of k.
[61] Direct measurements of rTin algal suspensions using
micron-sized spherical tracers give rT≈25 −30 µm [27].
[62] Note that when evaluating the experimental errors of
Da/Db, we computed Da/Dbfor each experimental run
and then calculated the standard deviation of the ratios
of all runs. Since the variations of Daand Dbare corre-
lated, the errors of Da/Dbare significantly smaller than
that evaluated by treating Daand Dbas independent
variables.
[63] R. Zwanzig , Nonequilibrium Statistical Mechanics (Ox-
ford University Press, Oxford, UK, 2001).
[64] At such a small time scale, fluid inertial certainly plays an
important role [65]. It is not clear how the fluid inertial
differentially affects the motion of ellipsoids along the
major and minor axis and modifies τa0/τb0.
[65] B. Lukic, S. Jeney, C. Tischer, A. J. Kulik, L. Forro, and
E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).