(Some)
Numbers and Maths
in Biology
Jörn Dunkel E17-412
dunkel@math.mit.edu
http://bionumbers.hms.harvard.edu/
dunkel@math.mit.edu
Phylogenetic tree
source: wiki
dunkel@math.mit.edu
DNA
source: wiki
•
•
http://ghr.nlm.nih.gov/handbook/basics/dna
DNA contour length in bacteria: ~1.5mm
Length of DNA in nucleus of mammals: ~ 2m
dunkel@math.mit.edu
DNA = biopolymer pair
~ 3m per cell
!
~ 10^14 cells/human
!
> max. distance between
Earth and Pluto
(~50 AU = 7.5 x 10^12 m)
dunkel@math.mit.edu
DNA packaging in eukaryotes
dunkel@math.mit.edu
source: wiki
mass
dunkel@math.mit.edu
source: wiki
dunkel@math.mit.edu
Prokaryotes
http://www.sci.sdsu.edu/~smaloy/MicrobialGenetics/topics/chroms-genes-prots/genomes.html
dunkel@math.mit.edu
Typical length scales
http://www2.estrellamountain.edu/faculty/farabee/BIOBK/biobookcell2.html
dunkel@math.mit.edu
Species estimates
•
estimated number of eukaryotic species on Earth:
8.7 million (Nature, 2011)
•
•
•
•
undiscovered: 86% land spec. & 91%marine spec ~ 300,000 plant species
prokaryotic biomass ~ eukaryotic biomass
oldest known fossilized prokaryotes from 3.5
billion years ago
dunkel@math.mit.edu
Size-Complexity relation
dunkel@math.mit.edu
Unicellular organisms
Algae
Chlamydomonas reinhardtii (K. Drescher)
Bacteria
Caulobacter crescentus (Gitai lab, Princeton)
Text
!
size ~ 10µm
doubling time ~ 5-8h
size ~ 1µm doubling time ~ 2h
Amoeba
size ~ 1mm doubling time ~ 1d
dunkel@math.mit.edu
evolution from
unicellular to multicellular ?
dunkel@math.mit.edu
Volvox carteri
200 ㎛
10 ㎛
Chlamydomonas
reinhardtii
dunkel@math.mit.edu
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
dunkel@math.mit.edu
how do organisms
achieve locomotion ?
dunkel@math.mit.edu
lds Numbers in Biology
Reynolds numbers
number is dimensionless group that characterizes the ratio o
fined as
⇥U L
UL
Re =
=
µ
density of the medium the organism is moving through; µ is t
; is the kinematic viscosity; U is a characteristic velocity of
stic length scale. When we discuss swimming biological organ
eatures that are moving through water (or through a fluid with
hose of water). This means that the material properties µ and
ber is roughly determined by the size of the organism.
e characteristic size of the organism and the characteristic sw
rule-of-thumb, the characteristic locomotion velocity, U , in bi
y U L/second e.g. for people L 1 m and we move
at U 1
dunkel@math.mit.edu
E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
dunkel@math.mit.edu
labeled,
non-tumbling
E. lines
coli due
as they
through
a suspenexistence
of closed stream
to the swam
presence
of the wall.
(H) The flow field of an E. coli “pu
presence
ofthe
the
wall.tracer
(H) The
flow field
of an E.measured
coli “pusher”
decays
much
when
ab
dthe
by subtracting
best-fit
dipole
from
field.
The
presence
of faster,
the flagella
However,
the
force
sionsince
of fluorescent
particles.
far from
it is partially
cancelled
by the
the experimentally
flowFor
fieldmeasurements
of its “puller” image.
Results
w
of itswe
“puller”
image.
d. field
Atwalls,
distances
rfocused
<6µ
mon
theadipole
model
overestimates
the
bacterial
flow
field.
(E)
Experimentally
measured
flow to t
plane 50 µm from the top and bottom
tancesurfaces
2 µm parallel
to sample
the wall. chamber,
(F) Best fitand
force-dipole
model,
(G) residual
field.
Notewhere
the
Bacterial
fl
cell
body,
th
of the
recorded
∼ 2 and
terabytes
of flow
4
(non-tumbling
HCB
e flowmovie
fieldResults
ofdata.
an E. In
coli
“pusher”
decays
much
faster,
a bacterium
swims close
thecule
surface,
fortothe
length
of th
theflow
meas
fi
this
data we
identified
∼437)
10when
rare
events when
achieved
by
fitting
theByTo
measured
andminisbest-fit
force
d
decays
of
cellsBacterial
swam in the
for > surfaces.
1.5 s.
tracking
labeled,
no
flowfocal
fieldplane
far from
resolve the
the
atspeed
variable
location
fluid
tracers
in
each
of
the
rare
events,
relating
their
position
of
the
ced
decays
of
the
flow
u
with
cule flow
field
created
by
individual bacteria, we tracked gfp- sion of
fluor
m surfaces.
To
resolve
the
minisfield (r >field
8 µm).
and labeled,
velocity to
the
position
and
orientation
of
the
bacterium,
disp
non-tumbling
E.
coli
as
they
swam
through
a
suspenof
the
cell
body
(Fig.
1D)
illustra
walls,
we
fo
dividual
bacteria,
we
tracked
gfpthe
measured
and
best-fit
force
dipole
field
(Fig.
1C).
The
the
specific
fitting
r
and performing an ensemble average over all tracers, we reHowever,
sion
of
fluorescent
tracer
particles.
For
measurements
far
from
field
displays
the
characteristic
1/1
dipole
length
ℓ =of
decays
of
the
flow
speed
u
with
distance
r
from
thesurfaces
center
i asministhey the
swam
through aflow
suspensolved
time-averaged
field in the E. coli swimming
he
measured
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is consis
movie
data
However,
the
force
dipole flow
sign
oftothe
cell
(Fig.
1D) illustrate
that
flow
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±the
5 measured
ticles.
For
measurements
far
from
ckedplane
gfpcell
body
surfaces
of
the
sample
chamber,
and recorded
terabytes
ofresistive
2 ∼ 2their
and
force
t
µm/s.
As
E.
coli
rotate
about
their
swimming
direction,
cells
swam
field
displays
the
characteristic
1/r
decay
of
a
force
dipole.
measured
flow
to
the
side
of
the
4
suspenea 50
µmmovie
fromdata.
the top
and
bottom
for
the be
le
In
this
data
we
identified
∼
10
rare
events
when
note
that
in
the
flow field
in
three
dimensions
isbody,
cylindrically
fluid
tracer
However,
the
force
dipole
flow
significantly
overestimates
the
cell
where
the
flow
magnitu
sber,
fartime-averaged
from
achieved
and
recorded
∼
2
terabytes
of
cells swam
in the focal plane
forall>components
1.5 s. Byoftracking
thebehind the cent
µm
symmetric.
Our
measurements
capture
this
measured
flow to when
the side of
the
cell
body,ofand
behind
the
4
and
velocit
didentified
bottom
for
the
length
the
flagellar
bund
at
variab
fluid
drag
on
the fla
∼
10
rare
events
fluid
tracers
in
each
of
the
rare
events,
relating
their
position
cylindrically symmetric flow, except the azimuthal flow due to
cellof body,
where
the
floworientation
magnitude
u(r)
isof nearly
constant
and
perform
abytes
of
field
(r fo
>
achieved
by
fitting
two
opposite
and
velocity
to
the
position
and
of
the
bacterium,
the
rotation
the
cell
about
its
body
axis.
The
topology
ne for > 1.5fors.theBy
tracking
the
length
of the
flagellar
bundle.
The
force
dipole
fitthe
was
solved
the
nts
when
speci
the
measured
flow
field
(Fig.
1A)
is
the
same
as
that
of
a
and
performing
an
ensemble
average
over
all
tracers,
we
reat
variable
locations
along
the
swi
are events, relating
their
position
Bacterial
flowdown
field
achieved
by
fitting
two
opposite
force
monopoles
(Stokeslets)
dipole
len
plane
ckingforce
the
dipole
flow
(Fig.
1B),
defined
by
solved
the
time-averaged
flow
field
in
the
E.
coli
swimming
field
(r
>
8
µm).
From
the
best
and
orientation
of
the
bacterium,
dipole
flow
describ
at
variable
locations
along
the
swimming
direction
to
the
far
value
of E
F
position
µm/s.
As
plane down to 0.1% of the mean swimming
speed
V
=
22
±
5
0
the
specific
fitting
routines
fit
with good and
accuracy
h
i
e
average
over
all
tracers,
we
refield
(r
>
8
µm).
From
the
best
fit,
which
is
insensitive
to
and resist
A E. coli
r direction, their time-averag
ℓF swimming
acterium,
µm/s.
As
2rotate about their
ˆ
thisµm
approximation
u(r)in= the
3(r̂.coli
d) −
1 r̂, routines
A=
, r̂fitting
= length
, regions,
[1
]= we
dipole
ℓ
1.9
and
dipo
ow
field
E.
swimming
2
the
specific
fitting
and
obtain
the
note
that
|r|
8πηdimensions
|r| is cylindrically
s, we retime-averaged
flow field in three
symmetric.
a
wall.
Focusing
2
value
of
F
is
consistent
with
optic
dipole
length
1.9±µm
and dipole
force F = of
0.42
This
µm behin
mean
swimming
speed
V0 ℓ==22
5 capture
symmetric.
Our
measurements
all components
thispN.
wimming
and
applying
the s
cylindricall
and
resistive
force
theory
calculatio
fluid
drag
of
Fforce,
is consistent
with
optical
trapforce
measurements
[45]
where
F isvalue
the dipole
ℓ the
distance
separating
the
ut
direction,
their
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a
slight
rotatio
note
thatThe
in
the
best
fit,
the
cell
d
and
resistive
force
theory
calculations
[46].
It is
interesting
to
pair,
η the
viscosity
the
fluid,
dˆ the
orientation
vector
theof
flow
field
struct
the
rotation
ofisof
the
cell
about
itsunit
body
axis.
topology
ion,
their
three
dimensions
cylindrically
the measur
(swimming
direction)
the best
bacterium,
and
rbehind
the
distance
surfaces,
thebfi
note
thatflow
inofthe
fit, 1A)
theµm
cell
drag
Stokeslet
isfrom
0.1
the measured
field
(Fig.
is the
same
as that
oflocated
aof
the
center
the
cell
ndrically
nts
capture
all
components
of
this
force
dipole
Drescher,
Dunkel, Ganguly,
Cisneros,
Goldstein (2011)
PNAS
Bacterial
vector
relative
to
the
center
of
the
dipole.
Yet
there
are
some
ity
of
a
no-slip
µm behind
the center
of thefluid
cell body,
reflectingbundle.
the surf
force dipole
flow (Fig.
1B), defined
by
dunkel@math.mit.edu
drag possibly
on the flagellar
nts of this
E.coli
, except
flow bacterium.
due to
Fig. 1. Averagethe
flow fieldazimuthal
created by a single freely-swimming
(A) Experimentally measured flow field far from a surface. Stream lines indicate local
direction of flo
dipole
Bacterial motors
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
source: wiki
Chen et al (2011) EMBO Journal
dunkel@math.mit.edu
Torque-speed relation
200 nm fluorescent bead attached to a flagellar motor
26 steps per revolution
30x slower than real time
2400 frames per second
position resolution ~5 nm
Berry group, Oxford
dunkel@math.mit.edu
Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
10 ㎛
speed ~100 μm/s
dunkel@math.mit.edu
Chlamy
Merchant et al (2007) Science
dunkel@math.mit.edu
Sperm near surfaces
A
C
B
Fig. 1. Surface scattering of bull spermatozoa is governed by ciliary contact interactions, as evident from the scattering sequences of individual cells at two
temperature values: (A) T = 10 °C and (B) T = 29 °C. The background has been subtracted from the micrographs to enhance the visibility of the cilia. The cyanPolin, Goldstein
PNAS
colored line indicates the corner-shapedKantsler,
boundaryDunkel,
of the microfluidic
channels(2012)
(see Movies
S1 and S2 for raw imaging data). The horizontal dotted line in the last
Surface + shear flow
!
Kantsler et al 2014 (submitted)
Amoeba
Eukaryotic motors
Sketch: dynein molecule carrying cargo down a microtubule
http://www.plantphysiol.org/content/127/4/1500/F4.expansion.html
Yildiz lab, Berkeley
dunkel@math.mit.edu
Molecular
ncisco, San
csf.edu
Downloaded fro
Downloaded from www.sciencemag.org on October
h among
ach head
nd-overernately
nd stays
her head
contrast,
th of the
ce as the
diffusion
nd to the
tracking
r probes)
Walking modes
dot) on the lever arm in the hand-over-hand model (left) and the inchworm model (right). The FIONA assay
Myosin V: Walking or inchworming? Predicted movement for the heads and a dye molecule label (green
The author is in the Department of Cellular and Molecular
has
revealed
that
myosin
V,
along
with
kinesin
and
VI, walksmodel
hand-over-hand.
dot)
on the lever
armmyosin
in the hand-over-hand
(left) and the inchworm model (right). The FIONA assay
Pharmacology, University of California at San Francisco, San
Francisco, CA 94107, USA. E-mail: yildiz@cmp.ucsf.edu
792 2006
0 FEBRUARY
VOL 311
has revealed that myosin V, along with kinesin and myosin VI, walks hand-over-hand.
10 FEBRUARY 2006
VOL 311 SCIENCE www.sciencemag.org
SCIENCE
www.sciencemag.org
Published by AAAS
CREDIT:
P. HUEY/SCIENCE
CREDIT:
P. HUEY/SCIENCE
another
e motor
The bias
e power
hes to the
irection.
t is a fungo transntensive
ovement
tional change in the forward head (head 1) and mining the center of its emis- are pleased to present the prize- point of the myosin. The trajecFIONA,
tracked
of organic
site. Again,
the bylabeled
headtoryalternately
moved
thisUsing
conformation
pulls the Irear
head (headthe
2) movement
sion pattern. However,
of moving spots
created
winning essay
Ahmet Yildiz,
forward, while head 1 stays fixed on the track. dyes are not very bright and the a regional winner for North America three classes of steps. I observed
the
motor proteins myosin V, kinesin, and twice as far as the stalk moved and stopped as
In the next step, head 2 stays fixed and pulls signal disappears quickly by who is the Grand Prize winner of 74-or 0-nm displacements for
myosin
VI,Alternatively,
which were
a single
the other
head
moved
Cy3head 1 forward.
in the labeled
inchworm with
permanent
photobleaching.
dye on the first IQ domain,
alterthe Young
Scientist
Award. (7). Unexpectedly,
model (2) only the forward head catalyzes ATP This limited previous singlenating 52-and 23-nm steps for
dye
in
the
head
region
as
follows.
calmodulin
showed
significant
flexibility when
and always leads while the other head follows molecule tracking experiments to a precision of dye on the fifth IQ domain and alternating 42-and
(seeMyosin
figure below).
around
30 nm (4). I have extended
photostabil33-nm
steps for dye
on the immobile
sixth IQ domainin
(5)
V. Bifunctional rhodamine
(Br)–labelit hadtheATP
bound,
whereas
it was
In both of these mechanisms, the motor ity and brightness of single organic dyes 20 times (see figure below, left).
needs two heads to be able to stay on the track by effectively deoxygenating the assay solution
Kinesin. A human kinesin was specifically
as it moves and its step size depends on the and using reducing agents, and I have achieved labeled on the head region with a single Cy3
length of the legs. However, myosin VI with 1.5-nm localization and collected 1.4 million pho- molecule. As the stalk took 8-nm steps, the
short legs (8 nm) was observed to take the same tons from single organic dyes. The technique, head was observed to take alternating 16-nm
long steps (30 nm) as myosin V. Moreover, a sin- named fluorescence imaging with one-nanome- and 0-nm steps (6).
gle-headed processive motor has suggested that ter accuracy (FIONA), has improved spatial resoluMyosin VI. Myosin VI was labeled with a
two heads are not necessary for processive tion in single molecule fluorescence by ~20-fold.
single Cy3 molecule on a calmodulin-binding
motion. These observations lead to another
Using FIONA, I tracked the movement of site. Again, the labeled head alternately moved
mechanism: biased diffusion of the motor the motor proteins myosin V, kinesin, and twice as far as the stalk moved and stopped as
along the actin/microtubule lattice (3). The bias myosin VI, which were labeled with a single the other head moved (7). Unexpectedly, Cy3is provided by the initial push of the power dye in the head region as follows.
calmodulin showed significant flexibility when
stroke, and the motor most likely attaches to the
Myosin V. Bifunctional rhodamine (Br)–label- it had ATP bound, whereas it was immobile in
next binding site in the forward direction.
Understanding motor protein movement is a fundamental step in understanding how cargo transport works within a cell, but despite intensive
research, the mechanism underlying movement
remained highly controversial.
The most direct way to distinguish among
these models is to measure how much each head
moves when the motor walks. The hand-overhand model predicts that a head alternately
moves twice the stalk displacement and stays
stationary in the next step while the other head
takes a step (see figure, left panel). In contrast,
the inchworm model predicts that both of the
heads move forward the same distance as the
stalk (see figure, right panel). The diffusion
model states that heads randomly bind to the
track. Current nanometer-precision tracking
techniquesV:
(optical
traps and
probes)
Myosin
Walking
orcantilever
inchworming?
Predicted movement for the heads and a dye molecule label (green
dunkel@math.mit.edu
0). Based
showed
oncluded
for kinen asym-
chnique,
Accuraking the
accuracy
NA, the
step is
rescence
a totalope. The
is a dif280 nm,
ponds to
ted with
plied the
lks in a
alternatcements,
(11).
s experiwith a
ch head
Fig. 1B)
d as the
e immoent conglutamic
cond homer with
ines and
43C and
B). Subthe hoof fluo-
(13). The dye’s position was monitored as the
kinesin moved on microtubules that were immobilized on a coverslip (13). Three different conposition
time. aHowever,
if the
observed
structs versus
were used:
homodimer
with
glutamic
17-nm
steps
arise
from
the
convolution
of
twohoacid mutated to cysteine (E215C), a second
sequential
17 nm,
nm. . .), thenwith
a
modimersteps
with (i.e.,
T324C,
and 0a heterodimer
dwell-time
of the number
of steps
one head histogram
lacking solvent-exposed
cysteines
and
versus
step-time
duration
will
be
the
convolution
the other head containing cysteines at S43C and
of T324C,
two exponential
(11).(Fig.
This1B).
yields
which areprocesses
2 nm apart
Sub2
exp(–kt),
thestoichiometric
dwell time probability,
P(t
)
$
tk
labeling was used for the howhich
is
zero
at
t
$ 0, quantal
rises initially,
andofthen
modimers, and single
bleaching
fluofalls,
when
k
is
the
stepping
rate
constant.
rescence confirmed that only a single dye In
was
contrast,
if
the
17-nm
steps
arise
from
a
single
present on each kinesin analyzed (fig. S1B). The
process,
then the
dwell-time
histogram
would
heterodimer
was
labeled with
an excess
ofbe
dye
and both single- and double-quantal bleaching
was observed (13).
In the absence of ATP, kinesins were stationary. In the presence of 340 nM ATP, discrete
steps were observed for the three different kinesin constructs (Fig. 2). A total of 354 steps from
35 kinesins were observed. We typically collected 4000 photons per 0.33-s image. Traces from
relatively bright kinesins ("5000 photons per
image) are shown in Fig. 2; a histogram of 143
steps from 26 molecules is shown in Fig. 3A.
The precision of step-size determination was 1.5
to 3 nm, based on measurement of the distance
between the average positions of the PSF centers
before and after a step (11, 14). The average step
size derived from the step-size histogram (Fig.
3A) is 17.3 # 3.3 nm. We did not observe
8.3-nm steps or odd multiples of 8.3 nm. These
data therefore strongly support a hand-over-hand
mechanism and not an inchworm mechanism.
The hand-over-hand mechanism predicts that
these 17-nm steps alternate with 0-nm steps,
which are not directly observable in a graph of
REPORTS
expected to yield an exponential decay (the
Poisson-distributed rate). The dwell-time histogram of 347 steps for E215C and T324C (Fig.
3B) is well fit by the above convolution function
(with k $ 1.14 # 0.03 steps per s), and not by
the single-step decaying function. The rise near
t $ 0 is not due to instrument artifacts: An
exponential process for myosin V stepping (with
dyes located to show every step) at very similar
rates yields the expected monotonic decay with
the same instrument (11). We also have immo-
tional human kinesin, were mutated to cysteines for fluorescent
dye labeling as described in the
text. The bound nucleotide
(adenosine
diphosphate)
is
shown as a space-filling model in
cyan. This figure was made with
MolMol (22).
Kinesin walks hand-over-hand
Fig. 1. (A) Examples of two alternative classes of mechanisms
for processive movement by kinesin. The hand-over-hand model (left) predicts that a dye on
the head of kinesin will move
alternately 16.6 nm, 0 nm, 16.6
nm, whereas the inchworm
mechanism (right) predicts uniform 8.3-nm steps. The inchworm model was adapted with
slight modification from (9). (B)
The positions of S43 (red), E215
(green), and T324 (blue) on the
rat kinesin crystal structure
[from (6), Protein Data Base
2KIN]. These residues, whose
numbers correspond to conventional human kinesin, were mutated to cysteines for fluorescent
dye labeling
as described
in the
Fig. 2. Position
versus time
for kinesin
motility. The blue and green traces are from E215C
nucleotide
homodimer text.
kinesin;The
the bound
red trace,
from the heterodimer S43C-T324C kinesin. The numbers
(adenosine
is
correspond to
the step sizediphosphate)
# %&. The uncertainties
were calculated as described (11). Red lines
shownpositions
as a space-filling
in between steps (plateau) and when the step occurs
represent average
of each model
duration
cyan.
This figure
was made with
(jumps) based
on data
analysis.
MolMol (22).
Yildiz et al (2005) Science
www.sciencemag.org SCIENCE VOL 303 30 JANUARY 2004
dunkel@math.mit.edu
67
Intracellular transport
Chara corralina
http://damtp.cam.ac.uk/user/gold/movies.html
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wiki
dunkel@math.mit.edu
Actin-Myosin
F-Actin
Myosin
helical filament
myosin-II
myosin-V
dunkel@math.mit.edu
our lecture course:
!
generic models of
micro-motors
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Polymers & filaments (D=1)
Dogic Lab, Brandeis
Drosophila oocyte
Physical parameters
(e.g. bending rigidity) from fluctuation
analysis
Goldstein lab, PNAS 2012
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Actin in 2D
F-Actin
helical filament
Dogic Lab (Brandeis)
dunkel@math.mit.edu
Actin in flow
PRL 108, 038103 (2012)
PHYSICAL REVIEW LETTERS
FIG. 1 (color online). Experimental setup. (a) Microfluidic cross-flow geometry controlled by a pressure differenc
and outlet branches. (b) Close-up of the velocity field near the stagnation point, showing a typical actin filament. (c
of an actin filament and definition of geometric quantities used in the analysis.
Kantsler & Goldstein (2012) PRL
were stored at !80 " C. Polymerization to form filamentous
actin (F-actin) was achieved by addition of 1=10th volume
of eigenfunctions W ðnÞ (and eigenvalues &
conditions Wxxdunkel@math.mit.edu
ð)L=2Þ ¼ Wxxx ð)L=2Þ ¼
our lecture course:
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• polymer models
• how to relate fluctuations to
mechanical properties
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Cell membranes (D=2)
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source: wiki
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Cell membranes (D=2)
Illustration by J.P. Cartailler. Copyright 2007, Symmation LLC.
transport:
stochastic
escape problems
shape: differential
geometry
red blood cells affected by
sickle-cell disease
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red blood cells affected by sicklecell disease
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Dynamics of a vesicle in general flow
viscosities of the inner and outer fluids. Another analytical
microfluidic 4-roll mill device (23,24) manufactured in silicone
approach based on a quasi-spherical vesicle approximated by a
elastomer by soft lithography (Fig. 1). Particle tracking veloci1
J.spherical
Deschamps,
V. Kantsler,
Segre,
and V. Steinberg
harmonics
expansion E.
used
a perturbation
scheme
metry (PTV) measurements show fair agreement with numerical
around the Lamb solution of the Stokes flow near a spherical
Department
of Physics
Complex
Systems,
Weizmann
Institute
body in external
shearof
flow
(1). Further
refinement
of the
modelof Science, Rehovot, 76100 Israel
resulted in dynamic equations for vesicle shape and inclination
Author contributions: V.K. and V.S. designed research; J.D. performed research; J.D., V.K.,
Edited
Harry
Swinney,rather
University
of Texas
at motion
Austin, and
Austin,
andand
approved
May
2009
(received
for review March 11, 2009)
V.S. analyzed
data;6,
and
V.S. wrote
the paper.
angle by
(3,4)
and L.
described
well both
the TT
the TX, E.S.,
The
authors
declare
no
conflict
of
interest.
transition line "c(!), as verified by the recent experiments
An
approach
to quantitatively
study vesicle
dynamics
as well
as
stronger
vesicle shape deformations (3,17). Moreover, a new
1 Direct Submission.
(16–18).
However,
theV.
recent
experimental
key
finding
of a new
This article is a PNAS
J.
Deschamps,
Kantsler,
E.
Segre,
and
V.
Steinberg
biologically-related
micro-objects
in a fluid
flow,(TR),
whichled
is based
on correspondence
aspectshould
is be
the
dependence
on $̇ of the separate regions of
1To whom
type of unsteady motion,
which we dubbed
trembling
addressed.
E-mail: victor.steinberg@weizmann.ac.il.
to reconsider
bothof
theoretical
models
(17).
TRadiffers
from
TU
the
combination
a dynamical
trap
and
control
parameter,
the
existence
of
TR
and
TU
(17).
Precisely these features changed
This article contains supporting information online at www.pnas.org/cgi/content/full/
Department
of
Complex
Systems, Weizmann
Institute
of Science, Rehovot, 76100 Israel
by oscillations
in Physics
0902657106/DCSupplemental.
% of to
lessof
than
!/2 (rather
2!) and by
ratio
of the vorticity
the
strain
rate, isthan
suggested.
The flow
is
the idea of vesicle dynamics as smooth and shape-preserved
Dynamics of a vesicle in general flow
A
continuously
varied
betweenUniversity
rotational,
shearing,
and elongaand called
for2009
an (received
adequatefor
theoretical
description.
Edited by Harry
L. Swinney,
of Texas
at Austin,
Austin, TX,motion
and approved
May 6,
review March
11, 2009)
11444 –11447
! PNAS ! July 14,
2009 mill
! vol.device,
106 ! no.
28 dynamical trap, that
www.pnas.org"cgi"doi"10.1073"pnas.0902657106
tional
in a microfluidic
4-roll
the
Several theoretical
models were suggested to describe the dystates in shear
flow,shape
their regions
of existence,
and Moreover, a new
allows
scanning to
of quantitatively
the entire phasestudy
diagram
of motions,
i.e., as namics
An approach
vesicle
dynamics
well asof all 3stronger
vesicle
deformations
(3,17).
transitions
have
verified recently,
1 of separate regions of
tank-treading
(TT), tumbling
(TU), andin
trembling
(TR), which
using ais based
biologically-related
micro-objects
a fluid flow,
on (5,7,13).
aspectAsiswethe
dependence
on only
$̇ ofthethe
them presented in ref. 5 describes adequately the experimental
single vesicle even at ! " #in/#out " 1, where #in and #out are the
B combination of a dynamical trap and a control parameter,
the
existence
of ofTR
TUwhich
(17).isPrecisely
these features changed
data the
(20). The
main result
thisand
model,
based on the
viscosities of the inner and outer fluids. This cannot be achieved in
ratioshear
of the
the strain
rate,TTisand
suggested.
flow is
theof !#
idea
vesicle
as smooth
#1, of
second
orderdynamics
spherical harmonics,
andand shape-preserved
pure
flow,vorticity
where theto
transition
between
either TU orThe approximation
continuously
varied
rotational,
shearing,
and elongamotion
and
called forsolution,
an adequate
theoretical description.
neglecting thermal
noise,
is a self-similar
which reduces
As a result,
it is found that
the vesicle
TR
is attained only
at !>1.between
the
number
of
the
dimensionless
control
parameters
to
just
2:
S to describe the dydynamical
states
in
a
general
are
presented
by
the
phase
diagram
tional in a microfluidic 4-roll mill device, the dynamical trap, that
Several theoretical models were suggested
3
$
$
$
/ 3&! and
%'
4(1 & 23
30!
, where
&
' 7!$̇#
in
a space scanning
of only 2 dimensionless
control
parameters.
The findings
of all
3 states
in"/32)
shear!/
flow,
their
regions
of existence, and
allows
of the entire
phase
diagram
of motions,
i.e.,outR namics
is the bending elasticity [taken further as & " 25 kBT'10(12 erg
are in semiquantitative accord with the recent theory made for a
transitions (5,7,13). As we have verified recently, only the 1 of
tank-treading (TT), tumbling (TU), and trembling (TR), using
a
(21)]. The phase diagram of the vesicle dynamical states is
quasi-spherical vesicle, although vesicles with large deviations
them
presentedbyinthe
ref.
5 describes
adequately
the experimental
" #in/experimentally.
#out " 1, where
#out 2-dimensional,
are the
single
vesicleshape
evenwere
at !studied
in andof
parameterized
variables
(S,%), and
indefrom
spherical
The#
physics
data
(20).
The
main
result
of
this
model,
which
is based on the
viscosities
of the inner and outer fluids. This cannot be achieved
in of other geometrical parameters. To scan all 3 regimes
pendent
TR
is also uncovered.
of TU
motion
to trace transitions
among
in aorder
shear spherical
flow,
approximation
of !#
#1, them
second
harmonics, and
pure shear flow, where the transition between TT and either
or and
$̇
and
"
,
that
is
change
the
viscosities
of
one
should
vary
both
neglecting thermal noise, is a self-similar solution, which reduces
!>1. As
result, itremains
is found
TR isnderstanding
attained only
the at
rheology
of abiofluids
a that
great the vesicle
Fig. 1. (A) Schematic of the microfluidic 4-roll mill device; Q1 and Q2 are the flow discharges, whose ratio defines the flow type. The flow is driven by gravity,
inner
and
outer
which is an impossible task to realize on
and the ratio between the pressure drop P0 and the pressure difference between
2challenge,
inlets #P determines
Q1/Q
(B)
(1 ! !/s)/(1%
!/s) as
function
of the
reduced
whose
relies,
inapresented
a large
part,
onthe
detailed
thefluids,
number
of the dimensionless control parameters to just 2: S
dynamical
states
in2.progress
af $general
are
by
phase diagram
an
individual
vesicle.
Because
of topology of the phase diagram
pressure drop (1- #P/P0). Inset: s as a function of (1 ! #P/P0). Large filled squares, P0 $ P; open squares, P0 $ 4/3P; small filled circles, P0 $ 5/3P, open circles, P0 $
3/ $
studies
of
the
dynamics
of
a
single
cell.
Vesicles
are
a
model
$!/$30!, where &
!$̇#outRpossibility
3&! and
4(1vesicle
& 23"is/32)
' 7remaining
2P (in our specific configuration P&750 Pa). The solid line is the 3D FEMin
simulation
of theof
flow.
Experimental
imperfections due to soft
lithography
lead to
a space
only
2 dimensionless
control
parameters.
The findings
(5,20),
the
only
with%a '
single
to
observable quantitative discrepancy with simulations.
system used to study the dynamic behavior of biological cells,
(12
is the
bending
elasticity
[taken
erg
are in semiquantitative accord with the recent theory made
fortransitions
a
scan
from
TU to TR
by varying
$̇. further as & " 25 kBT'10
similar in some respects to red blood cells, and their dynamics in
This limitation
overcome
a general of
flow,
the dynamical states is
(21)].is The
phasein diagram
thewhere
vesicle
quasi-spherical
vesicle,
although
vesicles
with large
shear
flow
been
the subject
theoretical
(1–8), deviations
simulations and high flow uniformity (Fig. 2 and SI Appendix).
The have
experiments
were
performedofinintensive
the following
way. A
velocity
gradient
can
be
written
as
'
V
"
s
&
(
)
,
where
sik
i k
ik
ikj the
j
2-dimensional, parameterized
by
variables
(S,%), and indefromvesicle
spherical
shape
studied
experimentally.
The physics of
was
The accessible range of s was [0.05–0.32] s!1 and that of !/s numerical
with given
R andexperimental
#,were
measured
initially
in
situ in
the same
(9–13),
and
(14–18)
investigations.
is
the
symmetric
strain
tensor,
)
the
vorticity
vector,
and
s
"
j
!
/s
and
s
in
the
1.18–9.23]. To reduce error bars on S and ", the geometricalA vesicle
device, was
followed
at
a
prescribed
value
of
pendent of other geometrical parameters. To scan all 3 regimes
is a droplet of viscous fluid encapsulated by a
TR isobservation
also uncovered.
2
$tr(sik
window. A feedback in the flow velocity was used to
parameters R and # of each vesicle were measured in situ from
)/2 the strain rate. The corresponding control parameters
phospholipid
bilayer
membrane
suspended
in
a
fluid
of
either
motion
and
transitions
among
hold a vesicle in the field of view for up to 10 TU/TR periods.
a 3D reconstruction of its shape (20) in TT motion at low S, with
for vesicles inofgeneral
flow
(5)to
aretrace
S '14
!s#outR3/3$
3&! andthem in a shear flow,
the same
or different
viscosity
as the inner
one.
Both
the volume
To explore
the whole
space of parameters
(S, "),
vesicles
with
mean errors of 1.7% on R and of 6.1% on # within the range [0.3,
$̇ and
", we
that
is change
the viscosities of
one
vary
$!()/s)/
$30both
% " 4 (1 & 23
"/32)should
!. In this
paper,
report
the
nderstanding
of biofluids
various
values of
R and
# the
wererheology
loaded and
individually
observed,
2.5]. Since only vesicles with " $ 1 were used in this work, there
and the
surface
area
ofthe
vesicle
are
conserved.
The remains
former a great
inner
and
outer
fluids,
which
is
an
impossible
task
to realize on
phase diagram in such general flow. This approach uses an
!/s was varied
in steps progress
during the experiment
by changing
s no error contribution from this parameter (20). Errors from
and
challenge,
whose
relies, in
atolarge
part, on detailed
means
that
the
vesicle
membrane
is
considered
be
imperme#P, which is the pressure difference between 2 inlets (see Fig. 1).
non-uniformity of the velocity field were below 2%; that cumuadditional control
parameter vesicle.
)/s, whichBecause
is fixed toof
unity
in shearof the phase diagram
an individual
topology
at
time
scale of
of the
experiment,
studies
ofwayon
the
dynamics
a(S, single
cell. and
Vesicles
are a flow
model
In least
this
thethe
space
of parameters
")
was populated
with the latter
ates to overall mean errors of 8.7% on S and of 2% on ".able,
(s " ) "
$̇
/2),
to
study
vesicle
dynamics
[it
was
suggested
(5,20), the only remaining possibility with a single vesicle is to
means
the membrane
dilatation
is neglected
since
is 2D
systemthat
used
to study the
dynamic
behavior
ofitbiological
cells,
first
by G. I. Taylor to study emulsions in a 4-roll mill (22)]. The
scan transitions from TU to TR by varying $̇.
fluid
(1,2).
theoretical,
and computational
efforts
similar
inExperimental,
some respects
to red blood
cells, and their
dynamics
ratio in
can be easily varied continuously in the experiment,
during the last decade led to the observation and characterizaThis from
limitation
is overcome
in from
a general
flow, where the
TT to either
TU or TR and
TU
shear
beendynamics
the subject
of intensive
theoreticalevidencing
(1–8), transitions
tion
of 3flow
stateshave
in vesicle
in shear
flow. The existence
velocity
cangiven
be written
as 'i"V. k The
" sik & (ikj)j, where sik
to TR on the
same gradient
vesicle with
R, !, and
numerical
and experimental
(14–18)
of
the first 2,(9–13),
tank-treading
(TT) and tumbling
(TU),investigations.
and the
experimentalispath
the phase
diagram
depends
the
the across
symmetric
strain
tensor,
)j theonvorticity
vector, and s "
A vesicle
is athem
droplet
of viscous
fluid
by a
transition
between
were already
predicted
by aencapsulated
phenom2way )/s and s are varied. The possibility to
initial state and
the
$
tr(sik)/2 the strain rate. The corresponding control parameters
enological
modelbilayer
of Keller
and Skalak
(19) and its
phospholipid
membrane
suspended
in afurther
fluid ofobserve
either all dynamical
statesinwith
the same
vesicle,
evenSfor
" "!s# R3/3$3&! and
for vesicles
general
flow
(5) are
'14
out
extensions
(2,11,12).
Twoviscosity
control parameters,
theone.
excess
areathe volume
the same
or
different
as
the
inner
Both
1 used in the current experiment, $
complements
the previous
2-4! and the viscosity contrast " " # /#
$
% the
" 4shear
(1 &
23dynamics
"/32) !(
)/s)/ 30
!.the
Inother
this paper, we report the
! " A/R
out, determine
based on
flow
(2–4,7–20).
On
and the surface area of the vesicle arein conserved.
The views
former
the transition line "c(!) between TT and TU, which is indepenphase diagram
such
flow. This approach uses an
hand, the experimental
approachin
used
heregeneral
will be advantageous
means
that
therate
vesicle
membrane
is considered
to be imperme$̇ in the
approximation
of a fixed vesicle
dent
of the
shear
additional
control
parameter
)
/s,
which
is fixed to unity in shear
study the dynamics of other flexible microobjects, including
able, with
at least
on the
time scale
of% the
and theto
latter
withexperiment,
respect to the flow
shape,
the vesicle
inclination
angle
biological membranes
and
cells,
in flow.
flow (s "
) red
" $̇blood
/2), to
study
vesicle dynamics [it was suggested
direction
as thethe
onlymembrane
dynamical variable
(2,9,10,16–19).
Here,
R it is 2D
means that
dilatation
is neglected
since
first by G. I. Taylor to study emulsions in a 4-roll mill (22)]. The
isfluid
the effective
vesicle radius, theoretical,
related to theand
volume
via V "
Results
and Discussion
(1,2).
Experimental,
computational
efforts
ratio can be easily varied continuously in the experiment,
4/3!R3, A is the vesicle surface area, and #in and #out are the
Measurements of the vesicle dynamics were conducted in a
during the last decade led to the observation and characterizaviscosities of the inner and outer fluids. Another analytical
transitions
from TT to either
TU or TR and from TU
microfluidic evidencing
4-roll mill device
(23,24) manufactured
in silicone
tion of 3based
states
vesicle dynamics
in approximated
shear flow. The
approach
onin
a quasi-spherical
vesicle
by a existence
on the(Fig.
same
vesicletracking
with given
elastomer byto
softTR
lithography
1). Particle
veloci- R, !, and ". The
of the first
2, tank-treading
(TT)a and
tumbling
(TU), and
the
spherical
harmonics
expansion used
perturbation
scheme
metry (PTV)experimental
measurements show
agreement
numerical
pathfairacross
the with
phase
diagram depends on the
transition
between
them
were
already
by a phenomaround
the Lamb
solution
of the
Stokes
flow predicted
near a spherical
U
APPLIED PHYSICAL
SCIENCES
U
dunkel@math.mit.edu
Volvox inversion
http://www.damtp.cam.ac.uk/user/gold/movies.html
dunkel@math.mit.edu
our lecture course:
!
• ‘differential geometry’ of membranes
dunkel@math.mit.edu
Stationary patterns
dunkel@math.mit.edu
Turing model
A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London. B 327, 37–72 (1952)
dunkel@math.mit.edu
wiki
The matching of
zebrafish stripe
formation and a
Turing model
Kondo S, & Miura T (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329 (5999), 1616-20
dunkel@math.mit.edu
odel equations
must be invariant under ⇤ ⇤ ⇤, implying that U (⇤) =
in Equation (2). Intuitively, the transformation ⇤ ⇤ ⇤
sider the simplest isotropic fourth-order
model
for at
a the
non-conserved
or
the observer
position
midplane of thescalar
film (watchin
watching
it from
below).
scalar order-parameter ⇤(t, x),2dgiven
by
Swift-Hohenberg
model
The situation can be rather different, however, if
2
⇧t ⇤ = F (⇤) + 0 ⇤
⇤,microorganisms close to a liquid-solid interface, such(1)
as th
2
2 slide (Figure 2). In thi
cells in the2 vicinity of a 22glass
⇤
⇥
=
U
(⇥)
+
⇥
⇥
(⇥
)
t
0 and ⇤ =2 ⌅ is⇥the d-dimensional
⇧t = ⇧/⇧t denotes the
time derivative,
trajectory of a swimming cell can exhibit a preferred han
an. The force F is derived from a Landau-potental
U (⇤)
example, the bacteria
Escherichia coli [40] and Cauloba
Scalar field theory
F =
⇧U
,
⇧⇤
U (⇤) =
b the
c 4 components of the Levi-Civita tensor,
⌃a ij 2denotes
3 Cartesian
⇤ + ⇤convention
+ ⇤for, equal indices throughout.
(2)
a summation
2
3
4
derivative terms on the rhs. of Equation (1) can also be obtained by variational
s from a suitably defined energy functional. In the context of active suspensions,
d, for example, quantify local energy fluctuations, local alignment, phase
ces, or vorticity. We will assume throughout that the system is confined to
spatial domain ⇥ ⇥ Rd of volume
⇤
|⇥| =
dd x,
(3)
g with periodic boundary conditions in simulations.
r completeness, one should note that in the case of a conserved order-parameter
the field equations would either have to take the current-form ⇧ ⌅ = ⌅ · J(⌅)
Active patterns
PRL (2013)
B. subtilis
tracer
bright field
fluorescence
3D bacterial suspension
bright field
fluorescence
PRL (2013)
3D suspension
Experiment: quasi-2D slice
PRL (2013)
Theory: 2D slice
Vector field theory
incompressibility
E=(
0
2⇥
2
)(⇥ v + ⇥v )
†
†
Active nematics
Dogic lab (Brandeis) Nature 2012
autonomous motility, which are not observed in their passive
ana- relative po
1
77 Massachusetts AvenueActive
E17-412,
Cambridge,
M
Department
of Mat
nematics
logues. Taken together, these observations exemplify
how
assemmicrotubul
(Dated:
October
23,
2013)
BASICS
77biomimetic
Massachusetts
Av
blages of animate microscopic objects exhibit
collective
between
mi
Active nematics
PACS
numbers:1
a
Jörn Dunkel1, ⇤
b
d
In 2d,
the
symmetric
order-parameter
tensor
Q(t
Department
of Mathematics, Massachusetts
In
+
+
PACS numbers:
with
77 Massachusetts Avenue E17-412, Cambridge,
PEG
Depletion
force
BASICS
Qij = Qji ,
(Dated: October
23,can
2013)
This
be
Tr Q = 0,
BASICS
numbers:
In 2d, thePACS
symmetric
can
be order-parameter
represented as tensor Q(t, x, y)
Microtubules
with
✓
◆
In
2d,
the
symmetric
order-parameter
tens
Motor Time
+
µ
Kinesin clusters
force
with
Q
=
.
where
the
u
Q
=
Q
,
Tr
Q
=
0,
(1)
BASICS
This
can
b
ij
ji
c
Dogicµ
lab (Brandeis) Nature 2012
S=
> 0,0, we
Q
=
Q
,
Tr
Q
ij
ji
can be represented as
whereas
n
=
Defining
In 2d, the
order-parameter
tensor Q(t, x, y)
✓ Q-tensor
◆
nosymmetric
head or tail
order-parameter
canµbe represented
as
p
th
Q=
.
2+µ
2 ,(2) ◆
✓
=
µ
µ
Q=
Qij = Qji ,
Tr Q = 0,
(1) . where the
Defining
the eigenvalues of Q are given by µ
S > 0, we
n be represented as pDefining
We
start
whereas
n
±
2
2
=✓
+◆
µ ,
(3)
⇤ =±
energy dens
p
µ
(rQ)
(rrQ)
2
= (@k Qij )(@k Qij )
(11b)
= (@k @n Qij )(@k @n Qij ).
(11c)
Matrix field theory
Note that the potential cannot contain odd-power terms
2k+1
since Tr Q
= 0 in 2D. Consider the corresponding
field equation
F
@
Q
+
v
@
Q
=
(12)
t
ij
k
k
ij
To obtain closed equation, we mustQ express v = (vk ) in
ij
terms of Q. We discuss two possible choices
where v is the advection velocity and
vk = D @n Qnk
(16a
2
4
F
a @ Tr Q
b
@
Tr
Q
vk +
= D @n (Qnj Qjk )
(16b
=
Qij
2 @Qij
4 @Qij
2 a response coefficient2 with unit
where the constant
D
is
@(rQ)
@(rrQ)
2
4
@
+
@
@
.
(13)
2
k
k
n
length /time.
the
crucial
di↵erence
between
th
2 Note
@(@k Q
)
2
@(@
@
Q
)
ij
k n ij
two closure conditions: Eq. (16a) assumes that active LC
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