# In-silico modeling of early-stage biofilm formation

Pin Nie  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Francisco Alarcon Oseguera  Departamento de Estructura de la
Materia, Fisica Termica y Electronica, Facultad de Ciencias Fisicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain  Departamento de
Ingeniería Física, División de Ciencias e Ingenierías, Universidad de
Guanajuato, Loma del Bosque 103, 37150 León, Mexico  Iván López-Montero
Instituto de Investigación Hospital 12 de Octubre (i+12), 28041 Madrid, Spain
Departamento de Química Física, Universidad Complutense de Madrid, 28040
Madrid, Spain  Belén Orgaz  Departamento de Farmacia Galénica y Tecnología
Alimentaria, Universidad Complutense de Madrid, 28040 Madrid, Spain  Chantal
Valeriani  Departamento de Estructura de la Materia, Fisica Termica y
Electronica, Facultad de Ciencias Fisicas, Universidad Complutense de Madrid,
28040 Madrid, Spain  Massimo Pica Ciamarra<EMAIL_ADDRESS>Division of
Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371, Singapore  CNR–SPIN,
Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126,
Napoli, Italy

(August 27, 2024)

###### Abstract

Several bacteria and bacteria strands form biofilms in different environmental
conditions, e.g. pH, temperature, nutrients, etc. Biofilm growth, therefore,
is an extremely robust process. Because of this, while biofilm growth is a
complex process affected by several variables, insights into biofilm formation
could be obtained studying simple schematic models. In this manuscript, we
describe a hybrid molecular dynamics and Monte Carlo model for the simulation
of the early stage formation of a biofilm, to explicitly demonstrate that it
is possible to account for most of the processes expected to be relevant. The
simulations account for the growth and reproduction of the bacteria, for their
interaction and motility, for the synthesis of extracellular polymeric
substances and Psl trails. We describe the effect of these processes on the
early stage formation of biofilms, in two dimensions, and also discuss
preliminary three-dimensional results.

Biofilms are self-organized bacteria communities comprising the bacteria and a
matrix of extracellular polymeric substances (EPS) Vert  _et al._ (2012).
Biofilms are certainly the most resilient form of life on Earth, as they
survive in both hot, salty, acid and alkaline waters, as well as at extremely
low temperature. Biofilms colonize their host environment, including humans,
in which case they are frequently the cause of persistent infections. Their
resilience mainly originates from the EPS matrix, which might account for up
to 90% of the dry biofilm weight. Besides allowing for a spatial and social
supracellular organization Flemming  _et al._ (2016), the matrix provides a
physical scaffold that keeps the cells together and protects them from
antimicrobial compounds (antibiotics) Organization  _et al._ (2014). EPS also
play a prominent role in the early stage biofilm formation, by promoting the
attachment of bacteria on surfaces Berne  _et al._ (2018).

The social need for research in biofilms is enormous. Biofilm grows on the
surface of a tooth, causing dental plaque Marsh (2006). More worryingly, they
grow on medical devices Francolini and Donelli (2010) such as prosthetic heart
valves, orthopaedic devices, skull implants, and might trigger virulent
rejection reaction. Pseudomonas aeruginosa, for example, can enter the blood
circulation Micek  _et al._ (2005) through open wounds to infect organs of the
urinary and respiratory systems. In a different context, biofilm cause
billions of dollars in damage to metal pipes in the oil and gas industry Xu
and Gu (2015); Ashraf  _et al._ (2014). Sulfate-reducing bacteria Enning and
Garrelfs (2014), for example, transform molecular hydrogen into hydrogen
sulfide which, in turn, produces sulfuric acid that destroys metal surfaces
causing catastrophic failures. In the water supply system, biofilm can grow in
pipes, clogging them due to their biomass Mazza (2016). It is of enormous
interest to develop surfaces to which bacteria are not able to attach. To
date, no surface able to reliably inhibit the formation of biofilms is known
Mon (1995).

On the other hand, one might also tame biofilms to benefit from them. For
example, we could exploit biofilms in environmental biotechnology, e.g., in
wastewater treatment Lazarova and Manem (1995), or for in situ immobilization
of heavy metals in soil Flemming  _et al._ (1996). Biofilms naturally grow by
consuming organic materials in the fluid. Microorganisms (typically bacteria
and fungi) can be used for microbial leaching, e.g., to metals from ores.
Copper, uranium, and gold are examples of metals commercially recovered by
microorganisms Mazza (2016).

The life cycle of a biofilm is traditionally described as consisting of five
phases: reversible attachment, irreversible attachment, growth, maturation and
dispersion. The first three phases identify the early-stage biofilm formation.
Understanding this phase is of particular interest, as it might allow for the
design of mechanisms able to prevent the formation of a biofilm. There is
mounting evidence that, in this phase, mechanical forces play a crucial role
in this stage Allen and Waclaw (2019), affecting the growth dynamics as
bacteria diffuse on the surface to be colonized, interacting among themselves
and with a chemical environment affected by their secretions. These include
EPS, and in particular, Psl exopolysaccharide, which promotes surface
attachment.

The observation that biofilms are formed by different bacteria and bacteria
strands, under highly variable external conditions, suggests that schematic
models could provide critical insights into biofilm formation. Indeed, several
models have been introduced in the literature Rudge  _et al._ (2012); Winkle
_et al._ (2017); Mattei  _et al._ (2018), e.g. to investigate biofilm jamming
Delarue  _et al._ (2016), nematic ordering Dell’Arciprete  _et al._ (2018);
Acemel  _et al._ (2018), role of psl trails Zhao  _et al._ (2013), nutrient
concentration Rana  _et al._ (2017), phase separation Ghosh  _et al._ (2015),
front propagation Farrell  _et al._ (2017).

In this manuscript, we introduce a flexible computational model for the
investigation of the early-stage biofilm formation. As in previous models, we
describe a biofilm as a collection of growing and self-replicating rod-shaped
particles. We do, however, also consider the role of Psl trails reproducing
previous experimental results Zhao  _et al._ (2013), and model for the first
time the growth of an EPS matrix, The article is structure as follows. In Sec.
I we introduce the numerical model, detailing all of the features we consider
as well as those we decided to neglect. We then examine the behavior of the
model, investigating different scenarios in increasing order of complexity:
Growth of non-motile cells, Sec II; competition between growth rate and
motility, Sec. III; multi-species biofilm, Sec. IV; role of Psl trails V;
formation of the EPS matrix, Sec VI. We conclude discussing the transition
from two- to three-dimensional colonies VII, and future research directions.

##  I Numerical model

Modelling the biofilm early-stage formation is a challenging task, as one need
to accounts for several biological and out-of-equilibrium processes. The
microscopic model also needs to be supplemented by several parameters, e.g. to
describe motility, reproduction, eps production, etc. We describe in the
following the main features of the computational model we have implemented.
While the model is general, we have calibrated the values of its many
parameters by referring to previous experimental investigation of the pathogen
Pseudomonas aeruginosa, whenever possible.

We describe in the following the implementation of different features of the
model, in order of complexity, which are schematically illustrated in Fig. 1.

Figure 1: Schematic illustration of the of the considered model. a) Bacteria
are modeled as a collection of particles. Isolated bacteria undergo a run and
tumble motion, we realize adding a propelling force and a torque, in a viscous
background. b) Consecutive particle making a bacterium interact via a harmonic
spring of rest length $l_{0}$. We model bacterial grow making $l_{0}$ time
dependent. A bacterium reproduces when its size doubles. c) Bacteria may
deposit a psl trail (red dots) as they move on the surface. These immobile psl
particles attract the particles making up a bacterium, effectively exerting a
net force and torque. Because of this, moving bacteria preferentially follow
existing psl trails. d) Bacteria may produce eps, we model as small particles.
Permanent bonds are formed between the EPS particles, and between the EPS
particles and those making up the bacteria. This polymerization process leads
to the formation of a EPS matrix.

###  I.1 Isolated non-reproducing bacterium

We model a bacterium as a spherocylinder, which we construct by lumping
together $7$ point particles. Point particles of different bacteria interact
via a Weeks-Chandler-Anderson (WCA) potential. This is a Lennard-Jones
potential with energy scale $\epsilon$ and diameter $\sigma$, we cut at its
minimum $d_{\rm b,b}=2^{1/6}\sigma$. This distance fixes the transverse width
of the bacteria that, in our units, is $w=d_{\rm b,b}=0.6\mu$m. Consecutive
particles of a bacterium interact via a Harmonic spring with stiffness $k_{\rm
b}=250\epsilon/w^{2}$ and initial rest length $l_{0}$, we fix so that the
bacterium aspect ratio is $[(n-1)l_{0}+w]/w=3$. These value for the size of a
bacterium mimic that of Pseudomonas aeruginosa. Bending rigidity is provided
introducing Harmonic angular interactions, with rest angle $\pi$ and stiffness
$k_{\rm a}=20\epsilon$, between any three consecutive particles. The value of
the stiffness coefficient is high enough for the bending deformation of the
bacteria to be negligible, for the range of parameters we will consider.

We assume the bacteria to follow an overdamped dynamics, which we realize by
applying to each particle making up a bacterium a viscous force $-\gamma v$
proportional to its velocity. Here $\gamma$ is a viscous friction coefficient.
We further assume the bacteria to perform a run and tumble motion. During a
‘run’ period, whose duration is a random number drawn from an exponential
distribution with time constant $t_{\rm run}=3$ min, we apply to the particles
making a bacterium a force $F=v_{\rm run}/\gamma$, where $v_{\rm run}=0.12\mu
m/s$ is the velocity of the particles in the running state. During a ‘tumble’
period, whose duration is a random number drawn from an exponential
distribution with time constant $t_{\rm tumble}=0.5$ min, we apply to the
bacterium a torque $T$, which fixes a rotational velocity. The equations of
motion are solved with a Verlet algorithm with timestep $5\cdot 10^{-3}s$. The
dynamical properties of a bacteria depend on the species, mutant, as well as
on the experimental condition. The values described above reasonable reproduce
the time dependence of the mean square displacement curves of Ref. Conrad  _et
al._ (2011), conducted in the early stage of formation of P. aeruginosa
biofilms. In particular, the diffusion coefficient results $D\simeq
0.7\mu^{2}/s$.

###  I.2 Growth and reproduction

We model the growth of bacterium by making time-dependent the rest lengths of
the springs connecting the beads making-up bacterium. Precisely, the rest
lengths grow linearly in $\min(t-t_{\rm b},1.2t_{r})$, where $t$ is the actual
time and $t_{\rm b}$ the time of birth of the bacterium, with a grow rate set
such that an isolated bacterium double its length in $t_{r}$, where for each
bacterium $t_{r}$ is taken from an exponential distribution with mean $\langle
t_{r}\rangle=1$h. The maximum value of the rest length has a cutoff to avoid
the unbounded growth of the pressure of a bacterium not able to grow, e.g. as
in a dense environment. A bacterium reproduces when its length equals twice
the original one. We implement the reproduction by replacing a bacterium with
two daughter cells, which occupy the same volume as the original one. The
polarity of the daughter cells is that of their father.

###  I.3 Psl exopolysaccharide trails

When moving on a surface, bacteria may secrete Psl exopolysaccharide. Psl
promotes attachment, effectively acting as a glue Zhao  _et al._ (2013).
Describing this process requires keeping track of the spatial location visited
by the moving bacteria. From a computational viewpoint, we do that by
superimposing to the computational domain a square grid, with grid size
$l\simeq w/20$, where $w$ is the width of the bacteria. As the bacteria move
on the surface, we record how many times each cell is visited. Specifically,
considering our coarse-grained description of the bacteria as a collection of
particles, we focus on the position of the central one. We indicate with
$n_{v}({\bf r},t)$ the number of times the grid cell in ${\bf r}$ has been
visited; this number originates from the superimposition of the trails left by
all bacteria. We assume $n_{v}({\bf r},t)$ to be proportional to the amount of
Psl deposited by the bacteria in ${\bf r}$.

To model the interaction between the bacteria and the trail pattern, we add to
the energy of our model the following term:

$V_{\rm trail}(t)=\sum_{b}\sum_{r_{i}\in b}\sum_{r}n_{v}({\bf r},t)v_{\rm
Gauss}({\bf r}-{\bf r_{i}}),$ (1)

where the first sum runs over all bacteria, the second one over the particles
of a bacterium, and the third one over the cells of the grid we use to record
the trail pattern. The interaction between each cell element and each particle
of our bacteria is given by an attractive potential, whose amplitude is
proportional to the number of times the grid element has been visited. We
model this attractive potential with an attractive Gaussian potential $v_{\rm
Gauss}$, with a width equal to half of the bacterial width. Notice that the
trail interaction acting on each bacterium exerts a torque, whose net effect
is that of aligning the bacteria to the trail.

In this model, the interaction potential is characterised by a typical energy
scale, $\epsilon$. We do not find literature data discussing the strength of
this interaction. Also, the rate of which bacteria deposit Psl has not been
discussed in the literature. Nevertheless, we understand that if bacteria
deposit Psl too frequently, and if the attraction is too strong, then the
bacteria will quickly bind to the deposited Psl, and will stop diffusing Tsori
and De Gennes (2004); Sengupta  _et al._ (2009). This self-trapping appears to
be unrealistic. On the order side, if the deposition rate is too small, then
the bacteria deposit Psl in uncorrelated locations, not on a trail. This
scenario also appears unrealistic. We have, therefore, arbitrarily chosen
simulation parameters for which the concept of a trail is well defined.

###  I.4 Extracellular Polymeric Substances

EPS production is essential to the growth of biofilm in vivo, as it bridges
bacteria cell together and to the hosting surface Xiao and Koo (2009). In the
early stage formation, EPS production appears to cooperate with bacterial
motility, e.g. twitching motility Conrad  _et al._ (2011), as bacteria need to
be close in space to agglomerate. Indeed, motility suppression may hinder the
formation of microcolonies and biofilms Recht  _et al._ (2000), at least if
the bacteria do not explore their environment via other physical processes,
e.g. diffusion or drift in a flow.

The theoretical and numerical description of the role of EPS is arduous and
limited. Here, we develop a numerical model for EPS along the line of the only
literature work explicitly modelling EPS particles Ghosh  _et al._ (2015) we
are aware of, but also introducing substantial advancements. Considering EPS
as polymer coils, Ref. Ghosh  _et al._ (2015) has modelled EPS as point
particles interacting via a purely repulsive potential. These particles have
been considered as passive and not able to form bonds to give rise to an EPS
matrix. In this condition, EPS and bacteria have been found to phase separate,
a result rationalized invoking a depletion-like interaction Ghosh  _et al._
(2015). Regardless, the features of the observed phase separation depend on
the rate at which EPS particles are produced. More recent results have also
highlighted the interplay between motility and depletion-like interactions
Porter  _et al._ (2019).

The main novelty of our approach is in the introduction of a polymerization
dynamics, allowing EPS particles to bond among themselves and with the
bacteria, to create an EPS matrix. Specifically, we describe EPS particles and
their dynamics as follows:

  1. 1.

Extracellular polymeric substances (EPSs) are represented as small spheres,
whose size is half of the width of the bacteria, $\sigma_{\rm eps}=D/2$.

  2. 2.

EPS particles interact among them with a purely repulsive WCA potential, with
energy scale $\epsilon$, as the particles of different bacteria.

  3. 3.

EPS particles are inserted by the bacteria in their surrounding, at a rate
$\tau_{\rm eps}^{-1}$. An EPS particle is inserted only if it does not
interact with any other particle or bacteria. This ensures numerical
stability. Hence, EPS production is suppressed in crowded conditions.

  4. 4.

Every $\Delta t$, where $\Delta t$ is a random variable taken form an
exponential distribution with average value $\Delta_{t}^{*}$, we look for all
possible pair of interacting EPS particles. If two EPS particles are
interacting, we add an harmonic bond $v(r)=10^{2}\epsilon(r-\sigma_{\rm
eps})^{2}$ between them, provided that they are not already bonded, with a
probability $p_{b}$.

  5. 5.

Similarly, every $\Delta t$ we add a bond between an EPS particle and a
bacteria particle in interaction, provided that they are not already bonded,
with probability $p_{b}$. In this case, the bond energy is
$v(r)=10^{2}\epsilon\left[r-\left(\frac{\sigma_{\rm
eps}+D}{2}\right)\right]^{2}$.

The steps 1-3 above essentially reproduce the model of Ref. Ghosh  _et al._
(2015). On the other hand, steps 4-5 describe the dynamics of a polymerization
process. The ratio between the mass $m_{\rm eps}$ of an EPS particle and the
mass $M$ of a bacterium is $m/M\ll 1$. EPS particles motion follow a Langevin
dynamics, with parameters fixed so that a particle has thermal velocity
$\sqrt{2k_{B}T/m_{\rm eps}}=0.18\mu$m/s, and a diffusion coefficient roughly
100 time smaller than that of bacteria in dilute conditions. This means that
the bacteria de-facto move in a bath of almost immobile EPS particles.

The EPS model has two parameters, $\Delta_{t}^{*}$ and $p_{b}$, and the rate
at which bonds are formed between possible pair of particles is
$p_{b}\Delta_{t}^{*}$. It isn’t easy to estimate these parameters from the
experiments. Besides, we notice that the EPS production rate depends on the
growing condition. Here, we decided to fix $\Delta_{t}^{*}=1$min
$=\tau_{r}/60$, and have investigated the dependence of the growing dynamics
on the bond probability $p_{b}$. We consider the bond between bacterial and
EPS particles to be permanent.

###  I.5 What is not in the model

This model takes into account all of the processes that appear to be relevant,
such as motility, reproduction, production of Psl trail, EPS matrix, etc.

Some features, we believe to be less relevant, are for now neglected. For
instance, we neglect hydrodynamic interactions, which after the initial
docking of the bacteria should be minor, due to the small Reynolds number.
Indeed, bacteria swim in bulk with velocity $\simeq 30\mu m/s$, and on surface
with velocity $\simeq 1\mu/s$. The Reynolds number is
${\Re}=\frac{\rho_{f}vL}{\nu}$, where $\rho_{f}$ is the density of the fluid,
$\nu$ is its viscosity ($\nu=10^{-3}Pas$ for water), $v$ is the relative
velocity of the particle with respect to the fluid, $L$ is the typical length
of a bacterium (around $1\mu m$). Thus, for bacteria swimming in bulk, the
Reynolds number is $\sim 3\times 10^{-5}$, and for Bacteria on the surface,
the Reynolds number is $\sim 10^{-6}$. Bacterial motion is thus in a low
Reynolds number regime where viscous forces dominate over inertial ones.

Furthermore, we do not consider the diffusion of nutrients and hence the
possibility that the growth rate and the motility properties might spatially
vary. In the early-stage formation in which the biofilm is essentially two-
dimensional, we do not expect diffusion of nutrients to be sensibly affected
by the forming biofilm. Indeed, experimental results suggest that the growth
rate in the interior and the periphery of a biofilm are comparable Zachreson
_et al._ (2017).

##  II Growth in the absence of motility

Figure 2: Growth of a colony of non-motile bacteria, imaged every $4$h. The
colour code reflects the angle between the bacteria and a fixed spatial
direction. Hence, patches with the same colour correspond to regions with the
same nematic director. See here for the corresponding animation.

We begin illustrating our model at work with the simplest possible example.
The growth of a colony of non-motile bacteria, in the absence of Psl and EPS.
In this scenario, we do expect the number of bacteria to grow exponentially
with time. Saturation occurs at large times due to finite-size effects. This
jamming transition induced by reproduction has been considered before Delarue
_et al._ (2016).

We illustrate the expanding colony in Fig. 2, where a fixed time interval
separate consecutive snapshots. The number of bacteria $n$ present at each
time is specified in each panel. The direct visualization of the colony
suggests that the bacteria tend to align with each other. Nematic ordering is
indeed commonly observed in experiments Volfson  _et al._ (2008);
Dell’Arciprete  _et al._ (2018); Yaman  _et al._ (2019); the order is short-
ranged due to the emergence of buckling instabilities Boyer  _et al._ (2011).

To investigate this issue, we colour code each bacterium according to the
angle its director forms with a given axis (modulus $\pi$, given that in the
absence of motility the bacteria are not polar). As the colony grows, we see
the emergence of domains with the same colour corresponding to regions of
local nematic alignment.

##  III Motility vs. growth rate

Figure 3: Growth of microcolonies of bacteria having different typical
velocity $v_{\rm run}$ and fixed average reproduction time, $\tau_{r}=1$h. See
these links for the corresponding animations: slow, medium, fast.

The motility properties of bacteria are highly variable. Different species
have different motility properties. For each species, motility depends on the
mutant, e.g. depending on the presence of type-4 pili or of the flagella.
Besides, motility depends on the external environment, e.g. on the presence of
nutrients. Because of this variability, it is interesting to consider the
dependence of the early-stage formation on the motility properties, in our
numerical model.

Here we consider that, once a bacterium adheres to the surface and seeds a
microcolony, the subsequent evolution depends on the competition of two
physical processes, reproduction and motility. To clarify the origin of this
competition, we start by considering the time dependence of the radius of a
microcolony, assuming the bacteria to have no motility. In this condition, a
colony expands as bacteria duplicate and push against each other.

To model this situation, we assume the colony to have a constant number
density $\rho$, number of bacteria per unit area, so that then the number $n$
of bacteria in a colony of radius $R$ is $n(R)=\rho 4\pi R^{2}$. How does $R$
evolves with time? To predict $R(t)$, we assume the bacteria to reproduce with
a constant rate $\tau_{r}^{-1}$, so that $\frac{dn}{dt}=\frac{n}{\tau_{r}}$.
From this assumption, we get

$\frac{n}{\tau_{r}}=\frac{dn}{dt}=8\pi\rho R\frac{dR}{dt}.$ (2)

Hence, the radial expansion velocity of the colony is

$v_{R}=\frac{dR}{dt}=\frac{R}{2\tau_{r}}.$ (3)

Interestingly, this model predicts that the expansion velocity grows linearly
with the cluster size. One might expect this to occur in the early stage
development of a microcolony. At a later time, the bacteria deep inside the
colony stop reproducing because of the limited nutrient diffusing to the core
or because of the high mechanical pressure.

If the bacteria are motile, then another typical velocity scale enters into
the problem: the characteristic bacteria velocity $v_{\rm run}$. It turns out
that $v_{R}$ and $v_{\rm run}$ compete. Precisely, when $v_{\rm run}\gg
v_{R}$, bacteria swim away from each other before they reproduce. Conversely,
they reproduce when still close. Since $v_{R}$ grows with the bacteria colony,
there is a characteristic colony radius $R\simeq 2v_{\rm run}{\tau_{r}}$ above
which the radial velocity profile due to the reproduction overcomes the
swimming velocity of the bacteria. When this occurs, the colony starts
becoming compact.

As an example, we illustrate in Fig. 3 the developing of three different
microcolonies, which only differ in the magnitude of the typical velocity of
bacteria. At small velocities, the microcolony is nearly compact at all times.
At large velocities, bacteria spread on the surface at short times, as
apparent in the configuration reached at $8\tau_{r}$ in the case of
intermediate velocities, but then become part of a dense microcolony. At even
larger velocities, compact shape is attained at a longer time, possibly not
yet achieved in our simulation with $v_{\rm run}=50$.

It is interesting to notice that, in this picture, a compact colony emerges in
this picture when the reproduction rate dominates over the motility of the
particles. In this respect, while microcolony formation visually resemble the
activity drive phase separation of active system of spherical Redner  _et al._
(2013); Wysocki  _et al._ (2014); Fily and Marchetti (2012); Buttinoni  _et
al._ (2013); Palacci  _et al._ (2013); Theurkauff  _et al._ (2012); Ginot  _et
al._ (2018); Nie  _et al._ (2020a, b) or dumbbells particles Suma  _et al._
(2014); Petrelli  _et al._ (2018), the underlying physical driving force is
different.

It is, however, arduous to understand the experimental relevance of these
findings. Indeed, one might expect that before a compact shape is attained,
the colony stops expanding in two dimensions, and start growing in the
vertical one. We discuss such a transition in Sec. VII. Besides, in the
picture we are considering, there are no bacteria in the planktonic state
joining the colony, and no bacteria move from the colony to the planktonic
state. We do not consider these processes in our numerical model, despite it
would be trivial to include them, as the rates of attachment and detachment
have not yet been thoroughly experimentally characterized.

##  IV Coexistence of different species

Figure 4: Early stage formation of a two-species biofilm. Blu bacteria (left
in the figures) are non-motile, while red bacteria are motile (right in the
figures). The motile bacteria are faster in the bottom row. See here for an
animation.

Biofilms are often multispecies Røder  _et al._ (2016). Our computational
model allows considering the coexistence of bacteria with different
properties. Here, as an example, we consider that of bacteria with different
motility properties.

Fig. 4 illustrates the growth of a colony of immotile bacteria (blue, on the
left), and a colony of motile ones (red, on the right). On the top row, we
consider the case in which the colony of motile bacteria becomes compact
before the two colonies start interacting. Hence, when the two microcolonies
enter in contact, both of them are compact. As a consequence, a sharp
interface between the two colonies develops. Notice that this interface is not
straight, but slightly curved. This curvature reflects the anisotropy of the
microcolony of non-motile bacteria, which is ellipsoidal at short times.

In the bottom row of Fig. 4 illustrates a case in which the motile bacteria
are fast so that when the two colonies start interacting, their microcolony is
not compact. In this case, the interface between the two colonies is rough. A
close look suggests that the interface might have a wavy appearance
reminiscent of the viscous fingering Saffman–Taylor instability which develops
when fluids with different viscosity pushed against each other. In this
respect, we notice that such instability has been reported at the interface of
cell populations growing at different rates Mather  _et al._ (2010), and in a
variety of other contexts Pica Ciamarra  _et al._ (2005a, b).

##  V The role of psl

Figure 5: Early stage biofilm formation in the presence of Psl production. The
red lines are the trails left by the bacteria as they explore the surface.
Bacteria interact through an attractive force with the trails. The attraction
to a particular location is space is proportional to the number of times this
location has been visited by the bacteria. See here for the corresponding
animation.

While exploring a surface, bacteria may leave a Psl trail, to which other
bacteria are subsequently attracted. Psl trails thus resemble pheromones
trails left by ants. The statistical features of the motion of particles
attracted by substances they secrete, generally known as reinforced random
walks, have been extensively investigated in the literature Allen and Waclaw
(2019). For the case of a single bacterium attracted to its own secreted
substance, for instance, Tsori and de Gennes Tsori and De Gennes (2004)
suggested the presence of self-trapping in one and two spatial dimensions, not
in three. More recent numerical simulations indicate that there is no self-
trapping, but rather a prolonged sub-diffusive transient Sengupta  _et al._
(2009). Here, we consider the growth of a microcolony, seeded by a single
bacterium, in the presence of Psl production.

In Fig. 5, we illustrate a representative time evolution of a bacterial
colony. Besides drawing the bacteria, we illustrate the corresponding trails,
which are clearly visible at short times, before trails of different bacteria
overlap. Qualitatively, these results are analogous to that experimentally
reported in Ref. Zhao  _et al._ (2013).

To be more quantitative, we have determined the time evolution of the
probability distribution of the number of times a particular space location
has been visited. Here, by location, we intend grid elements of side length
equal to 1/20th of the bacterial width. This visit frequency distribution
quantity favourably compares to experimental results. Fig. 6a,b presents
experimental results for this probability distribution Zhao  _et al._ (2013);
Gelimson  _et al._ (2016). The probability distribution decays as a power law,
with a large exponent that decreases as times evolve. In panel c of the same
figure, we present our numerical results for the same quantity. The numerical
model well reproduces the experimental results, both as concern the presence
of a power-law decay in the probability distribution, as well as the value of
the decay exponent and its time dependence.

Figure 6: Experimental and numerical results for the time evolution of the
probability distribution of the number of times a point (pixel) has been
visited by a bacterium. Panels a and b report experimental results from Ref.
Gelimson  _et al._ (2016) (with permission) and Ref. Zhao  _et al._ (2013)
(with permission), respectively. Panel c illustrates the results of our
numerical model.

##  VI EPS matrix

Since EPS come into the focus of the research community only recently, the
current knowledge of its role in early-stage biofilm development pales when
compared to the extensive understanding of biofilm formation in the absence of
EPS production, in particular for non-motile bacteria. The role of EPS has not
been considered in earlier literatureWingender  _et al._ (1999), as
“traditionally, microbiologists used to study and to subculture individual
bacterial strains in pure cultures using artificial growth media. Under these
in vitro conditions, bacterial isolates did not express EPS-containing
structures or even lost their ability to produce EPS”. However, it is nowadays
clear that EPS is of fundamental importance, as it allows for a spatial and
social supracellular organization Flemming  _et al._ (2016), while providing a
physical scaffold that keeps the cells together and protect them from
antimicrobial compounds and heavy metals Nadell  _et al._ (2015), and can also
retain water Wingender  _et al._ (1999). EPS also appears to play a prominent
role in the early stage biofilm formation, by promoting the attachment of
bacteria on surfaces Berne  _et al._ (2018).

Figure 7: Effect of the bonding probability on the number of bacteria. Panel
(a) illustrates the time dependence of the number of bacteria on the surface.
Different curves refer to different values of the bonding probability,
$p_{b}$. Panel (b) shows the dependence of the asymptotic steady state number
of bacteria on the bonding probability $p_{b}$. The fitting line is an
exponential one, $n_{\infty}+(n_{0}-n_{\infty})e^{-p_{b}/p_{b}^{*}}$.

In our numerical model, two control parameters affect the role of EPS. First,
there is the rate at which individual bacteria secrete EPS particles in their
surrounding, provided that these new particles do not interact with other EPS
particles or bacteria. We keep this rate to 1/60th or the reproduction rate.
Secondly, there is the probability $p_{b}$ that two EPS particles, or an EPS
and a bacterium, for a bond if close enough.

Here, we investigate the dynamics and the steady-state as a function of the
bonding probability $p_{b}$. Fig. 7a illustrates the time dependence of the
number of bacteria, for different values of $p_{b}$. At short times, $t<2h$,
the production of EPS does not quantitatively affect the dynamics, as
different curves collapse on each other. At larger times, the population grows
exponentially but then saturates. This saturation is not a finite-size effect.
This is a critical result, as it clarifies that in the presence of EPS a
microcolony stops spreading, in two dimensions. Indeed, we do expect a
transition towards a three-dimensional condition. Fig. 7b shows that the
asymptotic number of bacteria decreases exponentially with the bonding
probability. If $p_{b}$ is very high, growth stops with just a few bacteria on
the surface. This finding is reminiscent of early speculations for isolated
non-reproducing particles Tsori and De Gennes (2004).

Figure 8: Evolution of system of bacteria (red) which produce EPS (blue). The
EPS particles can bond to each other, and to the bacteria, with probability
$p_{b}$. Different rows correspond to different values of the bonding
probability $p_{b}$, as indicated.

To rationalize these results, we provide snapshots illustrating the time
evolution of the investigated system in Fig. 8. In this figure, the columns
correspond to different times, the rows to different values of the bond
probability $p_{b}$, as indicated. In all case, at long times, we do see the
formation of small clusters of bacteria. These bacteria are glued together
through the EPS particles. For small values of $p_{b}$, these clusters only
form when there are many EPS particles in the systems. Conversely, for a
larger value of $p_{b}$, few EPS particles can glue the bacteria together.
Bacteria are therefore self-trapped by the EPS particles the produce Tsori and
De Gennes (2004); Sengupta  _et al._ (2009). The exponential dependence of the
number of bacteria on $p_{b}$ observed in Fig. 7b is not simply recovered in a
mean-field approximation, starting from rate equations from the total number
of bacteria and the number of trapped bacteria. Spatial correlations, which
are apparent in Fig. 8, appear therefore to play an important role in
determining the size of the final population.

##  VII From two- to three-dimensional microcolonies

Figure 9: Evolution of a three dimensional microcolony of not-motile bacteria.
The microcolony develops with the bacteria embedded in an EPS gel matrix.

All investigations reported so-far have been restricted to the early stage
formation of a biofilm, which is essentially a two-dimensional process.
However, biofilms then develop as structured three dimensional aggregates.
Here, without the aim of being quantitative, we demonstrate that the numerical
approach we have developed is also able to describe this transition. To this
end, we extended the model to allow the bacteria to move in the vertical
direction.

In the absence of EPS, the transition for two- to three-dimensional colonies
has been suggested to originate from extrusion driven by the compression of
the cells-Farrell  _et al._ (2017); Grant  _et al._ (2014) \- alike in
epithelial cell tissues. In the presence of EPS, a different mechanism appear
to be at work. Indeed, while the bacteria are still on the plane, EPS
particles move also in the vertical direction, and their polymerization leads
to a three dimensional network. The stress induced in this network by the
continuous growth and reproduction of the bacteria, leads to upward-forces
acting on the bacteria, which force them out of the horizontal plane. A small
tilt of the bacterium is enough to seed the transition from a two to a three
dimensional biofilm.

Fig. 9 illustrates the developing of a three dimensional biofilm, for non-
motile bacteria. Clearly, the bacteria result embedded in a growing EPS
matrix. We leave to future studies the quantitative investigation of three
dimensional investigation, also because of their high computational cost.

##  VIII Conclusions

In this manuscript, we have illustrated a computational model for the
simulation of the early-stage biofilm formation. The model reproduces results
reported in previous numerical studies, such as the emergence of local nematic
order, as well as the role of Psl trails. Our model, however, shows for the
first time that it is possible to describe in numerical setting the production
of EPS as the growth of the extracellular matrix, in a coarse-grained fashion.

The main limitation of our model, and of related ones, appears the presence of
many parameters. Specifically, the issue concerns the absence of a proper
experimental measure of them, for most species. This renders a quantitative
comparison with experimental results difficult. Nevertheless, the universality
of the discussed phenomenology suggests that our model could suffice to
pinpoint the key physical processes at work in the early-stage formation of a
biofilm.

In this respect, our work suggests that not only the production of Psl trail
Zhao  _et al._ (2013), but also that of EPS, might induce the formation of
microcolonies. Specifically, EPS leads to the formation of an extracellular
matrix which traps the bacteria in what are de-facto microcolonies (see red
regions in Fig. 8. Besides, we have originally observed that the incipient EPS
matrix appears to foster the transition from a two- to a three-dimensional
morphology.

## Acknowledgements

N.P. and M.P.C. acknowledge support from the Singapore Ministry of Education
through the Academic Research Fund MOE2017-T2-1-066 (S), and are grateful to
the National Supercomputing Centre (NSCC) for providing computational
resources.

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