# Frustrated “run and tumble” of swimming E-coli bacteria in nematic liquid
crystals

Martyna Goral1,2, Eric Clément1, Thierry Darnige1, Teresa Lopez-Leon2, Anke
Lindner1 1Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR
7636, CNRS, ESPCI Paris-PSL, 75005 Paris, France
2Laboratoire Gulliver, UMR 7083, CNRS, ESPCI Paris-PSL, 75005 Paris, France

###### Abstract

In many situations bacteria move in complex environments, as for example in
soils, oceans or the human gut-track microbiome. In these natural
environments, carrier fluids such as mucus or reproductive fluids show complex
structure associated with non-Newtonian rheology. Many fundamental questions
concerning the the ability to navigate in such environments remain unsolved
due to the inherent complexity of the natural surroundings. Recently, the
interaction of swimming bacteria with nematic liquid crystals has attracted
lot of attention. In these structured fluids, the kinetics of bacterial motion
is constrained by the orientational molecular order of the liquid crystal (or
director field) and novel spatio-temporal patterns arise from this
orientational constraint, as well as from the interactions with topological
defects. A question unaddressed so far is how bacteria are able to change
swimming direction in such an environment. In this work, we study the swimming
mechanism of a single bacterium, E. coli, constrained to move along the
director field of a lyotropic chromonic liquid crystal (LCLC) that is confined
to a planar cell. In such an environment, the spontaneous “run and tumble”
motion of the bacterium gets frustrated: the elasticity of the liquid crystal
prevents flagella from unbundling. Interestingly, in order to change
direction, bacteria execute a reversal motion along the director field, driven
by the relocation of a single flagellum to the other side of the bacterial
body, coined as a ”frustrated tumble”. We present a detailed experimental
characterization of this phenomenon, exploiting exceptional spatial and
temporal resolution of bacteria and flagella dynamics during swimming,
obtained using a two color Lagrangian tracking technique. We suggest a
possible mechanism behind the frustrated run and tumble motion, accounting for
these observations.

††preprint: APS/123-QED

##  I Introduction

Microorganisms, such as bacteria, are not only of great importance in bio-
medicine or food industry, but they also constitute very interesting active
fluids [1]. Exotic phenomena emerge from the interplay between bacterial
dynamics and the properties of the fluid in which they swim; for example, the
suspension of bacteria into a Newtonian fluid leads to surprising non-
Newtonian properties and in particular a decrease of the effective viscosity
below the viscosity of the suspending fluid [2, 3], while the fluid in turn
can induce complex swimming behaviors in the bacterial suspension, as enhanced
swimming velocities in colloidal suspensions [4] or straightening of bacteria
trajectories in polymer solutions [5].

In Newtonian aqueous media, model peritrichous bacteria, as E. coli, follow
trajectories that resemble a random walk with no specific spatial orientation.
This random walk results from the so called ”run and tumble” dynamics
described in a seminal work by Berg et al. [6, 7].

Ideally, it consists of a run phase, where the bacterium swims in straight
line, that is periodically interrupted by a tumble phase, where the swimming
direction changes. In the run phase, the bacterium is propelled by a bundle of
flagella that turns counter-clockwise (CCW). The tumble is triggered when at
least one of the flagella starts turning clockwise (CW), provoking the opening
of the bundle.

The reversal in rotation direction and the associated torques acting on the
flagellum leaving the bundle have been shown to lead to polymorphic
transformations, where the flagella geometry as well as the handedness can
change [8, 9, 10]. A typical polymorphic transformation concerns the change
from a ”normal” left-handed flagellum as those comprised in the bundle to a
”curly” form of opposite handedness as in the single CW rotating flagellum. In
order to keep the total torque equal to zero, the body responds to the
consequent opening of the bundle by changing its orientation. Eventually, the
flagellum rotation switches back to CCW as well as to a ”normal”
configuration, leading to the reorganisation of the bundle and to a new run
phase. The frequency of tumbles is controlled by a chemotaxis signaling
network, enabling navigation toward or away from certain regions in the medium
[11, 12].

The swimming pattern of peritrichous bacteria have been intensively studied in
Newtonian aqueous media. Yet bacterial natural environments are usually
complex and non-Newtonian; examples range from crystalline mucus to structured
soils [13, 14, 15, 16, 17, 18, 19, 20, 21]. Many intriguing observations have
been made in complex fluids as polymer solutions or colloidal suspensions and
include enhanced bacterial mobility [5, 4], straightening of bacteria
trajectories [5, 4], suppression of tumbles [5] or even the appearance of new
swimming patterns as slow random walks [22]. They are all attributed to the
complex interactions of the motile bacteria with the complex fluid, either
through the effective fluid properties or the fluid inhomogeneities on the
scale of bacterial body or flagella [23].

Recently, the investigation of bacteria motility has been extended to fluids
presenting a structure at the molecular or supra-molecular scale, as for
example liquid crystals (LCs) [24, 25, 26, 27, 28, 29, 30]. LCs are
anisotropic viscoelastic materials displaying long range partial orientational
and positional order. Their physico-chemical properties have been well
established over decades of intensive research. Liquid crystals are responsive
to external conditions – including electro-magnetic fields, substrates and
interfaces, temperature and concentration gradients, or pH – what makes them
adaptable and controllable. Besides, they are often used as models for
biological systems, providing conditions close to those of natural bacterial
environments [31].

Recent studies on bacterial motion in LCs have shown that the swimming
patterns resembling an isotropic random walk, originating from ”run and
tumble” dynamics, typically observed in Newtonian aqueous solutions is
suppressed by the elasticity of LCs. Instead, the bacteria are constrained to
follow the direction of molecular alignment of the LC or director field [24,
25, 32, 33]. Intricate collective dynamics result from elastic and
hydrodynamic interactions between bacteria and topological defects, opening
very interesting perspectives for future research. However, it remains unclear
how changes in swimming direction occur in these anisotropic media and how
such changes are linked to the ”run and tumble” mechanism.

In this paper, we investigate the swimming mechanism of E. coli bacteria
dispersed in a lyotropic chromonic liquid crystal (LCLC) that is uniformly
aligned in a planar cell. We observe linear trajectories aligned with the
nematic field, consistently with previous observations. Interestingly, we
observe recurring re-orientation episodes, where the bacteria change their
swimming orientation while maintaining their direction aligned with the
director field. These ”reversals” entail the relocation of at least one
flagellum on the other side of the bacterial body. We show that this
configurational change is in most cases associated with a polymorphic
transformation in the flagellum and that it occurs at a frequency similar to
that of tumbling. We propose a possible mechanism for bacteria locomotion in
the reverse mode. Our results suggest that this novel reversal swimming
pattern emerges from a frustrated ”run and tumble” resulting from the elastic
constrains imposed by the liquid crystal.

##  II Materials and Methods

###  II.1 Bacterial strains and culture

We use a wild type strain of E. coli (ABAD62, derived from K12, see more
details in reference [34] Supplemental Material) that allows us to visualize
the body as well as the bacteria flagella. It has an encoded GFP production
for the bacterial body and special flagella receptors for fluorescent dye and
can thus be used to stain bacteria with two different colors. Bacteria were
first inoculated in a growth culture medium LB with ampicillin antibiotic over
night at 30° C and 250 RPM. The following morning 10% of the bacterial
solution was suspended in TB growth medium for 4-5 hours, until it reached an
optical density of 0.5. The bacteria were centrifuged at 5000 RPM for 6
minutes twice in Berg’s motility buffer (BMB) and rinsed with BMB in between.
At this point Alexa dye (Maléimide Alexa Fluor™ 647 C2 purchased from
ThermoFisher scientific) was added to the bacterial solution and the sample
was put on a rotation plate for 2 to 3 hours. Finally the bacteria were washed
(centrifuged and re-suspended in motility buffer) three times to remove the
excess of dye, re-suspended and ready to use. The final bacterial suspension
has an OD of 15.

###  II.2 Solution preparation

The liquid crystal chosen was disodium 5,5’-[(2-hydroxy-1,3-propanediyl)
bis(oxy)] bis[4-oxy-4H-1- benzopyran-2-carboxylate] (from Sigma Aldrich), also
known as DSCG, because of its compatibility with bacterial life. It is a
lyotropic chromonic liquid crystal, whose properties result from the
spontaneous stacking of disk-like molecules in water. Concentration or
temperature can trigger phase transitions from the isotropic state, to the
nematic, columnar, or crystalline state, as concentration increases or
temperature decreases. Lyotropic chromonic liquid crystals are characterized
by unusual values of their viscoelastic constants. In particular, they exhibit
an anomalously low elastic constant associated to twist deformation, which
sometimes results in spontaneous chiral symmetry breaking [35, 36].

Our liquid crystalline suspensions were prepared by dispersing the DSCG in
bacterial motility buffer. To help the dispersion of DSCG in the motility
buffer, we used a vortex shaker and gentle heating. We established the phase
diagram of DSCG in the motility buffer (see Appendix) in order to find the
sweet spot where the nematic phase appears at conditions of temperature and
viscosity compatible with the swimming of E. coli. According to the phase
diagram, the transition from an isotropic to a fully nematic phase can be
obtained by decreasing temperature. This transition occurs at lower
temperatures for smaller DSCG concentrations. Higher liquid crystal
concentrations lead to higher viscosities where bacteria swimming is slowed
down, whereas smaller temperatures also lead to a reduced bacterial mobility
[37]. A compromise has thus to be found between these two competing effects.
To facilitate the experimental observations we decided to work at room
temperature and the appropriate conditions thus necessitate to choose the
lowest DSCG concentration resulting in a full nematic phase between 21° C and
25° C, corresponding to $\Phi$ =12 wt %. The DSCG viscosity could not be
measured with a simple rheometer since it is anisotropic and time dependent as
it has been shown by Habibi et al. [36]. They used micro-rheology to
characterize this liquid crystal and measured viscosities in the direction
parallel to the LC director. In the other directions the fluid has a complex
and strongly time dependant behavior and they were not able to derive an
associated effective viscosity from their measurements.

For a concentration of $\Phi$ =12 wt%, the viscosity is $\mu$ = 18.5 mPa.s at
room temperature. Note that the viscosity increases very rapidly with
concentration and for $\Phi$ =13.3 wt %, $\mu$ = 63.5 mPa.s. It has been shown
that the viscosity to immobilize flagellated bacteria like E. coli is around
60 mPa.s at room temperature [38] and indeed in our experiments, E. coli
ceased to be motile from $\Phi$ =13 wt % onward, which is in agreement with
the corresponding viscosity.

To prepare a solution of liquid crystals containing bacteria, 5% of the
bacterial solution is added to the liquid crystal. The initial liquid crystal
concentration has to be 12.63 wt% in order to obtain final concentrations of
12 wt%. Bacteria concentrations are keep small enough to be able to observe
one bacteria at a time and to avoid any interaction between bacteria.

For the Newtonian viscous fluid we used Ficoll 400 (F8016 reference),
purchased from Sigma-Aldrich, taking the form of a powder that we dispersed in
motility buffer. The presence of small polymer fragments has been shown to
increase bacteria mobility as it can be metabolized by the latter [22, 23],
preventing a clear link between the viscosity of the surrounding fluid and the
bacteria swimming speed to be established. We thus assured that small polymer
fragments were removed by performing a dialysis in the laboratory on the
Ficoll 400 solution, during a week. Finally, experiments were performed in
dialyzed Ficoll solutions, with measured constant viscosity of
$\mu_{Ficoll400-dialyzed}=18\ $\mathrm{m}\mathrm{P}\mathrm{a}\,\mathrm{s}$$ at
room temperature. The viscosities were measured with an Anton Paar MCR 501
rheometer, with a conical geometry CP60-0.5.

###  II.3 Sample preparation and bacteria tracking

The observation chamber was formed by either two glass slides or a glass slide
and a cover slip. The glass slides were well controlled diamond-scratched
glass surfaces [39] enabling perfect LC alignment. The cover slips on the
other hand were rubbed manually on velvet to create micro-scratches, which
enabled a better image resolution compared to the glass slides. Observation
through the cover slips was possible using a water objective with small
working distance (C-Apochromat 63×/1.2W). The glass surfaces were separated by
adding PS beads (20 $\mu$m ) to the liquid crystal bacteria solution. The
samples were sealed to prevent external flows and evaporation, with either
epoxy glue or nail polish, both bio-compatible. Bacteria were observed with an
inverted microscope (Zeiss-Observer, Z1), Colibri LED illumination, a camera
(Hamamatsu Orca-flash 4.0), equipped with a beam splitter and a 3D tracking
device [40, 34]. Using the 3D tracking we were able to record bacteria
trajectories over long times while having access to recorded images of the
bacterial body and flagella. For data analysis, custom Matlab scripts were
written to detect bacteria reversals.

##  III Results

Experimental observations were performed tracking individual E. coli bacteria
swimming either in the Newtonian solution of Ficoll 400 or in the DSCG liquid
crystal. At the chosen experimental conditions of temperature and
concentration, the DSCG has the same viscosity as Ficoll 400, that is, $18\
$\mathrm{m}\mathrm{P}\mathrm{a}\,\mathrm{s}$$. The bacteria are confined to a
20 micron-high flat cell with planar LC alignment so that their trajectories
are considered confined to a nearly two dimensional plane. Bacterial body and
flagella are colored fluorescently and can be observed simultaneously during
tracking. 16 tracks have been recorded in Ficoll and 64 in the LC of a typical
duration of $40\pm 37\ $\mathrm{s}$$.

###  III.1 Reversal kinematics :
experimental observations

We will first describe typical swimming trajectories observed in the isotropic
viscous fluid and in the liquid crystal, focusing on trajectories with a
change in swimming direction. Fig. 1(a) shows snapshots of a bacterium
describing a typical trajectory in the Ficoll viscous solution. From these
snapshots, separated by a time interval of 0.5 s, it can be seen very clearly
how the flagella bundle opens up and several individual flagella become
visible, corresponding to a tumble event. At the end, the bundle is fully
formed again. It is also clear that the bacterium orientation changes during
the tumble, in agreement with the classical picture of run and tumble motion.
Fig. 1(b) shows a typical 3D-trajectory indicating bacterial motion without a
particular spatial orientation, as expected.

Figure 1(c) shows snapshots of a bacterium moving in the liquid crystal. One
remarks immediately that the bacterium moves along a single direction,
corresponding to the direction of alignment of the LC director. Interestingly,
the bacterium, that initially swims downwards, starts swimming upwards at some
point: it changes its orientation by 180°, while keeping the same swimming
direction. We will refer to this change in orientation as a ”reversal”. While
the bacterium is swimming in the forward direction, it is pushed by a well-
formed bundle. After some time, one observes a single flagellum appearing
gradually on the other side of the bacterial body, a phenomenon that coincides
with the reversal. We will refer to this flagellar configuration as ”bi-
polar”. The single flagellum then disappears and another 180° reorientation is
observed. The bacterium recovers its ”normal” configuration with a single
flagella bundle. We will discuss this flagellum relocation in more detail
below, but from these observations it is already clear that the alignment
between flagella and body is maintained throughout the reversal, in stark
contrast with the bundle opening observed during a classical tumble in the
isotropic medium. Note that such a reversal behavior, triggered by the
relocation of a single flagellum to the other side of the bacterial body has
been observed in most of our experiments, even if sometimes more complex
behaviors were also detected (see Fig. 2 (e)). In a large majority of ”bi-
polar” flagella configurations, the bacterium is pushed by the single
relocated flagellum and swims in the opposite direction compared to the
”normal” configuration. A typical trajectory is shown in Figure 1(c), which
confirms the existence of a remarkable bacterial dynamics in the liquid
crystal, showing that bacteria are able to reorient in this constraining
environment, by moving back and forth along the director.

We have analyzed the swimming velocities of the bacteria in both media,
obtaining the velocity distributions shown in Figure 1(c). The resulting mean
velocity for a bacterium swimming in the viscous Newtonian fluid is $1.22\pm
1.24\ $\mu\mathrm{m}\,\mathrm{s}^{-1}$$, whereas we find $2.32\pm 0.9\
$\mu\mathrm{m}\,\mathrm{s}^{-1}$$ for the liquid crystal. In the LC, the
histogram contains values from all possible swimming mechanisms (normal, bi-
polar or more complex, rare configurations), explaining the large width of the
velocity distribution. A more detailed analysis will be performed below (see
Fig.3). These two average swimming velocities are comparable within the error
bars, and are found to be significantly lower than the values reported for the
same bacteria strain in a buffer solution, 26 $\mu\mathrm{m}\,\mathrm{s}^{-1}$
[34]. Our fluids are about 20 times more viscous than the buffer solution.
According to Martinez et al. and Qu et al. , the swimming velocity is expected
to be inversely proportional to the fluid viscosity under operating conditions
where a constant motor torque is expected, as it is the case here [23, 22].
Based on the experimental results by Martinez et al. [23] establishing an
exact relationship between viscosity and swimming velocity, the typical
velocity predicted for our E-coli bacteria, in a fluid of viscosity of 18mPas,
is around $4$\mu\mathrm{m}\,\mathrm{s}^{-1}$$. While this prediction correctly
captures the order of magnitude of the swimming speed observed in our
experiments, we note however that the velocities observed in our experiments
are a least a factor two smaller. This is most likely due to the strong
confinement induced by the experimental cell (height $20$\mu\mathrm{m}$$),
previously shown to lead to decreased swimming velocities [41]. The velocity
probability distribution functions also show that some bacteria are able to
reach higher speeds in the LC than in the Newtonian viscous fluid. Similar
observations on increased swimming speed in complex fluids have also been made
in colloidal suspensions [4] or polymer solutions [5], consistently with our
results.

In the following we will rationalize this novel and surprising mechanism of
change in swimming direction, investigating first what triggers a reversal,
then, how a flagellum can relocate from one side of the bacterial body to the
other, and finally, how a single flagellum could be able to push the entire
bacterium.

###  III.2 Reversals triggered by the initiation of a tumble

The reversals observed in the LC solution occur at random positions within the
sample and we have checked that they do not coincide with tactoid boundaries
or any visible defects in the solution. Visual observation under crossed
polarizers indeed indicates a uniform director field in the liquid crystal.
From a mesoscopic point of view, the liquid crystal is composed of elongated
aggregates, with an average length of 8 nm, which are too small to represent
an obstacle for the bacteria [42]. We can thus exclude mechanisms such as
those recently proposed for bacterial reversals observed in mucus [43],
Carbopol solutions [44] or homeotropic liquid crystals [45]. In these systems,
where a spatial heterogeneity is present, bacterial reversals have been
attributed to the encounter of ”obstacles” along the bacteria swimming path.
The reversals occur when the bacterial body is suddenly stopped and the
compression on the flagellar bundle causes buckling, disassembly and
reorganization on the other side of the bacterium [43]. In Zhou et al. [45]
the reversal occurs perpendicular to the nematic director, playing the role of
the encountered ”obstacle” by limiting displacement along the original
swimming direction.

We hypothesize that the observed reversals in the LC are triggered by the
initiation of a tumble motion, and thus one flagellar motor changing rotation
direction to CW, causing a flagellum to leave the bundle and to relocate to
the other side of the bacterial body. Upon switching back to CCW, the
flagellum relocates back to its initial position and the bundle is formed
again. Control experiments with ”smooth swimmers”, mutant E. coli with
genetically suppressed tumbling, showed no reorientation, supporting the idea
that reversals are indeed linked to the initiation of a tumble. We will
discuss next if the observed timescales for the reversals in the LC are
compatible with this picture, taking advantage of the excellent resolution on
the flagella dynamics of our observations. We will compare the observed
timescales with those obtained in the Newtonian medium, where we know that
classical run and tumble motion is preserved. We will also verify that our
observations are compatible with previous observations in buffer solutions and
equally viscous surroundings.

Two important time scales can be discussed, first the frequency of tumbles
(corresponding in first approximation to the inverse of the run times), and
second, the tumble duration. In our picture, the tumble frequency corresponds
to the reversal frequency (or the inverse of the duration of swimming in a
”normal” configuration) whereas the tumble duration would correspond to the
time of swimming in the ”bi-polar” configuration. Berg et al. [6, 7]
identified a typical run time of 1s for E. coli in buffer solution
(corresponding to a tumbling frequency of about 1Hz) and a tumbling time of
0.1s. It has recently been shown that the run time distribution is very large,
consistently with the large distribution of the CCW motor rotation duration
and that a correct determination of the latter requires thus the analysis of a
large number of long bacteria trajectories [46, 47]. The number of
trajectories of limited length obtained in this work is thus not sufficient to
get a clear picture on the run time distributions, but we can state that
typical run times in Ficoll, as well as ”normal” mode swimming times in the
LC, are longer than 10s, a value significantly larger than that reported by
Berg in the buffer solution. Qu et al. [22] have stated that typical CW and
CCW motor rotation intervals should be relatively insensitive to viscosity, as
long as flagella rotation speeds remain not too low (typically above 50 Hz).
This would indicate an unchanged tumbling frequency as a function of
viscosity. However, their experiments have been performed for relatively low
viscosities (below 5 mPas), and do thus not correspond to the regime
investigated here. Recent observations by Patterson et al. [5] indicate that
run times can be increased for high solvent viscosities where bacteria operate
close to a stall regime (flagella rotation frequencies below 50Hz) leading to
low swimming velocities and low flagella rotation rates, in agreement with our
observations.

Figure 1(f) shows the time distributions for tumbling, in Ficoll, and swimming
in the ”bi-polar” mode, in the LC. Note that we refer to a tumble as the
period over which the bacteria bundle is not fully formed, a definition that
differs from the one used by Berg et al. [7], using a criterion based on the
reduction of swimming speed in combination with bacterial reorientation. The
trajectories in Ficoll have been analyzed by hand, inspecting the flagella
bundle. As in the LC the normal and bi-polar swimming modes correspond to
opposite swimming orientations (see Fig. 3 (c)) automated detection was
possible. The average values are respectively $t_{bi-polar,DSCG}=4,2s\pm 3.8s$
and $t_{tumble,Ficoll}=3s\pm 2s$, comparable to each other within the error
bars. This is a first indication that the ”bi-polar” swimming phase could
indeed correspond to a tumble phase, as similar time scales are observed for
both modes.

It is less straight forward to compare these values to the values reported in
the literature. Mainly, because of the different criteria used to establish
the tumble time, which necessarily lead to different measured values. Indeed,
in most investigations, the criteria used to establish the tumble time were
not based on the unbundling of the flagella, as clear observation of the
flagella dynamics was lacking. Only the recent work by Junot et al. [34]
provided typical durations of flagella unbundling in a buffer solution,
finding an average value of 0.8s. Qu et al. [22] have argued that even if the
duration of the CW rotation of the motor remains unchanged, the time to reform
the bundle is strongly increased by the viscosity of the surrounding fluid.
They claim that the rotation frequency of the flagella is reduced (as well as
the swimming speed) due to the increased viscosity, and they hypothesize that
20 rotations are necessary to completely reform a bundle once the flagella has
recovered its CCW rotation direction. In viscous surroundings, the tumbling
time, defined as the time during which the bundle is not fully formed, is thus
set by the time to reform the bundle, rather than by the time the flagella
motor turns CW. We observe typical rotation frequencies between 5Hz and 10Hz
(Fig. 3 (e)) leading to a bundle formation time of typically 2-4s, in good
agreement with our experimental observations. Note that the bundling time
resulting from these fluid structure interactions dominates largely the time
the flagella motor turns clockwise, so that possible increased durations of CW
rotation in the high load regime [48] possibly play a minor role.

Our analysis of the observed run and tumble times, as well as the time
distributions of ”normal” and ”bi-polar” swimming, shows comparable values for
the tumbling frequency in the Newtonian fluid and the reversal frequency in
the liquid crystal, even if no thorough statistics are available for this
variable. This suggests that reversals are triggered by the initiation of a
tumble motion (and thus the CW rotation of a flagellum). In addition, tumbling
times in the viscous fluid are similar to ”bi-polar” swimming in the LC,
supporting the hypothesis that the duration of a flagella being relocated on
the other side of the bacterial body would essentially corresponds to the
duration of a tumble.

Figure 1: (a) Snapshots of a free swimming E. coli bacterium performing a run
and tumble motion in a 3D space, in a Ficoll solution of viscosity
$18$\mathrm{m}\mathrm{P}\mathrm{a}\,\mathrm{s}$$, at room temperature. The
time duration between two snapshots is 1.5 s. Both the flagella and the
bacterial body can be seen, during the run phase when the flagella form a
bundle, and during the tumble, when the flagella separate. The final swimming
direction is different from the initial one. (b) Trajectory of a tracked
bacterium changing its swimming direction in the Ficoll solution. The color
scale corresponds to the absolute velocity values, and the blue ellipsoid
represents the bacterial body when the track starts (only for scale
purposes).(c) Snapshots of a bacterium swimming in a liquid crystal (DSCG 12
wt%, $18$\mathrm{m}\mathrm{P}\mathrm{a}\,\mathrm{s}$$, room temperature).
Confined in a $20$\mu\mathrm{m}$$ high observation chamber. The nematic
director is along the vertical direction, as well as the bacterium alignment.
The time duration between two snapshots is 3 s. The bacterium reverses its
swimming direction successively from downwards to upwards. While the bacterium
swims upwards, the bundle is split in two, in a bipolar configuration. The
double arrow indicates the LC director (d) Trajectory of a bacterium reversing
by 180 degrees its swimming direction in DSCG. The reversals are marked on the
trajectory by crosses. The images show the body and flagella of the bacterium,
corresponding to the different configurations: for t = 1 s and t = 11 s normal
configuration and for t = 8 s bipolar configuration. The relocated flagellum
is pointed with a red ellipse and the arrows highlight the swimming direction.
(e) Probability distribution function (PDF) histogram of absolute velocities
for 16 tracked bacteria in Ficoll solution (red) and 64 tracked bacteria in
DSCG liquid crystal (blue). The corresponding average velocities are
$V=1.22\pm 1.24$\mu\mathrm{m}\,\mathrm{s}^{-1}$$ for Ficoll and $V=2.32\pm
0.9$\mu\mathrm{m}\,\mathrm{s}^{-1}$$ in the LC, represented by the dashed
lines.(f) Histogram of tumble duration for bacteria swimming in Ficoll and LC,
and the histogram for run duration in Ficoll.

This holds despite that fact that the unbundling and bundling mechanisms seem
to be different in the two media, as we will discuss below. The observed
tumble durations, defined as the duration during which the bundle is not fully
formed, are larger compared to those in buffer solution, in agreement with the
increased time needed to reform the bundle in a viscous surrounding. These
observations comfort the idea that, in the LC case, the run time equals the
duration of swimming in the ”normal” configuration, whereas the tumble
duration equals swimming in the bi-polar configuration. In the light of these
observations we will refer to the reversal motion in the LCs as a frustrated
run and tumble motion. In the next section, we will analyze in detail how a
flagellum leaves the bundle to relocalize on the other side of the bacterial
body.

###  III.3 Flagella relocation during a frustrated tumble

We have seen that, in the LC, the flagella bundle, as well as the relocated
flagellum, and the bacterial body, always remain aligned with the director
field. This is shown in more detail in Fig. 2 (a), depicting the
configurational transformations of a bacterium during the reversal phase in
the LC. From these snapshots, we can see more clearly how the flagella
relocation takes place, here during a transformation from a ”bi-polar” to a
”normal” configuration. The single flagellum folds along the bacterial body
and then straightens out again on the other side to merge with the full
bundle. This process involves rather strong deformations of the flagellar
filament structure, as it can be seen in more detail in Fig. 2 (d), where the
relocating flagellum adopts a hair pin configuration with a strong and
localized curvature at the bending location. We will first focus on the
alignment with the director and then discuss a possible mechanism on how a
flagellum is able to undergo such a strong deformation during relocation.

Alignment of the bacterial body and the flagella bundle with the swimming
direction induced by a surrounding complex media has already been observed,
even if only indirectly, in elastic fluids, colloidal suspensions or colloidal
liquid crystals [5, 4, 49]. Typically a misalignment between the flagella
bundle and the bacterial body leads to wobbling of the body, easily observed
under microscope [50]. As a consequence, the reduction of wobbling amplitude
and frequency has been used to conclude on the body-bundle alignment. Several
studies, this time performed in simple Newtonian fluids, have linked decreased
wobbling amplitudes, as well as more closed bundle forms, to more efficient
swimming [50, 51, 52]. The better alignment is thus expected to be at the
origin of the increased swimming speeds observed in complex fluids [5, 4, 49].

It is thus not surprising that the elasticity of the LCs leads to bacteria-
bundle alignment. Our observations constitute however for the first time
direct visual evidence of this effect. However a completely new observation is
the passage of the flagellum to the other side of the body involving the
strong deformations described above (see Fig. 2 (d)). While no definitive
explanation of the full mechanism of flagella reorientation can be given yet,
we will in the following discuss some possible scenarios.

First we will discuss whether the elasticity of the liquid crystal DSCG used
in our study is important enough to sustain the strong deformations induced in
the bacteria flagella. Bacterial flagella are know to be rather rigid with a
bending stiffness of the order of $B\sim 3.5\ 10^{-24}\
$\mathrm{N}\mathrm{m}^{2}$$ [9]. An estimate of the elastic energy $E_{bend}$
stored when deforming a flagellum as in Fig. 2 (d) can be obtained taking only
the localized deformation in the kink into account and by estimating the
radius of curvature of the hairpin $R$ to be around 1 $\mu\mathrm{m}$. On
finds $E_{bend}\sim B/R\sim 10^{-18}Nm$. This can be compared to the elastic
energy stored when deforming the liquid crystal director over a distance of
the same order of magnitude as the radius of curvature. For the liquid crystal
employed, in the one elastic constant approximation, at our experimental
conditions, the elastic constant $K$ has a value $K\sim 10pN$ [53]. The
elastic energy associated to the director distortions induced by the bending
of the flagellum can be estimated as $E_{elast}\sim KR\sim
10^{-17}$\mathrm{N}\mathrm{m}$$. These two energies are comparable in
magnitude. Another approach is to estimate the flagella deformation at which
the sum of the elastic and bending energies is minimal, leading to a distance
of $R\sim 1$\mu\mathrm{m}$$, in agreement with our observations. The liquid
crystal thus seems able to sustain the strong deformation of the bent
flagellum.

The passage of the flagellum from one side of the body to the other is likely
facilitated by a second mechanism: a polymorphic transformation. It is well
known that polymorphic transformations can take place when the rotation
direction of one flagellum in the bundle is changed from CCW to CW. Typically
during such a transformation a normal shaped flagellum changes towards a so
called ”curly” shape, of different radius, pitch and chirality [9]. This
transformation is initiated close to the bacterial body and then travels
towards the free end of the flagellum. Geometrical constraints do not allow
the connection between two different flagellar shapes to be straight at their
junction. A natural kink, not involving any elastic deformation, necessary
appears where the curly and normal flagellar segments meet. Such a kink
travels along the flagellum as the polymorphic transformation advances. This
has been observed for tethered cells by Darnton et al. [9].

Different types of flagellar shapes, with well defined geometries, have been
predicted by [54], and several experimentally observed, namely normal, curly
I, curly II and semicoiled [9, 55]. Curly shapes are characterized by a
smaller pitch and thus a more ”compact” visual appearance. The main difference
between curly I and II lies in the total flagella length. Using this
classification, we have measured the values of the pitch and radius of the
helices for the normal and curly shapes in our experiments. The normal shape
is characterized by a pitch and radius of respectively $3.37\pm 0.20\
$\mu\mathrm{m}$$ and $0.47\pm 0.20\ $\mu\mathrm{m}$$, while the curly shape
has a pitch and radius of $1.47\pm 0.20\ $\mu\mathrm{m}$$ and $0.25\pm 0.20\
$\mu\mathrm{m}$$, respectively. These pitch values are slightly larger than
those reported in the literature for normal shaped E. coli flagella, with a
pitch of 2.3 $\mu$m and radius of 0.4 $\mu$m [55]. Generally, it is hard to
distinguish between curly I and curly II shapes from the images, as their
radius and pitch only differ by 0.1 $\mu$m , which is lower than the
measurement error. However, we can conclude that we mostly observe curly II
shapes, since all measurements indicate that the curly and normal shapes
observed have the same length, $10\pm 0.20$\mu\mathrm{m}$$. Other shapes like
semicoiled and curly 1 are usually the most observed in literature [7, 8, 9].
The different discrete flagella shapes are also associated with different
handedness [54]. A curly flagellum can only exist as right handed, whereas a
normal flagellum is left handed.

For the first time, an example of such a polymorphic transformation during a
tumble of a freely swimming bacteria in the Ficoll solution is shown in Fig. 2
(b). Thanks to the high time resolution of the videos, the polymorphic
transformation of the flagellum after its detachment from the bundle is
clearly visible, as well as the angle $\theta$ between the normal-shaped
portion of the flagellum (close to the free end) and the curly portion (close
to the bacterial body). The fact that the shapes of the different filament
morphologies are known with precision allows us to identify them unambiguously
in our images. Close inspection of the flagella during their relocation in the
LC (Fig. 2 (a)) reveals that also in this case a polymorphic transformation
takes place. Indeed, one can observe two normal shaped flagella on the upper
side of the body, until one of them, highlighted with a green ellipse, starts
to change its shape to curly. This flagellum then migrates towards the bottom
side of the body, while remaining aligned with the LC, parallel to the
bacterial body. In the last snapshots, the bacterium is propelled upwards by
the relocated flagellum, which remains in the curly shape.

Fig. 2 (c) compares the angle of the kink during the polymorphic
transformations shown in Figs 2 (a) and (b), for the LC and the Ficoll
solution, respectively. In both cases, the angle remains nearly constant
during the transformation and the kink propagates at nearly constant speed.
Note that the bacterium in Ficoll is freely swimming in space, and that the
measured angle is a 2D projection. In the LC, the distinct flagellar segments
are always parallel to the rest of the bacterium, resulting in a very small
angle and giving the impression of sliding. The fact that this angle is
significantly smaller compared to the observations in Ficoll might result from
the combined action of the morphological transition and the elastic
confinement the LC exerts onto the flagella segments. The deformation kink
propagates at a speed of $v=7.3\pm 1\ $\mu\mathrm{m}\,\mathrm{s}^{-1}$$ in the
LC, and at a speed of $v=20\pm 1\ $\mu\mathrm{m}\,\mathrm{s}^{-1}$$ in Ficoll.
No literature values of the propagation speed of a polymorophic transformation
under free swimming conditions exist and we can thus not compare our results
to previous works. We have not performed a statistical analysis of these
transformations because or their rarity in the field of view (in the case of
Ficoll), but the examples shown represent the first quantitative measurement
of the transformation dynamics. Despite the difference in absolute values, the
two speeds observed are comparable in order of magnitude and point towards the
fact that similar processes could be at play in both fluids.

In some cases the flagella relocation in the LC can take place through more
complex mechanisms and transformations. One example is shown in Fig. 2(e).
Initially, the flagellum is in a curly I shape (Fig. 2(e)-(i)). Then, it
transits to a hybrid configuration where two different shapes coexist (Fig.
2(e)-(ii)): one can see in image (ii) that the segment colored in red has a
smaller pitch and radius than the segment colored in blue, the red segment
corresponding to a curly shape and the blue segment to a normal shape. As
expected, a finite angle is formed between the two distinct flagellar
segments. This hybrid state remains stable during an extended period of time.
We have rarely observed another shape than curly or normal, but in the image
(iii), one can see an almost straight flagellum, a shape known to be unstable
and never observed previously.

The observations presented in this section on the relocation of a flagellum
from one side of the body to the other side have demonstrated that this
relocation takes place while the bacterium flagella and body remain aligned
with the director field. The flagellum seems to ”slide” along the bacterial
body, a process that involves strong structural deformations. A combination
between a morphological transformation, typically triggering the propagation
of a kink along one of the flagella, and the elastic bending induced by the
LC, seem to be at the origin of the observed dynamics.

Figure 2: (a) Flagellar relocation during a reversal in LC, accompanied by a
polymorphic transformation. $\theta_{L}$ is the projected angle between the
relocated flagellum (relocated by the green ellipse) and the bundle, here
parallel (b) Snapshots of a polymorphic transformation occurring during a
tumble, from normal to curly for a free swimming E. coli bacterium in a
viscous and Newtonian solution Ficoll. As indicated by the blue line,
$\theta_{F}$ is the projected angle between a normal shaped portion of the
flagellum and a curly portion. (c) Length from the body base to the
deformation and kink projected angle $\theta$ over time, for a bacterium in
Ficoll and one in DSCG liquid crystal (d) Zoom on the local deformation of the
flagellum (in green). The arrow indicates the deformation propagation
direction, travelling at a speed of $7.3\pm
1$\mu\mathrm{m}\,\mathrm{s}^{-1}$$, and the double arrow the LC director. (e)
Photographs of the same bacterium, presenting the zoology of flagellar shapes,
and their illustrations : (i) curly II, in red (ii) unidentified curly in red
- normal in blue (iii) unidentified straight shape, in green.

###  III.4 Swimming mechanism

From our observations on flagella relocation and reversal initiation, a
coherent picture of the swimming mechanism of E. coli in the LC emerges. In
the normal run phase, the flagella bundle pushes the bacterial body. At the
onset of a tumble, a flagellum leaves the bundle, a morphological
transformation from a normal to a curly shape occurs and the flagellum
relocates to the other side of the bacterial body, now pushing the bacterium
in the opposite direction. In this section, we address the question whether
this swimming scenario is consistent with the zero force and torque condition
required for swimming at low Reynolds number [56]. Besides, we try to
understand how the action of a single flagellum can overcome the action of the
whole bundle, placed on the other side of the bacterial body, pushing it in
the opposite direction.

Let us first state that bacteria swimming with a flagella bundle cannot
reverse swimming direction by simply changing the rotation direction of the
flagella bundle. This is in contrast with polar bacteria [57], where a change
in the rotation direction of the single flagellum indeed leads to a change
from pushing to pulling the bacterial body. In the case of multiflagellated
bacteria, as is the case of E. coli, first a coordinated change in rotation
direction of all flagella would have to take place simultaneously, which is
never the case, but in addition, for the left-handed E. coli, hydrodynamic
interactions between flagella lead to flagella bundling for CCW rotations, but
to unbundling for CW rotation and propulsion would thus be lost upon a change
in rotation direction [58]. This is why in normal swimming conditions
multiflagellated bacteria perform the already described tumble mechanism to
change swimming direction.

Let us now discuss whether the flagellum relocated on the other side of the
bacterial body is able to push the bacterium from a hydrodynamic point of
view. The thrust produced by the rotating flagellum is compensated by the
viscous friction on the moving body, following the zero force condition. In
order to push the bacterial body forward, flagella have to be either left-
handed and turning CCW, or right-handed and turning CW. During a normal run
the left-handed flagella in the bundle indeed turn CCW to push the cell. The
single flagellum leaves the flagella bundle after the initiation of a tumble,
and therefore, after a change of rotation direction. For the bacterium to be
propelled by the relocated flagellum turning CW, the later must be right-
handed. Handedness cannot be determined from our observations, representing a
2D projection of the flagella, however we can identify its morphological
configuration from its geometrical properties. We have found the relocated
flagella to be curly in almost all cases, which means that it is right-handed
[54]. We can thus conclude that a curly flagellum rotating CW can indeed push
a bacterium.

In addition, the zero torque condition also needs to be fulfilled for the bi-
polar swimming configuration. During a normal run, all flagella turn in CCW
direction and the body counter-rotates CW, maintaining zero torque. A
schematic of the normal and bi-polar swimming configurations is represented in
Fig. 3(a), where we always observe from the direction of the flagella bundle
towards the bacterial body. The flagellum leaving the bundle rotates CW, and
therefore, in the same direction as the body. However once it has migrated to
the other side of the body, from the perspective of the flagella bundle, it
now rotates CCW, and thus, in the opposite direction to the bacterial body.
Even if the details of the torques applied via the flagella hooks onto the
bacterial body can be complex, from a geometrical point of view, the body and
the pushing flagella counter-rotate again. Our swimming configuration on the
bi-polar state is thus based on a coherent scenario in terms of sense of
rotation of the body and flagella.

In nearly all the cases where the flagellar configuration is bi-polar (around
70 bacteria tracks), we observe that the bacterium is pushed by the single
flagellum against the flagella bundle. To understand the mechanism behind, we
characterized the swimming speeds of the bacterium in the different
configurations. We analysed the 64 exploitable tracks. From 36 tracks, with an
average duration of $30\pm 33\ $\mathrm{s}$$, the bacterial trajectories did
not show a reversal, while in 28 tracks, of $53\pm 40\ $\mathrm{s}$$ average
duration, the bacteria performed reversals along their trajectory. For tracks
without a reversal, the bacterium can either be in a normal or bipolar
configuration for the whole duration of the experiment. We have used these
trajectories to obtain clear information about the swimming speed in one or
the other configuration. Reversals lead by definition to a slowing down of the
swimming velocity, and therefore, they yield an overall reduction of the
average velocity of the bacterium (see Fig. 1 (c)). A bacterium swims in
average with a velocity of $\overline{\rm v}=4\pm 1\
$\mu\mathrm{m}\,\mathrm{s}^{-1}$$, in the normal configuration, and
$\overline{\rm v}=2\pm 0.5\ $\mu\mathrm{m}\,\mathrm{s}^{-1}$$, in the bipolar
configuration. Swimming in the bipolar configuration thus seems slower, but
the difference is small. This is a surprising observation, as in the bi-polar
configuration the single flagellum push against the flagella bundle. Even if,
from a hydrodynamic point of view, the propulsion efficiency, at identical
rotation rate and flagellar shape, is not higher for a flagella bundle than
for a single flagellum, one would still expect the swimming speed in the
normal configuration to be significantly larger.

To gain more insight into this question, we have analyzed a typical example of
a bacterium swimming in the bi-polar configuration in the LC, and we have
measured the rotation frequencies of both the flagella bundle and the
individual flagellum, see Fig. 3(d). The frequency measurement has been done
from kymographs along a defined line, perpendicular to the flagella. When the
flagellum rotates, its wave propagates in space, thus the graph presents a
wavy pattern. The abscissa being time, by measuring the separation between two
peaks, one can get the frequency. Surprisingly, we find that the flagella
bundle is nearly immobile and rotates with a frequency below $1Hz$. The curly
individual flagella on the other side of the bacterial body, in contrast,
rotates at a frequency of nearly $10Hz$. For comparison, we measured rotation
frequencies for bacteria swimming in Ficoll. The results presented in Fig.
3(e) are obtained during the tumble and show that the curly and normal
flagella rotate at a similar frequencies, between $5$ and $10Hz$, comparable
to the observed rotation frequency of the single flagellum in the bi-polar
configuration. Without being able to fully explain this observation, namely
why the flagella bundle becomes nearly immobile, this is in agreement with a
displacement of the bacterial body pushed by the single flagellum against a
passive flagella bundle. It has been demonstrated that the swimming efficiency
for curly shaped flagella is slightly smaller compared to normal flagella
[59], which could be among the reasons why the bi-polar swimming velocity is
observed to be smaller than the normal swimming velocity. The proposed
mechanism for bi-polar swimming obtained from our experimental observation is
thus also consistent from a hydrodynamic point of view.

Figure 3: (a) Illustration of the reorientation scenario for a bacterium
swimming in a liquid crystal : first the bacterium is propelled by the bundle
turning CCW in its reference frame, performing a normal run; in the red part,
in a bipolar configuration, one flagellum changes its sense rotation to CW
(and handedness) and migrates to the other side of the cell, it can be
considered as a tumble. As the relocated flagellum is on the other side of the
body, in the reference frame of the bundle, it turns CCW. The bacterium
changed it swimming direction, it is propelled by the relocated flagellum.
Finally the bacterium goes back to performing the normal run (blue part). (b)
The side view is given for each step. (c) Swimming velocity $V_{x}$ projected
along the forward swimming direction (prior to the first reversal) Each color
corresponds to a different configuration (normal or bipolar). The general
velocity averages for normal (blue) and bipolar (red) configurations (of all
tracked bacteria) are $V=1.3\pm 0.55$\mu\mathrm{m}\,\mathrm{s}^{-1}$$ and
$V=1\pm 0.57$\mu\mathrm{m}\,\mathrm{s}^{-1}$$ respectively. (d) Snapshot of a
bacterium in LC, in bipolar configuration, and kymographs corresponding to the
curly (of the left) and normal bundle (right) flagella. The bacterium swimming
direction is from the left to the right, so that the curly flagellum is
pushing it forward, by rotating at a 10 Hz frequency, while the bundle is
inactive. (e) Rotation frequencies for several bacteria swimming in Ficoll,
during a tumble phase, for both curly and normal shapes flagella, comparable
to the values of rotating curly flagellum in LC.

##  IV Conclusion

In this report we described how wild-type E. coli bacteria explore space in a
structured fluid like a liquid crystal (LC). In such a fluid the swimming
direction is constrained to take place along the nematic director and we
observed that bacterial re-orientation occurs in the form of 180° reversals.
This is in stark contrast with typical trajectories of wild-type bacteria
exploring isotropic fluids that resemble a 3D random walk, stemming from the
classical run and tumble dynamics.

Precise experimental observations of the swimming bacteria, including full
monitoring of the flagellar dynamics, revealed that these reversals always
occur together with a specific flagella rearrangement, implying a flagellum
leaving the bundle and repositioning itself on the other side of the bacterial
body. We believe that this remarkable relocation is triggered by a reverse in
rotation direction of the repositioning flagellum, a mechanism that, in an
isotropic fluid, would lead to a classical tumble, following the opening of
the flagella bundle.

To support this hypothesis, we performed experiments in a Newtonian fluid of
viscosity comparable to that of the LC. Typical swimming times using the full
bundle (here called normal configuration) are comparable to typical run times
in the viscous fluid, while swimming times using the relocated single
flagellum (here called bi-polar configuration) are comparable to typical
tumble times. This suggests that the reversals stem from a frustrated tumble.

Precise and time resolved observations were made possible by the slowing down
of the bacterial dynamics in the viscous surroundings and an excellent
resolution of the flagellar structure revealed a precise scenario for the
repositioning dynamics. The repositioning of the flagellum on the other side
of the bacterial body takes place, while the bacterium flagella and body
remain aligned along the director field. The flagellum seems to ”slide” along
the bacterial body, a process that involves strong structural deformations, in
contrast to the typical opening of the flagella bundle observed in isotropic
media. A combination between a morphological transformation, typically
triggering the propagation of a kink along one of the bacterium flagella, and
the elastic bending induced by the LC, seems to be at the origin of the
observed flagellar dynamics. Relocation of the single flagellum on the other
side of the bacterial body coincides in nearly all cases with a change in
swimming direction, meaning that the single flagellum pushes the bacterial
body against the flagella bundle located on the opposite side. Close
inspection of bundle and flagella rotation frequencies indicate that, in these
cases, the flagella bundle becomes nearly immobile and thus passive to
propulsion. Meanwhile, the single flagella rotates at frequencies of
comparable magnitude to those observed in the corresponding viscous fluid.
While we have no clear explanation for the observed stalling of the flagella
bundle, our observations are consistent with a swimming mechanism at low
Reynolds number.

We believe that our experimental characterization of the novel bacterial
reversal dynamics observed in LCs will constitute the basis of future modeling
and research. Some observations represent an extended characterization of
representative examples instead of a full statistical analysis, but in most
cases they constitute the first observation ever made of such dynamics, in the
context of highly viscous Newtonian viscous fluid as well as structured LCs.
As such, these observations could also pave the way to further experimental
observations of comparable phenomena including bacteria flagella dynamics in
various complex environments.

###### Acknowledgements.

We thank Prof. Dong Ki Yoon, from Korea Advanced Institute of Science and
Technology (KAIS), who has kindly provided us with patterned glass slides for
the alignment of the LC. We thank Prof Wilson Poon’s group of the University
of Edimburgh, for providing us with the two-colours strain AD62. We also thank
Anumita Jawahar for measuring the phase diagram presented in the Appendix. We
thank Prof. Alexander Morozov for enlightening discussions on possible
reversal and swimming mechanisms. We thank Prof. Kenny Breuer for bringing the
possible role of morphological transformations to our attention and Guillaume
Sintes for helpful discussions on flagella dynamics and bacteria swimming.

Authors’ Contributions : MG, EC, TLL and AL have designed the study. MG has
performed the experiments and analyzed the data, TD has developed and
implemented the two color tracking technique. MG, TLL and AL have written the
paper.

Funding : This work has been funded by the Agence Nationale de la Recherche
(ANR) grant ANR-13-JS08-0006-01 and the European Research Council (ERC)
Consolidator Grant ‘PaDyFlow’, Grant Agreement no. 682367. We acknowledge the
Institut Pierre-Gilles de Gennes (equipement d’excellence, Investissements
d’avenir program ANR-10-EQPX-34).

##  Appendix A Appendixes

###  A.1 DSCG phase diagram

The DSCG liquid crystal (LC) was dispersed in a motility buffer, thus we
established the phase diagram for this system, shown in the Figure 5.

The study consisted in varying the temperature and concentration of several LC
mixtures. For each concentration (from 8 to 16 wt%), the temperature was
varied from 12°to 43° using a microscope TS64 thermal stage. The samples were
sealed with nail polish to prevent any water evaporation that could induce an
increase of the DSCG concentration. In order to ensure a thin layer of LC, no
spacers were used. Using a custom Matlab code, 4 regions on the sample were
selected for the FOV in order to increase the statistics. Images were taken
using a cross-polarizer and after waiting for several minutes until the LC
mixture relaxed. An example of the resulting images is shown in the Figure 4
with isotropic regions called tactoids (in black) getting bigger with
increasing temperature. A custom Python code was written to analyze the
images, detecting the isotropic and nematic phases on binarized images, in
order to get the corresponding area fractions. On the phase diagram, when the
isotropic fraction equals one, the LC is in a fully isotropic phase. For a
zero isotropic fraction the LC is in a fully nematic phase. In between, the
DSCG has mixed regions of isotropic and nematic, like shown in the Figure 4.
In our case, we want to avoid tactoids because they disturb bacterial motion
[53].

Figure 4: Crossed-polarized images of the temperature-induced nematic to
isotropic phase transition for 15.8 % DSCG dissolved in pure water (A) 34° C
(B) 36° C (C) 38° C. In black the isotropic phase and in white the nematic
phase. Figure 5: Area fraction of DSCG (dispersed in MB) isotropic phase as a
function of the temperature, for several concentrations.

###  A.2 Experimental set-up for sample preparation

See Figure 6.

Figure 6: Sample preparation for the study of bacterial swimming in aligned
LC.

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#  Note on the second eigenvalue of regular graphs

Igor Balla , Eero Räty, Benny Sudakov, István Tomon22footnotemark: 2 Einstein
Institute of Mathematics, Hebrew University of Jerusalem, Israel _e-mail_ :
<EMAIL_ADDRESS>Research supported by SNSF Project 184522.Umeå
University, _e-mail_<EMAIL_ADDRESS>ER is supported by a
postdoctoral grant 213-0204 from the Olle Engkvist Foundation.ETH Zurich, _e-
mail_<EMAIL_ADDRESS>Research supported in part by SNSF
grant 200021_196965.

###### Abstract

The goal of this expository note is to give a short, self-contained proof of
nearly optimal lower bounds for the second largest eigenvalue of the adjacency
matrix of regular graphs. This is a companion note to [4].

##  1 Introduction

The Alon-Boppana theorem [3] is a fundamental result in spectral graph theory,
which states that if $G$ is an $n$-vertex $d$-regular graph of diameter $D$,
then the second largest eigenvalue $\lambda_{2}$ of the adjacency matrix is at
least $2\sqrt{d-1}\cdot(1-o_{D}(1))$. However, in case $D\in\\{2,3\\}$, which
can happen for any $d\gg n^{1/3}$, the Alon-Boppana bound becomes meaningless.

In a recent paper [4], we addressed the growth rate of $\lambda_{2}$ for all
values of $1\leq d\leq(1/2-\varepsilon)n$ by relating it to the so called
_positive discrepancy_ or _minimum bisection_ , getting close to optimal
bounds. The goal of this short note is to give a self-contained proof of these
bounds. We believe that the behaviour of the smallest $\lambda_{2}$ among
$d$-regular graphs is fairly surprising, so these results deserve their own
exposition.

######  Theorem 1.1.

Let $G$ be a $d$-regular $n$ vertex graph, and let $\lambda_{2}$ be the second
largest eigenvalue of the adjacency matrix of $G$. Then for every
$\varepsilon>0$,

$\lambda_{2}\geq\begin{cases}\frac{\sqrt{d}}{2}(1-o(1))&\mbox{ if }1\leq d\leq
n^{2/3}\\\ \frac{n}{2d}-1&\mbox{ if }n^{2/3}<d\leq n^{3/4}\\\
(\varepsilon/4)^{1/3}\cdot d^{1/3}-1&\mbox{ if
}n^{3/4}<d\leq(1/2-\varepsilon)n,\end{cases}$

where the $o(1)$-term tends to 0 as $n\rightarrow\infty$.

We did not put much effort into optimizing the constant terms. However,
slightly more careful calculations yield that
$\lambda_{2}\geq\sqrt{d/2}(1-o(1))$ if $d=o(n^{2/3})$,
$\lambda_{2}\geq\frac{n}{d}(1-o(1))$ in case $d=o(n)$ and $n^{2/3}=o(d)$, and
$\lambda_{2}\geq d^{1/3}\cdot(1-o(1))$ if $d=o(n)$.

In [4], we discuss upper bounds as well (see also [1] for relations to
constructions of _equiangular lines_). Up to a constant factor, the random
$d$-regular graph shows that the first inequality in Theorem 1.1 is tight. Our
bounds are also tight if $d=\Theta(n^{3/4})$ or $d=\Theta(n)$. Finally, by
relaxing the condition of being regular to almost regular (i.e. every degree
is $d(1+o_{n}(1))$), the second inequality is also tight. Therefore, we
conjecture that our Theorem 1.1 is tight up to constant factors for every
$1\leq d\leq(1/2-\varepsilon)n$. Finally, the complete bipartite graph shows
that $\lambda_{2}\geq 0$ cannot be improved if $d=n/2$.

We highlight that the first two inequalities are also implicit in a work of
Balla [1] (see Lemmas 3.13 and 3.14), while the third inequality was first
obtained in [1]. This third inequality also follows from a recent result of
Ihringer [2] after a bit of work. We present essentially this latter argument.

##  2 Proof of the main theorem

Let $G$ be a $d$-regular graph on $n$ vertices with adjacency matrix $A$,
$1\leq d\leq(1/2-\varepsilon)n$, and let the eigenvalues of $A$ be
$d=\lambda_{1}\geq\dots\geq\lambda_{n}$. Let $K$ be the largest index such
that $\lambda_{K}\geq 0$. Since $G$ is not a complete graph its independence
number is at least $2$, so by interlacing $\lambda_{2}\geq 0$ and $K\geq 2$.
First, we prove the first two inequalities of Theorem 1.1 by considering the
sum, quadratic sum, and cubic sum of eigenvalues.

The next equalities and inequalities follow by expressing $\mbox{tr}(A)=0$,
$\mbox{tr}(A^{2})=dn$, and $\mbox{tr}(A^{3})\geq 0$ with the help of the
eigenvalues.

$d+n\lambda_{2}\geq
d+\sum_{i=2}^{K}\lambda_{i}=\sum_{i=K+1}^{n}|\lambda_{i}|,$ (1)

$dn-d^{2}-n\lambda_{2}^{2}\leq
dn-d^{2}-\sum_{i=2}^{K}\lambda_{i}^{2}=\sum_{i=K+1}^{n}\lambda_{i}^{2},$ (2)

$d^{3}+n\lambda_{2}^{3}\geq
d^{3}+\sum_{i=2}^{K}\lambda_{i}^{3}\geq\sum_{i=K+1}^{n}|\lambda_{i}|^{3}.$ (3)

We may assume that $\lambda_{2}\leq\sqrt{d/2}$, otherwise we are done. This
ensures that the left-hand-side of (2) is nonnegative. Let us use the simple
inequality that for any $x_{1},\dots,x_{m}\geq 0$, we have

$\left(\sum_{i=1}^{m}x_{i}\right)\cdot\left(\sum_{i=1}^{m}x_{i}^{3}\right)\geq\left(\sum_{i=1}^{m}x_{i}^{2}\right)^{2}.$
(4)

Then, we get from (1,2,3) that

$(d+n\lambda_{2})(d^{3}+n\lambda_{2}^{3})\geq(dn-d^{2}-n\lambda_{2}^{2})^{2}.$

After expanding and simplifying, we arrive to the inequality

$\lambda_{2}((d-\lambda_{2})^{2}+2n\lambda_{2})\geq d(n-2d).$

Here, the left-hand-side is at most $\lambda_{2}(d^{2}+2n\lambda_{2})\leq
2\lambda_{2}\max\\{d^{2},2n\lambda_{2}\\}$. By considering which term takes
the maximum in the previous inequality, we get that

$\lambda_{2}\geq\min\left\\{\frac{n}{2d}-1,\frac{\sqrt{d}}{2}\cdot\sqrt{1-2d/n}\right\\}.$

This proves the first two inequalities in Theorem 1.1.

Now let us turn to the third inequality, which we prepare with the following
lemma. We may assume that $\lambda_{2}\leq\sqrt{d}/2$, otherwise we are done.

######  Lemma 2.1.

$(1+\lambda_{2})\cdot|\lambda_{n}|\geq\frac{d}{4}.$

###### Proof.

Using that $\lambda_{2}\leq\sqrt{d}/2$, we have
$\sum_{i=2}^{K}\lambda_{i}^{2}\leq\frac{nd}{4}$, so

$\frac{nd}{4}\leq\sum_{i=K+1}^{n}\lambda_{i}^{2}\leq|\lambda_{n}|\sum_{i=K+1}^{n}|\lambda_{i}|=|\lambda_{n}|\left(d+\sum_{i=2}^{K}\lambda_{i}\right)\leq|\lambda_{n}|(d+n\lambda_{2})\leq|\lambda_{n}|n(1+\lambda_{2}).$

Here, the first inequality holds by (2) (since $d\leq n/2$), and the first
equality by (1).

∎

Set $p=\frac{d}{n}$, and define the matrix $M$ as

$M=(1-\alpha)I-\beta A+\alpha J,$

where $\alpha=\frac{p}{\lambda_{2}+p}$, $\beta=\frac{1}{\lambda_{2}+p}$, $I$
is the identity matrix, and $J$ is the all-1 matrix. Then the eigenvalues
$\mu_{1},\dots,\mu_{n}$ of $M$ are

$\mu_{1}=1-\alpha-d\beta+n\alpha=1-\alpha\geq 0$

and

$\mu_{i}=1-\alpha-\beta\lambda_{i}=\frac{\lambda_{2}-\lambda_{i}}{\lambda_{2}+p}\geq
0$

for $i=2,\dots,n$. Hence $M$ is positive semi-definite, and thus the Hadamard
product (also known as entrywise product) $M\circ M$ is also positive semi-
definite by the Schur product theorem. By simple calculations, the Hadamard
product $M\circ M$ is equal to

$M\circ M=(1-\alpha^{2})I+(\beta^{2}-2\alpha\beta)A+\alpha^{2}J,$

and hence its eigenvalues $\eta_{1},\dots,\eta_{n}$ are

$\eta_{1}=(1-\alpha^{2})+d\beta(\beta-2\alpha)+n\alpha^{2}=1-\alpha^{2}+n\alpha(\beta-\alpha)=1+\frac{p(1-p)n-p^{2}}{(\lambda_{2}+p)^{2}}$

and

$\eta_{i}=(1-\alpha^{2})+(\beta^{2}-2\alpha\beta)\lambda_{i}=1+\frac{(1-2p)\lambda_{i}-p^{2}}{(\lambda_{2}+p)^{2}}$

for $i=2,\dots,n$. Using that $M\circ M$ is positive semi-definite, we have
$\eta_{i}\geq 0$ for $i\in[n]$. Furthermore, as
$d<\left(\frac{1}{2}-\varepsilon\right)n$, the inequality
$1-2p>2\varepsilon>0$ also holds. Therefore, $\eta_{n}\geq 0$ implies that
$(\lambda_{2}+p)^{2}\geq(1-2p)|\lambda_{n}|+p^{2}\geq
2\varepsilon|\lambda_{n}|$. Multiplying both sides of the previous inequality
by $(1+\lambda_{2})$ and using the fact that $\lambda_{2}+p\leq\lambda_{2}+1$,
we obtain

$(1+\lambda_{2})^{3}\geq
2\varepsilon|\lambda_{n}|(1+\lambda_{2})\geq\frac{\varepsilon d}{2}.$

Here the last inequality follows by Lemma 2.1. This gives the third inequality
in Theorem 1.1.

## References

  * [1] I. Balla. _Equiangular lines via matrix projection._ preprint, arXiv:2110.15842 (2021).
  * [2] F. Ihringer. _Approximately Strongly Regular Graphs._ Discrete Math. 346 (3) (2023): 113299.
  * [3] A. Nilli. _On the second eigenvalue of a graph._ Discrete Mathematics, 91 (2) (1991): 207–210.
  * [4] E. Räty, B. Sudakov, I. Tomon. _Positive discrepancy, MaxCut, and eigenvalues of graphs._ preprint, arXiv:2311.02070.