Dancing Volvox: Hydrodynamic Bound States of Swimming Algae

1.1 MB·English

by  Knut Drescher

Dancing Volvox : Hydrodynamic Bound States of Swimming Algae
KnutDrescher1,KyriacosC.Leptos1,IdanTuval1,TakujiIshikawa2,TimothyJ.Pedley1,andRaymondE.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.
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A surface-mediated hydrodynamic attraction combined with lubrication forces between spinning,
0
bottom-heavy Volvox explains the formation, stability and dynamics of the bound states. These
0
phenomenaare suggested to underlieobserved clustering of Volvox at surfaces.
2

n PACSnumbers: 87.17.Jj,87.18.Ed,47.63.Gd
a
J

4 Long after he made his great contributions to mi- the waltzingisreminiscentofvortexpairsininviscidflu-
1 croscopyandstartedarevolutioninbiology,Antony van ids, the attraction and the minuet are not, and as the

Leeuwenhoek peered into a drop of pond water and dis- Reynolds number is 0.03, inertia is negligible.
] ∼
t coveredoneofnature’sgeometricalmarvels[1]. Thiswas While one might imagine that signalling and chemo-
f
o thefreshwateralgawhich,yearslater,intheverylasten- taxis could result in these bound states, a combination
s try of his great work on biological taxonomy, Linneaus of experiment, theory, and numerical computations is
.
at named Volvox [2] for its characteristic spinning motion used here to show that they arise instead from the in-
m about a fixed body axis. Volvox is a spherical colonial terplay of short-range lubrication forces between spin-
- green alga (Fig. 1), with thousands of biflagellated cells ning colonies and surface-mediated hydrodynamic inter-
d anchored in a transparent extracellular matrix (ECM) actions[13],knowntobeimportantforcolloidalparticles
n anddaughtercoloniesinsidethe ECM.Since theworkof [14, 15]and bacteria[16]. We conjecture thatflows driv-
o
Weismann[3],Volvoxhasbeenseenasamodelorganism ing Volvox clustering at surfaces enhance the probability
c
[ in the study of the evolution of multicellularity [4, 5, 6]. of fertilizationduring the sexual phase oftheir life cycle.

1 Because it is spherical, Volvox is an ideal organism Volvox carteri f. nagariensis EVE strain (a subclone
v for studies of biological fluid dynamics, being an ap- of HK10) were grown axenically in SVM [6, 17] in diur-
7 proximate realizationof Lighthill’s “squirmer” model [7] nal growthchambers with sterile air bubbling, in a daily
8 of self-propelled bodies having a specified surface veloc-
0
ity. Suchmodelshaveelucidatednutrientuptakeathigh
2
P´eclet numbers [6, 8] by single organisms, and pairwise
.
1
hydrodynamic interactions between them [9]. Volvocine
0
algae may also be used to study collective dynamics of
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0 self-propelled objects [10], complementary to bacterial
: suspensions(E.coli,B.subtilis)exhibitinglarge-scaleco-
v
herence in thin films [11] and bulk [12].
i
X
WhileinvestigatingVolvoxsuspensionsinglass-topped
r
a chambers we observed stable bound states, in which
pairsofcoloniesorbiteachothernearthechamberwalls.
Volvox is “bottom-heavy” due to clustering of daughter
coloniesinthe posterior,soanisolatedcolonyswimsup-
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
whilespinning,andthey“waltz”abouteachotherclock-
wise (Fig. 1a) at an angular frequency Ω 0.1 rad/s. FIG. 1: (Color online) Waltzing of V. carteri. (a) Top view.
∼
When Volvox have become too heavy to maintain up- Superimposed images taken 4 s apart, graded in intensity.
swimming, two colonies hover above one another near (b) Side, and (c) top views of a colony swimming against a
the chamber bottom, oscillating laterally out of phase in coverslip, with fluid streamlines. Scales are 200 µm. (d) A
linear Volvox cluster viewed from above (scale is 1 mm).
a “minuet” dance. Although the orbiting component of
2
FIG. 2: (Color online) Dual-view apparatus.
FIG. 3: (Color online) Swimming properties of V. carteri as
a function of radius. (a) upswimming speed, (b) rotational
frequency,(c)sedimentationspeed,(d)reorientationtime,(e)
◦ density offset, and (f) components of average flagellar force
cycle of 16 h in cool white light ( 4000 lux) at 28 C
◦ ∼ density.
and 8 h in the dark at 26 C. Swimming was studied
in a dual-view system (Fig. 2) [18], consisting of two
identicalassemblies,eachaCCDcamera(Pike145B,Al-
lied Vision Technologies, Germany) and a long-working density offset ∆ρ = ρc ρ between the colony and wa-
−
distancemicroscope(InfiniVarCMS-2/S,Infinity Photo- ter. Bottom-heaviness implies a distance ℓ between the
Optical,Colorado). Dark-fieldilluminationused102mm centers of gravity and geometry, measured by allowing
diametercircularLEDarrays(LFR-100-R,CCSInc.,Ky- Volvox to roll off a guide in the chamber and monitoring
oto)withnarrowbandwidthemissionat655nm,towhich the axis inclination angle θ with the vertical. This an-
Volvoxisinsensitive[19]. Thermalconvectioninducedby gle obeys ζrθ˙ = (4πR3ρcgℓ/3)sinθ, where ζr = 8πηR3
−
theilluminationwasminimizedbyplacingthe2 2 2cm is the rotational drag coefficient, leading to a relaxation
sample chamber, made from microscope slides×he×ld to- time τ = 6η/ρcgℓ [21]. The rotational frequencies ωo of
gether with UV-curing glue (Norland), within a stirred, free-swimmingcolonieswereobtainedfrommovies,using
temperature-controlled water bath. A glass cover slip germ cells/daughter colonies as markers.
glued into the chamber provided a clean surface (Fig. Figure 3 shows the four measured quantities
1b) to induce bound states. Particle imaging velocime- (U,V,ω ,τ) and the deduced density offset ∆ρ. In the
o
try (PIV) studies (Dantec Dynamics, Skovelund, Den- simplest model [6], locomotion derives from a uniform
mark) showed that the r.m.s convective velocity within force per unit area f = (f ,f ) exerted by flagella tan-
θ φ
the sample chamber was <5 µm/s. gential to the colony surface. Balancing the net force
Four aspectsofVolvox∼swimming areimportantinthe dSf ˆz=π2fθR2 againstthe Stokesdragandnegative
·
formation of bound states, each arising, in the far field, bRuoyancyyieldsfθ =6η(U+V)/πR. Balancingtheflag-
from a distinct singularity of Stokes flow: (i) negative ellar torque dSR(ˆr f) ˆz = π2fφR3 against viscous
buoyancy (Stokeslet), (ii) self-propulsion (stresslet), (iii) rotational toRrque 8πη×R3ωo·yields fφ = 8ηωo/π. These
bottom-heaviness(rotlet), and spinning (rotlet doublet). components are shown in Fig. 3f, where we used a lin-
Duringthe48hourlifecycle,thenumberofsomaticcells ear parameterization of the upswimming data (Fig. 3a)
is constant; only their spacing increases as new ECM is to obtain an estimate of U over the entire radius range.
addedto increasethe colonyradius. This slowlychanges The typical force density fθ corresponds to several pN
the speeds of sinking, swimming, self-righting, and spin- per flagellar pair [6], while the relative smallness of fφ is
◦
ning, allowing exploration of a range of behaviors. The a consequence of the 15 tilt of the flagellar beating
∼
upswimming velocityU wasmeasuredwith side viewsin plane with respect to the colonial axis [22, 23].
the dual-viewapparatus. Volvox densitywasdetermined Usingthemeasuredparametersitispossibletocharac-
by arresting self-propulsion through transient deflagella- terize both bound states. Fig. 4c shows data from mea-
tion with a pH shock [6, 20], and measuring sedimenta- suredtracksof60pairsofVolvox,astheyfalltogetherto
tion. The settling velocity V = 2∆ρgR2/9η, with g the form the waltzing bound state. The data collapse when
accelerationofgravityandηthefluidviscosity,yieldsthe theinter-colonyseparationr,normalizedbyR¯,themean
3
of the two participating colonies’ radii, is plotted as a
functionofrescaledtimefromcontact. The waltzingfre-
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-fieldfeatures ofmutually-advectedsin-
gularities and near-field effects given by lubrication the-
ory, which will now be considered in turn.
Infalling: When swimming against an upper surface,
thenetthrustinducedbytheflagellarbeatingisnotbal-
anced by the viscous drag onthe 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.
SeenincrosssectionwithPIV,thevelocityfieldofasin-
gle colony has precisely this appearance (Fig. 1b). This
flowproducestheattractiveinteractionbetweencolonies; FIG. 4: (Color online) Waltzing dynamics. Geometry of (a)
Squires has proposeda similar scenario in the contextof twointeractingStokeslets(sideview)and(b)nearbyspinning
electrophoretic levitation [24]. colonies. (c)Radialseparation r,normalized bymean colony
The motion of a pointlike object at x , with axis ori- radius, as a function of rescaled time for 60 events (black).
i
Running average (green) compares well with predictions of
entation p and net velocity v from self-propulsion and
i i thesingularitymodel(red). InsetshowsorbitingfrequencyΩ
buoyancy,due tothe fluidvelocityuandvorticity u
∇× as a function of mean spinning frequency ω¯, and linear fit.
generated by the other self-propelled objects, obeys
x˙ = u(x )+v , (1)
i i i Appealing to known asymptotic results [25] we obtain
1 1
p˙ = p (zˆ p )+ ( u) p . the torque = (2/5)ln(d/2R)ζrω and a lateral force
i i i i T −
τ × × 2 ∇× × =(1/10)ln(d/2R)ζ ω/R on the sphere, where ω <ω
r o
F
Assuming that for the infalling, v = p˙ = 0, and that is the spinning frequency of a colony in the bound state.
i i
u(x )areduetoStokesletsofstrengthF =6πηR(U+V), Therotationalslowingoftheself-propelledcolonyhasan
i
Eq. 1 may be reduced, in rescaled coordinates r˜= r/h effect on the fluid that may be approximatedby a rotlet
and t˜=tF/ηh2 with h=R¯, to [24] ofstrength atitscenter. Fromtheflowfieldofarotlet
T
perpendicular to a horizontal no-slip wall [13] and the
dr˜ 3 r˜
= . (2) lateral force , we then deduce the orbiting frequency
dt˜ −π(r˜2+4)5/2 F
d
Integration of (2) shows good parameter-free agreement Ω 0.069 ln ω¯ . (3)
≃ (cid:18)2R(cid:19)
with the experimental trajectories of nearby pairs (Fig.
4c). Large perturbations to a waltzing pair by a third Typical values of d and R give a slope of 0.14 0.19
≃ −
nearbycolonycandisruptitbystronglytiltingthecolony forthe Ω ω line,consistentwiththe experimentalfitof
−
axes, suggesting that bottom-heaviness confers stability. 0.19 0.05(Fig. 4c). The nonzerointerceptis likely due
±
This is confirmed by a linear stability analysis [23]. to lubrication friction against the ceiling [23].
Orbiting: As Volvox colonies move together under the A second and more complex type of bound state, the
influence of the wall-induced attractive flows (Fig. 1b), “minuet,” is found when the upswimming just balances
orbitingbecomesnoticeableonlywhentheirseparationd the settling (at R 300 µm, see Fig 3a), and Volvox
is <30 µm; their spinning frequencies alsodecrease very colonieshoveratafi≃xeddistanceabovethechamberbot-
str∼ongly with decreasing separation. This arises from tom. In this mode (Fig. 5) colonies stacked one above
viscous torques associated with the thin fluid layer be- the other oscillate back and forth about a vertical axis.
tween two colonies (Fig. 4b). We assume that in the Themechanismofoscillationistheinstabilityoftheper-
thinfluidlayer,thespinningVolvoxcoloniescanbemod- fectly aligned state due to the vorticity from one colony
eled as rigid spheres, ignoring the details of the over- rotating the other, whose swimming brings it back, with
lapping flagella layers. For two identical colonies, ig- the restoring torques from bottom-heaviness conferring
noring the anterior-posterior “downwash,” and consid- stability. Studies of the coupled dynamics of x and p
i i
ering only the region where the fluid layer is thin, the showthatwhentheorientationalrelaxationtimeτ isbe-
plane perpendicular to the line connecting their cen- low a threshold the stacked arrangement is stable, while
ters is a locus of zero velocity, as with a no-slip wall. for τ larger there is a Hopf bifurcation to limit-cycle dy-
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FIG.5: (Coloronline)“Minuet”boundstate. (a)Sideviews3sapartoftwocoloniesnearthechamberbottom. Yellowarrows
indicatetheanterior-posterioraxespi atanglesθi tovertical. Scalebaris600µm. (b)Bifurcationdiagram,andphaseportrait
(inset), showing a limit cycle, with realistic model parameters F =6πηRV, R=300 µm, h1 =450 µm, h2 =1050 µm.
namics (Fig. 5b). In these studies, lubrication effects lularity and cellular differentiation (Cambridge Univer-
were ignored, x˙ was restricted to be in one horizontal sity Press, Cambridge, 1998).
i
dimension only, and x were at fixed heights h above [5] D.L. Kirk, Bioessays 27, 299 (2005).
i i
[6] C.A. Solari, et al., Proc. Natl. Acad. Sci. (USA) 103,
the wall. The flow u was taken to be due to vertically
1353 (2006); M.B. Short, et al., Proc. Natl. Acad. Sci.
oriented Stokeslets at x , of magnitude F, equal to the
i (USA) 103, 8315 (2006); C.A. Solari, J.O. Kessler, and
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sp
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(2004).
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