A mathematical model of quorum sensing regulated EPS production in biofilm communities
 Mallory R Frederick^{1}Email author,
 Christina Kuttler^{2},
 Burkhard A Hense^{3} and
 Hermann J Eberl^{1}Email author
https://doi.org/10.1186/1742468288
© Frederick et al; licensee BioMed Central Ltd. 2011
Received: 2 August 2010
Accepted: 10 April 2011
Published: 10 April 2011
Abstract
Background
Biofilms are microbial communities encased in a layer of extracellular polymeric substances (EPS). The EPS matrix provides several functional purposes for the biofilm, such as protecting bacteria from environmental stresses, and providing mechanical stability. Quorum sensing is a cellcell communication mechanism used by several bacterial taxa to coordinate gene expression and behaviour in groups, based on population densities.
Model
We mathematically model quorum sensing and EPS production in a growing biofilm under various environmental conditions, to study how a developing biofilm impacts quorum sensing, and conversely, how a biofilm is affected by quorum sensingregulated EPS production. We investigate circumstances when using quorumsensing regulated EPS production is a beneficial strategy for biofilm cells.
Results
We find that biofilms that use quorum sensing to induce increased EPS production do not obtain the high cell populations of lowEPS producers, but can rapidly increase their volume to parallel highEPS producers. Quorum sensinginduced EPS production allows a biofilm to switch behaviours, from a colonization mode (with an optimized growth rate), to a protection mode.
Conclusions
A biofilm will benefit from using quorum sensinginduced EPS production if bacteria cells have the objective of acquiring a thick, protective layer of EPS, or if they wish to clog their environment with biomass as a means of securing nutrient supply and outcompeting other colonies in the channel, of their own or a different species.
Background
Biofilms, quorum sensing, and EPS
Biofilms are microbial communities encased in a layer of extracellular polymeric substances (EPS), adhered to biotic or abiotic surfaces. Bacteria preferentially reside in biofilms, rather than in isolation as planktonic cells. In a biofilm, bacteria are protected by the EPS matrix from external stresses, and carry out a wide range of reactions which are relevant in many disciplines, such as environmental engineering, food processing, and medicine [1].
Quorum sensing is generally interpreted as a cellcell communication mechanism used by several bacterial taxa to coordinate gene expression and behaviour in groups, based on population densities [2]. Initially, bacteria cells produce and release low amounts of signalling molecules, called autoinducers (e.g., acylhomoserine lactones (AHL) in Gramnegative bacteria). Concurrently, the cells measure the environmental concentration of the autoinducer. When a critical concentration is reached, changes in gene expressions are induced. In most bacterial autoinducer systems, the autoinducer synthase gene itself is upregulated, initiating positive feedback, and the bacteria subsequently produce AHL molecules at an increased rate. As a number of traits in bacterial biofilms relevant for human and plant health are regulated via autoinducers [3, 4], a comprehensive understanding of quorum sensing systems is highly desirable. EPS is composed of organic molecules such as polysaccharides, proteins, and lipids. The EPS matrix provides several functional purposes for the biofilm, such as protecting bacteria from environmental threats, providing mechanical stability, and degrading macromolecules to be used by the cells [5]. EPS is thought to indirectly store nutrients, which could later be converted to an available form and used as an energy source during periods of low nutrient availability [6–9].
Modelling of biofilms and quorum sensing
Biofilms are complex systems that can be viewed simultaneously as microbial ecological communities and as mechanical objects. Traditional onedimensional biofilm models were formulated as free boundary value problems of semilinear diffusion reaction systems (see [10]). Newer models take the spatially heterogeneous structure of biofilms into account and are formulated as spatially multidimensional models. A host of mathematical modelling techniques has been proposed to model biofilms, including stochastic individual based models, stochastic cellular automata models, and a variety of deterministic partial differential equation models. Some examples for such approaches are: [11–25]. These models of biofilm structure are usually coupled with diffusionreaction models for growth controlling substrates such as nutrients and oxygen. This leads to hybrid models which are mathematically difficult to analyse and often only amendable to computational simulations. In most biofilm models, EPS is not explicitly included but implicitly subsumed in the variables that describe biomass and biofilm structure. Some early exceptions are the onedimensional model of [26], the hybrid individualcontinuum model of [11], the hydrogel model of [20], and the diffusionreaction model [27].
For our study we build on the prototype biofilm model of [16], in which the biofilm structure is described by a determinstic, densitydependent diffusionreaction equation with two nonlinear diffusion effects: porous medium degeneracy and a superdiffusion singularity. This model has been extended to explicitly account for EPS in [27] based on [26], and to model quorum sensing in [28]. In the current study, we combine both effects.
Although the various multidimensional biofilm models are based on fundamentally different assumptions, such as ecological vs. mechanical properties of biofilms, and although they utilise different mathematical concepts, such as discrete stochastic vs deterministic continuous descriptions, they have been shown to predict similar biofilm structures in [10]. More recently it was formally shown that the prototype densitydependent diffusionreaction biofilm model, on which our study is based, can be derived from a spatially discrete lattice model that is related to cellular automata biofilm models [29]. In [28], it was also shown that the same prototype densitydependent diffusionreaction model can likewise be derived from a the same hydrodynamic description of biofilms that underlies the biofilm model introduced by [15]. Thus, the densitydependent diffusion model of biofilms can be understood as bridge between ecological and continuum mechanical views in biofilm modelling. The idea of using nonlinear diffusion processes, in the form of modified CahnHilliard equations, to describe the propagation of the biofilm/water interface, is also used in current, physically more involved phase field models, as introduced in [24].
Initial mathematical models of quorum sensing describe the phenomena in suspended bacteria cultures [30–32]. These models focus on predicting the rapid switch in proportions of down and upregulated subpopulations of bacteria in a batch culture, which is the characteristic positivefeedback feature of quorum sensing systems. Papers [33–35] extended the work of early models to study quorum sensing in a growing biofilm, identifying key physical kinetics parameters required for induction. More recent models describe growth in two dimensions [28], and include the effects of hydrodynamics [28, 36, 37]. A variety of applications motivate development of specific quorum sensing and biofilm models. For example, papers [34, 35] determine the critical depth the biofilm must grow to, as a function of pH, in order for induction to occur. The models of [38–40] detail biochemical pathways in quorum sensing systems, also describing antiquorum sensing treatments for applications in the medical field. The role of convective and diffusive transport of signal molecules in intercolony communication within biofilm communities is investigated in [28].
These models share a common element: autoinducer molecules (e.g., AHL) are produced by downregulated bacteria, and AHL production is greatly enhanced when the characteristic switch (change from low to high quorum sensing activity) rapidly occurs throughout the biofilm.
Much mathematical modelling research has been conducted to understand when biofilms partake in quorum sensing activity, for example, determining population thresholds [30, 31], critical biofilm depth [34, 35], and the influence of the hydrodynamic and nutritional environment [28, 36, 41]. There have, however, been few studies that look at the reverse effect  the effect of quorum sensing induction on biofilms. Once biofilm cells are upregulated, AHL is produced at an increased rate, but the question of whether the biofilm behaves differently, grows differently, or undergoes some other functional change, remains largely unanswered.
We expand on the works of [38–40, 42]. Study [42] analyzes the effectiveness of the modelled antiquorum sensing therapies by comparing growth rates of the biofilms, and states that quorum sensing activity may be detected by EPS production and associated enhanced biofilm growth. Based on the findings of [43], it is assumed in [42] that EPS production is regulated by quorum sensing, and models significantly enhanced EPS production by upregulated cells. With our model, we will study in detail how the process of quorum sensingregulated EPS production impacts biofilm growth and development in a twodimensional patchy biofilm community with slow background flow, under various environmental conditions. Our objective is to understand the relationship between quorum sensing, biofilm growth, and EPS production, and investigate the benefits a biofilm receives by using quorum sensingregulated EPS production.
To validate the claim that quorum sensing controls EPS production, and to what degree, we turn to the experimental literature. In many studies, quorum sensing has been found to impact the quality of EPS. For example, study [44] showed differences in biofilm appearance with and without expression of the pelA gene, which is essential for the production of the EPS matrix. In a later study of the quorum sensingregulated expression of the PelA enzyme, it was shown that pelgenes are required for EPS production [45]. On the other hand, [46] found many factors which affect the quality of the EPS matrix to be regulated by quorum sensing in the early development stage, such as channel production within the biofilm, swarming activity, and lipid production. Also, many studies have shown the connection between quorum sensing and mucosity [47–49]. Quorum sensing regulates components of EPS (e.g., EPS II, polysaccharides) which contribute to the mucosity, thus impacting the biofilm matrix. These studies support the idea that the amount of EPS production per cell might be influenced by quorum sensing, but do not show to what degree.
There are some examples of bacteria species, mostly plant pathogens, in which a quantitative increase of EPS production by quorum sensing regulation has been demonstrated. In [50], quorum sensing was found to regulate alginate production in Pseudomonas syringae. Alginate is an important component of EPS, and without quorum sensing, alginate levels were 70% lower. However, the impact on biofilm thickness is not described, so conclusions cannot be drawn regarding whether overall EPS is significantly reduced by the drop in alginate levels.
In [51] it is concluded that the amount of EPS production per cell in a Pantoea stewartii biofilm is increased by quorum sensing, though the degree of production is not given. Similarly, in [52] is claimed that quorum sensing upregulated EPS production in the plant pathogen Erwinia amylovora, but do not provide quantitative data. However, images are shown, from which the upregulated EPS may be estimated as a factor five to ten increase. This is supported by the experiments in [53], in which an approximately tenfold increase of EPS production in a Pantoea stewartii biofilm upon QS induction was discovered. Though many studies have established connections between quorum sensing activity and qualitative changes in EPS or other structural components, there are very few quantitative studies which investigate the amount of EPS produced through quorum sensing regulation. We choose to use the direct values for change in EPS production as reported in [53] as an estimate for the difference in downregulated and upregulated cell production rates in our system.
Aim of study
In previous research, we developed a twodimensional model of quorum sensing in patchy biofilm communities in an early development stage to study how the hydrodynamic environment and nutrient conditions contribute to biofilm growth, spatiotemporal quorum sensing induction patterns, and flowfacilitated intercolony communication [28].
 1.
How does quorum sensingregulated EPS production impact the growing biofilm?
 2.
Why is it beneficial for the biofilm to regulate EPS production using a quorum sensing mechanism?
Answers to these questions will be sought through numerical experiments that simulate the growth of biofilms in microfluidic chambers.
Mathematical Model and Simulation Design
Model assumptions
We formulate a mathematical model that describes quorum sensing in a growing biofilm community in a narrow conduit which consists of several colonies, mimicking conditions that occur in soil pores or plant/blood vessels. The biofilm is assumed to consist of bacterial cells and EPS, and it is described by the local densities of its constituents. The biofilm proper is the region in which these densities are not zero; it is surrounded by the bulk liquid. The biofilm expands due to cell growth and EPS production, both of which are coupled to the availability of a carbon nutrient. The nutrient is assumed to be dissolved. In the aqueous phase surrounding the biofilm, the nutrient is transported by bulk flow convection and by Fickian diffusion. In the biofilm itself it diffuses, although at reduced rate due to the increased diffusive resistance of the EPS and cells. Nutrients are degraded in the biofilm by the growing cells for growth and EPS production.
We distinguish between down and upregulated bacterial cells. Upregulation and downregulation are controlled by the local concentration of AHL. Upregulation occurs locally when and where the AHL concentration exceeds a threshold. If the AHL concentration in a (partially) upregulated biofilm colony drops below this critical threshold, the upregulated cells become downregulated. AHL is also assumed to be dissolved. AHL is transported by convection and diffusion in the surrounding aqueous phase, and by diffusion in the biofilm, also at a reduced rate. After AHL is produced by the bacteria, it diffuses into the aqueous phase. Upregulated cells produce AHL at a higher rate (by one order of magnitude) than downregulated cells, and decay abiotically, at a rate much slower than they are produced.
We assume that up and downregulated cells grow at the same rate, but upregulated cells produce EPS at much higher rates (tenfold). Moreover, we assume that the average cell size for down and upregulated bacteria is the same, i.e., the maximum cell and EPS density is the same for both cell types. The increased production of EPS implies an increased nutrient consumption of upregulated cells. Based on the parameters for EPS production kinetics and stoichiometry of [26], we estimate with a simple rule of proportions that upregulated cells consume approximately twice the amount of nutrients that downregulated cells consume. We do not distinguish between the EPS that is produced by each type of bacteria, but combine them into one EPS fraction.
In addition to bacteria that engage in quorum sensing, i.e., switch between down and upregulated states, we also consider nonquorum sensing bacteria species, which behave as either downregulated or upregulated cells, in regards to parameters for growth, consumption, and EPS production. These nonquorum sensing cells carry an AHLreceptor mutation and cannot be upregulated or produce any AHL. Although they are technically mutant cells, we will refer to these nonquorum sensing bacteria as different species throughout the paper.
We formulate this model in the framework of the densitydependent nonlinear diffusion model for biofilms which was originally introduced for a prototype single species biofilm in [16], and has since been extended to multispecies systems. Quorum sensing was first included in this model in our earlier study [28]. In the current study, we expand on this model by explicitly accounting for EPS, which was previously implicitly subsumed in the biomass fractions. Our model of EPS production is based on the onedimensional biofilm model of [26]. Some authors suggest that under conditions of severe nutrient limitations, EPS could be broken down and converted into nutrients by the cells [6–9, 54]. Following [26], we include this process as an option in our model and investigate whether it affects quorum sensing activity and biofilm composition.
Governing equations
The mathematical model for biofilm growth, quorum sensing, and EPS production, based on the above assumptions, is formulated as a differential mass balance for the bacterial biomass fraction, EPS, growthpromoting nutrient substrate and quorum sensing molecules.
Following the usual convention of biofilm modelling, the density of the particulate substances (bacterial cells and EPS), is expressed in terms of the volume fraction that they occupy [10]. We denote the volume fraction locally occupied by downregulated quorum sensing cells by M_{0} [], the volume fraction of upregulated quorum sensing cells by M_{1} []. Their densities are accordingly M_{0} *M_{ max } and M_{1}*M_{ max }, where the constant M_{ max } [gm^{3}] is the maximum biomass density, in terms of mass COD per unit volume.
Summary of the cell types and functions used in the model.
Cell Type  Description 

M _{0}  downregulated QS, low EPS producer 
M _{1}  upregulated QS, high EPS producer 
M _{2}  nonQS, low EPS producer 
M _{3}  nonQS, high EPS producer 
Similarly, EPS density is expressed in terms of its variable volume fraction EPS [] and the constant maximum EPS density EPS_{ max } [gm^{3}], as EPS * EPS_{ max }.
The dissolved growth controlling nutrient substrates and the dissolved quorum sensing molecules are described in terms of their concentrations C [gm^{3}] and AHL [nM].
The twodimensional computational domain Ω consists of a liquid phase with no biomass, Ω_{1}(t) = {(x, y) ∈ Ω : M(t, x, y) = 0}, and the solid biofilm phase, Ω_{2}(t) = {(x, y) ∈ Ω : M(t, x, y) > 0}.
These regions change as the biofilm grows.
The diffusion coefficient can be assumed to be the same for all bacterial fractions and the EPS because we do not distinguish the cells with respect to size and growth behaviour, and because EPS and cells diffuse together. The biomass motility coefficient d_{ m } [m^{2}d^{1}] is positive but much smaller than the diffusion coefficients of the dissolved substrates. Exponents a > 1 [] and b > 1 [] ensure biofilm expansion when M approaches 1 (implying all available space is filled by biomass), and little or no expansion provided M is small. This choice of diffusion coefficient ensures a separation of the biofilm and its surrounding aqueous phase, and that the maximum cell density will not be exceeded. The latter effect is of the type of a superdiffusion singularity, the former of the type of the porous medium equation degeneracy.
where D_{ C }(0) and D_{ AHL }(0) are the diffusion coefficients in water, and D_{ C }(1) and D_{ AHL }(1) are the diffusion coefficients in a fully developed biofilm [m^{2}d^{1}]. Although these diffusion coefficients depend on the biomass density as well, they do so in a non critical way. Since D_{ C, AHL }(0) and D_{ C, AHL }(1) are positive constants within one order magnitude, substrate diffusion is essentially Fickian.
The model includes diffusive transport of carbon substrate and AHL in the biofilm, and both convective and diffusive transport in the surrounding aqueous phase of the biofilm. The convective contribution to transport of C and AHL in the aqueous phase is controlled by the flow velocity vector w = (u, v), where u and v [md^{1}] are the flow velocities in the x and y directions. The flow in the aqueous phase is described by a thinfilm approximation to the incompressible NavierStokes equations [56]. In order to drive the flow in the channel, we specify the volumetric flow rate in terms of the nondimensional Reynolds number Re. The growth and decay processes incorporated into our model are:

growth of bacterial cells, controlled by the local availability of carbon substrate, in equations (1)(4): the maximum specific growth rate is denoted by κ_{3} [d^{1}], dependency on C is described by standard Monod kinetics where κ_{2} [gm^{3}] is the half saturation concentration.

natural cell death, at rate κ_{4} [d^{1}], in equations (1)(4),

upregulation of downregulated biomass, i.e. the conversion of M_{0} cells into M_{1} cells in equations (1) and (2), as a consequence of AHL concentration inducing a change in gene expression, and a constant rate of backconversion. The parameter κ_{5} [d^{1}nM^{n}] is the quorum sensing regulation rate  the rate at which downregulated bacteria become upregulated, and vice versa. τ [nM] is the threshold AHL concentration locally required for quorum sensing induction to occur. The coefficient n (n > 1) describes the degree of polymerisation in the synthesis of AHL. We model the dimerisation process for AHL, assuming that dimers of receptorAHL complexes are necessary for the transcription of the AHLsynthase gene. Assuming mass action law kinetics, this process gives n = 2, however, the value of n used here is slightly higher, as further synergistic effects are lumped into this parameter as well [28].

production of EPS by the bacterial cells at rates proportional to the bacterial growth rates, in equation (7): the EPS production rate is
in [d^{1}] where the yield coefficients Y_{ i }(EPS) [] describe the amount of EPS produced per unit bacterial biomass of type M_{0,1,2,3}.

nutrient consumption by bacterial biomass in (5): the maximum specific substrate consumption rates are denoted by
in [gm^{3}d^{1}], where M_{ max } is the maximum cell density, and Y_{ i } [] are the yield coefficients that incorporate both, the amount of nutrient required for biomass growth and for EPS production

abiotic AHL decay, at rate σ [d^{1}], in equation (6)

AHL production by both quorumsensing cell types M_{0} and M_{1} in (6) at different rates: the AHL production rate of downregulated quorum sensing bacteria is α [nM/(gm^{3}d^{1})], and the increased production rate of upregulated quorum sensing bacteria is α + β [nM d^{1}]

when carbon becomes limited, EPS may be used as a food source, in equations (5) and (7). This process is represented by an inhibition term, in which EPS is transformed into carbon at rate δ [d^{1}], with inhibition constant κ_{6} [gm^{3}]; the rate in (5) is related to δ by a yield coefficient and a constant conversion factor, see [26]; to neglect the EPS consumption process, we let δ = 0 and .
Model parameters in the high nutrient case.
Parameter  Description  Source  Value 

κ _{10}  Rate of C consumption by M_{0}  W  923 
κ _{11}  Rate of C consumption by M_{1}  W  1846 
κ _{12}  Rate of C consumption by M_{2}  W  923 
κ _{13}  Rate of C consumption by M_{3}  W  1846 
κ _{2}  Monod half sat. const.  W  0.02 
κ _{3}  Max specific growth rate of bacteria  H  1.0 
κ _{4}  Bacterial lysis rate  W  0.2083 
κ _{5}  Quorum sensing upregulation rate  F  2.5 
κ _{6}  Monod half sat. const.  H  0.04 
δ  EPS conversion to C rate (C equation), if included  H  0.28 
EPS conversion to C rate (EPS equation), if included  H  11.2  
σ  Abiotic degradation rate of AHL  F  0.1109 
α  Constitutive production rate of AHL  F  920 
β  Induced production rate of AHL  F  9200 
n  Degree of polymerisation  F  2.5 
γ _{0}  M_{0} EPS production rate  H  0.84 
γ _{1}  M_{1} EPS production rate  H  8.4 
γ _{2}  M_{2} EPS production rate  H  0.84 
γ _{3}  M_{3} EPS production rate  H  8.4 
M _{ max }  Maximum cell density  H  24·10^{3} 
EPS _{ max }  Maximum EPS density  H  4·10^{3} 
D_{ C } (0), (1)  Substrate diffusion coefficients  ES  0.67, 0.54 
D_{ AHL }(0), (1)  AHL diffusion coefficients  HR  0.52, 0.26 
a  Diffusion coefficient parameter  ES  4.0 
b  Diffusion coefficient parameter  ES  4.0 
d _{ M }  Biomass motility coefficient  ES  6.67e09 
H/L  Channel aspect ratio  ES  0.1 
Computational approach
The numerical solution of the densitydependent diffusionreaction model is computed using a semiimplicit finite differencebased finite volume scheme, formulated for the concentrations in the centers of the grid cells. Time integration uses a nonlocal (semiimplicit) discretization in the fashion of nonstandard finite difference methods. The timestep size is variable and chosen in order to ensure stability, positivity, boundedness (by 1), and a finite speed of interface propagation [59]. In our application, the computational domain is discretized on a uniform rectangular grid of size 2000 × 200.
In each time step six sparse, banded diagonal linear algebraic systems (one for each of M_{0}, M_{1}, M_{2/3}, C, AHL, and EPS) are solved with the stabilized biconjugate gradient method. The flow field is calculated using the analytical approximation of [56].
The numerical method was first introduced for singlespecies biofilms in [59] and then extended to biofilm systems with several microbial species in [60] and [61], the latter also containing a stability analysis. A computational convergence study can be found in [62]. These results carry over qualitatively to the study at hand. The method is implemented in OpenMP for execution on multicore and shared memory multiprocessor architectures; the parallelization behaviour is documented in [61]. The simulations were conducted on an Intel Itanium based SGI Altix 450, typically using 12 cores concurrently.
Simulation setup
Summary of the simulation experiments.
Biofilm Name  Biofilm Type  Nutrient Case  EPS consumption 

QS  M_{0}, M_{1}  high, low  yes, no 
M_{2} nonQS  M _{2}  high, low  yes, no 
M_{3} nonQS  M _{3}  high, low  yes, no 
M_{2} mixed  M_{0}, M_{1}, M_{2}  high  yes, no 
M_{3} mixed  M_{0}, M_{1}, M_{3}  high  yes, no 
Our biofilm model is on a mesoscopic scale, and so the computational domain is considered to be a small portion, or open subdomain, existing within a larger reactor. The boundary conditions we choose describe both the reactor type and the operating conditions in which the experiment is conducted, and connect the computational domain to the outside physical environment. Our computational domain is representative of a microfluidics chamber which receives fluid at the left (inflow) boundary from a large, well mixed reactor. Carbon is supplied into the channel from the upstream boundary, but no AHL may enter into the flow channel from upstream. AHL and carbon in the dissolved liquid phase Ω_{1} may exit the system via convective transport.
Specifically, the following boundary conditions are imposed on our domain Ω = [0, L]×[0, H]:

For M_{0}, M_{1}, M_{2}, M_{3} and EPS, no flux conditions everywhere (n is the direction of the outward normal): ∂_{ n }M_{0} = 0, ∂_{ n }M_{1} = 0, ∂_{ n }M_{2} = 0 ∂_{ n }M_{3} = 0, ∂_{ n }EPS = 0 on ∂ Ω

For C and AHL, no diffusive flux conditions everywhere except for on inflow, where we specify the bulk concentration: C = 1, A = 0 for x = 0, ∂_{ n }C = 0, ∂_{ n }A = 0 everywhere else.
The initial conditions used are:

An inoculation of the bottom surface of the channel with 16 colonies, each with a density of 0.3. Biofilm colonies are placed randomly along the channel, at an offset from the channel entrance and exit, to avoid unphysical boundary effects. This random placement mimics experimental difficulties in controlling where bacteria settle. The type of cell inoculated depends on the biofilm being grown: either quorum sensing (16 M_{0} cells), nonquorum sensing (16 M_{2} or M_{3}), or mixed (8 M_{0} and 8 M_{2}, or 8 M_{0} and 8 M_{3})

AHL = 0, EPS = 0; initially, biomass consists of cells only, but EPS and AHL production begins immediately upon the start of the simulation

C = 1.
The simulations finish when an imposed stopping criterion is met: the biofilm height reaches 80% of the channel height. This ensures the simulation stops before clogging effects take place; when the biofilm height approaches the top of the channel, local flow velocities and shear forces increase to the level that detachment processes would no longer be negligible, leading to a breakdown of the biofilm growth model.
Analysis
The occupancy and total cell and EPS biomass measures will be used to compare the growth and composition of the biofilm over time.
We will use the following abbreviations: quorum sensing (QS), nonquorum sensing (nonQS).
Results
The results of the simulation experiments summarized in Table 3 will be described in the following sequence:

Example simulation of QS controlled EPS production in a biofilm: an example simulation of a quorum sensing biofilm under high nutrient conditions.

Simulations without the EPS consumption process: simulations of biofilms that do not include the process of EPS consumption. First, QS and M_{2} and M_{3} nonQS biofilms are compared under high and low nutrient conditions. Second, M_{2} and M_{3} mixed biofilms are regarded.

Simulations with the EPS consumption process: the experiments of the previous section are repeated, but the process of EPS consumption is included. QS and M_{2}, M_{3} nonQS biofilms under high and low nutrient conditions are described first, followed by M_{2} and M_{3} mixed biofilms.

Effect of random colony placement in mixed biofilms: a discussion on the effects of random initial colony placement in M_{2} and M_{3} mixed biofilms on quorum sensing induction.
Example simulation of QS controlled EPS production in a biofilm
To simulate growth of a QS biofilm, the bottom surface of the channel is inoculated with sixteen M_{0} colonies. A high supply of substrate enters the channel from the inflow boundary, and the process of EPS consumption is neglected.
AHL accumulates over time in the channel as it is produced by the growing colonies. Molecules produced by the colonies diffuse into the liquid region, and are transported downstream by convection and diffusion, causing AHL concentrations to increase in the main flow direction. The maximum AHL concentration found at the downstream boundary is a typical effect of flow facilitated convective transport [37]. In Figure 1(b), the switch to QS is occuring. Upregulation occurs locally when the nondimensional AHL concentration reaches 1.0. Positive feedback in the quorum sensing system is then initiated  upregulated cells produce AHL at ten times the downregulated rate, leading rapidly to large increases in AHL concentrations, and further upregulation of cells throughout the domain.
The downstream colonies begin to upregulate first, followed by the upstream colonies. This is an observation of flowfacilitated intercolony communication  AHL molecules produced by the large, upstream colonies are transported by convection and diffusion, contributing to upregulation in the smaller downstream colonies [28].
The biofilm in Figure 1(c) is fully upregulated, and EPS production has increased by a factor of ten. The biofilm grows and expands rapidly, until the flow channel becomes clogged with biomass and the maximum predetermined biofilm height is obtained.
Prior to induction (Figure 2(a)), the biofilm composition by mass is approximately 15% EPS, 85% cells. Following induction (Figure 2(b)), EPS production rates are upregulated, resulting in a change in biofilm composition to 6065% EPS. The large, merged, upstream colonies experienced the greatest increase in volume  in order for colonies to have increased growth due to upregulated EPS production rates, both upregulated cells and adequate nutrients are required. So although the downstream colonies are first to upregulate, these colonies lack the nutrients needed to expand quickly.
Simulations without the EPS consumption process
Quorum sensing and nonquorum sensing biofilms
In the first simulation experiment, QS biofilms (M_{0}, M_{1} cells), the lowEPS producing M_{2} nonQS biofilms and highEPS producing M_{3} nonQS biofilms were grown under high and low nutrient conditions, using the same initial distribution of colonies. Under the low nutrient condition, the concentration of incoming carbon (C_{ bulk }) is lowered, which affects the following nondimensional parameters: κ_{10} = 1846, κ_{11} = 3692, κ_{12} = 1846, κ_{13} = 3692, κ_{2} = 0.04, .
The composition of the biofilm over time is shown in Figures 4(g, h). After the initial time period in which EPS production begins, the lowEPS producing nonQS biofilm (M_{2}) remains composed of less than 20% EPS by mass, whereas the high EPS producing nonQS biofilm (M_{3}) is approximately 65% EPS. The QS biofilm switches its composition by mass after induction from predominantly bacteria cells to EPS. In summary, the QS biofilms obtained greater cell populations than M_{3} nonQS biofilms, and occupied more volume in the channel than M_{2} nonQS biofilms.
The quorum sensing mechanism is used to switch behaviours from the M_{2} like mode of low EPS production and faster cell growth to the M_{3} like mode of high EPS production at the expense of slower bacterial growth. This transition takes place rapidly after induction occurs and is almost completed after a period of time that is about twice as long as the characteristic time scale of biomass growth. Eventually the entire biofilm behaves like a highEPS producing M_{3} biofilm.
Mixed biofilms
In some cases, the upstream colonies utilized so much of the incoming nutrient that cells in the downstream colonies experienced cell death (this is evident by the decreasing curves in the cell biomass plots Figures 6(b, c, e, f)). In accordance with the results of the previous section, populations of highEPS producing biofilms are lower than populations of lowEPS producing biofilms.
Simulations with the EPS consumption process
An additional complexity was considered in our biofilm simulations, which accounts for the utilization of EPS by bacteria cells as a secondary source of the carbon nutrient when limitations occur. The experiments of the subsections "Quorum sensing and nonquorum sensing biofilms" and "Mixed biofilms" were repeated, but with consideration of this biological process.
Quorum sensing and nonQS biofilms
QS and nonQS biofilms were grown in the channel using the same initial distribution of colonies as the first simulation experiment, under high and low nutrient conditions. The trend discussed there is present here as well: QS biofilms have higher cell populations than M_{3} nonQS biofilms, and higher occupancies than M_{2} nonQS biofilms. Cell populations were approximately equivalent to those in the biofilms that did not consume EPS, however, occupancy was considerably lower, due to EPS lost through consumption.
When QS biofilms induced, they increased their volume from that of a M_{2} nonQS biofilm to match the size of a M_{3} nonQS biofilm. These trends are also present under the low nutrient regime, though cell populations and occupancies are lower than when the high nutrient regime was used.
After QS induction occurs, the biofilm undergoes a transition from a low EPS producing M_{2} like state to a state that is essentially entirely M_{3} like, i.e. it becomes a fast EPS producer at the expense of slower bacterial growth. After this transition is complete, within a time period that compares to the characteristic timescale of biomass growth, no remainders of the downregulated past are evident. This corresponds to the observations of the previous sections.
Mixed biofilms
Effect of random colony placement in mixed biofilms
The positive feedback feature of QS systems leads to a very rapid upregulation of QS cells in the biofilm  after induction occurs, M_{1} cell populations quickly rise. The switching time represents this upregulation period. In Figures 10(a, b), the M_{2} and M_{3}mixed biofilms that exclude EPS consumption have a switching time between 7.5 and 8.5. The M_{2} and M_{3} mixed biofilms that consume EPS (Figures 10(c, d)) switch between 7.5 and 8.0. Though the variation in switching time is small, we can conclude that the random initial placement of colonies does have a small effect of the time at which QS induction occurs. After induction, nutrient limitations arise, and the variability in total M_{1} cell biomass increases. Nutrient shortages occur not only because the biofilm is larger, but also because the upregulated M_{1} cells have a higher consumption rate. In each simulation, nutrient limitations cause M_{1} cell growth rates to decrease, and in some cases, the cell populations remain constant or even decline.
Discussion
The first question that motivated our study was: how does QSregulated EPS production impact the growing biofilm? In particular, what does the biofilm look like over time, with respect to distribution and composition of cells and EPS? The answers to these questions were obtained by studying biofilm composition and the upstream clogging effect.
It was found that colony growth was so greatly enhanced with highEPS producing nonQS biofilms, and in induced QS biofilms, that these biofilms rapidly fill the channel with biomass. When highEPS producing cells are located in a region where nutrient supply is abundant, these cells have access to the nutrients required for enhanced EPS production. If these cells are located in a region of nutrient scarcity, induced EPS production reactions cannot occur, and so colonies cannot undergo enhanced expansion. In all the simulations, spatial gradients of biofilm size were prevalent  large, merged upstream colonies have full access to available nutrients, grow and fill the channel, and cause the downstream colonies to remain small. At low EPS production rates, the biofilm is composed by majority of cells, at high EPS production rates, the biofilm is predominantly EPS. When QS biofilms induce, they switch their proportions of cellular and EPS biomass. The composition differs greatly if the EPS consumption process is modelled. In regions of nutrient deficiencies, EPS may be completely consumed by the bacteria cells, to the extent that the biofilm consists only of cells (generally in the mid to downstream region), and only the portion of the biofilm closest to the inflow boundary contains a protective layer of EPS.
We were also interested in investigating, is it beneficial for the biofilm to regulate EPS production using a QS mechanism? Several factors are considered in determining whether one biofilm was more successful than another, including occupancy and total cell biomass. HighEPS producing biofilms have higher occupancies than lowEPS producing biofilms, as a result of increased levels of EPS. They may furthermore benefit, for example, by protection from environmental hazards, such as detachment, antibiotics or grazers. LowEPS producing nonQS cells which have merged with upregulated QS colonies (in mixed simulations) therefore experience benefits from the thick EPS layer as well. If EPS consumption is occurring, nonQS cells in mixed colonies also benefit by consuming the additional nutrients produced.
However, in all simulations, lowEPS producing cells on average outnumbered the QS cells. This occurred in comparing QS with low EPSproducing nonQS biofilms, and mixed biofilms, with and without the EPS consumption process. These findings show that EPS production does not provide a benefit to the biofilm in regards to achieving a high cell population, which is one potential objective of the bacteria cells residing in a biofilm. To produce EPS at the induced rate, bacteria cells have a high nutrient demand, and if this demand is not met, colony growth cannot occur. Another cause of low populations of highEPS producing cells is that these expanding colonies, with high proportions of EPS, quickly clog the channel. In contrast, lowEPS producing nonQS biofilm colonies take much longer to grow to the point that the channel is clogged, but the biofilm is composed primarily of cells.
In the mixed biofilm simulations, success of QS or nonQS populations was determined by proximity to the nutrient source, largely an effect of the random initial distribution. In a mixed environment, clogging of the channel by upstream colonies may prevent downstream colonies from receiving nutrients, in some cases causing their populations to suffer declines. Resource requirements are considerably higher for highEPS producing cells. If an adequate nutrient supply can be secured, either by locating a colony near the nutrient source or utilizing EPS as a secondary food source, then it is possible for QS cell populations to be greater than lowEPS producing cell populations.
Under the investigated conditions, we found that overall, QSregulated EPS production rarely provides a benefit to a biofilm with the objective of achieving a high cell population, in comparison to biofilms with low EPSproducing nonQS cells. However, maximizing offspring generation is not necessarily the best strategy under all conditions. EPS production would be beneficial if the objective of the bacteria cells in the biofilm is to clog the channel. A QS biofilm colony located near the nutrient source may use QS to increase its volume, clog the channel, and secure its supply of nutrients while starving downstream colonies, and potentially force the downstream colony to deplete their EPS supply. This is a competitive advantage for a colony, whether it is located in a QS biofilm (indicating intraspecies competition), or in a mixed biofilm (interspecies competition). In any spacelimited environment, such as fine sediment or small vessels in higher organisms, channel clogging may be considered a beneficial strategy for bacteria cells to use. In the study [51], it was shown that biofilms developing in plants stems used QS to clog the plant vessel used for water and nutrient transportation, the socalled xylem, securing nutrients for themselves. Individual colonies in patchy biofilm communities have an antagonistic interaction (some colonies benefit at the expense of others) through utilization of an exploitative competition strategy  the upstream colonies reduce nutrient resources in the environment, depleting the resource required by the downstream colonies. If cells wish to outcompete another cell population in a biofilm, of their own species or a different species, (QScontrolled) EPS production can provide an advantage. The drawback is that nutrient supply must be abundant, as highEPS producing cells have a high nutrient demand.
Downstream cells are upregulated (partly) by AHL produced by upstream colonies. However, it is not obvious how upstream colonies could contribute to a potential benefit (e.g. protection) for downstream cells through EPS production. In this sense, downstream cells seem to be fooled into a premature EPS production, which they might not even be able to realize due to nutrient depletion.
The results showed that on a population level, highEPS producing biofilms suffered lower cell populations. However, the spatial distributions in the 2D visualizations demonstrated that for individual colonies, clogging the channel with biomass may be a beneficial strategy. Individual cells within microcolonies are genetically related, and have the ultimate objective of maintaining their own genes and reproducing in the future. To optimize survival of their genes in the colonies, and ensure survival of their offspring, bacteria cells would be interested in suppressing other colonies. In the event of a structural reorganization or detachment/reattachment, the upstream colony has an advantage, in that their genetically similar offspring will continue to succeed.
The nutrient supply, random placement of colonies, and QS had various impacts on the resulting biofilms. QS and highEPS producing nonQS biofilms had almost equivalent total cell biomass under both the high and low nutrient regimes, indicating that total cell population at the end of the simulation is predominantly controlled by nutrients and mass transfer in the aqueous phase, and not quorum sensing, which is also supported by the experimental findings of [63]. It was found that the random placement of colonies in the initial inoculation had a small impact on when induction occurred. Nutrient limitations succeeded induction, and led to greater variability in total cell biomass. The impact of QS was most prominent in regards to biofilm occupancy and composition. QS induction occurs very rapidly, and likewise, the QS biofilms were able to rapidly increase their occupancy and change their composition from that of a low EPSproducing to high EPSproducing nonQS biofilm.
We argued that EPS production benefits cells under certain conditions. The question arises of why EPS production would be associated with quorum sensing, that is, why would cells wait until upregulation to produce EPS at higher rates, versus always producing EPS at enhanced levels? A minimum thickness of an EPS layer is required for effective protection of cells from a hazard, such as antibiotics and washout. If a cell population is small, much energy would be expended to produce a layer of EPS which may not provide adequate protection. Coupling EPS production with quorum sensing ensures that a sufficient colony size is obtained such that the cost to produce EPS is returned by the benefits of a protective layer. QS may then be considered a mechanism for switching to a mode of highEPS production once a certain number of cells is present, ensuring the bacteria can protect themselves efficiently. Rather than existing in a permanently upregulated state, which requires additional nutrients, QS biofilms can quickly switch to the upregulated state and increase their size when necessary.
Conclusions and further work
In this study, we developed a mathematical model of QS in biofilms which incorporated an effect of QS on biofilm growth: upregulated cells produce EPS at an increased rate. Through simulations, we investigated QS, nonQS, and mixed biofilms under various nutrient regimes and the biological process of EPS consumption. Our main results are:

QSinduced EPS production allows a biofilm to switch rapidly from an initial colonization stage, in which the focus lies on rapid cell production, to a stage in which the focus lies on producing large amounts of EPS as protection against environmental threats such as grazers, mechanical washout, biocides, and transient periods of nutrient limitation.

lowEPS producing biofilms generally obtained higher cell populations, and highEPS producing biofilms obtained greater occupancies. The QS biofilms rapidly changed from a state of cell growth to a state of EPS production; in other words, quorum sensing is used as a signal for the biofilms to switch from a colonization mode to a protection mode.

highEPS producing biofilm colonies rapidly filled their spacelimited environment with biomass, and consisted of high proportions of EPS biomass. When EPS is consumed, however, extreme nutrient deficiencies lead biofilms to deplete their EPS supply, resulting in biofilms comprised purely of cells, with little to no EPS present.

a biofilm will benefit from using quorum sensinginduced EPS production if bacteria cells have the objective of acquiring a thick, protective layer of EPS, or if they wish to clog their environment with biomass as a means of securing nutrient supply and outcompeting other colonies in the channel, of their own or a different species.
Limitations of our model and simulation setup include the narrow simulation space. We chose to model biofilm growth in narrow channels, and because highEPS producing biofilm colonies expand rapidly, the observation period of QS induced biofilms is brief. Other experimental setups that allow for longterm biofilm development could be of interest; our model is intended to describe only the initial stages of biofilm growth in a reactor system representative of spacerestricted environments with fluid flow, such as soil pores.
We used our model to observe connections between biofilm growth, quorum sensing, and EPS production. Our study could be extended to test biofilms that use quorumsensing induced EPS production under the scenarios which we speculated biofilms would benefit from high levels of EPS, for example, when grazers or antibiotics are present. Future work in biofilm and quorum sensing modelling is required to continue to investigate how biofilms respond to quorum sensing induction.
Authors contributions
This study was conducted as part of the M.Sc. thesis research of MRF at the University of Guelph, and on research visits to the HelmholtzCenter Munich. HJE was the thesis supervisor and guided the biofilm modelling aspects, CK and BAH guided the quorum sensing modelling aspects and the ecological interpretation of the results. All authors contributed to the writing of the article, and read and approved the final manuscript.
Acknowledgements
This study was supported in parts by the Advanced Foods and Materials Network, a Network of Centers of Excellence (NCE) and the National Science and Engineering Research Council (NSERC). The computing equipment used for this study was provided by the Canada Foundation for Innovation (CFI) with a grant in the Leaders Opportunity Funding program, awarded to HJE. We thank the technical staff of the Shared Hierarchical Academic Research Network (SHARCNET), for their technical support.
Declarations
Authors’ Affiliations
References
 Lewandowski Z, Beyenal H: Fundamentals of Biofilm Research. 2007, Boca Raton: CRC PressView ArticleGoogle Scholar
 Fuqua C, Greenberg EP: Listening on bacteria: Acylhomoserine lactone signalling. Nature Reviews Molecular Cell Biology. 2002, 3: 685695. 10.1038/nrm907.View ArticlePubMedGoogle Scholar
 Demuth DR, Lamont RJ: Bacterial celltocell communication. 2006, Cambridge: Cambridge University PressView ArticleGoogle Scholar
 Whitehead NA, Barnard AML, Slater H, Simpson NJL, Salmond GPC: Quorumsensing in gramnegative bacteria. FEMS Microbiology Reviews. 2001, 25: 365404. 10.1111/j.15746976.2001.tb00583.x.View ArticlePubMedGoogle Scholar
 Flemming HC, Neu TR, Wozniak DJ: The EPS Matrix: The "House of Biofilm Cells". Journal of Bacteriology. 2007, 189 (22): 79457947. 10.1128/JB.0085807.PubMed CentralView ArticlePubMedGoogle Scholar
 Sutherland IW: Biofilm Exopolysaccharides. Microbial Extracellular Polymeric Substances  Characterization, Structure, and Function. Edited by: Wingender J, Neu TR, Flemming HC. 1999, BerlinHeidelberg: SpringerVerlag, 7392.View ArticleGoogle Scholar
 Wolfaardt GM, Lawrence JR, Korber DR: Function of EPS. Microbial Extracellular Polymeric Substances  Characterization, Structure, and Function. Edited by: Wingender J, Neu TR, Flemming HC. 1999, Springer, 170200.Google Scholar
 Laspidou CS, Rittmann BE: A unified theory for extracellular polymerica substances, soluble microbial products, and active and inert biomass. Journal of Water Research. 2002, 36 (11): 27112720. 10.1016/S00431354(01)004134.View ArticlePubMedGoogle Scholar
 Wang ZW, Liu Y, Tay JH: Biodegradability of extracellular polymeric substances produced by aerobic granules. Applied Microbiology and Biotechnology. 2007, 74 (2): 462466. 10.1007/s002530060686x.View ArticlePubMedGoogle Scholar
 Wanner O, Eberl H, Morgenroth E, Noguera D, Picioreanu C, Rittmann B, van Loosdrecht MCM: Mathematical Modeling of Biofilms. 2006, London: IWA PublishingGoogle Scholar
 Alpkvist E, Picioreanu C, van Loosdrecht MCM, Heyden A: Threedimensional biofilm model with individual cells and continuum EPS matrix. Biotechnology and Bioengineering. 2006, 94 (5): 961979. 10.1002/bit.20917.View ArticlePubMedGoogle Scholar
 Chambless JD, Hunt SM, Stewart PS: A threedimensional computer model of four hypothetical mechanisms protecting biofilms from antimicrobials. Applied Environmental Microbiology. 2006, 72 (3): 20052013. 10.1128/AEM.72.3.20052013.2006.PubMed CentralView ArticlePubMedGoogle Scholar
 Chopp DL: Simulating bacterial biofilms. Deformable Models. Edited by: Suri JS, Farag AA. 2007, BerlinHeidelberg: Springer, 131. full_text.View ArticleGoogle Scholar
 Cogan NG: A twofluid model of biofilm disinfection. Bull Math Biol. 2008, 70 (3): 800819. 10.1007/s1153800792803.View ArticlePubMedGoogle Scholar
 Dockery J, Klapper I: Finger formation in biofilm layers. SIAM Journal on Applied Mathematics. 2001, 62 (3): 853869.Google Scholar
 Eberl HJ, Parker DF, van Loosdrecht MCM: A new deterministic spatiotemporal continuum model for biofilm development. Journal of Theoretical Medicine. 2001, 3 (3): 161175. 10.1080/10273660108833072.View ArticleGoogle Scholar
 Hermanowicz SW: A simple 2D biofilm model yields a variety of morphological features. Mathematical Biosciences. 2001, 169 (1): 114. 10.1016/S00255564(00)000493.View ArticlePubMedGoogle Scholar
 Kreft JU, Picioreanu C, Wimpenny JWT, van Loosdrecht MCM: Individualbased modelling of biofilms. Environmental Microbiology. 2001, 147: 28972912.Google Scholar
 Picioreanu C, van Loosdrecht MCM, Heijnen JJ: A new combined differentialdiscrete cellular automaton approach for biofilm modeling: Application for growth in gel beads. Biotechnology and Bioengineering. 1998, 57 (6): 718731. 10.1002/(SICI)10970290(19980320)57:6<718::AIDBIT9>3.0.CO;2O.View ArticlePubMedGoogle Scholar
 Cogan NG, Keener JP: The role of the biofilm matrix in structural development. IMA Mathematical Medicine and Biology. 2004, 21 (2): 147166. 10.1093/imammb/21.2.147.View ArticleGoogle Scholar
 Pizarro G, Griffeath D, Noguera D: Quantitative cellular automaton model for biofilms. Journal of Environmental Engineering. 2001, 127 (9): 782789. 10.1061/(ASCE)07339372(2001)127:9(782).View ArticleGoogle Scholar
 Pizarro GE, Teixeira J, Sepulveda M, Noguera DR: Bitwise implementation of a 2D cellular automata biofilm model. Journal of Computing in Civil Engineering. 2005, 19 (3): 258268. 10.1061/(ASCE)08873801(2005)19:3(258).View ArticleGoogle Scholar
 Tatek YB, Slater GW: A simulation model of biofilms with autonomous cells: Analysis of a twodimensional version. Physica A. 2006, 362 (2): 382402. 10.1016/j.physa.2005.08.011.View ArticleGoogle Scholar
 Zhang T, Cogan NG, Wang Q: Phase field models for biofilms. I. Theory and onedimensional simulations. SIAM Journal on Applied Mathematics. 2008, 69 (3): 641669. 10.1137/070691966.View ArticleGoogle Scholar
 Zhang T, Cogan NG, Wang Q: Phasefield models for biofilms. II. 2D numerical simulations of biofilmflow interaction. Communications in Computational Physics. 2008, 4 (1): 72101.Google Scholar
 Horn H, Neu TR, Wulkow M: Modelling the structure and function of extracellular polymeric substances in biofilms with new numerical techniques. Water Science & Technology. 2001, 43 (6): 121127.Google Scholar
 Eberl HJ: A deterministic continuum model for the formation of EPS in heterogeneous biofilm architectures. Proceedings of the International Conference Biofilms. 2004, 2426 October 2004; Las Vegas, : Structure and Activity of BiofilmsGoogle Scholar
 Frederick MR, Kuttler C, Hense BA, Müller J, Eberl HJ: A mathematical model of quorum sensing in patchy biofilm communities with slow background flow. Canadian Applied Mathematics Quarterly. 2011, 18 (3): 267298.Google Scholar
 Khassehkhan H, Hillen T, Eberl HJ: A nonlinear master equation for a degenerate diffusion model of biofilm growth. Lecture Notes in Computer Science. 2009, 5544: 735744. full_text.View ArticleGoogle Scholar
 Ward JP, King JR, Koerber AJ, Williams P, Croft JM, Sockett RE: Mathematical modelling of quorum sensing in bacteria. IMA Journal of Mathematics Applied in Medicine and Biology. 2001, 18 (3): 263292. 10.1093/imammb/18.3.263.View ArticlePubMedGoogle Scholar
 Dockery JD, Keener JP: A mathematical model for quorum sensing in P. Aeruginosa. Bulletin of Mathematical Biology. 2001, 63 (1): 95116. 10.1006/bulm.2000.0205.View ArticlePubMedGoogle Scholar
 Müller J, Kuttler C, Hense BA, Rothballer M, Hartmann A: Cellcell communication by quorum sensing and dimensionreduction. Journal of Mathematical Biology. 2006, 53: 672702.View ArticlePubMedGoogle Scholar
 Ward JP, King JR, Koerber AJ, Croft JM, Sockett RE, Williams P: Early development and quorum sensing in bacterial biofilms. Journal of Mathematical Biology. 2003, 47: 2355.View ArticlePubMedGoogle Scholar
 Chopp DL, Kirisits MJ, Parsek MR, Moran B: A mathematical model of quorum sensing in a growing P. aeruginosa biofilm. Journal of Industrial Microbiology and Biotechnology. 2002, 29 (6): 339346. 10.1038/sj.jim.7000316.View ArticlePubMedGoogle Scholar
 Chopp DL, Kirisits MJ, Parsek MR, Moran B: The dependence of quorum sensing on the depth of a growing biofilm. Bulletin of Mathematical Biology. 2003, 65 (6): 10531079. 10.1016/S00928240(03)000570.View ArticlePubMedGoogle Scholar
 Janakiraman V, Englert D, Jayaraman A, Baskaran H: Modeling growth and quorum sensing in biofilms grown in microfluidic chamber. Annals of Biomedical Engineering. 2009, 37 (6): 12061216. 10.1007/s1043900996718.View ArticlePubMedGoogle Scholar
 Vaughan BL, Smith BG, Chopp DL: The influence of fluid flow on modeling quorum sensing in bacterial biofilms. Bulletin of Mathematical Biology. 2010, 72 (5): 114365. 10.1007/s1153800994858.View ArticlePubMedGoogle Scholar
 Anguige K, King JR, Ward JP, Williams P: Mathematical modelling of therapies targeted at bacterial quorum sensing. Mathematical Biosciences. 2004, 192: 3983. 10.1016/j.mbs.2004.06.008.View ArticlePubMedGoogle Scholar
 Anguige K, King JR, Ward JP: Modelling antibiotic and antiquorum sensing treatment of a spatially structured Pseudomonas aeruginosa population. Journal of Mathematical Biology. 2005, 51: 557594. 10.1007/s0028500503168.View ArticlePubMedGoogle Scholar
 Anguige K, King JR, Ward JP: A multiphase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm. Mathematical Biosciences. 2006, 203 (2): 240276. 10.1016/j.mbs.2006.05.009.View ArticlePubMedGoogle Scholar
 Kirisits MJ, Margolis JJ, PurevdorjGage BL, Vaughan B, Chopp DL, Stoodley P, Parsek MR: Influence of the hydrodynamic environment on quorum sensing in Pseudomonas aeruginosa biofilms. Journal of Bacteriology. 2007, 189 (22): 83578360. 10.1128/JB.0104007.PubMed CentralView ArticlePubMedGoogle Scholar
 Ward J: Mathematical modelling of quorum sensing control in biofilms. The Control of Biofilm Infections by Signal Manipulation. Edited by: Balaban N. 2008, BerlinHeidelberg: SpringerVerlag, 78107.Google Scholar
 Davies DG: The Involvement of CelltoCell Signals in the Development of a Bacterial Biofilm. Science. 1998, 280: 295298. 10.1126/science.280.5361.295.View ArticlePubMedGoogle Scholar
 Sakuragi Y, Kolter R: Quorumsensing regulation of the biofilm matrix genes (pel) of Pseudomonas aeruginosa. Journal of Bacteriology. 2007, 189 (14): 53835386. 10.1128/JB.0013707.PubMed CentralView ArticlePubMedGoogle Scholar
 Friedman L, Kolter R: Genes involved in matrix formation in Pseudomonas aeruginosa PA14 biofilm. Molecular Microbiology. 2004, 51: 675690. 10.1046/j.13652958.2003.03877.x.View ArticlePubMedGoogle Scholar
 Shrout JD, Chopp DL, Just CL, Hentzer M, Givskov M, Parsek MR: The impact of quorum sensing and swarming motility on Pseudomonas aeruginosa biofilm formation is nutritionally conditional. Molecular Microbiology. 2006, 62 (5): 12641277. 10.1111/j.13652958.2006.05421.x.View ArticlePubMedGoogle Scholar
 Morohoshi T, Nakamura Y, Yamazaki G, Ishida A, Kato N, Ikeda T: The Plant Pathogen Pantoea ananatis Produces NAcylhomoserine Lactone and Causes Center Rot Disease of Onion by Quorum Sensing. Journal of Bacteriology. 2007, 189 (22): 83338338. 10.1128/JB.0105407.PubMed CentralView ArticlePubMedGoogle Scholar
 Marketon MM, Glenn SA, Eberhard A, Gonzalez JE: Quorum Sensing Controls Exopolysaccharide Production in Sinorhizobium meliloti. Journal of Bacteriology. 2003, 185 (1): 325331. 10.1128/JB.185.1.325331.2003.PubMed CentralView ArticlePubMedGoogle Scholar
 Minogue TD, Carlier AL, Koutsoudis MD, von Bodman SB: The cell densitydependent expression of stewartan exopolysaccharide in Pantoea stewartii ssp. stewartii is a function of EsaRmediated repression of the rcsA gene. Molecular Microbiology. 2005, 56: 189203. 10.1111/j.13652958.2004.04529.x.View ArticlePubMedGoogle Scholar
 Quinones B, Dulla G, Lindow SE: Quorum Sensing Regulates Exopolysaccharide Production, Motility, and Virulence in Pseudomonas syringae. Molecular PlantMicrobe Interactions. 2005, 18 (7): 682693. 10.1094/MPMI180682.View ArticlePubMedGoogle Scholar
 Koutsoudis MD, Tsaltas D, Minogue TD, Beck von Bodman S: Quorumsensing regulation governs bacterial adhesion, biofilm development, and host colonization in Pantoea stewartii subspecies stewartii. Proceedings of the National Academy of Sciences. 2006, 103 (15): 59835988. 10.1073/pnas.0509860103.View ArticleGoogle Scholar
 Gao Y, Song J, Hu B, Zhang L, Liu Q, Liu F: The luxS Gene Is Involved in AI2 Production, Pathogenicity, and Some Phenotypes in Erwinia amylovora. Current Microbiology. 2009, 58 (1): 110. 10.1007/s002840089256z.View ArticlePubMedGoogle Scholar
 Beck von Bodman S, Majerczak DR, Coplin DL: A negative regulator mediates quorumsensing control of exopolysaccharide production in Pantoea stewartii subsp. stewartii. Proceedings of the National Academy of Sciences. 1998, 95 (13): 76877692. 10.1073/pnas.95.13.7687.View ArticleGoogle Scholar
 Zhang X, Bishop PL: Biodegradability of biofilm extracellular polymeric substances. Chemosphere. 2006, 50: 6369. 10.1016/S00456535(02)003193.View ArticleGoogle Scholar
 Stewart PS: Diffusion in biofilms. Journal of Bacteriology. 2003, 185: 14851491. 10.1128/JB.185.5.14851491.2003.PubMed CentralView ArticlePubMedGoogle Scholar
 Eberl HJ, Sudarsan R: Exposure of biofilms to slow flow fields: the convective contribution to growth and disinfections. Journal of Theoretical Biology. 2008, 253 (4): 788807. 10.1016/j.jtbi.2008.04.013.View ArticlePubMedGoogle Scholar
 Fekete A, Kuttler C, Rothballer M, Hense BA, Fischer D, BuddrusSchiemann K, Lucio M, Müller J, SchmittKopplin P, Hartmann A: Dynamic Regulation of NAcylhomoserine Lactone Production and Degradation in Pseudomonas putida IsoF. FEMS Microbiology Ecology. 2010, 72 (1): 2234. 10.1111/j.15746941.2009.00828.x.View ArticlePubMedGoogle Scholar
 Kaplan HB, Greenberg EP: Diffusion of autoinducer is involved in regulation of the Vibrio fischeri luminescence system. Journal of Bacteriology. 1985, 163 (3): 12101214.PubMed CentralPubMedGoogle Scholar
 Eberl HJ, Demaret L: A finite difference scheme for a doubly degenerate diffusionreaction equation arising in microbial ecology. Electronic Journal of Differential Equations. 2007, CS15: 7795.Google Scholar
 Eberl HJ, Khassehkhan H, Demaret L: A mixedculture model of a probiotic biofilm control system. Computational and Mathematical Methods in Medicine. 2010, 11 (2): 99118. 10.1080/17486700902789355.View ArticlePubMedGoogle Scholar
 Muhammad N, Eberl HJ: OpenMP Parallelization of a Mickens TimeIntegration Scheme for a MixedCulture Biofilm Model and its Performance on Multicore and Multiprocessor Computers. LNCS. 2010, 5976: 180195.Google Scholar
 Muhammad N: Computational analysis of parameter uncertainties in a 2d mixedculture biofilm model. 2010, PhD Thesis, University of GuelphGoogle Scholar
 Purevdorj B, Costerton JW, Stoodley P: Influence of Hydrodynamics and Cell Signaling on the Structure and Behavior of Pseudomonas aeruginosa Biofilms. Applied and Environmental Microbiology. 2002, 68 (9): 44574464. 10.1128/AEM.68.9.44574464.2002.PubMed CentralView ArticlePubMedGoogle Scholar
 Hobbie RK, Roth BJ: Intermediate physics for medicine and biology. 2007, New York: SpringerGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.