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Interplay of motility and polymer-driven depletion
forces in the initial stages of bacterial
aggregation
Michael K. Porter,
a
Asher Preska Steinberg
a
and Rustem F. Ismagilov
*
ab
Motile bacteria are often found in complex, polymer-rich environments in which microbes can
aggregate
via
polymer-induced depletion forces. Bacterial aggregation has many biological implications;
it can promote biofilm formation, upregulate virulence factors, and lead to quorum sensing. The steady
state aggregation behavior of motile bacteria in polymer solutions has been well studied and shows that
stronger depletion forces are required to aggregate motile bacteria as compared with their nonmotile
analogs. However, no one has studied whether these same trends hold at the initial stages of
aggregation. We use experiments and numerical calculations to investigate the polymer-induced
depletion aggregation of motile
Escherichia coli
in polyethylene glycol solutions on short experimental
timescales (
B
10 min). Our work reveals that in the semi-dilute polymer concentration regime and at
short timescales, in contrast to what is found at steady state, bacterial motility actually enhances
aggregate formation by increasing the collision rate in viscous environments. These unexpected findings
have implications for developing models of active matter, and for understanding bacterial aggregation in
dynamic, biological environments, where the system may never reach steady state.
Introduction
Bacteria thrive in a wide range of biological and ecological
contexts and play important roles in the human gut, soil,
wastewater sludge, and other complex environments. In these
environments, bacterial motility has implications for biofilm
formation and virulence. For example, in the gut,
Salmonella
typhimurium
uses its flagella to burrow through the intestinal
mucus layer and penetrate the host epithelium, causing
infection.
2
Certain species of
Pseudomonas aeruginosa
require
motility to form biofilms, such as those found on medical
devices.
3
Microbial motility is further influenced by polymers,
which are abundant in many environments.
4
For example, in
the gut, polymers are secreted by the host
5–8
and dietary fibers
are ingested regularly.
1
In wastewater treatment plants, sludge
used to collect microbes and particulate waste also contains
polymers.
9,10
Polymers are well known to aggregate bacteria, which is
important because aggregation precedes biofilm formation,
11
correlates with altered gene expression,
3,12
including antibiotic-
resistance genes,
13
and induces phenotypic changes such as
quorum sensing.
14
Polymers can bind to bacteria
via
specific
chemical interactions and cause aggregation through aggluti-
nation. These interactions are found in biological settings
such as the gut, where mucins,
5
immunoglobulins,
6
and other
host-secreted proteins
7
can aggregate bacteria
via
chemically
mediated interactions. Polymers can also aggregate microbes
via
depletion forces. This mechanism of aggregation does not
depend on microbes binding to specific chemical functional
groups but is instead only a function of the physical parameters
of the polymer (molecular weight (MW), hydrodynamic radius)
and bacteria size.
15–17
Inthepresenceofnon-adsorbingpolymers
such as polyethylene glycol (PEG), bacteria aggregate through
depletion interactions, which o
ccur when two bacteria approach
each other at a close enough distance that the polymer is excluded
from the space between the bacteria, a region called the depletion
zone.
13
The difference in polymer concentration between the
depletion zone and the bulk solution results in an osmotic
pressure difference that generates an attractive force between
bacteria. Because depletion forces depend on the physical
properties of polymers and bacteria, these forces can drive
bacterial aggregation irrespective of bacterial surface chemistry.
Inthecaseofnonmotilebacteria(whicharenotauto-aggregating)
in solutions of non-adsorbing polymers, the only driving
forces for aggregation are polymer-induced depletion forces.
17,18
a
Division of Chemistry & Chemical Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA.
E-mail: rustem.admin@caltech.edu
b
Division of Biology & Biological Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA
Electronic supplementary information (ESI) available. See DOI: 10.1039/
c9sm00791a
Received 17th April 2019,
Accepted 29th July 2019
DOI: 10.1039/c9sm00791a
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However, for motile bacteria in solutions of non-adsorbing
polymers, there are forces due to the motility of the bacteria
and due to depletion forces; the competition between these
forces determines the steady-state aggregation behavior.
18
At sufficient polymer concentrations and long time scales,
when the system reaches steady state, polymer-induced deple-
tion attractions between bacteria can result in aggregation
(
i.e.
, phase separation
18
). The addition of motility has been
found to require significantly stronger depletion attraction
to achieve the aggregation as compared with nonmotile
bacteria.
18
Active matter is an area of intense research, and
the field is currently working on a unified theoretical frame-
work to understand these systems.
19,20
In particular, the aggre-
gation behavior of motile bacteria in polymeric solutions at
long time scales is widely studied;
18,21–24
however, to our
knowledge there are no published studies on aggregation at
short time scales.
The initial stages of aggregate formation are of particular
interest because microbial responses to aggregation (
e.g.
,
upregulation of quorum sensing and virulence pathways) occur
on short time scales (tens of minutes) and would be influenced
by the initial stages of aggregation. Furthermore, the initial
stages of aggregation are particularly relevant for biological
systems where the system may never reach steady state; for
example, in the gastrointestinal (GI) tract where food and
ingested material are constantly in transit.
1,25
In these environ-
ments, aggregate formation is constantly disrupted because of
shear, peristaltic contractions, and other forces, making early
aggregate formation relevant to these biological systems.
In this study, we investigate how motility influences the
polymer-induced depletion aggregation of bacteria at short
time scales (
t
B
10 min). We quantify this experimentally by
using confocal fluorescence microscopy to measure the size
distribution of bacterial aggregates in PEG solutions with
molecular weights and concentrations relevant to the murine
small intestine.
1
Furthermore, we develop an understanding of
which physical parameters influence the initial formation of
these aggregates. We use a physical model for motile bacteria in
PEG solutions that focuses on the balance of depletion and
swim forces as well as the effective diffusivity of the bacteria.
Experimental
Bacteria cell culture
Overnight cultures of naturally motile
E. coli
K12 MG1655
(ATCC 47076) were prepared in liquid lysogeny broth (LB)
culture incubated at 35
1
C to mid-exponential phase. These
cultures were combined to reach the desired cell concentration
for the experiment (10
9
cells per mL). Cells were first centri-
fuged at 4.8 kG for 10 min and then resuspended in motility
buffer (MB; 10 mM potassium phosphate buffer, 0.1 mM EDTA,
pH 7.0) to stain with SYTO 9 (liv
e, ex/em 480/500 nm). Following
staining, cells were centrifuged again to wash out excess stain. To
obtain nonmotile
E. coli
, cells were treated with 0.5% (75 mM)
sodium azide in MB after washing.
Confocal microscope imaging and bacterial aggregation in
polyethylene glycol (PEG)
All images and z-stacks were obtained using a Zeiss LSM 800
confocal fluorescence microscope (488 nm excitation, detection
at 490–540 nm). Each stack was 200

200

45
m
m in volume
and contained about 135 slices. Nonmotile and motile
E. coli
K12 were prepared and stained using the method described
above. PEG solutions were prepared by dissolving four times
the overlap concentration into MB (for motile conditions) or by
using MB with 0.5% sodium azide (for nonmotile condition).
A range of 10 kDa, 100 kDa, and 1 MDa PEG solutions were
achieved by serial dilution. A 5
m
L aliquot of each respective
PEG solution was combined with 0.5
m
L
E. coli
for a final cell
concentration of 10
9
cells per mL. We pipetted 2
m
L of each
combined suspension into an imaging chamber made from
SecureSeal imaging spacer (Electron Microscopy Sciences;
0.12 mm depth and 9 mm diameter) and a glass slide, and
the top of the chamber was immediately sealed with a #1.5
glass coverslip. A single z-stack of each PEG dilution sample
was taken approximately 10 min after the imaging chamber was
sealed. Each biological replicate was conducted with a new
bacterial cell culture.
FIJI macro imaging and empirical bootstrapping
All imaging analysis was performed as previously described
in ref. 1.
Measuring mean-squared displacement (MSD) of
E. coli
E. coli
K12 were cultured and prepared as described above. The
final cell concentration was diluted to 5

10
8
cells per mL
when added to each PEG solution in MB. A Leica DMI6000 with
a Visitech Infinity3 confocal microscope was used to obtain 20 s
videos of the cells at about 16 frames per s.
Videos were analyzed using an ImageJ plugin developed
by the MOSAIC group for 2D/3D particle tracking using an
algorithm developed in ref. 26. At least 1000 bacteria were
analyzed per condition. Data out
put from the ImageJ plugin was
further analyzed and the mean square displacement (MSD) was
calculated using MATLAB code used in ref. 27 in conjunction with
an in-house script. (Script will be provided by request.)
Estimating overlap concentration for PEG
The polymer overlap concentration
c
P
* was estimated using the
following relation:
28,29
c
P

¼
MW
4
p
3
N
Avo
R
g
ð
0
Þ
3
(1)
where MW is the polymer molecular weight in kDa,
N
Avo
is
Avogadro’s number, and
R
g
(0) is the radius of gyration given in m.
Estimating solution viscosity
We estimated the solution viscosity
via
a virial expansion:
28
Z
=
Z
s
(1 + [
Z
]
c
P
+
k
H
[
Z
]
2
c
P
2
+

)
(2)
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where
Z
s
is the solvent viscosity in Pa s,
c
P
is the polymer mass
concentration in kg m

3
,[
Z
] is the intrinsic viscosity measured
to be 452.8 mL g

1
(using literature measurements for PEG
1 MDa),
30
and
k
H
is the Huggins parameter for PEG 1 MDa,
approximated to be 0.4. Using this equation,
Z
at
c
P
= 6.5 mg mL

1
is
B
7 mPa s compared with
B
1mPasat
c
P
=0.4mgmL

1
.
Literature measurements report similar values for the zero shear
rate viscosity (
Z
0
) of PEG 1 MDa at high concentrations, where
Z
0
=10mPasat
c
P
=5mgmL

1
(our experiments were
conducted our quiescent conditions,
i.e.
, no shear).
31
Results and discussion
Measuring
E. coli
aggregation at short time scales
To understand how motility affects the depletion-driven aggre-
gation of bacteria at short time scales, we measured the initial
formation of bacterial aggregates. As a model organism, we
used
E. coli
K12 MG1655. This naturally motile strain of
E. coli
displays ‘‘run and tumble’’ dynamics, and can be rendered
nonmotile by treating with 0.5 wt% sodium azide.
32,33
This
method of rendering cells nonmotile has been used before in
research focusing on the flagellar motility of
E. coli
.
32
Plating
showed that the azide treatment killed nearly all of the cells 1 h
after the initial treatment (Fig. S1, ESI
). To ensure we selected
biologically relevant physical parameters for our experiments,
we used data from our previous gel permeation chromato-
graphy experiments on luminal fluid from the murine small
intestine to determine the range of the polymer molecular
weights and concentrations.
1
In these previous experiments,
the polymers we found in the murine small intestine ranged in
size from a few kDa to a few MDa.
1
Therefore, we chose to tune
the depletion potential in this work with polymers within this
size range. As our test polymer, we chose to use PEG solutions
in a motility buffer (MB, 0.1 mM EDTA, 10 mM potassium
phosphate, pH 7.0). PEG is a linear, chemically inert
polymer
33,34
that is well characterized in inducing depletion
forces in passive colloid solutions.
35–42
We used a range of
1 MDa PEG concentrations (0.05–6.5 mg mL

1
) to adjust the
depletion potential and rheology of the solution, which
span both the dilute and semi-dilute polymer concentration
regimes (the transition between these regimes is denoted by the
overlap concentration,
c
P
* = 1.6 mg mL

1
(see calculations in
Experimental)). We also measured bacterial aggregation in
10 kDa and 100 kDa PEG near their respective overlap concen-
trations (
c
P
* = 8.5 mg mL

1
and 85 mg mL

1
for 100 kDa and
10 kDa PEG, respectively). We previously detected polymers of a
similar size and concentration in the murine small intestine.
1
A motility buffer control was implemented to confirm that the
cells were not auto-aggregating and that the aggregation mea-
sured in each sample containing PEG was the result of the PEG
in the solution.
To quantify the initial aggregation of bacteria, we measured
the volume-weighted average aggregate sizes (
N
) using fluores-
cence confocal microscopy (Fig. 1a). After mixing the
E. coli
with the PEG, the bacterial suspension was placed into an
imaging chamber, and sealed with a glass coverslip to eliminate
drifting and evaporation effects. Z-stacks of cells in solutions of
PEG at various concentrations were obtained after 10 min to
focus on the behavior on short timescales (Fig. 1b and c).
Separate experiments were performed for motile and nonmotile
cells. Imaging at short timescales also reduces the effects of
sedimentation from gravity.
43
Imaging analysis was performed
(using an ImageJ pipeline that we developed previously
1
)to
count each object, measure the volumes of each aggregate, and
normalize by the singlet volume to obtain the volume-weighted
average aggregate size.
In these experiments, we found that nonmotile bacterial
aggregation in the presence of 1 MDa PEG (Fig. 2a) was
qualitatively consistent with depletion-driven aggregation with
similar trends observed at half (Fig. 2b) and double (Fig. 2c) the
bacterial concentration. Additionally, we tested the effect of
changing PEG MW and found trends that were qualitatively
consistent with depletion-driven aggregation (Fig. S2, ESI
); the
extent of aggregation generally decreased with MW. Nonmotile
Fig. 1
Measuring the volume-weighted average size of bacterial aggregates in PEG solution. (a) Cartoon depicting experimental setup of motile
or nonmotile
E. coli
(green) in buffer or PEG. (b) Representative slices of nonmotile
E. coli
taken in motility buffer (MB) with no aggregation and in
(c) 0.8 mg mL

1
PEG 1 MDa showing aggregation.
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E. coli
aggregated less in the presence of PEG 100 kDa (Fig. S2b,
ESI
) as compared to in the presence of PEG 1 MDa (Fig. 2a)
and no aggregation was observed in the presence of PEG
10 kDa. Somewhat counterintuitively, as the 1 MDa PEG
concentration increased, aggregate size increased up to a limit
and then started decreasing. We previously measured similar
aggregation profiles with particles in PEG solutions;
1
the shape
of those profiles was at least in part attributed to the increase in
solution viscosity, which hindered the Brownian motion of the
particles, limiting the inter-particle collisions that initiate
aggregation. We suspect that a similar mechanism may be at
play for the nonmotile bacteria in this study. The depletion
attractions increase as a function of polymer concentration due
to the contribution of osmotic pressure, which is mirrored by
the increasing aggregate size of the nonmotile bacteria in the
dilute PEG concentration regime (eqn (3) and (4)). We observed
that as the PEG concentration increased and approached the
semi-dilute concentration regime, the aggregate sizes became
larger. In the semi-dilute regime, the depletion attractions
continued to increase with the increase in PEG concentration,
buttherangeoftheseattractionsdecreasedwithPEGconcen-
tration. At PEG concentrations far above the overlap concentration,
we observed a decrease in aggregate sizes. The use of sodium azide
to render
E. coli
nonmotile could alter cell surface properties in a
manner that affects their depletion-driven aggregation. However,
due to their lack of aggregation in the motility buffer control and
the aggregation curve being qualitatively similar to bioinert
particles in similar polymer solutions,
1
we assume that this
effect is minimal. We hypothesized that at higher PEG concen-
trations, the enhanced viscosity decreased inter-bacterial colli-
sions, thus hindering aggregate formation.
Motile bacteria demonstrated different aggregation trends
compared with their nonmotile analogs. In the PEG 1 MDa
solutions, we observed no aggreg
ationinthedilutePEGconcen-
tration regime, but at
c
P
=0.8mgmL

1
,motile
E. coli
abruptly
began to aggregate, and continued to aggregate through the semi-
dilute polymer concentration regime (Fig. 2a). Similar trends were
observed when we halved (Fig. 2b) or doubled (Fig. 2c) the
bacterial concentrations. In the lower MW PEG solutions,
we found minimal aggregation for the motile bacteria in PEG
100 kDa (Fig. S2b, ESI
) and no aggregation for motile bacteria in
PEG 10 kDa (Fig. 2a). Because the main physical difference
between the motile and nonmotile bacteria is their motion, we
hypothesized that the differences in depletion attractions required
to aggregate motile
versus
nonmotile bacteria at short time
scales are due to cell motility. Previously, researchers have
modeled the steady-state depletio
n aggregation of motile bacteria
by assuming that the swim force produced by bacterial motility
directly counteracts t
he depletion force.
18
We hypothesized that a
similar framework could be applied here; the swim force coun-
teracts the depletion force at lo
w PEG concentrations, negating
any attractive force to aggregat
ethecells.Therefore,astronger
depletion force, or higher PEG concentration, is necessary to
aggregate the motile bacteria to the same extent as the nonmotile
Fig. 2
A comparison of the aggregation of motile and nonmotile
E. coli
K12 at a range of concentrations of 1 MDa PEG. Volume-weighted average
aggregate sizes (Vol Wt Avg Size) of nonmotile and motile
E. coli
K12 for serial dilutions of 1 MDa PEG using a bacteria concentration of 1

10
9
CFU mL

1
(a) and for
E. coli
at half (b) and twice (c) this concentration. Aggregate sizes were measured 10 min after cells were mixed with PEG. Volume-weighted
average sizes in terms of bacteria per aggregate (
N
) are plotted against polymer mass concentration (
c
P
)inmgmL

1
. Vertical error bars are 95% empirical
bootstrap confidence intervals using the bootstrapping protocol described in [Imaging analysis] in Methods of ref. 1. Data for each PEG concentrati
on
were compiled from at least four (a), at least three (b), or at least two (c) biological replicates in these experiments (where a replicate is a new bacte
rial
culture). For each concentration of PEG, each replicate was obtained from one z-stack that was comprised of about 135 slices.
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bacteria. Support for this hypothesis was experimentally demon-
strated at
c
P
= 0.4 mg mL

1
whereby we observed the nonmotile
bacteria aggregate but the motile do not.
Effective potentials describe
E. coli
aggregation in the dilute
polymer concentration regime
To investigate the interplay of swim and depletion forces that
give rise to the differences in the observed aggregation behavior
between motile and nonmotile bacteria, we began by using
effective potentials to describe ag
gregationinthedilutepolymer
concentration regime. We focused on explaining the discrepancy
in aggregation at
c
P
= 0.4 mg mL

1
, where we observed substantial
aggregation of nonmotile bacter
ia but no aggregation of motile
bacteria(Fig.3a).InThediluteregime,nonmotilebacterial
aggregation increased with the P
EG polymer concentration from
the osmotic pressure contribution in the depletion potential
(Fig. 3b). The effective potential between nonmotile bacteria can
be described by the Asakura–Oosawa depletion potential:
15,16,44
U
dep
ð
r
Þ¼
þ1
for
r

0

2
p
P
P
aR
P

r
2

2
for 0
o
r
o
2
R
P
0for
r
4
2
R
P
8
>
>
>
>
<
>
>
>
>
:
(3)
where
U
dep
is the depletion potential (in
J
),
P
P
is the osmotic
pressure of the polymer solution (in Pa),
a
is the radius of the
bacteria (approximated as a sphere, in m),
R
P
is the characteristic
polymer size (in m), and
r
is the separation distance between two
bacteria surfaces (in m). The contribution of polymer concen-
tration to the depletion potential is implicit in the osmotic
pressure. PEG behaves as a polymer in good solvent in buffer,
45
and we can use the following cro
ssover equation for the osmotic
pressure of a polymer solution which spans the dilute and semi-
dilute polymer conce
ntration regimes:
46,47
P
P
¼
N
Avo
kT
MW
c
P
1
þ
c
P
c
P


1
:
3
!
(4)
where
N
Avo
is Avogadro’s number,
k
is the Boltzmann constant,
T
is the temperature (in Kelvin), MW is the molecular weight of
the polymer (in kDa),
c
P
is the polymer mass concentration (in
kg m

3
), and
c
P
*istheoverlapmassconcentration(inkgm

3
).
We use the concentration-dependent radius of gyration to esti-
mate the characteristic polymer size:
48,49
R
P
c
P
ðÞ¼
R
g
ð
0
Þ
MW
N
Avo
kT
d
P
P
d
c
P


1
=
2
(5)
where
R
P
(
c
P
) is the concentration-depe
ndentradiusofgyrationor
the characteristic polymer size (in m), and
R
g
(0) is the radius of
gyration (in m).
R
g
(0) was estimated using literature values of the
hydrodynamic radius of PEG
30
and the Kirkwood–Riseman
relation.
50,51
We estimated
R
g
(0) to be 62.6 nm for 1 MDa PEG,
and using this value and the molecular weight of the polymer,
we estimated
c
P
* to be 1.6 mg mL

1
(see Experimental for
calculation). Combining eqn (3)
–(5)givesusanexpressionfor
the depletion potential that closely approximates the Asakura–
Oosawa potential in the dilute po
lymer concentration regime and
the potential derived by Joanny, Leibler, and De Gennes in the
semi-dilute regime.
52
Similarcrossoverequationsforthedeple-
tion potential have been previously used to quantitatively describe
experimentally observed depletion-driven colloid aggregation in
polymer solutions that span the dilute and semi-dilute concen-
tration regimes.
1,53
As the polymer concentration increases, the
depletion potential also increase
s. We observed that the aggregate
size of the nonmotile
E. coli
increases with PEG concentration in
the dilute polymer concentration regime, which suggests that
depletion forces drive aggregation.
We found that motile bacteria do not aggregate at low PEG
concentrations until a certain PEG concentration threshold is
reached (Fig. 3c). To estimate the effective potential of motile
bacteria and the effect of the swim force on the aggregation of
motile bacteria, we used a previously established theoretical
framework.
18
We began by considering the forces that bacteria
experience in solution. This model accounts for the swim force
that arises from bacterial motility and the polymer-induced
depletion force. The swim force can be described from the ellipsoid
approximation to the Stokes–Einstein drag coefficient:
17,54
F
swim
¼
4
p
Z
b
ln 2
b
=
a
ðÞ
1
=
2
V
(6)
where
Z
is the solution viscosity in Pa s (see Experimental for details
of estimate),
a
and
b
are the lengths of the semi-minor and semi-
major axes for
E. coli
(in m), and
V
is the speed (in m s

1
). For
E. coli
,
a
and
b
are approximated to be about 0.5
m
mand2
m
m,
55
respectively, and
V
is assumed to be constant at about 10
m
ms

1
.
4
The effective force is then calculated using a force balance on the
bacteria, assuming that the swim force directly counteracts the
depletion force:
F
eff
¼
F
swim

@
U
dep
ð
r
Þ
@
r
(7)
where
F
eff
is the effective force (in Newto
ns) and the depletion force
is given by the negative first derivative of
U
dep
(
r
) with respect to
r
.
To find the effective potential
U
eff
, we integrate eqn (5) with respect
to
r
:
U
eff
¼
1
for
r

0

F
swim
r
þ
U
dep
ð
r
Þþ
U
0
for 0
o
r
o
r

0for
r


0
8
>
>
>
<
>
>
>
:
(8)
The integration constant,
U
0
(
J
), is defined as described previously.
18
In a condition where two bacteria are swimming in the exact
opposite direction, there is a separation distance
r
*wherethe
effective force acting on each bacterium is zero. Beyond this range,
the swim force overwhelms the depletion potential, and the
effective potential on the bacteria is zero.
U
0
is defined such that
U
eff
is zero beyond
r
*.
In the dilute polymer concentration regime, we observed
that the aggregation trends for both motile and nonmotile
bacteria were qualitatively consistent with expectations based
on the changes of the minima of their respective effective
potentials (
U
eff
(
r
= 0)) at each PEG concentration (Fig. 3d). For
nonmotile bacteria, the effective potential consists of only the
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depletion potential, which increases with polymer concentration.
Our observations of the aggregation of nonmotile bacteria were
qualitatively consistent with what is predicted from the depletion
potential. In contrast, for motile bacteria, theoretical calculations
suggest that swim force will exceed the depletion force at low
polymer concentrations, resulting in no effective potential. Our
experimental observations were consistent with these calculations;
we saw no aggregation in motile bacteria at PEG 1 MDa
Fig. 3
Effective potentials describe aggregation of motile and nonmotile
E. coli
in the dilute concentration regime of PEG 1 MDa. (a) The volume-
weighted average aggregate size (Vol Wt avg size,
N
) are plotted for both motile and nonmotile
E. coli
at
c
P
= 0.4 mg mL

1
PEG. The box plots depict the
95% empirical bootstrap confidence intervals of the Vol Wt avg size calculated using the method described in the ‘‘Imaging analysis’’ section of the
Methods of ref. 1. The line bisecting the box is the 50th percentile; the upper and lower edges of the box are the 25th and 75th percentile respectively;
and the whiskers are the 2.5th and 97.5th percentiles. Data were compiled from at least three biological replicates. (b) A schematic of nonmotile bact
eria
(orange) in PEG (purple) solution at 0 min and at (c) 10 min. The PEG is excluded from the inter-bacterial volume inducing an effective potential (brown
dotted line) due to depletion (pink arrows). (d) A schematic of motile bacteria (blue) in PEG (purple) solution at 0 min and at (e) 10 min. Although the PE
G
induces the same depletion potential (pink arrows) at a given concentration, the swim force (white arrows) from the bacterial motility decreases bot
h the
well depth and the range, reducing their effective potential (blue dotted line) and preventing aggregation in the dilute PEG concentration regime. (f
) The
effective potential at contact (
U
min
/
kT
) is plotted for motile and nonmotile
E. coli
against PEG concentration (mg mL

1
). The vertical black dotted lines at
c
P
= 0.4 and
c
P
= 1.6 mg mL

1
denote the potential minima taken from the complete potentials plotted in (g) and (h), respectively. (g) The full effective
potential (
U
eff
/
kT
) is plotted against distance from the bacterial surface for both motile and nonmotile
E. coli
at
c
P
= 0.4 mg mL

1
PEG. (h) The full
effective potential (
U
eff
/
kT
) is plotted against distance from the bacterial surface for both motile and nonmotile
E. coli
at
c
P
= 1.6 mg mL

1
PEG.
Paper
Soft Matter
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