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-Nàcher & Smets, 2014). Also, the bulk calcium ion (Ca2+) 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 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 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 Ca2+ concentrations in P. aeruginosa biofilms. 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.
2.Methods

Biofilm growth conditions

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 (OD600) 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 μm diameter, at a final concentration of 2.5\(\times\)106 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 μm diameter were used (Fluoresbrite 18660, Polysciences, Germany).
Biofilms were grown in borosilicate glass capillaries with a 1 mm internal side and 150 μm 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 Ca2+ addition of 100 mg/L as CaCl2. Anin-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).

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 μm above the capillary bottom. A z-stack with a depth step size of 0.5 μm 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 μm \(\times\)250 μm (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 μm 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).

Creep curves and mechanical behavior analysis