# Simulation and Modeling of the Adhesion of Staphylococcus aureus onto Inert Surfaces under Fluid Shear Stress

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}and Python. Overall, COMSOL accurately predicted the experimental trend of higher rates of bacterial adhesion with decreasing shear stress but poorly characterized the plateauing phenomena observed over time. Python provided a robust mathematical representation of the non-linear relationship between cell concentration, shear stress, and time but its polynomial regression approach was not grounded on theoretical physical concepts. These insights, combined with advancements in AI and machine learning, underscore the potential for synergistic computational techniques to enhance our understanding of bacterial adhesion to surfaces, offering a promising avenue for developing novel therapeutic strategies.

## 1. Introduction

^{2}. This biomechanical force shaped by fluid flow, vessel geometry, and fluid viscosity, is crucial in the physiological context of bacterial adhesion [13].

^{2}in brachial arteries and from 1 to 6 dyn/cm

^{2}in most veins [14,15]. In this study, the kinetics of S. aureus cells’ adhesion to abiotic surfaces were investigated under varied fluid shear forces. The use of inert glass surfaces in this study provides a simple and controlled methodology to examine bacterial adhesion, allowing a focus on understanding the fundamental principles of adhesion without the interference of surface chemistry variations [16,17]. Cell adhesion assays were performed in relevant hydrodynamic conditions with a BioFlux 200 microfluidic system. Theoretical and empirical studies have shown that cell spatial distribution plays a critical role in the physiological properties of bacteria in their natural milieus [18,19]; thus, the effect of shear stress on the overall organization and pattern of bacterial adhesion in the microfluidic system was investigated using MATLAB and COMSOL Multiphysics

^{®}software. Given the propensity of COMSOL Mutliphysics

^{®}, a well-known commercial fine element modeling package, to serve as a tool of choice for modeling physiological transport phenomena [20,21], the aim of this study was to develop a simple COMSOL multiphysics model to evaluate the adhesion kinetics of free-floating bacteria in hydrodynamic milieus.

## 2. Materials and Methods

#### 2.1. Bacterial Strains and Cultures

^{®}, BD; Franklin Lakes, NJ, USA), in a shaking flask incubator at 37 °C with continuous rotation, as previously described [23,24,25]. As applicable, cells were diluted with phosphate-buffered saline (PBS; 138 mM NaCl, 2.7 mM KCl [pH 7.4]) to achieve a bacterial concentration of $1\times {10}^{7}$ cells/mL—as determined with a cell counter (Beckman Coulter Multisizer 4). As previously described, PBS hindered further bacterial growth and ensured a controlled, physiologically relevant environment devoid of nutrients that could interfere with the adhesion process [23,24,25,26,27].

#### 2.2. Adhesion Assay under Hydrodynamic Conditions

^{7}cells/mL in phosphate buffer saline (PBS) at 37 °C, was pipetted to the BioFlux plate’s input well. To mimic physiologically and dynamically relevant conditions, adhesion assays were investigated at wall shear forces ranging from 1- to 5 dyn/cm

^{2}through the pressure interface of the BioFlux system. The flow system was connected to a Zeiss AXIO Observer microscope for image acquisition.

#### 2.3. Determination of Bacteria Surface Concentrations, Spatial Analysis and Adhesion Rates

^{2}) of bacteria cells are determined by taking the number of cells detected from each Excel file and dividing by the area of the image captured. The surface concentrations are then plotted against time for all shear stress values. The process is repeated for at least three experimental replicates per assay plate; and each plate experiment is repeated multiple times, and the results are averaged.

#### 2.4. Multiphysics Simulation in COMSOL

^{®}version 6.1 application was used to develop a two-dimensional multiphysics model that would simulate the adhesion of S. aureus cells. The choice of a two-dimensional channel simplified the computational analysis while retaining biological relevance and ensuring experimental relevance. The dimensions of the analytical channels were 400 μm by 70 μm (width × height) to mimic the dimensions of the BioFlux well plates. Similarly, the flow parameters used for the simulations were taken from the BioFlux 200 system and from the literature, when applicable. The applied wall shear stress levels (τ) ranged from 1 to 5 dyn/cm

^{2}to simulate different fluid flow conditions that the bacteria may encounter in vivo [29]. The density of the fluid in the system (ρ) was held constant at 1000 kg/m

^{3}, which represented the approximate density of phosphate-buffer saline [30]. The viscosity of the fluid (η) was calculated from γ, the fluid shear rate, with the following equation:

#### 2.5. Data-Driven Modeling in Python

#### 2.6. Statistical Analysis

## 3. Results

#### 3.1. Two-Dimensional Spatial Distributions of Adhered Bacteria Cells Were Independent of Hydrodynamic Shear Stress

^{2}(Figure 1). Then, the spatial distribution of bacteria was investigated by monitoring the mean spatial distribution between cells and the average distance between neighboring cells at varying wall shear stresses. The mean spatial distribution between cells depicted a steady trend when subjected to various shear stress levels over a 60-minute period during BioFlux assays (Figure 2). Remarkably, the average distance remained consistent at approximately 26 µm across all shear stress conditions, showing no significant change as time progressed. This uniformity suggested that the average spatial distribution of cells on the surface was not significantly influenced by the different shear stresses applied within the range of 1 to 5 dyn/cm

^{2}. This constant spatial distance implied a homogeneity in cell distribution post-adhesion, indicating that once cells adhere to the surface, their spatial organizations did not dynamically change with time under the shear stress conditions tested.

^{2}) reaching a plateau earlier than the lower shear stresses. By the end of the observation period, higher shear stresses (4 and 5 dyn/cm

^{2}) converge to a 33% higher mean distance than lower shear stresses (1–3 dyn/cm

^{2}). This trend explains how as the shear stress increases the cells are more loosely packed and the spatial clustering between the cells is lower.

#### 3.2. Bacterial Surface Coverage Decreased with Increasing Wall Shear Forces

^{2}, there was an initial steep ascent followed by a plateau at a final surface concentration. Whereas at a wall stress of 5 dyn/cm

^{2}, the surface concentration of bacterial cells increased moderately until it plateaued at a 2.78-times-lesser surface concentration level than at 1 dyn/cm

^{2}. Overall, the final surface concentration showed a declining trend as wall shear stress increased, although an outlier was observed at shear stress 3 dyn/cm

^{2}. The bacterial surface concentration at 3 dyn/cm

^{2}was not only greater than that at 2 dyn/cm

^{2}from the outset but also continued to diverge further over time (Figure 4). This observation suggested that the cells experienced a more favorable adhesion environment at 3 dyn/cm

^{2}compared to 2 dyn/cm

^{2}. This outlier suggested that cell adhesion in BioFlux assays was not solely governed by the magnitude of the shear stress but also by other factors that could be influencing cell adhesion kinetics at specific shear stress levels.

^{2}. Overall, a decrease of almost 2-fold in the rate of adhesion was observed as the shear stress level increased from 1 to 5 dyn/cm

^{2}. For instance, as the shear stress increased from 1 dyn/cm

^{2}to 2 dyn/cm

^{2}, there was a 16% reduction in the adhesion rate. However, an unexpected increase was observed at 3 dyn/cm

^{2}. The adhesion rate at 3 dyn/cm

^{2}demonstrated a local peak that featured higher values than what was observed at both 2 dyn/cm

^{2}and 4 dyn/cm

^{2}(Figure 5) and was consistent with the surface coverage data (Figure 2). Taken together, these results could have implications for understanding cellular responses to mechanical forces in various biological and biomedical applications.

#### 3.3. Bacterial Adhesion May Be Simulated in COMSOL with Leaking Wall Boundary Conditions

^{®}software using two distinct wall boundary conditions. In the first scenario, the COMSOL setup used the flow parameters defined in Table 1 coupled with non-leaking, no-slip wall boundary conditions. The channel with the dimensions of 400 µm by 70 µm mimicked the boundary wall and flow conditions of the BioFlux 200 microfluidic environment. All the walls of the channel featured a “no-slip” boundary condition. Vertical walls at the extremes of the channel act as the inlet and outlet. The data showed a fully developed velocity profile with maximum velocity towards the center of the channel and decreasing velocity as we move closer to the walls (Figure 6A). The velocity profile corroborated the presence of a fully developed laminar flow in the Bioflux microfluidic channel during in vitro adhesion experiments (Figure 3 and Figure 4). COMSOL simulations also accounted for S. aureus cells that could be seen as 1 µm diameter circles (Figure 6C) floating in the channel and moving with the fluid. However, no bacterial adhesion to the walls of the microfluidic channel was observed during the hour-long fluid flow simulation under these in silico conditions. Instead, the bacteria cells were observed to be sliding off the wall in the direction of flow. Since there was no adhesion observed under the above COMSOL scenario, the simulation setup was altered to exhibit relevant adhesion of cells.

#### 3.4. COMSOL Simulations Corroborated Results from Microfluidics Studies

#### 3.5. Python Modeling Reproduced the Non-Linear Relationship between Bacterial Adhesion and Wall Shear Stress

^{−2}− 8.4290 × 10

^{−2}τ + 4.6928 × 10

^{−2}τ

^{2}− 6.185 × 10

^{−3}τ

^{3}) + (2.6465 ×

10

^{−2}− 4.278 × 10

^{−3}τ + 2.46 × 10

^{−4}τ

^{2})t + (−2.11 × 10

^{−4}+ 7 × 10

^{−6}τ)t

^{2}+ 1 × 10

^{−6}t

^{3},

^{2}, the model predicted an initial steep increase in surface concentration, followed by a plateau, aligning well with the experimental observations; conversely, at 5 dyn/cm

^{2}, the model predicted a more gradual increase in surface concentration, eventually plateauing at a level 2.89 times lower than at 1 dyn/cm

^{2}, consistent with the experimental trend (Figure 4 and Figure 9). The model had an R

^{2}value of 0.96277, indicating an overall good fit with the data which was skewed at intermediate shear stress conditions by the outlier trend of adhesion observed at 3 dyn/cm

^{2}(Figure 4).

^{2}was generated. This model gave the following equation for surface concentration, C(t,τ):

^{−3}− 3.9318 × 10

^{−2}τ + 3.9432 × 10

^{−2}τ

^{2}− 6.185 × 10

^{−3}τ

^{3}) + (3.0093 ×

10

^{−2}– 8.027 × 10

^{−3}τ + 8.71 × 10

^{−4}τ

^{2})t + (−2.06 × 10

^{−4}+ 7 × 10

^{−6}τ)t

^{2}+ 1 × 10

^{−6}t

^{3},

^{2}value of 0.9982, suggesting that ignoring the outlier at shear stress 3 dyn/cm

^{2}yielded a more accurate mathematical representation of the change in bacteria surface concentration with time at wall shear stresses of 1, 2-, 4-, and 5 dyn/cm

^{2}under the investigated experimental conditions. Taken together, these data suggest that the Python models developed through machine learning polynomial regression accurately reflect the observed empirical trends and capture the complex interplay between shear stress and bacterial adhesion over time.

## 4. Discussion

^{®}simulations.

^{2}, suggesting that factors beyond sheer mechanical forces are at play in bacterial adhesion. For the most part, COMSOL simulations further supported BioFlux results, indicating a clear decreasing trend in the maximum rate of adhesion as shear stress increased. The consistent findings across both experimental and simulation approaches underscore the complex nature of cell adhesion under fluid shear stress and hint at the existence of non-specific interactions that promote adhesion under fluid flow [34].

^{2}value, it does not reveal the causal mechanisms underlying the bacterial adhesion phenomena observed in the microfluidics system.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Representative phase contrast images of S. aureus cells adhering to the glass surface of the Bioflux microfluidic system. Images of bacteria adhering to surfaces within the microfluidic system were captured using an AXIO Observer microscope at wall shear forces of 2 dyn/cm

^{2}at times 0 (

**A**), 20 (

**B**), 40 (

**C**), and 60 (

**D**)—which correspond to the start of the assay and three subsequent twenty-minute intervals.

**Figure 2.**Impact of shear stress on the average spatial distance in the Bioflux microfluidic system. A MATLAB script was used to calculate the average distance between all the adhered cells over time at each shear stress value. Three experimental replicates were performed on separate days, with a total of 180 images generated for analysis. These images represented cell adhesion at twenty-minute intervals for triplicate areas in each experiment. The data are the average of all experimental replicates at each shear stress and time point.

**Figure 3.**Impact of shear stress on the minimum average distance between adhered cells over time. A MATLAB script was used to calculate the average distance between the closest adhered cells over time at each shear stress value. Three experimental replicates were performed on separate days, with a total of 180 images generated for analysis. These images represented cell adhesion at twenty-minute intervals for triplicate areas in each experiment. The data are the average of all experimental replicates at each shear stress and time point.

**Figure 4.**Impact of shear stress on the surface concentration of cells in BioFlux system. S. aureus cells in PBS suspension at 37 °C flowed through the microplates at wall shear stress values between 1 and 5 dyn/cm

^{2}. The data show how the number of cells that adhered per unit area, i.e., surface concentration of cells, varies over 60 min. Images of adhered cells were captured at five-minute intervals for triplicate areas in each experiment, and three experimental replicates were performed on separate days.

**Figure 5.**Impact of shear stress on the maximum rate of adhesion in BioFlux system. Data show how the maximum rate of adhesion varies over wall shear stress values ranging from 1- to 5-dyn/cm

^{2}. Data represent average values from three experimental replicates, with triplicate runs for each experiment. Stars represent statistical significance (p < 0.05) from p-value test between the two columns.

**Figure 6.**COMSOL simulation scenarios for bacterial cell adhesion under hydrodynamic milieus. A microfluidic channel was simulated in COMSOL, using the parameters described in Table 1. Two scenarios were investigated: (

**A**,

**C**) the channel had fixed walls with no slip, and wall 4 served as the only outlet; and (

**B**,

**D**) the channel had fixed walls with no slip, wall 4 still served as the main outlet but walls 2 and 3 acted as leaking walls. S. aureus cells were represented by green spheres of 1 µm diameter. Images show representative flow and adhesion profiles at the wall shear stress value of 1 dyn/cm

^{2}captured at time t = 5 min during a sixty-minute run.

**Figure 7.**Impact of shear stress on the surface concentration of bacteria cells in COMSOL. The graph shows how the number of cells adhered per unit area (i.e., the surface concentration) varies with respect to time, in a 400 by 70 μm rectangular channel. These data are a result of simulating the adhesion of solids with 1 µm diameter in presence of a 10% leaking wall velocity at different wall shear stress values ranging from 1 to 5 dyn/cm

^{2}.

**Figure 8.**Impact of shear stress on the maximum rate of bacterial adhesion in COMSOL. The graph shows the trend of adhesion rates for the first 20-min interval of the COMSOL simulation at different shear stress values ranging from 1 to 5 dyn/cm

^{2}.

**Figure 9.**Comparison of the Python theoretical model to empirical data in the microfluidic system. The graph juxtaposes the curves predicted from the Python polynomial model in Equation (1) (“Model:”) with the averages of cell surface concentrations from the adhesion assays in the microfluidic system at wall shear stress values ranging from 1 to 5 dyn/cm

^{2}.

**Figure 10.**Comparison of the Python theoretical model to empirical data in the microfluidic system. The graph juxtaposes the curves predicted from the Python polynomial model in Equation (2) (“Model:”) with the averages of cell surface concentrations from the adhesion assays in the microfluidic system at wall shear stress values of 1-, 2-, 4-, and 5 dyn/cm

^{2}.

**Table 1.**Relevant biological and physical properties of the simulation system at different shear stresses.

Symbol | Value | Unit | Description | ||||
---|---|---|---|---|---|---|---|

t | 1 | 2 | 3 | 4 | 5 | dyn/cm^{2} | Shear Stress |

U_{0} | 1337.959 | 2675.736 | 4002.267 | 5340.136 | 6678.005 | μm/s | initial velocity |

P_{0} | 0.4 | 0.8 | 1.19 | 1.59 | 1.99 | psi | inlet pressure |

_{0}is the initial velocity of fluid flow through the system, calculated using U

_{0}= Q/A, where Q is volumetric flowrate from BioFlux 200 and A is the cross-sectional area of the channel. P

_{0}is the pressure at the system inlet, also determined from the BioFlux 200 system.

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**MDPI and ACS Style**

Shaikh, S.; Saleem, A.N.; Ymele-Leki, P.
Simulation and Modeling of the Adhesion of *Staphylococcus aureus* onto Inert Surfaces under Fluid Shear Stress. *Pathogens* **2024**, *13*, 551.
https://doi.org/10.3390/pathogens13070551

**AMA Style**

Shaikh S, Saleem AN, Ymele-Leki P.
Simulation and Modeling of the Adhesion of *Staphylococcus aureus* onto Inert Surfaces under Fluid Shear Stress. *Pathogens*. 2024; 13(7):551.
https://doi.org/10.3390/pathogens13070551

**Chicago/Turabian Style**

Shaikh, Sarees, Abdul Nafay Saleem, and Patrick Ymele-Leki.
2024. "Simulation and Modeling of the Adhesion of *Staphylococcus aureus* onto Inert Surfaces under Fluid Shear Stress" *Pathogens* 13, no. 7: 551.
https://doi.org/10.3390/pathogens13070551