††thanks: Email address.

# Microdomains and Stress Distributions in Bacterial Monolayers on Curved
Interfaces

Blake Langeslay1 Gabriel Juarez2<EMAIL_ADDRESS>1Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
2Department of Mechanical Science and Engineering, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA

###### Abstract

Monolayers of growing non-motile rod-shaped bacteria act as active nematic
materials composed of hard particles rather than the flexible components of
other commonly studied active nematics. The organization of these granular
monolayers has been studied on flat surfaces but not on curved surfaces, which
are known to change the behavior of other active nematics. We use molecular
dynamics simulations to track alignment and stress in growing monolayers fixed
to curved surfaces, and investigate how these vary with changing surface
curvature and cell aspect ratio. We find that the length scale of alignment
(measured by average microdomain size) increases with cell aspect ratio and
decreases with curvature. Additionally, we find that alignment controls the
distribution of extensile stresses in the monolayer by concentrating stress in
low-order regions. These results connect active nematic physics to bacterial
monolayers and can be applied to model bacteria growing on droplets, such as
marine oil-degrading bacteria.

##  I Introduction

The role of mechanical forces in bacterial growth is of increasing interest to
both biologists and physicists Persat  _et al._ (2015); You  _et al._ (2018);
Duvernoy  _et al._ (2018); Boyer  _et al._ (2011); Grant  _et al._ (2014);
Volfson  _et al._ (2008). Bacteria colonize a wide variety of interfaces
(liquid-solid, liquid-air, liquid-liquid) with vastly different properties
Marshall (1986); Krajnc  _et al._ (2022); Conrad (2020); Niepa  _et al._
(2017). To thrive under these diverse conditions, cells must contend with
physical forces such as surface tension and hydrodynamic interactions to
stably adhere to the interface over many generations. Bacteria growing on a
flat interface as a monolayer of cells have been successfully modeled as an
active nematic material. This is a widely studied class of active liquid
crystal whose components align parallel to one another and have end-to-end
symmetry You  _et al._ (2018); Dell’Arciprete  _et al._ (2018). Rod-shaped
bacteria cells have the required symmetry and, when in a dense monolayer,
align due to steric interactions between cells. Extensile activity can be
produced by motility Copenhagen  _et al._ (2021) or by the growth and division
of the cells Dell’Arciprete  _et al._ (2018). In either case, forces in the
monolayer are strongly coupled to alignment, as both motility and growth exert
forces along the axis of orientation.

The active nematic model of bacterial monolayers has proven powerful in
predicting the internal forces and behavior of real systems such as monolayers
of gliding Myxobacteria and chaining biofilms Copenhagen  _et al._ (2021);
Yaman  _et al._ (2019). In particular, the behavior of cells near topological
defects has been successfully tied to the behavior of other well-studied
active nematic systems. Topological defects are singularities in the director
field of liquid crystal alignment, points around which there is a net rotation
of the director; this rotation is defined as the defect’s charge. In active
nematic materials, defects are almost always limited to charges of $\pm 1/2$
Giomi  _et al._ (2013). These singularities drive much of the unique behavior
of active nematics. Comet-shaped +1/2 defects move as motile particles that
generate complex flows Giomi  _et al._ (2013); Thampi  _et al._ (2013), and
material accumulates at positively charged defects and depletes at negatively
charged defects, allowing 2D materials to escape into the third dimension by
multilayering or buckling Guillamat  _et al._ (2022); Saw  _et al._ (2017);
Endresen  _et al._ (2021); Turiv  _et al._ (2020); Kawaguchi  _et al._ (2017).
This last effect has been observed to drive transitions from monolayers to
multilayered 3D structures Copenhagen  _et al._ (2021); Yaman  _et al._
(2019).

Previous work on monolayers of rod-shaped bacteria has indicated that when the
cells are hard rods (rather than the flexible rods observed in species such as
Myxobacteria), their alignment behavior changes. In this case, working with
bacteria such as E. coli, cells have been observed to segregate into
microdomains, regions of near-parallel local alignment analogous to grains in
crystalline or granular materials You  _et al._ (2018). Rather than having a
continuous, gradually changing alignment field, these systems exhibit sharp
changes in alignment across the boundaries between microdomains. This
represents a fundamentally different type of liquid crystal behavior from that
observed in active nematics composed of microtubules or flexible cells. In
these systems, microdomains and their boundaries can replace topological
defects as a way of mapping alignment You  _et al._ (2018).

The alignment behavior of simulated and microtubule-based active nematic
materials is well known to change based on the curvature of their substrate.
Topological defects respond to curvature, with $+1/2$ ($-1/2$) defects
accumulating at regions of positive (negative) Gaussian curvature Alaimo  _et
al._ (2017); Ellis  _et al._ (2018); Nestler and Voigt (2022). However, it is
currently unknown what effect curvature might have on the alignment of a more
granular hard-cell system. To understand how stresses behave in a curved
growing monolayer (for example, one growing on a droplet of oil in water), it
is crucial to first understand the alignment behavior of its cells.

To investigate this issue, we simulated the growth of rod-shaped cells on
spherical surfaces. This enabled an analysis of how alignment responds to
curvature and how stresses in turn respond to alignment. We find that both
cell aspect ratio and surface curvature play a role in controlling the length
scale of alignment, with higher curvature substrates limiting alignment.
Additionally, we find that regions of high stress are predicted by the
orientational order, a general measure of low local alignment. These results
enable predictions of how granular monolayer behavior might vary on
differently curved surfaces.

##  II Methods

###  II.1 Experiments of bacterial growth at flat liquid interfaces

Cell cultures of _A. borkumensis_ were grown for 24 hours in ATCC medium 2698
at 30 ∘C in an orbital shaker at 180 RPM. Cells were non-motile and rod-shaped
with an average length of 2.7 µm and width of 0.7 µm. To observe bacterial
growth at oil-water interfaces, a custom microfluidic device was used. A flat
oil-water interface was pinned to a microscope slide by a thin copper TEM grid
(18 µm, SPI Supplies, 2010C-XA) with square apertures 205 µm wide. The cell
culture was injected above this, allowing cells to adsorb on the interface.
Then, a microfluidic chamber was constructed around the grid to house the
interface and allow for constant flow of growth media (diluted 10:1 with
artificial seawater) at 2 µL min-1. This flow prevented additional cells from
settling on the grid during the experiment.

Time-lapse phase contrast microscopy was used to image the growing cell colony
8 hours using a $60\times$ objective (NA $=0.6$ and a depth of field of 2 µm).
A single square aperture in the TEM grid was imaged in an experiment, selected
for clarity and lack of visible contaminants. Images were recorded with a 50
ms exposure time at 2-minute intervals for 24 hours.

###  II.2 Simulations of bacterial growth on flat and curved substrates

Molecular dynamics simulations were conducted to obtain precise quantitative
data on the physical characteristics of growing bacterial monolayers on flat
and curved surfaces. Each cell was modeled as a spherocylinder with a diameter
$d_{0}$ and length $l$ between the endcaps. To simulate cell growth, the
length of the cylinder $l$ increased linearly with time up to a maximum length
$l_{0}$ while the diameter remained fixed at $d_{0}$. The changes in position
$\vec{x}$ and orientation $\theta$ of each cell were modeled by the overdamped
Newton’s equations as follows:

$\frac{d\vec{x}}{dt}=\frac{1}{l\zeta}\vec{F}$ (1)
$\frac{d\theta}{dt}=\frac{1}{l^{3}\zeta}\tau$ (2)

where $\zeta$ is the effective viscosity of the interface based on the
interaction between the bacterium and its surroundings and $\vec{F}$ and
$\tau$ are the total force and torque on the cell due to cell interaction
forces. The interactions between cells were modeled as Hertzian forces, with
the force on cell $i$ due to cell $j$ calculated as follows:

$\vec{F_{ij}}=Yd_{0}^{1/2}h_{ij}^{3/2}\vec{N_{ij}}$ (3)

where $Y$ is proportional to the Young’s modulus of a cell, $h_{ij}$ is the
overlap distance between the two cell bodies, and $\vec{N_{ij}}$ is the vector
normal to cell $j$ at the point of contact Orozco-Fuentes and Boyer (2013);
You  _et al._ (2018); Hertz (1882).

Cell growth was modeled using a time-independent rate $g_{0}$. To prevent the
synchronized division of cells, each cell was assigned random a value between
$g_{0}/2$ and $3g_{0}/2$ for the growth rate. Whenever a cell length $l$
exceeded the maximum length $l_{0}$, cell division would occur where the cell
would split into two identical cells, each with a length $(l_{0}-d_{0})/2$.
The new cell would be initialized with the same orientation $\theta$ as the
original cell, however, the new cell would be assigned a different randomized
growth rate.

Simulations of cell growth on flat substrates were initialized with a single
parent cell, whereas simulations of growth on spherical substrates were
initialized with two parent cells, one located at the north pole and the other
at the south pole of the sphere. This resulted in two hemispherical colonies
growing until contact was made at the equator, after which, the distribution
of cells rapidly became homogeneous across the entire surface. Data was
collected on fully-covered spheres once they had reached a packing fraction of
$\phi=1.05$, where $\phi$ was defined as the area fraction of the surface
covered by cells. In all simulations, the center of volume of cells was
constrained to be attached to both flat and spherical substrates and the cell
orientation was constrained parallel to the surface (or to the tangent plane
at the point of contact, for spherical substrates). Therefore, no out-of-plane
motion was allowed.

Simulation model parameter values were chosen to be representative of a
generic gram-negative, rod-shaped bacterium, including those for the A.
borkumensis cells used in experiments at flat liquid interfaces You  _et al._
(2018). Therefore, the values were set to the following: cell diameter $d_{0}$
to 0.7 µm, Young’s modulus $Y$ to 4 MPa, drag per length $\zeta$ to 200 Pa h,
and growth rate $g_{0}$ to 2 µm h-1. A simulation time step of $5\times
10^{-6}$ hours was used. To study the effect of different cell aspect ratios,
the maximum growth length allowed was varied between $2<l_{0}<5$ µm. Cell
elongation was parametrized with the aspect ratio $\alpha$, defined here as
$\alpha=(l_{0}+d_{0})/d_{0}$, therefore, ranging from $3.9<\alpha<8.1$. To
study the effect of varying substrate curvature $\kappa$, spherical substrates
with radius $R=10,12,15,20,\ \text{and}\ 30$ µm were used. Here, substrate
curvature is defined as the Gaussian curvature, or $\kappa=R^{-2}$, for a
spherical surface.

##  III Results

Figure 1:  Formation of a bacterial monolayer due to growth at flat and curved
substrates. (a) Micrographs of _A. borkumensis_ cells growing on a flat liquid
interface at (top) early times and (bottom) up to full surface coverage. (b,
c) Simulations of cell growth on (b) flat and (c) spherical substrates with
$R=20$ µm at (top) early times and (bottom) up to full surface coverage.
Colors correspond to the cell orientation angle. All scale bars shown
represent 20 µm. See Supplementary videos SV1, SV2, and SV3.

In both experiments and simulations, bacteria grow and divide at the surface,
forming a monolayer that eventually covers the entire available surface area,
shown in Fig. 1. At early times, single cells grow and divide to form
colonies, shown in Fig. 1 (top row). At later times, as the cells continue to
grow and divide, their contact forces and torques cause them to align with
their neighbors to form a liquid crystal with nematic symmetry, shown in Fig 1
(bottom row). As cells grow, the area covered by the colony increases
exponentially with time until the available surface area is fully covered.
These behaviors are consistently observed in experiments of growth on flat
interfaces and in simulations of growth on flat and spherical substrates.

To establish correspondence between experiments and simulations, the
distribution of topological defects in cell monolayers on flat interfaces and
substrates, respectively, are compared. First, to identify topological
defects, a director field was established. In experiments, the director field
was determined using a custom image-processing algorithm based on the
brightness gradient of the phase contrast images. In simulations, the director
field was generated directly from the position and orientation of each cell.
Then, to locate topological defects within the director field, each point on
the grid (or each cell in simulations) was tested for a net rotation of the
surrounding director field (with net rotations of $\pm\pi/2$ corresponding to
defect charges of $\pm 1/2$) DeCamp  _et al._ (2015). Nearby points with
similar net rotations were then separated into clusters, with the centroid of
each cluster corresponding to a defect of the associated charge.

The distribution of topological defects in experiments at flat interfaces
corresponds to simulations of growing cells with the same dimensions as A.
borkumensis on flat substrates, shown in Fig. 2 (top row, flat) and
Supplementary Fig. S1. This is determined using the mean defect separation,
defined as the average over the three nearest-neighbor distances for all
defects. In experiments, the mean defect separation is $8.9\pm 3.1$ µm, while
in simulations, the mean defect separation is $8.17\pm 2.9$ µm.

Based on the good qualitative and quantitative agreement between experiments
and simulations, the effect of cell aspect ratio $\alpha$ and substrate
curvature $\kappa$ on orientational order, the degree of alignment, and stress
within the cell monolayer was investigated using simulations. First, the
orientational order S was evaluated to measure the degree of local alignment
in the monolayer. This was calculated at each individual cell i using the
following equation:

$S_{i}=\sum_{j}\frac{1}{2}(3\cos^{2}(\theta_{i}-\theta_{j})-1)$ (4)

where $\theta_{j}$ is the orientation of each cell within a search radius of
cell $i$. For the purpose of this analysis, the search radius was set equal to
the division length $l_{0}$ of the cells. Here, $S\rightarrow 1$ represents
ordered regions while $S\rightarrow 0$ represents disordered regions.

Simulations reveal regions of high order separated by regions of low order,
shown in Fig. 2 (top row). These regions of high cell alignment emerge for all
combinations of substrate curvature and cell aspect ratio, including flat
substrates. For a given curvature, increasing the cell aspect ratio from
$\alpha=4.9$ to $\alpha=6.7$ increases the size of these regions, shown in
Fig. 2 ($R=20$ µm). Additionally, topological defects tend to coincide with
areas of low order, shown in Fig. 2 (top row). This follows from the
definition of defects, as the net rotation of the director field around the
defect requires imperfect alignment. Therefore, increasing the cell aspect
ratio also increases the distance between $\pm$ 1/2 defect separation, as
shown in Supplementary Fig. S2.

Figure 2:  Orientational order, topological defects, and microdomains in
simulations of monolayers with varying cell aspect ratio $\alpha$ and
substrate radii $R$. (Top row) Orientational order S and topological defects
with charge $\pm 1/2$ shown in red/blue, respectively. Topological defects
tend to occur in regions of low orientational order for all aspect ratios and
substrate radii. (Bottom row) Microdomain representation of the cell
monolayers shown in the top row. Microdomain boundaries tend to coincide with
regions of low orientational order for all aspect ratios and substrate radii.
For visualization purposes, spherical substrates ($R=10$ µm and $R=20$ µm) are
not shown to scale while the field of view for the flat simulations (leftmost
column) is $50\times 50$ µm.

Next, the mean microdomain area $\langle A\rangle$ was calculated to
characterize the degree of cell alignment in the monolayer. Boundaries between
microdomains are one-dimensional discontinuities in the alignment field rather
than point defects, separating regions of near-parallel alignment. Cells were
sorted into microdomains using the following two criteria: (i) cells were in
contact with one another and (ii) their orientation differed by less than 0.2
radians.

Microdomains, represented as differently colored regions, correspond to
regions of high-aligned cells, shown in Fig. 2 (bottom row). Borders between
large microdomains correspond to regions of low order, reflecting the
discontinuity in alignment between microdomains. Similarly, for a given
curvature, increasing the cell aspect ratio increases the size of a single
microdomain, shown in Fig. 2 ($R=20$ µm). The area of a microdomain is given
as $A=\phi\sum{A_{i}}$, where $A_{i}$ are the areas of each cell component
within the microdomain. For all substrate curvatures and aspect ratios
investigated here, the distribution of microdomain areas within the monolayer
is described by $P(A)\propto$ exp($A/\langle A\rangle$), where $\langle
A\rangle$ is the mean microdomain area for that monolayer, shown in
Supplementary Fig. S3. You  _et al._ (2018). The mean microdomain area for a
system is used to characterize the area scale of its cell alignment.

The mean microdomain area increased with increasing cell aspect ratio, shown
in Fig. 3(a). For the lowest curvature ($\kappa=0.001$, $R=30$ µm) the effect
of cell aspect ratio was the most dramatic, with an increase in the mean
domain area of seven-fold. For the highest curvature ($\kappa=0.01$, $R=10$
µm), however, the effect of cell aspect ratio was less prominent, producing an
increase in the domain area of a factor of two. In fact, the dependence of
mean domain area on aspect ratio could be described with a power law relation
in the range of aspect ratios investigated, with scaling exponents increasing
from 1.2 for $\kappa=0.01$ up to 2.9 for $\kappa=0.001$. The increase in
alignment at higher $\alpha$ is consistent with previous work on bacterial
monolayers, which has shown that more elongated (higher aspect ratio) cells
produce stronger alignment in flat monolayers You  _et al._ (2018).

Figure 3:  The effect of cell aspect ratio and substrate curvature on
microdomain area. (a) Log-log plot of mean microdomain area $\langle A\rangle$
for different cell aspect ratios $\alpha$ on five different substrate
curvatures. Microdomain area increases with the aspect ratio for all
curvatures. (b) Semilog plot of mean microdomain area $\langle A\rangle$ for
different substrate curvatures $\kappa$ and four different cell aspect ratios.
Microdomain area decreases with substrate curvature for all aspect ratios. (c)
Summary of the mean microdomain area $\langle A\rangle$ across the simulated
parameter space of aspect ratio and substrate curvature.

The mean microdomain area decreased with increasing substrate curvature, shown
in Fig. 3(b). For low aspect ratios ($\alpha\leq 5$), an inverse exponential
form with $\langle A\rangle\propto$ exp($-\kappa/\kappa_{0}$) accurately
describes the relation between area and curvature. Specifically, for $\alpha$
of $3.9$ and $4.9$, the value of $\kappa_{0}$ was equal to $0.039$ µm-2 and
$0.020$ µm-2, respectively. For high aspect ratios ($\alpha>5$), the inverse
exponential does not accurately describe the relation between area and
curvature, demonstrating a qualitative change in the system’s alignment
behavior. In all cases, however, more curved substrates produced consistently
lower microdomain areas.

Measurements of mean domain area across a range of aspect ratios and
curvatures were combined to show the system’s response over the
$\alpha-\kappa$ parameter space, shown in Fig. 3(c). The combined effect of
cell aspect ratio and substrate curvature on microdomain area is evident. The
largest domain areas are observed at high cell aspect ratios and low substrate
curvatures. The smallest domain areas, however, are observed at low cell
aspect ratios and high substrate curvatures.

Lastly, the parallel component of the Virial stress $\sigma_{\parallel}$ on
each cell was measured to determine the force distributions in the monolayer.
The Virial stress $\bm{\sigma}_{i}$ on a cell is given as follows:

$\bm{\sigma}_{i}=\frac{\phi}{a_{i}}\sum_{j}\bm{r}_{ij}\bm{F}_{ij}$ (5)

where $a_{i}$ is the area of cell $i$, $\bm{r}_{ij}$ is the vector from the
center of cell $i$ to the point of contact with cell $j$, and $\bm{F}_{ij}$ is
the force from cell $j$ on cell $i$. When $\bm{\sigma}$ is calculated in the
basis of vectors parallel and perpendicular to the cell’s orientation, it can
be decomposed into parallel ($\sigma_{\parallel}$), perpendicular
($\sigma_{\perp}$), and shear ($\tau$) components:

$\bm{\sigma}=\begin{bmatrix}\sigma_{\parallel}&\tau/2\\\
\tau/2&\sigma_{\perp}\end{bmatrix}$

The parallel stress $\sigma_{\parallel}$ corresponds to the force in the
direction of the cells’ growth. Because of the extensile nature of the system,
$\sigma_{\parallel}$ is always negative (corresponding to a compressive force)
You  _et al._ (2018).

Figure 4:  Parallel component of the Virial stress $\sigma_{\parallel}$ in
growing cell monolayers on curved surfaces. (a) Visualization of the
normalized parallel stress in cell monolayers on a curved surface with $R=20$
µm and for cell aspect ratios of $\alpha=4.9$ and $6.7$. (b) Scatter plot of
the normalized parallel stress of individual cells and their local
orientational order for the case of $R=20$ µm and $\alpha=4.9$. The trendline
shows the data binned by orientation order and the errorbars represent the
standard error. (c) Relation between the normalized parallel stress magnitude
and the orientational order for differing aspect ratios. Average parallel
stress magnitude increases by up to 30% in the lowest-order bin ($S<0.1$). (d)
Relation between the parallel stress magnitude and the orientational order for
differing surface curvatures. (e) Deviation from mean parallel stress at
topological defects ($+1/2$ in red, $-1/2$ in blue) and in low-order regions
($S<0.1$ in magenta). Low-order regions have a greater deviation from the mean
stress than topological defects for all curvatures and aspect ratios.

The parallel component of the Virial stress, normalized by the mean stress,
for each cell in the monolayer on a curved substrate with $R=20$ µm is
visualized and shown in Fig. 4(a). At the cell level, the normalized stress
varied over a wide range of values. For example, for $\alpha=4.9$ and $R=20$,
the mean stress was $\langle\sigma_{\parallel}\rangle=-0.043$ N m and the
normalized stress varied from a minimum of zero up to a maximum of $\sim
3.8\langle\sigma_{\parallel}\rangle$. For an aspect ratio of $\alpha=6.7$ and
the same surface curvature, the mean stress was
$\langle\sigma_{\parallel}\rangle=-0.035$ N m and the normalized stress varied
from a minimum of zero up to a maximum of $\sim
4.4\langle\sigma_{\parallel}\rangle$. Stress in growing monolayers with
varying values of $\alpha$ and $\kappa$ can be seen in Supplementary videos
SV4-SV7.

A scatter plot of the normalized parallel component of Virial stress and the
orientational order for a representative simulation ($R=20$, $\alpha=4.9$) is
shown in Fig. 4(b), grey points. It is evident that, at the individual cell
level, stress values can vary over a wide range with respect to the average
value of stress. While no trend is immediately visible in the scattered data,
binning the data by the orientational order $S$ reveals a relationship between
the normalized stress $\sigma_{\parallel}/\langle\sigma_{\parallel}\rangle$
and orientational order $S$. First, five evenly spaced bins were determined by
identifying the minimum $S_{min}$ and maximum $S_{max}$ values of the
orientational order for each simulation. Then, the mean orientational order
and mean parallel stress were computed for each bin. The result is shown in
Fig. 4(b), red line. For this example ($R=20$, $\alpha=4.9$), the binned data
shows that cells in regions of low-order experience a higher magnitude of
parallel stress compared to cells in high-order regions.

The trend of higher-magnitude stress at lower-order regions is a robust
observation for all simulations with varying cell aspect ratio and substrate
curvature. Plots of normalized parallel stress binned by order and averaged
across all simulations with the same aspect ratio or surface curvature are
shown in Fig. 4(c) and 4(d), respectively. For all cases, the average
normalized stress decreases with increasing orientational order. Specifically,
in regions of high orientational order ($S>0.5$), the local average of
normalized stress approaches unity, or
$\sigma_{\parallel}\approx\langle\sigma_{\parallel}\rangle$. In regions of low
orientational order, however, ($S<0.1$), the local average of normalized
stress is 15% to 35% larger than the global average stress, or
$\sigma_{\parallel}\geq 1.15\langle\sigma_{\parallel}\rangle$. The reason for
the large standard error for $\alpha=8.1$ and $S<0.1$ is that the high aspect
ratio cells are more aligned on average, resulting in a sparse number of cells
in low-order regions to analyze.

The stress in other active nematics such as those composed of epithelial cells
is higher near $+1/2$ topological defects, which themselves are low-order
regions by definition Guillamat  _et al._ (2022). To investigate whether this
effect was responsible for the correlation of high stress and low order in
bacterial monolayers, stress near topological defects was compared to the
previously calculated stress in low-order regions ($S<0.1$). The mean value of
the parallel component of the Virial stress ($\sigma_{\parallel}$) was
computed for all cells within close proximity ($r=l_{d}/2$) of a $\pm 1/2$
topological defect within the monolayer. Then, this value was compared to the
average normalized parallel stress of the bin with the lowest orientational
order ($S<0.1$) and is shown in Fig. 4(e). For all cases, the average
deviation from the mean stress near $\pm 1/2$ defects is less than 10%, or
$\sigma_{\parallel}\leq 1.1\langle\sigma_{\parallel}\rangle$. In contrast, the
average deviation from the mean stress in regions of low orientational order
($S<0.1$) is always greater than 15%, or $\sigma_{\parallel}\geq
1.15\langle\sigma_{\parallel}\rangle$, for all cell aspect ratios and
substrate curvatures. This shows that the observed high stress-low order
correlation in this system is not caused by stress concentration near
topological defects, and is instead a distinct effect.

##  IV Discussion

Previous work on monolayers of growing hard-rod cells has shown that, in the
case of a flat substrate, microdomain size increases with increasing cell
aspect ratio You  _et al._ (2018). Our work is consistent with this result,
and further confirms that the trend holds for monolayers growing on curved
substrates as well. Additionally, while previous work demonstrated the
relation between microdomain size and aspect ratio for an unconfined growing
colony You  _et al._ (2018), our results show that the same relation is true
for a growing colony confined to a finite substrate area (the surface of a
sphere). Together, these show that microdomain formation and its dependence on
cell aspect ratio are robust collective behaviors in growing hard-rod
monolayers under a variety of conditions.

We also find a new dependence of microdomain area on the curvature of the
surface. Regardless of cell aspect ratio, higher curvature substrates resulted
in smaller microdomains. This decrease in alignment is attributable to the
surface’s curvature restricting cells from lying parallel to each other.
Microdomains can only maintain close cell alignment over a small enough area
to be approximated as locally flat, and a microdomain spanning a much larger
area on a curved surface will eventually be geometrically required to
fracture. As curvature increases, the maximum area that can be treated as
locally flat decreases, and accordingly the domain area decreases as well.

In a continuous (non-granular) active nematic composed of flexible epithelial
cells, extensile stress is highest at positively charged topological defects
Guillamat  _et al._ (2022). This causes the cell layer to deform at these
points, producing “mounds” localized near the defectsGuillamat  _et al._
(2022). Similar deformations are seen in other continuous active nematics at
deformable 2D interfaces, such as microtubules on the surface of a vesicle or
actin fibers in the morphogenesis of multicellular organisms Keber  _et al._
(2014); Maroudas-Sacks  _et al._ (2021), and this behavior has been further
confirmed and studied in numerous simulations and theoretical works Metselaar
_et al._ (2019); Ruske and Yeomans (2021); Santiago (2018); Vafa and Mahadevan
(2008); Hoffmann  _et al._ (2022); Alert (2022). In a granular active nematic,
our simulations show that extensile stress concentrates at all low-order
regions rather than at the locations of point $+1/2$ topological defects. This
can be expected to result in different forms of deformation under growth
stress.

Our results are relevant for systems of bacteria growing at liquid-liquid
interfaces, such as at the oil-water interface of a droplet. For example,
previous work has shown that cell growth confined to the surface of a droplet
produces tube-like protrusions similar to those produced by continuous active
nematics Hickl and Juarez (2022); Prasad  _et al._ (2022). However, a complete
theoretical description of droplet deformation by cell growth is still
lacking. Our results suggest that these protrusions should nucleate at low-
order sites such as boundaries between microdomains, rather than nucleating
exclusively at $+1/2$ defects. By extension, this allows us to predict how
interfacial curvature influences deformations due to the underlying
microstructure. For example, higher curvature (smaller) droplets will produce
more closely spaced protrusions since the characteristic size of their
microdomains decreases.

Our results also emphasize the importance of constituent particle properties
on collective behavior. While a hard-rod monolayer exhibits many of the same
properties as a nematic composed of microtubules or other flexible components
Dell’Arciprete  _et al._ (2018), its internal forces are not well predicted by
topological defects. Furthermore, a growing self-organized monolayer produces
microdomains that are distinct from the continuous alignment fields of other
active nematic systems. The partially granular nature of a bacterial monolayer
is clearly of great importance to understanding its collective behavior.

In conclusion, we have shown that stress distributions in a hard-rod bacterial
monolayer vary predictably based on cell aspect ratio and substrate curvature.
Specifically, our experimentally validated simulations show that stress in a
hard-rod monolayer concentrates at low-order regions, which occur at the
boundaries of microdomains. The length scale of these microdomains increases
with cell aspect ratio and decreases with substrate curvature. These results
demonstrate that while a bacterial monolayer can be effectively modeled as a
continuum active nematic, in some cases when its cells act as hard rods, the
alignment and stress distributions behave in distinctly different ways.

##  V Conflicts of Interest

There are no conflicts of interest to declare.

##  VI Acknowledgements

This work used the eXtreme Science and Engineering Discovery Environment
(XSEDE), which is supported by National Science Foundation grant number
#ACI-1548562. In particular, we used the Pittsburgh Supercomputing Center’s
Bridges-2 resources under allocation ID PHY210132. We thank Vincent Hickl for
assistance with experiments on growing bacterial monolayers at liquid
interfaces.

## References

  * Persat  _et al._ (2015) A. Persat, C. Nadell, M. Kim, F. Ingremeau, A. Siryaporn, K. Drescher, N. Wingreen, B. Bassler, Z. Gitai, and H. Stone, Cell 161, 988 (2015).
  * You  _et al._ (2018) Z. You, D. J. G. Pearce, A. Sengupta, and L. Giomi, Phys. Rev. X 8, 031065 (2018).
  * Duvernoy  _et al._ (2018) M.-C. Duvernoy, T. Mora, M. Ardré, V. Croquette, D. Bensimon, C. Quilliet, J.-M. Ghigo, M. Balland, C. Beloin, S. Lecuyer, and N. Desprat, Nature Communications 9, 1120 (2018).
  * Boyer  _et al._ (2011) D. Boyer, W. Mather, O. Mondragón-Palomino, S. Orozco-Fuentes, T. Danino, J. Hasty, and L. S. Tsimring, Physical Biology 8, 026008 (2011).
  * Grant  _et al._ (2014) M. A. A. Grant, B. Wacław, R. J. Allen, and P. Cicuta, Journal of The Royal Society Interface 11, 20140400 (2014).
  * Volfson  _et al._ (2008) D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, Proceedings of the National Academy of Sciences 105, 15346 (2008).
  * Marshall (1986) K. Marshall, Advances in Colloid and Interface Science 25, 59 (1986).
  * Krajnc  _et al._ (2022) M. Krajnc, P. Stefanic, R. Kostanjšek, I. Mandic-Mulec, I. Dogsa, and D. Stopar, npj Biofilms and Microbiomes 8, 25 (2022).
  * Conrad (2020) J. C. Conrad, Journal of Industrial Microbiology and Biotechnology 47, 725 (2020).
  * Niepa  _et al._ (2017) T. H. R. Niepa, L. Vaccari, R. L. Leheny, M. Goulian, D. Lee, and K. J. Stebe, Scientific Reports 7, 17864 (2017).
  * Dell’Arciprete  _et al._ (2018) D. Dell’Arciprete, M. L. Blow, A. T. Brown, F. D. C. Farrell, J. S. Lintuvuori, A. F. McVey, D. Marenduzzo, and W. C. K. Poon, Nature Communications 9, 4190 (2018).
  * Copenhagen  _et al._ (2021) K. Copenhagen, R. Alert, N. S. Wingreen, and J. W. Shaevitz, Nature Physics 17, 211 (2021).
  * Yaman  _et al._ (2019) Y. I. Yaman, E. Demir, R. Vetter, and A. Kocabas, Nature Communications 10, 2285 (2019).
  * Giomi  _et al._ (2013) L. Giomi, M. J. Bowick, X. Ma, and M. C. Marchetti, Phys. Rev. Lett. 110, 228101 (2013).
  * Thampi  _et al._ (2013) S. P. Thampi, R. Golestanian, and J. M. Yeomans, Phys. Rev. Lett. 111, 118101 (2013).
  * Guillamat  _et al._ (2022) P. Guillamat, C. Blanch-Mercader, G. Pernollet, K. Kruse, and A. Roux, Nature Materials 21, 588 (2022).
  * Saw  _et al._ (2017) T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans, and B. Ladoux, Nature 544, 212 (2017).
  * Endresen  _et al._ (2021) K. D. Endresen, M. Kim, M. Pittman, Y. Chen, and F. Serra, Soft Matter 17, 5878 (2021).
  * Turiv  _et al._ (2020) T. Turiv, J. Krieger, G. Babakhanova, H. Yu, S. V. Shiyanovskii, Q.-H. Wei, M.-H. Kim, and O. D. Lavrentovich, Science Advances 6, eaaz6485 (2020).
  * Kawaguchi  _et al._ (2017) K. Kawaguchi, R. Kageyama, and M. Sano, Nature 545, 327 (2017).
  * Alaimo  _et al._ (2017) F. Alaimo, C. Köhler, and A. Voigt, Scientific Reports 7, 5211 (2017).
  * Ellis  _et al._ (2018) P. W. Ellis, D. J. G. Pearce, Y.-W. Chang, G. Goldsztein, L. Giomi, and A. Fernandez-Nieves, Nature Physics 14, 85 (2018).
  * Nestler and Voigt (2022) M. Nestler and A. Voigt, Communications in Computational Physics 31, 947 (2022).
  * Orozco-Fuentes and Boyer (2013) S. Orozco-Fuentes and D. Boyer, Phys. Rev. E 88, 012715 (2013).
  * Hertz (1882) H. Hertz,  _On the Contact of Rigid Elastic Solids and on Hardness_ (MacMillan, 1882).
  * DeCamp  _et al._ (2015) S. J. DeCamp, G. S. Redner, A. Baskaran, M. F. Hagan, and Z. Dogic, Nature Materials 14, 1110 (2015).
  * Keber  _et al._ (2014) F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Science 345, 1135 (2014).
  * Maroudas-Sacks  _et al._ (2021) Y. Maroudas-Sacks, L. Garion, L. Shani-Zerbib, A. Livshits, E. Braun, and K. Keren, Nature Physics 17, 251 (2021).
  * Metselaar  _et al._ (2019) L. Metselaar, J. M. Yeomans, and A. Doostmohammadi, Phys. Rev. Lett. 123, 208001 (2019).
  * Ruske and Yeomans (2021) L. J. Ruske and J. M. Yeomans, Phys. Rev. X 11, 021001 (2021).
  * Santiago (2018) J. A. Santiago, Phys. Rev. E 97, 052706 (2018).
  * Vafa and Mahadevan (2008) F. Vafa and L. Mahadevan, Physical Review Letters 129, 098102 (2008).
  * Hoffmann  _et al._ (2022) L. A. Hoffmann, L. N. Carenza, J. Eckert, and L. Giomi, Science Advances 8, eabk2712 (2022).
  * Alert (2022) R. Alert, Journal of Physics A: Mathematical and Theoretical 55, 234009 (2022).
  * Hickl and Juarez (2022) V. Hickl and G. Juarez, Soft Matter 18, 7217 (2022).
  * Prasad  _et al._ (2022) M. Prasad, N. Obana, S.-Z. Lin, K. Sakai, C. Blanch-Mercader, J. Prost, N. Nomura, J.-F. Rupprecht, J. Fattaccioli, and A. S. Utada, bioRxiv (2022).