††thanks: D.A.G and R.R. contributed equally to this work.††thanks: D.A.G and
R.R. contributed equally to this work.††thanks: Corresponding author.
<EMAIL_ADDRESS>
# Polymers in Two-Dimensional Bacterial Turbulence

David A. Gagnon  Department of Mechanical Engineering & Applied Mechanics,
University of Pennsylvania, Philadelphia, PA 19104 Department of Physics and
Institute for Soft Matter Synthesis & Metrology, Georgetown University,
Washington, D.C. 20057 Ranjiangshang Ran  Department of Mechanical Engineering
& Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 Paulo
E. Arratia  Department of Mechanical Engineering & Applied Mechanics,
University of Pennsylvania, Philadelphia, PA 19104

###### Abstract

We experimentally investigate the effects of polymer additives on bacterial
turbulence in two-dimensional (2D) films of swarming Serratia marcescens. We
find that even minute amounts ($\leq 20$ ppm) of polymers can suppress
velocity fluctuations and increase the size and lifetime of large-scale
coherent flow structures. In addition, we report an upscale transfer of
enstrophy and energy using the recently developed filtering techniques. Unlike
in classical 2D turbulence, both enstrophy and energy fluxes move primarily
towards large scales in bacterial turbulence; such fluxes are greatly modified
by polymer additives.

Relatively simple life forms, like bacteria and protozoa, can exhibit complex
behaviors, such as swarming Kearns (2010); Ariel  _et al._ (2015), quorum
sensing Waters and Bassler (2005); Daniels et al. (2006), and biofilm
formation Costerton  _et al._ (1999); Hall-Stoodley  _et al._ (2004); Desai
and Ardekani (2020). At sufficiently high cell densities, microorganisms can
communicate chemically Budrene and Berg (1995); Kalinin  _et al._ (2009) and
hydrodynamically Sokolov  _et al._ (2007); Sokolov and Aranson (2012), and
move together in a coordinated manner known as collective motion Koch and
Subramanian (2011). An intriguing phenomenon is the emergence of turbulent-
like features in bacterial suspensions; examples include large-scale coherence
Dombrowski  _et al._ (2004); Sokolov  _et al._ (2007), strongly fluctuating
velocity Guasto  _et al._ (2010); Rushkin  _et al._ (2010), and anomalous
diffusivity Ariel  _et al._ (2015); Mukherjee  _et al._ (2021). Due to the
qualitative similarity to turbulence at high Reynolds numbers, these behaviors
are often referred as “bacterial turbulence” Wolgemuth (2008); Wensink  _et
al._ (2012); Dunkel  _et al._ (2013). Both theoretical and numerical studies
based on continuum theory Dunkel  _et al._ (2013); Wensink  _et al._ (2012);
Saintillan and Shelley (2008) as well as discrete swimming particles
Hernandez-Ortiz  _et al._ (2005); Underhill  _et al._ (2008); Leoni and
Liverpool (2010) have shown that hydrodynamic effects alone can capture some
of the main features of bacterial turbulence, even without biochemical
interactions.

Microorganisms often live in fluid environments where (bio)polymers are
present Lauga (2007). For instance, bacteria can secrete slime to reduce
friction while swarming across a solid surface Angelini  _et al._ (2009), and
produce protective exopolymeric matrix during the formation of biofilms
Seminara  _et al._ (2012). How the presence of polymer molecules in the fluid
media affect the swimming behavior of single microorganisms has been
investigated for the past decade or so; both enhancement Teran  _et al._
(2010); Martinez  _et al._ (2014); Patteson  _et al._ (2015); Zöttl and
Yeomans (2019) and hindrance Shen and Arratia (2011); Liu  _et al._ (2011);
Thomases and Guy (2014); Qin  _et al._ (2015) in swimming speed have been
found depending on the often nonlinear interaction between the swimmer
kinematics, velocity fields, and fluid rheological properties.

Less explored, however, are the effects of polymers on the collective behavior
of swimming microorganisms. A numerical study on the collective dynamics of
rod-like swimmers shows that fluid elasticity can suppress velocity
fluctuations and break down large-scale flow structures Li and Ardekani
(2016). Simulations based on mean field theory suggest that fluid elasticity
can mediate hydrodynamic interactions and lead to larger coordinated
structures Bozorgi and Underhill (2011). Recently, large oscillatory vortices
are found in bacterial suspensions inside droplets containing viscoelastic
fluids (DNA suspensions); the observed spatial-temporal order is found for
fluids with sufficiently high levels of elasticity Liu  _et al._ (2021).
Despite recent advances, there is still a dearth of investigations on the
effects of polymers on the collective motion of microswimmers, particularly in
the (ultra) dilute regime where polymer molecules have a relatively minor
effect on the bulk fluid properties. As a result, our understanding of the
collective behaviors of living organisms in polymeric fluids remains
incomplete.

Figure 1: PIV flow fields of swarming Serratia marcescens in the buffer and
polymeric solutions. (a, b) velocity magnitude fields and (c, d) vorticity
fields, for the PBS buffer (left) and polymeric solutions with 20 ppm of CMC
(right). Solid lines are streamlines computed from instantaneous velocity
fields.

In this manuscript, we experimentally investigate the effects of polymer
additives on the collective dynamics of swarming bacteria in quasi-2D liquid
films. Our results show that even minute amounts of polymers ($\leq 20$ ppm)
can significantly enhance bacterial collective motion and promote large-scale
coherence. Velocimetry data show that the size and lifetime of the flow
structures are nearly doubled in the presence of polymers, and velocity
fluctuations are suppressed. Energy spectra show a power law of $-5/3$,
reminiscent of the inverse energy cascade scaling in 2D Newtonian turbulence.
Surprisingly, our calculations show that the primary directions of both energy
and enstrophy fluxes are inverse (upscale) in bacterial turbulence. The
inverse enstrophy flux increases substantially with the addition of polymers,
which is a potential mechanism for the increase in large-scale coherence.

Swarming experiments are performed on ATCC 274 strain of Serratia marcescens,
a rod-shaped bacterium that is on average 2 m long and 0.8 m in diameter. When
cultivated on soft agar plate (see Supplemental Material Sup  for details),
the bacteria differentiate into swarmer cells with additional (10 to 100)
flagella and elongated bodies of $\geq 5~{}\umu$m Alberti and Harshey (1990).
Polymeric solutions are prepared by diluting a carboxymethyl cellulose (CMC,
$7\times 10^{5}$ MW) stock solution in phosphate-buffered saline (PBS) to
final concentrations of 5, 10, and 20 ppm. Note that the highest polymer
concentration is much below the overlap concentration $c^{*}$ ($\leq
0.2\%\,c^{*}$). A 2-L drop of PBS or CMC solution containing swarming S.
marcescens Sup  is placed in a thin-film apparatus Patteson  _et al._ (2015);
Qin  _et al._ (2015); Sup , and stretched into an approximately 1-cm2 large
and 40-m thin film. Images are taken using bright-field microscopy and a CMOS
camera (Flare 4M180) at 24 frame/s. Velocity fields of swarming bacteria are
obtained using particle imaging velocimetry (PIV, see Gagnon  _et al._
(2020)), with a total number of 6400 or $80\times 80$ interrogation windows,
each of a size of $25\times 25$ pixel or $7.0\times 7.0$ m2.

Figure 1 shows experimental velocity and vorticity fields for the buffer (PBS)
and the 20 ppm CMC solution (see movies in Sup ); instantaneous streamlines
are plotted on top of the fields to better visualize structures. The flow
fields show that the addition of polymers significantly increases the swimming
speed of S. marcescens; the maximum velocity magnitude is nearly doubled from
10 m/s in the buffer to 20 m/s in the polymeric solution [Fig. 1(a) and 1(b)].
While it has been previously found that a single microbe can swim faster in
polymeric solutions Teran  _et al._ (2010); Martinez  _et al._ (2014); Zöttl
and Yeomans (2019); Patteson  _et al._ (2015), the flows induced by bacterial
collective motion here are not merely scaled up with a higher swimming speed.
If that were the case, one would expect the flow structures to remain roughly
of the same size. Here, on the other hand, we find that the flow structures
length-scale increases with the addition of polymers, as shown by the
vorticity fields [Fig. 1(c) and 1(d)]. This indicates that bacterial
collective motion in these ultra-dilute polymeric fluids have distinct
underlying flow structures from those in Newtonian fluids.

Figure 2: (a) PDFs of the velocity magnitude $u$ for different polymer
concentrations $c$. The black solid curve is a Rayleigh distribution fit for
the 20 ppm CMC data. Inset: root mean square (RMS) velocity $\bar{u}$ versus
$c$. (b) PDFs of the velocity components $u_{x}^{*}$ (circles) and $u_{y}^{*}$
(squares), normalized to have a zero-mean and a unity-variance. Solid curves
are Gaussian and super-Gaussian fits to the data. Inset: kurtosis of the PDFs
of $u_{x}^{*}$ and $u_{y}^{*}$ as a function of $c$.

The effects of polymers on flow structures in swarming bacteria are further
quantified by calculating the probability density functions (PDFs) of the
velocity magnitude fields. We find that the addition of 20 ppm of CMC ($\leq
0.2\%\,c^{*}$) more than doubles the maximum swimming speed [Fig. 2(a)] and
roughly triples the mean speed $\bar{u}$ [Fig. 2(a), inset]. We note that
these PDFs are not simply rescaled, rather they follow different statistical
distributions. As polymer is added to the swarms, the PDFs of the velocity
magnitudes tend to a Rayleigh distribution [black curve in Fig. 2(a)]. The
Rayleigh distribution is a feature of the magnitude of a vector with two
independent Gaussian components. This result suggests that the PDFs of
velocity components are Gaussian for the 20 ppm CMC case (and non-Gaussian for
the buffer case).

To test this hypothesis, we compute the PDFs of the in-plane velocity
components $u_{x}$ and $u_{y}$ for the 0 (PBS) and 20 ppm CMC cases [Fig.
2(b)]. For better contrast, the velocity components $u_{x}^{*}$ and
$u_{y}^{*}$ are normalized to have a mean of zero and a standard deviation of
unity. Importantly, we find no noticeable difference between the PDFs of x-
and y-velocity components, suggesting the in-plane motion of bacteria is
statistically isotropic. In the buffer case (0 ppm), the velocity
distributions are broadened, with heavy tails at high velocities. A
generalized Gaussian function fitting, $N\exp(-c|u_{x,y}^{*}|^{\beta})$,
reveals that the PDFs are super-Gaussian with $\beta\approx 1.4$. In contrast,
such tails are absent in the polymeric case (20 ppm), and the PDFs are
approximately Gaussian with $\beta\approx 2.0$. The polymer additives seem to
reduce the velocity distributions tails by suppressing velocity fluctuations.

The suppression of tails in the velocity PDFs can be characterized by the
kurtosis of velocity components [Fig. 2(b), inset]. The kurtosis is 3 for a
Gaussian distribution, greater than 3 for super-Gaussian and less than 3 for
sub-Gaussian distributions. We find that as the polymer concentration
increases, the kurtosis of velocity components decreases from $\sim 4.5$ in
the buffer to $\sim 3$ in the polymeric fluid (20 ppm). The decrease in
kurtosis suggests that polymers suppress outlier velocities and weaken the
intermittency of fluctuations in 2D bacterial turbulence. This is likely due
to polymer molecules mediating the hydrodynamic interaction between nearby
bacteria, which reduces the likelihood of local swimming velocity deviating
from the mean swarming velocity. The decrease in velocity fluctuation with
polymers is consistent with the observation in previous numerical simulations
Li and Ardekani (2016). Note that polymer additives have an opposite effects
in classic 2D turbulence, where the sudden release of polymer elastic energy
increases intermittency and the kurtosis of velocity distributions Amarouchene
and Kellay (2002); Boffetta  _et al._ (2003); Kellay (2004).

Figure 3: (a, b) Spatial correlation functions of (a) velocity and (b)
vorticity. Inset: vorticity integral length scale $L_{\omega}$. (c, d)
Temporal correlation functions of (c) velocity and (d) vorticity. The time lag
$\tau$ is normalized by the eddy turnover time $L_{\omega}/\bar{u}$ in (c),
and by the enstrophy time scale $\bar{\Omega}^{-1/2}$ in (d). Inset:
normalized vorticity integral time scale $T_{\omega}\,\bar{\Omega}^{1/2}$.

Polymer mediation on local bacteria interaction may result in a long-range
hydrodynamic effect, which could explain the increase in structure size shown
in Fig. 1. The flow structure size can be quantified by the spatial
correlation functions of velocity $\mathbf{u}$ and vorticity $\omega$, defined
as:
$C_{u}(r)=\langle\mathbf{u}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r})\rangle/\langle
u^{2}\rangle$ and
$C_{\omega}(r)=\langle\omega(\mathbf{x})\,\omega(\mathbf{x}+\mathbf{r})\rangle/\langle\omega^{2}\rangle$.
Here $\langle\cdot\rangle$ denotes the spatiotemporal ensemble average. As
polymer is added, we find that the velocity fields are increasingly correlated
over a distance of $250~{}\umu$m [Fig. 3(a)]. Similar increases in spatial
correlations are also found in the vorticity fields [Fig. 3(b)]. The average
vortex size can be estimated by the integral length scale of vorticity
$L_{\omega}$, defined by the convergent integral:
$L_{\omega}=\int_{0}^{\infty}C_{\omega}(r)\,dr$. Inset of Fig. 3(b) shows that
the average vortex size increases by roughly $50\%$, from $\sim 30~{}\umu$m in
the buffer to $\sim 45~{}\umu$m in the 20 ppm CMC solution. This increase in
structure size with polymers is in contrast to previous numerical studies Li
and Ardekani (2016), where the induced polymer stress breaks down large-scale
flow structures, but the cluster size of swimmers is increased. A much lower
swimmer concentration was used in the simulations than in the current
experiments, which may explain this discrepancy.

The lifetime of flow structures are examined by the temporal correlation
functions of velocity $\mathbf{u}$ and vorticity $\omega$, defined as:
$C_{u}(\tau)=\langle\mathbf{u}(t)\cdotp\mathbf{u}(t+\tau)\rangle/\langle
u^{2}\rangle$ and
$C_{\omega}(\tau)=\langle\omega(t)\,\omega(t+\tau)\rangle/\langle\omega^{2}\rangle$.
To compensate for the increases in flow speed and structure size with
polymers, the time lag $\tau$ for velocity correlation is rescaled by the eddy
turnover time, $L_{\omega}/\bar{u}$. An increase in velocity temporal
correlations is found [Fig. 3(c)] up to half of an eddy turnover time ($\sim
5$ s), suggesting that polymers increase the average lifetime of flow
structures. The vorticity fields are also increasingly correlated in time with
the addition of polymers [Fig. 3(d)]. Here, the time lag $\tau$ is normalized
by the enstrophy time scale, $\bar{\Omega}^{-1/2}$, where enstrophy is defined
by the mean square vorticity, $\bar{\Omega}=\langle\omega^{2}\rangle/2$. The
mean lifetime of flow structures can be measured by the vorticity integral
time scale, defined as: $T_{\omega}=\int_{0}^{\infty}C_{\omega}(\tau)\,d\tau$.
We find that the normalized mean lifetime is more than doubled in the 20 ppm
case compared to the buffer [Fig. 3(d), inset]. Overall, these results imply
that the interplay between polymer stresses and bacteria interaction leads to
a longer flow memory in 2D bacterial turbulence.

Figure 4: Energy spectra $E(k)$ for different polymer concentrations. Solid
curves are exponential fits to low-wavenumber data, $\exp(-kL_{c})$. A power
law of $k^{-5/3}$ (dashed line) is drawn for comparison. Inset: characteristic
lengthscale $L_{c}$ obtained from the exponential fits, versus polymer
concentration.

Next, we explore how the presence of polymer affect the transfer of (kinetic)
energy across different length scales in bacterial turbulence. We compute the
energy spectra, $E(k)=2\pi k\langle|\hat{\mathbf{u}}(\mathbf{k})|^{2}\rangle$,
where $\hat{\mathbf{u}}(\mathbf{k})$ denotes the Fourier transform of the
velocity field. An increase in spectral power at all wavenumbers with polymers
is found (Fig. 4) due to the enhancement in bacterial swimming speed. At large
scales, energy spectra is found to follow an exponential decay,
$\exp(-kL_{c})$, for all polymer concentrations. Exponential decay in energy
spectra has been observed in flows dominated by viscous dissipation Alexakis
and Biferale (2018) and expected here since the Reynolds number $Re\ll 1$. The
length scale associated with the exponential decay, $L_{c}$, increases with
polymer concentration (Fig. 4, inset). This is consistent with our
observations in the spatial correlations [Fig. 3(a) and 3(b)], although the
value of $L_{c}$ is larger ($\sim 350$ to 450 m) than the integral length
scale, since $L_{c}$ is primarily associated with the exponential decay at the
smallest wavenumbers.

A well-defined power law of $k^{-5/3}$ for the energy spectra is found at
small-scales of all cases (Fig. 4). In classic Newtonian 2D turbulence, the
power law of $k^{-5/3}$ suggests the existence of an inverse energy cascade
Batchelor (1969); Kraichnan (1967). In bacterial turbulence, kinetic energy is
injected by bacteria at the smallest length scale of the flow, which admits an
inverse flux of energy towards large scales. The addition of polymers does not
affect this inverse energy cascade scaling. Note that the power law of
$k^{-5/3}$ is different from the $k^{-8/3}$ scaling that has been found in
previous experimental studies Wensink  _et al._ (2012). This discrepancy is
likely due to the difference in experimental geometry; while our experiments
are performed in free-standing films, previous studies were conducted in
microfluidic channels with no-slip boundary conditions Wensink  _et al._
(2012); Dunkel  _et al._ (2013).

We further examine the spatial fluxes of energy and enstrophy using a recently
developed filtering technique Rivera  _et al._ (2003); Chen  _et al._ (2003,
2006). Central to this technique is to obtain a low-pass filtered velocity
field, $u^{(k)}$, containing the information only at wavenumbers smaller than
$k$. Here, the low-pass filter is chosen to be a Gaussian function 111The low-
pass filtering is performed in Fourier space through multiplication of a
Gaussian function, $\hat{G}^{(k)}(\boldsymbol{q})=\exp(-q^{2}/24k^{2})$. More
details are available in: Xiao  _et al._ (2009).. The local energy flux is
defined as: $\Pi_{E}^{(k)}=-\tau^{(k)}_{ij}(\partial u^{(k)}_{i}/\partial
x_{j})$, where $\tau^{(k)}_{ij}=(u_{i}u_{j})^{(k)}-u_{i}^{(k)}u_{j}^{(k)}$ is
a stress tensor arising from the coupling between the resolved large scales
and the filtered small scales Kelley and Ouellette (2011). By definition, the
local energy flux is positive for energy flow to smaller scales ($>k$) and
negative for energy flow to larger scales ($<k$). The local enstrophy flux is
defined in a similar manner, with the same sign convention:
$\Pi_{\Omega}^{(k)}=-\sigma^{(k)}_{j}(\partial\omega^{(k)}/\partial x_{j})$,
where $\sigma^{(k)}_{j}=(\omega u_{j})^{(k)}-\omega^{(k)}u_{j}^{(k)}$ is a
vector describing the spatial transport of vorticity due to the elimination of
smaller scale vortices Chen  _et al._ (2003).

The mean energy flux, $\langle\Pi_{E}^{(k)}\rangle$, is shown in Fig. 5(a),
for 0 (buffer) and 20 ppm CMC cases. To compensate for the increase in kinetic
energy and enstrophy with polymers, $\langle\Pi_{E}^{(k)}\rangle$ is
normalized by the factor $\bar{u}^{2}\bar{\Omega}^{1/2}$ to be dimensionless.
Results show that in the buffer (0 ppm) case the net flow of energy is upscale
(inverse) at all wavenumbers, consistent with the $k^{-5/3}$ inverse energy
cascade scaling. In the polymeric liquid (20 ppm), however, energy moves
primarily towards larger scales, with a reverse of direction towards smaller
scales at low $k$. This polymer-induced forward energy transfer is likely due
to polymers extracting energy from large-scale flows by getting stretched, and
releasing it at smaller scales through relaxation. The result seems
unexpected, since the downscale transfer of energy should aid the breakdown of
large-scale flow structures, while an increase in vortex size with polymers is
observed. This is explained below by the changes in the mean enstrophy flux.

Figure 5: (a) Mean energy flux $\langle\Pi_{E}^{(k)}\rangle$ and (b) mean
enstrophy flux $\langle\Pi_{\Omega}^{(k)}\rangle$ in Fourier space, for 0 and
20 ppm CMC solution; both fluxes are normalized to be dimensionless.

The mean enstrophy flux, $\langle\Pi_{\Omega}^{(k)}\rangle$, normalized by
$\bar{\Omega}^{3/2}$ to be dimensionless, is shown in Fig. 5(b). We find that
the transfer of enstrophy is primarily upscale with slight downscale flux at
low $k$ values for both 0 and 20 ppm CMC cases. This inverse transfer of
enstrophy in bacterial turbulence may have been expected (it has been
suggested for sperm cells Creppy  _et al._ (2015)) since both energy and
enstrophy are injected at the smallest length scale. Remarkably, the addition
of polymers to the media enhances the inverse transfer of enstrophy towards
larger scales, which may explain the increase in vortex size. In other words,
the origin of larger coherent structures is not the accumulation of kinetic
energy at large scales; rather, it is due to a enhanced transfer of vorticity
or enstrophy in the presence of polymers.

In summary, we investigate the effects of polymers on 2D bacterial turbulence.
Our experiments show that polymers can significantly affect bacterial
collective motion by suppressing velocity fluctuations and increasing the size
and lifetime of coherent flow structures. Further analysis on the energy and
enstrophy fluxes reveals that polymers enhance the inverse transfer of
enstrophy, which can be responsible for the large-scale coherence. Our work
extends the studies of active turbulence in Newtonian fluids to a more general
case of viscoelastic fluids. These results can be helpful in understanding the
collective dynamics of microswimmers in more common viscoelastic fluid
environments, such as spermatozoa in cervical mucus and bacteria in biofilms.

We thank Daniel Blair and Christopher Browne for helpful discussions. This
work was supported by NSF Grant No. DMR-1709763.

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