Dilute bacterial
suspensions
18.S995 - L04
dunkel@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 run & tumble motion
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
Chen et al (2011) EMBO Journal
source: wiki
movie: V. Kantsler
Bacterial run & tumble motion
Rep. Prog. Phys. 72 (2009) 096601
E Lauga and T R Powers
(a)
(b)
(c)
(d)
Figure 15. Bundling of bacterial flagella. During swimming, the bacterial flagella are gathered in a tight bundle behind the cell as it moves
through the fluid ((a) and (d)). During a tumbling event, the flagella come out the bundle (b), resulting in a random reorientation of the cell
before the next swimming event. At the conclusion of the tumbling event, hydrodynamic interactions lead to the relative attraction of the
flagella (c), and their synchronization to form a perfect bundle (d).
necessary conditions for phase locking [243–246]. An early
are bundled together tightly behind the cell (figure 15(a)
for more movies, see also
http://www.rowland.harvard.edu/labs/bacteria/movies/index.php
dunkel@math.mit.edu
n(y) [2N cells]
n (y) [N cells]
combined with Percoll (2:3 ratio) to match the medium and
ion 200 !m).
ls by the closest wall. Our model allows us to predict
cell buoyant densities [18]. A droplet of the cell mixture is
th
a constant
ulting steady-state cell distribution and is exploited
week ending
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T
T
E
R
S
deposited
between
two
glass
cover
slips,
previously
ear
the
walls
18
JULY 2008
PRL
038102 (2008)
ain an estimate for
the101,
flagellar
propulsive force in
cleaned in a mixture of ethanol saturated with potassium
umanE.spermaming
coli.
experimental
is illustrated
Fig.
1.
hydroxide,
rinsed in
with
ultrapure
water,
and allowedby Surfaces
nimal
sperma-procedure
Hydrodynamic
Attraction
offiltered
Swimming
Microorganisms
(smooth-swimming
strain
is grown
to airHCB-437
dry. The[17])
cover
slips
are separated
by a distance
H, 4,*
nce
of three1
2
2,3
Allison
P.
Berke,
Linda
Turner,
Howard
C.
Berg,
and
Eric
Lauga
exponential
phase
in
T
broth
(1%
tryptone,
0.5%
controlled
by layers
ofS Iother
cover
slips
(No.
1.5),
and Massachusetts 02139,
ions of model 1Department
week ending
of Mathematics,
Massachusetts
Institute
ofETechnology,
77
Mass.
Avenue,
Cambridge,
USA
P
H
Y
C
A
L
R
V
I
E
W
L
E
T
T
E
R
S
washed 3PRL
times
by
centrifugation
(2200
g
for
2
18
JULY
2008
101,
038102
(2008)
Rowland Institute at Harvard, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA
] supported an 3
), and then resuspended
a motility
medium
Department ofinMolecular
and Cellular
Biology, Harvard University, 16 Divinity Avenue, Cambridge, Massachusetts 02138, USA
4
ynamic
interverified
by caliperpH
measurement.
Aofphase-contrast
60 University of California
Department
Mechanical
Aerospace Engineering,
120
H = 100 µm San Diego,
M
potassium
phosphate,
7.0,
0.1 mM
EDTA).andmicro9500 Gillman
Drive, La Jolla, California 92093-0411, USA
scope
Optiphot-2) using 600!
magnification
exp. (N cells)
mputations
for (Nikon
(a)
0 (polyvinylpyrrolidone)
is
added
(0.005%)
to
pre11 December 2007; published 17 July 2008) theory
(depth of field 4:3 !m) and equipped (Received
with a shuttered
exp. (2N cells)
ber swimmers
dsorption
ofCCD
cellsvideo
to glass,
and
the swimming
finalElectronics
suspension
isenvironments
Cells
in confined
the steady-state 80
camera
(Marshall
V1070)
set for are attracted
40 by surfaces. We measure
theory
distribution
of
bacteria (Escherichia coli) between two glass plates. In agreement with
ned
(2:3 ratio)
to
match
thesmooth-swimming
medium
andpopulation
an exposure
of 1 ms=frame
is used
to image the
ngewith
in Percoll
swim-
earlier
studies,signal
we find
strong
of the cell concentration at the boundaries. We demonstrate
of swimming
Theof
video
is asent
toisincrease
aHMacG4
oyant densities
[18]. A cells.
droplet
the
cell
mixture
theoretically that hydrodynamic interactions of the swimming cells with solid surfaces lead to their
40
equipped
with
an
LG-3
frame
grabber
(Scion Image) and
20
(b)
ted
between
two
glass
cover
slips,
reorientation
in the previously
direction parallel to the surfaces, as well as their attraction by the closest wall. A
of swimming
IMAGEJ software [National
Institutes
Health distribution
(NIH), of swimming cells, which compares favorably with our
model is derived
forpotassium
theof
steady-state
d in a mixture
of
ethanol
saturated
with
Bethesda,
MD]. Wemeasurements.
capture 2-second
movies
20toframes
We exploit
our at
data
estimate the flagellar propulsive force in swimming E. coli.
distribution
of
0
xide, rinsed per
with
ultrapure
filteredthewater,
and
allowed cells by
0
second
and measure
number
of swimming
0 PACS20numbers:4047.63.Gd,6087.17.Jj,80
between
twoslipscells
DOI: 10.1103/PhysRevLett.101.038102
87.18.Ed100
dry.
The cover
areswimming
separated
by afaster
distance
counting
at speed
than 1H,
body
length
y
y (µm)
obtain
results
per
second.
We
start
5
!m
above
the
lower
glass
surface;
lled by layers of other
coverofslips
(No.microorganisms
1.5), and involved in the cells
The(c)
majority
swimming
(d)by the closest wall. Our model allows us to predict
n (y)
we thenhuman
bring the
plane and
of focus
up 10
andinrepeat
the
]. We demonthe60resulting steady-state
cellµmdistribution and is exploited
functions
diseases
are!m
found
geometrically
H = 200
measurement
until
we
reach
within
5
!m
of
the
upper
to obtain an estimate for the flagellar propulsive force in
confined environments. Spermatozoa in the female reprocell profile
is surface.
exp.
glass
Experiments
are
then
repeated
with
other
swimming
E. coli.
ductive
tract1 swim
in online).
constricted domains
[1]. Bacteriaof the
FIG.
(color
Representation
experimental
protheory
cell samples
cover slips.
uments based
Our
experimental
procedure
is illustrated in Fig. 1.
make and
theirsets
wayof through
host cells and tissues [2] and
40are mixed with a
cedure.
(a)
Smooth-swimming
E.
coli
cells
In our
protocol,in two
parameters canbiofilms
be varied:
the dis-[3].
E. coli (smooth-swimming strain HCB-437 [17]) is grown
aggregate
antibiotic-resistant
on surfaces
etween swimdensity-matched
fluid.
(b) The
cell
mixture
is midexponential
deposited between
tance H between
the
cover slips
(we
H "motility
100 or
to
phase in T broth (1% tryptone, 0.5%
Despite
the two
ubiquitous
nature
of chose
biological
near
e interactions
Htwo
200 !m)
and
thenot
cell
density
of (separation
the
finalthemixture,
i.e.,
the H). (c)
NaCl),
3 times of
by centrifugation (2200 g for
surfaces,
much
is known
about
physical
consequenglass
plates
distance
Thewashed
distribution
20
8 min), and then resuspended in a motility medium
ces overall
of locomotion
in a confined
environment
size of the
cell population
(when
H " 100[4,5].
!m, Perhaps
we
ection parallel
swimming
cells
is
imaged
as
a
function
of
the distance y from
(10 mM potassium phosphate, pH 7.0, 0.1 mM EDTA).
the simplest
observed
effect of
locomotion
wallsofis the
performed
additional
experiments
doubling
the near
number
tial condition
the lower
surface.
(d)
Example
imagePVP-40
obtained
from data is added (0.005%) to pre(polyvinylpyrrolidone)
accumulation
of swimming
oninsurfaces.
In an
1963,
cells). The
experimental
results are cells
shown
Fig. 2; of
verti0
adsorption
glass, and150
the final 200
suspension is
Rothschild
measured
distribution
of bull
spermatozoa
0
50of cells to
100
attractioncal of
acquisition
inthethe
first
above
the glassvent
surface.
errors
bars
represent
statistics
on
tenlayer
different
experiy
combined with Percoll y(2:3
ratio) to match the medium and
swimming
between
glass
200in!m).
(µm)
ments and
horizontal
errortwo
bars
theplates
depth(separation
of field. As
cell buoyant densities [18]. A droplet of the cell mixture is
cell that
distribution
was
withnear
a constant
(d)nonuniform,
Ref. [6],The
we find
the cell profile
peaks strongly
the
between Experimental
two glass cover
slips, previously
density in the center strongly increasing near the wallsFIG.deposited
2 (color online).
data: number
of swimwalls,
with
a
nearly
constant
cell
density
about
20
!m
038102-1
2008
The spermaAmerican
Physical
Society
ofofethanol
saturated
potassium
[6]. Similar results were later©
obtained
for human
mingcleaned
cells
ninasaamixture
function
the distance
to thewith
bottom
cover
Goals
•
minimal SDE model for microbial swimming
•
wall accumulation & density profile
dunkel@math.mit.edu
1.3
Dilute microbial suspensions
A minimalist model for the locomotion of an isolated microorganism (e.g., alga or bacterium) with position X(t) and orientation unit vector N (t) is given by the coupled system
of Ito SDEs
1.3
Dilute microbial suspensions
p
dX = V N dt + 2D ⇤ dB(t),
(1.45a)
A minimalist model for the locomotion ofT anpisolated microorganism (e.g., alga or bacdNX(t)
= and
(1 orientation
d)DR N dt unit
+ vector
2DR (IN (t)
NN
) ⇤ dWby(t).
terium) with position
is given
the coupled (1.45b)
system
of Ito SDEs
Here, V is the self-swimming speed of the organism, DT the translational di↵usion coefp
ficient, and DR dX
the rotational
di↵usion
(I N N ) is an orthogonal projector
= V N dt + 2Dcoefficient,
(1.45a)
T ⇤ dB(t),
p
with d-dimensional unit matrix I, and B and W are two independent d-dimensional BrowdN =Eq.
(1(1.45a)
d)DRdescribes
N dt + locomotion
2DR (I N
Nto
) ⇤translational
dW (t).
(1.45b)
nian motion processes.
due
di↵usion
and
self-swimming
in the direction
of the
N , Dandthe
Eq.translational
(1.45b) models
changes
in
Here,
V is the self-swimming
speed
of orientation
the organism,
di↵usion
coefT
orientation
on thedi↵usion
d-dimensional
unit sphere.
ficient,
and DasR di↵usion
the rotational
coefficient,
(I N N ) is an orthogonal projector
To confirm that Eq. (1.45b) conserves the unit length of the orientation vector, |N |2 = 1
with d-dimensional unit matrix I, and B and W are two independent d-dimensional Browfor all t, it is convenient to rewrite Eqs. (1.45) in component form:
nian motion processes. Eq. (1.45a) describes locomotion due to translational di↵usion and
p orientation N , and Eq. (1.45b) models changes in
self-swimming in dX
the direction
of
the
2DT ⇤ dBi (t),
(1.46a)
i = V Ni dt +
p
orientation as di↵usion on the d-dimensional unit sphere.
dNEq.
(1 conserves
d)DR Nj dtthe
+ unit
2Dlength
(1.46b)
j =
R ( jk of N
j Norientation
k ) ⇤ dWk (t).vector, |N
To confirm that
(1.45b)
the
|2 = 1
forFor
allthe
t, itconstraint
is convenient
(1.45) we
in component
|N |2to=rewrite
1 to beEqs.
satisfied,
must have form:
d|N |2 = 0. Applying the dp see Eq. (A.12), to F (N ) = |N |2 , one finds indeed
dimensional version of Ito’s formula,
dXi = V Ni dt + 2DT ⇤ dBi (t),
(1.46a)
that
p
dNj = (1 d)DR Nj dt + 2DR ( jk Nj Nk ) ⇤ dWk (t).
(1.46b)
d|N |2 = 2Nj ⇤ dNj + @Ni @Nj Nk Nk DR ( ij Ni Nj ) dt
i
For the constraint |N |2 = 1h to be satisfied, wepmust have d|N |2 = 0. Applying
the d(1 d)Dsee
dt +(A.12),
2DR to
( jkF (NN)j N
⇤ dW
(t) finds
+
j ⇤ formula,
R NjEq.
dimensional version= of2N
Ito’s
=k )|N
|2 , kone
indeed
that
@Ni ( jk Nk + Nk jk ) DR ( ij Ni Nj ) dt
= 2(1 d)DR dt +
d|N |2 = 2N
Ni Nj ) dt
j ⇤ dNj + @Ni @Nj Nk Nk DR ( ij
http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.0020044
h + ik jk ) DR ( ij p
i
( jk ik
Ni Nj ) dt
elf-swimming
in the direction
, and
Eq. (1.45b)
models Browchanges in
with d-dimensional
unit matrixofI,the
andorientation
B and W areN
two
independent
d-dimensional
rientation
as di↵usion
onEq.
the(1.45a)
d-dimensional
unit sphere.
nian motion
processes.
describes locomotion
due to translational di↵usion and
Toself-swimming
confirm thatinEq.
conserves
the unit length
the(1.45b)
orientation
the(1.45b)
direction
of the orientation
N , andofEq.
modelsvector,
changes|N
in|2 = 1
BM on
the
unit
sphere
as di↵usion
the d-dimensional
or allorientation
t, it is convenient
toonrewrite
Eqs. (1.45)unit
in sphere.
component form:
2
To confirm that Eq. (1.45b) conserves
the
unit
length
of
the
orientation
vector,
|N
|
=1
p
for all t, it isdX
convenient
to rewrite Eqs.
component form:
2DT(1.45)
⇤ dBiin(t),
(1.46a)
i = V Ni dt +
p
p
dNjdX=
(1 d)DR N2D
( jk Nj Nk ) ⇤ dWk (t).
(1.46b)
j dtT+
⇤ dB2D
(1.46a)
i = V Ni dt +
i (t),R
dNj 2 = (1
p
d)DR Nj dt + 2DR (
jk
Nj Nk ) ⇤ dW2k (t).
(1.46b)
or the constraint |N | = 1 to be satisfied, we must have d|N | = 0. Applying the dimensional
Ito’s
see Eq. we
(A.12),
to F (N
|N Applying
|2 , one finds
For the version
constraintof|N
|2 =formula,
1 to be satisfied,
must have
d|N)|2 =
= 0.
the d-indeed
hat dimensional version of Ito’s formula, see Eq. (A.12), to F (N ) = |N |2 , one finds indeed
that
d|N |2 =2 2Nj ⇤ dNj + @Ni @Nj Nk Nk DR ( ij Ni Nj ) dt
i
d|N | = 2Njh⇤ dNj + @Ni @Nj Nk Nk pDR ( ij Ni Nj ) dt
= 2Nj ⇤ (1h d)DR Nj dt + p 2DR ( jk Nj Nk ) ⇤ dWik (t) +
= 2Nj ⇤ (1
d)DR Nj dt +
2DR (
jk
Nj Nk ) ⇤ dWk (t) +
@Ni@(N jk( N
Ni N
k + Nk jk ) DR ( ij N
iN
j ) dt
jk Nk + Nk jk ) DR ( ij
j ) dt
i
= =2(12(1 d)D
dtdt++
RR
d)D
( jk( jk
) )DDRR(( ijij N
NiiNNj )j )dtdt
ik +ik ikjkjk
ik +
= =0. 0.
(1.47)(1.47)
To understand the dynamics (1.46), it is useful to compute the orientation correlation,
To understand
the dynamics (1.46), it is useful to compute the orientation correlation
hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)],
hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)],
(1.48)
where we have assumed (w.l.o.g.) that N (0) = ez . Averaging Eq. (1.46b), we find that
(1.48)
= 2Nj ⇤ (1
d)DR Nj dt +
2DR (
jk
Nj Nk ) ⇤ dWk (t) +
@Ni ( jk Nk + Nk jk ) DR ( ij Ni Nj ) dt
= 2(1 d)DR dt +
( jk ik + ik jk ) DR ( ij Ni Nj ) dt
= 0.
Orientation correlations
(1.47)
To understand the dynamics (1.46), it is useful to compute the orientation correlation,
hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)],
(1.48)
where we have assumed (w.l.o.g.) that N (0) = ez . Averaging Eq. (1.46b), we find that
d
E[Nz (t)] = (1
dt
d)DR E[Nz (t)],
(1.49)
implying that, in this model, the memory loss about the orientation is exponential
hN (t) · N (0)i = e(1
12
d)DR t
,
(1.50)
Mean square displacement
which is approximately true for many microorganisms. Another relevant quantity is the
mean square displacement E[X(t)2 ], assuming that X(0) = 0. Using Ito’s formula,
d|X|2 = 2Xj ⇤ dXj + @Xi @Xj Xk Xk DT ij dt
= 2Xj ⇤ dXj + ( jk ik + ik jk ) DT ij dt
= 2Xj ⇤ dXj + 2d DT dt
p
= 2Xj [V Nj dt + 2DT ⇤ dBj (t)] + 2d DT dt,
(1.51)
averaging and dividing by dt, gives
d
E[X 2 ] = 2V E[X(t)N (t)] + 2d DT .
dt
(1.52)
The expectation value on the rhs. can be evaluated by making use of Eq. (1.50):
Z t
E[X(t) · N (t)] = E
dX(s) · N (t)
0 Z t
= VE
ds N (s) · N (t)
0
Z t
= V
ds hN (t) · N (s)i
Z0 t
= V
ds e(1 d)DR (t s)
0
⇥
V
=
1
(d 1)DR
e
(1 d)DR t
⇤
.
By inserting this expression into Eq. (1.52) and integrating over t, we find
= V
0
Z
t
ds hN (t) · N (s)i
Mean square
displacement
= V
ds e
Z0 t
(1 d)DR (t s)
0
⇥
V
=
1
(d 1)DR
e
(1 d)DR t
⇤
.
By inserting this expression into Eq. (1.52) and integrating over t, we find
2
⇥
2V
2
E[X ] =
(d
2
2
(d 1) DR
1)DR t + e
(1 d)DR t
⇤
1 + 2dDT t.
(1.53)
If DT is small, then at short times t ⌧ DR 1 the motion is ballistic
E[X 2 ] ' V 2 t2 + 2dDT t,
(1.54)
At large times, the motion becomes di↵usive, with asymptotic di↵usion constant
E[X 2 ]
2V 2
lim
=
+ 2dDT .
t!1
t
(d 1)DR
(1.55)
Inserting typical values for bacteria, V ⇠ 10µm/s and DR ⇠ 0.1/s, and comparing with
DT ⇠ 0.2µm2 /s for a micron-sized colloids at room temperature, we see that active swimming and orientational di↵usion dominate the di↵usive dynamics of microorganisms at
long times.
13
n(y) [2N cells]
n (y) [N cells]
combined with Percoll (2:3 ratio) to match the medium and
ion 200 !m).
ls by the closest wall. Our model allows us to predict
cell buoyant densities [18]. A droplet of the cell mixture is
th
a constant
ulting steady-state cell distribution and is exploited
week ending
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T
T
E
R
S
deposited
between
two
glass
cover
slips,
previously
ear
the
walls
18
JULY 2008
PRL
038102 (2008)
ain an estimate for
the101,
flagellar
propulsive force in
cleaned in a mixture of ethanol saturated with potassium
umanE.spermaming
coli.
experimental
is illustrated
Fig.
1.
hydroxide,
rinsed in
with
ultrapure
water,
and allowedby Surfaces
nimal
sperma-procedure
Hydrodynamic
Attraction
offiltered
Swimming
Microorganisms
(smooth-swimming
strain
is grown
to airHCB-437
dry. The[17])
cover
slips
are separated
by a distance
H, 4,*
nce
of three1
2
2,3
Allison
P.
Berke,
Linda
Turner,
Howard
C.
Berg,
and
Eric
Lauga
exponential
phase
in
T
broth
(1%
tryptone,
0.5%
controlled
by layers
ofS Iother
cover
slips
(No.
1.5),
and Massachusetts 02139,
ions of model 1Department
week ending
of Mathematics,
Massachusetts
Institute
ofETechnology,
77
Mass.
Avenue,
Cambridge,
USA
P
H
Y
C
A
L
R
V
I
E
W
L
E
T
T
E
R
S
washed 3PRL
times
by
centrifugation
(2200
g
for
2
18
JULY
2008
101,
038102
(2008)
Rowland Institute at Harvard, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA
] supported an 3
), and then resuspended
a motility
medium
Department ofinMolecular
and Cellular
Biology, Harvard University, 16 Divinity Avenue, Cambridge, Massachusetts 02138, USA
4
ynamic
interverified
by caliperpH
measurement.
Aofphase-contrast
60 University of California
Department
Mechanical
Aerospace Engineering,
120
H = 100 µm San Diego,
M
potassium
phosphate,
7.0,
0.1 mM
EDTA).andmicro9500 Gillman
Drive, La Jolla, California 92093-0411, USA
scope
Optiphot-2) using 600!
magnification
exp. (N cells)
mputations
for (Nikon
(a)
0 (polyvinylpyrrolidone)
is
added
(0.005%)
to
pre11 December 2007; published 17 July 2008) theory
(depth of field 4:3 !m) and equipped (Received
with a shuttered
exp. (2N cells)
ber swimmers
dsorption
ofCCD
cellsvideo
to glass,
and
the swimming
finalElectronics
suspension
isenvironments
Cells
in confined
the steady-state 80
camera
(Marshall
V1070)
set for are attracted
40 by surfaces. We measure
theory
distribution
of
bacteria (Escherichia coli) between two glass plates. In agreement with
ned
(2:3 ratio)
to
match
thesmooth-swimming
medium
andpopulation
an exposure
of 1 ms=frame
is used
to image the
ngewith
in Percoll
swim-
earlier
studies,signal
we find
strong
of the cell concentration at the boundaries. We demonstrate
of swimming
Theof
video
is asent
toisincrease
aHMacG4
oyant densities
[18]. A cells.
droplet
the
cell
mixture
theoretically that hydrodynamic interactions of the swimming cells with solid surfaces lead to their
40
equipped
with
an
LG-3
frame
grabber
(Scion Image) and
20
(b)
ted
between
two
glass
cover
slips,
reorientation
in the previously
direction parallel to the surfaces, as well as their attraction by the closest wall. A
of swimming
IMAGEJ software [National
Institutes
Health distribution
(NIH), of swimming cells, which compares favorably with our
model is derived
forpotassium
theof
steady-state
d in a mixture
of
ethanol
saturated
with
Bethesda,
MD]. Wemeasurements.
capture 2-second
movies
20toframes
We exploit
our at
data
estimate the flagellar propulsive force in swimming E. coli.
distribution
of
0
xide, rinsed per
with
ultrapure
filteredthewater,
and
allowed cells by
0
second
and measure
number
of swimming
0 PACS20numbers:4047.63.Gd,6087.17.Jj,80
between
twoslipscells
DOI: 10.1103/PhysRevLett.101.038102
87.18.Ed100
dry.
The cover
areswimming
separated
by afaster
distance
counting
at speed
than 1H,
body
length
y
y (µm)
obtain
results
per
second.
We
start
5
!m
above
the
lower
glass
surface;
lled by layers of other
coverofslips
(No.microorganisms
1.5), and involved in the cells
The(c)
majority
swimming
(d)by the closest wall. Our model allows us to predict
n (y)
we thenhuman
bring the
plane and
of focus
up 10
andinrepeat
the
]. We demonthe60resulting steady-state
cellµmdistribution and is exploited
functions
diseases
are!m
found
geometrically
H = 200
measurement
until
we
reach
within
5
!m
of
the
upper
to obtain an estimate for the flagellar propulsive force in
confined environments. Spermatozoa in the female reprocell profile
is surface.
exp.
glass
Experiments
are
then
repeated
with
other
swimming
E. coli.
ductive
tract1 swim
in online).
constricted domains
[1]. Bacteriaof the
FIG.
(color
Representation
experimental
protheory
cell samples
cover slips.
uments based
Our
experimental
procedure
is illustrated in Fig. 1.
make and
theirsets
wayof through
host cells and tissues [2] and
40are mixed with a
cedure.
(a)
Smooth-swimming
E.
coli
cells
In our
protocol,in two
parameters canbiofilms
be varied:
the dis-[3].
E. coli (smooth-swimming strain HCB-437 [17]) is grown
aggregate
antibiotic-resistant
on surfaces
etween swimdensity-matched
fluid.
(b) The
cell
mixture
is midexponential
deposited between
tance H between
the
cover slips
(we
H "motility
100 or
to
phase in T broth (1% tryptone, 0.5%
Despite
the two
ubiquitous
nature
of chose
biological
near
e interactions
Htwo
200 !m)
and
thenot
cell
density
of (separation
the
finalthemixture,
i.e.,
the H). (c)
NaCl),
3 times of
by centrifugation (2200 g for
surfaces,
much
is known
about
physical
consequenglass
plates
distance
Thewashed
distribution
20
8 min), and then resuspended in a motility medium
ces overall
of locomotion
in a confined
environment
size of the
cell population
(when
H " 100[4,5].
!m, Perhaps
we
ection parallel
swimming
cells
is
imaged
as
a
function
of
the distance y from
(10 mM potassium phosphate, pH 7.0, 0.1 mM EDTA).
the simplest
observed
effect of
locomotion
wallsofis the
performed
additional
experiments
doubling
the near
number
tial condition
the lower
surface.
(d)
Example
imagePVP-40
obtained
from data is added (0.005%) to pre(polyvinylpyrrolidone)
accumulation
of swimming
oninsurfaces.
In an
1963,
cells). The
experimental
results are cells
shown
Fig. 2; of
verti0
adsorption
glass, and150
the final 200
suspension is
Rothschild
measured
distribution
of bull
spermatozoa
0
50of cells to
100
attractioncal of
acquisition
inthethe
first
above
the glassvent
surface.
errors
bars
represent
statistics
on
tenlayer
different
experiy
combined with Percoll y(2:3
ratio) to match the medium and
swimming
between
glass
200in!m).
(µm)
ments and
horizontal
errortwo
bars
theplates
depth(separation
of field. As
cell buoyant densities [18]. A droplet of the cell mixture is
cell that
distribution
was
withnear
a constant
(d)nonuniform,
Ref. [6],The
we find
the cell profile
peaks strongly
the
between Experimental
two glass cover
slips, previously
density in the center strongly increasing near the wallsFIG.deposited
2 (color online).
data: number
of swimwalls,
with
a
nearly
constant
cell
density
about
20
!m
038102-1
2008
The spermaAmerican
Physical
Society
ofofethanol
saturated
potassium
[6]. Similar results were later©
obtained
for human
mingcleaned
cells
ninasaamixture
function
the distance
to thewith
bottom
cover
‘Hydrodynamic’ fields
Concentration profile between two walls An interesting question that is relevant
from a medical perspective concerns the spatial distribution of bacteria and other swimming
microbes in the presence of confinement. Restricting ourselves to dilute suspensions10 , we
may obtain a simple prediction from the model (1.45) by considering the FPE for the
associated PDF p(t, x, n). Given p and the total number of bacteria Nb in the solutions,
we obtain the spatial concentration profile by integrating over all possible orientations
Z
c(t, x) = Nb
dn p(t, n, x).
(1.56a)
Sd
The associated mean orientation field reads
Z
u(t, x) = Nb
dn p(t, n, x) n.
(1.56b)
Sd
The FPE for the Ito-SDE (1.45) can be written as a conservation law
@t p =
(@xi Ji + @ni ⌦i ),
(1.57a)
where
Ji = (V ni DT @xi )p
⌦i = DR (1 d)ni p @nj [(
ij
ni nj )p] .
(1.57b)
(1.57c)
Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57)
the stationary concentration profile c of a suspension that is confined by two quasi-infinite
Sd
The associated mean orientation field reads
Z
u(t, x) = Nb
dn p(t, n, x) n.
Concentration field
(1.56b)
Sd
The FPE for the Ito-SDE (1.45) can be written as a conservation law
@t p =
(@xi Ji + @ni ⌦i ),
(1.57a)
where
Ji = (V ni DT @xi )p
⌦i = DR (1 d)ni p @nj [(
ij
ni nj )p] .
(1.57b)
(1.57c)
Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57)
the stationary concentration profile c of a suspension that is confined by two quasi-infinite
parallel walls, which are located z = ±H. That is, we assume that the distance between
the walls is much smaller then their spatial extent in the (x, y)-directions, 2H ⌧ Lx , Ly .
To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n
with
Z
dn @ni ⌦i = 0.
(1.58)
Sd
This yields the mass conservation law
@t c =
r · (V u
DT rc).
(1.59)
To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk ,
@t (nk p) =
@xi (nk Ji )
n k @ n i ⌦i .
(1.60)
b
where
Sd
The associated meanJorientation
field
DTreads
@xi )p
(1.57b)
i = (V ni
Z
⌦i = DR (1 d)ni p @nj [( ij ni nj )p] .
(1.57c)
u(t, x) = Nb
dn p(t, n, x) n.
(1.56b)
Sd
Focusing on the three-dimensional case, d = 3,
we are interested in deriving from Eq. (1.57)
the
stationary
profilecan
c ofbe
a suspension
is confinedlaw
by two quasi-infinite
The
FPE for concentration
the Ito-SDE (1.45)
written as athat
conservation
parallel walls, which are located z = ±H. That is, we assume that the distance between
@t p spatial
=
(@
⌦i ),(x, y)-directions, 2H ⌧ Lx(1.57a)
xi Ji +in@nthe
the walls is much smaller then their
extent
, Ly .
i
To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n
where
with
Z
Ji = (V ni DT @xi )p
(1.57b)
dn @ni ⌦i = 0.
(1.58)
⌦i = DR Sd(1 d)ni p @nj [( ij ni nj )p] .
(1.57c)
Orientation (velocity) field
This
yields on
thethe
mass
conservation law
Focusing
three-dimensional
case, d = 3, we are interested in deriving from Eq. (1.57)
the stationary concentration profile c of a suspension that is confined by two quasi-infinite
@t c z== ±H.
r · (V
u is,
DTwe
rc).
(1.59)
parallel walls, which are located
That
assume that the distance between
the
is much
thenequation
their spatial
in the Eq.
(x, y)-directions,
Towalls
obtain
also ansmaller
evolution
for u,extent
we multiply
(1.57a) by nk ,2H ⌧ Lx , Ly .
To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n
with
@t (nk p) =
@xi (nk Ji ) nk @ni ⌦i .
(1.60)
Z
and note that
dn @ni ⌦i = 0.
(1.58)
Sd
nk @ni ⌦i = @ni (nk ⌦i ) (@ni nk )⌦i = @ni (nk ⌦i )
This yields the mass conservation law
ik ⌦i .
10
(1.61)
The simplifying assumption that microbes can be considered as non-interacting is usually justified
when their volume filling fraction is less
@t cthan
= 1%. r · (V u DT rc).
(1.59)
To obtain also an evolution equation for
14 u, we multiply Eq. (1.57a) by nk ,
This yields the mass conservation law
Orientation
@c =
r(velocity)
· (V u D rc). field
t
T
(1.59)
To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk ,
@t (nk p) =
@xi (nk Ji )
n k @ n i ⌦i .
(1.60)
and note that
nk @ni ⌦i = @ni (nk ⌦i )
10
(@ni nk )⌦i = @ni (nk ⌦i )
ik ⌦i .
(1.61)
The simplifying assumption that microbes can be considered as non-interacting is usually justified
This their
allows
us tofilling
rewrite
(1.60)
as than 1%.
when
volume
fraction
is less
@t (nk p) =
=
@xi (nk Ji ) + ⌦k @ni (nk ⌦i )
14
@xi [V nk ni p DT @xi (nk p)] +
DR
2nk p @nj [( kj nk nj )p]
@ni (nk ⌦i )
=
@xi [V nk ni p DT @xi (nk p)] 2DR nk p
@nj (nk ⌦j + ( kj nk nj )p).
(1.62)
Multiplying by Nb and integrating over n with appropriate boundary conditions gives
@ t uk =
@xi [V Nb hnk ni ip
DT @xi uk ]
2DR uk ,
where we have abbreviated
hni nk · · ·i =
Z
Sd
dn p(t, n, x) ni nk · · · .
(1.63)
To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find
@t u '
2DR u + DT r2 u.
(1.64)
This yields the mass conservation law
Orientation
@c =
r(velocity)
· (V u D rc). field
t
T
(1.59)
To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk ,
@t (nk p) =
@xi (nk Ji )
n k @ n i ⌦i .
(1.60)
and note that
nk @ni ⌦i = @ni (nk ⌦i )
10
(@ni nk )⌦i = @ni (nk ⌦i )
ik ⌦i .
(1.61)
The simplifying assumption that microbes can be considered as non-interacting is usually justified
This their
allows
us tofilling
rewrite
(1.60)
as than 1%.
when
volume
fraction
is less
@t (nk p) =
=
@xi (nk Ji ) + ⌦k @ni (nk ⌦i )
14
@xi [V nk ni p DT @xi (nk p)] +
DR
2nk p @nj [( kj nk nj )p]
@ni (nk ⌦i )
=
@xi [V nk ni p DT @xi (nk p)] 2DR nk p
@nj (nk ⌦j + ( kj nk nj )p).
(1.62)
Multiplying by Nb and integrating over n with appropriate boundary conditions gives
@ t uk =
@xi [V Nb hnk ni ip
DT @xi uk ]
2DR uk ,
where we have abbreviated
hni nk · · ·i =
Z
Sd
dn p(t, n, x) ni nk · · · .
(1.63)
To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find
@t u '
2DR u + DT r2 u.
(1.64)
Z
Stationary profiles
hni nk · · ·i =
Sd
dn p(t, n, x) ni nk · · · .
(1.63)
To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find
@t u '
2DR u + DT r2 u.
(1.64)
To find the stationary density and orientation profiles, we look for solutions of the form
c = ⇢(z) and ux = uy = 0, uz = u(z). According to Eqs. (1.59) to (1.63), the functions ⇢
and uz must satisfy
0 = V u DT c 0 ,
0 =
2DR u + DT u00 ,
(1.65)
(1.66)
and it is physically plausible that they also fulfill the symmetry12 requirements ⇢(z) =
⇢( z) and u(z) = u( z). Hence, solution takes the form
u(z) = A sinh(z/⇤),
V⇤
⇢(z) = A
[cosh(z/⇤)
DT
(1.67a)
1] + ⇢0 ,
(1.67b)
p
where ⇤ = D? /(2DR ).
The cosh-profile (1.67b) agrees qualitatively with experimental measurements for dilute
bacterial suspensions [BTBL08, LT09].
11
Ad hoc simplifications of this type are usually referred to as ‘truncation (of the moment hierarchy)’ or
‘closure conditions’ - they are (almost) always unavoidable when one tries to derive continuum equations
from ODEs, SDEs or FPEs. Closure conditions are not unique, for example, we could also have adopted
the mean field approximation Nb2 hnk ni i ' uk ui , which leads to a nonlinear set of equations for (c, u).
12
Neglecting gravity is valid approximation, provided the density of the microbes roughly matches that
But …
E. coli
RT (χ2 = 0.018)
E. ecoli (smooth)
P. aeruginosa
RT (χ2 = 0.016)
B. subtilis
RT (χ2 = 0.092)
BD (χ2 = 0.021)
BD (χ2 = 0.026)
BD (χ2 = 0.024)
BD (χ2 = 0.13)
RT (χ2 = 0.074)
ρ
1
0.5
ρ± /ρ
0
1
Density profiles seem ok … what about fluxes?
0.5
0
0
5
10 0
(z − zm )(µm)
5
(z − zm )(µm)
10 0
5
(z − zm )(µm)
10 0
5
(z − zm )(µm)
joint work with Peter Lu, Rik Wensink & Jeff Guasto
10
But …
E. coli
RT (χ2 = 0.018)
E. ecoli (smooth)
P. aeruginosa
RT (χ2 = 0.016)
B. subtilis
RT (χ2 = 0.092)
BD (χ2 = 0.021)
BD (χ2 = 0.026)
BD (χ2 = 0.024)
BD (χ2 = 0.13)
5
5
5
5
RT (χ2 = 0.074)
ρ
1
0.5
ρ± /ρ
0
1
0.5
0
0
(z − zm )(µm)
10 0
(z − zm )(µm)
10 0
(z − zm )(µm)
10 0
(z − zm )(µm)
10
rotation
translation
PT 2 tD
rotation
PR translation
PT
time ⌧ = tv0 /`
û =PR
DR (T
t
+
û
(1)
bw ⇥ û)
R
G
Laboratoire
de
Physique
des Solides, Université
Paris-Sud & CNRS,
UMR 8502, 91405 Orsay, France
Let us start with the coupled Langevin
equations
for
September
4, 2014)
0.04 ± 0.005
0.1 ± 0.01
0.11 ± 0.005 0.31 ±
0.0051 = k(Dated:
with
thermal
energy.
The bacteriummensionless
form
B T the
the
center-of-mass
position
r
and
orientation
vector
û
of
We propose two simple statistical models to describe the density and orientation profiles of E. Coli
wall
forces
emerge
from a simple segment potential,
0.02 ±
0.005 collisions
0.07 ± are
0.005
0.07 and
± 0.001
0.34
0.001
and
other ±
swimming
bacteria near aP
flat hard wall. The models focus on purely Brownian dynamics
Mutual
bacteria
neglected
contribu`/2
a
swimming
bacterium,
linearized
in
time:
versus explicit ‘run and
tumble’
(RT) dynamics
of non-interacting bacteria near a flat structure2
F
=
r
+ ↵û|) with Yukawa
form
1
0.14
± 0.05
0.02 ± 0.005
0.10
± 0.001
0.09
±
0.001
bw
r
ons
of O(
t ) are negligible
for small
enough
time disless substrate. Using best fit values for ↵=
the e↵ective
translational and rotational di↵usivities both
`/2 u(|r
`
r
=P
models generate similar density profiles – all in quantitative agreement with the experimental ones
retizations
t. Fbw
(Tbw
denotes 0.04
the force
(torque)
u(x)
=di↵erent
u0 exp(
x/
)/x.
amplitude
u0 isInimmaterial
0.02 ± 0.005
0.08
± )0.005
± 0.001
0.005
–0.49
but yield±
distinctly
stationary
density
fluxesThe
with respect
to the wall normal.
global
p
1
Need to include run & tumbling
comparison, the RT dynamics provides a much more appropriate description of the non-equilibrium
xerted by the flat, repulsive wall r
on=the
bacterium,
v
whereas
the
screening
range
can be tuned to match the
0 v of
bacteria
close to2
confining
substrates.
Compared to purely Brownian dynamics it turns
DT · Fbw t structure
+
û
t
+
tD
r
0
T
G
=P
out that run-and-tumble
dynamics generates
qualitatively
di↵erent
stationary
orientational
wall
or-û100).
the swim velocity and DR and DT = DT I are the efpeak
position
near
the
wall
(z
⇡
1µm,
`/
⇡
80
m
p
der with distinct anti-nematic features and a narrow boundary layer of bacteria swimming towards
the wall. Eq. (2) then permits a two-parameter (PT , PR ) best fit
ctive rotational and translationalûbacterial
di↵usivities
= DR (T
2 tDR ûG
(1)
bw ⇥ û) t +
1 range
gnoring the intrinsic k / ? contributions for simplicof the bacterial density profiles overwith
the spatial
= Pécle
kB
DYNAMICS
rotation:
I. BROWNIAN DYNAMICS
(typical) bacterium length ` to define the inverse
1in the fitted
1 density proy). rG represents a random Gaussian displacement
zm < z < 10µm. Error-bars
numbers PT = DT /v0 ` and PR = DR `/v0 and rescale
wall
forces
em
timecomparing
⌧ = tv0 /` to recast
the equations
of
motion
in d
n 3D Cartesian space whereas
a
Gaussian
are
estimated
from
results
for
the
bare
Let us start with
thefiles
coupled
Langevin
equations
for
Mutual 1ûbacteria
collisions
are
neglected
and
contribuG denotes
P
Fbw the center-of-mass position r and orientation vector û of mensionless form:
2
r =tvector
⌧a negligible
isplacement
of the orientational
unit
the surbacterium
aspect
of the
bacterial
body
Fto
=q of an
r
bwthose
refswimming
bacterium,
linearized
in time:
)Faareon
for
small
enough
time
disangle ✓ between
thetions `of O(
1
1
1
`
r = PT `Fbw ⌧ + û ⌧ + 2 ⌧ PT rG
ce
of
the
unit
sphere.
Both
stochastic
variables
have
fective
hydrodynamic
aspect
ratio
ae↵ based on
p
q the total
✓
◆
✓
◆
he bacterium duringcretizations t. Fbw (T
)
denotes
the
force
(torque)
u(x)
=
u0I).
exp(
1
1
=bw
DT · Fbw t + v0 û t + 2 tDT rG
û = PR ( Tbw
⇥ û) Table
⌧ + 2 ⌧P
ûG
(2
Tbw
û and the flagellum
R
ero mean and unit variance. It is expedient
to use rthe
lengthpof the body
(see
BD:
(SF-SDE)
RT:
we assume this angleexerted by
û =
⇠ flat, repulsive
⇥ û
⌧ +⇥wall
⇥
⌧v0
û = D (T
û) t +ûon
2 tD the
û
(1) ⇥ û
the
bacterium,
whereas the scre
with
=
k T the thermal energy. The bacterium
Fa `
⌧tumble wall forces emerge
from
a simple
segment potentia
bacteria
neglected
is the swim velocityMutual
and
DRcollisions
andareD
Dcontribuefpeak
position
ne
P
T =and
T I areF the
(6)
=
r
u(|r
+
↵û|)
with
Yukawa
form
tions of O( t ) are negligible for small enough time disR
bw
R
G
1
2
B
r
bw
`/2
↵= `/2
cretizations
t. F (T ) denotes
the force (torque)
u(x) = u exp( x/ )/x.Eq.
The amplitude
u is immateria
fective rotational and
translational
bacterial
di↵usivities
(2) then
pe
by the flat,
repulsive wall on used
the bacterium,
v fit.
whereas the screening range can be tuned to match th
TABLE I: Mainexerted
bacterial
parameters
for the
)
(3)(ignoring
is the swim
D and(✓
D =, D
I are
thefor
efpeakAway
position near the wall
⇡ 1µm,
`/ ⇡ 80 100
the intrinsic
k velocity
/û?=andcontributions
simplicof (zthe
bacterial
with
tumbling
vector
û
'
)
û(0).
new
T
R
fective
rotational
and
translational
bacterial
di↵usivities
Eq.
(2)
then
permits
a
two-parameter
(P
, P ) best fi
culture
` (µm) a (ae↵ ) v0 (µm/s) ✓T
ttumble (s) trun (s)
2
(ignoring
the
intrinsic k / ?exhibits
contributions zero
for simplicof
the bacterial densityzprofiles
overzthe<
spatial
rang
ity).
r
represents
a
random
Gaussian
displacement
<
10µm
from
the
wall,
each
bacterium
net
translaG
m
E. coli
⇠ 3 ⇠ity).
2(5.6)
⇠ a20
68 displacement
⇠ 0.1 z < z < ⇠
1 Error-bars in the fitted density pro
r represents
random Gaussian
10µm.
3D Cartesian
space whereas
a Gaussian
files are estimated fromfiles
comparing
results
for the bar
tion
and ⇠û3reorients
linearly
a forward-directed
in 3D
Cartesian
space
whereas
ûGû denotes
a0 Gaussian
are
estima
tified from observaE. coli
(smooth)
⇠in
2(5.6)
⇠ 20towards
0denotes
1
displacement of the orientational unit vector on the surbacterium aspect of the bacterial body to those of an e
cone parameterized
byunit(✓sphere.
Near
the wall
bacof the
stochastic
variables
have
fective hydrodynamic
ratio a based on
the tota
T⇠, '
R ).unit
displacement
orientational
vector
on
surbacterium
aspec
B.
subtilis
⇠of
5 the
⇠face
6(11.5)
50Both
⇠ 40
⇠
0.1the
⇠ 0.5 aspect
of tumbling bactezero mean and unit variance. It is expedient to use the
length of the body and the flagellum (see Table I).
terial
dynamics
is
governed
by
a
nontrivial
interplay
beTABLE
II:
Best
fit
parameters
for
the
intrinsic
bacterial
di↵usivities.
Errors
are
estimated
from di↵erent
fit results
P.
aeruginosa
⇠
2
⇠
4(9.8)
⇠
40
⇠
110
⇠
0.1
⇠ 0.5
gle appears stronglyface of the unit sphere. Both stochastic variables have
fective hydrodyn
or
and ae↵ we draw
byistheexpedient
wall and ato
‘tumbling’
bw exerted It
In aaddition
zero tween
mean the
andtorque
unit Tvariance.
use the
length of the bo
TABLE I: Main bacterial parameters used for the fit.
bw
bw
0
0
0
T
R
T
m
T
G
R
m
G
e↵
torque Ttumble ⇠ ⇠RRT
✓T / ttumble
. ` (µm) a (a ) vBD
distribution on the
culture
(µm/s) ✓
t
(s) t
(s)
1
1
1 ⇠ 20
1
E.
coli
⇠
3
⇠
2(5.6)
68
⇠
0.1
⇠
1
The above
can be easily
implemented
for N in- PT
rotationscheme
PR translation
PT rotation
PR translation
acterium unit vector
E. coli (smooth)
⇠ 3 ⇠ 2(5.6)
⇠ 20
0
0
1
E.
coli
0.04
±
0.005
0.1
±
0.01
0.11
±
0.005
0.31
±
0.005
dependent bacteria with random
initial
B. subtilis
⇠ 5 conditions.
⇠ 6(11.5)
⇠ 50 Time
⇠ 40
⇠ 0.1
⇠ 0.5
e orthonormal frame
aeruginosa
⇠ 2 ⇠ 4(9.8)
40
⇠ 110
⇠ 0.5
E. coli (smooth)
0.02
0.005‘run’
0.07stage
± P.
0.005
0.07
± 0.001
± 0.001⇠ 0.1
intervals
for± the
are drawn
from a⇠0.34
Poisson
TABLE
I: 0.001
Main 0.09
bacterial
parameters used fo
1 0.10 ±
B. subtilis
0.14
±
0.05
0.02
±
0.005
±
0.001
distribution with ⌧run = ⇤ ln(1 U ) with U a ranP. aeruginosa
0.02 ± from
0.005 a 0.08
± 0.005
0.04
± 0.001
0.49
± 0.005
culture
` (µm)
a (a
✓T
dom number
uniform
distribution
in the
interval
e↵ ) v0 (µm/s)
'R e2 + cos ✓T û (4)
[0, 1] and ⇤ = f `/v0 a E.
dimensionless
tumbling
coli
⇠ 3 ⇠frequency.
2(5.6)
⇠ 20
68
wensink@lps.u-psud.fr
e↵
0
T
tumble
run
Need to include run & tumbling
E. coli
RT (χ2 = 0.018)
E. ecoli (smooth)
P. aeruginosa
RT (χ2 = 0.016)
B. subtilis
RT (χ2 = 0.092)
BD (χ2 = 0.021)
BD (χ2 = 0.026)
BD (χ2 = 0.024)
BD (χ2 = 0.13)
5
5
5
5
RT (χ2 = 0.074)
ρ
1
0.5
ρ± /ρ
0
1
0.5
0
0
(z − zm )(µm)
10 0
(z − zm )(µm)
10 0
(z − zm )(µm)
10 0
(z − zm )(µm)
10