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Dancing Volvox : Hydrodynamic Bound States of Swimming Algae
Knut Drescher1, Kyriacos C. Leptos1, Idan Tuval1, Takuji Ishikawa2, Timothy J. Pedley1, and Raymond E. Goldstein1
1Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK and
2Department of Bioengineering and Robotics, Tohoku University, Sendai 980-8579, Japan
(Dated: January 14, 2009)
The spherical alga Volvox swims by means of flagella on thousands of surface somatic cells. This
geometry and its large size make it a model organism for studying the fluid dynamics of multi-
cellularity. Remarkably, when two nearby Volvox swim close to a solid surface, they attract one
another and can form stable bound states in which they “waltz” or “minuet” around each other.
A surface-mediated hydrodynamic attraction combined with lubrication forces between spinning,
bottom-heavy Volvox explains the formation, stability and dynamics of the bound states. These
phenomena are suggested to underlie observed clustering of Volvox at surfaces.
PACS numbers: 87.17.Jj,87.18.Ed,47.63.Gd
Long after he made his great contributions to mi-
croscopy and started a revolution in biology, Antony van
Leeuwenhoek peered into a drop of pond water and dis-
covered one of nature’s geometrical marvels [1]. This was
the freshwater alga which, years later, in the very last en-
try of his great work on biological taxonomy, Linneaus
named Volvox [2] for its characteristic spinning motion
about a fixed body axis. Volvox is a spherical colonial
green alga (Fig. 1), with thousands of biflagellated cells
anchored in a transparent extracellular matrix (ECM)
and daughter colonies inside the ECM. Since the work of
Weismann [3], Volvox has been seen as a model organism
in the study of the evolution of multicellularity [4, 5, 6].
Because it is spherical, Volvox is an ideal organism
for studies of biological fluid dynamics, being an ap-
proximate realization of Lighthill’s “squirmer” model [7]
of self-propelled bodies having a specified surface veloc-
ity. Such models have elucidated nutrient uptake at high
Pe´clet numbers [6, 8] by single organisms, and pairwise
hydrodynamic interactions between them [9]. Volvocine
algae may also be used to study collective dynamics of
self-propelled objects [10], complementary to bacterial
suspensions (E. coli, B. subtilis) exhibiting large-scale co-
herence in thin films [11] and bulk [12].
While investigating Volvox suspensions in glass-topped
chambers we observed stable bound states, in which
pairs of colonies orbit each other near the chamber walls.
Volvox is “bottom-heavy” due to clustering of daughter
colonies in the posterior, so an isolated colony swims up-
ward with its axis vertical, rotating clockwise (viewed
from above) at an angular frequency ω ∼ 1 rad/s for a
radius R ∼ 150 µm. When approaching the chamber
ceiling, two Volvox are drawn together, nearly touching
while spinning, and they “waltz” about each other clock-
wise (Fig. 1a) at an angular frequency Ω ∼ 0.1 rad/s.
When Volvox have become too heavy to maintain up-
swimming, two colonies hover above one another near
the chamber bottom, oscillating laterally out of phase in
a “minuet” dance. Although the orbiting component of
the waltzing is reminiscent of vortex pairs in inviscid flu-
ids, the attraction and the minuet are not, and as the
Reynolds number is ∼ 0.03, inertia is negligible.
While one might imagine that signalling and chemo-
taxis could result in these bound states, a combination
of experiment, theory, and numerical computations is
used here to show that they arise instead from the in-
terplay of short-range lubrication forces between spin-
ning colonies and surface-mediated hydrodynamic inter-
actions [13], known to be important for colloidal particles
[14, 15] and bacteria [16]. We conjecture that flows driv-
ing Volvox clustering at surfaces enhance the probability
of fertilization during the sexual phase of their life cycle.
Volvox carteri f. nagariensis EVE strain (a subclone
of HK10) were grown axenically in SVM [6, 17] in diur-
nal growth chambers with sterile air bubbling, in a daily
FIG. 1: (Color online) Waltzing of V. carteri. (a) Top view.
Superimposed images taken 4 s apart, graded in intensity.
(b) Side, and (c) top views of a colony swimming against a
coverslip, with fluid streamlines. Scales are 200 µm. (d) A
linear Volvox cluster viewed from above (scale is 1 mm).

2FIG. 2: (Color online) Dual-view apparatus.
cycle of 16 h in cool white light (∼ 4000 lux) at 28◦ C
and 8 h in the dark at 26◦ C. Swimming was studied
in a dual-view system (Fig. 2) [18], consisting of two
identical assemblies, each a CCD camera (Pike 145B, Al-
lied Vision Technologies, Germany) and a long-working
distance microscope (InfiniVar CMS-2/S, Infinity Photo-
Optical, Colorado). Dark-field illumination used 102 mm
diameter circular LED arrays (LFR-100-R, CCS Inc., Ky-
oto) with narrow bandwidth emission at 655 nm, to which
Volvox is insensitive [19]. Thermal convection induced by
the illumination was minimized by placing the 2×2×2 cm
sample chamber, made from microscope slides held to-
gether with UV-curing glue (Norland), within a stirred,
temperature-controlled water bath. A glass cover slip
glued into the chamber provided a clean surface (Fig.
1b) to induce bound states. Particle imaging velocime-
try (PIV) studies (Dantec Dynamics, Skovelund, Den-
mark) showed that the r.m.s convective velocity within
the sample chamber was <∼ 5 µm/s.
Four aspects of Volvox swimming are important in the
formation of bound states, each arising, in the far field,
from a distinct singularity of Stokes flow: (i) negative
buoyancy (Stokeslet), (ii) self-propulsion (stresslet), (iii)
bottom-heaviness (rotlet), and spinning (rotlet doublet).
During the 48 hour life cycle, the number of somatic cells
is constant; only their spacing increases as new ECM is
added to increase the colony radius. This slowly changes
the speeds of sinking, swimming, self-righting, and spin-
ning, allowing exploration of a range of behaviors. The
upswimming velocity U was measured with side views in
the dual-view apparatus. Volvox density was determined
by arresting self-propulsion through transient deflagella-
tion with a pH shock [6, 20], and measuring sedimenta-
tion. The settling velocity V = 2∆ρgR2/9η, with g the
acceleration of gravity and η the fluid viscosity, yields the
FIG. 3: (Color online) Swimming properties of V. carteri as
a function of radius. (a) upswimming speed, (b) rotational
frequency, (c) sedimentation speed, (d) reorientation time, (e)
density offset, and (f) components of average flagellar force
density.
density offset ∆ρ = ρc − ρ between the colony and wa-
ter. Bottom-heaviness implies a distance ℓ between the
centers of gravity and geometry, measured by allowing
Volvox to roll off a guide in the chamber and monitoring
the axis inclination angle θ with the vertical. This an-
gle obeys ζr θ˙ = −(4πR3ρcgℓ/3) sin θ, where ζr = 8πηR3
is the rotational drag coefficient, leading to a relaxation
time τ = 6η/ρcgℓ [21]. The rotational frequencies ωo of
free-swimming colonies were obtained from movies, using
germ cells/daughter colonies as markers.
Figure 3 shows the four measured quantities
(U, V, ωo, τ) and the deduced density offset ∆ρ. In the
simplest model [6], locomotion derives from a uniform
force per unit area f = (fθ, fφ) exerted by flagella tan-
gential to the colony surface. Balancing the net force
∫
dS f · zˆ = π2fθR2 against the Stokes drag and negative
buoyancy yields fθ = 6η(U +V )/πR. Balancing the flag-
ellar torque
∫
dS R(rˆ × f) · zˆ = π2fφR3 against viscous
rotational torque 8πηR3ωo yields fφ = 8ηωo/π. These
components are shown in Fig. 3f, where we used a lin-
ear parameterization of the upswimming data (Fig. 3a)
to obtain an estimate of U over the entire radius range.
The typical force density fθ corresponds to several pN
per flagellar pair [6], while the relative smallness of fφ is
a consequence of the ∼ 15◦ tilt of the flagellar beating
plane with respect to the colonial axis [22, 23].
Using the measured parameters it is possible to charac-
terize both bound states. Fig. 4c shows data from mea-
sured tracks of 60 pairs of Volvox, as they fall together to
form the waltzing bound state. The data collapse when
the inter-colony separation r, normalized by R¯, the mean

3of the two participating colonies’ radii, is plotted as a
function of rescaled time from contact. The waltzing fre-
quency Ω is linear in the mean spinning frequency of
the pair ω¯. These two ingredients of the waltzing bound
state, “infalling” and orbiting, can be understood, re-
spectively, by far-field features of mutually-advected sin-
gularities and near-field effects given by lubrication the-
ory, which will now be considered in turn.
Infalling: When swimming against an upper surface,
the net thrust induced by the flagellar beating is not bal-
anced by the viscous drag on the colony, as the colony is
at rest, resulting in a net downwards force on the fluid.
The fluid response to such a force may be modeled as a
Stokeslet normal to and at a distance h from a no-slip
surface [13], forcing fluid in along the surface (Fig. 1c)
and out below the colony, with a toroidal recirculation.
Seen in cross section with PIV, the velocity field of a sin-
gle colony has precisely this appearance (Fig. 1b). This
flow produces the attractive interaction between colonies;
Squires has proposed a similar scenario in the context of
electrophoretic levitation [24].
The motion of a pointlike object at xi, with axis ori-
entation pi and net velocity vi from self-propulsion and
buoyancy, due to the fluid velocity u and vorticity ∇×u
generated by the other self-propelled objects, obeys
x˙i = u(xi) + vi , (1)
p˙i =
1
τ pi × (zˆ× pi) +
1
2
(∇× u)× pi .
Assuming that for the infalling, vi = p˙i = 0, and that
u(xi) are due to Stokeslets of strength F = 6πηR(U+V ),
Eq. 1 may be reduced, in rescaled coordinates r˜ = r/h
and t˜ = tF/ηh2 with h = R¯, to [24]
dr˜
dt˜ = −
3
π
r˜
(r˜2 + 4)5/2 . (2)
Integration of (2) shows good parameter-free agreement
with the experimental trajectories of nearby pairs (Fig.
4c). Large perturbations to a waltzing pair by a third
nearby colony can disrupt it by strongly tilting the colony
axes, suggesting that bottom-heaviness confers stability.
This is confirmed by a linear stability analysis [23].
Orbiting: As Volvox colonies move together under the
influence of the wall-induced attractive flows (Fig. 1b),
orbiting becomes noticeable only when their separation d
is <∼ 30 µm; their spinning frequencies also decrease very
strongly with decreasing separation. This arises from
viscous torques associated with the thin fluid layer be-
tween two colonies (Fig. 4b). We assume that in the
thin fluid layer, the spinning Volvox colonies can be mod-
eled as rigid spheres, ignoring the details of the over-
lapping flagella layers. For two identical colonies, ig-
noring the anterior-posterior “downwash,” and consid-
ering only the region where the fluid layer is thin, the
plane perpendicular to the line connecting their cen-
ters is a locus of zero velocity, as with a no-slip wall.
FIG. 4: (Color online) Waltzing dynamics. Geometry of (a)
two interacting Stokeslets (side view) and (b) nearby spinning
colonies. (c) Radial separation r, normalized by mean colony
radius, as a function of rescaled time for 60 events (black).
Running average (green) compares well with predictions of
the singularity model (red). Inset shows orbiting frequency Ω
as a function of mean spinning frequency ω¯, and linear fit.
Appealing to known asymptotic results [25] we obtain
the torque T = −(2/5) ln(d/2R)ζrω and a lateral force
F = (1/10) ln(d/2R)ζrω/R on the sphere, where ω < ωo
is the spinning frequency of a colony in the bound state.
The rotational slowing of the self-propelled colony has an
effect on the fluid that may be approximated by a rotlet
of strength T at its center. From the flow field of a rotlet
perpendicular to a horizontal no-slip wall [13] and the
lateral force F , we then deduce the orbiting frequency
Ω ≃ 0.069 ln
( d
2R
)
ω¯ . (3)
Typical values of d and R give a slope of ≃ 0.14 − 0.19
for the Ω−ω line, consistent with the experimental fit of
0.19± 0.05 (Fig. 4c). The nonzero intercept is likely due
to lubrication friction against the ceiling [23].
A second and more complex type of bound state, the
“minuet,” is found when the upswimming just balances
the settling (at R ≃ 300 µm, see Fig 3a), and Volvox
colonies hover at a fixed distance above the chamber bot-
tom. In this mode (Fig. 5) colonies stacked one above
the other oscillate back and forth about a vertical axis.
The mechanism of oscillation is the instability of the per-
fectly aligned state due to the vorticity from one colony
rotating the other, whose swimming brings it back, with
the restoring torques from bottom-heaviness conferring
stability. Studies of the coupled dynamics of xi and pi
show that when the orientational relaxation time τ is be-
low a threshold the stacked arrangement is stable, while
for τ larger there is a Hopf bifurcation to limit-cycle dy-

4FIG. 5: (Color online) “Minuet” bound state. (a) Side views 3 s apart of two colonies near the chamber bottom. Yellow arrows
indicate the anterior-posterior axes pi at angles θi to vertical. Scale bar is 600 µm. (b) Bifurcation diagram, and phase portrait
(inset), showing a limit cycle, with realistic model parameters F = 6piηRV , R = 300 µm, h1 = 450 µm, h2 = 1050 µm.
namics (Fig. 5b). In these studies, lubrication effects
were ignored, x˙i was restricted to be in one horizontal
dimension only, and xi were at fixed heights hi above
the wall. The flow u was taken to be due to vertically
oriented Stokeslets at xi, of magnitude F , equal to the
gravitational force on the Volvox.
Hydrodynamic bound states, such as those described
here, may have biological significance. When environ-
mental conditions deteriorate, Volvox colonies enter a
sexual phase of spore production to overwinter. Field
studies show that bulk Volvox concentrations n are < 1
cm−3 [26], with male/female ratio of ∼ 1/10, and ∼ 100
sperm packets/male. Under these conditions, the mean
encounter time for females and sperm packets is a sub-
stantial fraction of the life cycle. The kinetic theory mean
free path λ = 1/
√
2nπ(R+Rsp)2×10/100, with R = 150
µm for females, and Rsp = 15 µm for sperm packets, is
λ ∼ 1 m, implying a mean encounter time > 2 h [27].
This suggests that another mechanism for fertilization
must be at work, with previous studies having excluded
chemoattraction in this system [28]. At naturally occur-
ing concentrations, more than one Volvox may partake
in the waltzing bound state, leading to long linear ar-
rays (Fig. 1d). In such clusters, formed at the air-water
interface, the recirculating flows would decrease the en-
counter times to seconds or minutes, clearly increasing
the chance of sperm packets finding their target. Studies
are underway to examine this possibility.
We thank D. Vella, S. Alben and C.A. Solari for key
observations, A.M. Nedelcu for algae, and support from
the BBSRC, DOE, and the Schlumberger Chair Fund.
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