# American Institute of Mathematical Sciences

June  2017, 14(3): 625-653. doi: 10.3934/mbe.2017036

## Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects

 1 Biomedical Physics, Dept. Physics, Ryerson University, 350 Victoria Street Toronto, ON, M5B 2K3, Canada 2 Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36,8010 Graz, Austria 3 Dept. Mathematics and Statistics, University of Guelph, 50 Stone Road East, ON, N1G 2W1, Canada

* Corresponding author: Blessing O. Emerenini

Received  August 20, 2015 Accepted  October 26, 2016 Published  December 2016

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.

Citation: Blessing O. Emerenini, Stefanie Sonner, Hermann J. Eberl. Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects. Mathematical Biosciences & Engineering, 2017, 14 (3) : 625-653. doi: 10.3934/mbe.2017036
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##### References:
Schematic of the biofilm system cf [18]: The aqueous phase is the domain $\Omega_1(t) = \{x\in \Omega: M(t;x) =0 \}$, the biofilm phase $\Omega_2(t) = \{x\in \Omega: M(t;x) >0 \}$. These regions change over time as the biofilm grows. Biofilm colonies form attached to the substratum, which is a part of the boundary of the domain
2-D structural representation of the microbial floc growth for autoinducer production rate $\alpha=30.7$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances t. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale
1-D Spatial representation of the development and dispersal of bacterial cells from the microbial floc: The snapshots are taken at different computational time $t$, with an autoinducer production rate $\alpha=30.7$ and a dispersal rate of $\eta_1=0.6$
Temporal plots of simulations computed for a non-quorum sensing producing microfloc (Non-QS) and a quorum sensing producing microfloc using seven different constitutive autoinducer production rate $\alpha = \{92.0,46.0,30.7,23.0,18.4,15.3,13.1\}$ and fixed maximum dispersal rate $\eta_1=0.6$. Shown are (a) the total sessile biomass fraction $M_{tot}$ in the floc, (b) the floc size $\omega$ (c) dispersed cells $N_{tot}$, (d) relative variation $R$, and (e) signal concentration $A_{ave}$
2-D structural representation of the microbial biofilm growth for autoinducer production rate $\alpha=30.7$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances $t$. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale
2-D structural representation of the microbial biofilm growth for autoinducer production rate $\alpha=92.0$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances $t$. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale
Temporal plots of simulations computed for a non-quorum sensing producing biofilm (Non-QS) and a quorum sensing producing biofilm using seven different constitutive autoinducer production rate $\alpha = \{92.0,46.0,30.7,23.0,18.4,15.3,13.1\}$ and fixed maximum dispersal rate $\eta_1=0.6$. Shown are (a) the total sessile biomass fraction $M_{tot}$ in the floc, (b) the floc size $\omega$ (c) dispersed cells $N_{tot}$, (d) relative variation $R$, and (e) signal concentration $A_{ave}$
Comparison of the sessile biomass $M_{tot}$ and the dispersed cells $N_{tot}$ under different boundary conditions for the signal molecule $A$: Homogenous Dirichlet conditions and Neumann conditions. The left panel is for a microbial floc while the right panel is for a biofilm
Parameters used in the numerical simulations
 Parameter Description Value Source $k_1$ half saturation concentration (growth) $0.4$ [44] $k_2$ lysis rate $0.067$ assumed $\sigma$ nutrient consumption rate $793.65$ [19] $\eta_1$ maximum dispersal rate varied [18] $\lambda$ quorum sensing abiotic decay rate $0.02218$ [39] $\alpha$ constitutive autoinducer production rate varied - $\beta$ induced autoinducer production rate $10 \times \alpha$ [19] $m$ degree of polymerization $2.5$ [19] $d_1$ constant diffusion coefficients for $N$ $4.1667$ assumed $d_2$ constant diffusion coefficients for $C$ $4.1667$ [15] $d_3$ constant diffusion coefficients for $A$ $3.234$ [15] $d$ biomass motility coefficient $4.2 \times 10^{-8}$ [13] $a$ biofilm diffusion exponent $4.0$ [13] $b$ biofilm diffusion exponent $4.0$ [13] $L$ system length $1.0$ [15] $H$ system height $1.0$ assumed
 Parameter Description Value Source $k_1$ half saturation concentration (growth) $0.4$ [44] $k_2$ lysis rate $0.067$ assumed $\sigma$ nutrient consumption rate $793.65$ [19] $\eta_1$ maximum dispersal rate varied [18] $\lambda$ quorum sensing abiotic decay rate $0.02218$ [39] $\alpha$ constitutive autoinducer production rate varied - $\beta$ induced autoinducer production rate $10 \times \alpha$ [19] $m$ degree of polymerization $2.5$ [19] $d_1$ constant diffusion coefficients for $N$ $4.1667$ assumed $d_2$ constant diffusion coefficients for $C$ $4.1667$ [15] $d_3$ constant diffusion coefficients for $A$ $3.234$ [15] $d$ biomass motility coefficient $4.2 \times 10^{-8}$ [13] $a$ biofilm diffusion exponent $4.0$ [13] $b$ biofilm diffusion exponent $4.0$ [13] $L$ system length $1.0$ [15] $H$ system height $1.0$ assumed
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