Phototaxis in Volvox
e 22, 2010 U vol. 107 U no. 25 U 11147–11650
In This Issue
PNAS
Jörn Dunkel
Proceedings of the National Academy of Sciences of the United States of America
o the light
www.pnas.org
20.416
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
led to higher organisms with central nervous systems. — T.J.
Tuesday, January 14, 14
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
swimming
on helical
paths relies
alongonly
which
their eyespot
acts asanda
of v0 . Forby
therecording
simulations
webeating
used U ¼
390 μm∕s,of
theeach
mean
of
response
the
frequency
flagellum
of
is able continuously
to give detailedsearching
predictionsspace
of the for
swimming
light antenna
bright characterisspots (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
different
sources
and
orchestrate
coordinated
Author
contributions:
K.D., the
R.E.G.,
and I.T. designed
K.D.,
R.E.G., and I.T.
amplitude
of w0 . For
simulations
we usedresearch;
ωr ¼ 2.3
rad∕s,
Volvoxfrom
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.
as shown in Fig. 7 of the main text.
responses
(7,are
8).exposed to a light stimulus.
face
•TheThe
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 alga
text. To
determine
the relatives
time evolution
of theevolved
system of
the colonial
Volvox
and its
(9), have
a
This article is a PNAS Direct
1
1 ðcos θÞ, as shown by
Legendre
functions,
−P1 ðcosRoad,
θÞ þ 0.25P
2
coupled
equations,
we solved
the coupled
partial
differential
Department
of phototaxis
Applied
Mathematics
and
Theoretical
Physics,
University
of
Cambridge,
Wilberforce
Cambridge
CB3 0WA, United Kingdom
means of
high-fidelity
without
a central
nervous
sysTo whom correspondence should be addressed. E-mail: R.E.Goldstein@damtp.cam.ac.uk.
the
dashed
magenta
line
in
Fig.
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
articlesin
supporting
information
online
at www.pnas.org/lookup/suppl/
simple
dependence
forfor
v0 ðθÞ
gives
qualitatively
similar reby Harry L. Swinney,
University
of Texas,
Austin,
TX,t and
May contains
6,θ 2010
(received
review
January
28, 2010)
verEdited
in Mathematica
(Wolfram
Research)
between
times
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
Alongtothe
evolutionary
path for
from
cells
multicellular
or- the carteri
consists
thousands
ofstimulation.
biflagellated
method
then
solve the equation
IðtÞ
at every
timetostep.
We
the full model,
we used
a close
approximation
to atthethe
βðθÞ
ensured
convergence
of thenervous
results by
choosing
small enough
like somatic
cells
sparsely
distributed
of a passi
ganisms
with a central
system
areaspecies
of intermediate
www.pnas.org/cgi/doi/10.1073/pnas.1000901107
PNAS
∣
June
22,
2010
∣
vol.
107
∣
no.
25
∣ surface
11171–11176
shown in the inset in Fig. 5A of the main text. For the reduced
step
size
δt.
spherical
matrix,
small
of germ ce
complexity that move in ways suggesting high-level coordination,
model, we
used βðθÞextracellular
¼ 0.3, the mean
of theand
βðθÞaused
fornumber
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
about respectively.
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 s−1 , as displayed in
intensity
of
16
μmol
PAR
photons
m
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
that the surface somatic cells act as Proceedings
individuals
but
are orchestrated
axisofof
Volvox
would
relax back
to the
vertical atproperty
a rate at the coloni
of the
National
Academy ofThe
Sciences
thethis
United
States of
America
www.pnas.org
which allows trajectories of the organism
to be reconstructed.
level.bhYet
remains
to beweunderstood
what
by
their relative
position in the
spherical
extracellular
matrix ζ_and
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 of
program
measured
in ref. 19.must take to coordinate the cells and to yield hig
their
photoresponse
functiondefined
to achieve
cothe
maincommon
text, using
the “reduced model”
in thecolony-level
main
fidelity phototaxis in the presence of the steering constraints
Moving
to
the
lightthat range
ordination.
Analysis
models
the minimal
to the
text.
A decomposition
ofofthis
photoresponse
intofrom
spherical
harIn 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 disTo optimize
photosynthesis,
such
as the
Volvox
carteri
swim
with
the
measurements
of the phototactic ability as a function of
puted
“fullormodel”
during
a phototactic
is light,
shown
More
than
a century ago, Holmes (12) proposed that the s
toward
away
from
sunlight.
To execute
thisturn
motion,
knownthe colony
playsbyanthe
adaptive
down-regulation
in response
to
viscosity, we implemented a viscosity dependence in the model.
in Fig. S8, neglecting
bottom-heaviness.
phototaxis,
microorganismdirection
colonies must
facing
a source of light down-regulate their flagell
needs to as
spin
aroundthese
its swimming
andcoordinate
that the response
For this we matic
definedcells
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
appearthe
toviscosity
be
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
phototactic
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
spins
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 of
phototaxis
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)
Cellularquantitative
differentiation in Pleodorina
and thewith
organisation
dictive,
modelcalifornica
consistent
the principles of flu
was
slowed
by increasing
the
fluid
viscosity.
produced
by the
flagella
modeled
3. Sakaguchi H,fluid
Iwasavelocities
K (1979) Two
photophobic
responses
inand
Volvox
carteri. the
PlantmoCell
of colonial Volvocales (translated from German). Arch Protistenkunde 104:292–358.
dynamics, nor are there data on Volvox phototaxis that can b
tion with hydrodynamic equations. Using the model, the auPhysiol 20:909–916.
14. Hoops HJ (1993) Flagellar cellular and organismal polarity in Volvox carteri. J Cell Sci
compared with such a theory. Here we use a combination of e
4.adaptation
Schletz K (1976)
Phototaxis
in∣ Volvox—Pigments
involved
in the∣ perception
of and
light
104:105–117.
thors
identified
aflagella
theoretical
optimal
spinning
frequency
∣ evolution
∣ fluid
dynamics
multicellularity
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 in media with increased viscosities that inhibited
playMutants
key
the phototaxis
mechanism
Multicellular
colonyroles
Volvoxin
carteri.
16. Huskey
RJ (1979)
affecting
vegetative
cell orientation
in Volvox carteri.of
DevV. carteri. By qua
gametes
and Dinophyceae.
Physiol
Plant 11:118–153.
he most
primitive
“eyes”
evolved
longthe
before
brains
the organism’s
to spectral
spin.
According
authors,
ex- and even
Biol 72:236–243.
6. Mast SO (1917)
The relation ability
between
color and to
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
accurately
17. were
Herraez-Dominguez
JV, Gilphototaxis
Garcia de Leon
F, Diez-Sales O, Herraez-Dominguez M
organism.
J periments
Expthe
Zool simplest
22:471–528.
that it
actsByasbetter
a band
pass filter
allows adaptation
to differe
asOA,
before,
suggesting
in VTwo
. organisms
carteri,
flagellar
beating
linked
adaptations.
(2005)
Rheological
characterization
of understanding
two viscosity
grades ofthat
methylcellulose:
An ap7.peared
Sineshchekov
Jung K-H,
Spudich
(2002)
rhodopsins
mediate
phototaxis
tosenseare
on
Earth
(1,
2). JLthat
Many
are
ableandtospinning
and
to the
modeling
of the
behaviour.
Polym
Sci 284:86–91.
low- and high-intensity
in Chlamydomonas
reinhardtii.
Proc Natl
USAfindings proach
light
environments,
minimizes
the
influence
of fast light fluctu
how
simplelight
organisms
coordinate
multicellular
processes,
may provide
insight
into
keythixotropic
evolutionary
steps Colloid
respond
to light
stimuli,
an
ability
essential
toAcad
theScithe
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
and
photosynthesis,
avoidance
of for
photodamage,
the Phys
useRev Letttions,
19. Drescher K, correspond
et al. (2009) Dancing
Hydrodynamic
bound
swimming These measur
details are at http://www.math.uio.no/∼jks/matpiv/.
to Volvox:
the rotation
rate
of states
the of
organism.
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
isoptimized
phototaxis,
inagainst
which
motile
photosynthetic
microorganisms
ad20. 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.
protect
cholera
magmas
evolved
be and
mutually
tuned
and inoptimized
for phototax
tested
the
bacteria
as ainprophylactic
multicellularity:
Flagellar to
motility
germ-somadunkel@math.mit.edu
differentiation
Volvocalean
10.just
Huth their
K (1970)swimming
Movement and path
orientation
of Volvox
aureus
(translated
from
with
respect
toEhrbg.
incident
light
a finely
Whereas low-density
Vibrio
cholerae
Recent
studies
have
argued
that
hygreen
algae.
Am
Nat
167:537–554.
German). Z Pflanzenphysiol 62:436–450.
against V. cholerae virulence in an
Furthermore, we develop a mathematical theory that predic
bacterial accumulations
the hu- relies on sensory inputs from
tuned manner
(3, 4). Thisinsteering
droxyl ions in lunar minerals indicate
infant
mouse
model.
The
researchTuesday, January 14, 14
the phototactic fidelity of Volvox as the rotation rate and oth
Fidelity of adaptive phototaxis
1
In This Issue
T
PNAS
Knut Drescher
Princeton
Idan Tuval
Mediterranean
Institute for
Advanced Studies
Ray Goldstein
Cambridge
dunkel@math.mit.edu
Tuesday, January 14, 14
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 identified 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
that eventually led to higher organisms with central nervous systems. — T.J.
dunkel@math.mit.edu
Tuesday, January 14, 14
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
differentiated Pleodorina californica (D), to the germ–soma differentiated V.
Tuesday, January 14, 14
er-
Volvox carteri
somatic
cell
200 ㎛
cilia
daughter colony
from germ cell
http://www.youtube.com/watch?v=fqEHbJbuMYA
dunkel@math.mit.edu
Tuesday, January 14, 14
Volvox carteri
somatic
cell
cilia
200 ㎛
daughter colony
from germ cell
... and they can dance !
Drescher et al (2010) PRL
Tuesday, January 14, 14
dunkel@math.mit.edu
Volvox carteri
somatic
cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
Tuesday, January 14, 14
dunkel@math.mit.edu
Volvox carteri
200 ㎛
10 ㎛
Chlamydomonas
reinhardtii
dunkel@math.mit.edu
Tuesday, January 14, 14
Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
Tuesday, January 14, 14
10 ㎛
speed ~100 μm/s
dunkel@math.mit.edu
Chlamydomonas
Merchant et al (2007) Science
Tuesday, January 14, 14
dunkel@math.mit.edu
Model organism
for studying
meta-chronal waves
Brumley et al (2012) PRL
dunkel@math.mit.edu
Tuesday, January 14, 14
ertial (dynamic) pressure ⇤U 2 and viscous shearing str
µU/L can be characterized by the Reynolds number4
Swimming at low Reynolds number
R ⌅ U L⇤/µ = U L/⇥.
Example:
Swimming in water with ⇥ = 10
6
m2 /s
• fish/human: L ⌅ 1 m, U ⌅ 1 m/s ⇧ R ⌅ 106 .
R
• bacteria: L ⌅ 1 µm, U ⌅ 10 µm/s ⇧ R ⌅ 10
U L⇥/ ⇥ 1
Geoffrey Ingram Taylor
5
James Lighthill
If the Reynolds number is very small, R ⇥ 1, t
NSE (8) can be approximated by the Stokes equation
0 = µ ⌥2 u
0 = ⌥ · u.
⌥p + f ,
(10
(10
These equations+must
still be endowed
time-dependent
BCs with appropri
initial and boundary conditions, such as ,e.g.,6
Edward Purcell
u(t, x) = 0,
as
Shapere
& Wilczek
PRL
p(t, x)
= p⇥(1987)
,
Tuesday, January 14, 14
|x| ⇤⌃ .
(1
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’
Tuesday, January 14, 14
r
2
Volvox carteri
swimming speed
~ 100 ㎛/sec
PIV
100 ㎛
⇤✓⌧⇡⌧ ⌧ ⇡⇠ ⌅⌥ ↵⇡⌧# ⇥⇡◆ ⌧ ⇣
⌃ ⇠ ⌫ ⇡⇣⌘⌧ ⇣ ⌧ $ ⇣⇢⌥ ⇤⇣⇧⌘⌅ ⇣
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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
Tuesday, January 14, 14
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
(Inset),
create a fluid
flow from
the anterior
to the
e chamber.
of a
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
Ca
Tuesday, January 14, 14
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
Tuesday, January 14, 14
Photo-response at different intensities
0.25Hz
amplitude of the photoresponse for top-hat stimuli of frequency 0.25 Hz, at different stimulus light intensities.
dunkel@math.mit.edu
Tuesday, January 14, 14
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
act
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 coupled
For
a model
ofequapðtÞ 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 ðs it−
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
theu
The
step
function
Hðs
−
hÞ
is
ðs2 Heaviside
− s1 Þ −t∕τ
pðtÞ ¼
− e−t∕τ Þ:
[4]τ r is always a f
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],
peaking and
at a time
t ∼ τr lnðτ
be
are
stable
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
shownsetup.
inlong
Fig.(A)
2A.The
Fig. 1. Geometry of V.flagellar
carteriphotoresponse
and experimental
beating
flaIn Volvox,ifa sst
& , h&periodic
The rotation of Volvox
about
its axis andto
the (p
resulting
). However,
(p,
h)flow
relaxes
gella, two per somatic cell
(Inset),
create
a
fluid
from
the
anterior
to
the
illumination of the photoreceptors suggest an investigation of theof 1 ms (22) is
posterior, with a slightdependence
azimuthal of
component
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 (
, suggest
¼ s e to τ þ
s2 ð1 −
10µm from hðtÞ
cilium
Ca at the 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 angularForfrequency
bar: 100 dependence
μm.) (B) of 2
the above ω
model
this frequency
r . (Scale
Fig. 2. Characteristics of the adaptive photoresponse.
(A) The local flagella-axisstimulation.
generated fluid speed uðtÞ (Blue), measured with PIV
just above
the flagella
photoresponse
is R light
¼ j~p∕sent
s~j, where
p~ an
andoptical
s~ are the
Fourier
Studies
of the
flagellar the
photoresponse
utilize
down
fiber.
−t∕τ
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 a 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)
ωτ
2
1
[5]
Rðωs Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
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 ∣
Tuesday, January 14, 14
a
τr ∕τa is only ∼
approach implicitly averages over several neighboring flagella,1 − latter
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 theinlow
speed at
ju
[and
decrease
uðtÞ],fluid
peaking
Reynolds number associated with followed
flows generated
by V. carteri
scribed
by a slow
relaxation
backbytouðz
The rotation of
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
above the
om
theuðtÞ
anterior
pole. We
found thatwith
a step PIV
up injust
light intensity
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
ry smoothly with the stimulus
lightthe
intensity,
measured in of the stimulus lightRð
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 e
Although the kinetics down
and inbiochemistry
of photorecep
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 th
(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
¼
s
e
with
PIV
just above
the flagella
1
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.
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
followed by a slow relaxation back to zero, as i
þ
metry (PIV). This
current
in photoresponse
Volvox, theshown
decay
constant
like that of the
H ∣ www.pnas.org/cgi/doi/10.1073/pnas.1000901107
11172
flagellar
in Fig.
2A.
a
0
a
a
r
r
a
r
a
Tuesday, January 14, 14
The
rotation
of
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
ms (22);
bioche
termined
constant
β >∼75
0 quantifies
thethe
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
Thea Heaviside
step
function
Hðs
−1
hÞ
is−βpðtÞ,
used
to
ensur
responsible
for
adaptation
(24,
25).
A
system
o
and
longer
adaptation
time
:
a
short
response
time
τ
r
generated
by
V.
carteri
_
¼
−
wher
scribed
by
uðtÞ∕u
τ
p
¼
ðs
hÞHðs
−
hÞ
−
p;
n a time scale τa
0
down
in
light
stimulus
cannot
increase
u above
u0
retical curve (Red) is fromtions
Eq. 4. that
(B) The
times
τr (Squares)
is
consistent
with
the
measured
uðtÞ∕u
_
− h;
τ h ¼ s that
photoresponse
variable
is
flagella-induced
flow
& larg
¼
0
keeps
p
≥
0.
In
these
equations,
the
values
p
Rð
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
e
ured in terms of theciently
&, _
&in
down
light
stimulus
cannot
increase u above
h
).
However,
if
s
increases
fro
(p, h)termined
relaxes to (p
h
¼
s
−
h;
τ
a
constant
β
>
0
quantifies
the
ized by the flow speed0 to s 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
2
are stable and global attractors in the sense th
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
responsible
adaptation
(24,
down
in
light
stimulus
increase
above
ty on a time scale τa
1 − τr ∕τa ðs − s Þ
pðtÞ ¼with the
ðe
−
e & Þ:
tions
that
is
consistent
meas
− τ ∕τvalues p ¼
keeps p ≥ 0. In these equations, 1the
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
photoresponse
þ variable that is large when th
8–20), fluid inertia is negligible and the flagella-induced flow
τr p_ ¼ ðs −decrease
hÞHðsin−flagellar
hÞ − p;activity and
light-induced
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
_
termined
constant
β > 0 quantifies the amplitude
agella-generated flow speed uðtÞ, normalized by the flow speed
h
¼
s
−
h;
τ
auðtÞ∕u . For a model of pðtÞ that captures the tw
stics of the adaptive photoresponse. (A)over
The
localinflagella#30°
0
nder time-independent illumination u0 , and averaged
0
andflagella
τr , we require a second variable hðtÞ, which
eed
(Blue),
measured
above the
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
awhere
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
−t∕τa
1
r
Tuesday, January 14, 14
2
r
−t∕τa
−t∕τr
a
τa ,global
as for
Volvox,
is athe
sharp
areWhen
stableτr ≪
and
attractors
in
tha
as for Volvox,
there sense
is atransien
sharp tran
When
τr ≪ τa , there
†−
_
τ
p
¼
ðs
−
hÞHðs
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
∼
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
t
r dunkel@math.mit.edu
ciently long time followed
underby constant
light
stimul
a slow relaxation back to zero, as i
followed
by
a
slow
relaxation
back
to
zero,
as
in
th
&
&
flagellar
photoresponse
shown
in
Fig.
2A.
(p, h) relaxes to (p , h ). However, if s increases
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 diTuesday, January 14, 14 rection I. The time scale for turning τ t ∼ 3.3 s can be estimated by assuming
Let’s try to be more
quantitative ...
dunkel@math.mit.edu
Tuesday, January 14, 14
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
photoresponses were measured fall within the range of R indicated by the
Tuesday, January 14, 14
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 an
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
best attainable orientation towards the light is
al input for the hydrodynamic models offollowed
photoThe
measured
adaptive
response
ofgenerally
the to
flag
For Volvox,
which
by
a
slow
relaxation
back
near
the
anterior
pole
(Fig.
5),
yet
−t∕τa þ s ð1 − e−t∕τa Þ; the
mathematical
model Because
with predictive
hðtÞ
¼
s
e
escribed
further below.
of thepower.
low
fluid speed just above
the
(Fig.
2
posterior-anterior
axis
is
th
1 colony
2 surface
outlined
above
remains.
However,
havin
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 ash
by uðtÞ∕u
kinetics,
sponsive
region
complicates
heuristi
0
ðs
−
s
Þ
thenegligible
flagellar and
behavior
between the illuminated
and shaded sides
2direct1 an all-or-nothing
−t∕τa −when
r resp
could
only
photoresponse
variable
that
is
ertia is
the flagella-induced
flow
essential
for
high-fidelity
ph
pðtÞ
¼
ðelarge
e−t∕τ
Þ:the
The
rotation
of
Volvox
about
its
axis
of the
organism.
TheFig.
mechanism
achieveslight-induced
this asymmetry
is shaded
−toflagellar
τthe
the
side of
the
r ∕τilluminated
a
decrease
activity
and
e of the
flagellar
activity.
2A showsthat
a typical
of1in
adaptation.
Spinning
may
orientation is drawnsugg
in Fig.
species-dependent,
but it is
ofsuch
theofphototactic
photoreceptors
therea ishierarchy
no
change
flagellar activity.
Thede
of in
unsymmetrical
colony
he photoresponse,
measured
in instructive
terms illumination
of to
theconsider
τa , that
as for
Volvox,
there isthe
a sharp
transieo
When
τr ≪
ingredients. First, consider a nonspinning spherical
organism
termined
constant
β
>
0
quantifies
amplitude
organisms with a restricted
d flow speed uðtÞ, normalized by the flowdependence
speed
†
of
the
photoresponse
on
th
pðtÞ
[and
decrease
in
uðtÞ],
peaking
at
a
time
t
pðtÞ that captures
the tw
in uðtÞ∕u0 . For a modelofofeyespots,
such as Chlamy
endent illumination u0 , and averaged over #30°
followed
byrequire
a slow
relaxation
back
to hðtÞ,
zero,which
as in
stimulation.
For
the
above
model
this
and
τ
,
we
a
second
variable
onse.
(A)
The
local
flagellaalso
required
for
detecting
r
pole. We found that a step up in light intensity
flagellar
photoresponse shown in Fig. 2A.
dimensionless
representation
of the colonies,
hidden interna
In
Volvox
,
in
flagellar
activity
on
a
response
time
scale
τ
r
~
PIV just above the flagella
~
the photoresponse
is
R
¼
j
p
∕
s
j,
The rotation of Volvox
about
its
axis
andwhere
thethe
resu
responsible
for
adaptation
(24,
25).
A
system
of
near
the
anterior
pole
(Fig
covery to baseline activity on a time scale τa
illumination of the photoreceptors suggest an inves
ure of flagellar activity. The
tions
that
theabove
measured
uðtÞ∕u
transforms
ofispofconsistent
and
s,withrespectively.
R
outlined
remains.
Ho
dependence
the
photoresponse
on the
frequency
sponsive region
complicatesd
for
dataset.
Two time
stimulation.
For theτrabove
this
stics ofthis
the adaptive
photoresponse.
(A) The localneglecting
flagellathe Heaviside
in
E
p_ ¼ ðsmodel
−function
hÞHðs
− frequency
hÞ − p;
could
direct pan
all-or-n
eed
uðtÞ (Blue), adaptation
measured with PIVtime
just above the flagella
~ and
the photoresponse is R
¼ jp~only
∕s~j, where
s~ ar
a longer
_the
illuminate
light intensity, serves as a measure of flagellar activity. The
¼
sshaded
− h; to the
transforms of p andτa h
s,
respectively.
R is
well-app
.d(B)
The
τrμm∕s
(Squares)
in the
darktimes
is u0 ¼ 81
for this dataset. Two time
phototactic
is
d
τSI
neglecting the Heaviside
function orientation
in Eq. 1 ω
(see
s
a
a short response time τr and a longer adaptation time
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimffi
Þ
¼
Rðω
where the light
stimulus
sðtÞp
is ffiaffiffiffidimensionless
ght
intensity,
measured
in
s
retical curve (Red) is from Eq. 4. (B) The times τr (Squares)
2eyespot
2 Þð1
ωofsþ
τthe
Fig. 4.that
Heuristic
analysis
phototactic
fid
a the
photoreceptor
input
incorporates
ð1
ω
τ
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
p
Þ
¼
: ar
Rðω
s
r
y smoothly with the stimulus light intensity, measured in
s
phototaxis
models. Photoresponsive
regions
2
2
2
2
The Heaviside step
is used
ð1Hðs
þ aω−
þ ωiss τto
Þen
s τhÞ
r Þð1
thatfunction
actually displays
photoresponse
ina shad
r bars are standard deviations.
down in light stimulus
cannot
increase
above
are gray. (A)
If τa ¼ ∞,
ωr ¼ 0, anduthe
respon
posterior-anterior
axis k will
perfect
¼
keeps p ≥ 0. In these
equations,
theachieve
values
p& an
rection I. The time scale for turning τt ∼ 3.3 s ca
as.org/cgi/doi/10.1073/pnas.1000901107
are
stable
and
global
inthethe
senseside
that
that the attractors
fluid velocity on
illuminated
is
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 increases
(p,
h)
relaxes
to
(p
After the initial transient in A has decayed, the
normalized flagellar photoresponse for different frequencies of sinusoidal
t ≥ 0, flagellar
then for
t > 0 the issolution
is that
to1 sand
down-regulation)
in the region
stimulation, with minimal and maximal light intensities0 of
20 μmol
2 for
Frequency dependence of
photo-response
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
Tuesday, January 14, 14
an illustration, the configuration drawn in this p
the organism would
turn away from−t∕τ
the light, i
−t∕τ
a
a Þ;
hðtÞ ¼ iss1reached
e
þ steering
s2 ð1 −isestopped
ientation
the
at a
dunkel@math.mit.edu
with I. A remedy
against this orientational limita
best attainable
ðs2 orientation
− s1 Þ towards the light is
pðtÞ ¼
ðe−t∕τa − e−t∕τr Þ:
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-anteriorthat
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
maximal
intensities
ofapproxi1 and 20
μmol in a flagellar
is localized
small anterior
region, and
purple minimal
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
from
2B for
(E) Volvoxthe
is bottom-heavy,
because
th
line indicates
the mean), agreement,
showing that the using
response τtime
scalesτ a
and
colonyFig.SI Text).
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
Drescher et al.
PNAS ∣ June 22, 2010
best
attainable orie
Tuesday, January 14, 14photoresponses were measured fall within the range of R indicated by the
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
dunkel@math.mit.edu
Tuesday, January 14, 14
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
line indicates the mean), showing that the response time scales and colony
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 eyespo
also requi
In Volv
near the
outlined a
sponsive r
could onl
the shade
phototacti
Fig. 4. He
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 !
How about spatial structure ?
dunkel@math.mit.edu
Tuesday, January 14, 14
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
∼75
ms
(22);
the
biochemical
origin
of
τ
remains
an
a
varying
surface
If θ on
and
ϕ are
the
polar
and
of the
thousands velocity.
of individual flagella
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
spatially varyingadaptive
surface velocity.response
If θ and ϕ are the
polar
and
loc
and
spatially
varying surface velocity. If θ and ϕ are the polar and
The and
measured
of
the
flagella-generated
^
^
azimuthal
the
surface velocity
u the
ex
mposed
intoangles
u ¼on va θsphere,
þ wrespectively,
ϕ.
We
interpret
u
as
azimuthal
anglesveon a sphere, respectively, the surface velocity u
^
^
^ We interpret u as the vemay be decomposed
into u the
¼ vθ þcolony
wϕ. We interpret
u as the
ve-be decomposed
In
uid speed
just above
surface
(Fig.
2A) can
may
into ube
¼ vθ^ deþ wϕ.
edge of
the
flagellar
layerlayer
(32);
forpractical
practical
locity
at the
edge of the flagellar
(32); for
reasons
ra
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
u
are
made
just
above
that
layer.
ch
In the absence of
a light stimulus,
u ¼ u0isandlarge
we assumewhen
that
In the
the absence
of a light
stimulus,
u ¼ u0 and we assume that the
hotoresponse
variable
that
there
is
a
large
sh
is constant on the colony surface because of the
ratio
on¼
theu
colony
surface
because
of
thev0 ðθÞ∕w
ratio
v0 ðθÞ∕w
0 ðθÞ
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 fixed
ϕ gre
models. Photoresponsive
regions
are
colored
changes
in light intensity,
measurements
of vðθ;ϕ;tÞ atbecause
fixed
ϕ phototaxis
ðθÞ
is
constant
on
the
colony
surface
of
the
of
thousands
of
individual
the
colony
s
0
all
here isshow
nothat
such
change
in
flagellar
activity.
The
empirically
deshow
that
in
each
region,
the
surface
velocity
displays
a
photoactually displays a photoresponse is in shades of red, and s
in each region, the surface velocity displays a photo- that
adopt
a continuum
approximation
in which
there is a
response
of
the
form
shown
in
Fig.
2A
but
that
the overall
tational
order
of
somatic
cells
(9).
Following
step
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 the
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
allowing
the
quantities
β,
p,
and
h
to
depend
on
position:
azimuthal
angles
on
a
sphere,
respectively,
the
surface
allowing
the a
quantities
β, p,
h to that
dependcaptures
on position: therection
model
ofandpðtÞ
two I.time
scales
n uðtÞ∕u
The time
scale forτturning
∼ 3.3^ s can be estimate
0 . For
au ¼ vθ^τ þ
Th
may
be
decomposed
into
w
ϕ.
We
interpret
u
that
the
fluid
velocity
on
the
illuminated
side
is
reduced
to
0.7
each
region,
the
surface
velocity
displays
a
photo− βðθÞpðθ;ϕ;tÞ&:
[6]
¼ u0 ðθÞ½1
nd τr , we require
a ¼second
variable hðtÞ, which
we
define
as
aflagellar
locityand
atuðθ;ϕ;tÞ
the
edge
the
layer (32);
[6] value
uðθ;ϕ;tÞ
u0 ðθÞ½1 − βðθÞpðθ;ϕ;tÞ&:
using
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
Photoresponse
dependence
and 2A
colony
rotation.
(A) The
un
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
uthe
assum
ab
After
theabsence
initial transient
in
A has
decayed,
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
the
stimulus
s
on
the
colony
surface,
we
make
use
of
esponsible
for
adaptation
(24,
25).
A
system
of
coupled
equa^ defined
^ colony
try
the angle
ψðθ;ϕ;
IÞ
¼
n^ · I,
where
n^surface
isturned
the bec
ðθÞ∕w
is constant
on−
the
ratio
v0down-regulation)
flagellar
iscos
inψthe
region
that just
int
ation, with minimal and ^maximal light 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 of
(π),somatic
theinlight
is
directly
an
illustration,
the configuration
drawn
this cells
panel
surprising
s
(Blue
Circles).
The
theoretical
response
function
(Eq.
5,
hotons
m
quantities
β,
p,
and
h
to
depend
on
position:
precise
orientational
(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
a
r
t
a
t
−2 −1
the
organism
turn away from
the light, indicating
that
ne) 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
at
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
velocity
display
try in s can therefore be modeled by a factor Hðcos ψÞ. Superimde
posed on this factor may be another functional dependence on ψ
I. A remedy
against
thisshown
orientational
limitation
would
of the with
fluid
flow,
measured
10 μm
above
the beating
flagella,
following
athatbet
unction ofposed
colony
R. Themay
highly
phototactic
organisms
for which
r be another
response
of
the
form
inforward
Fig.
2A
but
dim
onradius
this factor
functional
dependence
on
ψ
to
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
falleyespot
within sensitivity
the range in
of the
R indicated
by
the
account
for the
forward Fig.
direction,
with
experiments
on
Chlamydomonas
(28)
supporting
a
depenmagnitude
varies
with
θ
(Fig.
5A).
We
thus
model
5.imation
Anterior-posterior
asymmetry.
(A) The
anterior-posterior
model.
(Inset)
βðθÞ is blue (with pcomponen
norv 0 ðθÞ used in the
isfto
localized
in The
a numerical
small
anterior
region,
and the
eyespots
d
Fig.and
5. Anterior-posterior
asymmetry.
(A)be
Thetransformed
anterior-posterior
component
e box,
theexperiments
distribution
of R
can
into
an approxidence
ðψÞ
¼
cos
ψ.
class
of
models
we
consider
for
the
with
on
Chlamydomonas
(28)
supporting
a
depenmalized
to unity),
and
thequantities
mean
β isabove
red.β,(B)the
The
probability
of flagella
to pos
allowing
the
p,
and
h
to
depend
on
of
the
fluid
flow,
measured
10
μm
beating
flagella,
following
a (PDF)
of the fluid
flow, measured
10 μm
above the
beating
flagella,
following
onent
or-nothing
as
they
move
from the
shaded
to
illu
probability
distribution
function
of ω
by
using
theanoisy
dimensionless
is response
therefore
r (Inset),
respond
to light scorrelates
with
the
size
of dashed
the somatic
cell
eyespots.the
Thethe
dence
f
ðψÞ
¼
cos
ψ.
The
class
of
models
we
consider
for
the
step
up
in
illumination
at
time
t
¼
0
s.
The
line
indicates
approx
W
step
up
in
illumination
at
time
t
¼
0
s.
The
dashed
line
indicates
the
approxwing
a
(D) Measurements
offlow
the occurs
eyespot
(Orange)
placement
yield κ
light-induced
decrease in fluid
beyond
the region
of flagellar
The
purple
box
in
A
marks
the
range
of
ω
in
this
PDF
(green
of ωimation
r ðRÞ. dimensionless
r
s
is
therefore
to v 0 ðθÞ used in the numerical model. (Inset) βðθÞ is blue (with p norusedofinthe
the
numerical
model.
(Inset)−βðθÞ
is blue (with
p no
imation
to v 0 ðθÞ
ve
βðθÞpðθ;ϕ;tÞ&:
uðθ;ϕ;tÞ
¼ udynamics.
^ bottom-heavy,
pprox0 ðθÞ½1
response
because(E)
nonlocality
of fluid
ψÞ:
[7]
sðθ;ϕ;
IÞ
¼
f
ðψÞHðcos
SI
Text).
Volvox
is
because
the
center
of
mass
dicates
the
mean),
showing
that
the
response
time
scales
and
colony
malized to unity), and the mean β is red. (B) The probability of flagella malized
to
to unity), and the mean β is red. (B) The probability of flagella t
p nor^
from the geometric center of the colony as indicated.
ψÞ:
[7]
sðθ;ϕ;
IÞ
¼
ðψÞHðcos
respond to are
lightmutually
correlates with
the size
offmaximize
the
somatic the
cell eyespots.
The
on
frequency
optimized
to
photoresponse.
ella to
above
specification
ofsize
the ofdynamics
the
Withto the
respond
light
correlates
with
the
cellsurface
Th
The
measured
βðθÞthe
is
shown
in somatic
theofinset
ineyespots.
Fig.it 5A.
light-induced decrease in fluid flow occurs beyond the region of flagellar
may
be
sufficient
for
Volvox
in
natural
environments,
because
s. The
velocity, the
angular
of the
colony
is (31)
light-induced
decrease
in velocity
fluid flow
occurs
beyond
the region of 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,
would
robustly
navigate
Volvox
closer
to
the
light,
even though
the we m
agellar
Zfluid dynamics.
response
because
of
the
nonlocality
of
wh
velocity,
the
angular
velocity
of
the
colony
is
(31)
1
3
^
^
erTuesday,
et al. January
PNAS
∣
June
22,
2010
∣
vol.
107
∣
no
^
organism
does
not
swim
directly
toward
the
light.
The
orientathe
angle
ψðθ;ϕ;
IÞ
defined
through
cos
ψ
¼
−
n
·
I,
wh
14, 14
g^ × k^ −
n^ × uðθ;ϕ;tÞdS;
[8]
ΩðtÞ ¼
an
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
^ definedinput
^the
;ϕ;IÞ
through
ψ ¼ −n^ · I,
where
n^ isdirectionality.
the
hotoreceptor
thatcos
incorporates
eyespot
o the
surface.step
When
ψ ¼ Hðs
0 (π),
the
light to
is ensure
directlythat a step
he
Heaviside
function
− hÞ
is used
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
angleanterior.
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
Tuesday, January 14, 14
Basic ingredients of
a‘full’ model
• self-propulsion
• bottom-heaviness
• photo-response kinetics
• photo-response spatial variation
dunkel@math.mit.edu
Tuesday, January 14, 14
onstant
β a>light
0 quantifies
the^inamplitude
of
the
decrease
posed
toangle
stimulus.
magnitude
varies
with
θ
(Fig.
5A).
We
thus
mode
e
instead
continuously
changes
with
the
angle
at
ugh
theSciences
measured
βðθÞ
is ^and
shown
the
inset
in
Fig.
5A.
th
the
atResearch
he
Council
(K.D.),
the
Engineering
and
Biological
Sciences
trythe
inksrotational
can
therefore
be of
modeled
by
a•factor
Hðcos
ψÞ.posteriorSuperimdrag
the
sphere
(20).
The
second
term
is
where
g
and
are
the
directions
of
gravity
and
the
sence
of
a
light
stimulus,
u
¼
u
and
we
assume
that
the
and
the
ht.
Together
with
an
appropriate
e
The
θ
dependence
of
the
0
^ sposed
For
aappropriate
model
of
pðtÞ
that
captures
thefactor
two
time
scales
τSciences
pled
equations
that
make
upthe
the
model
are
given
in
the
responsible
for
phototactic
steering,
w
^
allowing
the
quantities
β,
p,
and
h
to
depend
on
p
a and
where
g
and
k
are
directions
of
gravity
the
posterioran
o
define
the
stimulus
on
the
colony
surface,
we
make
use
of
rientads
program
of
the
Biotechnology
and
Biological
Research
Council,
on
this
may
be
another
functional
dependence
on
ψ
espots
receive
light.
Together
with
an
appropriate
responsible
for
phototactic
steering,
where
the
integral
is
taken
we
approximated
v
ðθÞ
by
anterior
axis,
respectively.
The
first
term
in
Eq.
8
arises
from
athis
directionality
leads
to
the
uðθ;ϕ;t
0
Þ∕w
ðθÞ
is
constant
on
the
colony
surface
because
of
the
require
a
second
variable
hðtÞ,
which
we
define
as
a
To
determine
the
time
evolution
of
the
system
of
responsi
^
^
0anterior
over
of
the
sphere
of
radiu
ty
leads
theaxis,
respectively.
The
first
insurface
Eq.
arises
from
ere
angle
ψðθ;ϕ;
IÞ
defined
cos
ψ¼
−
n^ term
· I,the
where
n^ in
is8Legendre
the
theto
Human
Frontier
Science
Program
(I.T.),
the
US
Department
of
Energy,
tothrough
account
for
the
eyespot
sensitivity
the
forward
direction,
1ð
e
if,
as
over
the
surface
of
the
sphere
of
radius
R.
In
a
reference
frame
functions,
−P
bottom-heaviness
and
represents
aFollowing
balance
between
thea depentorque 1
as
ement
(Fig.
4D),
this
directionality
leads
to
the
mmetry
between
illuminated
and
uations,
we
solved
the
coupled
partial
differential
ess
representation
of
the
hidden
internal
biochemistry
rientational
order
of
somatic
cells
(9).
step
with
experiments
on
Chlamydomonas
(28)
supporting
ve
and
the
Schlumberger
Chair
Fund
(R.E.G.).
illuminated
and
bottom-heaviness
and
represents
a
balance
between
the
torque
ðθÞ½1
−
βðθÞpðθ;ϕ;tÞ&:
uðθ;ϕ;tÞ
¼
u
where
the
Volvox
at the origin
with
a
normal
to the surface.
When
ψ ¼ is0 at
(π),
the
light
is
directly
over
the
0theisdashed
of
the
where
the
Volvox
the
origin
with
a
fixed
orientation,
the
light
magenta
line
ior-posterior
component
e
that
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
pðθ;ϕ;tÞ
and
hðθ;ϕ;tÞ
numerically
with
a
built-in
solThe
measured
βðθÞ
is
efor
for
adaptation
(24,
25).
A
system
of
coupled
equaentation
toward
the
light
has
been
dence
f
ðψÞ
¼
cos
ψ.
The
class
of
models
we
consider
for
the
^
^
he
light
has
been
a
response
asymmetry
between
illuminated
and
that
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
^
^
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.
ngle
at
simple
sin
θ
dependence
fo
ing
flagella,
following
a
where
th
hematica
(Wolfram
Research)
between
times
t
and
at
is
s
consistent
with
the
measured
uðtÞ∕u
and
the
rotational
drag
of
the
sphere
(20).
The
second
term
is
dimensionless
s
is
therefore
To
define
the
stimul
0signalling
2þ
and
the
rotational
drag
of
the
sphere
(20).
The
second
term
is Biol
yes
The
measured
βðθÞ
is
shown
in
the
inset
in
Fig.
5A
23.
Tamm
S
(1994)
Ca
channels
and
in cilia
and
flagella.
Trends
Cell
n
s
can
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
Superimne
indicates
the
approxThe
above
coupled
equations
can
be
solved
numerically
(see
The
above
coupled
equations
can
b
topriate
in
each
region,
the
surface
velocity
displays
a
photosults.
We
assume
that
w
s
until
perfect
orientation
toward
the
light
has
been
0
^
to
the
integral
in
the
equation
for
Ω,
we
used
an
Euler
e
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 nornatural
can
be
op4:305–310.
responsible
for
phototactic
steering,
where
the
integral
is
taken
^
To
define
the
stimulus
s
on
the
colony
surface,
on
this
factor
may
be
another
functional
dependence
on
ψ
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
• to
βðθÞ,
the responsivity
ofwe
th
_ ¼ equation
τ
p
ðs
−
hÞHðs
−
hÞ
−
p;
[1]
SI
Text),
e.g.,
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
for
IðtÞ
at
every
time
step.
We
ecreated
probability
of
flagella
to Jülicher
89.
unit
normal
to
the
sur
over
the
surface
of
the
sphere
of
radius
R.
In
a
reference
frame
by
the
24.
Friedrich
BM,
F
(2007)
Chemotaxis
of
sperm
cells.
Proc
Natl
Acad
Sci
USA
^
^
The
a
over
the
surface
of
the
sphere
of
radius
R.
In
a
reference
frame
0),
which
may
be
created
by
the
with
the
light
direction.
It
is
interesting
to
consider
two
special
^
ccount
for
the
eyespot
sensitivity
in
the
forward
direction,
the
angle
ψðθ;ϕ;
IÞ
defined
through
cos
ψ
¼
−
n
·
I,
w
the
full
model,
we
used
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
θh;
(Fig.
5A).
We
thus
model
uðθ;ϕ;tÞ
by
104:13256–13261.
above
(behind)
a
given
h
¼
s
−
[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
cnisms,
in
natural
environments
can
be
opa
where
the
Volvox
is
at
the
origin
with
a
fixed
orientation,
the
light
cases
of
the
model
class
outlined
above.
In
the
biologically
faithshown
in
the
inset
in
Fig.lig5
convection,
or
wind-driven
experiments
on
Chlamydomonas
(28)
supporting
a
depenunit
normal
to
the
surface.
When
ψ
¼
0
(π),
the
d. the region of flagellar
SI
Text),
cases
of
the
model
class
outlined
abov
as
been
25. Spiro
PA, Parkinson
JS, ^Othmer
(1997)
A on
model
of excitation
and
adaptation
in
^HG
^position:
velocity,
the
angular
velocity
of
the
colony
is
(31)
nphototaxis
the
quantities
β,
p,
and
h
to
depend
even
^
try
in
s
can
therefore
b
direction
evolves
as
d
I∕dt
¼
−Ω
×
I.
mics.
ful
“full
model,”
we
use
the
measured
βðθÞ
and
the
realistic
eyedirection
evolves
as
d
I∕dt
¼
−Ω
×
I.
model,
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
ful
“full
model,”
we
use
the
measured
bacterial
chemotaxis.
Proc
Natl
Acad
Sci
USA
94:7263–7268.
with
the
light
stimulus
sðtÞ
is
a
dimensionless
measure
of
the
e to a property
posed
on
this
factor
m
The
above
coupled
equations
can
be
solved
numerically
(see
spot
directionality
f
ðψÞ
¼
cos
ψ.
In
the
“reduced
model,”
we
1
3
full
model.
The
above
coupled
equations
can
be
solved
numerically
(see
ensionless
s isalso
therefore
try
in
s
can
therefore
be
modeled
by
a
factor
Hðcos
he
model
can
be used
to(1983)
determine
the
organism
^of−directionality
experiments
is
due
to
a
property
26.
Walsh
P,
Legendre
L
Photosynthesis
natural
phytoplankton
under
high
^
^
g
×
k
n
×
uðθ;ϕ;tÞdS;
[8]In
ΩðtÞ
¼
her
nearby
organisms,
convection,
or
wind-driven
spot
f
ðψÞ
¼
cos
ψ.
t
be
opptor
input
that
incorporates
the
eyespot
directionality.
pChlamydomonas
3
to
account
for
the
eye
cases
of
consider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
en
•
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
τ
8πR
ðθÞ½1
− βðθÞpðθ;ϕ;tÞ&:
[6]
uðθ;ϕ;tÞ
¼
u0fluctuations
r
a functional
ronments,
because
it
bhStone
velocity
U,
via
another
result
from
and
frequency
light
simulating
those
induced
by be
sea another
surface
waves.
Limnol depe
posed
on
this
factor
may
cellular
ancestor
Chlamydomonas
side
step
function
Hðs
−
hÞ
is
used
to
ensure
that
a
step
consider
only
a
light-shadow
respons
om
their
center
e
by
the
AlluC
1—and
use
the
mean
of
the
measured
βðθÞ—i.e.,
βðθÞ
¼
0.3.
with
experiments
on
.ight,
A
mechanism
that
can
counteract
phototaxis
even
For
the
simulations,
we
^
with
the
light
direction.
It
is
interesting
to
consider
two
special
even
though
the
with
the
light
direction.
It
is
interesting
to
consider
two
special
ψÞ:
[7]
sðθ;ϕ;
IÞ
¼
f
ðψÞHðcos
)
Oceanogr
28:688–697.
ful
“full
^
to
account
for
the
eyespot
sensitivity
in
the
forwa
^
where
g
and
k
are
the
directions
of
gravity
and
the
posteriorof
mass
is
offset
from
their
center
asymmetry.
(A)
The
anterior-posterior
component
,
because
it
ght
stimulus
cannot
increase
u
above
u
ure
srior-posterior
due
to
clustern light.
use
the
mean
of
the
measured
0of1—and
-driven
other
features
are
shared
between
the
models.
intensity
of
16
μmol
PAR
e
The
orientadence
f
ðψÞ
¼
cos
ψ.
T
cases
of
the
model
class
outlined
above.
In
the
biologically
faith27.
Jennings
HS
(1901)
On
the
significance
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
with
experiments
on
Chlamydomonas
(28)
support
sured
βðθÞ
is
shown
in
the
inset
in
Fig.
5A.
spot
dir
low,
measured
10
μmisZ
above
the
beating
flagella,
following
a
¼
0
and
h
¼
s
0.
In
these
equations,
the
values
p
eads
to
a
torque
ottom-heaviness
due
to
cluster1
n
A
phototactic
turn
of
a
hypothetical
non-bottom-heavy
Volvox
Fig.
2B
of
the
main
text.
n
be
overcome
if,
as
other
features
are
shared
between
th
h06.
the
above
specification
of
the
dynamics
of
the
surface
is
even
dimensionless
seyeis ther
35:369–378.
1 model,”
ful
“full
model,”
we
use
the
measured
βðθÞ
and
the
realistic
eyevs
terior-posterior
component
ful
“full
we
use
the
measured
βðθÞ
and
the
realistic
bottom-heaviness
and
represents
a
balance
between
the
torque
dence
freduced
ðψÞ
¼ cos
ψ.
The
class
of
models
we con
mination
at
time
t ¼unicellular
0 on
s.inThe
dashed
line
indicates
the
approxand
global
attractors
the
sense
that,
after
a
suffihares
with
its
ancestor
Chlamydomonas
UðtÞ
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
model
is
shown
in
Fig.
6,
indicating
an
or
(Fig.
4E)
and
leads
to
a
torque
y
consider
28.
Schaller
K,
David
R,
Uhl
R
(1997)
How
Chlamydomonas
keeps
track
of
the
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
ating
flagella,
following
a
ia- used in the
4πR
spot
directionality
f
ðψÞ
¼
cos
ψ.
In
the
“reduced
model,”
we
ðθÞ
numerical
model.
(Inset)
βðθÞ
is
blue
(with
p
nordimensionless
s
is
therefore
,
the
pair
g
time
under
constant
light
stimulus
s
es
with
the
angle
at
^ has
^ where
totaxis
inIÞ
Volvox
1 n
a^ center
flagellaless
Volvox
that
is
intricate
link
between
organism
rotation,
adaptation,
and
reached
the
right
phototactic
orientation.
Biophys
J 73:1562–1572.
as
^
Z
ψðθ;ϕ;
defined
through
cos
ψ
¼
−
·
I,
n
is
the
sðθ
Their
center
of
mass
is
offset
from
their
gae:
direction
with
the
vertical
on
a
and
the
rotational
drag
of
the
sphere
(20).
The
second
term
is
d
line
indicates
the
approxconsider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
simulated
by
the
reduced
model
is
sho
omonas
1—and
u
&appropriate
& ).
consider
only
a
light-shadow
response
asymmetry—i.e.,
f
ðψÞ
¼
nity),
and
the
mean
β
is
red.
(B)
The
probability
of
flagella
to
1
3
with
an
,
h
However,
if
s
increases
from
s
for
t
<
xes
to
(p
self-propulsion,
29.
Schaller
K,the
Uhl
Rresponsible
(1997) In
A microspectrophotometric
study
of
the
shielding
properties
The
axis
of
this
Volvox
wou
1 steering,
steering.
reality,
however,
Volvox
is
bottom-heavy,
which
is
parfor
phototactic
where
the
integral
is
taken
ws
trajectories
of
organism
to
be
reconstructed.
coer
^
set)
βðθÞ
is
blue
(with
p
northful
theory
of
phototaxis
in
Volvox
All
1—and
use
the
mean
of
the
measured
βðθÞ—i.e.,
βðθÞ
¼
0.3.
mal
to
the
surface.
When
ψ
¼
(π),
the
light
is
directly
^0use
^0×the
g
×
k
−
n
uðθ;ϕ;tÞdS;
[8]
ΩðtÞ
¼
intricate
link
between
organism
r
center
ght
correlates
with
the
size
of
the
somatic
cell
eyespots.
The
^
For
Volvox,
this
bottom-heaviness
is
due
to
clusterAll
1—and
the
mean
of
measured
βðθÞ—i.e.,
βðθÞ
¼
0.3.
onality
leads
to
the
_
3
other
fe
ψÞ:
sðθ;ϕ;
IÞ
¼
f
ðψÞHðcos
t
≥
0,
then
for
t
>
the
solution
is
of eyespot
and
cell
body
in
Chlamydomonas
. Biophys
Jdirection
73:1573–1578.
ζR.¼
−
sinðζÞ∕τ
.frame
For
the
ticularly
important
when
the
light
is
horizontal.
In
this
the
above
speci
With
over
the
surface
of
the
sphere
of
radius
In
a
reference
. photoresponse
τ
8πR
bh
on
of
the
photoresponse
pðθ;ϕ;tÞ
is
plotted
in
Fig.
6
of
e
probability
of
flagella
to
rbh
other
features
are
shared
between
the
ast
four
features:
self-propulsion,
decrease
in
fluidand
flow
occurs
beyond
the
region
ofmodels.
flagellar
clustersteering.
In
reality,
however,
Volvox
is
b
een
illuminated
hind)
a
given
surface
patch.
The
light-shadow
asymmeother
features
are
shared
between
the
models.
30.
Durham
WM,
Kessler
JO,
Stocker
R
(2009)
Disruption
of
vertical
motility
by
shear
where
the
Volvox
is
at
the
origin
with
a
fixed
orientation,
the
light
ells
in
the
posterior
(Fig.
4E)
and
leads
to
a
torque
measured
in
ref.
19.
case,
we
previously
observed
(33)
that
the
organisms
reach
a
final
velocity,
the
angular
v
A
pho
ext,
using
the
“reduced
model”
defined
in
the
main
somatic
cell
eyespots.
The
e
ause
of
the
nonlocality
of
fluid
dynamics.
A
phototactic
turn
of
a
hypothetical
non-bottom-heavy
Volvox
^¼
−t∕τ
−t∕τ
ard
theand
light
has
been
^ layers.
^non-bottom-heavy
nse
kinetics,
and
photoresponse
the
above
specification
of
thetorque
dynamics
of
With
a þ
a Þ;
torque
ticularly
important
when
the
light
dire
^
triggers
formation
of
thin
phytoplankton
Science
323:1067–1070.
re
g
k
are
the
directions
of
gravity
and
the
posteriordirection
evolves
as
d
I∕dt
¼
−Ω
×
I.
hðtÞ
s
e
s
ð1
−
e
[3]
A
phototactic
turn
of
a
hypothetical
Volvox
n
therefore
be
modeled
by
a
factor
Hðcos
ψÞ.
Superimset
by
the
balance
of
the
bottom-heaviness
and
the
angle
α
1
2
omposition
of
this
photoresponse
into
spherical
harnd the
region
ofby
flagellar
f
gn
the
swimming
direction
with
the
vertical
on
a
a
simulate
simulated
the
reduced
model
is
shown
in
Fig.
6,
indicating
an
In
order
to
compare
the1 re
31.
Stone
HA,
Samuel
ADT
(1996)
Propulsion
of
microorganisms
by
surface
distortions.
eus
velocity,
the
angular
velocity
of
the
colony
is
(31)
The
above
coupled
equations
can
be
solved
numerically
(see
al
on
a
rior
axis,
respectively.
The
first
term
in
Eq.
8
arises
from
case,
we
previously
observed
(33)
that
eynolds
number
simulated
by
the
reduced
model
is
shown
in
Fig.
6,
indicating
an
phototactic
torque.
We
therefore
define
the
“phototactic
abilðθ;ϕÞ
is given
inopFig.
S7.
The photoresponse
pdependence
comthis intricate
factor
may
be
another
functional
on
ψ
amics.
ΩðtÞaxis
¼of th
onments
can
be
ox
link
between
organism
rotation,
adaptation,
and
ðs
−
s
Þ
Phys
Rev
Lett
77:4102–4104.
SI
Text),
e.g.,
to
determine
the
angle
αðtÞ
of
the
organism
with
the
measurements
Z
∼
14
s
(20).
A
faithful
theory
of
phototaxis
in
Volvox
2
1
intricate
−t∕τ
−t∕τ
om-heaviness
and
represents
a
balance
between
the
torque
ed
and
angular
Volvox
set
by
the
balance
of
the
bottom
angle
α
a between
r Þ: turn
ity”
A
¼
ðswimming
speed
toward
the
lightÞ∕ðswimming
speedÞ.
τ
intricate
link
organism
rotation,
adaptation,
and
he
“full
model”
during
a
phototactic
is
shown
ficient
for
Volvox
in
natural
environments,
because
it
pðtÞ
¼
ðe
−
e
[4]
f
bh
y
be
created
by
the
nt
for
the
eyespot
sensitivity
in
the
forward
direction,
1
3
om
n,
32. Blake1 In
JR
(1971)
A
spherical
envelope
approach
to
ciliary
propulsion.
J
Fluid
Mech
with
the
light
direction.
It
is
interesting
to
consider
two
special
steering.
reality,
however,
Volvox
is
bottom-heavy,
which
is
par−
τReynolds
^ − η dunkel@math.mit.edu
viscosity,
implemented
a
he
fluid
velocity
r ∕τ
aIn
acts
when
the
posterior-anterior
axis
is
not
parallel
to
gravity
Both
models
predict
that
as isthe
viscosity
is3weincreased,
while
^
^
g
×
k
n
×
uðθ;ϕ;tÞdS
ΩðtÞ
¼
axis.
In
the
low
number
ulsion,
neglecting
bottom-heaviness.
re
include
at
least
four
features:
self-propulsion,
phototactic
torque.
We
therefore
def
steering.
reality,
however,
Volvox
bottom-heavy,
which
is
parstly
navigate
Volvox
closer
to
the
light,
even
though
the
steering.
tion,
or
wind-driven
46:199–208.
^
cases
of thethe
model
class
outlined
above.
In
the biologically
faithe
riments
on
Chlamydomonas
(28)
supporting
a
depenticularly
important
when
light
direction
is
horizontal.
In
this
^
where
g
and
k
are½u&thη
τ
8πR
vironments,
because
it
Forτ this
we defined
u¼
totactic steering
fixed,
the phototactic
keeping the internal parametersbhτ and
Hydrodynamic model
Tuesday, January 14, 14
‘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
showsofthe
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.
of the fidelity of phototaxis in Volvox and that a quantitative
understanding
can be obtained
if a realistic
of the fidelity
of phototaxis
in eyespot
Volvoxdirectionand that a qu
dunkel@math.mit.edu
ality
and anterior-posterior
response
asymmetry
are
included. eyespot
understanding
can
be
obtained
if
a
realistic
These models further illustrate that if all somatic cells were
SICAL
and therefore a reduced phototactic torque. The sharp transition
Fig. 7 occurs
when the phototactic
ore a inreduced
phototactic
torque.torque
Thebecomes
sharp comparable
transition
to the other
torques in thetorque
system. The
simulations
neglected
curs when
the phototactic
becomes
comparable
torques
Tuesday,
Januarydue
14, 14to ambient fluid motion and included only the
‘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
Tuesday, January 14, 14
Squirmer model
movie provided by K. Drescher
Tuesday, January 14, 14
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
dunkel@math.mit.edu
Tuesday, January 14, 14
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
line indicates the mean), showing that the response time scales and colony
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 eyespo
also requi
In Volv
near the
outlined a
sponsive r
could onl
the shade
phototacti
Fig. 4. He
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 !
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
Tuesday, January 14, 14
ality of the population swimming behavior (see SI Text), whereas
features: self-propulsion, steering. In reality, however, Volvox is bottom-heavy, which is par-The model thus yields insight
in the model it is solely a measure of αf . The data from several
etics, and photoresponse
ticularly important when the light direction is horizontal. In thisthe phototactic torque and illu
populations
are
in Fig.
7 and(33)
arethat
found
be in quantitacase,
weshown
previously
observed
the to
organisms
reach a finaltorque must be significantly
tive agreement
with
the
full
model
for
realistic
parameters
(given
and theachieve high-fidelity phototax
angle αf set by the balance of the bottom-heaviness torque
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
adaptation are
the
key ingredients
qualitative
ulated if the fluid velocity
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
autonomous cells on its anter
ing for the effects of each in η reduces ωr , which leads to a reduced photoresponse (Fig. 3A)
Phototactic ability decreases with rotation frequency
000901107
Tuesday, January 14, 14
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 wa
viscosity are in excellent agre
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
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
Outlook & 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
Tuesday, January 14, 14