Phototaxis in Volvox
e 22, 2010 U vol. 107 U no. 25 U 11147–11650
In This Issue
PNAS
18.S995 - L28
Proceedings of the National Academy of Sciences of the United States of America
www.pnas.org
o the light
otosynthesis, algae such as Volvox carteri swim
from sunlight. To execute this motion, known
hese microorganism colonies must coordinate
housands of flagellated cells despite the organcentral nervous system. Using analytical and
ods, Knut Drescher et al. (pp. 11171–11176)
at V. carteri spins about its swimming direction
hat likely coevolved with the organism’s flagelmaximize photoreactivity. To characterize the
g of the organisms, the authors measured the
produced by the flagella and modeled the modynamic equations. Using the model, the aua theoretical optimal spinning frequency and
ng experimentally by observing how well the
media with increased viscosities that inhibited
Multicellular colony Volvox carteri.
ability to spin. According to the authors, the exonstrated that with a decreased rotation rate the algae were unable to execute phototaxis as accurately
esting that in V. carteri, flagellar beating and spinning are linked adaptations. By better understanding
anisms coordinate multicellular processes, the findings may provide insight into key evolutionary steps
dunkel@mit.edu
perimentally.
spot at different
moments
in time
(6).The
Many
species do
this for
by
Details of the
Mathematical
Model.
mathematical
model
beating
direction (17).
Instead
of quantifying
average photo•
U, the translational
swimming
speed,
which fixes thethe
amplitude
phototaxis
of Volvox
on measured
parameters
anda
swimming
on helical
paths relies
alongonly
which
their eyespot
acts as
response
of
of v0 . Forby
therecording
simulationsthe
webeating
used U ¼frequency
390 μm∕s,of
theeach
meanflagellum
of
is able continuously
to give detailedsearching
predictionsspace
of the for
swimming
light antenna
brightcharacterisspots (3).
the populations we investigated experimentally.
and the ability
toward the
light. It
based on a knowlHigher tics
eukaryotes
haveto aturn
nervous
system
tois integrate
visual • ωr , the rotation rate without a light stimulus, which fixes the
edge of the fluid velocity at the edge of the flagellar layer of
information
from different sources and orchestrate coordinated
Author
contributions:
K.D.,the
R.E.G.,
and I.T. designed
R.E.G., and I.T.
amplitude
of w0 . For
simulations
we usedresearch;
ωr ¼ 2.3K.D.,
rad∕s,
Volvox and how this fluid velocity changes when parts of the surperformed
research;
K.D.
analyzed
data;
and
K.D.,
R.E.G.,
and
I.T.
wrote
the paper.
responses
(7,are
8).exposed to a light stimulus.
as shown in Fig. 7 of the main text.
face
•The
The
θ dependence
of theofsurface
authors
declare no conflict
interest.velocity. For the simulations
Multicellular
organisms
of that
intermediate
such
asTuval
The coupled
equations
make
thecomplexity,
model 1are
given
in the
Knut
Drescher,
Raymond
E.upGoldstein
, and
Idan
we
approximated
v
ðθÞ
by
a superposition of two associated
0 Submission.
main text.
determine
the relatives
time evolution
of theevolved
system ofa
the colonial
alga To
Volvox
and its
(9), have
This article is a PNAS Direct
1
Legendre functions, −P1 ðcos θÞ þ 0.25P12 ðcos θÞ, as shown by
equations,
we solved
the coupled
partial
differential
Department
of phototaxis
Applied
Mathematics
and
Theoretical
Physics,
University
of Cambridge,
Wilberforce
Road,
Cambridge
CB3 0WA, United Kingdom
means ofcoupled
high-fidelity
without
a central
nervous
sysTo whom
correspondence
addressed.
E-mail: R.E.Goldstein@damtp.cam.ac.uk.
the dashed
magenta should
line inbeFig.
5 of the
main text. Using a
equations
for
pðθ;ϕ;tÞ
and
hðθ;ϕ;tÞ
numerically
with
a
built-in
soltem and, in many cases, even in the absence of intercellular
This
articlesincontains
supporting
information
online
at www.pnas.org/lookup/suppl/
simple
dependence
forfor
v0 ðθÞ
givesJanuary
qualitatively
similar reby Harry L. Swinney,
University
of Texas,
Austin,
May 6,θ 2010
(received
review
28, 2010)
verEdited
in Mathematica
(Wolfram
Research)
between
timesTX,t and approved
communication
through
cytoplasmic
connections
(10).
Volvox
doi:10.1073/pnas.1000901107/-/DCSupplemental.
sults.
We
assume
that
w
has
the
same
θ
dependence
as v0 .
0
t þ δt. Due to the integral in the equation for Ω, we used an Euler
• βðθÞ,
responsivity
of theof
fluid
flow to light
For Chlamydomona
^ single
method
thenevolutionary
solve the equation
IðtÞ
at everycells
timeto
step.
We
Alongtothe
path for
from
multicellular
or- the carteri
consists
thousands
ofstimulation.
biflagellated
the full model, we used a close approximation to the βðθÞ
ensured
convergence
of thenervous
results by
choosing
small enough
like somaticPNAS
cells∣ June
sparsely2010
distributed
the
of a passi
ganisms
with a central
system
areaspecies
of intermediate
www.pnas.org/cgi/doi/10.1073/pnas.1000901107
∣ vol.
107the
∣ at
no.
25 ∣ surface
11171–11176
shown in the inset in Fig.
5A of the22,
main text.
For
reduced
step
size
δt.
matrix,
small
of germ ce
complexity that move in ways suggesting high-level coordination,
model, wespherical
used βðθÞextracellular
¼ 0.3, the mean
of theand
βðθÞ aused
for number
the
In addition to finding the angle of the Volvox axis with the light
inside the sphere (Fig. 1A). During development the flagella or
yet have none. Instead, organisms of this type possess many autonfull model.
direction, the model can also be used to determine the organism
such that
rotates
aboutrespectively.
its swimming direction, th
omous cells
endowed
with
programs
that
haveStone
evolved
• τr and τa , ent
the response
andVolvox
adaptation
time scales,
swimming
velocity
U, via
another
result
from
andto achieve
For the simulations,
we used
the its
values
measured
for a light
trait that gave
Volvox
name
(11). Coordination
of the somat
concerted
to 2010
environmental
Here experiment
June 22,
U vol. 107 U no.stimuli.
25 U 11147–11650
Samuel
(18) responses
−2
−1
intensity
of
16
μmol
PAR
photons
m
s
,
as
displayed
in
cells resembles orchestrating a rowboat with thousands of ind
and theory are used to develop
a quantitative understanding of
Z
Fig.
2B
of
the mainrowers
text. but without a coxswain (9). Nature’s solution is
1
pendent
how cells of such
organisms
coordinate to achieve
phototaxis,
UðtÞ ¼
uðθ;ϕ;tÞdS;
[S1]
• τbh , the bottom-heaviness time scale, is defined by considering
response program at the single-cell level that produces an acc
by using the colonial 4πR
alga2 Volvox carteri as a model. It is shown
a flagellaless Volvox that is tilted at an angle ζ from the vertical.
rate
steering
mechanism,
an
emergent
at the coloni
thatallows
the surface
somatic
cellsorganism
act as Proceedings
individuals
but
are orchestrated
axisofof
Volvox
relax back
to the
vertical atproperty
a rate
of the
National
Academy ofThe
Sciences
thethis
United
States would
of America
www.pnas.org
which
trajectories
of the
to be reconstructed.
level.bh .Yet
remains
to beweunderstood
what
by
their relative
position in the
spherical
extracellular
¼ − sinðζÞ∕τ
Foritthe
simulations,
used τbh ¼ 14
s, asform the respon
A solution
of the photoresponse
pðθ;ϕ;tÞ
is plotted
in Fig. 6 ofmatrixζ_and
program
measured
in ref. 19.must take to coordinate the cells and to yield hig
photoresponse
functiondefined
to achieve
cothetheir
maincommon
text, using
the “reduced model”
in thecolony-level
main
fidelity phototaxis in the presence of the steering constraints
text.
A decomposition
ofof
this
photoresponse
intofrom
spherical
harMoving
to
the
lightthat range
ordination.
Analysis
models
the minimal
to the
In order to compare the results from this mathematical model
m
monics
Y l ðθ;ϕÞ
is given
in Fig. that,
S7. algae
The
photoresponse
p coma viscous environment.
biologically
faithful
shows
because
flagellar
beating
To optimize
photosynthesis,
such
as the
Volvox
carteri
swim
withdisthe measurements of the phototactic ability as a function of
puted
by
the
“full
model”
during
a
phototactic
turn
is
shown
More than aaviscosity
centurydependence
ago, Holmes
(12) proposed that the s
or away
from sunlight. To execute
this motion,
knownthe viscosity,
plays an toward
adaptive
down-regulation
in response
to light,
colony we implemented
in the model.
in Fig. S8, neglecting
bottom-heaviness.
phototaxis,
microorganismdirection
colonies must
matic cells
facing
a source of light down-regulate their flagell
needs to as
spin
aroundthese
its swimming
andcoordinate
that the response
For this we defined
u ¼ ½u&
The initial
conditions
of
the
model
were
a
horizontal
light
diw ηw ∕η and τbh ¼ ½τ bh &w η∕ηw , where η is
the beating of thousands of flagellated cells despite the
organactivity,
a hypothesis
latervalues
confirmed
kineticsan
andupward-pointing
natural spinningposterior-anterior
frequency of the colony
appear
toviscosity
be
the
and the subscript
w denotes
in water.by several investigato
rection,
axis, and
ism’s lack of a central nervous
system. Using analytical
and
(13–16). Although this control principle will initially turn the co
mutuallyempirical
tuned methods,
to give
the
maximum
photoresponse.
These
Knut
Drescher
et al. (pp.
11171–11176)
ony towards
the light, the
colony
might
adapt (14, 15) to the lig
further
predict
ability
decreases
dra-H, Hegemann
1.models
Kirk DL, Kirk
MM (1983)
Protein
patterns
during
asexual
life
cycle
of
11. Harz
P (1991) Rhodopsin-regulated
calcium
currents
in Chlamydomonas.
demonstrate
that Vsynthetic
.that
carterithe
spinsphototactic
about
itsthe
swimming
direction
Volvox carteri.
Dev
Biol
96:493–506.
Nature
351:489–491.
a frequency
likelydoes
coevolved
theat
organism’s
flagel-frequency,
before good alignment with the light direction has been reache
maticallyatwhen
the that
colony
notwith
spin
its natural
2. Solari CA, Ganguly S, Kessler JO, Michod RE, Goldstein RE (2006) Multicellularity and
12. Schaller K, Uhl R (1997) A microspectrophotometric study of the shielding properties
lar
kinetics
to
maximize
photoreactivity.
To
characterize
the
Surprisingly,
this observation
has not been synthesized into a pr
atheresult
confirmed
by ofphototaxis
assaystransport.
in which
colony
functional
interdependence
motility and molecular
Proc Natl
Acad rotation
of eyespot and
cell body in Chlamydomonas.
Biophys J 73:1573–1578.
flagellar
beating
of
the
organisms,
the
authors
measured
the
Sci USA
103:1353–1358.
13. Gerisch G (1959)
Cellular quantitative
differentiation in Pleodorina
and the with
organisation
dictive,
modelcalifornica
consistent
the principles of flu
was
slowed
by increasing
the
fluid
viscosity.
produced
by the
flagellainand
modeled
the mo3. Sakaguchi H,fluid
Iwasavelocities
K (1979) Two
photophobic
responses
Volvox
carteri. Plant
Cell
of colonial Volvocales (translated from German). Arch Protistenkunde 104:292–358.
dynamics, nor are there data on Volvox phototaxis that can b
Physiol 20:909–916.
14. Hoops HJ (1993) Flagellar cellular and organismal polarity in Volvox carteri. J Cell Sci
tion with hydrodynamic equations. Using the model, the au4.adaptation
Schletz K (1976)
Phototaxis
in∣Volvox—Pigments
involved
in the ∣perception
of light
104:105–117.compared with such a theory. Here we use a combination of e
∣ evolution
∣ fluid
dynamics
multicellularity
thors
identified
aflagella
theoretical
optimal
spinning
frequency
and
15.
Coggin
SJ, Kochert G (1986) Flagellar development and regeneration in Volvox carteri
direction. Z tested
Pflanzenphysiol
77:189–211.
the finding experimentally by observing how well the
periment and theory to show that adaptation and colony rotatio
(Chlorophyta). J Phycol 22:370–381.
5. Halldal P (1958) Action spectra of phototaxis and related problems in Volvocales, Ulva
algae swam inPhysiol
mediaPlant
with
increased viscosities that inhibited
playMutants
key
the phototaxis
colonyroles
Volvoxin
carteri.
16. HuskeyMulticellular
RJ (1979)
affecting
vegetative
cell orientationmechanism
in Volvox carteri.of
DevV. carteri. By qua
gametes
and Dinophyceae.
11:118–153.
he most
primitive
“eyes”
evolved
long
before
brains
the
organism’s
ability
to
spin.
to the
authors,
ex- and even
Biol 72:236–243.
6. Mast SO (1917) The relation between spectralAccording
color and stimulation
in thethe
lower
tifying
the flagellaras photoresponse
of V. carteri in detail, we sho
before
forms
nervous
system
organization
ap-unable
demonstrated
thatof
with
a decreased
rotation
rate the algae
to execute
17. were
Herraez-Dominguez
JV, Gilphototaxis
Garcia de Leonaccurately
F, Diez-Sales O, Herraez-Dominguez M
organism.
J periments
Expthe
Zool simplest
22:471–528.
that characterization
it actsByasbetter
a of
band
pass grades
filterofthat
allows adaptation
to differe
(2005)
Rheological
two viscosity
methylcellulose:
An ap7.peared
Sineshchekov
Jung K-H,
Spudich
(2002)
rhodopsins
mediate
phototaxis
tosenseare
asOA,
before,
suggesting
in VTwo
. organisms
carteri,
flagellar
beating
linked
adaptations.
understanding
on
Earth
(1,
2). JLthat
Many
are
ableandtospinning
and
to the
modeling
of the
thixotropic
behaviour.
Polym
Sci 284:86–91.
low- and high-intensity
in Chlamydomonas
reinhardtii. Proc Natl
Acad Sci USAfindings proach
light
environments,
minimizes
influence
of fast light fluctu
how
simplelight
organisms
coordinate
may provide
insight
into
key
evolutionary
stepsColloidthe
respond
to light
stimuli,
an abilitymulticellular
essentialprocesses,
to the the
optimization
18. Stone HA, Samuel ADT (1996) Propulsion of microorganisms by surface distortions.
99:8689–8694.
that eventually led to higher organisms with central nervous systems. — T.J.
and maximizes the response to stimuli at frequencies th
77:4102–4104.
8.of
MatPIV
is an open source PIVthe
software
toolbox written
Matlab. Downloads and
photosynthesis,
avoidance
of for
photodamage,
and the Phys
useRev Letttions,
19. Drescher K, correspond
et al. (2009) Dancing
Hydrodynamic
bound
swimming
details are at http://www.math.uio.no/∼jks/matpiv/.
to Volvox:
the rotation
rate
ofstates
the oforganism.
These measur
of
light as
a regulatory
signal.
One
of the
more
striking
responses
a probiotic
form
of Escherichia
coli,Phys Rev
probiotic
may
Water
in early lunar
algae.
Lett 102:168101.
9. Yoshimura Modified
K, Kamiya R (2001)
The sensitivity
of Chlamydomonas
photoreceptor
is
ments
suggest that the response kinetics and colony rotation ha
called
Nissle,
to express CAI-1,
andCA, Kessler
20. Solari
JO, Michod RE (2006) A hydrodynamics approach to the evolution of
for the frequency
of cell
body rotation.
Plant Cell
Physiol
42:665–672.
isoptimized
phototaxis,
inagainst
which
motile
photosynthetic
microorganisms
adprotect
cholera
magmas
evolved
be and
mutually
tuned
andin optimized
for phototax
multicellularity:
Flagellar to
motility
germ-somadunkel@math.mit.edu
differentiation
Volvocalean
10. Huth K (1970) Movement and orientation of Volvox aureus
(translated
from
the
bacteria
as ainprophylactic
just
theirPflanzenphysiol
swimming
pathVibrio
withcholerae
respect tested
toEhrbg.
incident
light
a finely
low-density
Recent
studies
have
argued
that
hygreen
algae.
Am
Nat
167:537–554.
German). ZWhereas
62:436–450.
against V. cholerae virulence in an
Furthermore, we develop a mathematical theory that predic
Fidelity of adaptive phototaxis
1
In This Issue
T
PNAS
Knut Drescher
MPI Marburg
Idan Tuval
Mediterranean
Institute for Advanced Studies
Ray Goldstein
Cambridge
dunkel@math.mit.edu
Why is Volvox interesting ?
• germ-soma differentiation
‘technique’
In Thisreproduction
Issue
• interesting asexual
• metachronal waves
• locomotion
• phototaxis
June 22, 2010 U vol. 107 U no. 25 U 11147–11650
PNAS
Proceedings of the National Academy of Sciences of the United States of America
www.pnas.org
Moving to the light
To optimize photosynthesis, algae such as Volvox carteri swim
toward or away from sunlight. To execute this motion, known
as phototaxis, these microorganism colonies must coordinate
the beating of thousands of flagellated cells despite the organism’s lack of a central nervous system. Using analytical and
empirical methods, Knut Drescher et al. (pp. 11171–11176)
demonstrate that V. carteri spins about its swimming direction
at a frequency that likely coevolved with the organism’s flagellar kinetics to maximize photoreactivity. To characterize the
flagellar beating of the organisms, the authors measured the
fluid velocities produced by the flagella and modeled the motion with hydrodynamic equations. Using the model, the authors identified a theoretical optimal spinning frequency and
tested the finding experimentally by observing how well the
algae swam in media with increased viscosities that inhibited
Multicellular colony Volvox carteri.
the organism’s ability to spin. According to the authors, the experiments demonstrated that with a decreased rotation rate the algae were unable to execute phototaxis as accurately
as before, suggesting that in V. carteri, flagellar beating and spinning are linked adaptations. By better understanding
how simple organisms coordinate multicellular processes, the findings may provide insight into key evolutionary steps
dunkel@math.mit.edu
Evolution of multicellularity
m in Applied Mathematics, and ¶BIO5 Institute, University of Arizona,
Providence, RI 02912
EudorinaJanuary 22, 2006)
Volvox oved AprilChlamydomonas
18, 2006 (received for review
reinhardtii
elegans
carteri
nes
lar
a
an
ent
lly
ges
nd
By
uid
in
ary
Gonum
Pleodorina
Volvox on
Fig. 1. Volvocine
green algae arranged
pectorale
californica according to typical
aureus colony radius R.
The lineage ranges from the single-cell Chlamydomonas reinhardtii (A), to
us,
Short
et al, PNAS 2013 Gonium pectorale (B), Eudorina elegans (C), to the somaundifferentiated
ng
dunkel@math.mit.edu
Volvox carteri
somatic cell
200 ㎛
cilia
daughter colony
from germ cell
http://www.youtube.com/watch?v=fqEHbJbuMYA
dunkel@math.mit.edu
Asexual reproduction & inversion
2014 Goldstein lab
dunkel@math.mit.edu
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
from germ cell
... and can dance
Drescher et al (2010) PRL
dunkel@math.mit.edu
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
dunkel@math.mit.edu
Volvox carteri
200 ㎛
10 ㎛
Chlamydomonas
reinhardtii
dunkel@math.mit.edu
Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
10 ㎛
speed ~100 μm/s
dunkel@math.mit.edu
Chlamydomonas
Merchant et al (2007) Science
dunkel@math.mit.edu
Model organism
for studying
meta-chronal waves
Brumley et al (2012) PRL
dunkel@math.mit.edu
Superposition of singularities
2x stokeslet =
symmetric dipole
stokeslet
rotlet
-F
F
r̂ · F
p(r) =
+ p0
2
4⇥r
(8⇥µ) 1
vi (r) =
[ ij + r̂i r̂j ]Fj
r
flow ~
r
1
F
r
2
‘pusher’
r
2
Volvox carteri
swimming speed
~ 100 ㎛/sec
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100 ㎛
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⇧ dunkel@math.mit.edu
⌥ ⌃ ⇧ ⌥⇤ ⇧ ⇥ ⇧ ⇥ ⌥↵ '74 ⇤BA
How does Volvox achieve phototaxis ?
Approach:
• light response of individual cells
• effects of size & spinning frequency
• mathematical modeling
• check predictions of model
dunkel@math.mit.edu
Fig. S1.
Experimental setup
Spectra of growth and stimulus light sources.
Fig. S1.
Spectra of growth a
as
tiv
th
fu
th
SI
aw
τr
(F
Fig. S2. (A) Schematic diagram of the sample chamber. (B) Photograph of a micr
the focal plane and pointing toward the fiber. (Scale bar: 200 μm.)
re
(2
Fig. 1. Geometry of V. carteri and experimental setup. (A) The beating flaIn
gella,(B)
twoPhotograph
per somatic cell
create a fluid
flow from
the anterior
to the
e chamber.
of a(Inset),
micropipette
holding
a V. carteri
colony
and the
ofo
posterior,
withμm.)
a slight azimuthal component that rotates Volvox about its
ber. (Scale
bar: 200
tim
posterior-anterior axis at angular frequency ωr . (Scale bar: 100 μm.) (B)
2
L
Studies of the flagellar photoresponse utilize light sent down an optical fiber.
to
dunkel@math.mit.edu
Spectra of light sources
Fig. S1.
Spectra of growth and stimulus light sources.
Fig. S1.
Spectra of grow
Fig. S2. (A) Schematic diagram of the sample chamber. (B) Photograph of a
the focal plane and pointing toward the fiber. (Scale bar: 200 μm.)
f the sample chamber. (B) Photograph of a micropipette holding a V. carteri colony and th
ard the
(Scaleofbar:
200and
μm.)
Fig.fiber.
S1. Spectra
growth
stimulus light sources.
bright-field 𝝀>620, 100 fps
dunkel@math.mit.edu
Photo-response at different intensities
0.25Hz
mplitude of the photoresponse for top-hat stimuli of frequency 0.25 Hz, at different stimulus light intensities.
dunkel@math.mit.edu
photoresponse
variable
that is large
18–20), fluid
inertia
isofnegligible
andFig.
the
flagella-induced
flowdecrease
light-induced
in flagellar activity and
vanishes when
is a direct
measure
the flagellar activity.
2A shows
a typical
there is no such change
in flagellar activity.decrease
The empirically
detime trace of
of the
photoresponse,
measured in
terms
of shows
the
light-induced
in
flagellar
acti
a direct measure
the
flagellar
activity.
Fig.
2A
a
typical
illumination
of the
termined constant β > 0 quantifies the amplitude of the decrease
flagella-generated flow speed uðtÞ, normalized by the flow speed
there
such
in τflagellar
acti
For a model
of pðtÞ is
thatno
captures
the change
two time scales
in uðtÞ∕u
me trace under
of time-independent
the photoresponse,
in#30°
terms
of 0 .the
, and averaged over
a
illumination u0 measured
dependence
of
the
and τr , we require a second variable hðtÞ,
which we define as a
from the anterior pole. We found that a step up in light intensity
termined
constant
β
>
0
quantifies
the
a
agella-generated
flow speed uðtÞ, normalized by the flow
speedrepresentation of the hidden internal biochemistry
dimensionless
elicits a decrease in flagellar activity on a response time scale τr ,
responsible
for adaptation
(24, 25).
system
of coupledofequaFor
a model
pðtÞ that
captu
in uðtÞ∕u
stimulation.
For
th
followedadaptive
by a recovery
tophotoresponse.
baseline activity
a time
scale
averaged
over
#30°
0. A
nder
illumination
u0 ,onand
sticstime-independent
of the
(A)τa The
local
flagellations that is consistent with the measured uðtÞ∕u0 is
and τr , we require a second variable hð
om the
anterior
pole.measured
We found that
a step
in light
intensity
peed
uðtÞ
(Blue),
with
PIVupjust
above
the flagella
photoresponse
τr p_ ¼ ðs − hÞHðs − hÞthe
− p;
[1]
dimensionless
representation
of the hidd
icits a decrease in flagellar activity on a response time scale τr , τ h_ ¼ s − h;
[2]
a
nollowed
light intensity,
serves
as
a
measure
of
flagellar
activity.
The
responsible
for
adaptation
(24,
transforms
of25).
p Aan
by a recovery to baseline activity on a time scale τa
where the light
stimulus
sðtÞ that
is a dimensionless
measurewith
of the the measur
tions
is consistent
ed in the dark is u0 ¼ 81 μm∕s for this dataset.
Two
time
Adaptive photo-response
neglecting the Hea
photoreceptor input that incorporates the eyespot directionality.
The Heaviside step function Hðs − hÞ is used to ensure that a step
_because
¼ ðsit−
hÞHðs −wit
h
associated
down in light stimulus cannot increase u above uτ0r, p
keeps p ≥ 0. In these equations, the values p& ¼ 0 and h& ¼ s1tivity upon a s
_ a suffiare stable rand global attractors in the sense that,τafter
s −the
h; ability of
ah ¼
ciently long time under constant light stimulus s1 , the pair
(p, h) relaxes to (p& , h& ). However, if s increases from s1 for t <further below.
for t > 0 the
the solution
0 to s2 for t ≥ 0, then
where
light isstimulus sðtÞtheisresponses
a dimen
: a short response time τr and a longer adaptation time
oretical curve (Red) is from Eq. 4. (B) The times τ (Squares)
ry smoothly with the stimulus light intensity, measured in
or bars are standard deviations.
Rð
SI Text), allow
that incorporates
th
[3]
away from the
The
Heaviside
step
function
Hðs − hÞ is u
ðs2 − s1 Þ −t∕τ
−t∕τ
τ
pðtÞ ¼
−e
Þ:
[4] r is always a
1 − τr ∕τ
down
ina ðelight
stimulus
cannot
(Fig.increase
2B), con
nas.org/cgi/doi/10.1073/pnas.1000901107
p ≥is a0.sharp
Intransient
theseincrease
equations,
the va
th
Volvox, there
in Although
When τr ≪ τa , as forkeeps
†
pðtÞ [and decrease in
uðtÞ],stable
peaking and
at a time
t ∼ τr lnðτ
be
are
global
attractors
in the
a ∕τ r Þ,rents have
followed by a slow relaxation back to zero, as in the measured(22), their con
ciently
time
under
constant ligh
flagellar
shownsetup.
inlong
Fig.(A)
2A.The
Fig. 1. Geometry of V.
carteriphotoresponse
and experimental
beating
flaIn Volvox,ifa sst
& , h&periodic
The rotation of Volvox about
its axis andto
the (p
resulting
). However,
relaxes
gella, two per somatic cell (Inset), create a(p,
fluidh)
flow
from the anterior
to the
illumination of the photoreceptors suggest an investigation of theof 1 ms (22) is
posterior, with a slightdependence
azimuthalofcomponent
rotates
about
its t > 0 the solu
t ≥Volvox
0, then
for
0 to that
s2 for
the photoresponse
on the
frequency
of sinusoidal
time for Ca2þ
a Þ;
hðtÞphotoreceptor
¼ s1 e−t∕τa þ s2 ð1 − e−t∕τinput
a
r
1𝜇m tracers L ∕D ∼ 0.2 s (
hðtÞ ¼ s e to τþ, suggest
10µm from cilium Ca ats2theð1 ba−
every somatic cell, we measured the fluid motion produced by theðs −
eses
23). A
s Þ(22, −t∕τ
u(t)
=
average
-30°
...
+30°
flagellar beating by using particle image velocimetry (PIV).
pðtÞ This
¼
ðe of th
like that
posterior-anterior
at angular
ωr . (Scale
bar: 100 dependence
μm.) (B) of 2
Forfrequency
the above model
this frequency
Fig. 2. Characteristics of the adaptive photoresponse.
(A) The local flagella-axisstimulation.
generated fluid speed uðtÞ (Blue), measured with PIV
just above
the flagella
photoresponseutilize
is R ¼
j~p∕s~sent
j, where
p~ and
s~ are the
Fourier
Studies
of the
flagellarthe
photoresponse
light
down
an optical
fiber.
−t∕τa
during a step up in light intensity, serves as a measure of flagellar activity. The
transforms of p and s, respectively. R is well-approximated
by
r
1
baseline flow speed in the dark is u0 ¼ 81 μm∕s for this dataset. Two time
2þ
neglecting the Heaviside function in Eq. 1 (see SI Text) to give
scales are evident: a short response time τr and a longer adaptation time
τa . The fitted theoretical curve (Red) is from Eq. 4. (B) The times τr (Squares)
ωs τa
2
1
ffi:
[5]
Rðωs Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
and τa (Circles) vary smoothly with the stimulus light intensity, measured in
2
2
2
2
ð1 þ ωs τr Þð1 þ ωs τa Þ
terms of PAR. Error bars are standard deviations.
11172 ∣
a
r ∕τa is only ∼
approach implicitly averages over several neighboring flagella,1 − τlatter
and, by measuring the fluid velocity just above the flagellarDrescher
tips, et al.unknown.
www.pnas.org/cgi/doi/10.1073/pnas.1000901107
we obtain a natural input for the hydrodynamic
τa , as of
forphotoVolvox, there
a sh
When τr ≪models
The is
measu
dunkel@math.mit.edu
tactic turning described further pðtÞ
below.
Because
of thein low
speed at
ju
[and
decrease
uðtÞ],fluid
peaking
The rotation of V
illumination of the
dependence of the
stimulation. For th
the photoresponse
transforms of p an
neglecting
therevers
Hea
the responses to step up and step down
stimuli are
eynolds number associated with flows generated by V. carteri
scribed by uðtÞ∕u0 ¼ 1 − βpðtÞ, where pðtÞ is a
photoresponse variable that is large when th
8–20), fluid inertia is negligible and the flagella-induced flow
light-induced decrease in flagellar activity and
a direct measure of the flagellar activity. Fig. 2A shows a typical
there is no such change in flagellar activity. The
me trace of the photoresponse, measured in terms of the
associated
adaptation;
there
wasβ >no
changethein
flage
termined
constant
0 quantifies
amplitude
agella-generated flow speed uðtÞ,
normalized bywith
the flow
speed
stics of the adaptive photoresponse.
(A)over
The
localinflagellaFor a model ofThis
pðtÞ that
captures theun
tw
,
and
averaged
#30°
nder time-independent illumination
u
tivity0 upon a step down in uðtÞ∕u
light0 .intensity.
response
andflagella
τr , we require a second variable hðtÞ, which
eed
(Blue),
measured
om
theuðtÞ
anterior
pole. We
found thatwith
a step PIV
up injust
light above
intensity the
the
ability
of
V
.
carteri
to
turn
toward
the
light,
as
ex
dimensionless
representation
of
the
hidden
intern
icits
a decrease
in flagellar
activity
a response
scale τr , activity. The
light
intensity,
serves
as a on
measure
oftime
flagellar
responsible
adaptation (24,
A system
o
further
Atscale
very
light for
intensities
and25).long
stimu
llowed by a recovery to baseline
activitybelow.
on a time
τa high
ed in the dark is u0 ¼ 81 μm∕s for this dataset. Two
time
tions that
is consistent with the measured uðtÞ∕u
Adaptive photo-response
: a short response time τr and a longer adaptation time
τr p_ ¼ ðs − hÞHðs − hÞ − p;
SI
Text),
allowing
Volvox
to
avoid
photodamage
by sw
retical curve (Red) is from Eq. 4. (B) The times τr (Squares)
_ ¼ s − h;
h
τ
a
away from
light. Irrespective
in
Rðω
ry smoothly with the stimulus
lightthe
intensity,
measured in of the stimulus light
a second,
several s
τr is always a fraction ofwhere
the light whereas
stimulus sðtÞ τisa ais
dimensionless
or bars are standard deviations.
input that incorporates
the eyespo
(Fig. 2B), consistent withphotoreceptor
early
observations
(14,
15).
The Heaviside step function Hðs − hÞ is used to en
Although the kinetics down
andinbiochemistry
of photorecept
light stimulus cannot
increase u above
&
keeps
p
≥
0.
In
these
equations,
the
values
p
rents have been studied in Chlamydomonas (2, 21) and¼
nas.org/cgi/doi/10.1073/pnas.1000901107
are stable and global attractors in the sense tha
(22), their connection tociently
the flagellar
photoresponse
is u
long time under
constant light stimul
& , h& ). However, if s increases
2þ
(A) The beating fla(p,
h)
relaxes
to
(p
In Volvox, a step stimulus elicits a Ca current whose tim
0 to s2 for t ≥ 0, then for t > 0 the solution is
m the anterior to the
of 1 ms (22) isFig.too
short to ofaccount
for
the measured
τr .
2.
Characteristics
the
adaptive
photoresponse.
(A)
The
local
flagella
tes Volvox about its
−t∕τ þ s ð1 − e−t∕τ Þ;
2þgenerated fluid speed uðtÞ (Blue), measured
hðtÞ
¼with
s1 ePIV
just above
the flagella
2
time
for
Ca
to
diffuse
the
length
of
the
flagellum
L
e bar: 100 μm.) (B) 2+
during a step up in light intensity, serves as−5
a measure
of flagellar activity. The
2
2
s (for
∼speed
15 μm,
D∼
cm
which
is
L ∕D ∼ 0.2(?)
ðs2for
−∕s),
s1 Þ dataset.
𝜏r :fiber.
Ca -diffusion
baselineL
flow
in the dark
is u10
¼ 81 μm∕s
this
Two
time
−t∕τ
−t∕τ
own an optical
pðtÞ ¼
ðe
−e
Þ:
a
longer
adaptation
time
scales are evident: a short response time τ and
1 −triggers
τr ∕τa
,
suggesting
that
the
photocurrent
an in
to
τ
r
𝜏a : unknown
τ . The fitted theoretical curve (Red) is from Eq. 4. (B) The times τ (Squares
and τ of
(Circles)
smoothly with
stimulus light
intensity,
measured inh
thevary
flagella,
previous
Ca2þ at the base
τa , the
as for
Volvox,with
there
is
a sharp
tran
When
τr ≪ consistent
terms
of
PAR.
Error
bars
are
standard
deviations.
n produced by the
pðtÞ [and
decrease in uðtÞ],
at a time
eses (22, 23). Although the
dependence
of τapeaking
on light
inte
dunkel@math.mit.edu
a
0
a
a
r
r
a
r
a
þ
followed by a slow relaxation back to zero, as i
The
rotation
of
V
eses
(22,
23).
Although
the
dependen
light-induced decrease in flagellar activity and
2A
shows
a
typical
elocimetry (PIV). This
illumination
of
the
current
in
Volvox
like
that
of
the
H
there is no such change in flagellar activity. The
in terms of the
dependence
of the
neighboring flagella,
latter
is
only
∼75
ms
(22);
the
bioche
termined constant β > 0 quantifies the amplitude
by the flow speed
stimulation.
For
th
bove
the
flagellar
tips,
unknown.
. For a model of pðtÞ that captures the tw
in uðtÞ∕u
eraged over #30°
the
photoresponse
models
of photoand τ , weThe
require
a secondadaptive
variable hðtÞ,
which
measured
response
pamic
in light
intensity
transforms of p an
dimensionless
representation
of
the
hidden
intern
Because
of τthe
fluid speed
just above
the
colony
su
, low
onse
time scale
neglecting the Hea
The
Heaviside
step
function
Hðs
−
hÞ
is
used
to
ensur
responsible
for
adaptation
(24,
25).
A
system
and
a
longer
adaptation
time
:
a
short
response
time
τ
r
generated
by
V.
carteri
scribed by uðtÞ∕u0 ¼τ 1p_ ¼−ðs −βpðtÞ,
hÞHðs − hÞwher
− p; o
n a time scale τa
light
stimulus
cannot
increase
u above
u0
retical curve (Red) is fromtions
Eq.down
4. that
(B) in
The
times
τr (Squares)
is
consistent
with
the
measured
uðtÞ∕u
_
h;
τ h¼s−
photoresponse
variable
that
is
flagella-induced
flow
& larg
¼
0
keeps
p
≥
0.
In
these
equations,
the
values
p
Rðω
ry smoothly with the stimulus light intensity, measured in
where
the ðs
light
stimulus
sðtÞ−
is
a dimensionless
light-induced
decrease
in
flagellar
a
Fig.
a typical
are stable
and global
attractors
in
the
sense
that,
a
or
bars2A
areshows
standard
deviations.
_
τ
p
¼
−
hÞHðs
hÞ
−
p;
r
photoreceptor
input that incorporates the eyespo
long istime
under
constant
light
there
noThe
such
change
in
flagellar
Heaviside
step function
Hðs
− stimulus
hÞ is used toa
en
ured in terms of theciently
& , _h&in
stimulus
cannot
increase u above
). light
However,
if s increases
fro
(p, h) relaxes to (pdown
h
¼
s
−
h;
τ
a
termined
constant
β
>
0
quantifies
the
ized by the flow speed0 to s2 for t ≥ 0, then
keeps p ≥ 0. In these equations, the values p ¼
nas.org/cgi/doi/10.1073/pnas.1000901107
for t > 0 the solution is
are stable and global attractors in the sense tha
alongmodel
of constant
pðtÞ that
cap
in uðtÞ∕u0 . For
nd averaged over #30°
ciently
time
under
light
stimul
where the light stimulus
sðtÞ
is, ha). dimensionless
−t∕τ
−t∕τ
However,
if s Þ;
increases
(p,
h)
relaxes
to
(p
and
τ
,
we
require
a
second
variable
hðtÞ
¼
s
e
þ
s
ð1
−
e
2 t > 0 the
ep up in light intensity
t ≥ 0, then for
is
s for1incorporates
photoreceptorr input0 tothat
thesolution
eyespo
dimensionless
representation
of
the
hi
response time scaleThe
τr , Heaviside
step function
Hðs
−
hÞ
is
used
to
ðs2 − shðtÞ
e
þ s −t∕τ
ð1 − e
Þ;en
1 Þ ¼ s−t∕τ
pðtÞfor
¼ cannot
ðe
− e u 25).
Þ: A
s:
stimulus
input
variable
responsible
adaptation
(24,
down
in
light
stimulus
increase
above
ty on a time scale τa
1 − τr ∕τa ðs − s Þ
pðtÞ ¼
ðe
− e & Þ:
h: hidden biochemistry
variable
tions
that
is
consistent
with
the
meas
− τ ∕τvalues
keeps p ≥ 0. In these equations, 1the
p ¼
that
consistent
the0 ¼measured
uðtÞ∕u
eynolds number associated with flows tions
generated
by V.iscarteri
1 − βpðtÞ, where
pðtÞ 0is is
a
scribed with
by uðtÞ∕u
þ variable that is large when th
photoresponse
8–20), fluid inertia is negligible and the flagella-induced flow
τr p_ ¼ ðs −decrease
hÞHðsin−flagellar
hÞ − p;
light-induced
activity and
a direct measure of the flagellar activity. Fig. 2A shows a typical
there is no such change in flagellar activity. The
me trace of the photoresponse, measured in terms of the
_ ¼ constant
termined
β > 0 quantifies the amplitude
agella-generated flow speed uðtÞ, normalized by the flow speed
h
s
−
h;
τ
a
stics of the adaptive photoresponse. (A)over
The
localinflagellauðtÞ∕u0 . For a model of pðtÞ that captures the tw
#30°
nder time-independent illumination u0 , and averaged
0 above the
andflagella
τr , we require a second variable hðtÞ, which
eed
(Blue),
measured
om
theuðtÞ
anterior
pole. We
found thatwith
a step PIV
up injust
light intensity
photo-response
dimensionless
representation
of the hidden intern
lightτr , activity.
stimulusThe
sðtÞ is a dimensionless
me
r ofthe
icits
a decrease
in flagellar
activity
a where
response
time
scale
light
intensity,
serves
as a on
measure
flagellar
variable
responsible for adaptation (24, 25). A system o
llowed by a recovery to baseline activity
on
a
time
scale
τ
a
photoreceptor
input
that
incorporates
themeasured
eyespot
d
ed in the dark is u0r ¼ 81 μm∕s
for this dataset.
Two
time
tions
that
is consistent with the
uðtÞ∕u
Photo-response model
r
a
&
a
&
&
a
2
−t∕τa
a
1
2
2
−t∕τa
1
r
a
r
−t∕τa
−t∕τr
≪ τa ,global
as for
Volvox,
there in
is athe
sharp
transien
areWhen
stableτr and
tha
there sense
is a sharp tran
Whenattractors
τr ≪ τa , as for Volvox,
†−
_
τ
p
¼
ðs
−
hÞHðs
∼
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
t
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
r dunkel@math.mit.edu
ciently long time followed
underby aconstant
light
stimul
slow relaxation back to zero, as i
Heuristic response model
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
outlined above remains. However, having only a small photoresponsive region complicates the heuristic picture: If the eyespots
could only direct an all-or-nothing response as they move from
the shaded to the illuminated side of the sphere, the best possible
phototactic orientation is drawn in Fig. 4C. Such a mechanism
Fig. 4. Heuristic analysis of the phototactic fidelity. A–C illustrate simplified
phototaxis models. Photoresponsive regions are colored green, the region
that actually displays a photoresponse is in shades of red, and shaded regions
are gray. (A) If τa ¼ ∞, ωr ¼ 0, and the responsive region is as drawn, the
posterior-anterior axis k will achieve perfect antialignment withdunkel@math.mit.edu
the light di-
Let’s try to be more
quantitative ...
dunkel@math.mit.edu
For Volvox, which generally
near the anterior pole (Fig. 5), yet the
mathematical model with predictive power.
posterior-anterior
axis is
th
outlined
above
remains.
However,
havin
In general, phototactic orientation is due to an asymmetry of
photoresponse
kinetics,
as sh
sponsive
region complicates
the heuristi
the flagellar behavior between the illuminated and shaded sides
could essential
only direct for
an all-or-nothing
high-fidelityresp
ph
of the organism. The mechanism that achieves this asymmetry the
is shaded
to the illuminated
side ofmay
the
of adaptation.
Spinning
phototactic orientation is drawn in Fig.
species-dependent, but it is instructive to consider a hierarchy of
Frequency dependence of
photo-response
of unsymmetrical colony de
ingredients. First, consider a nonspinning spherical organism that
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
normalized flagellar photoresponse for different frequencies of sinusoidal
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
organisms with a restricted
of eyespots, such as Chlamy
also required for detecting
In Volvox colonies, the
near the anterior pole (Fig
outlined above remains. Ho
sponsive region complicates
could only direct an all-or-n
the shaded to the illuminate
phototactic orientation is d
Fig. 4. Heuristic analysis of the phototactic fid
phototaxis models. Photoresponsive regions ar
that actually displays a photoresponse is in shad
are gray. (A) If τa ¼ ∞, ωr ¼ 0, and the respon
posterior-anterior axis k will achieve perfect ant
rection I. The time scale for turning τt ∼ 3.3 s ca
that the fluid velocity on the illuminated side is
value and using Eq. 8 without bottom-heaviness
photoresponse may decay before the optimal o
After the initial transient in A has decayed, the
flagellar down-regulation) is in the region that
an illustration, the configuration drawn in this p
the organism would turn away from the light, i
ientation is reached the steering is stopped at a
dunkel@math.mit.edu
with I. A remedy
against this orientational limita
al input for the hydrodynamic models offollowed
photoFor Volvox,
which
The
measured
adaptive
response
ofgenerally
the to
flag
by
a
slow
relaxation
back
near
the
anterior
pole
(Fig.
5),
yet
ass
−t∕τa þ s ð1 − e−t∕τa Þ; the
mathematical
model Because
with predictive
hðtÞ
¼
s
e
tiv
escribed
further below.
of thepower.
low
1 colony2 surface
fluid speed just above
the
(Fig.
2
posterior-anterior
axis
is
th
outlined
above
remains.
However,
havin
the
In general,
phototactic
orientation
is due toscribed
anphotoresponse
asymmetry
of ¼ photoresponse
shown
inthe
Fig.
r associated
with flows
generated
by V.flagellar
carteri
1−
βpðtÞ,
where
pðtÞ
isas afur
by uðtÞ∕u
kinetics,
sh
sponsive
region
complicates
heuristi
0
the
ðs
−
s
Þ
SI
2
1
thenegligible
flagellar and
behavior
between the illuminated
and shaded sides
−t∕τa −when
−t∕τr resp
could
only
direct
an
all-or-nothing
photoresponse
variable
that
is
large
the
ertia is
the flagella-induced
flow
essential
for
high-fidelity
ph
aw
pðtÞ
¼
ðe
e
Þ:
The
rotation
of
Volvox
about
its
axis
τ
of
the
organism.
The
mechanism
that
achieves
this
asymmetry
is
∕τ
1
−
τ
the
shaded
to
the
illuminated
side
of
the
r
a
(Fi
light-induced
decrease
in
flagellar
activity
and
of adaptation. Spinning may
e of the flagellar activity. Fig. 2A shows a typical
orientation is drawnsugg
in Fig.
species-dependent,
but it is
ren
ofsuch
theofphototactic
photoreceptors
therea ishierarchy
no
change
flagellar activity.
Thede
of in
unsymmetrical
colony
he photoresponse,
measured
in instructive
terms illumination
of to
theconsider
(22
≪
τ
,
as
for
Volvox,
there
is
a
sharp
transie
When
τ
In
ingredients. First, consider a nonspinning spherical
organism
r constant
a that β >organisms
termined
0
quantifies
the
amplitude
FT
depends
on
of
with
a
restricted
d flow speed uðtÞ, normalized by the flowdependence
speed
†
of
the
photoresponse
on
th
tim
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
t
pðtÞ thatinput
captures
the twLto
in uðtÞ∕u0 . For a modelofofeyespots,
suchsignal
as Chlamy
endent illumination u0 , and averaged over #30°
followed For
by a slow
relaxation
back
to zero,this
as inCa
stimulation.
the
above
model
and
τ
,
we
require
a
second
variable
hðtÞ,
which
every
somatic
cell,
we
measured
the
fluid
motion
produced
by the
onse.
(A)
The
local
flagellaalso
required
for
detecting
r
ese
pole. We found that a step up in light intensity
flagellar beating
by using particle
image velocimetry
(PIV). This
flagellar
photoresponse
shown
in
Fig.
2A.
lik
approach implicitly
averages
over several
neighboring intern
flagella,
dimensionless representation
ofvelocity
the
hidden
lat
In
Volvox
colonies,
the
,
in
flagellar
activity
on
a
response
time
scale
τ
and,
by
measuring
the
fluid
just
above
the
flagellar
tips,
r
~
PIV just above the flagella
un
~
The rotation of Volvox
about
its
axis
and
the
resu
the photoresponse
is
R
¼
j
p
∕
s
j,
where
we obtain a natural (24,
input for the
hydrodynamic
models
of photo- of
responsible for adaptation
25).
ABecause
system
near
the
anterior
pole
(Fig
tactic
turning
described
further
below.
of
theinves
low
covery to baseline activity on a time scale τa
flu
illumination
of
the
photoreceptors
suggest
an
Reynolds
number
associated
with
flows
generated
by
V.
carteri
scr
ure of flagellar activity. The
tions that
is pconsistent
with
the
measured
uðtÞ∕u
transforms
of
and
s,
respectively.
outlined
above
Ho
ph
(18–20),
fluid
inertia
is
negligible
and the remains.
flagella-induced
flow R
dependence of the photoresponse
on the
frequency
ligh
is a direct measure of the flagellar activity.
Fig. 2A shows
a typical
the
time
trace
of
the
photoresponse,
measured
in
terms
of
the
sponsive
region
complicates
for
dataset.
Two time
stimulation.
For
the
above
model
this
frequency
d
ter
stics ofthis
the adaptive
photoresponse.
(A) The localneglecting
flagellaflagella-generated
flow
speed
uðtÞ,
normalized
by
the
flow
speed
the Heaviside
in
E
τr p_time-independent
¼ ðs −function
hÞHðsu ,−
hÞ
−over
p;
in
and
averaged
#30°
under
illumination
could
direct
all-or-n
an
eed
uðtÞ (Blue), adaptation
measured with PIV time
just above the flagella
~an
the photoresponse from
is the
R
¼ pole.
jp~only
∕Wes~j,foundwhere
and
s~ ar
anterior
that a step upp
in light
intensity
a longer
dim
elicits_a decrease in flagellar activity on a response time scale τ ,
shaded
toactivity
the
illuminate
res
light intensity, serves as a measure of flagellar activity. The
¼
− h;to baseline
by
as
recovery
a time
scale τ
transforms of p andτfollowed
s,the
respectively.
Ronis
well-app
ah
tio
.d(B)
The
(Squares)
in the
darktimes
is u0 ¼ 81τ rμm∕s
for this dataset. Two time
phototactic
orientation
is
τSIad
neglecting the Heaviside function in Eq. 1 ω
(see
s
a short response time τr and a longer adaptation time
Þ
¼
Rðω
where the light
stimulus
sðtÞp
is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a dimensionless m
ght
intensity,
measured
in
s
retical curve (Red) is from Eq. 4. (B) The times τr (Squares)
wh
2eyespot
2 Þð1
ωofsþ
τthe
Fig. 4.that
Heuristic
analysis
phototactic
fid
a the
ph
photoreceptor
input
incorporates
ð1
ω
τ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
p
Þ
¼
:
Rðω
s
r
y smoothly with the stimulus light intensity, measured in
Th
s
phototaxis models. Photoresponsive
regions
ar
2
2
2
2
do
The
Heaviside
step
function
Hðs
−
hÞ
is
used
to
en
ð1
þ
ω
τ
Þð1
þ
ω
τ
Þ
s r
s ina shad
that actually displays a photoresponse
is
kee
r bars are standard deviations.
are
are gray. (A)
If τa ¼ ∞,
ωr ¼ 0, anduthe
respon
down in light stimulus
cannot
increase
above
cie
(p,
& ant
posterior-anterior axis k will achieve perfect
0t
¼
keeps p ≥ 0. In these
equations,
the
values
p
rection
I.
The
time
scale
for
turning
τ
∼
3.3
s
ca
t
as.org/cgi/doi/10.1073/pnas.1000901107
are stable and global
inthethe
senseside
thais
that the attractors
fluid velocity on
illuminated
01107
value
and using
Eq. 8 withoutlight
bottom-heaviness
ciently long time
under
constant
stimulu
photoresponse
may decay before the optimal o
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
& , h& ). However,
if s increasesWh
(p,
h)
relaxes
to
(p
After the initial transient in A has decayed, the
normalized flagellar photoresponse for different frequencies of sinusoidal
pðt
fol
t ≥ 0, flagellar
then for
t > 0 the issolution
is thatfla
to1 sand
down-regulation)
in the region
stimulation, with minimal and maximal light intensities0 of
20 μmol
2 for
Frequency dependence of
photo-response
r
Fig. 1. Geometry of V. carteri and experimental setup. (A) The beating flagella, two per somatic cell (Inset), create a fluid flow from the anterior to the
posterior, with a slight azimuthal component that rotates Volvox about its
posterior-anterior axis at angular frequency ωr . (Scale bar: 100 μm.) (B)
Studies of the flagellar photoresponse utilize light sent down an optical fiber.
2
0
r
a
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
an illustration, the configuration drawn in this p
illu
dei
the organism would
turn
away
from
the
light,
−t∕τ
−t∕τ
a
a
Fig.
2.
Characteristics
of
the
adaptive
photoresponse.
(A)
The
local
flagellahðtÞ
¼ iss1reached
e
þ
s2with
ð1PIV−
estopped
Þ; stima
ientation
the
steering
generated fluid
speed uðtÞ (Blue),
measured
justis
above
the flagella atthe
up in light intensity, serves as a measure of flagellar activity. The
tra
dunkel@math.mit.edu
withduring
I. Aa step
remedy
against this orientational limita
baseline flow speed in the dark is u ¼ 81 μm∕s for this dataset. Two time
ne
0
scales are evident: a short response time τr and a longer adaptation time
outlined above remains. However, hav
sponsive region complicates the heuris
could only direct an all-or-nothing res
the shaded to the illuminated side of th
phototactic orientation is drawn in Fi
Spinning frequency vs size
Fig. 4. Heuristic analysis of the phototactic f
phototaxis models.
Photoresponsive
regions
Fig.
4. Heuristic
a
that actually displays a photoresponse is in sha
models
are gray. (A) If τa phototaxis
¼ ∞, ωr ¼ 0, and
the resp
posterior-anterior that
axis k will
achieve perfect
actually
displaa
rection I. The time scale for turning τt ∼ 3.3 s
areongray.
(A) If side
τa
that the fluid velocity
the illuminated
value and using Eq.
8 without bottom-heavine
posterior-anterior
photoresponse may decay before the optima
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
rection
I. has
The
timet
After the initial transient
in A
decayed,
normalized flagellar photoresponse for different frequencies of sinusoidal
flagellar down-regulation)
is influid
the region
th
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
that the
veloc
−2
−1
an illustration, the configuration drawn in thi
PAR photons m s (Blue Circles). The theoretical response function (Eq. 5,
value and using Eq
the organism would turn away from the light
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
ientation
is reached
the steering is stopped
photoresponse
maa
16 μmol PAR photons
m−2 s−1 . dependence
(B) The rotation frequency
ωr of V.
carteri
Fig. 3. Photoresponse
frequency
and colony
rotation.
(A) The
with I. A remedy against this orientational lim
as a function of colony radius R. The highly phototactic organisms for which
After the initial tra
normalized flagellar
photoresponse
for
different
frequencies
of
sinusoidal
best attainable orientation towards the light
photoresponses were measured fall within the range of R indicated by the
down-reg
stimulation, with
minimal
maximal
intensities
ofapproxi1 and 20
μmol in a flagellar
is localized
small anterior
region, and
purple box,
and the and
distribution
of R canlight
be transformed
into an
response as they move from the s
distribution function (PDF) of ωr (Inset), by using the noisy
an illustration, the
s−1 (Blue
Circles). The theoretical response functionor-nothing
(Eq.
5,
PAR photons mate
m−2 probability
(D) Measurements of the eyespot (Orange) pl
curve of ωr ðRÞ. The purple box in A marks the range of ωr in this PDF (green
organism
woul
Red Line) shows
quantitative
2B for
SI Text).
(E) Volvoxthe
is bottom-heavy,
because
th
line indicates
the mean), agreement,
showing that the using
response τtime
scalesτ a
andfrom
colony Fig.
r and
the geometric
center of the
as
frequency
optimized
to maximize
the photoresponse.
ientation
is colony
reache
16 μmol PARrotation
photons
m−2are
s−1mutually
. (B) The
rotation
frequency
ωr of V.from
carteri
dunkel@math.mit.edu
with I. A remedy ag
as a function of colony radius R. The highly phototactic organisms for which
the flagellar behavior between the illuminated and shaded sides
of the organism. The mechanism that achieves this asymmetry is
species-dependent, but it is instructive to consider a hierarchy of
ingredients. First, consider a nonspinning spherical organism that
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
normalized flagellar photoresponse for different frequencies of sinusoidal
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
photoresponses were measured fall within the range of R indicated by the
purple box, and the distribution of R can be transformed into an approximate probability distribution function (PDF) of ωr (Inset), by using the noisy
curve of ωr ðRÞ. The purple box in A marks the range of ωr in this PDF (green
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
normalized flagellar photoresponse for different frequencies of sinusoidal
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
photoresponses were measured fall within the range of R indicated by the
essential f
of adaptat
of unsym
organisms
of eyespot
also requi
In Volv
near the
outlined a
sponsive r
could onl
the shade
phototacti
Fig. 4. Heu
phototaxis
that actuall
are gray. (A
posterior-a
rection I. Th
that the flui
value and u
photorespo
After the in
flagellar do
an illustrati
the organis
ientation is
with I. A re
best attaina
Optimal response !
dunkel@math.mit.edu
How about spatial structure ?
dunkel@math.mit.edu
ses
23). Although
the dependence
of τsurface,
intensity is
a on light
nds(22,
of individual
flagella
on
the
colony
we
þ current in Volvox, the decay constant of the
of
ke
that
of
the
H
nuum approximation in which there is a temporally
ad
atter
is only
∼75velocity.
ms (22);Ifthe
biochemical
origin
of
τa remains
an
varying
surface
θ
and
ϕ
are
the
polar
and
of the thousands of individual flagella on the colony surface, we
az
of the thousands of individual flagella on the colony surface, we
nknown.
adopt a continuum approximation in which there is a temporally
ma
adopt
a continuum u
approximation in which there is a temporally
gles on
a
sphere,
respectively,
the
surface
velocity
and spatially varying surface velocity. If θ and ϕ are the polarand
and
loc
spatially
varying
surface
velocity.
If
θ
and
ϕ
are
the
polar
and
The
measured
adaptive
response
of
the
flagella-generated
^ We the
azimuthal
surface velocity
u the
ex
mposed
intoangles
u ¼on va θ^sphere,
þ wrespectively,
ϕ.
interpret
u as
azimuthal
anglesveon a sphere, respectively, the surface velocity u
^
^
may be decomposed
into u the
¼ vθ þcolony
wϕ. We interpret
u as the
ve-be decomposed
In
^ We interpret u as the veuid speed
just above
surface
(Fig.
2A) can
may
into ube
¼ vθ^ deþ wϕ.
edge of
the
flagellar
layerlayer
(32);
forpractical
practical
ra
locity
at the
edge of the flagellar
(32); for
reasons
locity atreasons
the edge of the flagellar layer (32); for practical reasons
pðtÞ
is a dimensionless
cribedexperimental
by uðtÞ∕u
pr
measurements
of uβpðtÞ,
are madewhere
just above that
layer.
0 ¼1−
experimental
measurements of u are made just above that layer.
measurements
of
ustimulus,
are made
just
above
that
layer.
ch
In
the
absence
of
a
light
u
¼
u
and
we
assume
that
the
In
the
absence
of a light
stimulus,
u ¼ u and we assume that the
0
hotoresponse
variable
that
is
large
when
there
is
a
large
sh
on¼
theu
colony
surface
because
of
thev ðθÞ∕w
ratio
v0 ðθÞ∕w
ðθÞ
is constant on the colony surface because of the
ratio
0 ðθÞ is constant
e
of
a
light
stimulus,
u
and
we
assume
that
the
0 cells (9). activity
re
Fig.
4.vanishes
Heuristic
analysis
of the cells
phototactic
fidelity.step
A–C illust
orientational
order
of
somatic
(9). Following
precise orientational
order of
Followingprecise
step
ght-induced
decrease
insomatic
flagellar
and
when
ma
changes
inthe
light
intensity,
measurements
offlagella
vðθ;ϕ;tÞ on
at
ϕ gre
models. Photoresponsive
regions
arefixed
colored
changes
in light intensity,
measurements
of vðθ;ϕ;tÞ at
fixed
ϕ phototaxis
ðθÞ
is
constant
on
the
colony
surface
because
of
the
of
thousands
of
individual
the
colony
0
here isshow
nothatsuch
change
flagellar
activity.
The
empirically
deshow
that
each region,
surface
velocity isdisplays
a photoin each
region, thein
surface
velocity displays
a photothat in
actually
displaysthe
a photoresponse
in shades
of red, andalls
adopt
a continuum
approximation
in which
there is a
tational
order
of
somatic
cells
(9).
Following
step
response
of
the
form
shown
in
Fig.
2A
but
that
the overall
response
of
the
form
shown
in
Fig.
2A
but
that
the
overall
are
gray.
(A)
If
τ
¼
∞,
ω
¼
0,
and
the
responsive
region is
ermined
constant
β
>
0
quantifies
the
amplitude
of
the
decrease
and
spatially
varying
surface
velocity.
If
θ
and
magnitude
varies with θ axis
(Fig.k 5A).
We thusperfect
model antialignment
uðθ;ϕ;tÞ ϕbyare th
magnitude varies with θ (Fig. 5A). We thus model uðθ;ϕ;tÞ
by posterior-anterior
will achieve
wi
ght
intensity,
measurements
of
vðθ;ϕ;tÞ
at
fixed
ϕ
azimuthal
angles
on
a
sphere,
respectively,
the
surface
allowing
the
quantities
β,
p,
and
h
to
depend
on
position:
allowing
the a
quantities
β, p,of
and
h to that
dependcaptures
on position: therection
model
pðtÞ
two I.time
scales
n uðtÞ∕u
The time
scale forτturning
∼ 3.3 s can be estimate
0 . For
au ¼ vθτ^ þ
^ We interpretThu
may
be
decomposed
into
w
ϕ.
thatphotothe fluid
velocity
on the
illuminated side is reduced
to 0.7
each
the
surface
velocity
displays
a
ðθÞ½1
−
βðθÞpðθ;ϕ;tÞ&:
[6]
uðθ;ϕ;tÞ
¼
u
nd
τr , region,
we require
a
second
variable
hðtÞ,
which
we
define
as
a
locityand
at using
the edge
the flagellar
layer (32);
[6] value
uðθ;ϕ;tÞ ¼ u0 ðθÞ½1 − βðθÞpðθ;ϕ;tÞ&:
Eq. 8 of
without
bottom-heaviness.
(B)for
If τ practi
< τ th
,a
the
form frequency
shown
in Fig.
but
that
the
overall
experimental
measurements
are
just above
photoresponse
decay
before
the
optimal
orientation
has
un
Photoresponse
dependence
and 2A
colony
rotation.
(A) The
imensionless
representation
of
the
hidden
internal
biochemistry
The
measured
βðθÞ
is may
shown
in the
inset of
in u
Fig.
5A.made
The
measured
βðθÞ
is
shown
in
the
inset
in
Fig.
5A.
In the
of
light
stimulus,
uwe
¼ make
u
assum
ab
After
theabsence
initial
transient
in
A has
decayed,
the
largest
alized flagellar
photoresponse
for
different
frequencies
of
sinusoidaluðθ;ϕ;tÞ
To define
stimulus
s ona the
colony
surface,
usewe
of photo
0 and
aries
with
θ
(Fig.
5A).
We
thus
model
by
To define
theadaptation
stimulus
s light
on the
colony
surface,
we
make
use
esponsible
forand
(24,
25).
A
system
ofψðθ;ϕ;
coupled
equa^ defined
^ colony
try
the of
angle
IÞ
¼
n^ · I,
where
n^surface
is turned
the bec
ðθÞ∕w
is
constant
on−
the
ratio
v0down-regulation)
flagellar
iscos
inψthe
region
that just
int
ation, with minimal
maximal
intensities
of
1
and
20
μmol
0 ðθÞthrough
^
^
the angle ψðθ;ϕ;IÞ defined through cos ψ ¼ −n^ · I, where n^ unit
is thenormal
po
to the
surface.
Whenorder
ψ ¼ 0 drawn
(π),somatic
theinlight
directly
an
illustration,
the configuration
this is
panel
surprising
quantities
β,
p,
and
h
to
depend
on
position:
s
(Blue
Circles).
The
theoretical
response
function
(Eq.
5,
hotons
m
precise
orientational
of
cells
(9).
Foll
is
ons that
is
consistent
with
the
measured
uðtÞ∕u
unit normal to the surface. When ψ ¼ 0 (π), the light is directly
0 a given surface patch. The light-shadow asymmeto
above (behind)
Front-back asymmetry
0
0
0
a
r
t
0
a
t
−2 −1
the
organism
turn away from
the light, indicating
that
ine) shows quantitative agreement, using τr and τa from Fig. 2B for
changes
in would
light intensity,
measurements
of vðθ;ϕ;tÞ
above (behind)
a
given
surface
patch.
The
light-shadow
asymmewi
try
in
s
can
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
Superimientation
is reached
the
steering
is stopped
atvelocity
acomponent
suboptimal
o
mol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
Fig.
5.
Anterior-posterior
asymmetry.
(A)
The
anterior-posterior
show
that
in
each
region,
the
surface
display
de
try in s can therefore be modeled by a factor Hðcos ψÞ. Superimposed on this factor may be another functional dependence on ψ
of the with
fluid flow,
measuredagainst
10 μm above
the beating flagella,
following
a
I. A remedy
thisshown
orientational
limitation
would
unction of colony radius R. The highly
phototactic organisms for which
r be another
response
of
the
form
inforward
Fig.
2A
but
thatbe
dim
posed on this factor
may
functional dependenceto
on
ψ
account
for
the
eyespot
sensitivity
in
the
direction,
0
step
up
in
illumination
at
time
t
¼
0
s.
The
dashed
line
indicates
the
approxbest
attainable
orientation
towards
the
light
is
drawn,
if
the
p
responsestowere
measured
within sensitivity
the range in
of the
R indicated
by
the
with
experiments
on
Chlamydomonas
(28)
supporting
a
depenaccount
for thefalleyespot
forwardFig.
direction,
magnitude
varies
with
θ
(Fig.
5A).
We
thus
model
5.imation
Anterior-posterior
asymmetry.
(A) The
anterior-posterior
numerical model.
(Inset)
βðθÞ is blue (with pcomponen
norto v 0 ðθÞ used in the
is
localized
in
a
small
anterior
region,
and
the
eyespots
d
Fig.and
5. Anterior-posterior
asymmetry.
(A)be
Thetransformed
anterior-posterior
component
e box,
the
distribution
of
R
can
into
an
approxidence
f
ðψÞ
¼
cos
ψ.
The
class
of
models
we
consider
for
the
with experiments on Chlamydomonas (28) supportingofa the
depenmalized
to unity),
and
thequantities
mean
β is
red.β,(B)the
The
probability
of flagella
to pos
allowing
the
p,
and
h
to
depend
on
fluid
flow,
measured
10
μm
above
beating
flagella,
following
a (PDF)
of the fluid
flow, measured
10 μm
above the
beating
flagella,
following
or-nothing
as
they
move
from the
shaded
to
the illu
onent
probability
distribution
function
of of
ω
by
using
theanoisy
dimensionless
is response
therefore
r (Inset),
respond
to light scorrelates
with
the
size
of dashed
the somatic
cell
eyespots.
Theapprox
dence
f
ðψÞ
¼
cos
ψ.
The
class
models
we
consider
for
the
step
up
in
illumination
at
time
t
¼
0
s.
The
line
indicates
the
W
step
up
in
illumination
at
time
t
¼
0
s.
The
dashed
line
indicates
the
approxwing
(D)
Measurements
of
the
eyespot
(Orange)
placement
yield
κ¼
The purple boxs in
A
marks
the
range
of
ω
in
this
PDF
(green
of ωraðRÞ. dimensionless
light-induced
decrease
in
fluid
flow
occurs
beyond
the
region
of
flagellar
r
isnumerical
therefore
ðθÞ
used
in
the
model.
(Inset)
βðθÞ
is
blue
(with
p
norimation
to
v
ðθÞ
used
in
the
numerical
model.
(Inset)
βðθÞ
is
blue
(with
p
no
imation
to
v
ve
0
ðθÞ½1
−
βðθÞpðθ;ϕ;tÞ&:
uðθ;ϕ;tÞ
¼
u
^ bottom-heavy,
pprox0 ψÞ:
response
because(E)
ofVolvox
the
nonlocality
of fluid dynamics.
[7]of mass
sðθ;ϕ;
IÞ
¼ f ðψÞHðcos
SI0 Text).
is
because the center
dicates
thetomean),
showing
that
time scales
and to
colony
malized
unity), and
the mean
β isthe
red.response
(B) The probability
of flagella
malized
to
unity),
and
the
mean
β
is
red.
(B)
The
probability
of
flagella t
p nor^ ¼
from the geometric center of the colony as indicated.
respond to are
lightmutually
correlates with
the size
offmaximize
the
somatic the
cell eyespots.
The
on frequency
optimized
to
photoresponse.
ψÞ:
[7]
sðθ;ϕ;
IÞ
ðψÞHðcos
above
specification
theofdynamics
ofinset
the
Withto the
ella to
respond
light
correlates
with
theof
size
the
somatic
cellsurface
eyespots.
Th
The
measured
βðθÞ
is
shown
in
the
in
Fig.
5A.
light-induced decrease in fluid flow occurs beyond the region of flagellar
may be sufficient
for
Volvox
in
environments,
because
s. The
velocity,
the
angular
of natural
the
colony
is (31)
light-induced
decrease
in velocity
fluid
flow
occurs
beyond
the region
of it
flagella
dunkel@math.mit.edu
response
because
the nonlocality
of fluid dynamics.
the ofabove
specification
of the dynamics of the
surface
With
To
define
the
stimulus
s
on
the
colony
surface,
we m
would
robustly
navigate
Volvox
closer
to
the
light,
even
though
the
agellar
response because of the nonlocality of Zfluid dynamics.
τ p_ ¼
ðs − hÞHðs − hÞ − p;
[1]
− βðθÞpðθ;ϕ;tÞ&:
[6]
uðθ;ϕ;tÞ ¼ u ðθÞ½1
[2]
τ h_ ¼ s − h;
d βðθÞ is shown in the inset in Fig. 5A.
he stimulus
s on
the colony
we make use
of of the
here
the light
stimulus
sðtÞ issurface,
a dimensionless
measure
^ defined through cos ψ ¼ −n^ · I,
^ where n^ is the
θ;ϕ;
IÞ
hotoreceptor
input that incorporates the
eyespot directionality.
r
owards the light is drawn, if the photorespon
Eye-spot
measurements
erior region, and the eyespots display an a
move
from
anterior
polethe
shaded to the illuminated sid
The amplitude of the photoresponse for top-hat stimuli of frequency 0.25 Hz, at different stimulus light int
espot (Orange)
placement
yield κ ¼ 57° $ 7° (se
𝜽=50°
𝜽=0
heavy, because the center of mass (Pink) is offs
of the colony as indicated.
∣
Fig. S4.
The amplitude of the photoresponse for top-hat stimuli of frequency 0.25 Hz, at different stimulus light intensities.
20𝜇m
June 22, 2010 ∣
vol. 107 ∣
no. 25 ∣
111
teri somatic
cells at the anterior pole have their orange eyespots facing away from the fluid-mechanical anterio
Fig. S5. (A) The V. carteri somatic cells at the anterior pole have their orange eyespots facing away from the fluid-mechanical anterior pole. (B) The somatic
cellsθand
at polarthe
angle anterior.
θ ¼ 50° from the
anterior.bars:
(Scale bars:
μm.) (C)
Illustration
of the eyespot
in the somatic
cells and the relation
olar angle
¼eyespots
50° from
(Scale
2020μm.)
(C)
Illustration
of placement
the eyespot
placement
in theto somatic c
the posterior-anterior axis k. In contrast to this schematic drawing, V. carteri colonies consist of thousands of somatic cells, as shown in Fig. 1A of the main text
axis k. Inand
contrast
toin this
as measured
ref. 20.schematic drawing, V. carteri colonies consist of thousands of somatic cells, as shown in F
f. 20.
dunkel@math.mit.edu
Basic ingredients of
a‘full’ model
• self-propulsion
• bottom-heaviness
• photo-response kinetics
• photo-response spatial variation
dunkel@math.mit.edu
magnitude
varies
with
θand
(Fig.
5A).
We
thus mode
posed
toangle
a light
stimulus.
e
instead
continuously
changes
with
the
angle
at
ugh
the
measured
βðθÞ
is
shown
in
the
inset
in
Fig.
5A.
th
the
at
he
Sciences
Research
Council
(K.D.),
the
Engineering
Biological
Sciences
^
try
in
s
can
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
Superimand
the
rotational
drag
of
the
sphere
(20).
The
second
term
is
he
and
the
^
where
g
and
k
are
the
directions
of
gravity
and
the
posteriorsence
of
a
light
stimulus,
u
¼
u
and
we
assume
that
the
ht.
Together
with
an
appropriate
•
The
θ
dependence
of
the
^
0
For
a
model
of
pðtÞ
that
captures
the
two
time
scales
τ
^ that
pled
equations
make
upthe
the
model
are
given
in
the
where
g
and
k
are
directions
of
gravity
the
posteriorresponsible
for
phototactic
steering,
w
allowing
the
quantities
β,
p,
and
h
to
depend
on
p
a and
ds
an
appropriate
o
define
the
stimulus
s
on
the
colony
surface,
we
make
use
of
program
of
the
Biotechnology
and
Biological
Sciences
Research
Council,
rientaposed
on
this
factor
may
be
another
functional
dependence
on
ψ
espots
receive
light.
Together
with
an
appropriate
responsible
for
phototactic
steering,
where
the
integral
is
taken
awe
approximated
v
ðθÞ
by
anterior
axis,
respectively.
The
first
term
in
Eq.
8
arises
from
this
directionality
leads
to
the
uðθ;ϕ;t
0
Þ∕w
ðθÞ
is
constant
on
the
colony
surface
because
of
the
responsi
require
a
second
variable
hðtÞ,
which
we
define
as
a
To
determine
the
time
evolution
of
the
system
of
^
^
anterior
respectively.
The
first
term
insurface
Eq.
8R.
arises
from
0as
ere
over
the
of
the
sphere
of
radiu
ty
leads
theaxis,
theto
Human
Frontier
Science
Program
(I.T.),
the
US
Department
of
Energy,
^
^
tothrough
account
for
the
eyespot
sensitivity
in
the
forward
direction,
angle
ψðθ;ϕ;
IÞ
defined
cos
ψ
¼
−
n
·
I,
where
n
is
the
1ð
e
if,
over
the
surface
of
the
sphere
of
radius
In
a
reference
frame
Legendre
functions,
−P
as
bottom-heaviness
and
represents
aFollowing
balance
between
theaover
torque
ment
(Fig.
4D),
this
directionality
leads
to
the
1
mmetry
between
illuminated
and
ess
representation
of
the
hidden
internal
biochemistry
uations,
we
solved
the
coupled
partial
differential
rientational
order
of
somatic
cells
(9).
step
bottom-heaviness
and
represents
a
balance
between
the
torque
ve
and
the
Schlumberger
Chair
Fund
(R.E.G.).
with
experiments
on
Chlamydomonas
(28)
supporting
depenilluminated
and
ðθÞ½1
−
βðθÞpðθ;ϕ;tÞ&:
uðθ;ϕ;tÞ
¼
u
the
where
the
Volvox
at
the
origin
with
a
normal
to the surface.
When
ψ ¼ is0 at
(π),
the
light
is
directly
0theisdashed
of
the
where
the
Volvox
the
origin
with
a
fixed
orientation,
the
light
magenta
line
he
ior-posterior
component
that
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
or
pðθ;ϕ;tÞ
and
hðθ;ϕ;tÞ
numerically
with
a
built-in
solThe
measured
βðθÞ
is
for
adaptation
(24,
25).
A
system
of
coupled
equadence
f
ðψÞ
¼
cos
ψ.
The
class
of
models
we
consider
for
the
entation
toward
the
light
has
been
that
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
^
^
fengle
a
response
asymmetry
between
illuminated
and
he
light
has
been
^
^
n
light
intensity,
measurements
of
vðθ;ϕ;tÞ
at
fixed
ϕ
e
(behind)
a
given
surface
patch.
The
light-shadow
asymmedirection
evolves
as
d
I∕dt
¼
−Ω
×
I.
direction
evolves
as
d
I∕dt
¼
−Ω
×
I.
at
ing
flagella,
following
a
simple
sin
θ
dependence
fo
where
th
at
hematica
(Wolfram
Research)
between
times
t
and
is
s
consistent
with
the
measured
uðtÞ∕u
dimensionless
s
is
therefore
and
the
rotational
drag
of
the
sphere
(20).
The
second
term
is
To
define
the
stimul
2þdrag of the 0
and
the
rotational
sphere
(20).
second
term
is Biol
yes
23.
Tamm
S
(1994)
Ca
channels
and
signalling
in cilia
and
flagella.
Trends
Cell
The
measured
βðθÞ
isThe
shown
inbeen
the
inset
inthat
Fig.
5A
ne
indicates
the
approxn
s
can
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
SuperimThe
above
coupled
equations
can
be
solved
numerically
(see
The
above
coupled
equations
can
b
tte
in
each
region,
the
surface
velocity
displays
a
photosopriate
until
perfect
orientation
toward
the
light
has
sults.
We
assume
w
0
^
to
the
integral
in
the
equation
for
Ω,
we
used
an
Euler
direction
the
angle
ψðθ;ϕ;
IÞaxis
defi
responsible
for
phototactic
steering,
where
the
integral
is
taken
ents
can
be
op)dβðθÞ
is
blueenvironments
(with
p norresponsible
for
phototactic
steering,
where
the
integral
is
taken
4:305–310.
natural
can
be
op^
To
define
the
stimulus
s
on
the
colony
surface,
we
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
on
this
factor
may
be
another
functional
dependence
on
ψ
•
βðθÞ,
the
responsivity
of
th
_
τ
p
¼
ðs
−
hÞHðs
−
hÞ
−
p;
[1]
SI
Text),
e.g.,
to
determine
the
angle
^
ψÞ:
[7]
sðθ;ϕ;
IÞ
¼
f
ðψÞHðcos
of
the
form
shown
in
Fig.
2A
but
that
the
overall
r
to
the
then
solve
the
equation
for
IðtÞ
at
every
time
step.
We
he
probability
flagella
to Jülicher
89.
unit
normal
to
the
sur
over
the
surface
of
the
sphere
of
radius
R.
In
a
reference
frame
created
the
24.ofby
Friedrich
BM,
Flight
(2007)
Chemotaxis
of^issperm
cells.
Proc
Natl
Acad
Sci
USA
over
the
surface
of
the
sphere
of
radius
R.
In
a
reference
frame
^
The
a
0),
which
may
be
created
by
the
with
the
direction.
It
interesting
to
consider
two
special
^
ccount
for
the
eyespot
sensitivity
in
the
forward
direction,
the
full
model,
we
used
the
angle
ψðθ;ϕ;
IÞ
defined
through
cos
ψ
¼
−
n
·
I,
w
with
the
light
direction.
It
is
interestin
matic
cell
eyespots.
The
nvergence
of
the
results
by
choosing
a
small
enough
ed
and
Rev
_
edororientation
varies
with
θin
(Fig.
5A).
We
thus
model
uðθ;ϕ;tÞ
by
104:13256–13261.
above
(behind)
a
given
h
¼
s
−
h;
[2]
τ
wind-driven
the
above
specification
of
the
dynamics
of
the
surface
With
where
the
Volvox
is
at
the
origin
with
a
fixed
orientation,
the
light
natural
environments
can
be
opwhere
the
Volvox
is
at
the
origin
with
a
fixed
orientation,
the
light
a
cases
of
the
model
class
outlined
above.
In
the
biologically
faithshown
in
the
inset
in
Fig.lig
5
nisms,
convection,
or
wind-driven
experiments
on
Chlamydomonas
(28)
supporting
a
depend. the region of flagellar
unit
normal
to
the
surface.
When
ψ
¼
0
(π),
the
SI
Text),
cases
of
the
model
class
outlined
abov
as
25. Spiro
PA, Parkinson
JS, ^Othmer
(1997)
A on
model
of excitation
and
adaptation
in
n been
^HG
^position:
velocity,
the
angular
velocity
of
the
colony
is
(31)
^ −Ω
phototaxis
even
the
quantities
β,
p,
and
h
to
depend
try
in
s
can
therefore
b
direction
evolves
as
d
I∕dt
¼
×
I.
direction
evolves
as
d
I∕dt
¼
−Ω
×
I.
mics.
ful
“full
model,”
we
use
the
measured
βðθÞ
and
the
realistic
eyemodel,
we
used
βðθÞ
¼
0.3
e
f
ðψÞ
¼
cos
ψ.
The
class
of
models
we
consider
for
the
bient
vorticity
(30),
which
may
be
created
by
the
above
(behind)
a
given
surface
patch.
The
light-shad
hat
can
counteract
phototaxis
even
on
to
finding
the
angle
of
the
Volvox
axis
with
the
light
Z
bacterial
chemotaxis.
Proc
Natl
Acad
Sci
USA
94:7263–7268.
ful
“full
model,”
we
use
the
measured
with
the
sðtÞ isspot
a dimensionless
measure
of
the
elight
to astimulus
property
posed
on
this
factor
m
The
above
coupled
equations
can
be
solved
numerically
(see
The
above
coupled
equations
can
be
solved
numerically
(see
directionality
f
ðψÞ
¼
cos
ψ.
In
the
“reduced
model,”
we
1
3
full
model.
ensionless
s isalso
therefore
he
model
can
be
used
determine
the
organism
try
in
s
can
therefore
be
modeled
by
a
factor
Hðcos
^
experiments
is
due
to
property
26. Walsh
P,organisms,
Legendre
Lato
(1983)
Photosynthesis
of−directionality
natural
phytoplankton
under
highψ.
^
^
g
×
k
n
×
uðθ;ϕ;tÞdS;
[8]In
ΩðtÞ
¼
her
nearby
convection,
or
wind-driven
spot
f
ðψÞ
¼
cos
t
pbe
opptor
input
that
incorporates
the
eyespot
directionality.
Chlamydomonas
3
to
account
for
the
eye
cases
of
en
consider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
•
τ
and
τ
,
the
response
and
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
axis
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
axis
τbhStone
8πR
ðθÞ½1
− βðθÞpðθ;ϕ;tÞ&:
[6]
u0fluctuations
r
a functional
ronments,
because
itlight
velocityuðθ;ϕ;tÞ
U, via ¼
another
result
from
and
frequency
simulating
those
induced
by be
sea another
surface
waves.
Limnol depe
posed
on
this
factor
may
cellular
ancestor
Chlamydomonas
step
function
Hðs
−^ hÞ is
used
to
ensure
that
atostep
he
consider
only
a
light-shadow
respons
om
their
center
by
the
.side
A
mechanism
that
can
counteract
phototaxis
even
All
1—and
use
the
mean
of
the
measured
βðθÞ—i.e.,
βðθÞ
¼
0.3.
For
the
simulations,
we
uC
with
experiments
on
with
the
light
direction.
It
is
interesting
consider
two
special
with
the
light
direction.
It
is
interesting
to
consider
two
special
ight,
even
though
the
ψÞ:
[7]
sðθ;ϕ;
IÞ
¼
f
ðψÞHðcos
Oceanogr
28:688–697.
)
ful
“full
^
account
for
the
eyespot
sensitivity
in
the
forwa
where
g^toanterior-posterior
and
k areshared
the
directions
ofthe
gravity
andofthe
posteriorof
mass
is
offset
from
their
center
ure
asymmetry.
(A)
The
component
,
because
it
ght
stimulus
cannot
increase
u
above
u
srior-posterior
due
to
clustern
1—and
use
mean
the
measured
0
-driven
other
features
are
between
the
models.
intensity
of
16
μmol
PAR
dence
f
ðψÞ
¼
cos
ψ.
T
e
light.
The
orientacases
of
the
model
class
outlined
above.
In
the
biologically
faith27.
Jennings
HS
(1901)
On
the
significance
of
the
spiral
swimming
of
organisms.
Am
Nat
cases
of
the
model
class
outlined
above.
In
the
biologically
faitholled
laboratory
experiments
is
due
to
a
property
&
&
anterior
axis,
respectively.
The
first
term in Eq. 8 arises
from
Zabove
with
experiments
on
Chlamydomonas
(28)
support
sured
βðθÞ
is
shown
in
the
inset
in
Fig.
5A.
spot
dir
low,
measured
10
μm
the
beating
flagella,
following
a
¼
0
and
h
¼
s
0.
In
these
equations,
the
values
p
eads
to
a
torque
ottom-heaviness
is
due
to
clustern
1
A
phototactic
turn
of
a
hypothetical
non-bottom-heavy
Volvox
Fig.
2B
of
the
main
text.
nisbe
overcome
if,
as
other
features
are
shared
between
th
hterior-posterior
the
above
specification
of
the
dynamics
of
the
surface
even
35:369–378.
dimensionless
s
is
ther
1
ful
“full
model,”
we
use
the
measured
βðθÞ
and
the
realistic
eyevs
ful
“full
model,”
we
use
the
measured
βðθÞ
and
the
realistic
eyebottom-heaviness
and
represents
a
balance
between
the
torque
component
06.
mination
at
time
t ¼unicellular
0 on
s.inThe
dashed
line
indicates
the
approxdence
f
ðψÞ
¼
cos
ψ.
The
class
ofthe
models
we con
hares
with
its
ancestor
Chlamydomonas
and
global
attractors
the
sense
that,
after
a
suffiUðtÞ
¼
uðθ;ϕ;tÞdS;
[S1]
he
vertical
on
a
ne
the
stimulus
s
the
colony
surface,
we
make
use
of
),
the
strength
of
the
•
τ
,
the
bottom-heaviness
simulated
by
the
reduced
model
is
shown
in
Fig.
6,
indicating
an
or
(Fig.
4E)
and
leads
to
a
torque
ty
consider
28.
Schaller
K,
David
R,
Uhl
R
(1997)
How
Chlamydomonas
keeps
track
of
light
once
it
2
bh
city,
the
angular
velocity
of
the
colony
is
(31)
A
phototactic
turn
of
a
hypothetica
roperty
spot
directionality
f
ðψÞ
¼
cos
ψ.
In
the
“reduced
model,”
we
that
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
ia- used
eating
flagella,
following
a
4πR
spot
directionality
f
ðψÞ
¼
cos
ψ.
In
the
“reduced
model,”
we
ðθÞ
in
the
numerical
model.
(Inset)
βðθÞ
is
blue
(with
p
nor,
the
pair
ng
time
under
constant
light
stimulus
s
dimensionless
s
is
therefore
^
^
totaxis
in
Volvox
es
with
the
angle
atthe
a
flagellaless
Volvox
that
is
1 n
intricate
link
between
organism
rotation,
adaptation,
and
has
reached
the
right
phototactic
orientation.
Biophys
J 73:1562–1572.
^
^
Z
ψðθ;ϕ;
IÞ
defined
through
cos
ψ
¼
−
·
I,
where
n
is
the
ae:
Their
center
of
mass
is
offset
from
their
center
sðθ
gas
direction
with
vertical
on
a
and
the
rotational
drag
of
the
sphere
(20).
The
second
term
is
consider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
simulated
by
the
reduced
model
is
sho
omonas
d
line and
indicates
the1 approx1—and
u
& ,the
& ).
consider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
nity),
mean
β
is
red.
(B)
The
probability
of
flagella
to
3
h
However,
if
s
increases
from
s
for
t
<
xes
to
(p
with
an
appropriate
self-propulsion,
29. Schaller
K,the
Uhl
Rresponsible
(1997) In
A microspectrophotometric
study
of
the
shielding
properties
The
axis
of
this
Volvox
wou
steering.
reality,
however,
Volvox
is
bottom-heavy,
which
is
par1 steering,
ws
trajectories
of
organism
to
be
reconstructed.
cofor
phototactic
where
the
integral
is
taken
er
^
thful
theory
of
phototaxis
in
Volvox
set)
βðθÞ
is
blue
(with
p
norAll
1—and
use
the
mean
of
the
measured
βðθÞ—i.e.,
βðθÞ
mal
to
the
surface.
When
ψ
¼
(π),
the
light
is
directly
g^0use
×the
k the
−
n^0×the
uðθ;ϕ;tÞdS;
[8]¼ 0.3.
ΩðtÞ
¼
intricate
link
between
organism
r
center
For
Volvox,
this
bottom-heaviness
is
due
to
clusterght
correlates
with
the
size
of
the
somatic
cell
eyespots.
The
All
1—and
mean
of
measured
βðθÞ—i.e.,
βðθÞ
¼
0.3.
^
_
3
onality
leads
to
the
other
fe
t
≥
0,
then
for
t
>
solution
is
of
eyespot
and
cell
body
in
Chlamydomonas
.
Biophys
J
73:1573–1578.
ψÞ:
sðθ;ϕ;
IÞ ζR.
¼¼
f−
ðψÞHðcos
photoresponse
sinðζÞ∕τ
.frame
For
the
ticularly
important
when
the
light
direction
is
horizontal.
In
this
the
above
speci
With
9.
τ
over
the
surface
of
the
sphere
of
radius
In
a
reference
8πR
bh
on
of
the
photoresponse
pðθ;ϕ;tÞ
is
plotted
in
Fig.
6
of
rbh
e
probability
of
flagella
to
other
features
are
shared
between
the
st
four
features:
self-propulsion,
decrease
in
fluidand
flow
occurs
beyond
the
region
ofmodels.
flagellar
clustersteering.
In
reality,
however,
Volvox
is
b
hind)
a
given
surface
patch.
The
light-shadow
asymmeeen
illuminated
other
features
are
shared
between
the
models.
30.
Durham
WM,
Kessler
JO,
Stocker
R
(2009)
Disruption
of
vertical
motility
by
shear
measured
in
ref.
19.
ells
in
the
posterior
(Fig.
4E)
and
leads
to
a
torque
case,
we
previously
observed
(33)
that
the
organisms
reach
a
final
where
the Volvox
is atinthe
origin
with a fixedvelocity,
orientation,
the
light
the
angular
v
ext,
using
the
“reduced
model”
defined
the
main
A
pho
somatic
cell
eyespots.
The
ue
A
phototactic
turn
of
a
hypothetical
non-bottom-heavy
Volvox
ause
of
the
nonlocality
of
fluid
dynamics.
^¼
−t∕τ
−t∕τ
nse
kinetics,
and
photoresponse
ard
theand
light
has
been
torque
a þ
a Þ;
ticularly
important
when
the
light
dire
^
^ layers.
^non-bottom-heavy
triggers
formation
of
thin
phytoplankton
Science
323:1067–1070.
re
g
k
are
the
directions
of
gravity
and
the
posteriorthe
above
specification
of
the
dynamics
of
With
A
phototactic
turn
of
a
hypothetical
Volvox
n
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
SuperimhðtÞ
s
e
s
ð1
−
e
[3]
direction
evolves
as
d
I∕dt
¼
−Ω
×
I.
set
by
the
balance
of
the
bottom-heaviness
torque
and
the
angle
α
1
2
omposition
of
this
photoresponse
into
spherical
harf
ond
the
region
of
flagellar
gn
the
swimming
direction
with
the
vertical
on
a
a
simulate
simulated
by
the
reduced
model
is
shown
in
Fig.
6,
indicating
an
In
order
to
compare
the
re
31.
Stone
HA,
Samuel
ADT
(1996)
Propulsion
of
microorganisms
by
surface
distortions.
eus
1
al
on
a
rior
axis,
respectively.
The
first
term
in
Eq.
8
arises
from
The
above
coupled
equations
can
be
solved
numerically
(see
case,
we
previously
observed
(33)
that
velocity,
the
angular
velocity
of
the
colony
is
(31)
eynolds
number
simulated
by
the
reduced
model
is
shown
in
Fig.
6,
indicating
an
phototactic
torque.
We therefore
“phototactic
abilðθ;ϕÞ
is given
inopFig.
S7.
The photoresponse
pdependence
com- define the
this factor
may
be
another
functional
on
ψ
amics.
ΩðtÞ
¼
onments
can
be
ox
Phys Rev
Lett
77:4102–4104.
intricate
link
between
organism
rotation,
adaptation,
and
ðs
−
s
Þ
with
the
measurements
of
th
∼
14
s
(20).
A
faithful
theory
of
phototaxis
in
Volvox
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
axis
2
1
intricate
Z
om-heaviness
and
represents
a
balance
between
the
torque
−t∕τ
−t∕τ
ed
and
angular
Volvox
set
by
the
balance
of
the
bottom
angle
α
ity”
A
¼
ðswimming
speed
toward
the
lightÞ∕ðswimming
speedÞ.
τ
a
r
intricate
link
between
organism
rotation,
adaptation,
and
he
“full
model”
during
a
phototactic
turn
is
shown
ficient
for
Volvox
in natural
environments,
because
itdirection,
f[4]
pðtÞ
¼
ðe
−
e
Þ:
bh
nt
forcreated
the
eyespot
sensitivity
in
the
forward
y be
by
the
om
32.
Blake
JR
(1971)
A
spherical
envelope
approach
to
ciliary
propulsion.
J
Fluid
Mech
1
3
n,
steering.
In
reality,
Volvox
is
bottom-heavy,
which
is
parwithhowever,
the
lightaxis
direction.
Itparallel
isasinteresting
to
consider
two special
dunkel@math.mit.edu
viscosity,
we
implemented
a
∕τ
1
−
τ
he
fluid
velocity
acts
when
the
posterior-anterior
is
not
to
gravity
^
Both
models
predict
that
the
viscosity
η
is
increased,
while
r
a
axis.
In
the
low
Reynolds
number
ulsion,
neglecting
bottom-heaviness.
phototactic
torque.
We
therefore
def
e
include
at
least
four
features:
self-propulsion,
steering.
In
reality,
however,
Volvox
is
bottom-heavy,
which
is
par^
^
g
×
k
−
n
×
uðθ;ϕ;tÞdS
ΩðtÞ
¼
stly
navigate
Volvox
closer
to
the
light,
even
though
the
steering.
tion, or wind-driven
Hydrodynamic model
‘Simple’ squirmer model
ny behavior during a phototurn. A–E show the colony axis k (Red Arrow) tipping toward the light direction I (Aqua Arrow). Colors re
Fig. 6. Colony behavior during a phototurn. A–E show the colony axis k (Red Arrow) tipping toward the light direction I (Aqua Arrow). Colors represent the
ðtÞ of amplitude
the down-regulation
of flagellar
beating
in ain simplified
model
of phototactic
showsof the
location
of colonies
in A–E
pðtÞ of the down-regulation
of flagellar
beating
a simplified model
of phototactic
steering. steering.
F shows the Flocation
colonies
in A–E along
the
swimming trajectory.
ajectory.
L
of the fidelity of phototaxis in Volvox and that a quantitative
and therefore a reduced phototactic torque. The sharp transition
can be obtained
if a realistic
of the fidelity
of phototaxis
in eyespot
Volvoxdirectionand that a qu
ore a inreduced
phototactic
torque.torque
Thebecomes
sharp comparable
transition understanding
Fig. 7 occurs
when the phototactic
dunkel@math.mit.edu
ality
and anterior-posterior
asymmetry
included. eyespot
to the other
torques in thetorque
system. The
simulations
neglected
understanding
can response
be obtained
if aarerealistic
curs when
the phototactic
becomes
comparable
‘Full’ squirmer model
m
Fig. S7. The photoresponse p may be decomposed into the spherical harmonics Y m
l ðθ;ϕÞ via the equation pðθ;ϕ;tÞ ¼ ∑l;m alm ðtÞY l ðθ;ϕÞ. The decomposition
was done for the photoresponse shown in Fig. 6 of the main text–i.e., using the reduced model. For this model, the dominant modes are the constant Y 00 , the
0
Y $1
1 modes that give a ϕ dependence similar to the light-shadow asymmetry, and Y 1 , which gives an anterior-posterior asymmetry that becomes important in
this model when the organism has turned significantly toward the light. B–G display the spherical harmonics on a sphere.
Fig. S8. The behavior of the photoresponse pðθ;ϕ;tÞ during a phototactic turn, using the full model defined in the main text, neglecting bottom-heaviness. A–
E show the colony axis (Red Arrow) tipping toward the direction of light (Aqua Arrow) over time. The color scheme illustrates the magnitude of p. F shows the
location of colonies in A–E along the swimming trajectory.
dunkel@math.mit.edu
Squirmer model
movie provided by K. Drescher
dunkel@math.mit.edu
the flagellar behavior between the illuminated and shaded sides
of the organism. The mechanism that achieves this asymmetry is
species-dependent, but it is instructive to consider a hierarchy of
ingredients. First, consider a nonspinning spherical organism that
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
normalized flagellar photoresponse for different frequencies of sinusoidal
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
photoresponses were measured fall within the range of R indicated by the
purple box, and the distribution of R can be transformed into an approximate probability distribution function (PDF) of ωr (Inset), by using the noisy
curve of ωr ðRÞ. The purple box in A marks the range of ωr in this PDF (green
except here
Fig. 3. Photoresponse frequency dependence and colony rotation. (A) The
normalized flagellar photoresponse for different frequencies of sinusoidal
stimulation, with minimal and maximal light intensities of 1 and 20 μmol
PAR photons m−2 s−1 (Blue Circles). The theoretical response function (Eq. 5,
Red Line) shows quantitative agreement, using τr and τa from Fig. 2B for
16 μmol PAR photons m−2 s−1 . (B) The rotation frequency ωr of V. carteri
as a function of colony radius R. The highly phototactic organisms for which
photoresponses were measured fall within the range of R indicated by the
essential f
of adaptat
of unsym
organisms
of eyespot
also requi
In Volv
near the
outlined a
sponsive r
could onl
the shade
phototacti
Fig. 4. Heu
phototaxis
that actuall
are gray. (A
posterior-a
rection I. Th
that the flui
value and u
photorespo
After the in
flagellar do
an illustrati
the organis
ientation is
with I. A re
best attaina
Optimal response !
dunkel@math.mit.edu
Phototactic ability decreases with
rotation frequency
Fig. S5. (A) The V. carteri somatic cells at the anterior pole have their orange eyespots facing away from the fluid-mechanical anterior pole. (B) The somatic
cells and eyespots at polar angle θ ¼ 50° from the anterior. (Scale bars: 20 μm.) (C) Illustration of the eyespot placement in the somatic cells and the relation to
the posterior-anterior axis k. In contrast to this schematic drawing, V. carteri colonies consist of thousands of somatic cells, as shown in Fig. 1A of the main text
and as measured in ref. 20.
Fig. S6. (A) Schematic diagram of the apparatus used for the population assay. B and C show distributions of the swimming angle with the light direction σ as
measured for a population at the viscosity of water (B) and at 40 times the viscosity of water (C).
dunkel@math.mit.edu
features: self-propulsion,
steering.
reality,
however, Volvox
bottom-heavy,
is par-The model thus yields insight
in the model
it isInsolely
a measure
of αf . isThe
data from which
several
etics, and photoresponse
ticularly important when the light direction is horizontal. In thisthe phototactic torque and illu
populations
are shown in Fig. 7 and are found to be in quantitacase, we previously observed (33) that the organisms reach a finaltorque must be significantly
tive agreement
the full model for realistic parameters (given
angle αwith
f set by the balance of the bottom-heaviness torque and theachieve high-fidelity phototax
in SI Text)phototactic
and in qualitative
agreement
thethe
reduced
model.abilhe low Reynolds number
torque. We
thereforewith
define
“phototactic
The success
the
reduced speed
modeltoward
highlights
that spinning and
mming speed and angular
ity” of
A¼
ðswimming
the lightÞ∕ðswimming
speedÞ.Conclusion
ulated if the fluid velocity
adaptation are
the
key ingredients
qualitative
Both
models
predict thatfor
as athe
viscosity ηunderstanding
is increased, whileWe have shown how accurate
(31). Phototactic steering
keeping the internal parameters τr and τa fixed, the phototactic
colonial organism lacking a cen
specifying the response of
ability decreases dramatically (Fig. 7). Qualitatively, an increase
ing for the effects of each
in η reduces ωr , which leads to a reduced photoresponse (Fig. 3A)autonomous cells on its anteri
00901107
Phototactic ability decreases with rotation frequency
tive flagellar photoresponse. T
scales of this photoresponse d
Drescher et al.
the characteristic spinning of
tion. Because the organisms
r
quency, the flagellar
orienta
seem to have coevolved to m
mathematical model of phot
the phototactic fidelity decrea
does not spin at its natural fre
assay in which spinning was
viscosity are in excellent agree
This work raises a number
Chief among them are the bioc
scale and the reason for displ
anterior part of the organism.
Fig. 7. The phototactic ability A decreases dramatically as ωr is reduced by
of phototactically active V. car
increasing the viscosity. Results from three representative populations are
the frequency response functi
shown with distinct colors. Each data point represents the average phototacself displays a coincidence o
tic ability of the population at a given viscosity. Horizontal error bars are stanperiod (34, 35), it is natural t
dard deviations, whereas vertical error bars indicate the range of population
same evolutionary lineage, or i
mean values, when it is computed from 100 random selections of 0.1% of the
tic organisms, can be underst
data. A blue continuous line indicates the prediction of the full hydrodynamic
The allometry of the adaptati
model; the red line is obtained from the reduced model. (Inset) αðtÞ from the
fordunkel@math.mit.edu
study. It is also of cons
full and reduced model at the lowest viscosity.
tuning
𝜔
via viscosity increase
Open questions
• not all somatic cells photo-responsive ... why ?
• what determines 𝜏 ?
• chemotaxis vs phototaxis
• effects of (intrinsic) noise
• Chlamydomonas behave similarly ... generic ?
• artificial steering devices
a
dunkel@math.mit.edu