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Polymer-induced phase separation in Escherichia coli suspensions†
Jana Schwarz-Linek,a Alexander Winkler,b Laurence G. Wilson,a Nhan T. Pham,a Tanja Schilling‡b
and Wilson C. K. Poon*a
Received 7th April 2010, Accepted 25th June 2010
DOI: 10.1039/c0sm00214c
We studied aggregation and phase separation in suspensions of de-flagellated Escherichia coli (AB1157)
in phosphate buffer induced by the anionic polyelectrolyte sodium polystyrene sulfonate. We also
performed Monte Carlo simulations of this system based on the Asakura–Oosawa model of colloid–
polymer mixtures. The results of these simulations, as well as comparison with previous work on
synthetic colloid–polymer mixtures, demonstrate that the role of the polymer is to cause a depletion
attraction between the E. coli cells. The implication of these results for understanding the role of
(predominantly anionic) extracellular polymeric substances (EPS) secreted by bacteria in various
natural phenomena such as biofilm formation is discussed.
Introduction
Understanding the effect of polymers on suspensions of bacteria
is important in diverse areas of microbiology. In nature, many
bacteria secrete significant amounts of exopolysaccharides and
other extracellular polymeric substances (EPS) like proteins and
DNA into their surroundings.1,2 These polymers are important in
bacterial aggregation, initial colonization of surfaces as well as
development, maturation and stability of biofilms.3 Aggregation
caused by the formation of complex polymeric structures like
fibriles consisting of polysaccharide–protein mixtures4 is
involved in spore formation of Myxococcus xanthus5 and plant
root colonisation by Azospirillum brasilense.6 Industrially, poly-
meric additives find widespread use, e.g. for concentrating
bacteria in biotechnology, and for removing them in water
treatment.7
Since bacteria usually carry a net negative surface charge,8
synthetic polymeric additives for inducing bacterial aggregation,
such as polyethylenimine, are mostly cationic polyelectrolytes,
which can bridge neighbouring cells.9 This is analogous to the
way synthetic colloids are bridged by oppositely charged poly-
electrolytes.10 It has been shown that similar electrostatic inter-
actions may be involved in biofilm formation. Microbially
synthesized PGA, a homopolymer of b-1,6-linked N-acetylglu-
cosamine, partly de-acetylated and therefore positively charged,
can cause cell–surface and cell–cell attachment in Staphylococcus
epidermidis as well as Escherichia coli.11,12
However, the majority of bacterial exopolysaccharides are
anionic.1 From the perspective of colloid science, a dispersion of
bacteria and anionic exopolysaccharides constitutes a mixture of
like-charge colloids (the bacteria) and polymers (the exopoly-
saccharides). Three generic mechanisms may be invoked to
aSUPA and School of Physics & Astronomy, The University of Edinburgh,Kings Building, Mayfield Road, Edinburgh, EH9 3JZ, UK. E-mail:[email protected]; Fax: +44 (0)-131-6507174; Tel: +44 (0)-131-6505297bInstitut f€ur Physik, Johannes Gutenberg Universit€at, 55099 Mainz,Germany
† Electronic supplementary information (ESI) available: Time course ofviable cells in MPB (with stills shown in Fig. 2). See DOI:10.1039/c0sm00214c
‡ Present address: Universit�e du Luxembourg, Luxembourg.
4540 | Soft Matter, 2010, 6, 4540–4549
explain polymer-induced aggregation in such mixtures.
Consider, for specificity, anionic polyelectrolytes and colloids
carrying a net negative charge (such as most bacteria).
First, the colloids may be amphoteric, and display a minority
of positive charges. On such heterogeneous charged surfaces,
conditions exist in which the negative polymer segments may
adopt ‘loopy’ configurations to contact the positive surface
patches.13 This could lead to bridging of nearby colloids, and
hence aggregation.x This mechanism relies on strong electrostatic
interactions, and therefore operates only when there is little salt
in the solvent to screen the Coulomb interaction. Secondly,
polyvalent cations can form salt bridges between negatively
charged bacteria and polymer, once again causing bridging.14
At high enough salt concentrations and in the absence of
polyvalent cations, a third mechanism can operate. Here, the
Debye screening length (k�1) may become significantly smaller
than the size of the colloids and polymers, but is still large enough
to prevent attractive van der Waals forces from operating. In
these ‘marginally screened’ mixtures of anionic polyelectrolytes
and negatively charged colloids, we have effectively neutral
particles with the size increased by k�1, and slightly expanded
polymer coils that are non-adsorbing to the particles. It has long
been understood that the polymers in this situation cause an
effective interparticle attraction by the depletion mechanism.15
Exclusion of polymer from the region between two nearby
particles leads to an unbalanced osmotic pressure pushing them
together.{ The range of the resulting depletion attraction
between particles is controlled by the size of polymer coils, while
its strength increases with the polymer concentration.
Depletion aggregation is well understood in uncharged
colloid–polymer mixtures, especially mixtures of hard-sphere
colloids and near-ideal linear polymers. In such model systems,
experiments, theory and simulations show satisfactory agree-
ment, especially in the limit where the polymers are substantially
x At higher polymer concentration, it is possible that this mechanism mayalso give complete surface covering of the colloids and therefore lead tosteric (re)stabilisation of the suspension.
{ Thus, we require the polymer to be larger than k�1. Otherwise, thesurfaces of neighbouring particles never come close enough to excludepolymer from the space between them.
This journal is ª The Royal Society of Chemistry 2010
Fig. 1 AFM images of viable AB1157 cells harvested in stationary phase
and washed using the protocol described in the text. Note that most cells
have lost their flagella; the few flagella that are imaged are not clearly
attached to cells. The width of the image is 10 mm.
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smaller than the colloids.16 Marginally screened like-charge
colloid–polymer mixtures display qualitatively identical
phenomenology,17–19 although quantitative explanation needs to
take account of residual electrostatic repulsive effects.20 Mixtures
of non-adsorbing polymers and colloidal rods have also been
studied.21 The depletion attraction between two rods is aniso-
tropic; being stronger for two rods aligned ‘side by side’ than ‘end
on’. High enough concentrations of polymer lead to nematic (or
orientational) ordering if the aspect ratio of the rods is high
enough.
There are few physico-chemical studies of bacterial aggrega-
tion induced by anionic polyelectrolytes. Recently, however,
Eboigbodin et al.22 investigated this phenomenon using a non-
pathogenic laboratory strain of E. coli (AB1157) and the anionic
polyelectrolyte sodium polystyrene sulfonate (NaPSS). They
found that sufficient concentrations of NaPSS caused bacteria to
phase separate out of solution, and suggested that this was due to
depletion. Interestingly, Eboigbodin et al. found that the
minimum concentration of NaPSS, c*P, needed for phase sepa-
ration of bacteria at cell density cC increased with the latter, i.e.
dc*P/dcC > 0. But depletion is due to ‘crowding’—bacteria (or
other colloids) aggregate to ‘make room’ for the highly entropic
polymers, so that less polymer is needed to cause aggregation in
a more concentrated suspension, i.e. dc*P/dcC < 0 if aggregation is
depletion induced. An alternative mechanism may therefore be in
operation.
Eboigbodin et al. worked in distilled water. Such low ionicity
conditions favour the adsorption of an anionic polyelectrolyte onto
a negatively charged surface with (minority) positive patches.13 As
bacterial surfaces are indeed amphoteric,23 this mechanism
could lead to the bridging of neighbouring cells. To bridge
more cells, more polymer is required, resulting in dc*P/dcC > 0
as found by Eboigbodin et al. Moreover, consistent with such an
electrostatic bridging mechanism, these authors reported that more
negatively charged cells were less aggregated by NaPSS. Thus it is
likely that electrostatic interactions rather than depletion dominate
in the system studied by Eboigbodin et al.kTo access depletion-induced phenomena in mixtures of E. coli
and NaPSS, the electrostatic contribution needs to be significantly
reduced. In this paper, we present such a study using the same
bacterial strain as Eboigbodin et al.22 We followed closely their
preparative protocol, with the important exception that we
worked in a modified phosphate buffer (MPB) with ionicity in the
region of 0.1 M rather than distilled water. We used cells with few
or no flagella in order to access the basic physics first: flagella
would complicate the colloidal interaction between cells, and the
associated motility may give rise to new physics (since the
‘particles’ in our colloid–polymer mixture would now be ‘active’).
Our mixture should belong to the ‘marginally screened’
regime. The Debye screening length in our MPB (see below for
the exact composition) is calculated to be 0.8 nm, much smaller
k A further complication in seeking to understand Eboigbodin et al.22 isthat by adding different amounts of NaPSS to distilled water, the pH andconductivity of the solutions are likely to vary. Over the range of thepolymer concentrations used, we measured a decrease in pH from 7.9to 5.3 going from 0.2 wt% to 10 wt% of polymer. Preliminarymeasurements show that the electrophoretic mobility of the cells andthe conductivity of the dispersions also change significantly over thisrange of polymer concentration.
This journal is ª The Royal Society of Chemistry 2010
than even the smaller of the two polymers we used (radius z 17
nm, Appendix 1) or the size of an E. coli cell (Fig. 1). Charge
effects must therefore be mostly screened out. For NaPSS, this is
confirmed by literature data24 showing that NaPSS behaves very
nearly as an ideal, uncharged random coil. On the other hand, we
know that the screening has not gone so far as to induce aggre-
gation on its own (salting out), since direct observation of pure
cell suspensions and polymer solutions over the whole range of
conditions we used did not detect any instability. The depletion
mechanism therefore should operate.
Indeed, we find that our system reproduces the phase separa-
tion and ‘transient gelation’ phenomena expected in a mixture of
neutral colloids and non-adsorbing polymers.25
We also performed Monte Carlo simulations of spherocy-
linders and idealized polymers within the framework of the
Asakura–Oosawa model15 with parameters matching our
experimental system. The shape of simulated and measured
phase boundaries agree; there is also semi-quantitative agree-
ment of simulated and measured phase boundaries. We therefore
conclude that depletion is indeed the dominant mechanism
causing aggregation and phase separation in E. coli–NaPSS
mixtures in 0.1 M phosphate buffer. Since bacterial exopoly-
saccharides are predominantly anionic in character, this result
has implications for a variety of important phenomena in
microbiology such as biofilm formation.
Some of our results have been announced before (Fig. 2, the
high-cell-concentration portion of Fig. 3(A), and Fig. 6) in a brief
comparative study26 of polymer-induced phase separation in E.
coli and an unrelated Gram-negative bacterium, Sinorhizobium
meliloti. We re-presented these results here for completeness.
Materials and methods
Chemicals and media
All chemicals were of analytical or higher grade and obtained
from Fisher Scientific (NaCl, H2SO4, H2O2 and ethanol), Fluka
Soft Matter, 2010, 6, 4540–4549 | 4541
Fig. 3 Phase diagram for viable (A) and non-viable (B) E. coli AB1157
in MPB with NaPSS1 (Mw ¼ 64700). For calculation of cell and poly-
mer volume fraction, fc and hp, see text below. Note all axes are log-
arithmic. Data points above an initial cell concentration of �5 � 109 cfu
ml�1 are obtained from time-lapsed video observations (see Fig. 2), and
indicate three different kinds of behaviour with time: A ¼ single phase,
+ ¼ two-phase coexistence, B ¼ transient gels. Data points at cell
concentrations of 109 cells ml�1 or lower indicate the position of the
phase boundary, as estimated by the composition of the upper phase in
phase coexisting samples with three initial cell concentrations for (A)
2.2 � 1010, 5.5 � 1010, 1.1 � 1011 cfu ml�1 and (B) 1.4 � 1010,
4.5 � 1010, 9.1 � 1010 cfu ml�1. The dashed line indicates the
approximate position of the equilibrium vapour–liquid phase boundary
according to the two datasets combined. The position of the phase
boundary estimated from the averaged simulation data plotted in Fig. 7
is also shown (:).
Fig. 2 Samples of viable E. coli AB1157 (cell density ¼ 9.6 � 1010 cfu
ml�1, corresponding to a cell volume fraction of �12.5%) dispersed in
phosphate buffer with NaPSS1 (Mw ¼ 64700). The polymer weight
fraction increasing from left to right, with samples 1 to 11 containing 0%,
0.1%, 0.2%, 0.3%, 0.4%, 0.5%, 0.75%, 1%, 2%, 5% and 10% of NaPSS1.
Times: (a) t ¼ 0, (b) t ¼ 30 min, (c) t ¼ 100 min, (d) t ¼ 24 h. Part (e)
shows the lowest portion in samples 2–5 at 24 h at higher magnification.
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(HK2PO4 and H2KPO4) and Sigma-Aldrich (EDTA dipotassium
salt, kanamycin and 3-aminopropyltriethoxysilane). Tryptone
and yeast extract were purchased from Difco (Bacto, BD) and
bacteriological agar (no. 1) from Oxoid.
Bacteria
The non-pathogenic strain E. coli AB1157 (DSM9036) was used.
All cultivations were performed in LB medium (tryptone 10.0 g l�1,
yeast extract 5.0 g l�1 and NaCl 5.0 g�1) using an orbital shaker at
30 �C and 200 rpm. (Note that this differs from Eboigbodin et al.,
who added 0.5% glucose (w/v) to their growth medium.) A pre-
culture inoculated from a single colony on LB agar (tryptone 10.0
g l�1, yeast extract 5.0 g l�1, NaCl 5.0 g�1 and agar 15 g l�1) was
grown for 5 h and used in a 1 : 100 dilution to start an overnight
culture. After reaching stationary phase (16 h) cells were harvested
4542 | Soft Matter, 2010, 6, 4540–4549
by centrifugation (10 min, 2700� g, Hermle Z323K) and prepared
for experiments (see below)). Optical density measurements at 600
nm (Cary 1E, Varian) normalized by viable plate counts on LB
agar of serial diluted samples (OD600nm¼ 1 corresponding to 1.55
� 109 cfu ml�1) were used to determine cell densities. Non-viable
bacteria were obtained by heating suspensions at 60 �C for 30–60
minutes. To confirm cells were non-viable, representative samples
were serial diluted and plated on LB agar where no growth was
observed after incubation at 30 �C for 48 h.
P1 phage transduction27 was used to create a non-flagellated
mutant (AB1157DfliF) using the appropriate E. coli K-12 single
knockout mutant available from the KEIO collection.28
Kanamycin (final concentration 30 mg l�1) was added to all
growth media for AB1157DfliF.
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A representative sample of bacteria treated using the standard
preparation protocol was characterised using tapping-mode
atomic force microscopy (Multimode Veeco with a NanoScope
IIIa controller equipped with a ‘‘J’’ scanner) with silicon canti-
levers (Veeco, spring constant k z 48 N m�1 and resonance
frequency between 130 and 150 kHz). Glass coverslips (Agar
Scientific, 13 mm diameter, no. 2 thickness) were treated prior to
experiments using the following protocol. After cleaning in
a 8 : 2 (v/v) solution of 96% H2SO4 and 30% H2O2 for 20 min,
coverslips were rinsed with distilled water and dried in a stream
of nitrogen. Subsequently, they were immersed in a 0.1%
3-amino-propyltriethoxysilane solution in 95 : 5 ethanol : water
for 20 min and rinsed with ethanol. 10 ml cell suspensions were
deposited on a coverslip and left to settle for 5 min after which
they were then rinsed gently with water and left to dry under
ambient conditions. Comparison with optical microscopy
suggests that our protocol did not significantly change cell shape
or size.
Polymer
Sodium polystyrene sulfonate (Aldrich) of two different molec-
ular weights was used: NaPSS1 (Mw ¼ 64700 g mol�1, Mw/Mn ¼3.1) and NaPSS2 (Mw ¼ 620500 g mol�1, Mw/Mn ¼ 4.6). The
molecular weights and polydispersities were determined by gel-
permeation chromatography (GPC) against PSS standards.
Dynamic light scattering returned a hydrodynamic radius of
rH ¼ 8.7 � 0.1 nm for NaPSS1 in MPB. Polymer stock solutions
were prepared at 20% (w/v) in MPB and filtered through a 0.2 mm
disposable syringe filter prior to use.
Phase behaviour studies
Cells were prepared using the following standard protocol. After
harvesting at stationary phase, the cell pellet was washed three
times with and re-suspended in MPB containing 6.2 mM
K2HPO4, 3.8 mM KH2PO4, 67 mM NaCl and 0.1 mM EDTA
(pH¼ 7.0). Between washes centrifugation was carried out for 10
min at 2700 � g (Hermle Z323K). Vigorous shaking was used to
re-suspend cells. By adjusting the final volume bacterial
suspensions could be concentrated up to 200-fold compared to
the original overnight culture. Observations were made in closed
1.6 ml disposable cuvettes with a total sample volume of 1 ml.
Polymer solution, cell suspension and MPB were mixed in
different ratios to achieve cell concentrations in the range of
5 � 109 cfu ml�1 to 2 � 1011 cfu ml�1 (corresponding to volume
fractions of �0.5% to 20%, based on a single cell being a
2 � 1 mm spherocylinder, see Fig. 1) and polymer concentrations
in the range of 0 to 10 wt% (with the overlap concentration
corresponding to �5%, based on using a coil radius of 17.5 nm,
see Appendix 1).
Samples were homogenised by thorough mixing prior to
incubation at 20 �C (MIR-153, Sanyo). OD600nm was measured
at the start and after a 24 h period. Aggregation was followed
inside the incubator using a camera (QImaging, Micro-publisher
3.3RTV) controlled by QCapture pro 5.0 software. Images were
captured every two minutes for the initial two hours, and
thereafter for varying periods up to and beyond 24 h. From these
This journal is ª The Royal Society of Chemistry 2010
images, we created videos using ImageJ (National Institutes of
Health).
Experimental results
An AFM image of viable AB1157 cells is shown in Fig. 1. It
illustrates that our preparative protocol removes most, if not all,
flagella from the majority of cells, so that these can be appro-
priately modelled as bare spherocylinders. From the images of
40 cells, we measured the mean cell width and length to be
D¼ 0.95� 0.2 mm and L¼ 1.95� 0.5 mm respectively. The mean
aspect ratio is L/D ¼ 2 � 0.2. These findings are consistent with
previous data collected under a variety of conditions, showing
that an aspect ratio of 2 : 1 is a lower bound,29 and that cells are
smallest and least polydisperse in the stationary phase.30 Images
of heat-treated, non-viable cells (not shown) reveal that they lose
all flagella, maintain their overall shape and size, but have rather
more irregular surfaces.
Cuvettes containing bacteria and NaPSS dispersed in MPB at
various concentrations were monitored by time-lapsed imaging.
Since bacterial suspensions appear turbid in our sample cells
even at the lowest concentrations studied here (5 � 109 cfu
ml�1), sedimentation and phase separation are easily visible.
Over the range of bacteria and polymer concentrations we
investigated, three types of behaviour were seen irrespective of
whether viable or non-viable cells were used. Here we show
time-lapsed stills from experiments using viable cells and
NaPSS1, Fig. 2 (see also Movie S1†). At zero and the lowest
concentration of added polymer (Fig. 2a–d, samples 1 and 2) we
observed a meniscus that moved down at a few mm within 24 h,
which is consistent with what is expected for micron-sized
objects (density z 1.08 g cm�3)31 sedimenting in MPB (density z1.00 g cm�3).32 In other words, we are seeing essentially single-cell
sedimentation.
At each cell concentration, there was a critical polymer
concentration above which samples rapidly became optically
inhomogeneous, with a region denser in bacteria building up
at the bottom (Fig. 2b–d). Eventually this resulted in
completely phase separated samples. As the polymer concen-
tration was increased, the phase separation process was found
to accelerate.
Three observations suggest that we are seeing thermody-
namic phase separation into equilibrium coexisting phases.
First the upper phase, whilst more dilute than the lower phase,
certainly contains bacteria. This was evident to the naked eye in
the case of sample 3, Fig. 2(e). Optical density measurements
(see below) confirmed that this was also the case for other
phase separated samples (such as 4–9 in Fig. 2(d)). Secondly, at
a fixed cell concentration, increasing polymer concentration
gave rise to upper phases containing decreasing number of
bacteria. Again, this was evident by visual inspection (compare
sample 3 with samples 4 and 5 in Fig. 2(e)), but also from OD
measurements reported below. This is what is expected upon
moving deeper into a thermodynamic two-phase coexistence
region. Finally, we should anticipate the volume of the lower
phase to increase correspondingly. This was indeed observed,
Fig. 2(e).
The phase coexistence here is of the ‘vapour–liquid’ kind. Two
phases with low and high concentrations of colloids coexist, with
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the arrangement of particles being disorderly in both cases. This
is analogous to the coexistence of vapour and liquid states at the
boiling point of atomic and molecular liquids. Evidence for the
fluid nature of the lower phase comes from the simple experiment
of tilting phase separated samples; the meniscus separating the
two phases remained horizontal in each case.
Upon further increase of the polymer concentration, the
acceleration in the phase separation process is abruptly arrested,
samples 10 and 11 Fig. 2(c). Instead, these samples remained
uniformly turbid throughout for extended periods before any
sedimentation became visible. After 24 hours, all the bacteria
were precipitated out in a solid-like mass. The interface between
this precipitate and the supernatant was often visibly ‘lumpy’,
and it did not remain horizontal when the sample was tilted.
These observations are typical of the phenomenon of delayed
sedimentation, which is widely seen in colloids with strong short-
range attractions.33 There is an emerging consensus that such
transient gelation is the result of arrested vapour–liquid phase
separation.34
All three types of behaviour were found to be reversible—the
same observations were made in samples re-suspended by
shaking.
A summary of our observations is given in Fig. 3(A) in the
form of a phase diagram.** Note that we report these results in
two sets of variables. The ‘experimental’ variables are the cell
concentration in cfu per ml and the polymer concentration in
weight percent. To facilitate theoretical comparison, we also
show the respective volume fractions which were calculated as
follows. The cell density in cfu per ml multiplied by a cell volume
of 1.3 mm3 gives the cell volume fraction, fc, while the polymer
weight (wp) and volume (hp) fractions are related by eqn (1)
(see below) using a polymer radius of 17.5 nm (see Appendix 1).
The latter procedure is appropriate for polydisperse polymers.35
The lowest cell concentration we could use in these experi-
ments was limited by the ability to detect phase separation
visually and distinguish it from slow sedimentation.
To access the phase boundary at concentrations lower than
5 � 109 cfu ml�1, a different approach was necessary. We
measured the optical density of the dilute, upper phase in samples
showing two-phase coexistence. Thus, for example starting at
9.6 � 1010 cfu ml�1 (OD600nm > 3), after 24 h the OD600nm of the
upper phase in samples 4–10 in Fig. 2(d) was measured to be
0.7797, 0.5969, 0.5277, 0.3193, 0.2332, 0.0868 and 0.0523
respectively, confirming the presence of cells in each case. Up to
one unit OD600nm these values are proportional to cell density,
with OD600nm ¼ 1 corresponding to 1.55 � 109 cfu ml�1, thus
giving an estimate of bacterial concentration in the upper phase.
The polymer concentration in the upper phase in each case will
be higher than the polymer concentration in the sample as
a whole, balancing a lower concentration of polymer in the
coexisting lower phase more concentrated in bacteria. However,
this difference is small for our samples, because of the small
volume of lower phase in each case. Indeed, the maximum
possible underestimate in taking the upper phase polymer
concentration to be the average concentration is given by the
** Note that this terminology is loose. Strictly speaking the reporting oftransient gelation has no place in a phase diagram, which, sensus strictus,only contains equilibrium thermodynamic information.
4544 | Soft Matter, 2010, 6, 4540–4549
percentage of the sample occupied by the lower phase—from
�5 wt% in sample 4 to 13% in sample 9. With this proviso, we
were able to calculate the cell density in the upper phase for each
sample (4–9). These are plotted in Fig. 3(A), and should give
a lower bound for the low-bacterial-concentration side of the
phase boundary. Results obtained by repeating this procedure
for samples with initial cell densities of 2.2 � 1010 and 1.1 �1011 ml�1 are also shown.
Experiments using non-viable cells gave results that were
qualitatively identical to those made using viable cells: the same
three kinds of behaviour were seen as polymer concentration was
increased. This is not surprising considering the similar cell
shapes and dimension in both cases. Only very small quantitative
differences were found, summarized in a second phase diagram,
Fig. 3(B). Similarly, a limited number of experiments using viable
cells of the AB1157DfliF mutant unable to synthesize flagella,
gave results which were identical to viable wild-type cells
prepared using the vigorous washing procedure (data not
shown).††
We repeated our experiments using a higher molecular weight
sodium polystyrene sulfonate (NaPSS2, Mw ¼ 620500). The
results for viable cells are summarised in Fig. 4 (with hp calcu-
lated using a polymer radius of 54.2 nm). Results for non-viable
cells (not shown) are identical within experimental uncertainties.
Phase separation was observed above a polymer concentration of
0.05 wt%. The phase boundary was flat but at lower wt%
compared to NaPSS1.
Simulations
To help elucidate the mechanism for polymer-induced phase
separation in the bacterial suspensions, we performed Monte-
Carlo simulations of our experimental system in the spirit of
what is perhaps the simplest possible model of colloid–polymer
mixtures, the Asakura–Oosawa (AO) model. In the original AO
model15 both colloids and polymers are modelled as spheres. The
colloids are hard spheres. The polymers are spheres of radius r
that are interpenetrable to each other but whose centres cannot
approach closer than a distance r from the surface of the hard
particles.
The AO model offers a reasonably quantitative account of
mixtures of nearly monodisperse hard-sphere colloids and
polymers in nearly q (or ideal) solvents.16 We replace the hard
spheres by hard spherocylinders. Again, the polymers are
modelled as spheres that can penetrate each other but cannot
overlap with the hard spherocylinders,36 Fig. 5.
Choosing r, the radius of the ‘AO spheres’ representing the
polymers, is less straightforward. Finding the ‘best’ value of r in
an AO-simulation to represent a particular experimental system
is beset with experimental uncertainties (e.g. the solvent quality is
often not precisely known). There is also the theoretical issue of
how to model the depletion of polymer segments next to
a particle surface by a single parameter, the ‘depletion layer
thickness’. The issue is nevertheless important. While in the
†† The electrophoretic mobilities of viable and non-viable cells (washedand re-suspended in MPB) of wild type as well as DfliF mutant areidentical to within � 5% (data not shown), suggesting that they haveapproximately equal surface charges.
This journal is ª The Royal Society of Chemistry 2010
Fig. 5 The Asakura–Oosawa model for a mixture of polymers and rod-
shaped bacteria. The bacteria are represented as spherocylindrical
particles (such as 1, 2 and 3). Each polymer coil is represented as a sphere
of radius r (inset). These spheres can interpenetrate each other, but the
centre of a sphere cannot come closer than a distance r to the surface of
the spherocylinder. Each particle is therefore surrounded by a ‘depletion
zone’ (delineated by thin dotted lines). There are no polymers within the
overlapping depletion zones of two neighbouring particles (hatched); the
resulting imbalance in osmotic pressure pushes the two particles together.
This is the origin of the depletion attraction. Consideration of the volume
of overlapping zones immediately suggests that the depletion attraction
between particles 1 and 2, in the ‘parallel’ configuration, will be the
greatest compared to that between particles in any other mutual orien-
tation (such as 2 and 3). The dotted background indicates the ‘free
volume’ available to the centres of the interpenetrable polymer spheres.
Fig. 4 Phase diagram for viable E. coli AB1157 in MPB with NaPSS2
(Mw¼ 620500). Note axes are logarithmic and the same as in Fig. 3. Data
points above an initial cell concentration of �5 � 109 cfu ml�1 are
obtained from time-lapsed video observations, and indicate two different
kinds of behaviour with time: A ¼ single phase, + ¼ two-phase coexis-
tence. Data points at lower cell concentrations indicate the position of the
phase boundary, as estimated by the composition of the upper phase in
phase coexisting samples from samples with three initial cell concentra-
tions: 1.0 � 1010, 5.2 � 1010 and 1.0 � 1011 cfu ml�1. The dashed
line indicates the approximate position of the equilibrium vapour–liquid
phase boundary according to the two datasets combined.
This journal is ª The Royal Society of Chemistry 2010
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simulations we control the volume fraction of interpenetrable
spheres, hp, the corresponding experimental variable is the weight
fraction of polymers, wp. These two parameters are related by
wp ¼h
r0vp
Mw
NA
(1)
where r0 is the density of the solution (�1 g cm�3), NA is Avo-
gadro’s constant, and vp is the volume of a polymer coil.
Crucially, vP ¼ (4/3)pr3, so that converting between hp and wp
depends strongly on r.
Appendix 1 explains why we chose to perform simulations
using 2r ¼ 35 nm, which is the largest possible value of r con-
strained by our experimental measurement of the hydrodynamic
radius using dynamic light scattering and theoretical information
available from the literature (see Appendix 1). Using the largest
possible value for r should give a lower bound for the phase
boundary in a representation where polymer concentration is
reported in weight fraction. Even with the choice of the largest
possible value for r, the size ratio D/r z 29 is still large, so that
many ‘polymers’ (interpenetrable spheres) are required per
‘bacterium’ (hard spherocylinder). Thus, special simulation
techniques are necessary and the simulations are restricted to
a small number of spherocylinders. Configuration space was
explored by local translation and rotation moves and by cluster
moves. In order to avoid checking for overlaps with spherocy-
linders at a distance larger than the range of interaction, we used
cell systems. Due to the large size ratio we needed two systems of
different cell size. One was on the order of the spherocylinder
diameter, D, and it was used for checking cylinder–cylinder
overlaps. The second one consisted of very small cells on the
order of the diameter of the interpenetrable spheres, 2r, and it
was used for detection of sphere–spherocylinder overlaps.
As there is a high probability of generating overlaps with the
surrounding spheres for every displacement of a spherocylinder
over a distance on the order of D, standard translation and
rotation moves lead to very small acceptance probabilities (or to
very small displacements).
One possible approach to this problem is to derive an
approximation to the effective interaction potential between the
colloids by integrating out the degrees of freedom of the poly-
mers. Simulations are then performed for the colloids only using
the effective interactions.37 Another possibility, which explicitly
includes the degrees of freedom of the polymers, is to apply
rejection-free schemes.38,39
However, rejection-free schemes in which a cluster of particles
is mirror reflected at a pivot point would be computationally
expensive at our target densities due to the large number of
particles involved. If a spherocylinder overlaps with another
spherocylinder or with a polymer sphere, we reject the move.
These spherocylinder moves are similar to the rejection free
moves introduced previously.38,39 However, our moves include
some probability for rejection. For the concentrations studied
here, the computational cost for the rejections is more than
compensated by the smaller number of distance computations
involved in one move.
We used three different types of cluster moves:
1. Translate the spherocylinder along the translation vector M.
Mirror reflect all spheres which overlap with the new position of
the spherocylinder at the center of M.
Soft Matter, 2010, 6, 4540–4549 | 4545
Fig. 7 Simulation results for predicted phase boundaries. Data points
indicate two different kinds of behaviour: A¼ single phase and +¼ two-
phase coexistence.
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2. Rotate the spherocylinder like in the ordinary MC moves.
Calculate the vector w ¼ u + v, where u and v are the directional
vectors of the cylinder before and after the rotation. Detect the
overlapping spheres and rotate them by p around the axis w.
3. Detect a cluster of spherocylinders and proceed like in
cluster move 1 for every individual translation vector of the
spherocylinders in the cluster.
Since the size ratio is very large, systems near the phase
boundary, or binodal, contain very high numbers of spheres. We
used a box size (measured in units of spherocylinder diameter) of
9 � 9 � 9 for spherocylinder volume fractions between f ¼5.03% and 9.3% and a box size of 12 � 12 � 12 for lower volume
fractions. For these system sizes simulations with up to 6 million
spheres were performed, which were computationally expensive
(ca. one month of CPU time on a Intel(R) Xeon(R)CPU E5345
running at 2.33GHz). Therefore, we could not compute free
energy differences, but rather had to extract an estimate for the
location of the binodal from visual observation of snapshots and
from measuring the cluster size distribution, P(S). A cluster
consists of connected spherocylinders where two spherocylinders
are called ‘‘connected’’ if their depletion zones overlap. The size
of a cluster is the number of spherocylinders which belong to it.
Typical histograms of P(S) are shown in Fig. 6. The continuous
distribution in Fig. 6(a) corresponds to a sample that remains
single phase, while the ‘twin-peaked’ distribution in Fig. 6(b) we
take as the signature of phase separation. Note that we cannot
access spherocylinder volume fractions below �1% because of
the very large number of polymer spheres such simulations
would entail.
Our simulation results are summarized in Fig. 7 plotted in
terms of the volume fractions of spherocylinders, fc, and poly-
mer spheres, hP. We estimate the phase boundary (or binodal) to
Fig. 6 Distribution of cluster sizes. P(S) gives the probability of
encountering a cluster of size S. Note that the vertical scale is logarithmic.
(a) Cluster size distribution for a sample that we consider to be in the
single-phase region of the phase diagram; P(S) is approximately expo-
nential. (b) Cluster size distribution for a sample in what we consider to
be a phase separated sample, displaying two peaks.
4546 | Soft Matter, 2010, 6, 4540–4549
be mid-way between the highest single-phase sample and the
lowest phase separated sample at each spherocylinder concen-
tration. To compare with experiments, we convert hP into
polymer weight fraction using eqn (1), and fC into cell concen-
tration by dividing by the volume of a 2 � 1 mm spherocylinder
(1.3 mm3). The phase boundary estimated from simulations in
this way is shown in Fig. 3.‡‡
Discussion
The observed phenomenology summarized in Fig. 3 can be
mapped onto that seen in mixtures of synthetic colloids and non-
adsorbing polymers, where depletion is known to be operative.
In a nearly monodisperse suspension with volume fraction
(40%, adding sufficient polymer leads to fluid–crystal phase
separation.16 Buried in the fluid–crystal coexistence region of the
phase diagram, there is a metastable vapour–liquid phase
boundary.16 Particles that are sufficiently polydisperse or non-
spherical in shape will not be able to crystallize. In such
a suspension where crystallization is suppressed, increasing
polymer concentration gives rise to vapour–liquid phase sepa-
ration instead.40 In both cases, the highest concentrations of
polymer lead to transient gelation.16
If the particles are sufficiently anisotropic, adding polymer
leads to coexistence of isotropic and nematic phases of the
particles. For spherocylinders, this requires an aspect ratio of
T4.21
Our E. coli cells may be approximated as somewhat poly-
disperse spherocylinders of aspect ratio z 2, which is too low for
the occurrence of a nematic phase. If depletion is the dominant
mechanism in our bacteria–polymer mixtures, we may therefore
expect that adding polymer should give rise to vapour–liquid
phase separation, which is exactly what we observed, Fig. 2.
‡‡ As the number of spherocylinders in the simulations is rather small, weoverestimate the densities at the binodal due to droplet evaporation.Under conditions in which the bulk would be phase separated it isentropically favourable to break up a cluster of rods into severalsmaller clusters. Therefore one needs a larger density of depletant ina finite system to produce a single droplet than in the bulk.
This journal is ª The Royal Society of Chemistry 2010
xx We have shown before26 that the phase boundary also moves withpolymer chain stiffness in a way that is consistent with depletion.
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Qualitatively, this provides direct evidence that phase separation
in our mixtures is depletion-driven.
On the other hand, the shape of the directly measured phase
boundaries in the range of cell concentration between 5 � 109
and 1011 cfu ml�1 does not immediately support a depletion
mechanism. As we have already explained, since depletion is
a crowding effect, we expect that less polymer should be needed
to cause phase separation at higher cell densities, i.e. the phase
boundary should have a negative slope. But the directly
measured phase boundaries at cell densities between 5 � 109 and
1011 cfu ml�1, Fig. 3, appear flat. However, the indirectly
measured phase boundary at cell densities below �109 cfu ml�1,
Fig. 3, does have a pronounced negative slope, suggesting that
the boundary at higher concentrations should indeed have
a negative slope, but of such small magnitude as to be hardly
measurable. Importantly, the simulated phase boundary at
higher cell densities is also essentially flat. Depletion therefore
remains the most likely mechanism in operation.
Our data obtained using the higher molecular weight polymer
NaPSS2, Fig. 4, provide further support for this claim. Within
the AO model, the depletion potential at contact between two
spherical particles of radius a induced by polymers of radius r, so
that x ¼ r/a, is given by41
U contactdep
kBT¼ 3
2
hfree
x(2)
where kBT is the thermal energy and hfree is the concentration of
polymers in the ‘free volume’ left by the spherical particles
(dotted region in Fig. 5). In the limit of x� 1, hfree z h/(1 � f),
where f is the volume fraction of particles, and h is the volume
fraction of polymer in the total sample volume. The weight
fraction (wp) of a polymer with molecular weight Mw has already
been given in eqn (1). If we assume that the depletion potential at
the phase boundary at any given f is constant, then eqn (1) and
(2) together give the result that the weight fraction of polymer
needed to cause phase separation should scale as Mw/r2. In an
ideal solvent, r z M1/2w , so that we expect the phase boundary in
the (f, wp) plane to be independent of molecular weight. In
a good solvent, r z M0.588w , so that we expect the phase boundary
to scale as M�0.176w ; in other words, a 10-fold increase in molecular
weight should drop the phase boundary in the (f, w) plane by
about one third. Experimentally, we find that the phase bound-
aries in Fig. 3A and B lie within the range 0.1% < w < 0.2%, while
the boundary in Fig. 4 lies within 0.05% < w < 0.1%. This
observed small shift downwards is consistent with depletion
operating with the polymers being in a slightly better-than-q
solvent. A theoretically better justified procedure, matching
second virial coefficients for the two cases rather than Ucontactdep (see
Appendix 2), does not change this conclusion.
Quantitatively, the simulated boundary occurs for NaPSS1 at
a polymer weight fraction (w z 0.03%) that is about a factor of
5 lower than the measured ones (at w z 0.15%). There are
two reasons why we may expect the simulated boundary to be
lower than the measured ones. First, recall that we used the
largest justifiable diameter for the polymer spheres,
r ¼ 1:8rHð2=ffiffiffiffippÞ ¼ 17:5 nm, in our AO simulations. So the
simulated boundary plotted in Fig. 3 should represent a lower
bound. We can estimate an upper bound for the AO phase
boundary by using the smallest possible r¼ 8.7 nm¼ rH. If, as we
This journal is ª The Royal Society of Chemistry 2010
have argued above, the phase boundary scales at least approxi-
mately as Mw/r2, then we may expect the boundary to shift from
w z 0.03% to w z 0.03 � (17.5/8.7)2 ¼ 0.12%. The apparently
near-perfect agreement of this value with the observed position
of the phase boundary is no doubt fortuitous. Indeed, we expect
that in reality, 8.7 nm < r < 17.5 nm, so that the AO boundary
will still be somewhat lower than the observed ones. Again,
matching second virial coefficients rather than contact potentials
(Appendix 2) does not change this conclusion.
The polyelectrolyte nature of NaPSS constitutes a second
reason why the observed phase boundaries may be expected to lie
above the simulated one. Each polymer molecule gives rise to
a large number of Na+ counter ions when it is dissolved. In MPB
preliminary conductivity measurements showed an increase from
10 to 30 mS cm�1 for polymer samples of 0.1 and 10% respec-
tively. This leads to increased screening of the polystyrene
sulfonate backbone and therefore shrinking of the polymer coils.
This decrease in r would again lead to an experimental phase
boundary at higher wP then predicted by a model in which
individual polymer coils remain at a constant size irrespective of
the polymer concentration.
We should mention that due to the finite system size and the
use of the canonical ensemble in the computer simulations, we
overestimate the location of the binodal. Such effects have been
observed in several other systems before,42 and were always on
the order of a few percent. We therefore assume that the finite
size correction in our case is also weak. But we have no
knowledge of the relevant interfacial energies to give a precise
estimate.
Conclusions
We used sodium polystyrene sulfonate, an anionic poly-
electrolyte, to induce phase separation and transient gelation in
suspensions of stationary phase grown, deflagellated E. coli
AB1157 suspended in modified phosphate buffer. We have
argued that under our experimental conditions, the polymer and
bacteria should both be ‘marginally screened’. Thus, the deple-
tion mechanism should operate, whereby the polymer induces an
effective attraction between bacteria. The phase separation and
transient gelation phenomenology we observed are analogous to
similar phenomena seen and extensively studied in mixtures of
non-adsorbing polymers and spherical colloids, where depletion
is undoubtedly the responsible mechanism. The phase boundary
was observed to move with polymer molecular weight in
a manner that is consistent with depletion.xx We therefore
suggest that depletion is the dominant mechanism causing phase
separation in our system. Simulations of depletion-induced phase
separation using the Asakura–Oosawa model provided further
support for this.
Previous studies22 of mixtures of NaPSS and E. coli used
distilled water as the suspension medium. We argued that the
phenomenology in this distilled-water-based system was domi-
nated by electrostatics. In particular, the positive slope of the
observed phase boundary led us to suggest that depletion was not
the operative mechanism. The experiments and simulations
Soft Matter, 2010, 6, 4540–4549 | 4547
{{ This is defined as I ¼P
z2i ci. The summation is over all N species of
ions. The i-th species has charge zi (in units of the electronic charge) and(molar) concentration ci. In MPB, the main contributing species are 0.01M potassium phosphate and 0.067 M NaCl.
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reported here strengthen the plausibility of this suggestion. As
a consistency check, we performed a limited number of experi-
ments in distilled water and the resulting phase diagram also
showed a positive slope for the phase boundary (data not
shown). Whether other mechanisms contribute to aggregation
and phase separation in the distilled water system remains
unknown and intriguing. The recent prediction13 of the possible
adsorption of anionic polyelectrolytes onto negative surfaces
that also carry a minority of positive charges at low ionicity offer
an interesting possibility—the adsorbed polymers can then
bridge neighbouring bacteria.
The majority of bacterial exopolysaccharides are anionic. Our
experiments suggest that such exopolysaccharides can induce
depletion attraction between bacteria irrespective of any specific
chemical or biochemical effects they may have (such as adhesion,
recognition by receptors, etc.) at physiological ionicity (�0.1 M).
In the microbiology literature, appeal is almost invariably made
to these latter, specific, effects to explain experimental observa-
tions, with depletion almost never considered. This is no longer
tenable in view of our results. Since depletion is generic, its role
must always be taken into account.43 Experiments using the
nitrogen-fixing bacterium S. meliloti and a variety of anionic
polyelectrolytes of biological and chemical origin confirm this
conclusion, since depletion was also shown to be the operative
mechanism in these mixtures.44 Since depletion also operates
between a bacterium and a surface,45 it may also play a role in the
initial stages in biofilm formation. Note that the non-motile state
of our cells may be of particular relevance: the biofilm phenotype
of several bacteria, for instance S. meliloti, E. coli and Pseudo-
monas aeruginosa is often associated with down-regulation of
motility.46–48
Finally, we comment on how the presence of flagella may
affect the phenomenology. Passively, if the flagella are not
bundled, they may provide a degree of ‘steric stabilization’,
hindering the surfaces of neighbouring bacteria from coming
close enough for depletion to take effect. Actively, flagella enable
motility which could also have a large effect. We can see this by
calculating the depletion force holding two bacteria together,
a lower bound of which can be estimated by
FdepzU contact
dep
2r(3)
This comes to �0.5 pN at the phase boundaries shown in
Fig. 3, which is comparable to the hydrodynamic drag on a �1
mm sphere moving at 20–30 mm s�1 (typically E. coli swimming
speed). We therefore expect motility to affect very significantly
the phenomenology reported here. Experiments to test and
quantify these predictions are underway in our laboratory.
Appendix 1: estimation of polymer radius in the AOmodel
In the AO model, polymers are treated as interpenetrable spheres
of radius r, such that the centre of a polymer cannot approach
closer than distance r to the surfaces of the hard particles present
in the system. It is clear that below the concentration at which
polymer coils overlap (the overlap concentration), r should scale
as the dimension of single coil. But there is no unique recipe for
mapping the various possible dimensional measures of polymers
4548 | Soft Matter, 2010, 6, 4540–4549
to r. Theory suggests that the thickness of the layer depleted of
polymer segments for an ideal and athermal polymer next to a
flat hard wall is given by r ¼ 2=ffiffiffiffipp
rgz1:13rg49 and r¼ 1.074rg
50
respectively, where rg is the radius of gyration of the individual
polymer coils. The athermal result applies only at zero polymer
concentration; the prefactor decreases as the concentration
increases. NaPSS1 has a hydrodynamic radius of rH ¼ 8.7 �0.1 nm and behaves like a neutral polymer in a good solvent
in 0.1 M NaCl, but becomes progressive more ideal at higher salt
concentrations.23 Our experiments are performed using poly-
disperse polymers (Mw/Mn ¼ 3.1) in MPB with a total ionic
strength{{ of I z 0.18 M. For linear monodisperse polymers in
good solvents rg/rH ¼ 1.6, while in q solvents rg/rH ¼ 1.5 and
1.7 for monodisperse and polydisperse coils with Mw/Mn ¼ 2,
respectively.51 For our (larger) polydispersity, we may then
expect rg/rH > 1.7. Using r/rg ¼ 1.13, rH¼ 8.7 nm and rg/rH¼ 1.8
(since we have Mw/Mn > 2), we estimate 2r z 35 nm.
In the simulations, we vary the volume fraction of the ‘poly-
mers’, h ¼ 4/3pr3r, where r is the number density of inter-
penetrable spheres. We convert h into weight fraction for
comparing with experiments using eqn (1). Theory suggests that
Mw is the appropriate average to use when the polymer is poly-
disperse.35
Appendix 2: mapping phase boundaries viacorresponding states
For soft and hard spheres interacting with a variety of attractive
potentials, a law of corresponding states exists. In particular, the
critical volume fraction, fc, and a ‘reduced second virial coeffi-
cient’, b2c, at the critical point, stay remarkably constant as the
details of the interaction potential are varied.52 The second virial
coefficient of a system of particles with interparticle potential
U(r) is given by
B2 ¼ 2p
ðN
0
dr r2�1� e�UðrÞ=kBT
�
For attractive hard spheres (diameter s), the reduced second
virial coefficient is given by
b2 ¼B2
BHS2
whereBHS2 ¼ 2/3ps3 is the second virial coefficient of the hard
spheres without attraction. For many different interparticle
potentials, fc z 0.2 and b2c z �1.5.53 A similar result likely
holds for hard spherocylinders interacting via an attractive
square well.52 We can use these results to predict how the phase
boundary should move when the polymer molecular weight is
changed, or if a different ‘AO polymer radius’ is used to convert
the simulations to experiments. The reduced second virial coef-
ficient of spherocylinders of length L and diameter D (aspect
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ratio a ¼ L/D) interacting via a square well attraction of depth 3
and range lD is given by52
b2 ¼B2
pD3¼ �
�2
3
�l3 � 1
�þ a
�l2 � 1
�þ 1
4a2ðl� 1Þ
��e3=kBT � 1
�23þ aþ 1
4a2
We can make use of this result if we approximate the depletion
attraction between bacteria by a square well with the same
contact energy, and have a means to estimate this contact value.
As an approximation, we replaced the spherocylinders by spheres
with the same volume, and then used the AO model for spherical
particles and polymers given in eqn (1) in the main text to esti-
mate 3. We first calculated the reduced second virial coefficient
along the experimental boundary in Fig. 3 (with 1 cfu ml�1
corresponding to f ¼ 1.3 � 10�12) and then assumed that the
same b2(f) to be the phase boundary also for the larger polymer.
Converting this back into polymer weight fraction using eqn (2),
we find that the weight fraction of polymer to cause phase
separation should drop by �25% and �50% under ideal and
good solvent conditions respectively. Next we calculated the
second virial coefficient along the simulated phase boundary
(Fig. 3(A), calculated using an AO polymer radius of
r ¼ 17.5 nm), and used a similar procedure to predict where
this boundary would be located if an AO polymer radius of
r ¼ rH ¼ 8.7 nm was used instead. This gives a 4.6-fold rise in the
boundary in the (f, wp) plane, only slightly different from the
4-fold change predicted by matching contact potentials rather
than second virial coefficients.
Acknowledgements
We thank Gail Ferguson and Graham Walker for providing the
E. coli strain (AB1157) used in this work. The EPSRC funded
WCKP and JSL (EP/D071070/1), LGW and NTP (EP/E030173)
and GD (studentship). We thank Catherine Biggs for intro-
ducing us to the subject. NaPSS molecular weight and poly-
dispersity analysis were funded by the EPSRC and performed at
Rapra Technology.
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