LowRe hydrodynamics &
microbial locomotion
18.354 - L15
dunkel@mit.edu
Why microbial hydrodynamics ?
5㎛
10 ㎛
• micro-machines
• hydrodynamic propulsion • > 50% global biomass • gut flora, biofilms, ...
• global food web
• > 50% global carbon
fixation
30 ㎛
Whitman et al (1998) PNAS
100 ㎛
!
Guasto et al (2012) Annu Rev Fluid Mech
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
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)
,
|x| ⇤⌃ .
(1
Zero-Re flow
E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
dunkel@math.mit.edu
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
Chlamydomonas alga
10 ㎛
~ 50 beats / sec
Goldstein et al (2011) PRL
10 ㎛
speed ~100 μm/s
dunkel@math.mit.edu
Volvox carteri
200 ㎛
10 ㎛
Chlamydomonas
reinhardtii
dunkel@math.mit.edu
Stroke
Sareh et al (2013) J Roy Soc Interface
dunkel@math.mit.edu
Volvox carteri
beating frequency
25Hz
Sareh et al (2013) J Roy Soc Interface
dunkel@math.mit.edu
Meta-chronal waves
Brumley et al (2012) PRL
dunkel@math.mit.edu
Dogic lab (Brandeis)
dunkel@math.mit.edu
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
dunkel@math.mit.edu
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)
,
|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’
r
2
B@024@A↵ ⇢G:8<3@820: >=:G38;4B7G:A8:=F0<4 A0;>:4 270;⌦
14@A ⌃✓ ;; @038CA ⌅⌘ ;; 74867B⌥ E4@4 >@4>0@43 >028⌦
⇤43 0<3 ⇤::43 5=::=E8<6 -.↵ ⇠:: 4F>4@8;4<BA E4@4 >4@⌦
5=@;43 0B @==; B4;>4@0BC@4 E8B7 B74 :0A4@ >@=D838<6 B74
=<:G :867B A=C@24↵ *4 5=2CA43 =< 0 >:0<4 ✓ ⇥; 8<A834
B74 270;14@ B= ;8<8;8H4 AC@5024 4⇥42BA 0<3 @42=@343
;=D84A 0B ✏✓ 5>A ⌃ 0AB20; &⇠⇣ $7=B@=<⌥↵ ⌧027 ;=D84
swimming speed ~ 100 ㎛/sec
E0A 0<0:GA43 E8B7 AB0<30@3 0:6=@8B7;A B= B@029 1=B7 24::A
0<3 B@024@A↵ =@ 4027 24:: AE8;;8<6 0:=<6 B74 5=20: >:0<4
5=@ ;=@4 B70< A ⌃⇧
1=3G :4<6B7A⌥ E4 2=::42B43 B74
1008<AB0<B0<4=CA
㎛
D4:=28BG =5 0:: B@024@A <=@;0:8H43 1G B74
AE8;;4@⇧A A>443 C> B= 0 38AB0<24 =5 ⌘ ↵ '74 @4AC:B⌦
8<6 ⇣⌅⇣ ⇤ ⌃ D4:=28BG D42B=@A E4@4 18<<43 8<B= 0 ✏⌅✓ ⇥;
A?C0@4 6@83 ⌃A7=E< 8< 86↵ ⌘ 14:=E⌥ 0<3 B74 ;40< =5
B74 E4::⌦@4A=:D43 0CAA80< 38AB@81CB8=< 8< 4027 18< E0A
B094< 0A 0 :=20: ;40AC@4 =5 B74 ⌅=E ⇤4:3↵
< 1=B7 4F>4@8;4<BA ⇥ E8:: 8<3820B4 B74 AE8;;4@⇧A
A>443 E78:4 ⌅⌃⇤⌥ 0<3 ⇧⌃⇤⌥ ⌫ ⌅⌃⇤⌥ ⇥ 0@4 B74 D4:=28BG
⇤4:3 8< B74 :01=@0B=@G 0<3 2=;=D8<6 5@0;4A @4A>42B8D4:G↵
⇥
⇠ BG>820: 4F>4@8;4<B0: ⌅=E ⇤4:3 0@=C<3 ⇤ ⇣ ⌘ 8A
⇢⌃
A7=E< 8< 86↵ ⌃0⌥↵ *4 ⇤B B74A4 ⇤4:3A B= 0 AC>4@>=⌦
A8B8=< =5 0 C<85=@; 10296@=C<3 D4:=28BG ⌃⇥ ⌥ 0 &B=94A:4B
⇧
⌃&B⌥ 0 AB@4AA:4B ⌃AB@⌥ 0<3 0 A=C@24 3=C1:4B ⌃A3⌥
<
0:
⇧
64
✓
74
<
8⌦
Volvox Chlamy
swimming
PHYSICAL
PRL 105,
168101 speed
(2010) ~ 50 ㎛/sec
PIV
⇧⌅ ⇧ ⌃⇤⌥ ⌫
/
⌦ ⌃
⌥ ⌃
⇥
⌃
/⇤/⇤⌥ ⇥ ⌃
/
⇣⌃ ⌥ ⌥
⇥
⇥
/⇤
⌃ ⌥
⌥⇤
⇤
⇤
⇣
⌅
/⇤/⇤ ⇥ ⌃
/
/ 8A B74 C>E0@3 D4@B820: C<8B
E74@4 8A B74 C<8B B4<A=@ ⌃
D42B=@ /⇤ ⌫ ⇤⌥ 0<3 ⇤ 8A ;40AC@43 5@=; B74 24<B4@
=5 B74 =@60<8A; ⌃ ⇥ ⇧ ⇥ ⌥↵ '74 =@84<B0B8=< =5 0:: ;C:B8⌦
>=:4A 8A ⇤F43 B= 14 0:=<6 B74 D4@B820: 0<3 E4 0@4 :45B
E8B7 A8F >0@0;4B4@A ⌃⌦ ⇧
⇧ ⌥ ⌃ ⇧ ⌥⇤ ⇧ ⇥ ⇧ ⇥ ⌥↵ '74 ⇤BA
al (2010)
PRL
34A2@814 Drescher
@4;0@901:GetE4::
B74 4F>4@8;4<B0:
⌅=E 0:;=AB
2
FIG. 4 (color
online).
Timeand
azimuthally-averaged
3D : v 1/r
from velocity vectors (blue [dark gray]). The spiraling nea
velocity field.
A color
scheme
indicates
flow speed magnit
2D
:
v
1/r
model: flagellar thrust is distributed among two Stokes
arrows), whose sum balances drag on the cell body (cen
separate colors in the inset, compared to results from the
... no dipoles !
flow may be important [30]. We are currently investi
Guasto
et similar
al (2010)
PRL
whether
conclusions
hold for the flow field a
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
Twitching motility
Type-IV Pili
Twitching motility
Pseudomonas
Amoeboid locomotion
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