18.095 IAP Maths Lecture Series
Over-damped dynamics
of
small objects in fluids
Jörn Dunkel
Physical Applied Math
Biofilm formation
Fluid dynamics of microorganisms
Goldstein group
Microbial transport
in porous media
Drescher lab
Guasto lab
Surface wrinkling
Biological pattern formation
Reis lab
Gregor lab
Typical length scales
http://www2.estrellamountain.edu/faculty/farabee/BIOBK/biobookcell2.html
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
Laminar (low-Re) flow
For Re→0
fluid flow becomes
reversible !
... except for thermal
fluctuations
Brownian motion
“Brownian” motion
Übersicht
Brownsche Bewegung - Historischer Überblick
Relativistische Diffusionsprozesse
Fazit
Jan Ingen-Housz (1730-1799)
1784/1785:
http://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html
Relativistische Diffusionsprozesse
Fazit
Robert Brown (1773-1858)
Linnean Society (London)
1827: irreguläre
voninPollen
irregular Eigenbewegung
motion of pollen
fluid in Flüssigkeit
http://www.brianjford.com/wbbrownc.htm
Jörn Dunkel
Diffusionsprozesse und Thermostatistik in der speziellen R
Brownian motion
Mark Haw
David Walker
!"#$%& $' (%$)*+,* -$.+$*
!"#$%&!$'!(%$)*+,*!-$.+$*
!"#$%&'()*+,-#./0102/3//4
!"#$%&' ((()*+&,-&)%".),#
!"#$%&'!((()*+&,-&)%".),#
RT
D!
6"#aC
A'7*"#:+B"#!C#90/#./3D14
5"#67,8&(7,#./0932/3114
:"#$;<*%='<>8?7#
./09@2/3/94
!"#$%&' (/0/1&2/,)"$!"#$%&'!(/0/1&2/,)"$-
!"#$%&' (/0/1&2/,)"$!"#$%&'!(/0/1&2/,)"$-
RT 1
D!
N 6" k P
32 mc 2
D!
243 "$R
5,,"#A'E8"#"#C#1F3#./3D14
5,,"#A'E8"#$"C#91G#./3DG4
confirmed that Fazit
there was an increase of the viscosity that was in
of the size of the suspended particles, and only depends on the to
occupy.
However, he
got a prize
stronger
increase. Initially, this in
Jean Baptistethey
Perrin
(1870-1942,
Nobelpreis
1926)
Nobel
too steep; in the publication Bancelin gives the result η = η0 (1 +
On 27 December, 1910 Einstein wrote from Zürich to his form
and collaborator Ludwig Hopf about the puzzling situation, and
“I have checked my previous calculations and arguments and
of
no error in them. You would becolloidal
doing aparticles
great service
in this
radius 0.53µm
ter if you would carefully recheck my investigation. Either
successive
is an error in the work, or the volume
of positions
Perrin’s suspended
30 seconds
stance in the suspended state isevery
greater
than Perrin believes
joined by straight line
Hopf indeed found an error in some
differentiation process, a
segments
Mouvement brownien et réalité moléculaire, Annales de chimie et de
physique VIII 18, 5-114 (1909)
formula (4). Einstein communicatedmesh
the result
to Perrin, and p
size is 3.2µm
(1911) a correction of his thesis in the Annalen. (By the way, this
Les Atomes, Paris, Alcan (1913)
is the second most quoted paper of Einstein.) New experiment
ns in this way his famous formula
sugar solutions now gave the excellent value
Experimenteller
kT Nachweis Rder atomistischen Struktur der Materie
D=
, k=
(35)
NA = 6.56 ×
1023
6πη0 a
NA
for
the
Avogadro
number,
in
good
agreement
with
the
results
of
o
constant).
Jörn Dunkel
und Thermostatistik in der speziellen Relativitäts
ods, in particular
with Diffusionsprozesse
Perrin’s determination
from the Brownian
as almost simultaneously discovered in Australia by William Suther-
uty of the argument is
experimental evidence for
11
atomistic
structure
of
matter
that the exterior force drops out. Similar
Mathematical theory
Norbert Wiener
(1894-1864)
MIT
Relevance in biology
•
•
•
intracellular transport •
•
tracer diffusion = important experimental “tool” intercellular transport microorganisms must beat BM to achieve directed
locomotion
generalized BMs (polymers, membranes, etc.)
dunkel@math.mit.edu
Nano-spheres in water
Rutger Saly
Polymer in a fluid
Dogic lab
(Brandeis)
Ring-polymer in a fluid
Dogic lab
(Brandeis)
Flow in cells
Flow & transport in cells
Drosophila
embryo
Goldstein lab (Cambridge)
Flow & transport in cells
Drosophila
embryo
Goldstein lab (Cambridge)
Intracellular transport
Chara corralina
Giant cell
http://damtp.cam.ac.uk/user/gold/movies.html
Flow around cells
Vesicles in a shear flow
Vasily Kantsler
model for
blood cells
dynamics
Swimming bacteria
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
source: wiki
Chen et al (2011) EMBO Journal
dunkel@math.mit.edu
1.10.1 Problem
Problem 1:
1: Order-o
Order-o
1.10.1
Solutions1.10 Solutions
Solutions
(a)
How heavy
heavy
is aa bacterium?
bacterium?
A
1.10.1
Problem
1: Order(a)
How
is
A
3
Solutions
⇢
=
1000
kg/m
we
find
3 (water),1:
Problem
1:
Order-of-magnitude
estimat
1.10.1
Problem
Order-o
=
1000
kg/m
(water),
we
find
Problem ⇢1:
Order-of-magnitude
estimat
(a)
How
heavy
is
a
bacterium?
A
Solutions
1.10
1.10
1.10 fast
How
1.10.1
1.10.1
1.10
must a cell swim
(a)
How heavy
heavy
is aa bacterium?
bacterium?
Assuming
aa volume
vv ⇢v
=
11⇠µ
1.10.1
Problem
1:
Order-of-magnitude
estima
m
=
⇢
=
1000
kg/m
(water),
we
find
(a)
How
is
Assuming
volume
=
µ
(a)
How
heavy
is
bacterium?
A
to
beat
Brownian
motion?
m
=
⇢v
⇠
Solutions
= 1000
1000 kg/m
kg/m
(water),1:
we Order-of-magnitude
find
Solutions
Problem
estimate
⇢⇢1.10.1
=
(water),
we
find
3
3
1.10
1.10
3
3
⇢
=
1000
kg/m
(water),
we find
(a) How heavy is a bacterium? Assuming
a volume
v= =
1⇠
µ
m
⇢v
1.10 Problem
Solutions
18
3
3
1.10.1
1:
Order-of-magnitude
estimates
18
3volume
m
=
⇢v
⇠
10
⇥
10
kg
=
1
fg
(b)
How
fast
must
a
bacterium
sw
⇢
=
1000
kg/m
(water),
we
find
1.10.1 Problem
1:
Order-of-magnitude
estimates
(a)
How
heavy
is
a
bacterium?
Assuming
a
v
=
1
µm
m
=
⇢v
⇠
10
⇥
10
kg
=
1
fg
(b)
How
fast
must
a
bacterium
sw
1.10 Solutions
m
=
⇢v
⇠
21
3
3
(a)
How heavy
heavy
is =
a bacterium?
bacterium?
Assuming
aawe
volume
vv kT
=
11 µm
aa 10
mass
3 and
⇢
1000
kg/m Assuming
(water),
find
perature,
=
4
⇥
J.
Stokes
21density
1.10.1
Problem
1: Order-of-magnitude
estimates
18
3
(a)
How
is
a
volume
=
µm
and
mass
density
perature,
= must
4 ⇥⇥1010
Stokessw
3
m=
⇢vkT
⇠fast
10
kg
=J.1 fg
(b)
How
a
bacterium
⇢
=
1000
kg/m
(water),
we
find
3
Problem
1:
estimates
(b)
How
fast must
must
a water
bacterium
swim
so
that
swimming
make
⇢1.10.1
=his
1000
kg/m
(water),
we Order-of-magnitude
find
is way
famous
3
18
3mass
(a)
How
heavy
isformula
a bacterium?
Assuming
a
volume
v
=
1
µm
and
a
density
(b)
How
fast
a
bacterium
swim
so
that
swimming
make
211 fg
water
m
=
⇢v
⇠
10
⇥
10
kg
=
21
18
3
3 perature,m kT
perature,
kT
4
⇥
10
J.
Stokes
(b)
How
fast
a
bacterium
sw
3=must
=
4
⇥
J.
Stokes
drag
coefficient
of
a
sphere
18 10
3volume
=
⇢v
⇠
10
⇥
10
kg
=
1
fg
(1.159a)
21
⇢ = 1000
(water),
we
find
(a)
Howkg/m
heavyperature,
is a bacterium?
Assuming
a
v
=
1
µm
and
a
mass
density
m kT
= fast
⇢v=
⇠ must
10
⇥a10bacterium
kg
1 fg swim
(1.159a) Smake
= 66
4 ⇥ 10
J.=Stokes
dragsocoefficient
(b)
How
that
swimming
21 of a sphere
R
=
3 kT
S
⇢ = 1000 kg/m water
(water), we find
perature,
kT
=
4
⇥
10
J.
Stokes
water
18
3
D
=
,
k
(35)
21kg = 1 fg
water
m=
=fast
⇢v =
⇠must
10
⇥
10
(1.159a)
perature,
kT
4
⇥
10
J.
Stokes
drag
coefficient
a
sphere
(b)
How
a
bacterium
swim
so
that
swimming
makes
8 of
(b)
How
fast
must
a
bacterium
swim
so
that
swimming
makes
sense?
At
room
temHence,
we
find
for
the
di↵usion
const
18
3
6πη
a
N
8 kg/s
(b) How fast must0 a bacterium
swim
swimming
sense?
At10
room
tem- const
=makes
6⇡⌘a
⇠ the
2⇥
S =
10 so ⇥that
10water
kg
= 1 we
fgS
(1.159a)
Hence,
find
for
di↵usion
S =
21 m = ⇢v ⇠A
21
6⇡⌘a
⇠
2
⇥
10
kg/s
perature, kT
kT =
= 44perature,
⇥ 10
10 21 J.
J. Stokes
Stokes
drag
coefficient
aa sphere
of
radius
a
=
11of
µm
kT = drag
4 ⇥coefficient
10 J. of
Stokes
drag
coefficient
ain
water
perature,
⇥
of
sphere
of
radius
a
=
µm
insphere
(b)
How
fast
must
a
bacterium
swim
so
that
swimming
makes
sense?
At
room
tem=
6
8
S
water
Hence,
we
find
for
the
di↵usion
constant
water
=find
6⇡⌘a
⇠ 2the
⇥aroom
10
kg/s
21
Hence,
we
for
di↵usion
const
nt). water
Hence,
we
fordrag
the
di↵usion
S makes
perature,
4 ⇥ a10bacterium
J. find
Stokes
coefficient
a
sphere
of
radius
=
1
µm
in
(b)
How kT
fast=must
swim
so
that
swimming
sense?
At
tem8 of constant
8 kg/s
= 6⇡⌘a
6⇡⌘a ⇠
⇠ 22 ⇥
10
8
S =
21
⇥
10
kg/s
S
=
6⇡⌘a
⇠
2
⇥
10
kg/s
2di↵usion
perature,
kT = 4Hence,
⇥ 10 we
J. Stokes
drag
coefficient
of
a
of
radius
a
=
1
µm
in consta
Hence,
we
find
for
the
water
Ssphere
ost simultaneously
discovered
in
Australia
by
William
Sutherfind
for
the
di↵usion
constant
2
D
⇠
0.2
µm
/s
Hence,
we find
find for
for the
the di↵usion
di↵usion constant
constant
D ⇠ length
0.2 µm ⇠
/s 1 s, Browni
water we
2 ⇥ 10 8 kg/s a run
Hence,
S = 6⇡⌘a ⇠ Assuming
Hence, we find= for
the
di↵usion
constant
Assuming
a run length 2⇠ 1 s, Browni
8
6⇡⌘a
⇠
2
⇥
10
kg/s
2
S
Hence, we find for
the di↵usion
constant
D
⇠
0.2
µm
/s
21/s
D⇠
⇠ 0.2
0.2approximately
µm
Assuming
a
run
length
⇠
s,
Brownian
motion
would
move
aaTh
mi
0.5
µm
per
second.
D
µm
/s
Assuming
a
run
length
⇠
1
s,
Brownian
motion
would
move
mi
approximately
0.5
µm
per
second.
Th
2 ⇠ 1 s, Brown
the argument
is for
that
the
exterior
drops
out.
Similar
Assuming
a
run
length
Hence, we find
the di↵usion
constant force
D⇠
0.2
µm /s
2
approximately
0.5
µm
per
second.
Thus
a
bacterium
should
swim
is
close
to
typical
swim
bacterial
spee
D
⇠
0.2
µm
/s
Assuming
a
run
length
⇠
1
s,
Brownian
motion
would
move
a
micron-sized
bacterium
by
approximately
0.5
µm
per
second.
Thus
a
bacterium
should
swim
Assuming
a
run
length
⇠
1
s,
Brownian
motion
would
move
a Th
mi
Assuming
a
run
length
⇠
1
s,
Brownian
motion
would
move
a
micron-sized
bacterium
by
is
close
to
typical
swim
bacterial
spee
Assuming
a
run
length
⇠
1
s,
Browni
2
approximately
0.5
µm
per
second.
iderations
between
systematic
and
fluctuating
forces
were
D
⇠
0.2
µm
/s
isAssuming
close
to typical
typical
swim
bacterial
speeds.
approximately 0.5
0.5
µm
per second.
second.
Thus
a bacterium
bacterium
should
swim
at
last
5-10
µm/s,
which
approximately
µm
per
Thus
a
should
swim
at
last
5-10
µm/s,
which
is
close
to
swim
bacterial
speeds.
a
run
length
⇠
1
s,
Brownian
motion
would
move
a Thu
mic
Assuming a run length
⇠ 1 s, Brownian
motion
would
moveThus
a micron-sized
bacterium
by swim
approximately
0.5
µm
per
second.
a
bacterium
should
approximately
0.5
µm
per
second.
close
to
typical
swim
bacterial
spe
isAssuming
close to
to typical
typical
swim bacterial
bacterial
speeds.motionis
by Einstein.
is
close
swim
speeds.
a
run
length
⇠
1
s,
Brownian
would
move
a
micron-sized
bacterium
by
approximately 0.5approximately
per second.
Thus
a bacterium
should
swim
at
lastthe
5-10 e↵ective
µm/s,
whichdi↵usio
0.5 µm
per
second.
Thus
a is
bacterium
should
swim a
isµm
close
to typical
swim
bacterial
speeds.
(c)
How
large
is close
toswim
typical
swim
bacterial
spee
approximately
0.5
µm
per
second.
Thus
a bacterium
should
at constant
last
5-10
µm/s,
which di↵usio
(c)
How
large
is
the
e↵ective
is close to typical
swim
bacterial
speeds.
(c)
How
large
is
the
e↵ective
di↵usion
of
bacteria
that
is
close
tolarge
typical
swim
bacterial
speeds.
(c)
How
is
the
e↵ective
di↵usion
constant
of ⌧bacteria
that
(c)
How
large
is
the
e↵ective
di↵usion
constant
of
bacteria
that
perform
run-and-tumble
is
close
to
typical
swim
bacterial
speeds.
motion
with
run
periods
⇠
1
s?
(c) How large ismotion
the e↵ective
di↵usion
constant
of
bacteria
that
perform
run-and-tumble
motion
withlarge
run is
periods
⌧ ⇠ 1 s?
⌧ ⇠ 1How
s?
(c)
the e↵ective
di↵usi
motion with run periods ⌧ with
⇠ 1 s?run periods
E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
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
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
cell
body,
of the
recorded
∼ 2 and
terabytes
of flow
4
flowmovie
fieldResults
ofdata.
an E. In
coli
“pusher”
decays
much faster,
when
swims close
thecule
surface,
fortothe
length
of t
theflow
mea
this
data
we
identified
∼
10
events when
(non-tumbling HCB 437)rarea bacterium
achieved
by
fittin
theByTo
measured
andminisbest-fit
force
cellsBacterial
swam in the
for > surfaces.
1.5 s.
tracking
decays
labeled,
n
flowfocal
fieldplane
far from
resolve the
the
at
variable
locatio
fluidcule
tracers
in
each
of
the
rare
events,
relating
their
position
of
the
c
decays
of
the
flow
speed
u
with
flow
field
created
by
individual bacteria, we tracked gfp- sion of
flu
m surfaces.
To
resolve
the
minisfield (r >field
8 µm).
and labeled,
velocity to
the
position
and
orientation
of
the
bacterium,
dis
non-tumbling
E.
coli
as
they
swam
through
a
suspenof
the
cell
body
(Fig.
1D)
illustr
walls,
we
dividual
bacteria,
tracked
gfp- overforce
the measured
andaverage
best-fit
dipole we
field
The
the 1C).
specific
fitting
and performing
anwe
ensemble
all tracers,
re- (Fig.
Howeve
sionswam
ofdecays
fluorescent
tracer
particles.
For
measurements
farcharacteristic
from
field
displays
the
1
dipole
length
ℓ =o
of
the
flow
speed
u
with
distance
r
from
thesurfaces
center
i
as
they
through
a
suspensolved
the
time-averaged
flow
field
in
the
E.
coli
swimming
he minismeasure
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is cons
movie
dat
However,
the
force
dipole
flow
sig
oftothe
cell
(Fig.
1D) illustrate
that
the
flow
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±
5 measured
ticles.
For
measurements
far
from
ckedplane
gfpcell force
bod
surfaces
of
the
sample
chamber,
and
recorded
∼
2
terabytes
of
2
and
resistive
µm/s.
As E.
colidisplays
rotate
about
their
swimming
direction,
their
cells
field
the
characteristic
1/r4 decay
oftoa the
forceside
dipole.
measured
flow
ofswam
thel
a 50
suspenµmmovie
from
the
top
and
bottom
for
the
data.
In field
this in
data
we dimensions
identified ∼is10cylindrically
rare events when
note that
in
the b
time-averaged
flow
three
fluid
trace
However,
the
force
dipole
flow
significantly
overestimates
the
cell1.5body,
where
the
flow
magnit
far from
achieved
ber,
and
recorded
∼
2
terabytes
of
cells
swam
in
the
focal
plane
for
>
s.
By
tracking
the
µm behind the ce
symmetric.measured
Our
measurements
capture
all
components
of
this
flow to when
the side of
the
cell
body,ofand
behind
the
4
and
veloci
didentified
bottom
for
the
length
the
flagellar
bun
at
varia
fluid
drag
on
the
∼
10
rare
events
fluid
tracers
in
each
of
the
rare
events,
relating
their
position
cylindrically symmetric flow, except the azimuthal flow due to
cell
body,
where
the
flow
magnitude
u(r)
is
nearly
constant
and
perfo
abytes
of
field
(r f
achieved
by
fitting
two
opposite
and
velocity
to
the
position
and
orientation
of
the
bacterium,
the rotation
of the
cell
about its the
body axis. The topology of
ne for
> 1.5fors.
By
tracking
the
length
of the
flagellar
bundle.
The
force
dipole
fit
was
solved
the
nts
when
the
spec
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
sw
are events, relating
their
position
Bacterial
flowdow
fiel
achieved
by1B),
fitting
two
opposite
force
monopoles
(Stokeslets)
dipole
le
plane
kingforce
the
dipole
flowtime-averaged
(Fig.
defined
by
solved
the
flow
field
in
the
E.
coli
swimming
field
(r
>
8
µm).
From
the
best
and
orientation
of
the
bacterium,
dipole
flow
descri
at
variable
locations
along
the
swimming
direction
to
the
far
value
of
position
µm/s.
As
plane down to 0.1% of the mean swimming
speed
V
=
22
±
5
0
the
specific
fitting
routines
fi
with good and
accura
h
i
e
average
over
all
tracers,
we
refield
(r
>
8
µm).
From
the
best
fit,
which
is
insensitive
to
and resi
A E. coli
r direction, their time-avera
ℓF swimming
acterium,
µm/s.
As
2rotate about their
ˆ
thisµm
approximatio
u(r)in= the
3(r̂.coli
d) −
1 r̂, routines
A=
, r̂fitting
= length
, regions,
[ℓ
1 ]= we
dipole
1.9
and
dip
ow
field
E.
swimming
2
the
specific
fitting
and
obtain
the
note
tha2
|r|
8πηdimensions
|r| is cylindrically
s, we retime-averaged
flow field in three
symmetric
a
wall.
Focusing
value
F is Fconsistent
with
opt
dipole
length
1.9±µm
and
dipole
force
= of
0.42
pN.
This
µm
beh
mean
swimming
speed
V0 ℓ==22
5 capture
symmetric.
Our
measurements
allof
components
this
wimming
and applying
the
cylindrica
and
resistive
force
theory
calculat
fluid
dra
of
Fforce,
is consistent
with
optical
trap
measurements
[45]
where
F isvalue
the dipole
ℓ the
distance
separating
the force
ut
direction,
their
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a
sligh
rotati
note
thatThe
in
the
best
fit,
the
cell
and
resistive
force
theory
calculations
[46].
It is
interesting
to
η the
viscosity
the
fluid,
dˆ the
orientation
vector
theof
flow
field
struc
the
rotation
ofisof
the
cell
about
itsunit
body
axis.
topology
on, pair,
their
three
dimensions
cylindrically
the measu
(swimming
direction)
the best
bacterium,
and
rbehind
the
distance
surfaces,
the
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
dipo
Bacteria
vector
relative
to
the
center
of
the
dipole.
Yet
there
are
some
ity
of
a
no-slip
sur
µm
behind
the
center
of
the
cell
body,
possibly
reflecting
the
force
dipole
flow
(Fig.
1B),
defined
by
fluid
drag
on
the
flagellar
bundle
tsexcept
of
this
flow
due
Dunkel,
Ganguly,
Cisneros,
Goldstein
(2011) to
PNAS
Fig. differences
1.Drescher,
Averagethe
flow
fieldazimuthal
created
by a single
freely-swimming
bacterium.
(A)
Experimentally measured flow field far from a surface. Stream lines indicate local
direction offl
dipole
close
to
the
cell
body
as
shown
by
the
residual
of
outward
streamlin
fluid
drag
on
the
flagellar
bundle.
flow. (B) Best fit force-dipole model, and (C) residual flow field, obtained by subtracting the best-fit dipole from the experimentally measured field. The presence of the flagella
E.coli
w due to
weak ‘pusher’ dipole
Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
10 ㎛
speed ~100 μm/s
dunkel@math.mit.edu
Chlamydomonas
Merchant et al (2007) Science
dunkel@math.mit.edu
Evolution of multicellularity
m in Applied Mathematics, and ¶BIO5 Institute, University of Arizona,
Providence, RI 02912
Eudorina
January 22, 2006)
Volvox oved AprilChlamydomonas
18, 2006 (received for review
reinhardtii
elegans
carteri
nes
lar
a
an
ent
lly
ges
nd
By
uid
in
ary
Gonum
Pleodorina
Volvox on
Fig. 1. Volvocine
green algae arranged
pectorale
californica according to typical
aureus colony radius R.
The lineage ranges from the single-cell Chlamydomonas reinhardtii (A), to
us,
Short
et al, PNAS 2013 Gonium pectorale (B), Eudorina elegans (C), to the somaundifferentiated
ng
dunkel@math.mit.edu
Volvox carteri
somatic cell
200 ㎛
cilia
daughter colony
dunkel@math.mit.edu
Asexual reproduction & inversion
2014 Goldstein lab
dunkel@math.mit.edu
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
dunkel@math.mit.edu
Volvox
Meta-chronal waves
Brumley et al (2012) PRL
Ecological implications
&
technical applications
Colloidal gel formation via spinodal decomposition (rapid demixing)
Peter Lu
Sedimentation
Mississippi
NASA Earth Observatory
Sedimentation
Particle separation
Falk Renth
Knotted water
Irvine lab (Chicago)
Brownian motion of small objects in fluids is biologically and technologically relevant (and interesting)
How can we describe these phenomena
mathematically ?
http://web.mit.edu/mbuehler/www/research/f103.jpg
x(t)
Basic idea
Split dynamics into
• deterministic part (drift)
• random part (diffusion, “noise”)
Stochastic Differential Equation
Over-damped dynamics
Newton:
Stokes:
Neglect inertia (Re=0):
How can we characterize randomness?
Probability space
A
B
[0, 1]
P[;] = 0
Expectation values of random variables
Excellent reviews of the topics discussed in this chapter can be found in Refs. [CPB08,
HTB90, GHJM98, HM09].
1.1
1.1.1
Random walks
Unbiased random walk (RW)
Consider the one-dimensional unbiased RW (fixed initial position X0 = x0 , N steps of
length `)
X N = x0 + `
N
X
Si
(1.1)
i=1
where Si 2 {±1} are iid. random variables (RVs) with P[Si = ±1] = 1/2. Noting that
E[Si ] =
E[Si Sj ] =
1
1
1 · + 1 · = 0,
2
2 
2
2 1
2 1
+ (1) ·
=
ij E[Si ] = ij ( 1) ·
2
2
1
(1.2)
ij ,
(1.3)
we find for the first moment of the RW
E[XN ] = x0 + `
N
X
E[Si ] = x0
(1.4)
i=1
d
By definition, forR some RV X with normalized non-negative probability density p(x) = dx
P[X  x],
we have E[F (X)] = dx p(x)F (x). For discrete RVs, we can think of p(x) as being a sum of suitably
1
Second moment (uncentered)
the second moment
E[XN2 ] = E[(x0 + `
N
X
Si ) 2 ]
i=1
= E[x20 + 2x0 `
N
X
Si + ` 2
i=1
= x20 + 2x0 · 0 + `2
= x20 + 2x0 · 0 + `2
= x20 + `2 N.
N X
N
X
Si Sj ]
i=1 j=1
N X
N
X
E[Si Sj ]
i=1 j=1
N X
N
X
ij
i=1 j=1
riance (second centered moment)
⇥
⇤
2
E (XN E[XN ])
= E[XN2 2XN E[XN ] + E[XN ]2 ]
= E[XN2 ] 2E[XN ]E[XN ] + E[XN ]2 ]
(1.5)
= x20 + 2x0 · 0 + `2
N X
N
X
i=1 j=1
VarianceX X
N
= x20 + 2x0 · 0 + `2
E[Si Sj ]
N
ij
i=1 j=1
= x20 + `2 N.
The variance (second centered moment)
⇥
⇤
2
E (XN E[XN ])
= E[XN2 2XN E[XN ] + E[XN ]2 ]
= E[XN2 ] 2E[XN ]E[XN ] + E[XN ]2 ]
= E[XN2 ] E[XN ]2
therefore grows linearly with the number of steps:
⇥
⇤
2
E (XN E[XN ]) = `2 N.
(1.5)
(1.6)
(1.7)
Continuum limit From now on, assume x0 = 0 and consider an even number of steps
N Let
= t/⌧ , where ⌧ > 0 is the time required for a single step of the RW and t the total time.
The probability P (N, K) := P[XN /` = K] to be at an even position x/` = K 0 after N
steps is given by the binomial coefficient
2
✓ ◆N ✓
◆
1
N
P (N, K) =
N K
2
2
✓ ◆N
1
N!
Excellent reviews of the topics discussed in this chapter can be found in Refs. [CPB0
HTB90, GHJM98, HM09].
1.1
1.1.1
Continuum limit
Random walks
Unbiased random walk (RW)
Consider the one-dimensional unbiased RW (fixed initial position X0 = x0 , N steps
length `)
X N = x0 + `
N
X
Si
(1.
i=1
where Si 2 {±1} are iid. random variables (RVs) with P[Si = ±1] = 1/2. Noting that
Let
E[Si ] =
E[Si Sj ] =
1
1
1 · + 1 · = 0,
2
2 
2
2 1
2 1
+ (1) ·
=
ij E[Si ] = ij ( 1) ·
2
2
1
(1.
ij ,
(1.
we find for the first moment of the RW
E[XN ] = x0 + `
N
X
i=1
E[Si ] = x0
(1.
= E[XN ]
= E[XN2 ]
2E[XN ]E[XN ] + E[XN ] ]
E[XN ]2
(1.6)
Continuum limit
therefore grows linearly with the number of steps:
⇥
⇤
2
E (XN E[XN ]) = `2 N.
(1.7)
Continuum limit From now on, assume x0 = 0 and consider an even number of steps
N = t/⌧ , where ⌧ > 0 is the time required for a single step of the RW and t the total time.
The probability P (N, K) := P[XN /` = K] to be at an even position x/` = K 0 after N
steps is given by the binomial coefficient
✓ ◆N ✓
◆
1
N
P (N, K) =
N K
2
2
✓ ◆N
1
N!
=
2
((N + K)/2)! ((N
K)/2)!
.
(1.8)
The associated probability density function (PDF) can be found by defining
p(t, x) :=
P (N, K)
P (t/⌧, x/`)
=
2`
2`
(1.9)
and considering limit ⌧, ` ! 0 such that
`2
D :=
= const,
2⌧
3
(1.10)
Continuum limit
yielding the Gaussian
p(t, x) '
r
1
exp
4⇡Dt
✓
x2
4Dt
◆
(1.11)
Eq. (1.11) is the fundamental solution to the di↵usion equation,
@t pt = D@xx p,
(1.12)
where @t , @x , @xx , . . . denote partial derivatives. The mean square displacement of the continuous process described by Eq. (1.11) is
Z
E[X(t)2 ] = dx x2 p(t, x) = 2Dt,
(1.13)
in agreement with Eq. (1.7).
Remark One often classifies di↵usion processes by the (asymptotic) power-law growth
of the mean square displacement,
E[(X(t)
X(0))2 ] ⇠ tµ .
• µ = 0 : Static process with no movement.
(1.14)
E[X(t)2 ] =
Z
dx x2 p(t, x) = 2Dt,
(1.13)
Different types of “diffusion”
in agreement with Eq. (1.7).
Remark One often classifies di↵usion processes by the (asymptotic) power-law growth
of the mean square displacement,
E[(X(t)
X(0))2 ] ⇠ tµ .
(1.14)
• µ = 0 : Static process with no movement.
• 0 < µ < 1 : Sub-di↵usion, arises typically when waiting times between subsequent
jumps can be long and/or in the presence of a sufficiently large number of obstacles
(e.g. slow di↵usion of molecules in crowded cells).
• µ = 1 : Normal di↵usion, corresponds to the regime governed by the standard Central
Limit Theorem (CLT).
• 1 < µ < 2 : Super-di↵usion, occurs when step-lengths are drawn from distributions
with infinite variance (Lévy walks; considered as models of bird or insect movements).
• µ = 2 : Ballistic propagation (deterministic wave-like process).
Continuous representation of Brownian trajectories ?
Brownian motion (constant drift)
1.2
1.2.1
SDEs and discretization rules
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
x(t)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
n motion
(i) B(0) = 0 with probability 1.
d discretization
(ii) B(t) is stationary,rules
i.e., for t > s
distribution as B(t
0 the increment B(t)
B(s) has the same
s).
stic process
by Eq.
(iii) X(t)
B(t) has described
independent increments.
That(1.19a)
is, for all tor,
> tequivalently,
> . . . > t > t , Eq. (1.20)
the random variables B(t ) B(t ), . . . , B(t ) B(t ), B(t ) are independently
d by the stochastic
di↵erential
equation
distributed (i.e.,
their joint distribution
factorizes).
(iv) B(t) has Gaussian distribution
with variance t for all t 2 (0, 1).
p
dX(t)
=is continuous
u dt +with2D
dB(t).
(1.25)
SDE
(v) B(t)
probability
1.
dt)
dt)
n
n
n 1
2
n 1
1
2
1
1
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
X(t) is increment of the stochastic particle trajectory X(t),
B(t) denotes an increment of the standard Brownian motion
1.2.1
SDEs and discretization rules
Wiener process
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
can also be represented by the stochastic di↵erential equation
p
dX(t) = u dt + 2D dB(t).
(1.25)
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
(or Wiener) process B(t), uniquely defined by the following properties3 :
(i) B(0) = 0 with probability 1.
(ii) B(t) is stationary, i.e., for t > s
distribution as B(t s).
0 the increment B(t)
B(s) has the same
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
the random variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
(iv) B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written in the form
p
Ẋ(t) = u + 2D ⇠(t).
(1.26)
The random driving function ⇠(t) is then referred to as Gaussian white noise, characterized
by
randomand
variables
B(tn ) B(tn rules
1 ), . . . , B(t2 )
1.2.1 theSDEs
discretization
B(t1 ), B(t1 ) are independently
distributed (i.e., their joint distribution factorizes).
The continuous stochastic process X(t) described by Eq. (1.19a) or, equivalently, Eq. (1.20)
has Gaussian
withdi↵erential
variance tequation
for all t 2 (0, 1).
can(iv)
alsoB(t)
be represented
bydistribution
the stochastic
p
(v) B(t) is continuous withdX(t)
probability
1.
= u dt + 2D dB(t).
(1.25)
Langevin equation
The probability distribution P governing the driving process B(t) is commonly known as
Here, dX(t) = X(t + dt) X(t) is increment of the stochastic particle trajectory X(t),
the Wiener measure.
whilst dB(t) = B(t + dt) B(t) denotes an increment of the standard Brownian motion
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically,
Eq. (1.25) is
3
(or Wiener) process B(t), uniquely defined by the following properties :
in the physics literature often written in the form
p
(i) B(0) = 0 with probability 1.
Ẋ(t) = u + 2D ⇠(t).
(1.26)
(ii) B(t) is stationary, i.e., for t > s
0 the increment B(t) B(s) has the same
The distribution
random driving
function
as B(t
s). ⇠(t) is then referred to as Gaussian white noise, characterized
by
(iii) B(t) has independent increments. That is, for all tn > tn 1 > . . . > t2 > t1 ,
h⇠(t)i
0 , B(tnh⇠(t)⇠(s)i
= (t B(t
s),1 ), B(t1 ) are independently
(1.27)
the random variables
B(t=
n)
1 ), . . . , B(t2 )
distributed (i.e., their joint distribution factorizes).
with h · i denoting an average with respect to the Wiener measure.
(iv)3 B(t) has Gaussian distribution with variance t for all t 2 (0, 1).
2
Note that, since X has dimensions of length and D has dimensions length /time, the Wiener process
B in Eq. (1.25) has units time1/2 .
(v) B(t) is continuous with probability 1.
The probability distribution P governing the driving process B(t) is commonly known as
the Wiener measure.
Although the derivative ⇠(t) = dB/dt is not well-defined mathematically, Eq. (1.25) is
in the physics literature often written
in the
form
Dirac’s
Delta-function
p
Ẋ(t) = u + 72D ⇠(t).
(1.26)
Overdamped dynamics
small
objects
(i) B(0) = of
0 with
probability
1. in fluids
Mean displacement
Jörn Dunkel⇤
⇤
(ii) B(t)
stationary,
i.e., for t > s
JörnisDunkel
distribution
as B(t s).
January
14, 2015
January 14, 2015
0 the increment B
(iii) B(t) has independent increments. That is, for all tn >
the random
variables B(tn ) B(tn 1 ), . . . , B(t2 ) B(t1 ),
p
distributed
(i.e., their joint distribution factorizes).
Ẋ(t)
= p 2D ⇠(s)
Ẋ(t) =
2D ⇠(s)
Direct integration with X(0) = 0 (iv) B(t) has Gaussian distribution with variance t for all t 2 (
Direct integration with X(0) = 0
t
p is Zcontinuous
(v) B(t)
with probability 1.
Z
X(t) = p2D t ds ⇠(s)
X(t)probability
= 2D 0 ds
⇠(s)
The
distribution
P governing the driving process B(
Averaging
Averaging
and, similarly,
and, similarly,
0
the Wiener measure.
the derivative
⇠(t) = dB/dt is not well-defined mat
⌧
Z Although
Z t
t
p
p
in Zthe
physics
literature
written
t ds
hX(t)i = ⌧p2D
⇠(s) =
ds h⇠(s)i
= 0 in the form
(1)
p 2D Z t often
hX(t)i =
2D 0 ds ⇠(s) = 2D 0 ds h⇠(s)i = 0
(1)
p
0
0
Ẋ(t) = u + 2D ⇠(t).
⌧ random

Z t driving function
Z t⇠(t) is then referred to as Gaussian
The
p
p
⌧

hX(t)22i = by p2D Z t ds ⇠(s) · p 2DZ t du ⇠(u)
hX(t) i
=
=
=
=
=
=
=
2D 0 ds ⇠(s) ·
2D 0 du ⇠(u)
⌧
Z t 0Z t
0
⌧
Z t Z t
h⇠(t)i = 0 ,
h⇠(t)⇠(s)i = (t s),
2D
ds
du ⇠(s) · ⇠(u)
2D 0 ds 0 du ⇠(s) · ⇠(u)
Z t0
withZZ htt·0i denoting
an average with respect to the Wiener measur
Z t
2D
ds
du h⇠(s) · ⇠(u)i
2D3 Note
h⇠(s) · ⇠(u)i
0 ds 0 du
of length and D has dimensions leng
Z0 t that,
Z0 t since X has dimensions
1/2
B in ZEq.
(1.25)
Z has units time .
2D t ds t du (s u)
2D 0 ds 0 du (s u)
Z0 t
0
Z t
X(t) =
2D
ds ⇠(s)
p0
Ẋ(t) = 2D ⇠(s)
Mean square displacement
aging
Direct integration with X(0) = 0
hX(t)i =
⌧
p
2D
Z
t
Z
p
p
t
Z
t
ds ⇠(s)
2D
X(t)
= 2D= ds
⇠(s) ds h⇠(s)i = 0
0
0
0
Averaging
similarly,
Z t
p
⌧
Z t
p
hX(t)i
=
2D
ds ⇠(s)
2
hX(t) i
and, similarly,
⌧
=
=
2
hX(t) i
=
=
=
=
⌧
2D
0
Z
0
t
Z
2D⌧p dsZ
=
2D
=
Z
⌧
0 2D
t
Z
ds
2D
0
Z
Z

Z t
p
= 2D
ds h⇠(s)i = 0
ds ⇠(s) · 0 2D
du ⇠(u)
t
t
0
t
ds
Z
t
0
du ⇠(s)·p⇠(u)Z
0 ds ⇠(s) ·
t
0
p
2D
dut h⇠(s) · ⇠(u)i
du ⇠(s) · ⇠(u)
Z t Z 0Z Zt 0
t
t
2D
= 2D ds ds dudu (s
h⇠(s) ·u)
⇠(u)i
0 0 0 0
Z tZ t Z t
= 2D ds ds
du (s u)
2D
0
0
0 Z t
2Dt
= 2D
0
=
2Dt
ds
t
du ⇠(u)
0
(1)