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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:w.poon@ed.ac.uk; 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

Soft Matter, 2010, 6, 4540–4549 | 4543

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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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