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\begin{document}

\title{Spatial distribution of mechanical properties in \emph{Pseudomonas
aeruginosa} biofilms, and their potential impacts on biofilm deformation}



\author[1]{Juan Pablo Pavissich}%
\author[2]{Mengfei Li}%
\author[2]{Robert Nerenberg}%
\affil[1]{Universidad Adolfo Ibanez}%
\affil[2]{University of Notre Dame}%


\vspace{-1em}



  
  \date{June 17, 2024}


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\selectlanguage{english}
\begin{abstract}
The mechanical properties of biofilms can be used to predict biofilm
deformation, for example under fluid flow. We used magnetic tweezers to
spatially map the compliance of \emph{Pseudomonas aeruginosa} biofilms
at the micron scale, then used modeling to assess its effects on biofilm
deformation. Biofilms were grown in capillary flow cells with Reynolds
numbers (Re) ranging 0.28 to 13.9, bulk dissolved oxygen (DO)
concentrations from 1 mg/L to 8 mg/L, and bulk calcium ion
(Ca\textsuperscript{2+}) concentrations of 0 and 100 mg
CaCl\textsubscript{2}/L. Higher Re numbers resulted in more uniform
biofilm morphologies. The biofilm was stiffer at the center of the flow
cell than near the walls. Lower bulk DO led to more stratified biofilms.
Higher Ca\textsuperscript{2+} led to increased stiffness and more
uniform mechanical properties. Using the experimental mechanical
properties, fluid-structure interaction models predicted up to 64\%
greater deformations for heterogeneous biofilms, compared to a
homogeneous biofilms with the same average properties. However, the
error depended on the biofilm morphology and flow regime. Our results
show significant spatial mechanical variability exists at the micron
scale, and that this variability can potentially affect biofilm
deformation. The average mechanical properties, provided in many
studies, should be used with caution when predicting biofilm
deformation.%
\end{abstract}%



\sloppy


\textbf{1. Introduction}

Biofilms are aggregates of microbial cells embedded in a matrix of
extracellular polymeric substances (EPS) (Hans-Curt Flemming \&
Wingender, 2010; Hall-Stoodley et al., 2004). They are ubiquitous in
clinical, environmental, and industrial systems and can cause human
infections, foul water filtration membranes, and promote corrosion in
pipes (Costerton et al., 1995; Hall-Stoodley et al., 2004). Biofilms
also can play beneficial roles, for example in environmental treatment
processes. Thus, biofilm removal may be sought in some cases, and its
retention in others.

Biofilm formation, persistence, deformation, and detachment are largely
determined by the biofilm mechanical properties (Kundukad et al., 2016;
Powell et al., 2013; Boudarel et al., 2018; Gloag et al., 2019). For
example, biofilms' viscoelastic nature help them dissipate stress from
fluid flow, preventing detachment (Stoodley et al., 1999). The
characterization of biofilm mechanical properties is key to predicting
biofilm deformation and detachment (Klapper et al., 2002).

Biofilm mechanical properties can be influenced by a variety of factors,
including nutrient concentrations and microbial growth rates (Paul et
al., 2012; Van Loosdrecht et al., 2002), microbial composition (Abriat
et al., 2020; Kim et al., 2020; Yannarell et al., 2019), biofilm age
(Hwang et al., 2014; Shen et al., 2016), hydrodynamic conditions
(Dunsmore et al., 2002; Thomen et al., 2017), multivalent cation
concentration (Ahimou et al., 2007; Jones et al., 2011; Lieleg et al.,
2011), temperature (Pavlovsky et al., 2015; 2013), and pH values (Chen
\& Stewart, 2000; Ho et al., 2013). For example, the microbial growth
rates and ecological stratification, which are determined by the
substrate profiles within the biofilm, can have a strong influence on
biofilm mechanical strength (Rochex et al., 2009). Higher bulk oxygen
concentrations and higher shear stresses were found to increase the
strength of biofilms (Stoodley et al., 2002; Ahimou et al., 2007a;
Pellicer-N\selectlanguage{ngerman}àcher \& Smets, 2014). Also, the bulk calcium ion
(Ca\textsuperscript{2+}) concentration caused biofilms to become thicker
and denser, and to significantly decrease biofilm detachment episodes
(Goode \& Allen, 2011).

While biofilms are commonly thought to be mechanically homogeneous, this
may be an artifact generated by the use of bulk-scale techniques for
their mechanical characterization (Safari et al., 2015). When microscale
techniques were used, mechanical properties have been found to vary
significantly within the biofilm (Böl et al., 2012).

Biofilms usually have temporal and spatial variations of mechanical
properties. It has been widely recognized that the structure of biofilms
becomes more stable over time. For example, older biofilms were found to
be less affected by bubble disruptions, whereas younger biofilms were
easily removed by air bubbles (Jang et al., 2017). Laspidou and Rittmann
(2004b) hypothesized that biofilm increases its density over time due to
consolidation, i.e., the filling of voids within the biofilm. In a study
of the viscoelasticity of \emph{Pseudomonas aeruginosa} biofilms (Gloag
et al., 2018), various temporal changes were observed in different
phenotypes. Increased stiffness or cohesive strength was observed over
biofilm depth in several studies (Ahimou et al., 2007; Derlon et al.,
2008; Olivier Galy et al., 2012). Spatial distribution of biofilm
stiffness was also found in \emph{P. aeruginosa} biofilms (Hunt et al.,
2004; Karampatzakis et al., 2017).

In order to understand the spatial distribution of biofilm mechanical
properties, microscale techniques can be used. Microscale techniques
include microindentation compression (Cense et al., 2006), dedicated
microcantilever (Aggarwal et al., 2010; Poppele \& Hozalski, 2003),
atomic force microscopy (AFM) indentation (Arce et al., 2009; Volle et
al., 2008), and microbead force spectroscopy (Lau et al., 2009).
However, all these methods are invasive and can compromise the biofilm
integrity.

In recent years, novel microrheological techniques have been developed
(Birjiniuk et al., 2014; Cao et al., 2016; Galy et al., 2012;
Karampatzakis et al., 2017; Thomen et al., 2017). In particular,
magnetic actuation with magnetic tweezers, coupled with magnetic
microparticles, may be ideal (Galy et al., 2014; Galy et al., 2012;
Zrelli et al., 2013). This technique can overcome the limitations of
other microscale techniques by using strong forces and in-situ
measurements. With the displacement of magnetic particles, which are
added to the media during initial growth, biofilm properties can be
mapped spatially. For example, Galy et al. (2012) used magnetic tweezers
and found that stiffness measurements at different locations in a
biofilm ranged over of two orders of magnitude, even locally. This
indicates the importance of localized spatial measurements in the study
of biofilm mechanical properties. Nevertheless, the mechanical
heterogeneity of biofilms has received little attention.

Mathematical models describing biofilm mechanical behavior can improve
our understanding of biofilm structures and properties (Böl et al.,
2012). Such models include the simulation of biofilm deformation under
applied stress (Li et al., 2020; Picioreanu et al., 2018; Picioreanu et
al., 2001; Towler et al., 2007). However, the mechanical properties are
typically assumed to be homogeneous and are based on large-scale
measurements. It is not clear whether mechanical heterogeneity is
significant, and whether it should be considered for accurate
predictions of biofilm deformation.

As discussed above, very few studies have addressed the spatial
distribution and variability of mechanical properties of biofilms in a
non-destructive fashion (Cao et al., 2016; Galy et al., 2012;
Karampatzakis et al., 2017). Microscale heterogeneities could have
important impacts on biofilm formation and behavior (Böl et al., 2012),
yet the impact of this variability has not been explored.

In this study, we used magnetic tweezers to determine the spatial
distribution of biofilm mechanical properties for different flow
conditions, dissolved oxygen and Ca\textsuperscript{2+} concentrations
in \emph{P. aeruginosa} biofilms\emph{.} \emph{P. aeruginosa} is
commonly used as a model species for biofilm research, as it is an
important human pathogen and also widespread in the environment. We then
used a fluid-structure interaction mathematical model to evaluate the
mechanical behavior of a biofilm (i.e., biofilm deformation) considering
average mechanical properties versus the heterogeneity observed in the
experiments.

\textbf{2.Methods}

\subsubsection*{Biofilm growth
conditions}

{\label{biofilm-growth-conditions}}

\emph{P. aeruginosa} ATCC strain 15692 (PAO1), tagged with green
fluorescent protein (GFP), was used for biofilm growth (Shrout et al.,
2006). The bacterial strain was grown overnight to an optical density of
0.4-0.5 (OD\textsubscript{600}) in Luria-Bertani (LB) broth at 37°C on
an orbital shaker. A 1 mL aliquot of the culture was mixed with red
fluorescent magnetic beads (Dynabeads M-270 Amine, Invitrogen, Carlsbad,
CA), with a 2.8 \selectlanguage{greek}μ\selectlanguage{english}m diameter, at a final concentration of
2.5\(\times\)10\textsuperscript{6} particles/mL. These beads are
superparamagnetic, hydrophilic, and contain surface amino groups that
form covalent bonds with cell and EPS components. The hydrophilic
surface ensures non-specific binding to the biofilm. The
superparamagnetic behavior allows for high levels of magnetization and
the absence of magnetization without an external magnetic field. As a
control, fluorescent non-magnetic particles of 1 \selectlanguage{greek}μ\selectlanguage{english}m diameter were used
(Fluoresbrite 18660, Polysciences, Germany).

Biofilms were grown in borosilicate glass capillaries with a 1 mm
internal side and 150 \selectlanguage{greek}μ\selectlanguage{english}m wall thickness (Friedrich and Dimmock Inc.,
Millville, NJ). The capillaries were inoculated with the mixed
suspension and kept under static conditions for 2 h before starting a
continuous flow of 10\% LB for the growth period. Flow was provided
using a syringe pump (Kd Scientific, KDS-220, Holliston, MA). Flow rates
were kept between 0.1 and 5 mL/h to maintain Reynolds numbers (Re)
between 0.28 and 13.9 (laminar flow). Biofilms were grown with dissolved
oxygen (DO) concentrations of either 8 or 1 mg/L, and with or without a
Ca\textsuperscript{2+} addition of 100 mg/L as CaCl\textsubscript{2}.
An\emph{in-situ} magnetic actuation technique was then applied to
non-destructively estimate the mechanical behavior of the biofilm after
5 days of growth. The general procedures were followed according to Galy
et al. (2012). The detailed information of the magnetic tweezers and
force calibration procedure is provided in Supporting Information
Material (SI).

\subsubsection*{Microscopy and particle
tracking}

{\label{microscopy-and-particle-tracking}}

A confocal laser scanning microscopy (CLSM) (Nikon 90i A1R, Melville,
NY) equipped with a 40x Plan Fluor WD 0.2 objective was used to image
the biofilm and particles large-spectrum fluorescence signals.
Typically, the first examined plane was located between 5 to 10 \selectlanguage{greek}μ\selectlanguage{english}m above
the capillary bottom. A z-stack with a depth step size of 0.5 \selectlanguage{greek}μ\selectlanguage{english}m was
collected to verify biofilm thickness and abundance of magnetic beads.
For a given plane, particle motion upon magnetic force application was
recorded at a scanning speed of 30 frames per second (fps) over a period
of 20 s, and further processed using a particle tracking software
(ImageJ, NIH, Bethesda, MD). About 3 to 4 fields of 250 \selectlanguage{greek}μ\selectlanguage{english}m
\(\times\)250 \selectlanguage{greek}μ\selectlanguage{english}m (with each field) were collected along the
capillary width before changing the plane (z-direction) (Figure 1).
After each field was obtained, the field view was changed in the
x-direction. Vertical planes were analyzed every 10 \selectlanguage{greek}μ\selectlanguage{english}m across the height
of the biofilm. About 10-15 particles were analyzed per field. A more
visual description of the procedure can also be found in Galy et al.
(2014).

\subsubsection*{Creep curves and mechanical behavior
analysis}

{\label{creep-curves-and-mechanical-behavior-analysis}}

\textbf{Hosted file}

\verb`image1.emf`  available at \url{https://authorea.com/users/364770/articles/485068-spatial-distribution-of-mechanical-properties-in-pseudomonas-aeruginosa-biofilms-and-their-potential-impacts-on-biofilm-deformation}

Using particle tracking, particle positions in each frame for the
acquired images were obtained. The image acquisition frequency yielded
particle trajectories or displacement as a function of time. Individual
particle creep curves, giving material strain versus time, were plotted
by converting the displacement curves into creep compliance curves
(total compliance as a function of time \emph{J(t)} ), knowing the probe
deflection \emph{d(t)} and the applied force \emph{F} , according to the
compliance formula (Schnurr, Gittes, MacKintosh, \& Schmidt, 1997): ,
which gives the relation between material strain and applied stress for
a probe of radius \emph{R} (the microbead) embedded in an incompressible
viscoelastic medium. The compliance (\emph{J(t)} ) of the material is
the inverse of elastic modulus and equal to the ratio of strain to
stress, i.e., the deformation for a given stress. Thus, a higher
compliance indicates a more flexible material. The absolute force from
the magnetic field acting on the magnetic particles was determined by a
purely viscous mixture of 99.5\% glycerol with a concentration of
2\(\times\)10\textsuperscript{8} particles/mL (see SI).

\subsubsection*{Modeling biofilm with heterogeneous mechanical
properties}

{\label{modeling-biofilm-with-heterogeneous-mechanical-properties}}

To study the impacts of mechanical heterogeneity on biofilm deformation,
a two-dimensional (2D) continuum biofilm model was implemented using
COMSOL Multiphysics (COMSOL v4.4, Comsol Inc, Burlington, MA) with
finite element method. A fluid-structure interaction (FSI) model using
the arbitrary Lagrangian-Eulerian (ALE) approach was applied to simulate
the deformation of biofilms under fluid flow. The biofilm was considered
either a purely linear elastic material with elastic compliance varying
with depth in the biofilm, as per the experimental data, or a homogenous
material, using the average of the experimental data. The system
included a solid biofilm subdomain, and a bulk liquid subdomain. Two
arbitrary biofilm structures were considered: a biofilm colony and a
mushroom-like tower. The spatial distribution of mechanical properties
was set in the biofilm structure using the experimental data obtained
for the growth under 0.1 mL/h flow rate, DO saturation, and no
Ca\textsuperscript{2+} addition, with an elastic modulus (E) range of
0.005-0.05 Pa, as calculated from the compliance (\emph{J} )
determinations. The averaged elastic modulus of the complete
heterogeneous biofilm structure was computed over the biofilm domain and
then implemented into the homogeneous model. Time-dependent simulation
was performed for 2-3 seconds until obtaining a steady-state solution.
Biofilm deformations at 1.5 s were reported. A detailed description of
the FSI model is provided in the SI.

\subsection*{3. Results and Discussion}

{\label{results-and-discussion}}

We used magnetic microbead actuation together with confocal fluorescence
microscopy to assess the spatial distribution of mechanical properties
in GFP-tagged \emph{P. aeruginosa} PAO1 biofilms. Creep curves showed
the type of mechanical response ranging from elastic to viscoelastic.
Non-magnetic beads did not show significant motion in the respective
control at distances greater than 5 \selectlanguage{greek}μ\selectlanguage{english}m. In the local viscoelastic
environments, the deformation observed for the loading time was
dominated by elastic compliance, i.e., the viscous part of the
deformation was lower that the elastic initial deformation. Also, about
20 to 30\% of the particles (depending on the condition) showed purely
elastic response. Thus, the analysis was focused on the spatial
distribution of the elastic compliance \emph{J} for each biofilm growth
condition. From \emph{J} , the elastic modulus (E) can be obtained as
its reciprocal and applied for mathematical simulations.

\subsubsection*{Effect of shear stress}

{\label{effect-of-shear-stress}}

The effect of flow-induced shear stress on the mechanical properties was
studied by growing biofilms under different flow rates. The distribution
of compliance values in different directions is shown in Figure 2, for
biofilms grown under 0.1, 1 and 5 mL/h (Re = 0.28, 2.8 and 13.9),
respectively. Biofilms grown at the higher flow rates of 1 and 5 mL/h
were thinner (approximately 80--150 \selectlanguage{greek}μ\selectlanguage{english}m) than at 0.1 mL/h (approximately
150--250 \selectlanguage{greek}μ\selectlanguage{english}m). The spatial distribution of compliance revealed
significant variability in mechanical properties. Higher heterogeneity
was observed for the lower flow rates, with compliance values differing
by as much as two orders of magnitude within the same biofilm (Figure
2). The distribution of compliance showed a significant correlation with
biofilm depth (Z), some dependence across the capillary width (X), and
was variable but not stratified across the capillary length (Y).

The lowest flow rate, 0.1 mL/h (Re=0.28), produced the greatest vertical
heterogeneity of compliance, with more elastic areas at the biofilm
exterior and more rigid areas at the base. For the 1 mL/h flow rate
(Re=2.8), a more evenly layered structure was observed. The biofilm
grown at 5 mL/h (Re=13.9) showed high rigidity throughout, with a mean
compliance value as low as 0.02 m\textsuperscript{2}/N (E=50 Pa).
Increasing the flow shear stress decreased the average vertical
compliance from values of 2 m\textsuperscript{2}/N to 0.02
m\textsuperscript{2}/N (elastic moduli range of 0.5--50 Pa), producing
more rigid structures. Similar results were observed in \emph{E.
coli}biofilms in a previous study using the same experimental technique
(Galy et al., 2012). These results were observed for low shear stresses
during growth (Re from 0.28--13.9 corresponds to a shear stress of
0.22--11 mPa). Thus, the shear stress during growth was not expected to
cause deformation, i.e., the elastic modulus is high compared to the
wall shear stress in the channel. This suggests that the observed
biofilm stiffening may be explained by a biological adaptation to low
shear rather than a physical remodeling due to shear stress (Galy et
al., 2012).

The compliances of biofilm basal layers (0-20 \selectlanguage{greek}μ\selectlanguage{english}m) were almost the same
for the different flow rates, despite greater differences in the outer
layers. This suggests the mechanical properties near the biofilm base
were not significantly affected by shear. This may be because the outer
layer of the biofilm is more affected by the bulk conditions (Even et
al., 2017). It also may be due to the lower biological activity at the
base, leading to lower net growth rates compared to the upper layer. The
deeper layers of biofilms are known to consolidate and became more dense
over time (Laspidou \& Rittmann, 2004a).

The compliance across the capillary width (x-direction) showed a lower
value near the center of the capillary channel. Result showing
increasing stiffness in the center of biofilms have been reported by
others (Karampatzakis et al., 2017). The variation of compliance along
the capillary width may be caused by the shear force distribution.
Stiffer biofilms in the middle of the flow channel correlate with the
higher shear stress condition, while lower shear near the edges led to a
softer biofilm. This is consistent with the findings from Thomen et al.
(2017) who found that the development of biofilm had a strong
relationship with the spatial distribution of shear stress.

The non-uniformity of biofilm mechanical properties can be correlated
with the differences in concentrations of biofilm constituents, due to
changes in EPS density or water content (Wilking et al., 2011). For
instance, different EPS polysaccharides, such as Psl and Pel (Yang et
al., 2011), produced by biofilm microorganisms such as \emph{P.
aeruginosa} can either increase the elasticity or increase the viscosity
of biofilms, respectively (Chew et al., 2014; Friedman \& Kolter, 2004).
Thomen et al. (2017) suggested that surrounding flows may partially wash
out extracellular molecules secreted by cells. Based on the elastic
results in our study, it is possible that shear flow removed bacteria
cells instead of extracellular molecules, leading a higher stiffness
under high shear stress. Although relatively low shear was used in the
experiments, it is likely that continuous and higher shear stresses
would result in an increasing production of EPS components such as Psl,
increasing the strength of the matrix. Previous work also found an
influence of hydrodynamics on the levels of quorum sensing molecules
(Timp et al., 2009). Quorum sensing impacts gene expression and often
promotes and regulation biofilm formation and EPS production. Therefore,
it is also possible that shear stress changed the biofilm elasticity due
to its impacts on quorum sensing.

\subsubsection*{Effect of dissolved
oxygen}

{\label{effect-of-dissolved-oxygen}}

The effects DO on biofilm mechanical properties were studied for two
flow rates, 0.1 mL/h (Re=0.28) and 5 mL/h (Re=13.9), using the
methodology previously described. Compared to the biofilms obtained
under DO saturation (8 mg/L), biofilms grown at a DO of 1 mg/L showed
similar thickness but were more compact (less rough) (Figure 3). The
vertical distribution of compliance values for these biofilms after 5
days of growth is shown in Figure 3.

At equal flow rates, biofilms displayed higher compliance values at
lower DO levels. Similar to biofilms at high DO, the biofilms at low DO
produced a wider range of compliance values flow for the lowest flow
rates (Re=0.28). The outer biofilm layers had a higher compliance than
the basal layers (Figure 3). Interestingly, the highest flow rate
(Re=13.9) also produced a layered biofilm, not as stiff and uniform as
the one with high DO. The average compliance at Re=13.9, around 0.6
m\textsuperscript{2}/N, was lower than for Re=0.28, around 1.2
m\textsuperscript{2}/N. These results show the same trend discussed
above, that increased shear stress leads to increased biofilm stiffness.

Higher compliances were observed in the superficial biofilm under low DO
conditions. Previous research revealed a correlation between DO and
growth rate and EPS production (Applegate \& Bryers, 1991; Laspidou \&
Rittmann, 2004b). Laspidou and Rittmann (2004) used modeling to infer
that consolidation in the bottom layers resulted in higher density and
lower porosity. Lower production of polysaccharides was also associated
with oxygen-limited conditions (Ahimou et al., 2007; Applegate \&
Bryers, 1991). Under different bulk DO conditions, oxygen depletion with
depth caused a variation of cell density and EPS production, which
increased the heterogeneity of mechanical properties. As mentioned
above, higher compliance in the superficial biofilm at low DO may be
caused by low growth rate of cells. Furthermore, several studies have
demonstrated that the oxygen gradients in biofilms are determined by the
biological activity (Bridier et al., 2015; Xu et al., 1998). Thus,
biofilm properties in the bottom layer are not strongly correlated with
the oxygen concentrations in the bulk. The consistency of compliance
values in the bottom layer was likely due to the lack of growing cells
and anoxic conditions, regardless of the DO in the bulk liquid.

\subsubsection*{Effect of divalent
cations}

{\label{effect-of-divalent-cations}}

The compliance profile for biofilms grown at the lowest growth flow rate
(Re=0.28) was determined for presence of Ca\textsuperscript{2+}, as
shown in Figure 4. In both cases the compliance values decreased,
especially in the DO-saturated condition, where instead of the widely
distributed compliance observed without Ca\textsuperscript{2+} addition,
the results showed a reduced spread in values and a more homogeneous
compliance profile. The change in heterogeneity can be correlated with
the chemical effects of Ca\textsuperscript{2+}, which form a bridge
between negatively charged moieties in the EPS (Flemming \& Wingender,
2001). Ca\textsuperscript{2+} has been shown to promote ordered protein
helices, and has a strong affinity for metal ions (Sehar et al., 2016;
Sutherland, 2001). The change of compliance was due to the
Ca\textsuperscript{2+}, which promoted a stronger and stiffer biofilm
matrix. Previous studies showed that Ca\textsuperscript{2+} could
stimulate the development of thick and compact biofilms, with increased
mechanical stability and elastic modulus (Ahimou et al., 2007; K\selectlanguage{ngerman}örstgens
et al., 2001; Shen et al., 2018). The results suggest that the
development of stiffer structures was likely to be a consequence of high
degrees of cross-linking.

The presence of Ca\textsuperscript{2+} led to much thicker biofilms when
the bulk DO was high, and slightly thicker biofilms when the bulk DO was
low. Sehar et al. (2016) concluded that the addition of
Ca\textsuperscript{2+} could increase the cell density and thicken the
biofilm. Thus, the formation of mechanically more stable biofilms in the
presence of Ca\textsuperscript{2+} may be due to the strong cationic
bridging between bacterial cells and EPS polymers. This is consistent
with results obtained with other multivalent cations that play the same
role as Ca\textsuperscript{2+} , such as Cu\textsuperscript{2+},
Mg\textsuperscript{2+}, and Fe\textsuperscript{3+} (Beech \& Sunner,
2004).

\subsection*{Modeling the impacts of mechanical property
heterogeneity}

{\label{modeling-the-impacts-of-mechanical-property-heterogeneity}}

Past research has mainly assessed the mechanical properties of biofilms
with macroscale tools (e.g., shear rheometry), characterizing an entire
biofilm with one reading. Other studies have characterized only the
outer layers of the biofilm (e.g., microindentation). Also, few modeling
studies have considered mechanical heterogeneity. To illustrate the
potential impacts of mechanical heterogeneity on biofilm behavior (i.e.,
biofilm deformation), as well as the potential differences in assuming
biofilms have homogeneous elastic properties, we used a 2D FSI model to
simulate the mechanical responses of two arbitrary biofilm morphologies.
We arbitrarily assigned the experimental observed z-variations of
mechanical properties from this study. Two biofilm morphologies were
considered: a biofilm cluster or colony, and mushroom-like biofilm tower
(Figure 5a and 5b).

The biofilm structures were modeled as purely linear-elastic solids,
i.e., without viscous behavior. The initial morphologies were arbitrary,
and it was assumed that the timescales were short enough that morphology
changes due to growth or decay could be neglected. Thus, the model
captures the short-term behavior when exposed to fluid flow. For each
case, the deformations were compared for uniform mechanical properties
and properties varying in the z direction per one of our experimental
results.

Figure 5a and 5b shows 2D plots of the fluid velocities, von Mises
stresses in the biofilm, and initial and final biofilm positions. The
velocity of biofilm colony simulation (Re=250) was two orders of
magnitude higher than that of biofilm tower simulation (Re=1) since the
tower shape is more easily deformed. The original and deformed biofilm
structures are shown as white outlines and colored surfaces in Figure 5a
and 5b.

To evaluate the impact of mechanical heterogeneity on biofilm
deformation, the displacement of the biofilm colony was plotted for as a
function of biofilm depth (z direction) (Figure 5a and c). For the
biofilm colony (Figure 5c), the heterogeneous biofilm suffered larger
deformation. The more deformable top layer in the heterogeneous biofilm
showed three times greater deformation (55 \selectlanguage{greek}µ\selectlanguage{english}m at the top) than the
homogeneous biofilm colony (20 \selectlanguage{greek}µ\selectlanguage{english}m at the top). This is a 64\% increase
in deformation. For the mushroom-like biofilm tower (Figures 5b and d),
results showed larger deformations (70 \selectlanguage{greek}µ\selectlanguage{english}m at top) for the homogeneous
biofilm than for the heterogeneous biofilm (55 \selectlanguage{greek}µ\selectlanguage{english}m at top), a difference
of 22.8\%.

The modeling simulations show that, even when the spatial distribution
of elastic parameters is the same, the effect of mechanical
heterogeneity on overall deformations can vary depending on the biofilm
morphology. The assumption of homogeneous mechanical properties can lead
to significant differences in deformation predictions.

Possible mechanisms and implications of biofilm mechanical heterogeneity

Our experimental results showed that \emph{P. aeruginosa} biofilms have
a heterogeneous distribution of mechanical properties, with a wide range
of values. Other researchers have explored the mechanical non-uniformity
both experimentally (Birjiniuk et al., 2014; Cao et al., 2016; Galy et
al., 2012; Karampatzakis et al., 2017) and theoretically (Bridier et
al., 2015; Even et al., 2017; Laspidou et al., 2014). The spatial
distribution of biofilm mechanical properties was also found to vary
with environmental conditions. The overall impacts of chemical,
biological, and physical factors from the environment and biofilm itself
explain the mechanical heterogeneity of the biofilm.

Stewart and Franklin (2008) concluded that mechanisms of biofilm
heterogeneity including, but were not limited to, microscale chemical
gradients, adaptation to local environmental conditions, stochastic gene
expression, and genotypic variation. Another possible mechanism is that
the growth of biofilms tended to be in the direction of minimal
mechanical resistance, which lead to consistent gradients of stiffness,
oxygen concentrations, nutrient concentrations, and growth rates (Even
et al., 2017).

Biofilm heterogeneity has important practical implications, especially
for biofilm control. Past research on the mechanical properties of
biofilms show a wide range of results, possibly due to the wide variety
of testing techniques at different length scales. In addition, spatial
heterogeneity of mechanical properties, and differences in biofilm
morphologies and structures, play key roles on the deformation of
biofilms under fluid flow (Even et al., 2017; Trejo et al., 2013). Our
modeling results show that the spatial heterogeneity of biofilm
mechanical properties could lead to significant differences of the
deformation, even with the same averaged value. For this reason,
previous studies (Picioreanu et al., 2018; Stoodley et al., 1999) which
back-calculated homogeneous mechanical properties using the experimental
deformation should be viewed with caution.

Our modeling results show that the variability of mechanical properties
can have different effects, depending on the biofilm morphology, spatial
distribution of mechanical properties, and hydrodynamic profiles.
Surface erosion and biofilm sloughing may occur more easily and
frequently on more non-uniform biofilm colonies and more uniform
mushroom-like biofilms.

In water and wastewater engineering, biofilms can play both beneficial
and detrimental roles. For example, the accumulation of biofilm in
biotreatment processes is critical to their good performance, while
development of biofilm fouling layers in membrane filtration systems can
greatly increase energy requirements. Thus, promoting accumulation of
beneficial biofilms and promoting detachment of detrimental biofilms is
an important practical need. Yet there are few tools available to
predict biofilm behavior. This research provides both results on the
variability of mechanical properties for a \emph{P. aeruginiosa}biofilm,
and also a tool to assess its impact on biofilm deformation.

In order to affect the mechanical stability of biofilms, the fluid flow
rates and shear, DO concentrations, and concentration of divalent
cations are important variables. Our results also suggest that biofilm
initially may be more flexible and easy to detach at the base, but then
become stiffer at the base as the biofilm matures. Thus, lighter and
more frequent biofilm removal treatments may be more effective than ones
that are less frequent and more intense. For longer term assessments of
biofilm behavior, models should include growth and decay processes, and
also the viscoelastic behavior of biofilms.

\subsection*{4. Conclusions}

{\label{conclusions}}

Magnetic tweezers were used to characterize the spatial distribution of
elastic properties of \emph{P. aeruginosa} biofilms for a range of
conditions. Spatial heterogeneity was observed in all three-dimensions,
and environmental conditions had a significant impact on the spatial
distribution. Higher shear resulted in a stiffer and more uniform
distribution, possibly due to mechanical adaptation. Also, stiffer
biofilms in the center of the flow cell were correlated with greater
hydrodynamic shear. Lower bulk DO led to a more heterogeneous biofilm,
probably due to the greater variation of biological activity with depth
in the biofilm. The addition of Ca\textsuperscript{2+} in bulk liquid
increased the average stiffness and resulted in more uniform biofilms.
Further research should address the viscoelastic behavior, and the
combination of elastic deformation, viscous deformation, and growth, as
all of these may significantly impact biofilm deformation and morphology
in the long term.

Using mathematical model for two different hypothetical biofilm
morphologies under fluid flow, it was shown that the spatial
heterogeneity of mechanical properties can lead to significant
differences in biofilm deformation. This demonstrates that biofilm
mechanical heterogeneity should be considered when predicting biofilm
deformation. Conversely, it also should be taken into consideration when
using biofilm deformation to infer its mechanical properties.

\textbf{Acknowledgements}

This research was funded by NSF Project 1605177. Partial support to JPP
was provided by FONDECYT 11181107 and NID PIA/BASAL FB0002, and to ML by
the CEST/Bayer Predoctoral Research Fellowship at the University of
Notre Dame. The authors would like to thank the Notre Dame Center for
Research Computing for the computational resources. We thank Joshua
Shrout for providing the \emph{P. aeruginosa} strain.

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