Self-Organisation of
Confined Active Matter
Hugo Wioland
Churchill College
This dissertation is submitted for the degree of
Doctor of Philosophy
December 2014

Preface
This dissertation is the result of my own work and includes nothing which
is the outcome of work done in collaboration except as declared in the
preface and specified in the text.
It is not substantially the same as any that I have submitted, or, is being
concurrently submitted for a degree or diploma or other qualification at the
University of Cambridge or any other University or similar institution ex-
cept as declared in the preface and specified in the text. I further state that
no substantial part of my dissertation has already been submitted, or, is
being concurrently submitted for any such degree, diploma or other quali-
fication at the University of Cambridge or any other University of similar
institution except as declared in the Preface and specified in the text.
In chapter 1, I describe the self-organisation of bacterial suspensions, il-
lustrated by experiments in special microchambers. These chambers have
been designed and prepared by Vasily Kantsler, a post-doc in Ray Gold-
stein’s group at the time and now an assistant professor at the University
of Warwick. I performed the experiments and the analysis.
In chapter 2, I present experimental results in flattened drops and race-
tracks. I have designed chambers and carried out all experiments. I anal-
ysed movies using Matlab (The MathWorks) plugin mPIV, which has been
previously adapted for time averaging by Aurelia Honerkamp-Smith, a
post-doc in Ray Goldstein’s group. Experimental results on flattened drops
have been published in Physical Review Letters [1].
Chapter 3 first introduces an experimental method I have developed to
simultaneously measure the directions of swimming and of motion of bac-
teria. Mutant strain amyE::hag(T204C) DS1919 3610 is a gift from Linda
Turner Stern, a research associate at the Rowland Institute, and Daniel
i
Kearns, an associate professor at Indiana University. This mutant has
been created in Daniel Kearns’ group and he also helped me on proto-
col technicalities. The second part of this chapter presents in silico studies.
Enkeleida Lushi, who visited the Newton Institute and is now a post-doc at
Brown University, wrote and performed all simulations. I only thoroughly
discussed the results with her. Experiments on fluorescent bacteria and
simulations in flattened drops have been published in Proceedings of the
National Academy of Sciences [2]. Experiments and simulations results in
racetracks will be submitted in the near future [3].
Chapter 4 introduces experimental and theoretical work on lattices of vor-
tices. For this project, I collaborated with Francis Woodhouse, a former
PhD student in Ray Goldstein’s group and now a research assistant pro-
fessor at The University of Western Australia, and Jörn Dunkel, a former
post-doc in Ray Goldstein’s group and now an assistant professor at MIT.
I carried out all experiments. Theories are the result of discussions be-
tween the three of us and I performed the direct analyses presented in this
manuscript.
ii
Acknowledgements
I would like to warmly thank Ray Goldstein for welcoming me in his lab,
first as a master student and then as a PhD student. It has been a great
pleasure to work in Cambridge and to discover marvellous organisms like
Bacillus, Chara, and Volvox.
I would also like to thank all members of Ray Goldstein’s and Eric Lauga’s
groups, in particular Kyriakos Leptos, for his regular biological advices,
Aurelia Honerkamp-Smith and Vasily Kantsler, for their help on experi-
ments, Francis Woodhouse and Jörn Dunkel, for great collaborations, Jo-
celyn Dunstan Escudero, Pierre Haas, Philipp Khuc Trong and François
Peaudecerf, for fruitful discussions.
Enkeleida Lushi has considerably contributed to this PhD, and I am thank-
ful for our collaboration. I also would like to thank Linder Turner Stern
and Daniel Kearns for their friendly help on experiments with fluorescent
bacteria.
Finally, I am extremely thankful to all colleagues who proofread this
manuscript, including Stephanie Höhn, Emily Riley, Kirsty Wan and oth-
ers cited above.
This work was supported by the EPSRC and ERC Advanced Investigator
Grant 247333.
iii
Summary
Active matter theory studies the collective behaviour of self-propelled or-
ganisms or objects. Although the field has made great progress in the past
decade, little is known of the role played by confinement and surfaces.
This thesis analyses the self-organisation of dense bacterial suspensions in
three different microchambers: flattened drops, racetracks and lattices of
cavities.
Suspensions of swimming bacteria are well-known to spontaneously form
macroscopic quasi-turbulent patterns such as jets and swirls. Confinement
inside flattened drops and racetracks stabilises their motion into a spiral
vortex and wavy streams, respectively.
We have quantitatively measured and analysed bacterial circulation and
discovered cells at the interfaces to move against the bulk. To understand
this phenomenon, we developed a method able to measure simultaneously
the directions of swimming and of motion. Experiments in drops reveal
that cells align in a helical pattern, facing outward and against the main
bulk circulation. Likewise, bacteria in racetracks share a biased orientation
against the overall stream.
Particle-based simulations confirm these results and identify hydrodynamic
interactions as the main driving force: bacteria generate long-range fluid
flows which advect the suspension in the bulk against its swimming direc-
tion, resulting in the double-circulation pattern.
We have finally injected dense suspensions of bacteria into lattices of cavi-
ties. They form a single vortex in each cavity, initially spinning clockwise
or counterclockwise with equal probabilities. Changing the topology of
the lattice and the geometry of connections between cavities allows us
to control the lattice state (random, ferromagnetic, antiferromagnetic, or
iv
unstable). Edge currents along interfaces and connections appear to de-
termine the lattice organisation. We finally propose an Ising model to un-
derstand experimental results and estimate Hamiltonian and interactions
parameters.
This work opens new perspectives for the study of active matter and, we
hope, will have a great impact on the field.
v
vi
Contents
1 Introduction 1
1.1 Active Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Bacterial Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Different Motility Mechanisms . . . . . . . . . . . . . . . . . 4
1.2.2 Swimming in Bacillus subtilis . . . . . . . . . . . . . . . . . 5
1.2.3 Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Dense Suspensions of Bacteria . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Collective Motion of Bacteria . . . . . . . . . . . . . . . . . 15
1.3.2 Modelling Active Matter . . . . . . . . . . . . . . . . . . . . 19
1.4 Bacteria under Confinement . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Effect of Interfaces on the Swimming of Single Microswimmers 24
1.4.2 Active Matter under Confinement . . . . . . . . . . . . . . . 26
1.4.3 Thesis Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Bacteria under circular and linear confinement 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Flattened drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Racetracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.1 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vii
2.4 Cross channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Bacterial Self-Organisation Mechanisms and the Role of Fluid Flows 65
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Experimental Observation of Fluid Flows . . . . . . . . . . . . . . . 66
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Lattice of Vortices 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Modelling the Square Lattice as an Ising System . . . . . . . . . . . . 99
4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.2 Direct Analysis of the Experiments . . . . . . . . . . . . . . 101
4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Conclusion 115
viii
Chapter 1
Introduction
“Order without power”1. It is in these words that Pierre-Joseph Proudhon, in his book
“The Confessions of a Revolutionary” [4], defines the anarchist society... or bacterial
communities.
Bacteria are certainly anarchists according to the etymological definition, “without
rulers”, as their behaviour is usually not dictated by a subset of the population. And yet
they can achieve strong ordering as observed in macroscopic colonies which bacteria
form [5, 6, 7] . In such structures, there exist no notion of government or property but
rather cooperation, collective decision making and public goods [8, 9, 10].
Comparing bacterial colonies to anarchist states is of course highly anthropomor-
phic and it would be more accurate to speak of self-organisation instead. Self-
organisation can be described as the process by which large scale order arises from
interactions between individual agents only. In the case of bacteria, interactions can
take various forms. Chemical signalling can be used to communicate on the cell phys-
iological status, measure the density of bacteria (quorum sensing) or inform on envi-
ronmental conditions [9, 11, 12, 13, 14]. Mechanical interactions such as direct steric
repulsion or hydrodynamic forces can affect the spatial organisation and dynamics of
colonies [5, 15, 16, 17]. Other complex mechanisms such as DNA exchange can also
play a critical role on population behaviour [18].
These interactions, taking place at the level of individual cells, can lead to vari-
ous collective behaviours: population migration [9, 19, 20], formation of structured
1“L’anarchie, c’est l’ordre sans le pouvoir.”
1
biofilms [5, 6, 7] and fruiting bodies [12, 21], collective adaptation of phenotype [11,
14], cannibalism [15, 22], and many more.
Self-organisation and collective behaviours can of course be found in many other
organisms, from the formation of fruiting bodies in the slime mold Dictyostelium [23],
foraging in social insects [24], to the motion of flocks of birds [25, 26]. These be-
haviours have not only fascinated biologists but also physicists. In particular a new
subfield has recently emerged, namely active matter, which seeks to understand the
physical properties of self-organising systems under motion [27].
1.1 Active Matter
Active materials are composed of a set of objects, organisms or molecules able to
move in their environment by internally consuming energy. Unlike classical passive
materials, they are by nature out of equilibrium. These objects then interact through
various forces and can self-organise into large stable or unstable patterns.
Examples of active matter range from submicron molecules to large mammals and
can either be extracts and full living organisms, or artificial objects. I present here a
non exhaustive list of commonly studied systems:
Flocks of birds and shoals of fishes Animals are able to move in groups, in these two
cases, the visual cues are predominant to explain the mechanisms of organisa-
tion [25, 26]. Birds for example have been shown to finely control their density
depending on the flock size [25]. Motion in flocks and shoals can be a defence
mechanism against predators.
Swarms of locusts Even though locusts are not truly social insects (’eusocial’), they
often live in swarms, and can collectively migrate [28]. Cannibalism has been
proposed to explain the motion of marching locust, which adapt their orientation
to avoid contact and being eaten [29].
Fruiting bodies of Dictyostelium The slime mold Dictyostelium is an eukaryotic or-
ganism that exhibits both unicellular and multicellular life styles. When nutrients
become scarce, cells attract each other by chemical signalling and aggregate into
2
a slug, precursor of the fruiting body [23]. At the top of the fruiting body, Dic-
tyostelium turns into spores able to survive starvation. A similar fruiting body
formation happens in Myxobacteria, also called “slime bacteria” [12, 21].
Swimming bacteria, algae and sperm cells Numerous microorganisms have evolved
to swim and explore their environment. Population of microswimmers can form
various circulation patterns: dense bacteria can generate turbulent swirls [7, 30,
31, 32]; algae, bioconvection [33]; and sperm cells, arrays of vortices [34]. The
behaviour of microswimmers will be described in further detail in the rest of this
thesis.
Eukaryote cytoskeleton The skeleton of eukaryotic cells is mainly composed of fila-
ments, including actin and microtubules, on which many proteins interact. These
proteins can be purified to create artificial active system in vitro. When micro-
tubules and kinesin motors are copurified, the latter slide filaments past each
other, resulting in a collective rearrangement exhibiting +12 and −
1
2 defects typ-
ical of liquid crystal assembly [35, 36]. An alternative setup is to place actin
filaments on a carpet of myosin motors, themselves bound to a surface. At high
concentration, filaments, gliding on the myosins, can organise into ordered clus-
ters (‘swarms’), travelling bands and migrate uniformly [37, 38].
Bristlebots Centimetre-long objects are in contact with a surface through asymmetric
bristles. Vibration of the surface or of the bots results in a directed motion.
Bristlebots interact through direct repulsion and have been shown to form stable
vortices under confinement [39, 40, 41].
Janus particles These micron-sized colloids have two sides with different chemical
properties. One half typically reacts with a compound present in the medium,
generating a chemical gradient that sets the particles into motion [42]. Janus
particles have been shown to decelerate at high density thus assembling into
dense immotile groups [43].
These systems exhibit a great variation in their form, propulsion mechanism, in-
teraction and resulting pattern formation. Nature teems with active matter and new
artificial forms are quickly being developed.
3
I decided to work with suspensions of swimming bacteria. They present several
advantages: they are simple to grow and prepare, can be controlled either by changing
external conditions (e.g. oxygen concentration) or by using mutants, are easy to image,
and most importantly can self-organise into a variety of dynamic patterns.
In the following sections, I first describe how bacteria explore their environment
with particular focus on the role of fluid flows. I then present experimental and theo-
retical results on dense bacterial suspensions. I finally introduce the topic of my PhD:
how confinement and interfaces modify the self-organisation of bacterial suspensions.
1.2 Bacterial Motion
1.2.1 Different Motility Mechanisms
Bacteria make an extremely diverse kingdom and as such have developed numerous
locomotion mechanisms, which can be listed into five main groups [44, 45, 46]:
Swimming Bacteria grow helical flagella that are spun by a rotor embedded in the
membrane. The rotation of the flagella creates a net force on the fluid and on
the body, thus propelling the cell [47]. Depending on the species, a bacterium
can have one or numerous flagella, placed at one or both ends, in the middle or
covering the entire body [48]. The swimming motion is described in more detail
below. Some bacteria have also been found to be covered in short cilia that beat
in waves similar to Paramecia locomotion [49, 50].
Swarming As for swimming, swarming is also driven by flagella. Yet, not all swim-
ming bacteria can swarm. This motion happens at the interface of a gel and air
and has mostly been studied on agar plates. In order to swarm, bacteria need to
grow numerous flagella (called peritrichous bacteria). They often produce sur-
factant, creating a thin film in which they can move. Bacteria can form dense
groups in which they are aligned and move in a common direction, thus giving
them the name swarms (figure 1.1a). The migration of swarming bacteria on an
agar plate can result in macroscopic patterns whose shape depends on nutrient
concentration and surface stiffness [19].
4
Twitching Some bacteria are equipped with long pili which they extend, bind to sur-
faces and retract to move towards the attachment point (figure 1.1b) [51]. As in
swarming, bacteria exhibit collective motion on agar plates, which can take the
form of dense groups.
Gliding This motion is the least understood and most probably corresponds to differ-
ent mechanisms depending on the specific species. In some bacteria, focal adhe-
sion points are placed on an helical pattern around the cell body and are actively
displaced from one end to the other. When attached to a surface, this motion
propels the bacterium in a forward and rotating fashion (figure 1.1c) [52, 53].
Sliding Bacteria growing on agar plates form colonies initially limited to a single
layer. When expanding, steric repulsion between bacteria leads to the outward
motion of the cells (figure 1.1d). This mechanism cannot exactly be called motil-
ity as, apart from the cell growth, it is passive. Yet it is necessary to understand
the migration and structure formation of bacterial colonies [16, 17].
1.2.2 Swimming in Bacillus subtilis
Bacillus subtilis is one of the most studied bacteria. It can grow multiple flagella and
is a model for swarming, swimming and active matter. It is also extensively used to
study the formation of biofilms, sporulation, chromosome replication, etc. Even if the
general mechanism for swimming is conserved among bacteria, the exact actuation
varies between species and I only describe here that of B. subtilis.
Flagellar Structure
Flagella are organelles growing at the outside of the cell. They are composed of three
main parts: a long filament, a rotor and a hook joining the two (figure 1.2).
The filament is a hollow tube assembled as a periodic array of subunits: the protein
flagellin. Each subunit migrates inside the flagella from the cell body to the tip where
it is added, the filament then grows from its free end rather than its base, independently
of the flagella length [54, 55].
To form the hollow filament, flagellin assemble into a closed array of eleven fib-
rils or protofilaments [56]. Flagellin can take two conformations, right and left type.
5
Figure 1.1: Locomotion mechanisms in various bacterial species. (a) Swarming:
flagellated bacteria move on a gel surface in dense groups, swarms. Surfactant helps in
making a thin liquid film for the cell to move. (b) Twitching: bacteria expand, bind and
retract pili to move towards the attachment point. (c) Gliding: in this example, focal
adhesion point are moved on an helical line around the cell body (purple arrow). This
traction results in the spinning and net motion of the bacteria. (d) Sliding: by growing
and dividing, bacteria push their neighbours outwards.
Figure 1.2: Structure of the flagella split into three main pieces: rotor, hook and
filament.
6
Figure 1.3: Filament macroscopic shape is determined by the combination of left
(purple) and right (green) conformations. When all fibrils (numbered 1 to 11) take
the same conformation, the filament forms a straight tube. In case of heterogeneous
conformations, supercoiling generates a helical shape. Each rhombus represent a single
flagellin protein.
Generally, all proteins on a given fibril have the same conformation. If all fibrils share
the same conformation, the filament takes a straight shape. However if fibrils become
heterogeneous, supercoiling of the filament modifies the macroscopic structure into a
helical shape (figure 1.3). A total of twelve theoretical combinations of right and left
conformations have been identified, each giving rise to a specific flagella geometry:
helical pitch, radius, and handedness [57].
Some of the theoretical shapes have been observed experimentally on swimming
Escherichia coli: normal, coiled, semi-coiled, curly [57, 58]. Notably, bacteria adapt
the shape of their flagella depending on their swimming behaviour (as explained in the
next subsection). For example, the “normal” conformation has been shown to be the
most hydrodynamically efficient for self-propulsion [47] and turns out to be the one
used for that purpose [58].
The spinning of a flagellum is actuated by a rotor embedded in the cell mem-
brane [54]. This motor is a masterpiece of engineering, made of about thirteen differ-
ent proteins (and twice as many are necessary for its assembly) [59]. It is driven by an
influx of protons towards the inside of the cell (protons are then expelled by an active
7
pump placed in the membrane). Protons going through the rotor actuate its rotation, up
to several hundreds Hertz, either clockwise (CW) or counterclockwise(CCW) [60, 61].
The exact details of how the flagella rotor works are still to be discovered.
The filament and the motor are finally attached to one another by a hook [54].
The hook is flexible in bending, such that the filament can point to either end or side
of the bacteria, but rigid in twisting to properly transmit torque from the rotor to the
filament [62].
Flagella Expression
Cells do not always need to swim and have developed mechanisms to regulate the pro-
duction of flagella [63, 64]. In biofilms for example, swimming bacteria can disrupt the
complex organisation [65] and cells collectively prevent the formation of flagella [6].
Likewise, when bacteria are placed in a fresh nutritious medium, they switch to a grow-
ing state without flagella and form chains of partially divided cells. Once they have
consumed most of the nutrients (at the end of the exponential phase), they revert to a
swimming phenotype to seek another source of food [64, 66].
To precisely control the formation of flagella, all genes involved are placed under
a unique master regulatory operon [54, 63, 64]. The gene expression is regulated de-
pending on the external conditions but also monitors the number of flagella. Some
bacterial species grow only one or two flagella [48]. Peritrichous bacteria like B. sub-
tilis and E. coli instead use multiple flagella. E. coli has about ten rotors at a time,
while B. subtilis can have up to thirty1. In peritrichous bacteria, flagella emanate from
the entire body surface. Their distribution is homogeneous over the cell length and
the distance between rotors is controlled such that, in B. subtilis, they are further away
from one another than if randomly distributed [48].
Swimming Behaviour
In order to propel themselves, bacteria simultaneously spin all flagella CCW (when
looking from the flagella to the body) [67]. Flagella are in a normal conformation
(left-handed) such that they exert a force expelling the fluid and pushing on the body.
1Not all rotors are associated with a functional flagellum, as the filament can be undevel-
oped. However more than half of them typically contribute to the swim [48, 63].
8
Figure 1.4: Bundling and unbundling of flagella during a run & tumble. Run: flagella
emerge from the entire cell body and form, in the model, a tight bundle. In experiments
I observed flagella to aggregate at the rear of the cell. Tumble: upon the reversal of
some flagella rotors, filaments unbundle. This state was obtained experimentally by
shinning a blue light onto the sample cell. Scale bar: 5µm. Membrane and flagella
were labelled respectively with fluorophores FM4-64 (false coloured red) and Alexa
Fluor 488 C5 Maleimide (false coloured green). Labelling and imaging methods are
explained in full details in chapter 3.
Since rotors cover the entire cell body, flagella bundle at one end of the rod-like body
in order to generate a net propelling force [58]. Bacteria do not have an intrinsic front
or back and the bundle can form at either end. Some cells have even been observed to
revert the bundle position after hitting a wall [68].
This bundle was first described in E. coli where all ∼ 8 flagella perfectly merge and
spin as a single object [58]. The formation of this bundle has been shown to result
from hydrodynamic interactions between each of the helical filaments [69, 70]. In
the case of wild-type B. subtilis (strain 168), which I used in my experiments, even
though flagella accumulate at one end, they do not form a tight bundle but rather a
large cylindrical or conical aggregate (figure 1.4). This is probably due to the shape
of cells, which are two to three times larger than E. coli and to the larger number of
flagella, which physically limits the formation of the bundle [71].
This swimming behaviour results in a straight motion called a run and lasts about
a second. In order to turn, bacteria revert the spinning direction of one or several
rotors, thus disassembling the bundle [58, 67]. The filament conformation is also
changed from a left-handed (normal) to a right-handed conformation (semicoiled and
curly 1) [67], such that the filament is still pushing on the cell body. Flagella point
outward in different directions, randomly turning the cell. This step is called a tum-
ble and only lasts a tenth of a second. A succession of runs and tumbles results in a
9
Figure 1.5: Trajectories of swimming bacteria. (a) Model of run and tumbles. Runs
are longer when bacteria are swimming up a gradient of nutrients, resulting in a chemo-
tactic behaviour. (b) Wild-type B. subtilis trajectories show an absence of tumbles.
Tracking was done manually on bacteria in and out of the plane of focus.
random walk motion (figure 1.5a) [72]. Run & tumble has been described in several
bacteria including E. coli and some B. subtilis strains [58, 67, 72, 73]. However I did
not observe clear tumbles with the strain 168 of B. subtilis but rather smooth turns
(figure 1.5b). E. coli has also been observed to lack tumbles when swimming close to
surfaces [74].
Bacteria can regulate the frequency of tumbles in order to navigate in their environ-
ment. For example they are able to swim towards nutrient and oxygen sources [72, 73].
Since cells are too small to directly sense chemical gradients, they integrate the infor-
mation over time instead: if the chemical concentration increases, i.e. if the bacterium
is swimming towards the source, it will down regulate tumbles, and likewise shorten
the run period when the concentration decreases [72]. This behaviour, called chemo-
taxis, allows bacteria to have net motion along chemical gradient (figure 1.5a).
1.2.3 Fluid Flows
Bacteria can only live in water and their motion is affected by flows, advecting and
rotating them. Moreover while swimming, they generate specific fluid flows that can
result in long-range interactions. These phenomena happen on very small scales (tens
of microns) at low Reynolds number.
10
Reynolds Number Re
The motion of an incompressible Newtonian fluid is governed by the Navier-Stokes
equation,
ρ
(
∂v
∂t
+ v · ∇v
)
= −∇p +µ∇2v + f, (1.1)
where v is the fluid velocity, ρ its density, p its pressure, µ its dynamic viscosity and
f any force acting on the fluid. This equation shows the balance between inertia (left-
hand side of the equation) and viscosity (µ∇2v). The Reynolds number Re measures
the ratio between the two, such that
Re =
inertial forces
viscous forces
=
ρVL
µ
, (1.2)
where V and L are typical velocity and length of the system. In the case of a B. subtilis
in water, ρ = 103 kg/m3; µ ≈ 10−3 Pa.s; V ∼ 10µm/s; L ∼ 1µm, and
Re ∼ 10−5 1. (1.3)
Thus viscous forces largely dominate such that, if a bacteria stops swimming, it will
keep on moving by inertia for less than an Ångström. Likewise, fluid flow around a
microswimmer is only determined by instantaneous forces, and equation 1.1 can be
reduced to the Stokes equation,
−∇p +µ∇2v + f = 0, (1.4)
with incompressibility, ∇ ·u = 0.
The motion of bacteria simply results from the balance of forces. The self-propulsion
speed for example can be computed from the force f applied by the flagella on the fluid,
and by the viscous drag [75], such that
f + 6piµav = 0, (1.5)
where a ∼ 1µm is the typical size of a bacterium.
11
Fluid Flow Generated by a Bacterium
A swimming bacterium then applies two forces to the surrounding fluid; one arising
from the spinning of the flagella, and one from viscous drag centred around the body.
Each force can be approximated by a point stokeslet [76, 77].
A point force f = f ex at {x,y} = {0,0}, in a low Re fluid results in a disturbance of
fluid (figure 1.6a),



vx =
f
8piµ
(
1
r
+
x2
r3
)
,
vy =
f
8piµ
xy
r3
,
vz =
f
8piµ
xz
r3
,
(1.6)
where r2 = x2 + y2 + z2 and v = (vx,vy,vz).
In the case of a bacterium and many other microswimmers, the two forces are close
(but not colocalised) and in opposite directions, generating a stresslet. The fluid flow
around two point forces f1 = f ex at {x,y} = {δx/2,0} and f2 = − f ex at {x,y} = {−δx/2,0}
(figure 1.6b) is given by



vx =
D
8piµ
(
−
x
r3
+
3x3
r5
)
,
vy =
D
8piµ
(
−
y
r3
+
3yx2
r5
)
,
vz =
D
8piµ
(
−
z
r3
+
3zx2
r5
)
,
(1.7)
where D = δx · f is the stresslet strength.
The fluid around a swimming B. subtilis has been predicted to move away at the
front and back and towards the side of the cell (figure 1.6c). Microswimmers having
this behaviour are called pushers [78]. The opposite, pullers, also exist if flagella are
placed at the front of the cell. The green alga Chlamydomonas is a good example of
such an arrangement, with two flagella pulling fluid at the front of the cell in a breast
stroke motion (figure 1.6d) [79].
The fluid flows around E. coli has been experimentally measured using passive
colloids as tracers. The experimental measurement matches the stresslet model, ex-
cept in the near vicinity of the cell (figure 1.6c) [78]. In the case of Chlamydomonas,
12
Figure 1.6: Fluid flows at low Reynolds number. (a) Streamlines around a point force.
(b) Streamlines around a force doublet. (c) Streamlines around a swimming bacterium.
(d) Streamlines around a Chlamydomonas cell.
13
fluid flows are more complex, as each of the two flagella applies its own force. How-
ever, in the far field of the cell, fluid flows are consistent with the puller model (figure
1.6d) [80].
Transport of Bacteria by Fluid Flow
Similar to other micron-sized objects, bacteria can also be moved by external fluid
flows. The contribution of fluid flows to the motion of a rod-like cell can be divided
into two parts: advection and reorientation [75, 81].
The first term simply states that the cell is pushed along streamlines. The motion
of the centre of mass X of a single bacterium in fluid flow is then described by
dX
dt
= UP + v, (1.8)
where U is the swimming speed, P the cell orientation and v the fluid flow at position
X.
The body of B. subtilis has a cylindrical shape, capped with half spheres. It is 4 to
8µm long and ≈ 1µm in diameter. A cell can be approximated as a prolate spheroid
which is reoriented following Jeffery’s equation [82]. When a swimming bacterium is
placed in a shear flow, it exhibits Jeffery’s orbits as theoretically predicted [83].
Moreover if the shear is not homogeneous, cells accumulate in high-shear re-
gions [83]. This phenomenon can be simply understood: size of orbits is inversely
proportional to the shear strength, resulting in a trapping effect where shear is the
greater. When bacteria are placed in a channel with parabolic flow, this effect result in
the accumulation of the cells close to surfaces [83].
Long-range Hydrodynamic Interactions
When passive colloids are placed in a bath of microswimmers, they move by Brownian
motion and advection by the fluid generated by the cells. This results in the enhanced
diffusion of colloids [84, 85].
Likewise, the motion of bacteria is also affected by that of other swimmers. The
combination of pusher disturbance, advection and reorientation can result in long-
range hydrodynamic interactions [86].
14
In the case of two swimmers, this force is often negligible compared with self-
propulsion and direct contact forces between swimmers [7, 78, 87, 88]. Yet when the
bacterial concentration increases, the cumulated hydrodynamic interactions between
multiple swimmers, could become significant and affect the suspension motion. This
question is still controversial in the field of active matter [89, 90, 91] and will be con-
sidered in detail in chapter 3.
1.3 Dense Suspensions of Bacteria
When B. subtilis are grown in a liquid medium (rather than on agar plates), they can
reach a maximum concentration of 108 cells/mL (volume fraction ∼ 0.01%, distance
between swimmers ∼ 10µm). At such a density, the suspension is said to be dilute: the
motion of one bacterium is weakly affected by that of others. If the concentration is
further increased, either artificially or by growing cells on agar, interactions lead to the
formation of collective motion, typical of active matter systems [30, 31, 32, 92, 93].
1.3.1 Collective Motion of Bacteria
Swimming bacteria are a common model to study active matter and their behaviour in
large quasi-2D and 3D chambers has been extensively described1. A classical approach
is to concentrate cells by centrifugation before injecting them into the setup [7, 92]. I
present here an alternative method developed by Vasily Kantsler, former post-doc of
the group, to study sperm cells (he prepared the PDMS chambers and I performed
experiments with B. subtilis).
The setup consists of two main chambers, a central one in which cells are concen-
trated and a larger one around it, used as a reservoir of bacteria (figure 1.7a). They are
connected by several radial and asymmetric channels, the walls of which are shaped
as ratchets, as designed by Denissenko et al. [94]. When swimming along the walls,
bacteria are reoriented to travel towards the central chamber (figure 1.7b).
1I call quasi-2D a chamber whose height is small enough for the bacterial motion to mostly
happen in the plan of focus. In the case of B. subtilis, I used chambers 15 to 25µm in height.
I also assume in both quasi-2D and 3D chambers that the medium and the suspension are
homogeneous along the vertical direction.
15
Figure 1.7: Collective behaviour arises in dense suspensions. (a) Sketch of the PDMS
chamber used to concentrate bacteria. Cells are injected through the inlet and swim
with a net migration from the reservoir to the central chamber. The reservoir can be
mm-large while the central chamber as a typical diameter of 100µm. (b) Ratchet-
like wall reorient the bacteria to swim towards the central chamber. (c) Motion in the
central chamber at the beginning of the experiment. Trajectories of dilute bacteria are
weakly correlated. Lines: cell trajectories, manually tracked. Circle: initial position.
(d) Motion after 75 minutes. The dense suspension forms large patterns like swirls and
jets. Arrows: bacterial flow measured by PIV (see Methods in chapter 2, here obtained
using PIVLab [95]). Scale bars: 20µm.
16
Effect of the Increasing Concentration
The experiments run over a few hours, starting with an homogeneous dilute suspen-
sion. Migration of the bacteria towards the central chamber results in a progressive
increase of the concentration, which allows us to observe the emergence of collective
motion.
At low density, bacteria interact weakly and mainly follow single-cell trajectories
(figure 1.7c). The cell concentration then slowly increases such that after 75 minutes,
collective motion appears. Bacteria frequently come into contact with one another.
Since they are rod-shaped, they locally align, akin to nematic liquid crystals. The
combination of self-propulsion, steric and hydrodynamic interactions then results in
collective motion of the suspension. This motion is said to be quasi-turbulent as it
forms large scale patterns such as swirls and jets, stable for only a fraction of a sec-
ond [7, 30, 31, 32, 92, 93, 96]. The length scale of these patterns is on the order of tens
of µm (the diameter of a swirl is typically ∼ 70µm), an order of magnitude larger than
bacteria (B. subtilis are 4 to 8µm long).
If the bacterial concentration is further increased, cells enter a jammed state where
the density is too large for them to move [7]. I did not observe this state in this experi-
ment.
Cell Concentration
Different methods have been used to measure cell concentration: direct counting, op-
tical density or plates counting [32, 88]. I did not find any method that was precise
and reproducible enough. By measuring the length and number of individual cells in
shallow chambers (1 and 2µm in height), I evaluated the volume fraction between 5
and 20% (∼ 1 to 4 · 1010 cells/mL). This measure only takes into account the body of
the bacteria. Yet flagella which are extremely thin, can extend over 10µm around the
body, thus yielding a much larger effective volume fraction. Similar values have been
given in previous publications [88].
Pattern Length Scale
A classical approach to quantitatively study the bacterial flow is to estimate its typical
length scales. Two measures have been regularly used, the size of the vortices and the
17
correlation length. The former is usually 50 to 90µm in diameter and can depend on
the chamber’s height (quasi-2D chambers have smaller swirl diameter [7, 96]).
The latter is computed from the two-points spatial correlation. It indicates over
which distance the suspension orientation is conserved. As the cell concentration in-
creases, collective motion appears and so grows the persistence length [31, 32]. These
questions are discussed in more detail in chapter 2, section 3.
Oxygen Concentration
Unlike E. coli, B. subtilis need oxygen to swim. A good way to control the velocity
of the suspension is then to change oxygen concentration. Sokolov et al. have shown
that the mean speed of the dense suspension scales almost linearly with the oxygen
concentration. Moreover, the correlation length quickly increases with the oxygen,
before reaching a plateau value for [O2]> 0.1mM and persistence length ≈ 40µm [93].
Likewise Dunkel et al., without measuring the oxygen concentration, have char-
acterised the turbulence of the suspension by first measuring the local vorticity ω =
∂xuy−∂yux, where u is the bacterial flow velocity. They then estimated the enstrophy,
the energy associated with the vorticity, defined as Ω =< ω2/2 >. The enstrophy is
used to quantify the turbulence of the suspension motion. They then compared Ω with
the kinetic energy E =< u2/2 > and found that the enstrophy scales linearly with the
energy: Ω = E/Λ2 where Λ ≈ 24µm compares to a quarter of a typical swirl diame-
ter [96].
Equations of motion at low Re are all linear, which explains this behaviour: de-
creasing the oxygen concentration reduces the swimmer speed and overall motion but
does not affect the pattern formation otherwise.
Collective Behaviour in Other Chambers
The motion in quasi-2D and 3D chambers is quite similar, except for the length scales,
which are larger in the latter. Other setups have been considered. In particular drops
and freely standing films can give rise to convection patterns.
I have only presented so far chambers that were homogeneous in bacteria and
chemicals in the medium. When a liquid/air interface is added , oxygen that is con-
sumed by the cells, can diffuse into the medium. This results in the formation of a
18
gradient of oxygen concentration, low in the bulk where it is consumed and large at
the interface where it diffuses. Microswimmers, by chemotaxis, accumulate in regions
of high oxygen concentration. Moreover bacteria are slightly denser than water, giving
rise to a Rayleigh-Taylor instability. This motion leads to the formation of bioconvec-
tion patterns on the scales of several millimetres [92, 97].
In the rest of this thesis, I only use solid or liquid interfaces and try to avoid as
much as possible gradients of oxygen in the chamber. I also limit the height of the
chamber to less than 30µm (quasi-2D chamber) to ensure that most of the bacterial
flow happens in the plane of focus (except when otherwise stated).
1.3.2 Modelling Active Matter
The field of active matter is dominated by theoretical papers. Models often quest for
the simplest unified theory that can reproduce the behaviour of many if not all active
systems.
Vicsek and Toner-Tu Models
Vicsek and Toner-Tu models are two facets of the same ordering mechanism, which
only assumes that self-propelled particles reorient themselves along the local mean
direction [98].
Vicsek et al. first proposed a simulation method in which birds, locusts or bacteria
are represented as a set of particles, moving at constant speed and whose orientation at
each time step is set as the average of nearby particles [99].
Toner and Tu proposed a continuum version for this model [100]. The motion of
particles is only represented by the mean local velocity u, which follows the set of
equations:
∂u
∂t
+ (u · ∇)u =αu−β|u|2u−∇P
+ DL∇(∇ ·u) + D1∇2u + D2(u · ∇)2u + f,
(1.9)
P =
∞∑
σn(ρ−ρ0)n, (1.10)
19
and
∂ρ
∂t
+∇ · (uρ) = 0. (1.11)
Equation 1.9 is designed as a Navier-Stokes equation with added terms to account
for the self-propulsion and collective behaviour of the suspension. The left-hand side
describes advection at swimming speed u. α and β are two positive terms representing
self-propulsion, respectively increasing and decreasing u, such that in a mean field
approach, swimmers move at constant speed
√
α/β. DL,1,2 are constants that represent
the diffusivity of u (or likewise the viscosity of the active fluid). DL,1,2 are taken
positive such that the velocity becomes uniform, representing the fact that swimmers
align their velocity with neighbours. If no noise f is included, this set of equation result
in all swimmers moving in a common direction at constant speed
√
α/β.
Finally, swimmers repel each other by steric interactions, such that the local con-
centration must be finite. This effect is taken into account by the effective pressure P
which depends on the cell density ρ, the mean density being conserved by equation
1.11. In equation 1.10, σn are positive constants and ρ0 represent the mean density.
Both models give rise to various collective behaviours depending on the alignment
strength, noise and particle density [98]. The most common patterns are as follows [99,
101]:
Disordered At low alignment strength or high noise, each particle moves indepen-
dently of its neighbours (in continuum models, u has a short spacial correlation).
Swarms At moderate concentration, particles cluster into dense groups. Each cluster
moves independently of others, but particles inside a single cluster share the
same orientation.
Travelling bands Particles form dense strips. Particles are usually oriented perpen-
dicular to the strip length.
Uniform The density is rather homogeneous and particles travel in a common direc-
tion.
These models are able to reproduce some behaviours observed in nature, from
schools of fish, locust swarms, and bacteria in agar plates, but not the quasi-turbulent
motion described above.
20
Meso-scale Turbulence Model
With this limitation in mind, Wensink et al.1 have proposed a variation of Toner-Tu
model to generate quasi-turbulence [7, 96]. The bacterial suspension velocity u follows
the dynamics (
∂
∂t
+λ0u · ∇
)
u =αu−β|u|2u−∇P +λ1∇u2
+Γ0∇2u +Γ2(∇2)2u,
(1.12)
and
∇ ·u = 0. (1.13)
As in equation 1.9, the suspension is set into motion by terms α and β, with typical
velocity
√
α/β. The pressure P ensures the fluid incompressibility (equation 1.13).
Here, λ0 can be different from 1 which, along with λ1, account for hydrodynamic
interactions between swimmers [98].
But the turbulent behaviour relies on the two last terms Γ0 and Γ2. Γ0 was already
introduce in Toner-Tu model, noted D1. However, unlike in equation 1.9, Γ0 can here
take positive or negative values. In the first case, it as the same effect as a diffusion or
viscosity term, making the velocity u uniform over the domain. If instead Γ0 < 0 any
disturbance in the velocity field becomes unstable, generating turbulence. This effect
is dampened by Γ2 (as would β for the velocity), resulting in quasi-turbulent patterns
of typical length scale 2pi
√
Γ2
−Γ0
[102].
This model was compared to experiments in quasi-2D and 3D chambers by mea-
suring the velocity and vorticity ω = ∇×u [7, 96]. Statistics and spacial distribution of
the two measures are comparable in experiments and in simulations.
This model could then be used to predict the suspension self-organisation under
different conditions, yet its aim is only to give a minimal description of turbulent mo-
tion in active matter and thus presents some limitations. First it only represents mi-
croswimmers through the average velocity u. Two other quantities could be important
to fully understand bacterial motion: their orientation and the fluid flow v. Moreover,
the model is only phenomenological, the macroscopic flow patterns are reproduced
but no insights into the physical interactions is gained. It is then unlikely to help in
1Jörn Dunkel, a post-doc in Ray Goldstein’s group at that time and now an assistant pro-
fessor at MIT, and Ray Goldstein contributed to this model and to this thesis.
21
understanding the microscopic phenomena driving bacterial self-organisation.
Q-tensor
Toner-Tu and the meso-scale turbulence models reproduce active matter starting from
the macroscopic behaviour, both assume some alignment mechanism and in the latter,
quasi-turbulence. An alternative approach is to start from the physical interactions and
try to derive the suspension motion [103, 104, 105].
The Q-tensor method was imported from nematic liquid crystal theories [106],
expanding it to include self-propulsion and long-range active hydrodynamic interac-
tions [104, 107]. The Q-tensor represents the distribution of particle orientation. At a
given position, unlike in the two previous models, not all particles are aligned in the
same direction. To model fluid flows, the stresslet approximation is used. As described
above, particles can have a pusher (e.g. bacteria, cytoskeleton extracts) or puller be-
haviour (e.g. Chlamydomonas). Different suspension patterns have been predicted
depending on this fluid disturbance [108].
The active nematic is the class of active matter modelled by Q-tensor theories and
can be simply understood as a nematic liquid crystal whose elements are self-propelled.
The archetype probably corresponds to extracts of microtubules set into motion by ki-
nesin motors: long thin filaments locally behave like liquid crystals but sliding powered
by motors generate turbulent flows characterised by +12 and −
1
2 defects [35, 36].
This approach may appear as the most appropriate to model bacterial suspensions.
However, a crucial limitation can come from the size of bacteria relative to that of the
suspension flows: a coarse-grained approach requires particles (bacteria) to be small
compared to the simulation grid. Yet this condition is not always met as I will show in
chapter 3.
Particle-based Simulations
A final approach is to directly model the motion of individual particles (as in the Vic-
sek model [99]) and the various interactions between them. The difficulty in these
simulations is not to write the governing equations but rather to make appropriate ap-
proximations so that computations run in a humanly acceptable time.
A widespread approximation is to look at the motion in a 2D domain, significantly
22
reducing interactions and the total number of particles in a given domain size. A main
concern is that, if steric interactions are included, particles cannot swim over one an-
other, as they do experimentally in 3D or even in quasi-2D chambers. Yet simulations
do not seem to suffer significantly from this approximation (chapter 3). Experimen-
tally, the main difference between truly 2D and quasi-2D or 3D chambers appears in
the hydrodynamic interactions, which varies in strength and flow pattern. Yet all these
differences do not rely on the domain itself but on the governing equations and can
easily be implemented [109, 110, 111].
Another recurring idea is to neglect fluid flows, supported by the fact that “ mea-
surements [78] ... suggest that hydrodynamic far-field interactions are negligible for
bacterial reorientation” [7]. Moreover, models with steric interactions only are able
to reproduce the quasi-turbulent state. Particle-based simulations have managed to re-
produce dense suspension behaviours: disordered, swarming, turbulent, jammed; and
to predict several others, including lanes. These states depend on the bacterial density
and aspect ratio [7, 90].
However these simulations do not prove that fluid flows are actually negligible in
dense suspensions. Indeed, models that only include hydrodynamic interactions, rather
than steric repulsion, also show the suspension motion to be unstable and result in a
quasi-turbulent behaviour [89, 91, 109].
Only few articles have considered full systems with both direct steric and long-
range interactions [90, 110]. The question of fluid flows, their significance and the
correct way to model them is still under debate. In chapter 3, I will discuss these
issues and present results supporting hydrodynamics as a key ingredient for bacterial
self-organisation.
1.4 Bacteria under Confinement
I have mostly presented so far results in 3D or quasi-2D chambers (periodic domains in
simulations). Most experiments that look at dense bacterial suspensions have actually
been performed in large homogeneous chambers1.
And yet bacteria and other active systems can regularly encounter interfaces and
1A notable exception is that of drop and freely standing films presented above [92, 97].
23
confinement. B. subtilis for example is a soil bacterium and its motion is most cer-
tainly affected by sand particles and liquid/air interfaces. Likewise, bacteria infecting
eukaryotic cells are trapped into an environment about ten fold their size [112].
1.4.1 Effect of Interfaces on the Swimming of Single Microswim-
mers
Even though very few papers have considered the effect of interfaces on dense bacterial
suspensions, the case of dilute bacteria has been well documented. Here, I consider
interfaces, solid or liquid, that only play a mechanical role but do not modify the chem-
ical environment (oxygen cannot diffuse through the interface for example).
Accumulation at Interfaces
When swimming bacteria are placed in a large chamber with solid walls, they quickly
accumulate at interfaces, emptying from the centre of the chamber [113, 114]. This
behaviour is not due to sedimentation as bacteria accumulate at top walls as well. Two
effects have been proposed to explain this attraction: long-range hydrodynamics and
direct steric interactions.
When a bacterium swims, it pushes fluid away from its front and back and towards
its side. If a cell is parallel to a wall, fluid flows result in a net force attracting the
swimmer to the interface [78, 113]. However this argument has been debated as the
force may be too weak compared to that of swimming or Brownian motion [115, 116].
Instead, the shape of the microswimmer could explain the trapping mechanism:
when considering the full volume explored by flagella, the front of the swimmer is
thinner than its back. When swimming at an interface, steric interactions then make
the cell point slightly towards the wall. The same argument holds for sperm cells [117]
and can explain why the green alga Chlamydomona escapes from walls: with flagella
at the front, where the triangular shape is reversed [116].
Swimming Behaviour at an Interface
Once a bacterium is trapped at an interface, its motion changes. First of all, Molaei
et al. have observed that E. coli cells reduce their tumbling rate by 50% and move in
24
Figure 1.8: Effect of surfaces and shear flow on the swimming of B. subtilis. (a) Body
and flagella spin in opposite directions, (b) creating a lubricated friction between the
swimmer and the surface, of opposite direction at the head and flagella (small arrows).
These two forces generate a torque in the vertical direction thus make the bacteria
follow a circular trajectory. (c) When a flow is applied, cells are reoriented by the fluid
flow (blue), friction with the surface (purple) and move by self-propulsion (black).
Balance between forces result in the bacterium swimming against the flow, at an angle
θ.
25
smoother trajectories [74]. Hydrodynamic forces that lead to unbundling are thought
to be dampened by the near surface.
Furthermore, when swimming, the body of the bacterium spins in opposite direc-
tion to its flagella, ensuring that the microswimmer is torque-free along its swimming
direction (figure 1.8a). The rotation of the body next to an interface generates a net
force. The same effect applies to the flagella but in opposite direction. This results
in a net torque orthogonal to the interface, the sign of which depends on the interface
type, free-shear or no-slip [118]. In the case of a solid wall, bacteria then have CW
circular trajectories, with a radius of curvature typically tens of µm in the case of E.
coli (figure 1.8b) [119, 120].
Swimming Against a Fluid Flow
When a fluid flow is then applied to the chamber, it reorients bacteria, setting their
swimming direction [121].
Both the body and flagella are subject to advection by the fluid and friction in the
vicinity of the surface. The difference in shape between flagella and body is such that
the latter is more inclined to advection, acting like a tail to reorient the cell against the
fluid.
As a secondary effect, the torque that leads to circular trajectories in the absence
of fluid flow still applies here. The exact angle between the swimmer and the fluid
flow results from the balance of this torque and that of reorientation against the flow.
Bacteria thus swim against the current at a fixed angle θ, as shown on figure 1.8c [122,
123, 124].
1.4.2 Active Matter under Confinement
Confinement has already been shown to have a strong effect on the self organisation
of some active systems. In chapters 2 and 3 I will compare and discuss in full detail
previous experiments with results I obtained. Here I briefly give examples of confining
geometries used experimentally and found in nature.
Circular confinement Probably the simplest geometry, active matter is placed in a flat
(2D or quasi-2D) cylinder. Examples range from bristle-bots [39, 40] to single
26
or multiple eukaryotic cells [125, 126]. Under certain conditions (confinement
size, shape of the boundary, concentration...), the self-propellers coordinate their
direction of motion to form a single vortex.
Racetrack A racetrack is a periodic channel. This geometry has been used to study
marching locusts in the lab [29] and rolling colloids [127]. As for circular con-
finement, interfaces promote the alignment of the particles such that all objects
or organisms move in a common direction. This motion is usually stable and can
take the form of a travelling wave, as seen in rolling colloids [127].
Sphere Changing the confinement from a 2D to a 3D chamber can have dramatic ef-
fects. Indeed, while for example racetracks have a preferred direction along the
channel length, spheres are symmetric along all three coordinate axis. A no-
table example is that of microtubules and kinesin motors in vesicles. Filaments
aggregate at the lipid bilayer and start flowing collectively. Their motion sets
the vesicle into motion, rolling over the substrate, but the direction of motion is
unstable due to the symmetry, resulting in the random walk of the vesicle [35].
1.4.3 Thesis Outlook
The effect of confinement on the self-organisation of dense bacterial suspensions has
been mostly unexplored. During my PhD, I have studied, mainly experimentally, how
B. subtilis behaviour changes with confinement geometry. The aim of this project was
not only to discover new active phenomena under confinement, but also to use these
setups to better understand general properties and behaviours of bacterial suspensions.
Chapter 2 presents experimental results in two main setups: flattened drops and
racetracks. I quantitatively measured the flow patterns with a strong emphasis on the
role of chamber geometry: drop diameter, racetrack width, chamber height. I also
found bacteria at the interface to move against the bulk circulation. This phenomenon
has not been reported in earlier experiments or models and was initially not understood.
Chapter 3 introduces a new simulation method developed in collaboration with
Enkelaida Lushi, the aim of which was to fully describe interactions between swim-
mers and with interfaces. An important point was to understand the role (if any) played
by hydrodynamic interactions. We prove that under these confinements, the suspen-
27
sion motion cannot be understood without considering fluid flows. Results were also
confirmed with a new experimental method that I developed, enabling us to observe
simultaneously the orientation and the direction of motion of individual bacteria.
Chapter 4 focuses on a new confining geometry: a lattice of flat circular cavities.
Each cavity is similar to a flattened drop, except for junctions between them, allowing
for interactions between vortices. While the direction of circulation in an isolated
vortex is random, a chain, square or triangular lattice can spontaneously turns itself
into an ordered array. I observed two critical patterns: ferromagnetic, when all vortices
share the same direction, and antiferromagnetic, when each vortices spins in opposite
direction to its neighbours. Finally, in collaboration with Francis Woodhouse and Jörn
Dunkel, I analysed this phenomenon using an Ising model in a decorated lattice. In
particular I obtained directly from experimental measurements simplified parameters
of the system Hamiltonian.
28
Chapter 2
Bacteria under circular and linear
confinement
2.1 Introduction
In the 1950’s, Yoshio Yotsuyanagi took 10cm long internodal cells from the algae
Nitella and Chara, and manually isolated the content such that part of the cytoplasm
would get encapsulated into vesicles [128, 129]. He then observed that the content
of these vesicles, while being subject to Brownian motion, could also actively move,
spontaneously creating protuberances, pulsations and sometimes circulation. This mo-
tion is most probably due to myosin motors carrying smaller vesicles along actin fila-
ment bundles, a mechanism used to generate cytoplasmic streaming in live cells [130].
However, how such an active fluid could self-organise was not understood.
Sixty years later, Francis Woodhouse, a PhD student in Ray Goldstein’s group at
that time and currently a research assistant professor at the University of Western Aus-
tralia, reproduced this self-organisation using a continuum model [131]. He showed in
particular that the motion of motors on actin filaments generate a stresslet disturbance.
I also managed to experimentally reproduce Yotsuyanagi observations (figure 2.1) but
this system presented several challenges to work with: the content of the cytoplasm
(proteins, vesicles...) is not well characterised, it is difficult to visualise constituents
(actin filaments, motors), and the myosin motors are very unstable once purified.
Instead, swimming bacteria such as B. subtilis present many advantages: they are
29
Figure 2.1: Rotating vesicle of Chara extract, prepared by manually squeezing out
the cytoplasm of an internodal cell. The extract has not been mixed into any buffer, the
surrounding medium is the vacuole content. The vesicle edge takes a polygonal shape
with a protrusion (purple arrow) which rotates CW, while the bulk circulates CCW
(green arrow). Diameter: 15µm. Timescale ≈ 1s.
a model system for active matter, better characterised than Chara cells, are easy to
grow and image, and the numerous mutants can be useful to understand the population
behaviour. Even though Chara cytoplasmic streaming and bacterial swimming have no
common biological basis, they both generate a stresslet disturbance in the surrounding
fluid and thus Woodhouse’s simulations can still be relevant for the study of bacterial
circulation.
In large chambers, dense bacterial suspensions self-organise into two main turbu-
lent patterns: vortices and jets. In this chapter, I describe how each flow pattern can
be stabilised by confinement in a specific microchamber. The last section presents an
opening to more complex confining topologies.
I first studied dense bacterial suspensions inside flattened drops, similar in shape
to Chara vesicles, and found that the cells would spontaneously self-assemble into a
single spiral vortex state. Bacteria also form a boundary layer moving in the opposite
direction to the bulk, a finding which was not predicted by Woodhouse’s model. I
then injected the dense suspension into periodic channels, which stabilise the bacterial
motion into a persistent directed stream.
I quantitatively studied the bacterial dynamics in these two systems, measuring in
particular the flow pattern dependence on the chamber size. Confinement not only sta-
bilised and controlled the suspension dynamics, but also gave us valuable information
on bacterial self-organisation and raised questions on the mechanism driving bacterial
30
Figure 2.2: Numerical results of Chara-like active matter confined into a circular
chamber. (a) nematic order orientation. Dark areas indicate vertical and horizontal
orientation, brighter ones, diagonal orientation. (b) spontaneously circulation stream-
lines. Darker lines reflect faster motion. Adapted from [131]
motion. These questions will be studied in details in the next chapter.
The last section of this chapter presents how bacteria self-organise in an 8-shaped
channel and shows that the bacterial motion can be finely controlled by the exact ge-
ometry of the chamber.
2.2 Flattened drops
Dense bacterial suspensions have been extensively studied in unconfined 3D or 2D
chambers. My first challenge was to design experimental chambers that could enclose
the suspension into a limited volume. Inspired by algal extract vesicles, I first consid-
ered flattened cylinders.
2.2.1 Protocol
Dense bacterial suspensions have been studied for many years [7, 30, 31, 32, 92, 93, 96]
and there are now standardised protocols to grow cells, image them and analyse their
motion. A common approach to make microchambers is to use microfluidic channels
made of Polydimethylsiloxane (PDMS). However, this technique is not appropriate to
form flattened cylinders. An issue I encountered is that the injection channels (inlets)
would affect the motion inside the small cavity. Instead I modified a method used to
study Xenopus frog egg extract in an oil emulsion [132, 133].
31
Figure 2.3: Protocol. (a) Bacteria were grown in TB and (b) concentrated 200 fold by
centrifugation. (c) The pellet was then mixed into mineral oil to form an emulsion, (d)
placed between two hydrophobic coverslips. Chamber height h was typically 25µm
and drop diameter d, 15−100µm.
Bacterial preparation
Wild type Bacillus subtilis (strain 168) were grown following a standard method [7,
32]. A stock of bacteria was maintained in a −80◦C freezer, mixed with 50% glycerol.
A few days prior to the experiment, a small amount was transferred on an agar plate
using a sterile inoculating loop. Bacteria were grown in a growth chamber at 35◦C for
12 to 24 hours. A day before the experiment, a monoclonal colony of bacteria was
transferred to 20mL of liquid medium and left to grow overnight at 35◦C on a shaker
to allow for a better mixing of the suspension and higher concentration of oxygen thus
increasing the growth rate. 100µL was then transferred to 20mL of fresh medium and
allowed to grow for 5 to 6 hours before being harvested for the experiment.
As a nutritious medium, I used Terrific Broth (TB, Sigma) which contains 12g/L of
tryptone, 24g/L of yeast extract, 8mL/L of glycerol, 9.4g/L of K2HPO4 and 2.2g/L of
KH2PO4. The medium is more concentrated than the widely used Luria Broth which
only contains 10g/L of tryptone, 5g/L of yeast extract and no glycerol. B. subtilis can
then reach higher concentration than in LB.
Bacteria adapt their phenotype depending on external conditions. In particular they
only start swimming when the nutrient concentration is sufficiently low [64]. After 5
to 6 hours of incubation, cells reached the end of the exponential phase, at which stage
the vast majority is swimming. Nishihara et al. quantified this effect and observed
up to 80% of cells swimming [66]. Prior to any experiment, I quickly confirmed the
suspension motility. Under a 40× objective, I would find less than ten immotile cells
in a field of view containing around a hundred swimmers.
32
I then harvested 10mL of the suspension. To further increase the cell density, the
suspension was centrifuged (1500g, 10min) and the pellet was directly used. I tried
different methods to estimate the density with difficulty to obtain an accurate measure.
By measuring the length and number of individual cells in a shallow chamber (1 and
2µm in height), I evaluated the volume fraction between 5 and 20%.
Drop formation
The dense bacterial suspension was then mixed into 4 times the volume of mineral oil
(Sigma) with added diphytanoyl phosphatidylcholine (10mg/mL, DiPhyPC, Avanti).
By gentle pipetting, the mixture forms an emulsion with drops filled with bacteria, 1µm
to 1mm in diameter. The lipid DiPhyPC reduced the coalescence of drops. Bacteria,
which only live in aqueous medium, did not swim into the oil phase.
I left the largest drops settle at the bottom of the tube and pipetted 10µL of the
emulsion over a 55×22mm coverslip. A second coverslip, 22×22mm, was placed over
the emulsion which then spread by surface tension to the edge of the top coverslip. No
spacer or seal was used and the height of the chamber was simply set by the volume
of the emulsion and the surface area of the smaller coverslip. This method has the
advantage to prevent evaporation and large macroscopic flow.
Additionally glass surfaces were made hydrophobic by a silane treatment before
being used: 5µL of Dichlorodimethylsilane (Sigma) were vaporised in a vacuum
chamber, alongside glass coverslips. The hydrophobicity modifies the shape of the
drops, such that the radius at the top and bottom of the chamber was smaller than in
the centre (figure 2.3d).
Imaging
The suspension quickly self-organises and I solely imaged the steady state.Two sec-
onds long movies were acquired on a high-speed camera (Photron Fastcam) at 125fps,
with bright field illumination, through a 100× oil-immersion objective, on an inverted
microscope (Zeiss Axio Observer Z1). I focused the focal plane in the centre of the
chamber to image the largest cross-section and avoid optical distortion from the wa-
ter/oil interface. Yet, the suspension’s motion did not depend on the height except for
the slower velocity close to the coverslips. The dense suspension quickly consumed
33
Figure 2.4: Measure of the suspension velocity using a standard PIV method. (a,b)
Two images of the suspension, taken at 1/60s interval are compared. Scale bar: 20µm.
(c) PIV flow field between the two images. For clarity, the arrow size is much larger
than the displacement between the two frames. The red arrow indicates the velocity
obtained by comparing subwindows (d) and (e). (f) Cross correlation CPIV(s) between
(d) and (e). White circle: origin, s = 0. Arrow: estimated displacement. CPIV is colour
coded with dark red for the largest value and dark blue for the smallest
the oxygen, and even though oxygen diffuses from the oil phase, bacteria stopped
swimming in less than 10minutes.
Flow measurement
To measure the bacterial flow, I used a standard particle image velocimetry (PIV)
method. PIV was initially developed to measure fluid flows through the motion of
tracer colloids. Here I did not add colloids to bacteria but simply used the speckle
made by the dense suspension (figure 2.4). I measured the bacterial flow, not the fluid
flow.
Images were first divided into subwindows, typically 8×8 pixels large (2×2µm),
with 50% overlap (i.e. centres are placed every 4 pixels). Each subwindow was then
34
cross-correlated with itself, on the next frame, at time t + dt,
CPIV(s, t) =
∑
n
I(n, t) · I(n + s, t + dt), (2.1)
where n = {nx,ny} are the pixel positions inside the subwindow, and I is the pixel inten-
sity. The maximum of CPIV at smax gives a first approximation to the average displace-
ment s˜ = smax + ds of the suspension in the subwindow. To obtain subpixel resolution,
I used a classical three-points Gaussian fitting in each coordinate direction [134],
dsi =
1
2
ln(CPIV(smaxi −1))− ln(CPIV(s
max
i + 1))
ln(CPIV(smaxi −1))−2ln(CPIV(s
max
i )) + ln(CPIV(s
max
i + 1))
, (2.2)
where i = x,y. This measure then gives the bacterial flow u(x, t) = s˜/dt, at the position
x in the centre of the subwindow.
Bacterial motion is often turbulent with a short time correlation, typically less than
1s [135, 136]. In the case of flattened drops however, the circulation was stable until
oxygen depletion, slowing down over time but without changing direction. To im-
prove the accuracy of the measurement, the correlation function CPIV was averaged
over the 250 frames (2 seconds) of a movie [137]. PIV calculations were performed on
a customised version of mPIV [134], a Matlab (The MathWorks) toolbox. All mod-
ifications were done in collaboration with Aurelia Honerkamp-Smith, post-doc in the
group, who used the same algorithm to measure the motion of fluorescent beads inside
and around vesicles.
Cell orientation
Akin to nematic liquid crystals, dense rod-like bacteria align locally. The suspension
is too dense to distinguish individual cells. However bacteria form a striated pattern,
indicating the mean local orientation.
To measure this orientation, I adapted an ImageJ plugin, OrientationJ [138], on
Matlab. The algorithm computes the local orientation tensor
J(n) =


〈Ix · Ix〉 〈Ix · Iy〉
〈Ix · Iy〉 〈Iy · Iy〉

 , (2.3)
35
where I is the pixel intensity, Ix and Iy its derivatives along each direction and 〈·〉 a
weighted average around pixel position n. The eigenvector of J of lowest eigenvalue
indicates the local orientation θ (figure 2.5a). Calculation gives [138]
θ =
1
2
arctan
(
2〈Ix · Iy〉
〈Ix · Ix〉〈Iy · Iy〉
)
. (2.4)
2.2.2 Results
Circulation
While the dynamics of dense bacterial suspension in unconfined environments is quasi-
turbulent, displaying transitory swirls and jets, confining the suspension into a flattened
drop can stabilise its motion into a vortex state (figure 2.5b). To quantify this effect, I
measured the vortex order parameter Φ, defined as
Φ =
∑
x
|u(x)·t(x)|
∑
x
||u(x)|| −pi/2
1−pi/2
, (2.5)
where u is the bacterial flow measured by PIV at all positions x inside the drop, and t
the local azimuthal unit vector (figure 2.5a). The parameter Φ measures how well the
bacterial circulation orders into a vortex. If all cells move along the azimuthal direction
with a constant speed, u = u t, Φ = 1. Instead, if the direction of motion is uncorrelated
with the azimuthal direction,
∑
x
|u(x) · t(x)|
∑
x
||u(x)||
= pi/2, (2.6)
and Φ = 0. Finally, if the flow is purely radial, u · t = 0 and Φ = −pi/21−pi/2 < 0.
Since the setup has a top/bottom symmetry (gravity is negligible), the suspension
should circulate CW or CCW with equal probability. To test this hypothesis, I mea-
sured the circulation direction on all 29 drops forming a steady vortex on a given day.
I obtained 14 movies with CCW circulation and 15, CW.
I measured Φ over drops of different diameters d (figure 2.5d) and found three
different circulation states. If the chamber is 25µm high, drops between d− ≈ 30µm
36
Figure 2.5: Circulation of dense suspensions of B. subtilis in flattened drops. (a)
Bright field image of a drop and definitions of r, t and θ. (b) Bacterial flow measured
by PIV. Not all PIV arrows are shown. (c) Enlarged region showing the boundary layer
to move in opposite direction to the bulk. (d) Vortex order parameter Φ in 25µm high
chambers. Each dot is a measurement over a two seconds long movie. Dense sus-
pensions (large blue dots) form a stable vortex state between d− and d+ (dotted lines).
Semi-dilute suspensions (smaller green dots) still exhibit a vortex-like circulation at
large diameter. (e) Φ for two chamber heights h. The vortex is maintained in small
drops by lowering h. Each point is an average over a 5µm bin. Error bars indicate
standard deviation. (f) Averaged azimuthal flow ut(r). The region of negative circula-
tion indicates the counterrotating boundary layer. (g) Boundary layer thickness lb, as
defined in (f), depending on the drop diameter d. Each dot is a measurement over a two
seconds long movie. While data present a large spread due to measurement inaccuracy,
no significant dependance on the drop diameter is observed. Adapted from [1]
37
and d+ ≈ 70µm exhibit a single vortex state, Φ > 0.7 (figure 2.6b). The suspension
reaches a steady state in less than a minute (the transition was not captured in these
experiments), the circulation is stable for ≈ 10 minutes until oxygen depletion, and
does not change its direction.
For drops slightly larger than d+, bacteria located less than 30µm away from the oil
interface still circulate along the azimuthal direction but some turbulence arises in the
centre of the drop. For the largest drops, d ∼ 100µm, the suspension forms a turbulent
state similar to observations in unconfined chambers (figure 2.6c).
In drops smaller than d−, the diameter is comparable to or smaller than the chamber
height and the drop is not flattened anymore. Bacteria circulate in all three directions
equally, in an unstable motion (figure 2.6a). The steady vortex state can be restored by
lowering the top coverslip: in 15µm high chambers, drops that are 25µm in diameter
still exhibit a stable circulation, Φ > 0.7 (figure 2.5e). Similarly if the chamber height
is increased to 40µm, no stable flow can be observed.
Flow patterns of stable vortices
I now focus on drops that self-organise into stable vortices, d− < d < d+ and h ≈ 25µm.
I first studied in more detail the bacterial flow by measuring the azimuthal flow profile
for each drop
ut(r) =< u(x) · t(x) >||x−x0||=r, (2.7)
where x0 is the centre of the drop, r the distance from the centre and t, the local
azimuthal unit vector, is chosen such that ut is on average positive (figure 2.5b,f).
The azimuthal flow ut is by definition antisymmetric (equation 2.7) and should
thus go to zero at the centre of the drop, r = 0. Figure 2.5f shows that experimental
estimates of ut seem to originate from ut(0) = 0, before increasing to reach a maximum
value about mid-way to the oil interface, r ∼ 10µm. The maximum velocity depends
on the drop size (the larger, the faster) and on the oxygen concentration (the later in
the experiment, the slower). The suspension could reach more than 50µm/s, an order
of magnitude faster than the single cell swimming speed v0 ∼ 5µm/s.
The azimuthal velocity ut then decreases and reaches negative values: bacteria
close to the oil interface move in opposite direction to the bulk circulation. This coun-
terrotating layer was not a visual effect or a PIV artefact as confirmed by imaging the
38
Figure 2.6: Circulation in drops of different diameter d and bacterial density. (a)
d < d−, bacteria circulate in all three dimensions, as indicated by the strong radial
flow in the bottom left quadrant. (b) d− < d < d+, single stable vortex. (c) d > d+, the
suspension becomes turbulent, here forming two vortices. (d) Semi-diluted suspension.
The centre of the drop is almost depleted in cells (dotted line). (a-d) arrows: bacterial
flow measured by PIV, only a fourth of the measurements are shown for clarity, except
in (a). Scale bars: 10µm (a,b) and 20µm (c,d).
39
Figure 2.7: Single individual bacteria can be distinguished when imaging the bottom
of the drop. Arrows: bacterial circulation direction. Rods: two bacteria tracked at the
oil interface, moving against the bulk circulation. Bar: 10µm. Images are taken every
0.2 second.
bottom of the drop where individual bacteria can be distinguished (figure 2.7).
I then estimated the thickness of the boundary layer lb from ut (figure 2.5f,g). For
a given drop diameter, lb can take values from 0 to 8µm. This large spread is mostly
due to inaccuracies in the measurement method: PIV subwindows are ≈ 2µm in size,
limiting the resolution, drops are not perfectly circular and the oil interface is often
difficult to accurately detect. However the average thickness of the boundary layer l˜b
did not vary with the drop diameter (figure 2.5g). Moreover l˜b ≈ 4µm is smaller than
the single cell length, 5 to 8µm for these B. subtilis [7], indicating that only cells in
contact with the oil interface move against the bulk circulation. Imaging at the bottom
of the chamber also confirmed this observation (figure 2.7).
Moreover, I measured the maximum velocity in the boundary layer. Bacteria move
at ∼ 5µm/s, independently of the drop diameter.
Occasionally I also observed microdrops in the oil phase, moving with the bound-
ary layer and against the bulk circulation. The oil is most certainly dragged at the drop
edge by the bacteria and water circulation. This motion was much slower than that
of the bacteria. Unfortunately I was not able to quantify this motion. I tried to add
microbeads but this operation is quite challenging as most beads end up in the water
phase.
40
Figure 2.8: Bacteria form a spiral vortex pattern. (a) Bright field image of a drop
shows the bacterial suspension to be oriented along a spiral pattern. (b) Right-side of
the same drop after processing with an edge detection filter (ImageJ function ’Con-
volve’). (c) Local orientation, averaged over 2 seconds. The outer ring is located at
the oil interface and colour coding indicates the local azimuthal direction. (d) Angle of
the cells at the interfaces. Dots: experiments with dense (circles) and diluted (squares)
suspensions. Lines: theoretical angle for 4 and 8µm long rectangles. (e) Model: at
constant cell length and concentration, θm increases with the curvature (decreases with
the drop diameter). Adapted from [1].
Bacterial orientation
On a static viewpoint, bacteria form a spiral pattern (figure 2.8a). I measured the mean
local orientation as the eigenvector of lower value of the orientation tensor computed
from the bacterial speckle. I then calculated the mean angle θ(r) between this orien-
tation and the azimuthal direction t (figure 2.5a) depending on the distance to centre
(figure 2.8). Bacteria mostly appear parallel to t in the centre of the drop, θ(0) = 0, and
diverge from t, to reach a maximum angle θm at the oil interface (figure 2.8c). Plot-
ted against the drop diameter, θm is smaller for larger drops, θm(d = 30µm) ≈ 30µm,
θm(d = 70µm) ≈ 15µm (figure 2.8d).
To gain insights into this dependence, I calculated the smallest angle θth that a set of
41
rectangles, the size of bacteria, can make at a curved interface (figure 2.8e). θth depends
on the rectangle size, density and on the interface curvature. At low density, rectangles
can align with the interface, θth = 0, and at highest density they align perpendicularly,
θth = pi/2. Likewise, the larger the curvature, i.e. the smaller the drop, the larger the
angle θth (figure 2.8e). Finally, longer rectangles yield a larger angle to the interface. I
calculated θth by solving on Matlab the set of following equations
tan(θth +η) =
`1 sinθth−d(1− cos(η))/2− `2 cos(θth +η)
`1 cosθth−d sin(η)/2 + `2 sin(θth +η))
, (2.8)
η =
`1`2
V f (`1 sin(θth) + `2 cos(θth)) (d− `1 sin(θth)− `2 cos(θth))/2
, (2.9)
where `1 is the cell length, `2 its width, d the drop diameter, η the angle between the
end of two neighbouring cells, seen from the drop centre (figure 2.8e). The volume
fraction V f is the only parameter that I cannot measure accurately in experiments. I set
the cell length `1 = 4 or 8µm, the cell width `2 = 1µm and manually fit V f with experi-
mental data (figure 2.8d). If V f = 40%, θth matches the trend observed in experiments.
This volume fraction appears much larger than my measurements (I estimate the over-
all suspension volume fraction around 10%) but not unrealistic: first, my experiment
measurement is quite inaccurate and could underestimate the actual volume fraction.
Then, flagella which are invisible under bright field illumination, take up space around
bacteria and could increase their effective volume fraction.
Semi-dilute suspensions
In addition, I performed experiments with semi-dilute suspensions: bacteria are diluted
after centrifugation but still exhibit collective motion in unconfined chambers. For
these experiments, I mixed the pellet with supernatant (TB) at ratio 1/4 to 3/4, to
reach a volume fraction between 1 to 10%. When placed inside flattened drops, the
suspension still forms a single vortex, except for bacteria accumulating at the edge of
the drop, leaving the centre empty (or with a few non-motile cells, figure 2.6d). The
thick layer enriched in cells seems homogeneous. Visually, the cell density depending
on the distance to the centre then appears as a step function: zero in the centre and
constant closer to the edge (figure 2.6d). Diluting the bacterial suspension prevents the
transition for a stable vortex to a turbulent state: the net flow Φ stays larger than 0.5 for
42
drops around 100µm in diameter. This result is not in contradiction with those from
the dense suspension as the turbulence first arises in the centre of the drop, which here
is depleted in cells.
Furthermore, the cell orientation at the oil interface, as quantified by the angle θm,
is unchanged (figure 2.8d). This result shows that this angle only depends on local
properties (cell concentration, interface curvature) but not on the bulk organisation.
A year after we published this result, Vladescu et al. presented measurements in
spherical drops filled with E. coli and found that, in this setup, the concentration profile
is not a step function but instead peaks at the oil interface and is constant in the bulk,
increasing with the overall bacterial concentration [139]. This difference may arise
from the shape of the drops which are not flattened in their experiment and from the
use of a different bacterial species.
2.2.3 Discussion
Active matter under circular confinement
Even though I am not aware of any other experiment that confined a dense bacterial
suspension, circular confinement has already been shown to affect the self-organisation
of other active systems. Here, I give a few examples, briefly summarising the interac-
tion mechanisms and the resulting motion:
Bristle-bots Centimetre-sized objects are set into motion by the vibration of a ta-
ble [39, 40, 41]. They have either elongated or circular shapes but are always
polar: the motion direction is set by the shape of the bristles in contact with
the table (figure 2.9a). Bristle-bots organise only by direct steric interaction
and through circular confinement. Under some conditions, these objects self-
organise into a single vortex state, without a boundary layer. Remarkably, the
elongated shape is not necessary to obtain this circulation [39]. Furthermore,
Kumar et al. [41] have filled the chamber with mm-large beads (figure 2.9b).
Bristle-bots there interact through indirect long-range forces, transmitted by this
granular medium, and still managed to form a similar vortex state.
Eukaryotic cells on patterned surfaces A large circular pattern is filled with a mono-
layer of crawling cells. Cells are constantly in direct contact with their neigh-
43
bours and show an initial random motion direction (figure 2.9c). They sponta-
neously start moving in a common direction, again without a countercirculating
boundary layer [125].
Rotating cell nucleus A single fibroblast cell is placed on a small circular pattern.
The actin cytoskeleton grows from the outer edge of the cell, and thus shows a
net motion towards the nucleus at the centre. Active stresses in the cytoskeleton
spontaneously break the cell symmetry, resulting in the rotation of the nucleus
(figure 2.9d). Once again, no double circulation was observed [126].
Chara vesicles As presented at the beginning of this chapter, when the cytoplasm of a
Chara cell is manually extracted, tens of microns sized vesicles form, containing
actin bundles and myosin motors. Some of these vesicles spontaneously create a
vortex state with edges moving in opposite direction to the bulk, as also observed
with dense bacterial suspensions [129, 131]. In Chara vesicles, the behaviour
is probably due to the motion of actin filaments at the edge, against myosins
which haul the bulk. In comparison, flattened drops of B. subtilis have similar
swimming bacteria at the oil interface and in the bulk.
Boundary layer
Even though all these active systems can self-organise into a vortex state, the flow
pattern in dense bacterial suspensions differs significantly: in all flattened drops, ir-
respective of their diameter, a single layer of bacteria (∼ 4µm) moves in the opposite
direction to the main bulk circulation.
Bristle-bots are probably the system closest to swimming bacteria: both can ex-
hibit elongated shapes with similar aspect ratio, are self-propelled and interact through
direct steric repulsion. The major difference comes from the fluid flows that bacte-
ria generate which can affect the organisation of the suspension at long-range. The
boundary layer then probably arises from these hydrodynamic interactions or from mi-
croscopic effects. Moreover, when bristle-bots are placed in a granular medium, even
though they generate granular flows, all objects still move in the same direction. The
exact details of the fluid flow patterns around swimming bacteria is thus important to
understand their self-organisation. This question will be studied in the next chapter.
44
Figure 2.9: Vortex states in different active systems. (a,b) Bristle-bots, are set into
motion by the vibration of a table [39, 40, 41] and can interact either by direct repulsion
or through a granular medium [41]. Bots are typically 5mm long and beads 1mm in
diameter. (c) Monolayer of crawling cells on a circular pattern [125]. (d) Flow of the
cytoskeleton and nucleus of a single cell on a small circular pattern [126].
45
Length scales
Turbulent vortices have been commonly observed in unconfined suspensions, with life-
times less than 1s [135, 136]. Confining the suspension into a flattened drop stabilises
such a vortex for more than 10 minutes, until oxygen depletion. I found two critical
diameters, d− ≈ 30µm that ensures the quasi-2D confinement and d+ ≈ 70µm after
which the suspension recovers its turbulent behaviour.
Notably d+ is comparable to the swirl diameter measured in quasi-2D chambers.
Patterns in unconfined suspensions are usually studied by their spatial correlation which
gives a typical length scale of 40µm as the radius of the turbulent swirls [7, 31, 32].
Measurements in racetracks (section 2.3) revealed a swirl diameter of 90µm. These
measurements indicate that confinement stabilises the suspension but does not increase
its typical length scale.
3-dimensional flow
A common interrogation is whether bacteria move in the vertical direction. When
the drop diameter is smaller than 30µm, of comparable length to the chambers height
h ≈ 25µm, bacteria circulate randomly and unsteadily in all three directions. When
the drop diameter is increased, most of the flow seems to be confined to the horizontal
plane. I tried to observe remaining vertical flow by adding 1µm fluorescent beads
which are advected by the fluid flows but also pushed by the bacteria. I imaged them
using a confocal spinning disc illumination, which highly restricts the thickness of the
focal plane (better z-resolution). I observed beads to move in the same direction as
the bacteria and to mostly stay in the focal plane, suggesting that the vertical flow was
negligible compared to the in-plane motion: uz ux,y.
2.3 Racetracks
Bacteria and other microswimmers have often been studied in microchannels, in which
they have been shown to accumulate at surfaces [113, 114] and, in the presence of an
external fluid flow, to move in a common direction [122, 123, 124]. Yet, there has
been no experimental realisations of dense bacterial suspensions in channels and only
46
Figure 2.10: Fluorescent 0.5µm beads, imaged in a drops of B.subtilis. Beads in the
bulk (purple triangle) circulate CW while beads at the oil interface (green arrow, grey
circle) move CCW, both are going in the same direction as the bacteria. Bar: 10µm.
1.2 second between images.
a few simulations and theoretical studies have considered this case, with contradictory
results [103, 105, 140].
Instead of using linear microchannels, I designed periodic racetracks, tens of µm
high and wide (as in flattened drops) but more than 1 mm long. In these microcham-
bers, bacteria are able to quickly self-organise into a persistent stream, stable for tens
of minutes. In this section, I quantitatively study the flow characteristics, show surfaces
to play a critical role and present a model of pattern formation.
2.3.1 Protocol
PDMS microchamber
The main difference between racetrack and flattened drop experiments is the design of
the microchambers. I fabricated racetracks in the laboratory with polydimethylsiloxane
(PDMS) following a standard microlithography protocol (figure 2.11). Channels were
drawn on computer and printed on a plastic sheet (mask) at high resolution by an
external company (Microlitho). I then spin-coated a thin layer of SU-8 (Microchem)
on a silicon wafer at 1500 to 2000rpm. SU-8 is a light-sensitive material that solidifies
under UV exposure. The wafer was prebaked at 95◦ for 5min and placed beneath
a UV lamp for 3× 8s, with the mask place on top such that SU-8 solidifies only at
the location of channels. After baking a second time at 95◦ for 6min, the uncurred
SU-8 was removed in a bath of solvent (PGMEA, 4min), such that only the negative
47
Figure 2.11: Protocol for PDMS chambers preparation. (a) A silicon wafer is coated
with a thin layer of SU-8. (b) Exposure to UV light through a mask solidifies negative
patterns in the SU-8 layer. (c) PDMS is then poured on the cleaned wafer to form the
microchannels. (d) PDMS is then bound to a coverslip and the bacterial suspension is
injected through hand-made inlets.
of the microchannels was left on the silicon wafer. A 5mm thick layer of PDMS
(Sylgard 184, Dow Corning), was then poured onto the patterns. PDMS is a polymer
that solidifies when mixed with a curing agent (at 10% w/w). After baking to accelerate
the solidification (70◦ for at least 3h), individual chambers were cut and punctured at
inlets and outlets. PDMS and a glass coverslip were washed with ethanol and dried
with pressurised nitrogen. PDMS was finally bound to the glass coverslip using an
oxygen plasma (2 × 10s exposure). Surface binding after this oxygen treatment is
irreversible and the chamber could sustain high pressure.
Racetracks
Microchannels were shaped into racetracks, 20µm in heigh, 20−120µm in width and
more than a millimetre in length. To simplify the analysis, racetracks had two straight
parts, rather than being perfectly circular (figure 2.12b). They were linked by thinner
channels (∼ 10µm wide), into a chain of racetracks ended by inlets (figure 2.12a).
Connecting channels were thin,≈ 10µm, and long,  200µm, avoiding interactions
between adjacent racetracks.
Suspension preparation and imaging
Wild type B. subtilis were grown in TB and concentrated by centrifugation following
the same protocol used for flattened drops (section 2.2.1). They were manually in-
jected using a syringe. To stop external flows, inlets were sealed with a broken and
closed syringe needle. The system reached steady state in less than 2 minutes, before
I took any movie. Five second long movies were acquired and analysed with the same
48
method. I used a 63× oil-immersion objective and recorded at 60 or 125frames/s. PIV
measurements were not averaged over time and computed every 10 frames only. No
evaporation was observed as I used the microfluidic chamber for less than 15 minutes.
2.3.2 Results
As in drops, confining a dense suspension into a racetrack stabilises the suspension mo-
tion but this time into a stream. Bacteria at interfaces also move in opposite direction
to the bulk (figure 2.13). However due to the lower magnification and the roughness of
PDMS surfaces, PIV measurements could not catch this feature. I analysed racetrack
experiments by measuring the net flow, correlation lengths and flow profiles, depend-
ing on the channel width.
Net flow
In thin channels the suspension exhibits a directed stream, going either CW or CCW.
To quantify this motion, I measured the normalised net flow Ψ, defined as
Ψ =
∣∣∣∣∣
∑
u · ex
∑
||u||
∣∣∣∣∣ , (2.10)
where u is the bacterial flow as measured by PIV, ex the local unit vector along the
channel main direction (figure 2.12c). Both sums span over all PIV subwindows inside
the channel and over ≈ 70 frames of a 5 second long movie. Ψ = 1 denotes a purely
longitudinal flow, e.g. u = ||u||ex, and Ψ = 0 an overall stationary suspension.
Ψ is further averaged over all movies for a given channel width w (figure 2.14).
Ψ shows a clear dependence on w, with strong streaming for thin racetracks and no
large scale circulation for wide ones (figure 2.14). The critical width w∗ ≈ 70µm at
which streaming ceases, Ψ < 0.2, is comparable to the critical diameter in flattened
drops d+ ≈ 70µm.
Flow patterns
In order to understand this transition from a stable circulation to the stationary state,
I now describe the different flow patterns occurring depending on the channel width
(figure 2.15a), which I quantitatively study in the next subsections.
49
Figure 2.12: Spontaneous circulation in a periodic channel. (a) Design of the mi-
crofluidic chamber. Bacteria are injected through the inlet which is then closed to avoid
external flows. (b) Close up on one racetrack. Arrow: net circulation. (c) Close up on
the straight section of the racetrack where all measurements are performed. The x-axis
is set along the channel length, in the main streaming direction. Arrows: bacterial flow
measured by PIV. Adapted from [3].
Figure 2.13: Individual bacteria can be distinguished when imaging the bottom of the
channel. The bulk is moving from the right to the left side, while bacteria at the glass
interface (coloured lines) swim to the right side. Bacteria were tracked and coloured
by hand. Bar: 10µm. Time between frames: 0.16s
50
Figure 2.14: Normalised net flow Ψ depending on the channel width w. Ψ shows
a clear transition from a strong stream to a stationary motion. The net flow ceases
around w∗ ≈ 70µm. Each point is an average taken over at least five movies, on at least
2 different days. ∞: unconfined quasi-2D chambers. Standard error < 0.1 (not shown).
Adapted from [3].
For channels thinner than the critical width w∗ ≈ 70µm, nearly all bacteria in the
bulk move in the same direction. Confinement dampens the turbulence such that the
suspension solely creates partial swirls. The flow pattern then appears as a wave.
Around the critical width, w ≈ w∗, the suspension is able to form full vortices.
Each side of the channel moves in opposite direction to the other, resulting in no net
flow. Moreover, vortices have a prefered direction, CW or CCW depending on the
circulation direction at the PDMS interface.
For the largest channels, the suspension recovers its turbulent behaviour, with little
or no correlation between opposite sides.
Correlation lengths
A classical approach to characterise flow patterns in dense bacterial suspensions is
to measure the spatial correlation. I adapted the usual isotropic formulation to take
advantage of the linear geometry of the racetracks:
C(s) =
∑
uy(x,y, t) ·uy(x + s,y, t)
∑
||uy(x,y, t)||2
(2.11)
where uy = u · ey is the flow perpendicular to the channel main direction. The sum
runs over all PIV sub-windows in the bulk of the channel, and over ≈ 70 frames of
a 5 second long movie. Note that this formulation only correlates flow along the ex
direction.
51
Figure 2.15: Bacterial flow patterns depending on the channel width. (a) Bacterial
flow measured by PIV. Channel width 35, 45, 55, 75 and 90µm. (b) Flow cross section
F(y) in experiments. Dots: measurements. Solid line: Ffit. Dotted lines: f
(1)
beta and f
(2)
beta.
Adapted from [3].
52
For all channels, C initially decreases from C(0) = 1 to negative values, oscillates
and fades at large distances, C(s→∞) = 0 (figure 2.16b). Such correlation behaviour
which includes oscillations, has been observed for unconfined suspension, using an
isotropic definition of the correlation function, but has never been fully analysed. Two
standard approaches have been used to extract the correlation length: fitting with a
decaying exponential or measuring the first zero or minimum of C. I combined the
two techniques to fit C with the function
Cfit =
[
Ae−s/Le + (1−A)
]
· cos(2pis/Lc), (2.12)
where the first and second term capture the amplitude decay and oscillations, respec-
tively. Lc is first estimated as the maximum of the Fourier transform of C. Le and A
are both estimated, knowing Lc, using the Matlab function fit.
Lc reflects the swirl size (the pattern spacial periodicity) and Le the decay length
scale (persistence length). They share a similar dependence on the channel width w
(figure 2.16a), increasing with w to reach a plateau around w ≈ 80µm, shortly after
streaming ceases, w∗ ≈ 70µm, and when the suspension is able to form full vortices.
Values at the plateau, Lc ≈ 90µm and Le ≈ 50µm, are comparable to measurements in
unconfined chambers (noted∞ on figure 2.16a)
Flow cross section
I then quantitatively measured the averaged flow along the channel direction
F(y) = 〈u(x,y, t) · ex〉(x,t) , (2.13)
where ex is chosen such that F is on average positive. In the case of a passive New-
tonian fluid driven by a pressure gradient, this profile should appear parabolic or
flat [141]. Instead, bacteria forming partial or full swirls, exhibit complex flow profiles,
highlighting the interplay between boundaries and bacterial collective motion.
Figure 2.15b shows different profiles depending on the channel width. For the
thinnest channels, w ∼ 30µm, F is quasi parabolic. F then flattens and splits into a
double peak function, at w ∼ 50µm. When the channel is large enough for the suspen-
sion to form full vortices, F takes a sinusoidal shape, and finally gets multiple extrema
53
Figure 2.16: Spatial flow correlation. (a) Flow correlation lengths depending on the
channel width w. Dots indicate experimental measurements averaged over at least five
movies, taken on at least two different days. Bars: standard error. Dotted lines: value
at plateau. ∞: unconfined quasi-2D chambers. (b) Example of a correlation function
C (dots) and its fitting function Cfit. Adapted from [3].
Figure 2.17: Distance ym between the PDMS interface and the maximum of f
(1)
beta, as
indicated in figure 2.15b by a thin black line. Each point is an average taken over at
least five movies, recorded on at least two different days. Error bars: standard error.
Adapted from [3].
54
Figure 2.18: Bacterial flow cross section F1(y, t) becomes unstable at large channel
width. (a) Flat and (b) sinusoidal profiles slightly change between frames. (c) For large
channel width, the suspension as well as the flow cross section become turbulent. Each
line represents the flow cross section calculated over a single frame of a five seconds
long movie.
in the largest channels, w ∼ 100µm.
To assess the stability of the bacterial circulation I then studied the flow profile over
time. I calculated the flow cross section on a single frame
F1(y, t) = 〈u(x,y, t) · ex〉(x) . (2.14)
For thin channels, w<w∗, the suspension produces a stable net circulation. F1 does not
change significantly with time and I never observe the overall circulation to switch di-
rection (figure 2.18a). For intermediate channel width, w ≈w∗, the suspension presents
a stable sinusoidal profile (figure 2.18b). The net flow is here negligible since each half
of the channel moves in opposite direction to the other. For the largest channels instead,
F1 quickly changes, highlighting the quasi-turbulent behaviour of the suspension (fig-
ure 2.18c).
In order to get insights into the transition from a parabolic to a double peak profile,
I measured the distance ym between the PDMS interface and the first maximum of F.
ym is straight forward to measure when F has two maxima. For parabolic and flat
55
profiles, F is fitted using two beta distribution functions,
Ffit(y) = f
(1)
beta + f
(2)
beta
= A1
( y
w
)α−1 (
1−
y
w
)β−1
+ A2
(
1−
y
w
)α−1 ( y
w
)β−1
,
(2.15)
where A1 and A2 are the amplitude of each beta distribution function, and α and β two
parameters setting their shape. The functions f (1)beta and f
(2)
beta are symmetric, except for
the amplitude: f (2)beta(y) = A2/A1 f
(1)
beta(w− y). I also set α = 2 to ensure that f
(1)
beta (resp.
f (2)beta) has a non-zero finite derivative at y = 0 (resp. y = w). I used beta distribution
functions for convenience, as they possess a single maximum and go to 0 at y = 0,w.
Ffit has no physical basis to bacterial suspensions or fluid mechanics, except when
β = 2 which gives a parabolic profile, as expected for a passive fluid [141].
The beta distribution functions fit the different behaviours: for the smallest chan-
nels, f (1)beta and f
(2)
beta mostly overlap; they then come apart to form the double peak
profile and adopt opposite signs (A1A2 < 0) when F is sinusoidal. At the largest width,
Ffit only captures the first and last extrema.
I then calculated ym, the distance at which f
(1)
beta reaches its maximum. ym solely
depends on β and w. Figure 2.15d shows that ym does not depend on the channel width
but rather stays at a fixed distance ≈ 15µm from the PDMS edge (with the exception
of the thinnest channels where ym = w/2). Each beta distribution function then appears
as pinned to opposite edges and to move apart as the channel gets wider. The overall
cross section flow is here described as the sum of two main streams, driven close to the
PDMS interface with a constant length scale ym. This result emphasises the role played
by boundaries in the overall circulation. As in flattened drops, bacteria at the surfaces
move against the bulk and may be essential for the suspension self-organisation.
Flow pattern motion
As bacteria move in the channel, the flow patterns they form do not stay static. I
measured the motion of the partial swirls in thin channels (w < w∗) by computing the
spatial correlation function Cswirl between consecutive frames of a movie, defined as
Cswirl(s) =
∑
uy(x,y, t) ·uy(x + s,y, t +∆t)
∑
||uy(x,y, t)||2
, (2.16)
56
where uy = u · ey and ∆t the time between consecutive frames. Sums run over all
PIV subwindows inside the channel, and over ≈ 70 frames of a 5 second long movie.
The distance sm at which Cswirl reaches its maximum value indicates the average dis-
placement of the pattern between two consecutive frames and the velocity of the swirl
motion, Vswirl = sm/∆t.
I compared this velocity with the average bacterial flow along the main channel
direction: Vbacteria =< ux >x,y,t. I found that Vswirl and Vbacteria had the same direction
and the swirls seem to move faster. The ratio Vswirl/Vbacteria has a median of 1.2 with
a standard error of 0.1. However the measurement of Vswirl is still rudimentary and
can be biased or inaccurate. It is difficult to conclude whether the swirls are moving
faster than the bacteria. Studying the dynamics of formation and motion of these partial
swirls could nevertheless be an interesting approach to understand bacterial suspension
behaviour.
This measurement could be improved by tracking single cells (as in Chapter 3)
and following singularities in the flow pattern, such as a vortex centre. The relative
motion would indicate whether the flow pattern moves faster or at the same speed as
the swimmers.
2.3.3 Discussion
In both racetracks and flattened drops, the bacterial suspension shows a similar be-
haviour in which confinement stabilises the turbulent dynamics into a persistent stream
or vortex, respectively. Both display bacteria swimming at interfaces moving in oppo-
site direction to the bulk. The racetrack experiment can then be understood as a simple
extension of the drops: in drops the suspension is confined in all three dimensions,
whereas racetracks can be more than a millimetre long. Yet, racetracks also give valu-
able information on the general suspension behaviour. Two particular points discussed
below are the length scale of the flow patterns and how interfaces drive the local or-
ganisation. I finally present a macroscopic model of the bacterial pattern formation
depending on the channel width.
57
Flow length scales
I have extracted three length scales, two from the correlation function and one from
the flow cross section. The bacterial circulation stabilised in racetracks leads to a new
picture of the correlation function, as the combination of an oscillating term and a
decaying one. In previous studies, the latter dominates and few or no oscillations were
observed, sometimes leading to a confusion between the persistence length (decay
length) and the swirl size (oscillation period). I have computed both of these values
depending on the channel width and showed that they reach a plateau value shortly
after streaming ceases.
These lengths have been measured depending on other parameters, such as the
oxygen concentration or the bacterial density. Of particular interest is the study by
Gachelin et al. of diluted to dense unconfined suspensions of E. coli. They showed
that the persistence length (decay length) increased with the cell concentration, quan-
tifying the appearance of collective motion. Additionally, with the added insight from
racetrack analysis, one can estimate the swirl size from the first zero of the correlation
function. Interestingly, all correlation functions in Gachelin’s article reach zero at the
same distance, s ≈ 55µm, indicating that the swirl size does not depend on the bacte-
rial concentration but is rather an intrinsic length associated with the swimmers. Cell-
driven fluid flows, which generate long-range hydrodynamic interactions, are probably
controlling this typical length [89].
I also measured the typical length scale between the PDMS interface and the max-
imum longitudinal velocity, ym ≈ 15µm. This distance can be linked back to the swirl
size Lc. Consider a typical vortex, with zero flow at the edge and in the centre. The
maximum azimuthal velocity is roughly half way between the centre and the edge, as
I found in flattened drops. If the edge of the vortex is placed at the PDMS interface,
then the vortex diameter is typically 4ym ≈ 60µm (figure 2.19b), smaller but compara-
ble with Linfc ≈ 90µm, the vortex size in unconfined chambers.
Model of pattern formation
From the experimental observations, I now propose a model for the flow pattern forma-
tion (figure 2.19). As a starting point, it considers the turbulent dynamics of bacterial
suspensions, which form swirls typically ∼ 90µm in diameter (figure 2.19a).
58
Figure 2.19: Model of pattern formation. The bacterial suspension forms swirls (green
arrows), whose organisation depends on confinement. Plots indicate the cross section
flow F. (a) Unconfined state. (b-e) Self-organisation in channels of decreasing width
w.
In the presence of interfaces, these swirls are initiated at the boundaries with their
centre located at about 2ym in the bulk (figure 2.19b). If the opposite wall is close
enough, w < w∗, only partial truncated swirls are generated. The continuity and in-
compressibility of the flow then imposes neighbouring swirls to be placed on opposite
sides, with inverse rotation directions, leading to the parabolic and double peaked flow
cross section (figure 2.19d,e). However, when the channel width is large enough for
a full vortex to form, continuity of the flow along the PDMS interface imposes the
vortices to share the same rotation direction, leading to the sinusoidal profiles (figure
2.19c). At even larger width, vortices are free to form and the suspension recovers its
turbulence, akin to that in unconfined chambers (figure 2.19a,b).
Bacteria and interfaces
PDMS and glass surfaces appear to play a central role in the formation of the stream.
Rather than confinement only, it is the combination of the two, interfaces and confine-
ment, that leads to the suspension behaviour.
In particular, the fact that bacteria close to the PDMS interface generate a strong
flow as revealed by the beta distribution functions, can be linked back to previous
experiments in which bacteria bound to surfaces generate bulk fluid flow [142, 143].
Similarly, bacteria in racetracks, even though sliding along the PDMS interface, can
generate fluid flows that reorganise cells in the bulk. A full analysis of fluid flow in
experiments and simulations is presented in the next chapter.
59
2.4 Cross channels
Bacteria like B. subtilis often live in porous habitats created by the environment, like
sand particles in soil, or by the bacteria themselves. For example, B. subtilis biofilms
have been shown to form channels, 90µm in diameter [16]. Motile populations can
then frequently encounter interfaces and confinement that affect their behaviour. How-
ever these environments are more complex than drops and racetracks and could consist
of a network of such structures. In the fourth chapter of this thesis, I show how a
bacterial suspension orders inside a lattice of vortices. Here, I briefly present results
for 8-shaped channels and show how the circulation at the junction depends on its
geometry.
Setup
The bacterial suspension and the microchannels were prepared as in racetrack exper-
iments (figure 2.11). Channels were 50µm in width, less than the critical width w∗,
and were shaped in a figure of 8. The only changing parameter was the angle ϕ of
incoming channels at the ×-junction (figure 2.20).
Results
As in circular racetracks, the suspension quickly starts circulating and reaches steady
state in about a minute. Bacteria can circulate in each branch either CW or CCW. The
flow pattern at the junction is of particular interest.
If the circulation in each branch is in opposite direction, bacteria simply pass the
junction without changing side (figure 2.20a).
If both branches circulate in the same direction, CCW for example, bacteria ap-
proach the junction from opposite sides, bottom left and top right on figure 2.20b,c.
Depending on the angle ϕ, bacteria can follow different paths through the junction. If
cells are moving from the bottom left channel,
• ϕ = pi/2: cells split equally to the top left and bottom right channels,
• ϕ < pi/2: cells predominantly go to the bottom right channel,
• ϕ > pi/2: cells predominantly go to the top left channel.
60
Thus, the path followed by the suspension is only dictated by the initial large scale
circulation and by the angle ϕ, such that cells follow the steeper corner. Moreover, I
observed that for the smallest angles ϕ, a stable vortex could form in the centre of the
junction
Cachile et al. have studied the same setup in the case of a passive Newtonian fluid
driven by pressure [144]. They came to the same conclusion: fluid predominantly
follows the steeper angle (figure 2.20d-f). They also observed vortex formation at the
centre of the junction but appearing at smaller angles than for bacteria: ϕfluid = 30◦,
ϕbacteria = 50◦. I have reproduced some of Cachile et al. results, as shown on the figure
2.20d-f, on FreeFem++, in collaboration with Pierre Haas, a PhD student in the group.
Discussion
Measuring the flow cross section in racetracks has shown that bacteria diverge from
the behaviour of passive fluids by generating various flow profiles from parabolic,
double-peaked, to sinusoidal. This difference appears to depend on the way flows
are generated, arising from the interfaces in the case of dense bacterial suspensions.
Yet, active and passive fluids display comparable streamlines at the junction of an
8-shaped channel, suggesting that flows are transmitted in the bulk in a similar fashion.
This result indicates that the bacterial circulation might here be driven by the fluid flow
they generate. Further experiments are necessary, measuring the fluid streamlines for
different angles ϕ and channel width w, in order to reveal all bacterial flow states.
2.5 Conclusion
In this chapter, I have presented how confinement modifies the spontaneous motion of
a dense suspension of B. subtilis. I have used a experimental approach and measured
bacterial flow on a mesoscopic scale.
This approach has already given us valuable information on the suspension dy-
namics, the role of interfaces and how bacteria self-organise and can be stabilised by
confinement in flattened drops, racetracks and 8-shaped channels. On one hand, previ-
ous experiments on dense bacterial suspensions have only considered unconfined 3D
or quasi-2D chambers. On the other hand, simulations that have studied circular and
61
Figure 2.20: Flow at an ×-junction depending on the angle ϕ, (a-c) of a dense bacterial
suspension and (d-f) of a passive Newtonian fluid. (a,d) Flow coming from bottom
branches simply goes through the junction without changing side. (b,e) ϕ = 80◦, flow
coming from opposite inlets and split almost equally beween outlets. (c) ϕ = 50◦ and
(f) ϕ = 30◦, flow coming from opposite inlets follow the steeper corner, and forms a
vortex in the centre of the junction. Bar and channel width: 50µm. Arrows: bacterial
and fluid flow, averaged over 10s in experiments. Arrow length is proportionnal to the
velocity except in (f) where it is constant.
62
linear confinement have failed to predict the suspension dynamics.
My results thus bring fresh information on bacterial behaviour and more generally
for the field of active matter. However, I have not explained yet how these organisa-
tions arise and in particular how the combined interactions between bacteria with one
another and with the fluid leads to the observed flows.
63
64
Chapter 3
Bacterial Self-Organisation
Mechanisms and the Role of Fluid
Flows
3.1 Introduction
Active matter is a field dominated by theoreticians. They have proposed many ap-
proaches to model these systems and yet none has been able to predict countercircu-
lating bacteria at interfaces or streaming patterns in racetracks. How is it possible that
theories that claim to reproduce bacterial population dynamics without confinement,
break down in the presence of interfaces?
An open question in the field is the role of fluid flows on the self-organisation of
bacterial suspensions. Many models, mostly particle-based, still do not include them,
arguing that they play a minor role. These models managed to reproduce most bacterial
states (dilute, turbulent, jammed...) but also revealed behaviours that are not observed
in large chambers (swarms). These simulations are actually more likely to reproduce
the behaviour on agar plates, where fluid streams are limited to thin surface layer [44,
46]. On the other hand simulations that directly or indirectly include hydrodynamic
interactions have neglected other aspects, such as steric interactions, self-propulsion, or
boundary conditions. Finally no experiment has simultaneously measured the bacterial
and fluid flows, and directly compared them (Wensink et al. have only looked at the
65
velocity distributions), most probably due to experimental limitations. We are therefore
placed in a situation in which studies are not able to prove but merely suggest the
mechanisms behind bacterial self-organisation.
This chapter is divided into two main sections. First, I will present a new method
to experimentally estimate the local fluid flow direction and velocity, directly from the
displacement of single bacteria. These results show that bacteria self-organise by the
action of fluid flows that they generate.
In the second section, I will present a simulation method, proposed and mainly car-
ried out by my collaborator Enkeleida Lushi. This model takes into account both direct
steric and long-range hydrodynamic interactions, as well as the effect of interfaces. Re-
sults from the simulations give us a detailed description of the bacterial organisation
from the microscopic to the macroscopic scale.
3.2 Experimental Observation of Fluid Flows
3.2.1 Introduction
Different groups (including Ray Goldstein’s) have claimed that micron sized colloids,
placed in dense bacterial suspensions, reflect the fluid rather than the bacterial flow. Yet
no proof, that I am aware of, for such assertions has been given so far (for example,
the bacterial flow and bead motion have not been simultaneously measured).
To measure fluid flows, beads has to be diluted compared with the bacterial concen-
tration in order to limit the disturbance. Bead diameter then has to be at least 0.5µm in
size to avoid strong Brownian motion. However, this is also the typical size of bacteria
(E. coli and B. subtilis are 0.5 and 1µm in width, respectively). At high cell density,
beads can thus be trapped between bacteria, if not directly sticking to one, in which
case their motion would reflect predominantly the bacterial flow.
As I will show later, bacteria and the surrounding fluid can locally move in the
same direction and at similar velocities. The error in measurement can be larger than
the difference between flows, affecting the results’ interpretation.
To avoid these issues I have developed a new method that allows me to simultane-
ously measure the motion (displacement) and the orientation (swimming direction) of
single cells. Given that the motion of one bacterium results from swimming, advection
66
by the fluid, and direct steric interactions with other cells, one can deduce the local
fluid flow direction and estimate its velocity.
3.2.2 Protocol
I used the mutant strain amyE::hag(T204C) DS1919 3610, grown in TB, under the
same conditions as wild type bacteria (section 2.2.1). This mutant has been constructed
in Daniel Kearns’ lab [145] and is a gift from Linda Turner Stern. The gene hag that
encodes the protein Flagellin, the main building block of the flagella, has been re-
placed by a version with a similar sequence except for a cysteine taking the place of a
thiamine. This mutation does not affect the motility but allows for an efficient fluores-
cent labelling. I followed the protocol by Guttenplan et al. to label cells [48]: 1ml of
the suspension was centrifuged (1000g, 2min) and resuspended in 50µL of phosphate
buffered saline (PBS) containing 5µg/mL Alexa Fluor 488 C5 Maleimide (Molecular
Probes) and incubated at room temperature for 5min. This fluorophore binds to the
cysteine added to the flagella protein Flagellin. Bacteria were then washed in 1mL
PBS and resuspended in PBS containing 5µg/mL FM4-64 (Molecular Probes). This
fluorophore incorporates itself into lipid bilayers and labels cell membranes. The ex-
cess of fluorophore was removed in a final wash and the suspension was resuspended
into 50µl PBS. A fraction of the labelled mutants was then gently mixed with unla-
belled wild type bacteria. B. subtilis does not swim in PBS because it lacks a carbon
source. Once transferred back into TB, they quickly recover their motility.
Images were acquired on a spinning disc confocal microscope (Zeiss Axio Ob-
server Z1) at 6 fps. Both fluorophores were excited at 488 nm and the emission was
filtered using a GFP emission filter (barrier filter 500-550 nm, Zeiss) for Alexa Fluor
488 C5 Maleimide and a DsRed emission filter (barrier filter 570-640 nm, Zeiss) for
FM4-64. I changed the filter between frames to alternatively visualise the cell body and
the flagella. The frame rate is limited to 6fps by the time it takes to switch between
filters. Results are presented as a sequence of three frames: membrane (false coloured
red), flagella (false coloured green) and again membrane (false coloured blue). Red and
blue correspond to the same fluorophore, FM4-64, and reflect different time points.
67
3.2.3 Results
Low Density
The mutant amyE::hag(T204C) has been previously used to study the motility phe-
notypes [48]. Here, I propose and show that imaging flagella and body successively
allows measurement of the orientation and motion of cells. I started with low density
bacteria which weakly interact and are set into motion by swimming only. B. subtilis
which are rod shaped, do not have permanent anterior (front) or posterior (back). The
rear is defined as the rod-end where flagella form a bundle. Bacteria can revert this
polarity, for example when hitting a surface [68].
I successively imaged the body (false coloured red in figures), flagella (false
coloured green) and body again (false coloured blue). The motion of bacteria is de-
duced from the displacement of the body between the first and third frames. The
orientation can be inferred from the bundling at the rear of the cell or from the average
position of the body to the flagella.
As expected for a dilute suspension, the motion results from the self propulsion
and cells exhibit a forward motion (figure 3.1).
Flattened Drops
I then mixed labelled bacteria into a dense suspension of wild-type cells and performed
experiments in drops and racetracks as described in the previous chapter.
In the case of drops, I have already shown that the suspension self-organises into
a spiral pattern, which indicates the orientation (geometric angle) but not the direction
of the bacteria (position of the front and back). Two main hypothesis then arise (figure
3.2):
• Self-propulsion driven motion: cells in the bulk and at the interface swim in
opposite direction, both with a forward motion.
• Advection driven motion: all cells are swimming in the same direction but ad-
vection by the fluid moves cells in the bulk against the boundary layer.
Due to the high cell density and the diffraction that results from it, fluorescent
bacteria are more difficult to image in drops than in dilute suspensions. Yet, I managed
68
Figure 3.1: Forward motion of two fluorescent bacteria in a dilute suspension. Left-
most: superimposed images of body (FM4-64, false coloured red), flagella (Alexa
Fluor 488, false coloured green) and body again (false coloured blue). Top: bacteria
moving from the left to the right side of the image. Bottom: bacteria moving from
the top right corner to the bottom left. In both cases flagella bundle indicates the same
swimming direction as the motion.
Figure 3.2: Two hypothesis for the drop organisation. (a) Cells in the bulk and at the
boundary swim in opposite direction. (b) All cells swim in the same direction but cells
are advected against their swim.
69
Figure 3.3: Fluorescent bacteria tracking in dense flattened drops. Leftmost image:
superimposed images of body (false coloured red and blue), flagella (false coloured
green) and bright field image (grey scale). (a) Cell at the oil interface has a forward
motion, moving and swimming to the top left corner. (b) Cell in the bulk exhibit a
backward motion, displacement of the body indicates that the cell move to the bot-
tom right corner, while bundling reveals a swimming direction to the top left corner.
Adapted from [2].
to track 24 cells, 16 of which were located in the bulk and 8 at the oil interface (which
is resolved from bright field images, figure 3.3).
I found that all cells at the interface perform a forward motion, moving in the same
direction as their swimming. In contrast, cells in the bulk travel against their swim-
ming direction showing that the fluid flow generated by the suspension was responsible
for the bulk circulation. Collisions between cells might also contribute to the overall
motion. However, since all cells are swimming in the same direction, steric interac-
tions are most probably responsible for the local alignment and cell density, but cannot
explain the macroscopic forward or backward motion.
Therefore, these results support the second hypothesis: all cells are swimming in
70
the same direction but long-range hydrodynamic interactions advect cells in the bulk
against the boundary layer. In the bulk, the fluid flows can then be estimated as the sum
of the bacterial flow and the single cell swimming speed ≈ 5µm/s. The azimuthal flow
profiles measured in the previous chapter showed bacterial flows up to 50µm/s and I
measured local flows up to 100µm/s, showing that, except close to the oil interface,
the flow is hugely dominated by advection.
Racetracks
I then performed the same experiment in racetracks, which equally exhibit a layer of
bacteria at the PDMS interface moving against the bulk circulation. I used 35 to 45µm
wide channels, displaying a strong stream (normalised net flow Ψ > 0.7) with limited
perpendicular flow uy.
I tracked 26 cells, 2 at the interface and 24 in the bulk. Both cells at the interface
moved against the bulk circulation, as indicated by bright field images (figure 2.13),
and performed a forward motion (identical swimming and motion directions). Among
cells in the bulk, 4 were aligned perpendicular to the flow, 4 were swimming with
the flow and a large majority of cells, 16, were swimming against the flow (backward
motion, figure 3.4). Suspensions in racetracks are more turbulent than in flattened
drops, which would explain why not all cells are swimming in the same direction.
3.2.4 Discussion
Results in flattened drops and racetrack chambers are consistent: while cells share a
common biased direction, they generate fluid flows, up to an order of magnitude faster
than the single cell swimming speed, advecting cells in the bulk against the suspension
swimming direction, leading to the formation of the countercirulating boundary layer.
Sokolov et al. [31] have previously measured the orientation, but not the direction,
of cells without confinement. They imaged the suspension in bright field but could not
visualise flagella. In dense suspensions, they found the cell orientation to be isotropic
and independent of the flow direction. This result already showed that self-propulsion
was not the main contribution to the cell motion. Indeed, steric repulsion between
swimmers could also play a strong role. However, in racetracks and even more so in
flattened drops, cells locally share the same swimming direction. Steric interaction
71
Figure 3.4: Fluorescent bacteria tracking in dense racetracks. (a) Bright field image of
a racetracks. Arrows: bacterial flow measured by PIV, not all measurements are shown.
(b) Forward motion at the PDMS interface. Both swim and motion are directed to the
right side. (c) Backward motion in the bulk. The cell moves to the left side of the
image, while bundle indicates a swimming direction to the top right corner. Adapted
from [3].
72
Figure 3.5: Effect of an interface on the bacteria generated fluid flow. (a) Unconfined
bacteria: the stresslet creates a ’pusher’ disturbance, fluid is symmetric, coming from
the side and propelled at the front and back. (b) No-slip boundary condition breaks the
symmetry to generate a net fluid flow against the swimmer direction. Black arrows:
bacteria motion. Blue arrows: fluid flow.
then mostly appears to align cells, akin to nematic liquid crystals. This phenomenon
might as well ensures that all cells locally move at the same speed but is unlikely to
substantially contribute to the overall motion.
Advection then seems to be leading bacterial flow. Left to be understood is how
bacteria generate fluid flows against their swimming direction, when stresslet distur-
bance is symmetric. Figure 3.5 presents a diagram of a cell at an interface, at an angle
θm (as also defined in drops). The interface breaks the symmetry, producing a net flow
in the bulk, moving against the cell swimming direction.
Finally, many simulations that aimed to reproduce dense bacterial suspensions have
neglected hydrodynamic interactions [7, 90, 99]. My results show that the fluid plays
a major role in the suspension circulation and that circulation in drops and racetracks
cannot be understood without considering the fluid flows. To understand in more de-
tails how the suspension self-organises, I now present a simulation model that includes
self-propulsion, steric as well as hydrodynamic interactions.
3.3 Simulation
3.3.1 Model
Bacteria are represented as elongated ellipsoids, of centre of mass Xi(t) and orienta-
tion Pi(t). Each swimmer moves by self-propulsion, direct repulsion and torque with
neighbours, advection and reorientation by the fluid flow. Fluid is set into motion by
the bacteria only, through a classical stresslet model. Finally, we take into account
boundary conditions using a system of images.
73
Bacterial Motion
The motion and reorientation of a swimmer is dictated by the following equations
∂tXi = U0Pi + v +Ξ−1i
∑
j,i
Fei j , (3.1)
∂tPi = (I−PiPTi )(γE + W)Pi + k
∑
j,i
(Tei j×Pi) , (3.2)
where U0Pi denotes the self-propulsion with constant speed U0 in the swimming direc-
tion Pi, v the advection by the fluid flow and Fei j the steric repulsion between swimmers.
Cells are then reoriented by shear flow [82], with 2E = ∇v +∇vT , 2W = ∇v−∇vT and
γ ≈ 0.9 for ellipses with aspect ratio 4, and by steric interactions with torque Tei j.
To estimate the steric interactions Fei j and T
e
i j, swimmers are represented as a set of
circular beads, which interact with neighbouring swimmers through a Lennard-Jones
potential [140]. Ξ = m||PiPTi +m⊥(I−PiP
T
i ) with the mobility parameters m⊥ = 2m|| ≈ 2,
and k ≈ 5 for ellipses.
There is no noise term and neither Brownian motion nor run-and-tumble behaviour
are taken into account.
Fluid Flow
The fluid flow is estimated by summing the contribution of each single swimmer into
a classical Stokes equation,
−∇2v +∇q = ∇ ·
∑
i
Sai δ(x−Xi) , (3.3)
∇ ·v = 0 , (3.4)
where q is the fluid pressure to account for the fluid incompressibility and Sai = αPiP
T
i
the active stress tensor resulting from the swimmer locomotion in 2D with non-dimen-
sional stresslet strength α ≈ −1 for a pusher swimmer of normalised length ` = 1 and
velocity U0 = 1 [146].
These equations do not take into account the presence of a dense suspension: the
overall fluid flow generated by bacteria is simply the sum of the fluid flow created
by each single swimmer in an empty medium, as if the medium could move through
74
neighbouring bacteria. Computing the fluid flow v around a suspension of bacteria
would require much larger computation power and probably would not be achievable
in a reasonable time. Yet we expect the result to be at least qualitatively similar, as
swimmers occupy 5 to 20% of the volume in experiments.
Domain and Boundary Conditions
Simulations are performed in 2D domains for two reasons. First, experimental cham-
bers are usually low enough to have a predominantly horizontal motion (with the no-
table exception of the smallest drops). Secondly, computing in 2D significantly reduces
the computation time.
Interfaces are reproduced using a system of images, mirrored on the other side of
the interface. In the case of circular confinement of radius R, if a swimmer is placed
at a distance less than 3` from the interface, an image is added, at a distance R2/||Xi−
Xcentre|| from the centre and with orientation Pi−2(Xi−Xcentre)/||Xi−Xcentre|| [120].
Periodic channels that reproduce the straight part of racetracks are made into a peri-
odic domain (x,y) ∈ ([0,L], [−w,w]), with mirrors at y = 0,w. Images of all swimmers
are placed at position Xi − 2(Xi · ey)ey, with orientation Pi − 2(Pi · ey)ey, creating the
boundary condition for both PDMS interfaces simultaneously.
Steric interactions and fluid flows are computed by taking into account these swim-
mers. Through direct repulsion, the system of images ensures that bacteria do not cross
interfaces and approximates a free-slip boundary condition for the fluid. This condi-
tion is not perfectly suitable to model oil and PDMS interfaces. The latter requires
a no-slip boundary condition. Given its large viscosity, oil has a strong resistance to
motion under such confinement (the motion of oil around drops is much slower than
that of the suspension). Computing more realistic boundary conditions necessitate to
use a more complex image system [77] and a greater computation power. However, I
do not expect this approximation to qualitatively change the result of the simulation,
except for the boundary layer thickness, as explained below.
3.3.2 Results
Enkeleida Lushi performed simulations in unconfined circular and straight periodic
channels. She managed to reproduce the different behaviours observed experimen-
75
tally. These results help in understanding how the bacteria self-organise and in partic-
ular confirm the mechanism forcing bulk bacteria to move against the boundary layer.
Note that we have not done any parameter fitting and all values (swimmer aspect ratio,
generated flows...) are either normalised or taken from the literature.
Reproducing Experimental Results
Before we actually analysed the simulation in order to understand the bacterial self-
organisation, we made sure they could reproduce the different bacterial motions ob-
served in vitro.
Lushi first simulated bacteria under circular confinement. She was able to obtain
the double circulation, peripheral bacteria moving in opposite direction to the centre
(figure 3.6). She measured the vortex order parameter Φ and found as in experiments
that swimmers stop forming a single vortex state around dsimu+ ∼ 17`. d
simu
+ can be
matched to experimental measurements by choosing ` ≈ 4µm, which is slightly shorter
than the actual B. subtilis length of 5 to 7µm. Likewise she also found that the vortex
order parameter remains high in semi-dilute suspensions, even for diameters larger
than dsimu+ (figure 3.6).
For the smallest diameters, less than d−, the unstable 3D circulation observed in
vitro could not be reproduced since simulations are performed in 2D. Instead, Φ de-
creases when the diameter is less than 6` and bacteria do not have enough space to
move.
The main difference between simulations and experiments appears in the thickness
of the counterrotating boundary layer. In experiments, I measured a constant boundary
width lb ≈ 4µm, consisting of a single layer of bacteria. In simulations, we found up
to three layers of swimmers, moving against the central circulation (figure 3.7). This
discrepancy may be due to the inaccuracy of parameters which we did not fit. Another
likely explanation comes from the inaccurate boundary condition. Simulations assume
no-slip at the oil interface, allowing for the fluid flow direction to revert at the edge
of the drop (figure 3.6): bacteria are advected with their swimming direction over a
few layers. Instead if a no-slip boundary condition was used, the fluid flow would be
negligible at the oil interface, reducing the number of cells moving against the bulk.
Furthermore, confinement in straight periodic channels reproduced the overall race-
76
Figure 3.6: Simulations reproduce the bacterial motion under circular confinement.
(a) Simulation steady state. Arrows: fluid flow. (b) Vortex order parameter Φ in simu-
lations (triangles) and experiments (circles) with dense (full symbols) and semi-dilute
suspensions (empty symbols). Adapted from [2]
track circulation. In this geometry, boundary bacteria moving against the bulk formed
a single layer of swimmers, in agreement with experiments. Lushi measured the net
flow and obtained a transition from streaming to an overall stationary behaviour, for
w∗simu ≈ 17` (figure 3.7). w
∗
simu can be matched to experimental measurements by set-
ting ` ≈ 4µm, which as in drops, is smaller but still comparable to B. subtilis length, 5
to 7µm.
In addition, Lushi also measured the bacterial flow cross section F(y) for varying
channel width and reproduced the different behaviours, from parabolic to sinusoidal
profiles (figure 3.7). Remarkably, since bacteria at the boundary move against the bulk,
she obtained negative flows at y = 0,w. This result is also expected for experiments but
the PIV resolution was here too low to capture the bacterial motion at the interface.
Self-Organisation and Hydrodynamics
We then studied in more detail how fluid flows affect and drive the suspension motion.
Over experiments, simulations have the advantage that we can easily add or exclude
hydrodynamic interactions and directly measure their effect. Before studying the con-
fined state, Lushi simulated bacteria in a periodic domain. As described in previous
publication without fluid flow [7], bacteria form swarms: aggregates that travel in a
77
Figure 3.7: Swimmer self-organisation in periodic channels. (a) Simulation steady
state. Swimmers (green ellipses, black circles indicate the front) generate fluid flows
(purple arrow) which advect bulk swimmers against the boundary layer. (b) Nor-
malised net flow Ψ depending on the channel width w in simulations and experiments.
Simulations are averaged over a time 5U0/` and experiment over at least 5 movies.
`: swimmer length. U0: single cell swimming speed. (c,d) Averaged bacterial flow
profile in simulations and experiments. All three plots on the left share the same y-
axis. Sketches of swimmers indicate the cell orientation and swimming direction at
the boundary (green arrows), as well as the fluid flow they generate (purple arrows).
78
Figure 3.8: Swimmer self-organisation in simulations with periodic boundary condi-
tions. (a) Without hydrodynamic interactions, swimmers collectively move in a large
swarm. (b) Adding fluid flows (blue arrows) generated by swimmers, disassemble the
swarm and leads to the formation of a turbulent state. Adapted from [2].
collective fashion. This behaviour has been observed experimentally for swarming
bacteria on agar plates (as opposed to swimming bacteria). Yet such an organisation
has not been found in 3D or quasi-2D chambers. Adding hydrodynamic interactions in
Lushi’s simulations disrupts these aggregates to form a more homogeneous suspension
(figure 3.8).
To fully understand the self-organisation of bacteria under circular confinement,
Lushi not only included or excluded hydrodynamic interactions but also changed the
swimmers’ shape (figure 3.9). In the following, I list the different parameters and
results we tried:
• Elongated swimmers with fluid flow: the most realistic case, which reproduced
the circulation as observed in vitro (figure 3.9a).
• Elongated swimmers without fluid flow (α = 0): bacteria assemble into a sin-
gle vortex but fail to form the counterrotating layer (figure 3.9b). This motion
matches that of elongated robots [39, 40].
• Circular swimmers with fluid flow (γ = 0, Tei j = 0): bacteria are fully circular and
do not align by steric or hydrodynamic interactions (figure 3.9c). Few transient
circulation was observed.
• Semi-circular swimmers with fluid flow (γ ≈ 0.9, Tei j = 0): swimmers react dif-
79
ferently depending on the interaction type. They are circular for steric (they do
not align like liquid crystals) but elongated for hydrodynamic interactions (they
are reoriented along shear flow). This case may correspond to spherical bacteria
whose flagella act as a ghostly tail . Such species have been shown to exhibit col-
lective motion similar to rod-like bacteria, such as E. coli and B. subtilis, but have
not been studied under confinement [136]. Simulations under circular confine-
ment show clear circulation, with counterrotating boundary layer (figure 3.9d).
Hydrodynamics are here sufficient to organise and drive the bacteria motion.
These different cases show that fluid flows are not only necessary but they are also
sufficient to reproduce and understand the bacterial circulation.
Figure 3.10 presents the evolution of a bacterial suspension over time from a ran-
dom disordered state to a single vortex. Swimmers were found to first organise at the
boundary from which they create a strong bulk fluid flow, moving in the opposite di-
rection to their swimming. This flow advects bacteria in the bulk before they organise
into a spiral pattern. Eventually, all cells share the same swimming direction but due to
advection, the bulk moves against the boundary layer. Moreover, simulations of semi-
dilute suspensions show that forces generated by the boundary layer are sufficient to
generate strong bulk fluid flow (figure 3.9e).
I also reproduced the emergence of order in flattened drops by exposing bacteria
to a blue light, to which they react by switching to a tumbling state, disorganising
the suspension [147].1 Once the light is switched off, cells recover their swimming
behaviour. As in simulations, bacteria first organise at the oil interface, setting into
motion the bulk before it is fully ordered (figure 3.10).
Finally, Lushi studied swimmers under linear periodic confinement, this time only
adding or excluding hydrodynamic interactions. She came to the same conclusion:
bacteria at the interface drive the bulk circulation through strong fluid flows. Swim-
mers likewise posses a biased swimming direction against the fluid flow, as found with
fluorescent bacteria. Yet, swimmers in the bulk are much less ordered than in drops, in
contrast to those at the boundary which generate most of the fluid flow.
1I discovered this effect when imaging fluorescent mutants. It has been reported previ-
ously [147] but I am not aware of any other paper using this method in dense bacterial suspen-
sions. Lu et al. instead used a photosensitiser that produces reactive oxygen species and alter
bacterial motion [148].
80
Figure 3.9: Self-organisation under circular confinement depending on interactions,
shape and density of swimmers. (a) Elongated swimmers with hydrodynamic interac-
tions. (b) Elongated swimmers without fluid flow. (c) Circular swimmers. (d) Circular
swimmers for steric interactions but align with shear flow. (e) Dilute suspension of
elongated swimmers. (f) Bacterial flow as measured in experiments. (a, c-e) Blue ar-
rows, lower-left quadrant: fluid flow. Arrows are magnified 1, 5, 1, 2 fold respectively.
(a-e) Black arrows, top-right quadrant: bacterial flow. Arrows are magnified 1.2, 4, 13,
1.2, 1.2 fold respectively. Adapted from [2].
81
Figure 3.10: Self-organisation in flattened drops from a disordered state. (a-c) Sim-
ulations. Blue arrows: fluid flow. (d-e) Experiment. The lower half is obtained by an
edge detection filter and contrast improvement. Adapted from [2].
82
3.3.3 Discussion
Simulations vs. Experiments
Simulations were in strong agreement with experimental results, reproducing the bac-
terial motion with and without confinement and validating the model of fluid flow
driven circulation. Flattened drops and racetracks are set into motion by the same
mechanism, in which most bacteria point in a common direction and generate back-
ward fluid flows. Due to the boundary condition, these flows are faster in the bulk
where advection becomes stronger than the single cell swimming speed, resulting in
the countercirculating layer observed in both flattened drops and racetracks.
Hydrodynamic interactions also had a strong effect on unconfined suspensions.
Simulations without fluid dynamics predicted a phase separation with dense bacterial
clusters, ’swarms’, and very dilute regions. This state has not been observed experi-
mentally in large chambers and disappears when hydrodynamics are added to simula-
tions. Fluid flows break apart clusters, resulting in a more homogeneous suspension.
We have not quantitatively studied this effect - since we wanted to focus on drops
and racetracks - but it has also been reported in the context of squirmers by Matas et
al. [91]. This effect appears universal among microswimmers.
When Can Hydrodynamics Be Neglected?
Numerous continuum models and particle-based simulations have aimed to reproduce
the motion of microswimmers. Even though several studies have shown hydrody-
namics to strongly modify population-level behaviour, they are still neglected in many
articles. In particular a phase diagram of bacterial motion state has been proposed de-
pending on the cell volume fraction and aspect ratio. Wensink et al. for example have
identified swarming and laning phases, appearing without fluid. Yet these states have
not been observed in 3D or quasi-2D chambers, which they aim to model.
Among all motility mechanisms, biologists have described distinctively swimming
and swarming [44]. They are both actuated by flagella but swimming refers to a motion
in a 3D volume (’suspension’), while swarming, that on agar plates. In the latter case,
bacteria produce surfactants that create a liquid film on the agar surface [46]. The layer
of fluid at the surfaces can be very thin, strongly dampening fluid flows. Furthermore,
surface tension at the edge of each swarm can promote their aggregation. Swimming
83
Figure 3.11: Mechanism for bacterial circulation decomposed into two previous ex-
periments. (a) Bacteria glued to a surface generate flows into the bulk, against their
swimming direction. (b) Swimming bacteria reorient and accumulate at the surfaces
by the action of a shear flow. (c) Self-organisation in racetracks, through long-range
hydrodynamic interactions. Blue arrows: fluid flow.
and swarming bacteria are then quite different systems which should be distinguished
and modelled carefully.
Previous Experiments
I am not aware of any publication reporting the arrangement of bacteria observed in
flattened drops and racetracks, in which the self-propulsion and advection create the
mentioned double circulation pattern. However, my experimental results can be under-
stood as a combination of previous experiments (figure 3.11).
First, different teams have attached swimming bacteria to surfaces, coordinating
their orientation and producing overall motion. Darnton et al. showed that these bacte-
ria create fluid flows in the surrounding medium and when attached to small particles,
would propel them [142]. Likewise Kim et al. covered walls of microfluidic chan-
nels with swimming bacteria and managed to use swimmers to generate a pump [143].
Cells push fluid in and out of the channel, setting up a net stream comparable to that in
thin racetracks.
Secondly, bacteria have often been studied in microchannels, but in a dilute regime.
Experiments showed that bacteria accumulate at the surfaces. When a shear flow is
then imposed, bacteria are not only advected but also reoriented against the fluid flow,
leading to upstream motion, as observed in flattened drops and racetracks.
Previous Simulations
Several previous theoretical studies have considered confinement, either circular or in
periodic channels, but none has been able to predict the behaviour I observed, and in
84
particular the boundary layer moving against the bulk flow.
As an inspiration for my experiments, Francis Woodhouse’s simulations of Chara
extracts, predicted a single vortex state with a spiral pattern, yet without the coun-
tercirculating boundary layer [131]. Indeed, he simulated an active system driven by
stresslet perturbations but did not include self-propulsion. He then correctly modelled
the orientation of bacteria and the fluid flows they generate. The double circulation can
then be restored by adding a swimming term, directed against the fluid circulation. At
the edge, the fluid flow becomes weaker than the cell velocity, creating the boundary
layer.
Several continuum models have studied periodic channels, yet without predicting
the circulation and flow patterns in the bulk. The following two models used active
nematics (Q-tensor) theory to represent the active matter. Ravnik et al. [103] anal-
ysed short periodic channels and thus could not capture the formation of partial swirls.
Instead, they predicted vortex-like circulation with the axis of rotation along ex. Un-
fortunately these flows were 1-2% the magnitude of the overall stream. Such a weak
component is overcome by noise and turbulence in experiments and cannot be mea-
sured. Fielding et al. [105] studied the circulation depending on the size of the swirls
relative to the channel width. Surprisingly they found opposite results to my experi-
ments: streaming appears only when the channel is much larger than the vortex size.
Likewise, Costanzo et al. simulated dense suspensions of swimmers in periodic
channels, using a particle-based model [140]. They considered the system with and
without added external fluid flow, and showed that this flow is necessary to obtain a
collective motion of the suspension, in contradiction to simulations presented in this
thesis. This difference may arise from the fact that Costanzo et al. did not take into
account the interface to compute the fluid flow (there are only steric boundary condi-
tions in their model), suggesting that mirror swimmers in Lushi’s simulations increase
the hydrodynamic interactions, leading to the suspension organisation.
3.4 Conclusion
Physical sciences is a domain built around universal laws and minimal models. Theo-
reticians working on active matter are almost exclusively physicists or applied mathe-
maticians and imported this approach to study these systems. Yet active matter is ex-
85
tremely diverse, in particular concerning interaction mechanisms, such as direct steric
repulsion, long-range fluid flows, and more complex behaviours including visual clues
in birds.
As a result, the most common model, Vicsek’s, and its coarse-grained counterpart,
Toner-Tu, seek to study active matter simply by assuming a local alignment mechanism
between its constituents. In the quest for minimality, the complexity of active systems
has been left behind and these models only capture few experimental behaviours. Re-
garding dense bacterial suspensions for example, many simulations have neglected
fluid flows. Yet such an approach is only relevant for some swarming bacteria or for
vibrated objects.
To disprove this model in the case of confined and unconfined suspensions, I have
shown using fluorescently labelled bacteria, that fluid flows are crucial to understand
cell motion. Enkeleida Lushi’s simulation is not the first, but still one of few, to fully
take into account self-propulsion, steric interactions between ellipsoid swimmers and
the fluid flow they create. Moreover, she includes boundary conditions for both fluid
and swimmers, through a system of images.
She then presented a minimal model that can still reproduce all particularities of
this active suspension. We obtained a detailed understanding of the dense bacterial
suspensions under confinement, confirming that fluid flows, predominantly generated
next to interfaces, drive the overall motion.
86
Chapter 4
Lattice of Vortices
4.1 Introduction
Swimming bacteria are a powerful model system for active matter theories. B. sub-
tilis are easy to prepare and use: cells grow quickly, are resistant to centrifugation
and chemical treatment, and the large variety of species and mutants allows to vary
experimental parameters.
Despite the apparent simplicity, the various behaviours they exhibit can be quite
complex, due to the combination of direct steric and long-range interactions. Experi-
ments in confined chambers have shown unexpected dynamics and more are likely to
be discovered. In this chapter, I present a setup that differs from natural habitats but
brings new ideas to active matter theories.
When a dense suspension of swimming bacteria is confined to a flattened drop, it
spontaneously forms a single stable vortex, spinning CW or CCW with equal probabil-
ity. If such vortices are brought close to one another, one can expect some interaction
to couple the directions of these vortices.
In the simplest case of two vortices, interacting through a given region called gap
(figure 4.1b), bacteria can be expected to move in the same direction in this gap, leading
to opposite vortices.
Experiments are performed in three different arrays of vortices (lines, square and
triangular lattices, figure 4.1c,d). Calculations of the mean absolute spin and of the
spin-spin correlation reveal four distinct states (random, antiferromagnetic, ferromag-
87
netic and disordered) that depend on the lattice topology and on the cavities geometry.
Finally, I present an Ising model that describes the network order and gives insights
into the mechanism driving the bacterial dynamics.
4.2 Experiments
4.2.1 Protocol
Wild type B. subtilis were grown in TB, as for flattened drops and racetracks (section
2.2.1), and concentrated by centrifugation. Chambers were prepared in PDMS (section
2.3.1), their design probably being the most challenging part of these experiments.
Microchambers
Chambers were about 18µm in height, and formed a lattice of flat circular cavities
connected by gaps (figure 4.1b). The distance between cavity centres was set to 60µm
(cavities were about 50µm in diameter) and I varied the topology of the lattice (line,
triangular, square, figure 4.1c,d) as well as the gap size (4 to 20µm).
All plots shown here have been obtained using these geometries. I also tried nu-
merous other lattice sizes (smaller and larger cavity diameters, gap depths, chamber
heights, etc), which behaved in a qualitatively similar way.
Imaging
The suspension was injected into the lattice and both inlets were sealed to prevent
external flows. I imaged the suspension using a 40× oil-immersion objective on an
inverted microscope (Zeiss Axio Observer Z1), recording at 60fps for 10s (Photron
Fastcam), 4 and 8 minutes after injection. The lattices were typically 15 cavities large,
but I only imaged a 6× 6 cavities region, taken in the centre of the chamber to avoid
boundary effects. I also avoided recording ill-filled cavities which could be semi-
diluted or in a jammed state. However, I did not choose specifically well-ordered
lattices.
88
Figure 4.1: Lattice of vortices setup. (a) Sketch of the full microfluidic chamber.
Bacteria are injected in the inlet and fill up the central lattice of cavities. For clarity,
only a 8×8 lattice is shown.(b) Bacterial flow inside a cavity. The suspension forms an
unstable vortex. Due to a lower magnification than in experiments in flattened drops,
the motion of bacteria at the PDMS interface against the bulk is here difficult to image
but is still present. Arrows: flow measured by PIV. The cavity is ≈ 50µm in diameter.
(c) Square lattice. (d) Triangular lattice.
89
4.2.2 Analysis
Notations
I analysed the lattice behaviour by computing the state of each vortex and around
each pillar. The sets of vortices and pillars in a given lattice are denoted by V and P
respectively. The symbol ∼ denotes adjacent vortices or pillars. EV = {v,v′ ∈ V |v ∼ v′}
and EP = {v ∈ V, p ∈ P|v ∼ p} are the sets of pairs of vortices and pairs of vortex/pillars
that are adjacent. Finally ∂i denotes the set of all adjacent sites to the vortex or pillar i,
such that ∂Vi = {v ∈ V |i ∼ v} and ∂
P
i = {p ∈ P|i ∼ p}.
Flow Measurement
Bacterial flow u(x,y, t) was measured by the same PIV method presented earlier (sec-
tion 2.2.1), without time averaging. Subwindows were 16× 16pixels large (16pixels
= 6.7µm), with 50% overlap, yielding about 15×15 measurements per cavity.
Spins Estimates
I assigned a spin σv(t) to each cavity v ∈ V , defined as the normalised angular momen-
tum
σv(t) =
∑
u(x,y, t)× r(x,y)
〈‖u‖〉x,y,t
∑
‖r(x,y)‖
, (4.1)
where sums are taken over all subwindows of coordinates {x,y} inside the cavity v, u
is the bacterial flow measured by PIV, r is the radial vector from the cavity centre to
{x,y}, and 〈‖u‖〉x,y,t is the averaged bacterial velocity taken over all cavities of a movie.
If the suspension forms vortices of consistent speed in each cavity, σv goes to ±1
depending on the vortex direction, CW (σv > 0) or CCW (σv < 0). If the bacteria do
not form vortices, σv ≈ 0. To quantify how well vortices form, I also computed the
average absolute spin for each movie: S = 〈|σv|〉v,t.
For square lattices, I also computed a similar spin around each pillar p ∈ P,
σp(t) =
∑
u(x,y, t) · t(x,y)
〈||u||〉
∑
1
, (4.2)
where t is the unit vector, tangential to the interface, and the sum runs over PIV sub-
windows closer than 5µm from the PDMS surface.
90
Spin-Spin Correlation
To quantify the interactions between adjacent vortices, the spin-spin correlation C was
measured over each movie, defined as
C =
∑
{v,v′}∈EV
σvσv′
∑
{v,v′}∈EV
1/2
(
σ2v +σ
2
v′
) . (4.3)
Thus C = 1 if all vortices experience the same direction (ferromagnetism) and C = −1
if the circulation direction alternates between adjacent cavities (antiferromagnetism).
4.2.3 Results
Lines of Vortices
I first injected the dense suspension into chains of cavities. The suspension forms
reasonably stable vortices, with S = 〈|σv|〉v,t > 0.6 (figure 4.2c), in agreement with
measurements in flattened drops (for a similar diameter in a flattened drop, the vortex
order parameter yields Φ ≈ 0.8).
Each vortex interacts through a gap, yielding different lattice states depending on
the gap size. For the smallest gaps, 5 to 10µm, interactions are too weak and no
correlation was measured between vortices. For larger gaps, 10 to 40µm, bacteria on
each side of the gap locally move in the same direction, yielding adjacent vortices of
opposite spins (figure 4.2a). The correlation then oscillates around C ≈ −0.6, such that
the bacterial suspension forms an antiferromagnetic state (figure 4.2b). For even larger
gaps, the cavity walls become almost straight and form a channel. In such cases, the
suspension moves in a persistent stream, like in racetracks (not shown).
Triangular Lattices
Bacteria are then injected into triangular lattices, in which each cavity is connected to
its six neighbours (figure 4.1c) [149]. Due to the cavities distribution, the suspension
cannot form a perfect antiferromagnetic state in which each vortex circulates in oppo-
site direction to its six neighbours. A state of lowest energy can still be calculated for
an antiferromagnetic triangular Ising net [149]. One solution is to have a frustrated
91
Figure 4.2: Self-organisation in chains of vortices. (a) Example of four lines. Vortices
are colour-coded according to the spin direction, purple for CW and green for CCW.
Cavity diameter: 50µm. Gap: 16µm. (b) Spin-spin correlation C depending on the
gap size. (c) Mean absolute spin S . Each point is averaged over at least four moves.
Bars: standard error.
arrangement with two thirds of the pairs of vortices in opposite direction (figure 4.3).
The resulting spin-spin correlation is then Cth = −1/3.
However, experiments with bacteria show that a ferromagnetic state in which all
vortices share the same direction is hugely favourable (figure 4.4a,b). For gaps between
9 to 17µm, C > 0.6 (figure 4.4c). When considered over all movies, the vortices did
not have a preferred direction: the mean spin over 88 movies what < σv >= 0.053, less
than the estimated standard deviation for a normal distribution around 0.076 (given
S = 0.71). Thus, spin-spin interactions only are responsible for the ferromagnetic state,
Figure 4.3: Two examples of antiferromagnetic triangular lattices of lowest energy.
The spin-spin correlation Cth = −1/3.
92
Figure 4.4: Self-organisation in triangular lattices. (a) Examples of two chambers.
Vortices are colour-coded according to the spin direction, purple for CW and green
for CCW. Cavity diameter: 55µm. Gap: 16 and 20µm. (b) Spin-spin correlation C
depending on the gap size. (c) Mean absolute spin S . Each point is averaged over at
least four movies. Bars: standard error.
and not an intrinsic spin associated with the cavities. I also observed that bacteria at the
PDMS interfaces would swim around pillars. Such a circulation is likely to promote
the ferromagnetic state.
The lattice then transitions for gaps ≈ 20µm to a random state: the absolute spin
decreases from S ≈ 0.7 to S ≈ 0.5, indicating a modest disorganisation of the vortices,
and the correlation becomes negligible, C ≈ 0 at gaps 23 and 33µm. At such gap sizes,
pillars become relatively small, their confinement effect being too weak to stabilise
vortices, thus the suspension enters a turbulent state.
Square Lattices
Square lattices are of particular interest as they exhibit both ferromagnetic and anti-
ferromagnetic states, even though they do not present a frustrated topology. I study
93
the spin-spin correlation and the mean absolute spin depending on the gap width and
identify four states (figure 4.5):
Random state For the smallest gaps, less than 2µm, the suspension self-organises
into stable vortices, S > 0.7, with independent orientations, C ≈ 0. This state is
difficult to obtain experimentally as bacteria get blocked in the gaps during the
injection process, resulting in either half-concentrated or jammed cavities. Yet
this state is simple to understand theoretically: small gaps prevent interactions
between vortices, resulting in spins of random sign. It is equivalent to flattened
drops experiments in which vortices are disconnected.
Antiferromagnetic state For small intermediate gaps, between 4 and 7.5µm, spins
are still high, S > 0.6, but their correlation becomes negative, C ≈ −0.2. The
bacterial suspension self-organises into vortices of alternating direction, as also
found in chains of cavities.
Ferromagnetic state For large intermediate gaps, between 7.5 and 18µm, the spins
decrease in magnitude, S ≈ 0.5, but most importantly the correlation changes
sign, C ≈ +0.1. The suspension is ordered into domains where vortices share the
same direction, as in triangular lattices.
Turbulent state For the largest gaps, more than 20µm, the confinement is not strong
enough to stabilise vortices, S ≈ 0.2, and the suspension recovers its turbulent
state, as observed in unconfined chambers. The correlation between cavities is
also lost, C ≈ 0.
4.2.4 Discussion
Experiments show that the bacterial circulation can be simply controlled by changing
the topology of the lattice and the opening between cavities. Yet a serious paradox
appears: how can the small geometrical differences modify the bacterial interaction to
yield opposite behaviours? Two criteria appear central: the gap size and the topology
of the lattice (i.e. the number of adjacent votices). Here, I present some clues to un-
derstand bacterial self-organisation, focusing in particular on the interaction between
bacteria in the bulk or at the PDMS interface.
94
Figure 4.5: Self-organisation in square lattices. (a-d) Four lattice states, for increasing
gap sizes. Vortices are colour-coded, purple for CW and green for CCW. (a) Ran-
dom state. (b) Antiferromagnetic state. (c) Ferromagnetic state. (d) Turbulent state.
Brighter vortices indicate smaller spin magnitude. (e) Spin-spin correlation C and (f)
average absolute spin S depending on the gap size. Each point is averaged over at least
seven movies. Bars: standard error.
95
Figure 4.6: Model for the vortex interaction. (a) Bacteria in the bulk move in the same
direction on either side of the gap, resulting in opposite spins. (b) Bacteria swim along
the PDMS surface, leading to spins of same sign.
Bulk and Interface Bacteria
The antiferromagnetic state seems to be driven by bacteria in the bulk of each cavity,
interacting through long-range fluid flows across the gap, forcing bacteria close to the
gap to move in the same direction, resulting in opposite spins (figure 4.6a). One would
expect this mechanism to become stronger for wider gaps. Instead, the antiferromag-
netic state in square lattices is only dominant in small gap chambers.
Au contraire the ferromagnetic state is probably driven by bacteria circulating
around pillars (figure 4.6b). This interaction also weakens for small gaps, as bacteria
from neighbouring pillars have to cross the gap in opposite directions, thus resulting in
a strong shear along the gap width.
The resulting lattice ordering can be understood as the competition between bac-
teria in the bulk and at interfaces. The former dominate at small gaps and the latter
at larger ones, as shown in the transition from an antiferromagnetic to a ferromagnetic
state in square lattices.
Likewise these interactions could also be favoured depending on the lattice topol-
ogy: for example small pillars in triangular lattices may promote the circulation of
bacteria at the PDMS interface. Furthermore, the more neighbours, the more pillars
around which bacteria can swim, possibly explaining why antiferromagnetic state was
only observed in lines of cavities and ferromagnetic state, in triangular lattices.
Chamber Geometry
I show that the vortex state can be controlled in square lattices by varying the gap
size. Obviously other parameters can affect the bacterial circulation. I tried numerous
96
shapes during preliminary experiments and here report some of their effects:
Gap thickness This is the distance between cavity edges i.e. the difference of dis-
tance between centres and their diameter. Experiments presented here have a
gap thickness around 10µm. I also tried chambers with a gap thickness ∼ 5µm
and only observed the ferromagnetic state. This effect can be understood from
the bacteria swimming around the pillars: when the gap thickness increases, so
does the total shear in the gap region, weakening the ferromagnetic interaction.
Chamber height I tried different microchamber heights, from 15 to 30µm. Higher
chambers form more stable vortices (larger S but also more persistent, see sub-
section on ‘Time’) and lattices with larger spin-spin correlation |C|. However
higher chambers also come with a strong downside: they are likely to have a
preferred vortex direction (CCW in movies). This is likely due to the asymmetry
of the chamber: glass at the bottom surface and PDMS at the top. The dense
suspension quickly consumes the oxygen diluted in the medium and PDMS acts
as a reservoir, creating a gradient of oxygen in the chamber. Thus bacteria swim
faster at the top. Moreover, single bacteria exhibit circular trajectories when
swimming close to surfaces (CW). The combined effect can then result in a pre-
ferred vortex direction, which itself produces artificial ferromagnetic states. In
all experiments presented here, I made sure that vortices had the same probabil-
ity to spin CW or CCW.
Cavity diameter I set the diameter to 50µm in experiments, in order to have stable
enough vortices (the critical diameter is w∗ = 70µm, as found in flattened drops).
Changing the diameter could potentially affect the stability of the vortices as well
as the interaction strength.
Suspension density I have only performed experiments with very dense suspensions.
This is actually a limitation of my experimental protocol, as I do not control
the density perfectly. The suspension density could vary by a factor of two,
potentially resulting in either semi-diluted chambers (showing inhomogeneous
density) or jammed cavities. I would not record any movie in such cases. When,
occasionally, some cavities were not fully filled, cells do not accumulate at the
97
Figure 4.7: Correlation time of the vortex spins depending on the gap size. The
correlation time is computed over each movie and then averaged over at least four
movies. Bars: standard error.
periphery as in flattened drops, but rather form a homogeneous suspensions with
little ordering.
Time
In the entire analysis, I have neglected the effect of time. I assume that the system
reaches its steady state in less than four minutes, and then average the measurements
over the 10 seconds of a movie. I measured the spin correlation over time for each
movie,
Ct(dt) =
∑
v∈V
(σ(v, t).σ(v, t + dt))
∑
v∈V
(12 (σ(v, t)
2 +σ(v, t + dt)2)
. (4.4)
I then fitted Ct with a decaying exponential to obtain the typical correlation time. Fig-
ure 4.7 shows the variation of the correlation time with the gap size in square lattices.
The persistence time Ct decreases with the gap size, from more than a minute with
gaps ≈ 2µm to a couple of seconds in the largest chambers.
I took ten second long movies, four and eight minutes after injection. The correla-
tion time shows that the suspension has most probably reached its steady state at this
stage. Moreover, for the smallest gaps g < 8µm, the time correlation indicates that
spins do not vary significantly over a single movie.
98
Large and Small Correlations
Surprisingly, the correlation in lines and triangular lattices is quite strong compared
with that in square lattices. I have managed to get larger |C| in square lattices but the
results would not be reproducible (note that I have not selected any of the movies in
the data set, they are all movies I have taken, with this particular chamber shape and
height).
However, the small C in square lattices can be understood from the competition
between the antiferromagnetic and the ferromagnetic interactions. While one of them
strongly dominates in linear and triangular lattices, they are almost balanced in square
lattices, resulting in the smaller |C|.
4.3 Modelling the Square Lattice as an Ising System
4.3.1 Model
Union-Jack lattice
To describe and gain insight into the self-organisation in lattices, I collaborated with
Francis Woodhouse and Jörn Dunkel, former PhD student and post-doc in the group,
respectively. We propose an Ising model in which each vortex is solely represented
by its spin. We only consider the case of square lattices as we have significantly more
experimental data and since this arrangement offers study of more varied behaviours.
We start by assuming that fundamental spin-spin interactions are negative: adjacent
vortices force each other to circulate in opposite directions. This idea contradicts with
the observed ferromagnetic state. This state is most probably due to bacteria swimming
around pillars, and thus, to overcome this contradiction, we also model the circulation
around pillar with a similar continuous spin.
We then obtain a decorated lattice, called centred-square or Union-Jack lattice [150,
151, 152]: each vortex interacts with its four adjacent pillars as well as four adjacent
vortices (figure 4.8). Moreover, we start by assuming that each spin lies in a double
well potential. As a basic principle, we expect that if vortex-vortex interactions domi-
nate, the lattice will form an antiferromagnetic state, while, if vortex-pillar interactions
dominate, the ferromagnetic state will be favoured.
99
Figure 4.8: Decorated lattice used to model square lattices. Pillars (P) are placed at the
centre of each four adjacent vortices (V). Vortex-vortex (solid lines) and vortex-pillar
interactions (dotted lines) are included while pillar-pillar interactions are neglected.
I first present a mathematical description and the system Hamiltonian of this Ising
model, before applying it to the study of experimental results.
Francis Woodhouse, Jörn Dunkel and I equally contributed to this section. Francis
Woodhouse and Jörn Dunkel came up with the idea of an Ising model in a decorated
lattice, we developed the model together, I performed most of the direct parameter fit-
ting and Francis Woodhouse is currently working on the complete Hamiltonian fitting
and simulations.
Lattice Hamiltonian
We consider the statistics of vortex and pillar spins as measured experimentally. We
call σ = {σv,v ∈ V} ∪ {σp, p ∈ P}, the set of spins at a given time (equations 4.1 and
4.2). We describe the lattice by its Hamiltonian H which is related to the probability p
to find spins in the configuration σ by
p(σ) =
1
Z
exp(−H(σ)) , (4.5)
where Z =
∫
σ
H(σ)dσ is the partition function. The Hamiltonian depends on vortex-
vortex and vortex-pillar interactions together with single vortex and single pillar po-
100
tentials
H(σ) = −JV
∑
{v,v′}∈EV
σvσv′ − JP
∑
{v,p}∈EP
σvσp +
∑
v∈V
VV(σv) +
∑
p∈P
VP(σp), (4.6)
where JV and JP denote the vortex-vortex and vortex-lattice interaction strengths re-
spectively and VV and VP are the single vortex and single pillar potentials. We choose
to use a double-well potential,
VV(σ) =
aV
2
σ2 +
bV
4
σ4, (4.7)
VP(σ) =
aP
2
σ2 +
bP
4
σ4, (4.8)
where aV , bV , aP and bP are constants which should depend on the lattice geometry
and the suspension behaviour.
Our goal is to estimate the Hamiltonian parameters from experimental statistics,
looking in particular at the role of gap size.
4.3.2 Direct Analysis of the Experiments
Here, I present a method to estimate, directly from the experimental measurement
statistics, some effective parameters of the lattice Hamiltonian. The idea is to measure
the probability to find a vortex v at a given spin σv, depending on the average spin of
its adjacent vortices. We first simplify the expression of the Hamiltonian by renormal-
isation of the lattice, and define an effective potential associated with single vortices,
from which we can estimate some parameters.
Pillar spin behaviour
Pillar spins {σp, p ∈ P} contribute to the lattice Hamiltonian in their single pillar poten-
tial,
∑
p∈P
VP(σp) , (4.9)
101
Figure 4.9: Distribution of pillar spins in two movies (a-b). Colour-coded 2D his-
togram of the pillar spins σ and averaged neighbour vortex spins ∂Vσ (equation 4.11).
Red: highest density, Purple: lowest density. In both cases, spins are clustered around
σ = −∂Vσ.
and vortex-pillar interactions,
−JP
∑
{v,p}∈EP
σvσp . (4.10)
The spin of a given pillar then only depends on that of the four adjacent vortices. We
thus study the distribution of pillar spins depending on the averaged neighbour vortex
spin
∂Vσp =
1
4
∑
{v,p}∈EP
σv. (4.11)
Figure 4.9 shows two examples of 2D histograms obtained from two different movies.
While spins appear more clustered in (a) than in (b), they both show the density to be
largest around σ = −∂Vσ: each pillar spin is the opposite of the average spin of its four
adjacent vortices.
Renormalised lattice
The system Hamiltonian H is too complex to directly fit from the distribution of vortex
and pillar spins. We thus use the clustering of pillar spins around −∂Vσ to renormalise
H and obtain a form that can be decomposed into effective single vortex potentials.
We first replace all instances of {σp, p ∈ P}, by −∂Vσp. We then assume that since
the value of the pillar spins is fixed, their potential VP(σk) is constant. We obtained
102
the simplified version of the Hamiltonian of the system
H(σ) = −JV
∑
{v,v′}∈EV
σvσv′ +
JP
4
∑
{v,p}∈EP
σv
∑
{v′,p}∈EP
σv′ +
∑
v∈V
VV(σv)
= −
(
JV −
1
2
JP
) ∑
{v,v′}∈EV
σvσv′ +
1
4
JP
∑
{v,v′′}∈E2V
σvσv′′
+
∑
v∈V
[(
1
2
aV + JP
)
σ2v +
1
4
bVσ
4
v
]
= −JeffV
∑
{v,v′}∈EV
σvσv′ +
1
4
JP
∑
{v,v′′}∈E2V
σvσv′′ +
∑
v∈V
[
1
2
aeffV σ
2
v +
1
4
bVσ
4
v
]
,
(4.12)
where E2V is the set of diagonally adjacent vortices. We further neglect interactions
between diagonal pairs of vortices, as this effect is of lesser strength and bacteria are
unlikely to keep their orientation from one side of a pillar to the other (going through
two gaps). We end up with a classical expression for a square lattice, except that JeffV =
JV− 12 JP reflects the effective interactions between vortices, through bulk and boundary
bacteria. Since JV , JP < 0, the sign of JeffV indicates which interaction dominates.
We can now express the lattice Hamiltonian as the sum of effective vortex potentials
H(σ) =
∑
v∈V
Veff(σv,∂Vσv), (4.13)
where
Veff(σv,∂Vσv) = −2JeffV σv∂
Vσv +
(
1
2
aeffV σ
2
v +
1
4
bVσ
4
v
)
. (4.14)
Estimation of the effective single vortex potential
This simplified version of H allows us to compute effective parameters from the spin
distribution σ.
First of all, we need to experimentally estimate Veff . We assume that the spin
distribution follows a Boltzmann distribution,
p(σv|∂Vσv) =
1
Zv(∂Vσv)
exp
(
−Veff(σv,∂Vσv)
)
, (4.15)
103
where p(σv|∂Vσv) =
#(σv,∂Vσv)
#(∂Vσv)
, and Zv =
∫
σv′
V(σv′ ,∂Vσv)dσv′ .
Equivalently,
Veff(σv,∂Vσv) = − ln(p(σv|∂Vσv))− ln(Zv(∂Vσv)). (4.16)
So far, we can measure p, except for spins with low probability which we will not
take into account in the parameter fitting and Zv is unknown but only depends on ad-
jacent spins ∂Vσv. In particular, Veff(0,∂Vσv) = 0 (equation 4.14) and we can then get
ln
(
Zv(∂Vσv)
)
= − ln
(
p(0|∂Vσv)
)
. We will actually use a slightly different approach to
estimate Zv, as described below. Moreover, Veff is symmetric in σv and antisymmetric
in ∂Vσv, we use these properties to estimate JeffV , a
eff
V and bV .
Vortex-vortex interaction
We first study the antisymmetric part of the effective potential. On the experimental
side, Veffanti can be directly estimated from the probability distribution
Veffanti =
1
2
[
Veff(σv,∂Vσv)−Veff(−σv,∂Vσv)
]
(4.17)
=
1
2
[
− ln(p(σv|∂Vσv)) + ln(p(−σv|∂Vσv))
]
. (4.18)
On the theoretical side, Veffanti only depends on the effective vortex vortex interaction
Veffanti(v) = −2J
eff
V σv∂
Vσv. (4.19)
Figure 4.10 shows the estimate of Veffanti, depending on the vortex spin (x-axis) and its
neighbours spins (colour-coded). For a given mean adjacent spin ∂Vσv (colour-coded
in the figure), data point are roughly aligned, with slope −2JeffV ∂
Vσv.
This construction allows us to subtract the partition function Zv and we can directly
estimate the effective interaction (figure 4.10)
JeffV =
ln(p(σv|∂Vσv))− ln(p(−σv|∂Vσv))
2σv∂Vσv
. (4.20)
104
Figure 4.10: The antisymmetric part of the effective Hamiltonian, reveals the vortex-
vortex interaction. (a-c) Veffanti as a function of the vortex spin σ. Colour coding reflects
the average neighbour spin ∂Vσ, the brighter the larger. Each plot shows a different
interaction mode: (a) weak interaction, (b) ferromagnetic and (c) antiferromagnetic.
105
Single Vortex Potential
The symmetric part of the effective potential allows us to neglect the interactions be-
tween neighbours, where
Veffsym =
1
2
[
Veff(σv,∂Vσv) +Veff(−σv,∂Vσv)
]
(4.21)
=
1
2
[
− ln(p(σv|∂Vσv))− ln(p(−σv|∂Vσv))
]
− ln(Zv(∂Vσv)) (4.22)
=
1
2
aeffV σ
2
v +
1
4
bVσ
4
v . (4.23)
As explained above, one can estimate Zv(∂Vσv) = p(0|∂Vσv)). Unfortunately, there are
often few data points around σv = 0, and for this reason, p(0|∂Vσv)) can be difficult
to estimate. Yet, Zv is the only term that depends on ∂Vσi and we therefore adjust the
estimated value of Veffsym such that Zv(∂
Vσv) = c, where c is a constant. We then fit aeffV ,
bV and c, such that (figure 4.11)
Veffsym =
1
2
aeffV σ
2
v +
1
4
bVσ
4
v + c (4.24)
=
1
2
[
− ln(p(σv|∂Vσv))− ln(p(−σv|∂Vσv))
]
. (4.25)
Additional information
We aim to measure aeffV , bV and J
eff
V for each individual gap width. To improve the
accuracy, we add up all statistics for chambers of similar gap size (each data point
corresponds to at least six movies).
Moreover we take advantage of the symmetry of the system Hamiltonian H by
the transformation: {σ,∂Vσ} → {−σ,−∂Vσ}. To obtain more data points and accurate
estimates, we transform all data set such that ∂Vσ > 0.
Finally, all fittings have been done using the function fit in Matlab.
106
Figure 4.11: The symmetric part of the effective Hamiltonian reveals the single vortex
potential. (a-c) Veffsym depending on the vortex spin σ. Colour coding reflects the av-
erage neighbour spin ∂Vσ, the brighter the larger. between panels (a) to (c), the Veffsym
transitions from a potential with a single minimum at σ = 0 to a double-well profile,
with σmin→±1. Lines: fitted potential.
107
4.3.3 Results
Four states
We first plotted the 2D distribution of spin σ to mean adjacent spin ∂Vσ for different
movies (figure 4.12). We found again the four behaviours corresponding to the square
lattice states identified previously. Figure 4.13 shows, for each state, an example of the
estimated effective single vortex potential Veff (points) and fitting (surface) following
the method described above.
Random The histogram shows two horizontal bands, revealing that spins are clus-
tered around σ = ±1 but are independent of their neighbour spins ∂Vσ. Fitting
of the energy shows that spins sit in a deep double-well potential, stabilising
the suspension into a strong vortex. Moreover the interaction strength JeffV is
negligible.
Antiferromagnetic The two bands are still present but broader, meaning vortex states
are less organised. Moreover, the distribution appears asymmetric: σ is more
likely to be of opposite sign of ∂Vσ. This reflects on the energy plot in a smaller
energy barrier in the double well. The interaction Jeffv becomes negative, reveal-
ing that vortex-vortex interaction dominates: |Jv| > 12 |Jp|.
Ferromagnetic Spin magnitude still decreases and this time is positively correlated
with ∂Vσ. The energy profile shows an almost flat single vortex potential Veffs , in
combination with a positive interaction Jeffv , causing the spins to share the same
direction.
Turbulent σ and ∂Vσ are clustered around 0 without much correlation. This results
in a single well potential with weak or no interactions.
Dependence on the gap size
We then quantified the variation of the effective parameters depending on the gap size.
To study the effective single vortex potential
Veffsym =
aeffV
2
σ2 +
bV
4
σ4, (4.26)
108
Figure 4.12: Distribution of vortex spins in four movies (a-d). Colour-coded 2D
histogram of the vortex spins σ and mean vortices neighbour spins ∂Vσ. Red: highest
density, Purple: lowest density. Each histogram indicates a different lattice state. (a)
Random: σ is clustered around ±1 and is independent of ∂Vσ. (b) Antiferromagnetic:
σ is clustered around +0.8 when ∂Vσ < 0 and −0.8 when ∂Vσ > 0. (c) Ferromagnetic:
σ is clustered around +0.6 when ∂Vσ > 0 and −0.6 when ∂Vσ < 0. (d) Disordered:
broad distribution of σ and ∂Vσ around 0 without correlation.
109
Figure 4.13: Effective vortex potential Veff in four movies, depending on the vor-
tex spin σ and the mean neighbour spin ∂Vσ. Points: estimates from experiments.
Surface: overall fit. Each plot represent a different lattice state. (a) Random: deep
double-well potential with no interaction. (b) Antiferromagnetic: double-well poten-
tial with negative interaction JeffV . (c) Ferromagnetism: weak double-well potential
with positive interaction JeffV . (d) Disordered: single-well potential with, here, a weak
positive interaction.
110
Figure 4.14: Parameter estimates for the effective potential Veff, depending on the gap
size. (a) Parameters of the single vortex potential: σmin, spin of lowest energy, and
Vmin, the energy barrier. (b) Effective interaction strength JeffV . Each point is computed
from the spin statistics obtained over at least six different movies.
rather than aeffV and bV , I considered the spin of lowest energy σmin and the energy
barrier
Vmin = Veffsym(0)−V
eff
sym(σmin). (4.27)
The interaction JeffV follows the behaviour of the spin-spin correlation C
(figure 4.14b). For small gaps, they are both negative, indicating that the bulk in-
teractions Jv between bacteria dominates. When the gap increases, JeffV changes sign
revealing that the interaction through the pillars Jp leads the lattice behaviour, in agree-
ment with observations of bacteria swimming around pillars in ferromagnetic cham-
bers. For the largest gaps, the interaction fades. This last result could not be predicted
from earlier measurements. Indeed, the turbulent state could simply be due to the lack
of stability of vortices. Our analysis also shows that interactions JV and JP either can-
cel each other or simply die out. Finally, I only obtained a few movies at very small
gap. Although they were in a random state, they do not appear in figure 4.14.
We then found that the spin of lowest energy σmin decreases with the gap size,
from ≈ 0.8 to ≈ 0.4. This shows that the double-well does not become a single-well
at large gaps and that even in this case, pillar still have a weak stabilising effect. Yet
the energy barrier Vmin quickly drops from 1.1 for the smallest gap to less than 0.2 for
gaps larger than 8µm. Even though spins still sit in a double well, the potential profile
is effectively flat. These results are in agreement with measurements of the absolute
spin S (figure 4.5f).
111
4.4 Conclusion
These measurements only give a limited understanding on the self-organisation of bac-
terial suspensions in lattices. Francis Woodhouse is currently working on fitting the full
Hamiltonian using a Langevin approach. This should allow us to have a full estimate
of parameters. Of particular interest are the interactions between vortices and pillars.
While I only compute JeffV = JV −
1
2 JP, Woodhouse’s method should give us access to
JV and JP independently.
We have so far neglected the time component of the lattice self-organisation. In-
deed, the probability of finding a spin state was defined as p = 1Z exp(−H(σ)). A more
accurate description should include an effective temperature, H = H¯Teff . Note that all
parameters of the system Hamiltonian are inversely proportional to the temperature:
JV =
J¯V
Teff
, aV =
a¯V
Teff
, etc.
Estimating Teff appears to be challenging in an active suspension of bacteria. For
example, the bacterial swimming speed increases with the oxygen concentration, re-
sulting in faster variations of the lattice. Yet we do not expect the system steady state,
and thus Teff , to be affected by faster swimming [93, 96]. Measuring Teff then requires
to renormalise spins as well as time. I suspect the effective temperature to vary with
the chamber geometry and the suspension state (e.g. bacterial concentration).
Nonetheless, experiments in lattice of vortices revealed unexpected complex bac-
terial organisation, in the form of antiferromagnetic and ferromagnetic arrangements.
Interactions appear to be driven by bacteria in the bulk and at the PDMS interface,
showing that confinement not only stabilises vortices but also control long-range or-
der.
Further experiments are required to understand this system. The effect of the cham-
bers height, cavity diameter and gap length on the stability of the circulation and on
the lattice ordering would be of particular interest. Further studies could also focus on
the dynamics of ferromagnetic and antiferromagnetic domain formation.
Of particular interest will be the analyses of hydrodynamic interactions and the
exact role of bacteria swimming along interfaces. Simulations, as those presented in
chapter 3, are probably the method of choice to study the suspension organisation at
the microscopic level.
Finally we have presented an Ising model to describe the lattice order. This ap-
112
proach has been successful in measuring effective interactions between adjacent cav-
ities and in estimating vortex potentials, and we expect to obtain more detailed and
accurate results in the near future. This Ising model is consistent with the current trend
in active matter to use statistical physics tools in order to describe self-propelled parti-
cle suspensions [153]. Examples include Boltzmann equations [154] and the definition
of an active pressure [155]. This Ising model could be useful to predict lattice ordering
in geometries that are currently difficult to realise experimentally. One could look at
3-dimensional square lattices with either spherical or flattened cavities for example.
113
114
Chapter 5
Conclusion
The number of publications on active matter has exploded in the past decade. The
field has evolved from simple experiments in unconfined chambers [135, 156] to well-
controlled systems in complex environments [32, 93, 127]. Likewise, theories have
matured from the minimal approach taken for the Vicsek simulations [99] to models
able to reproduce interactions accurately and intricate suspension behaviours [96, 154].
A guiding thread in this thesis is the role played by interfaces and confinement
in the self-organisation of dense bacterial suspensions. Solid and liquid interfaces are
well-known to affect the swimming behaviour of single cells, generating trapping [113,
114], modifying the bacteria-driven fluid flows [109] and cell trajectories [119, 120].
Yet dense suspensions under confinement have rarely been considered.
This thesis presented three different confining topologies: flattened drops, race-
tracks and lattices of cavities. Beyond merely listing new phenomena, we have tried
to explain the suspension behaviour from the microscopic level and to obtain a better
understanding of bacterial motion.
Flattened Drops
I have first proposed a novel experimental setup in which a concentrated suspension
is mixed with oil to form an emulsion and then squeezed between two coverslips.
In chambers that are 25µm in height and 30 to 70µm in diameter, the suspension
forms a single stable vortex state. Unlike unconfined suspensions which exhibit quasi-
turbulent motion, I found the bacterial flow in flattened drops to persist over several
minutes, until oxygen depletion. Furthermore, while the cell orientation and direction
115
of motion are uncorrelated in unconfined chambers [31], bacteria in drops align in a
helical pattern. Measurements of the mean cell orientation show the pattern to depend
on the drop diameter and thus on the curvature at the interface with the surrounding
oil.
But the most striking feature was bacteria at the edge of the drop moving against
the bulk circulation. This phenomena was observed in all drops, even with smaller or
larger diameters, and consisted of a single layer of cells at the very edge of the drop.
This observation strongly contrasts with the classical picture of collective motion, in
which all swimmers locally share a common direction of motion. Indeed, no previous
experimental or theoretical work had reported or predicted such a behaviour.
Hydrodynamic Interactions
To unravel the mystery of double circulation, I developed a new experimental tech-
nique which simultaneously measures the swimming and motion directions of bacteria
whose flagella and body have been fluorescently labelled. Measurements reveal that
all cells are aligned in the same direction, but while bacteria at the edge are moving
forwards, those in the bulk are moving against their swimming direction.
To fully understand this self-organisation, Enkeleida Lushi developed a simula-
tion method which includes steric and hydrodynamic interactions, along with realis-
tic boundary conditions. Simulations are able to reproduce the bacterial motion and
showed that cells close to the edge (and their interfacial images) generate a net fluid
flow in the drop, against the common swimming direction. The motion of swimmers
in the bulk is then dominated by advection, explaining the backward motion observed
with fluorescent bacteria.
The role of fluid flows in the self-organisation of bacterial suspensions has been the
subject of much debate. While some researchers support the idea that steric interac-
tions drive the quasi-turbulent motion in unconfined chambers [90], others have shown
the suspension to be unstable under hydrodynamic interactions only [89]. Moreover,
fluid flows have been shown to disassemble bacterial swarms [91]. Additionally, dense
suspensions can reach velocities an order of magnitude faster than single cells [93].
This acceleration has been attributed to short-range hydrodynamic effects, that in-
creases the swimming efficiency [7], but could as well result from advection by long-
range fluid flows. Experiments that show bacterial orientation and direction of motion
116
to be uncorrelated in unconfined chambers [31] and anticorrelated in drops, suggest
the role of hydrodynamic interactions to be more important than often represented.
Simulations Improvements
On a more technical side, simulations presented in this thesis do not propose a new
method in that steric repulsion, hydrodynamic interactions, and interfaces have already
been modelled, but they are the first to combine all these ingredients to obtain a more
realistic representation of the suspension. Improvements of the model can take several
directions. First, simulations were limited to a 2D domain. Expanding them into 3D
requires significantly improving the computation efficiency and probably introducing
some new approximations.
Moreover, we have only presented simple geometries, circular and linear, which are
easy to model with a system of images. Including more complex boundary geometries,
as in a lattice of vortices, would necessitate developing other computation methods.
Furthermore, hydrodynamic interactions are computed from the sum of fluid flows
generated by isolated swimmers. Yet bacteria take up to 20% of the volume fraction
and can significantly affect the flow pattern generated by neighbours. Considering
the presence of the suspension, its porosity or effective viscosity [157] will certainly
improve the accuracy of estimated fluid flows.
Finally, interactions at the microscopic level between flagella of different swim-
mers could modify the swimming mechanism. In that case, the fluid flows generated
in dense suspensions could differ significantly from those predicted by the idealised
stresslet model. Knowing which details are relevant to model bacterial suspensions
will certainly improve the efficiency of simulations and our understanding of active
matter.
Racetracks
The second experimental chambers I considered were racetracks. Dimensions are com-
parable to those of flattened drops (height around 20µm, width on the order of tens of
microns), except for the length which could be more than several millimetres. The
suspension is thus only confined in two directions, rather than three, as in the case of
drops. Yet the effect of interfaces remains comparable.
117
If the channel width is smaller than 70µm, the suspension forms a persistent wavy
stream, comparable to jets observed in unconfined bacterial suspensions. Swimmers
at the PDMS and glass interfaces move in opposite direction to the bulk circulation.
Moreover fluorescent bacteria and simulations show that cells inside the channel have
a biased orientation against the fluid flow. Racetrack experiments could then be solely
thought of as a confirmation of the mechanism driving self-organisation in drops.
Nevertheless, these experiments brought at least two significant results. The first
regards the correlation length. In order to quantify the self-organisation of active mat-
ter, researchers usually compute the spatial correlation of the flow. This function often
decreases monotonically with distance. The decay then gives a typical length scale
sometimes linked to the swirl radius. Confinement in racetracks shows that there are
actually two length scales, that of the vortices and a persistence length. Compari-
son with other papers shows that the two length scales can vary independently. For
example, the persistence length increases with the bacterial concentration while the
swirl size remains constant from dilute to the densest suspensions [32]. Further studies
which would independently measure the vortex size and persistence length, are neces-
sary to fully understand these relations. These new results could then be compared to
theoretical predictions to confirm the role of steric and hydrodynamic interactions in
bacterial self-organisation.
The second important finding refers to the effect of interfaces. Not only do they
stabilise the suspension motion into a vortex (in flattened drops) or a stream (in race-
tracks), but the flow profiles show that they also contribute significantly to the bulk
circulation. The swimming of a single bacterium in the bulk generates a stresslet dis-
turbance, but no net flow. However, when placed next to a surface, the no-slip or
free-sheer boundary condition yields a net fluid flow against the swimming direction.
Boundary currents thus play a central role in the bulk circulation, setting the direction
of bacteria and, on a larger scale, of vortices.
Lattices of vortices
The last experimental chambers also rely on boundary currents, yielding unexpected
circulation ordering. Bacteria were injected into a network of cavities, inside each of
which they could form a more or less stable vortex, interacting with adjacent cavities
through connections (gaps). Depending on the number of connections and on their
118
shape, interactions can result in an antiferromagnetic or ferromagnetic array of vor-
tices. These two states appear to be driven by distinct circulation patterns of bacteria at
PDMS surfaces. In the first case, bacteria predominantly stay in one cavity, long-range
interactions dominate, so that swimmers on each side of the gap move in the same
direction, yielding opposite spins. In the second, bacteria can cross the connection to
move continuously around pillars, generating spins of same sign.
Interactions with surfaces have a dramatic effect on other condensed matter sys-
tems. Surface texture can for example set the orientation of liquid crystals at the
boundary, affecting the organisation in the bulk [158]. Likewise, quantum Hall effect
is governed by currents of electrons at edges [159]. In confined bacterial suspensions,
motion at surfaces appears to dictate the overall circulation. The suspension behaviour
could then be predicted from edge currents, integrated for instance into a Green’s func-
tion.
We finally presented an Ising model that reproduces the four states observed in
square lattices. By renormalisation of the grid, we have managed to fit simplified
energy profiles and measure effective parameters. Further analysis should provide a
complete estimate of the Hamiltonian of the system (interaction strength, single vor-
tex potential, temperature, etc) and shed light on the self-organisation mechanism in
lattices of vortices.
Future work
Experiments under confinement have shown that many active behaviours are still to be
discovered. I wish for this approach to be considered by the community. I am already
aware of three groups that started looking at bacteria in drops after we published results
on flattened drops [1]. These new works consider different drops shapes (larger or
spherical) and other bacterial species (E. coli, Pseudomonas aeruginosa) [139]. Owing
to the variety between species in shapes (e.g. aspect ratio, length) and swimming
behaviours (e.g. number of flagella, run-and-tumble), the suspension could exhibit
surprising behaviours unobserved with B.subtilis.
Likewise, after I started experiments in racetracks, Bricard et al. [127] presented a
similar setup with rolling colloids instead of bacteria. These colloids are also subject to
steric and hydrodynamic forces but present a distinct collective motion. Using different
active systems allows various interactions to be examined. An other approach is to mix
119
active or passive material such as bacteria in liquid crystals [160] and bristle-bots in
granules [41].
A final perspective will be to reproduce systems akin to natural environments. One
could then focus on two aspects. First, natural environments are populated with numer-
ous microorganisms species. Chemical and physical interactions between them could
result in particular collective behaviour such as attraction or repulsion patterns, as in
the case of symbiotic or competing species [161]. Moreover, in clonal populations
of bacteria, individuals can present different phenotypes, swimming or vegetative for
example [64]. The heterogeneity of the microbial population could then result in com-
plex collective motion and phase transitions [91]. The second aspect will be to make
realistic boundary conditions and topologies. Natural environments could be presented
as a porous structure with numerous connected cavities and channels. Preliminary ex-
periments in 8-shaped racetracks show that bacterial trajectories are determined by the
precise chamber geometry. Baroque arrangements of cavities and channels could, thus,
modulate and control the collective motion of microorganisms.
120
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