A MATHEMATICAL MODEL OF QUORUM SENSING IN PATCHY … · QUORUM SENSING IN BIOFILMS 269 [11, 52], a comprehensive understanding of quorum sensing systems is highly desirable. 1.3 Modelling

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 18, Number 3, Fall 2010

A MATHEMATICAL MODEL OF QUORUM

SENSING IN PATCHY BIOFILM COMMUNITIES

WITH SLOW BACKGROUND FLOW

M. FREDERICK, C. KUTTLER, B. A. HENSE,J. MULLER AND H. J. EBERL

ABSTRACT. Quorum sensing is a type of cell to cell com-munication used to coordinate behavior in groups based on pop-ulation density. We present a mathematical model of this phe-nomenon in bacterial biofilms, consisting of a nonlinear diffu-sion reaction system. The model includes production and spa-tial spreading of biomass, conversion of down-regulated biomassinto up-regulated biomass as a consequence of the quorum sens-ing signalling molecule concentration, and convective and dif-

fusive transport of nutrients and quorum sensing molecules.In numerical experiments, using quantitative parameter dataderived from P.putida experiments, we investigate how the hy-drodynamic environment and nutrient conditions contribute tobiofilm growth and quorum sensing behaviour. Our main find-ings are the observation of flow-facilitated inter-colony commu-nication, and that at sufficiently high bulk flow rates, biofilmgrowth and signalling molecule production is enhanced, yetwashout of signalling molecules suppresses quorum sensing in-duction. A discussion of the ecological implications of theseresults is included.

1 Introduction

1.1 Bacterial biofilms Biofilms are microbial communities encasedin a layer of extracellular polymeric substances (EPS), adhered to bioticor abiotic surfaces. Biofilms are an extremely successful life form, andare able to flourish in any aqueous environment with adequate nutrientsavailable. In fact, bacteria preferentially reside in biofilms, rather thanin isolation as planktonic cells. In a biofilm, bacteria are protected by

AMS subject classification: MSC2000: 92D25, 92C17.Keywords: Biofilm, quorum sensing, mathematical model.

Copyright c©Applied Mathematics Institute, University of Alberta.

267

268 M. FREDERICK ET AL.

the EPS matrix from external stresses, and carry out a wide range ofreactions which are relevant in many disciplines, such as environmentalengineering, food processing, and medicine [33].

1.2 Quorum sensing in biofilms Quorum sensing was originally de-scribed as a diffusion-reaction based cell-cell communication mechanismused by several bacterial taxa to coordinate gene expression and be-haviour in groups, based on population densities [22]. Bacteria cells con-stantly produce and release low amounts of signalling molecules, calledautoinducers (e.g., acyl-homoserine lactones (AHL) in gram-negativebacteria). When a critical environmental autoinducer concentration isreached, the bacteria are rapidly induced and undergo changes in geneexpressions. In most bacterial autoinducer systems, the autoinducersynthase gene is up-regulated, initiating positive feedback, and the bac-teria subsequently produce AHL molecules at an increased rate. As theextracellular autoinducer concentration reflects the cell density, the timeat which this phenotypic change occurs is dependent on the cell density.Quorum sensing was originally described for and, until now, mainly in-vestigated in homogeneous planktonic cultures. However, most bacterialive attached to surfaces in spatially heterogeneous biofilms, usually asassemblages of microcolonies, or “patchy” biofilm communities [8, 29].Under these conditions, autoinducers reflect local, rather than global,cell densities. Redfield [39] assumed alternatively that autoinducers areused by single cells for the measurement of the diffusible space. Thisconcept, which does not involve cooperation, is called diffusion sensing.Hense et al. [24] unified both concepts with the idea of efficiency sensingand argued that induction of cells is influenced by both cell density andmass transfer limitations, as well as inhomogeneities of cell distribution.Although there are some experimental indications for the influence ofmass transfer limitations [14, 30, 41], this has never been analyzedsystematically for spatially heterogeneous growing biofilms in complexmatrices with varying nutrient conditions. In fact, most papers still as-sume that the purpose of autoinducers is for cell density assessment.The term “quorum sensing” originally referred to only the cell densitymeasurement, and even today it is usually defined in this sense. How-ever, the term is well established and indeed used to refer to generalmeasurements of autoinducers, disregarding whether mass transfer lim-itation or cell distribution contribute or even dominate. We adopt thisbroad use of terminology and refer to pure cell density measurementas “quorum sensing in the strict sense.” As a number of traits rele-vant for human, animal and plant health are regulated via autoinducers

QUORUM SENSING IN BIOFILMS 269

[11, 52], a comprehensive understanding of quorum sensing systems ishighly desirable.

1.3 Modelling of quorum sensing in biofilms We will model quo-rum sensing induction in patchy biofilms in a slow-flow environment,and investigate the effect of environmental conditions such as bulk flowvelocities and nutrient availability on the onset of quorum sensing. Wewill consider Gram-negative bacteria with their typical autoinducer sys-tem, which involves acyl-homoserine lactones (AHLs) as the signallingmolecule. The effect of nutrient availability on quorum sensing is qual-itatively easy to predict: increased food availability promotes growth,and accelerated growth in turn accelerates quorum sensing. On the otherhand, the contribution of bulk flow hydrodynamics is more difficult toestimate. Increased flow implies an increased convective supply of nutri-ents, implying growth is increased, and thus quorum sensing inductionis accelerated. Increased flow velocities also imply increased convectivetransport of AHL in the aqueous phase. Convective AHL transport canhave three effects on the onset of quorum sensing: (i) AHL moleculesthat are produced in a biofilm colony and diffuse into the aqueous phaseare transported downstream (i.e., locally removed), which delays localupregulation. (ii) Downstream colonies receive AHL molecules from up-stream. If the AHL concentration in the aqueous phase is higher thanin the biofilm colonies, AHL will diffuse into the colony and can pro-mote upregulation, even when local cell densities are relatively low. (iii)AHL is washed out from the channel by the flow field; these washed outAHL molecules have no effect on quorum sensing. Which of these threeeffects dominates, and how they balance each other, cannot be readilypredicted.

There are currently numerous mathematical models which can befound in the literature describing quorum sensing in both suspendedbacteria and in biofilms, and for various applications. Ward et al. [49],Dockery and Keener [12], and Muller et al. [35], proposed models of quo-rum sensing in a well-mixed bacterial population, describing growth andAHL production in a manner representative of a batch culture experi-ment. These models predicted a rapid switch in proportions of down-and upregulated sub-populations, which is a characteristic feature ofquorum sensing systems. Ward et al. [50] and Chopp et al. [4, 5] studiedmodels of quorum sensing activity in a growing one-dimensional biofilm,in which biofilm thickness varies in time, to identify the key biofilm andquorum sensing kinetics parameters responsible for induction. Othermodels detail the biochemistry of quorum sensing systems [1, 2, 3]. As


fluid flow in the liquid phase of the biofilm has the potential to greatlyimpact both biofilm growth and quorum sensing [38], more recent mod-els have incorporated hydrodynamic effects [26, 45].

Janakiraman et al. [26] studied biofilm growth and quorum sens-ing in microfluidic chambers experimentally and with a one-dimensionalmathematical biofilm thickness model. It was found that the flow rategreatly affects quorum sensing in the biofilm: at high flow rates, thebiofilm thickness is smaller due to detachment, and the transport rateof AHL out of the biofilm channel can be so high that the AHL concen-tration does not reach the threshold required for induction. Purevdorjand Costerton [38] remark on the importance of diffusive processes, inparticular, that the washout of signal molecules from the bulk fluid sur-rounding colonies would increase the concentration gradient between thebiofilm and the bulk liquid, driving the diffusive flux of signal moleculesout of the biofilm. Their work indicates that diffusion through thebiofilm in different hydrodynamic conditions is key to the AHL signalconcentration in microcolonies. Wanner et al. [48] also demonstratedthat hydrodynamics are required to accurately describe mass conversionand substrate degradation in biofilm systems, so including a flow fieldcomponent will in turn lead to an accurate description of quorum sens-ing in our developing biofilm. Flow may have the potential to disturbquorum sensing, interrupting bacterial behaviour. With this evidenceof the key roles a hydrodynamic environment plays, we will expand onthe works of Janakiraman et al. [26], Vaughan et al. [45], and previousmodels by investigating the contribution of flow to mass transfer andinter-colony communication in the simulation environment.

We will apply our model to a two-dimensional patchy biofilm pop-ulation. There are limitations to one-dimensional or two-dimensionalflat (slab-like) biofilm models. One-dimensional biofilm models predictthickness or height in time. A flat, two-dimensional biofilm (e.g., [30])may be described as a collection of cells in a rectangular grouping. Thebiofilm may grow thicker and horizontally expand in time, but cells arehomogeneously distributed in this setup, and so microcolony behaviourcannot be monitored. In our “patchy” biofilm community, multiple mi-crocolonies, or clusters of cells, are separately placed throughout a simu-lated environment. By using a two-dimensional patchy biofilm pattern,we are able to analyze multiple colonies, as well as the spatial varia-tions in the sub-populations within the biofilm. In addition, the dis-solved substrates will be spatially modelled, specifically, the depletionof carbon and accumulation of AHL. A limitation of one-dimensionalor two-dimensional flat biofilm models is that induction will not be


spatially resolved; once AHL reaches the threshold concentration forquorum sensing, the entire biofilm simultaneously becomes induced. Incontrast, cells arranged heterogeneously in patchy colonies do not reacthomogeneously, and will not necessarily undergo induction at the sametime. Furthermore, a comparatively low number of cells in a small colonycan be sufficient for induction in a patchy biofilm, because in additionto cell density, spatial distribution is also a factor involved in induction[35]. We are particularly interested in low flow velocities, which willenable us to focus on the contribution of flow to mass transfer and reac-tions in our system, as opposed to [26], in which the system behaviourwas largely controlled by detachment processes, and [45], in which thesystem is convection dominated.

To resolve the aforementioned limitations of existing models, we be-gin with a biofilm growth model based on the nonlinear density depen-dent diffusion-reaction equation originally proposed in [16] for a singlespecies biofilm with one limiting nutrient, but we distinguish betweenthe down- and upregulated bacterial sub-populations. Also, in contrastto the models which begin with a fully developed biofilm (e.g., [45]),we have a mechanistic biofilm growth model which starts with very fewsmall bacteria colonies initially adhered to a substratum.

We then add the processes of a quorum sensing system in a molec-ularly founded way. We take into account the dimerisation process forAHL, which appears for almost all investigated quorum sensing systems[23, 52]. When AHL binds to its receptor, this complex dimerizes,and becomes effective as a transcription factor. We model this non-linear AHL dose-response relationship in combination with the positivefeedback loop of induction, describing the switch-like behaviour in theupregulation of bacteria. The bacteria Pseudomonas aeruginosa is typ-ically selected for biofilm and quorum sensing studies. Though thisbacterium is highly relevant for quorum sensing, it has three quorumsensing systems, and so difficulties may arise in the definition of whenquorum sensing is “on” or “off” in the bacteria cells. Pseudomonas

putida is used in our study; with only one known quorum sensing system,this bacterium is practical for studying basic quorum sensing behaviour.The quantitative quorum sensing parameter values used in our numeri-cal simulations were derived from laboratory experiments on planktonicP. putida cultures [21].

We exclude dead cells, additional microbial species, attachment/detachment, and high flow velocities in our model. Though these as-pects may be present in a full biofilm and quorum sensing system, wechoose to specifically model quorum sensing induction patterns in patchy


biofilms in a slow-flow environment. This is a creeping flow regime withReynolds numbers smaller than one, but not so small that flow does notcontribute to mass transfer. At the low flow velocities considered here,mechanical effects such as detachment of the biofilm matrix need not beconsidered - flow induced shear forces will have a negligible effect on thedeformation of the biofilm [18]. We may then focus on the contributionof flow to mass transfer reactions in our system. Secondly, we focus onnarrow conduits, mimicking pore spaces in soils and fractured media.

In the following sections, we will derive our mathematical model, anddescribe the two different flow mechanisms we will use: one in which thevolumetric flow rate is constant throughout the domain, and in the other,the flow rate decreases as the channel fills with biomass. In numericalexperiments we compare the spatiotemporal effects of a growing biofilmon quorum sensing, under various environmental conditions, namely floweffects and nutrient availability. We discuss our key findings from the nu-merical simulations: the observation of mass transfer processes initiatingflow-facilitated inter-colony communication, and spatiotemporal induc-tion patterns for the various environmental conditions tested. Nutrientsupply directly influences both biofilm growth and the time at which quo-rum sensing induction occurs. Flow velocity influences biofilm growth,but is not directly related to the time at which quorum sensing occurs.Flow velocity, if high enough, may prohibit quorum sensing induction.An ecological interpretation of our results is discussed in Section 4.

2 Mathematical Model

2.1 Biofilm growth model We formulate a mathematical model forbiofilm development in terms of the dependent variable m, representingthe local biomass density. As in [13] and [53], we use the mass balancefor biomass as the starting point:

(1) mt + ∇(Um) = G(C, m),

where the function G(C, m) is the net biomass production rate, whichdepends on the concentration of a growth limiting substrate C. Theunknown vector-valued function U is the velocity with which the biomassmoves in the biofilm colony, caused by growth and spatial expansionof the biofilm. Equation (1) represents one equation for the unknownfunctions m, U, C. The growth limiting substrate C will be described bya transport-reaction equation below. In order to obtain U , we consider


the momentum equation,

(2) (Um)t + ∇(UUm) = −∇P − f(m)mU,

in which the new variable P is the unknown biomass pressure. The lastterm in this equation describes forces moderating biomass movement,such as kinematic friction; see [31, 54], in which f(m) = f = constant issuggested for some biological materials. In order to close this model, anadditional relationship must be established. In [13], closure is achievedby introducing the assumption that in the biofilm, the biomass densityis always at the maximum cell density, m = Mmax = constant, that is,the biofilm is assumed to be incompressible. Alternatively, Wood andWhitaker [53] suggested to close the model algebraically by imposinga constitutive relationship P = P (m), which recovers the traditionalWanner-Gujer [47] biofilm model in the one-dimensional setting. How-ever, the higher dimensional case remained open. We follow the Woodand Whitaker [53] approach. As in [13], we neglect the inertial forces,and the momentum equation (2) reduces to the Darcy-like equation,

−1

mf∇P = −

1

mfP ′(m)∇m = U.

Substituting this into (1) gives for the biomass density the quasilineardiffusion-reaction equation,

mt = ∇ (DM (m)∇m) + G(C, m),

with the density-dependent diffusion coefficient,

DM (m) :=1

fP ′(m).

DM (m), or alternatively, P (m), must have the following properties: (i)notable spatial expansion of the biofilm only takes place for large enoughm, and (ii) the biomass density m does not exceed the maximum possiblevalue Mmax. One choice that is known to satisfy these conditions [20]is

DM (m) = δma

(Mmax − m)b, a, b > 1, δ > 0,

which can be derived from power law assumptions P ∼ mη for m ≈ 0and P ∼ (Mmax − m)−λ for m ≈ Mmax.


The growth controlling substrate C in (1) is governed by the transport-reaction equation

Ct + ∇(UC) = ∇(d∇c) − F (C, m).

Commonly, Monod kinetics are assumed to describe growth of biomass

F (C, m) =κ3

Y

Cm

k2 + C, G(C, m) =

κ3Cm

κ2 + C− κ4m.

Here, κ3 is the maximum growth rate of the species, Y is the yieldcoefficient, κ2 is the Monod half saturation concentration, and κ4 is thecell death rate. The characteristic time-scale for spatial expansion ofthe biofilm is much larger than the characteristic time-scale for diffusionand consumption of substrate in the biofilm. Therefore, following [48],it is common practice to neglect the convective term induced by biomassmovement.

For convenience, in the following section, and for the remainder ofthe paper, we will use the volume fraction occupied by biomass ratherthan the biomass density, that is, we use M := m/Mmax as a depen-dent variable. Then, summarizing the above, the biofilm growth model,formulated for two space dimensions, is

∂tM = ∇(DM (M)∇M) +κ3CM

κ2 + C− κ4M(3)

∂tC = ∇(d∇C) −κ1CM

κ2 + C(4)

in which κ1 := µMmax/Y , and the density dependent diffusion co-efficient for biomass is DM (M) = δMa−b

maxMa(1 − M)−b. The spa-tial domain Ω in which our model is formulated is accordingly sub-divided into two not necessarily connected subdomains, the aqueousphase Ω1(t) := (x, y) ∈ Ω : M(t, x, y) = 0 and the solid biofilm phaseΩ2(t) := (x, y) : M(t, x, y) > 0. These regions change as the biofilmgrows.

The model (3), (4), which we derived here from the same startingpoint as the biofilm model of [13], is the density-dependent diffusion-reaction model for biofilm formation that was originally introduced in anad hoc fashion in [16]. More recently, this model was also derived froma semi-discrete master equation in [28], beginning with assumptionsresembling those that are used in cellular automata models of biofilms,such as [37].


The diffusion-reaction biofilm model (3), (4) has been studied in asmall series of papers, for analytical, numerical, and biological interest.In particular, an existence and uniqueness proof was given in [20], inwhich it was also shown that the solution remains bounded by 1 (themaximum biomass density is not exceeded). Moreover, it was shownthat there exists a η > 0, η << 1, such that the solution M < 1 − η,provided M = 0 somewhere on the boundary, and so M will be separatedfrom the singularity at M = 1. The model possesses a global attractorthat was studied in [9].

2.2 Quorum sensing model We now present a mathematical modelfor quorum sensing in a growing biofilm community in a narrow conduitwhich consists of several colonies. This model combines the diffusion-reaction biofilm model derived in Section 2.1 with the quorum sensingmodel for suspended populations of [35]. The dependent variables ofour model are the biomass volume fractions for down-regulated bacte-ria M0 and up-regulated (quorum sensing induced) bacteria M1, andconcentrations of AHL (AHL) and the dissolved carbon substrate (C).

The mathematical model is:

∂tM0 = ∇(DM (M)∇M0) +κ3CM0

κ2 + C− κ4M0

− κ5AHLnM0 + κ5τnM1

∂tM1 = ∇(DM (M)∇M1) +κ3CM1

κ2 + C− κ4M1

+ κ5AHLnM0 − κ5τnM1

∂tC = ∇(DC(M)∇C) −∇(wC) −κ1C(M0 + M1)

κ2 + C

∂tAHL = ∇(DAHL(M)∇AHL)) −∇(wAHL) − σAHL

+ αM0 + (α + β)M1

The total volume fraction occupied by the biofilm is M = M0 + M1.The diffusion coefficient for biomass (DM (M)) is described in Section2.1. The diffusion coefficients for C and AHL are lower in the biofilmthan in the surrounding aqueous phase [42]. We use DC(M) = DC(0)+M(DC(1)−DC(0)), DAHL(M) = DAHL(0)+M(DAHL(1)−DAHL(0)),where DC(0) and DAHL(0) are the diffusion coefficients in water, andDC(1) and DAHL(1) are the diffusion coefficients in a fully compressedbiofilm.


The convective contribution to transport of C and AHL in the aque-ous phase is controlled by the flow velocity vector w = (u, v), where uand v are the flow velocities in the x- and y-directions. The calculationof w is further described in Section 2.3. The model includes diffusivetransport of carbon substrate and AHL in the biofilm, and both con-vective and diffusive transport in the surrounding aqueous phase of thebiofilm.

The reaction processes incorporated into our model are:

• nutrient consumption by down- and up-regulated biomass. The max-imum specific substrate consumption rate is denoted by κ1.

• production of new up- and down-regulated biomass, controlled bynutrient availability. κ3 is the maximum specific growth rate of thebacterial biomass. Bacterial growth is described by Monod kinetics,where κ2 is the half-saturation constant.

• natural cell death; κ4 is the cell lysis rate.• conversion of down-regulated biomass into up-regulated biomass as

a consequence of AHL concentration inducing a change in gene ex-pression, and a constant rate of back-conversion. The parameter κ5 isthe quorum sensing regulation rate - the rate at which down-regulatedbacteria become up-regulated, and vice versa. τ is the threshold AHLconcentration locally required for quorum sensing induction to occur.The specific form of the switching term is derived from Hill kineticswith a quasi-steady state assumption [35]. The coefficient n (n > 1)describes the degree of polymerisation in the synthesis of AHL. Asmentioned in Section 1.3, we model the dimerisation process for AHL,and assume that dimers of receptor-AHL complexes are necessary forthe transcription of the AHL-synthase gene. Our model follows massaction kinetics to describe how AHL molecules and downregulatedcells are required for upregulation. The dimerisation process givesn = 2, however, we lump further synergistic effects into n as well,which leads to a slightly higher value of n.

• abiotic AHL decay at rate σ.• differential AHL production rates of down-regulated and up-regulated

cells. The AHL production rate of down-regulated bacteria is α, andthe increased production rate of up-regulated bacteria is α + β. dif-fusive transport in the surrounding aqueous phase of the biofilm

We nondimensionalise our model using the following rescalings:

x =x

L, y =

y

Land t = tκ3,


where L is the flow channel length, and 1κ3

is the characteristic time

scale. The new dimensionless concentration variables are: C = CCbulk

and

A = AHLτ , where Cbulk is the bulk substrate concentration (the amount

of substrate C supplied at the inflow boundary). The new reactionparameters are:

κ1 =Mmax

Y Cbulk, κ2 =

κ2

Cbulk, κ3 = 1, κ4 =

κ4

κ3,

κ5 =κ5τ

n

κ3, σ =

σ

κ3, α =

αMmax

κ3τ, β =

βMmax

κ3τ.

Finally, the diffusion coefficients become

DC =DC

L2κ3, DAHL =

DAHL

L2κ3and DM =

DM

L2κ3,

and we obtain the nondimensional system

∂tM0 = ∇(DM (M)∇M0)) +κ3CM0

κ2 + C− κ4M0 − κ5A

nM0 + κ5M1(5)

∂tM1 = ∇(DM (M)∇M1)) +κ3CM1

κ2 + C− κ4M1 + κ5A

nM0 − κ5M1(6)

∂tC = ∇(DC(M)∇C)) − ∇(wC) −κ1CM

κ2 + C(7)

∂tA = ∇(DA(M)∇A)) − ∇(wA) − σA + αM0 + (α + β)M1(8)

The parameters used in our simulations and their non-dimensionalvalues are listed in Table 1. The biofilm parameters were chosen fromthe range of standard values in biofilm modelling literature [48], andthe quorum sensing parameters were derived from experiments on thekinetics of suspended P. putida IsoF cultures and the AHL molecule3-oxo-C10-HSL [21].

In order to complete the model (5)–(8), boundary and initial con-ditions must be specified. The usual approach for dissolved substratessuch as A and C (e.g., [7, 15, 37, 45]) is to prescribe a bulk concentra-tion value at the inflow boundary, assume that the walls of the channelare impermeable to the substrate, and to specify a standard outflowcondition at the downstream boundary. Like [45], we assume that thefluid entering the flow channel does not contain AHL, while the growth


Parameter Description Nondimensional Value Reference

κ3 Maximumspecific growthrate

1 [48]

κ4 Lysis rate 0.067 [48]

κ5 Quorumsensing up-regulationrate

2.5 [21]

σ Abiotic degra-dation rate ofAHL

0.022 [21]

α Constitutiveproductionrate of AHL

13 [21]

β Induced pro-duction rate ofAHL

130 [21]

n Degree of poly-merisation

2.5 [21]

DC(0), DC(1) Diffusion coef-ficients for sub-strate

0.67, 0.54 [18]

DAHL(0), DAHL(1) Diffusion co-efficients forAHL

0.52, 0.26 [25]

a Diffusion coef-ficient parame-ter

4.0 [18]

b Diffusion coef-ficient parame-ter

4.0 [18]

dM Biomass motil-ity coefficient

6.67e-09 [18]

H/L Channel aspectratio

0.1 S

TABLE 1: Constant model parameters used in our simulations and theirnon-dimensional values. Values for the parameters varied in our simu-lation experiments are given in the text. References: [48] = Wanneret al. (2006), [18] = Eberl and Sudarsan (2008), [25] =Hobbie andRoth (2007), [21] = measured experimentally in Fekete ıet al. (2010),S = selected for our system.

promoting substrate is available at the normalised bulk concentration(e.g., [7, 15, 37]). We then have C = 1 and A = 0 for x = 0 and∂nC = 0, ∂nA = 0 everywhere else on the boundary, where ∂n denotes


the outward normal derivative. For the initial conditions, we assumethat at t = 0, both dissolved substrate concentrations attain their bulkvalue everywhere: C(0, x, y) = 1, A(0, x, y) = 0 for (x, y) ∈ Ω.

The walls of the flow channel are impermeable to biomass. Thus, ho-mogeneous Neumann conditions for the biomass are specified: ∂nM0 =∂nM1 = 0 at y = 0 and y = H/L. At the inflow and outflow boundariesthe situation for the biomass fractions M0 and M1 is less straightfor-ward, and additional assumptions must be made. First, we note thatin degenerate diffusion-reaction equations like (5), (6), initial data withcompact support lead to solutions with compact support (cf. [40]).In other words, the biofilm/water interface spreads with finite speed.This implies that where the biomass does not reach the boundary, thehomogeneous Dirichlet and the homogeneous Neumann conditions aresatisfied simultaneously. The boundary conditions are not active and donot affect the solution as long as biomass does not reach the boundary.In order to obtain a physically closed system for which we can formu-late boundary conditions, we assume a hydrophilic membrane at theupstream and downstream end of the flow channel, which is permeableto water and the substrates dissolved in the water, but impermeable tobiomass, which therefore is described by homogeneous Neumann condi-tions: ∂nM0 = ∂nM1 = 0 at x = 0 and x = 1. Moreover, we assumethat the channel is fed from the upstream boundary from a well-mixedreservoir, which is so large in comparison to the flow channel that thetransport of substrates across the upstream membrane does not affectthe bulk concentration values. Initially, at t = 0 no up-regulated cells arein the channel, M1(0, ·) ≡ 0. Down-regulated biomass with a specifiedvolume fraction M0 > 0 can only be found in small randomly chosenpockets along the substratum while M0(0, ·) = 0 everywhere else (seeSection 3.1 for further detail).

Existence proofs for the solutions of similar biofilm models with mul-tiple volume occupying substances have been presented in [10] and [27]for a purely diffusive setup, that is, in the absence of convective transportof dissolved substrates in the aqueous phase. The same concept of theproof, which does not allow a statement on uniqueness of the model so-lution, may be applied to the quorum sensing and biofilm growth modelintroduced here. In contrast to the models studied previously, we re-mark that here, by adding the equations for M0 and M1, the biofilmgrowth model (3), (4), can be recovered. With the arguments detailedin [20], we know that M = M0 + M1 and C are unique and separatedfrom the singularity M < 1. Therefore, the question of uniqueness ofthe solutions of the quorum sensing model reduces to the question of


uniqueness of the semi-linear system:

∂tM1 = ∇(DM (M)∇M1) +κ3CM1

κ2 + C− (κ4 + κ5 − κ5A

n)M1 + κ5AnM

∂tA = ∇(DA(M)∇A) − σA + αM + βM1,

in which we can treat C and M as known functions. This uniquenessproof has been carried out in detail in [46]

2.3 Hydrodynamic component In the aqueous phase, the dissolvedsubstrates C and A are convectively transported with flow velocityw = (u, v), which is yet to be determined. As an alternative to solvingthe complete Navier-Stokes equations numerically, we use the thin-filmapproximation as proposed in [18]:

−1

ρ

∂p

∂x+ ν

∂2u

∂y2= 0,

∂p

∂y= 0,

∂u

∂x+

∂v

∂y= 0,

where p is the hydrodynamic pressure field; constants ρ and ν are thedensity and kinematic viscosity of water. This approximation is valid inthe liquid region of long, narrow conduits for slow flow regimes. Unlikethe complete Navier-Stokes or the Stokes equations, this flow model canbe solved analytically, which allows for fast computation. In order todrive the flow in the channel, we specify the volumetric flow rate in termsof the nondimensional Reynolds number Re for the empty flow channel,given by

Re =HW

ν

where our characteristic length scale H is the channel height, ν is thekinematic viscosity of the bulk liquid, and W is the maximum flowvelocity of the unperturbed Poisseuille flow. The flow rate q may berelated to the Reynolds number [18]:

q =2

3ν Re.

We consider two flow mechanisms. For what we refer to as the regularflow mechanism, we assume the flow rate to be constant for all time.We refer to the bioclogging flow mechanism if we take into account thatmicrobial growth in the channel affects the flow throughput. Thullner et


al. [43] demonstrated that a relatively small amount of biofilm can causesignificant clogging effects and flow reductions. This is in particularrelevant for biofilm processes in microbial soil ecology. In this case, wecan conceptualize a narrow flow conduit, like our simulated channel,as a small pore in a larger network of such channels, i.e., in a porousmedium. Several empirical relations have been suggested to describehow microbial growth affects flow rates [44]. We choose the Clementet al. [6] model for simplicity, because it does not introduce additionalunknown parameters. In this case, flow resistance k is described by anapproximately cubic power law

k(h) = h19/6,

where the pore space h is defined to be the smallest gap between the topof an individual biofilm colony and the top of the flow channel, scaledto our channel height. The variable flow rate for bioclogging conditionsis then,

q = k(h)2

3ν Re.

In the absence of biomass, h = 1.0 and k(h) = 1.0, so the flow veloci-ties remain precisely the same as the regular flow mechanism. In time,biomass increases, and k → 0 as h → 0. The flow velocities will decreaseslowly at first, then notably as the channel fills.

Both the regular and bioclogging flow mechanisms for hydrodynamicswill be used and compared in the subsequent simulations to investigatethe effects of the flow field on our biofilm and quorum sensing system.

2.4 Computational aspects The numerical solution of the biofilmmodel follows the approach that was developed and discussed in [17,

19]. The computational domain is discretised on a uniform rectangulargrid of size 2000 × 200. We solve our system numerically using a semi-implicit finite difference-based finite volume scheme. In each time stepfour sparse, banded diagonal linear algebraic systems are solved with thestabilized biconjugate gradient method (one system for each of AHL,C, M0, and M1). Simulations run until an imposed stopping criterion ismet—either a maximum height of the biofilm in the channel is obtained,or in the cases in which the biofilm does not develop fully, a maximumsimulation time is reached.

The code was prepared for parallel execution on multi-core/multi-processor platforms as described in [34]. Simulations were carried outon a SGI ALTIX450 with 32 dual core Itanium processors; in mostsimulations 12 cores were used per compute job.


3 Results

3.1 Simulation experiment design In simulation experiments, wemeasure the effects of environmental conditions (nutrient concentrationsand channel hydrodynamics) on the onset of quorum sensing inductionin the biofilm system, and how production of autoinducer molecules inone colony affects induction in neighbouring colonies.

(a) t = 7.0, Maximum AHL = 0.13. All colonies are down-regulated.

(b) t = 8.0, Maximum AHL = 0.37. All colonies are still down-regulated.

(c) t = 9.0, Maximum AHL = 1.6. Some colonies have up-regulated.

(d) t = 9.5, Maximum AHL = 4.9. The entire biofilm is up-regulated.

FIGURE 1: Development of a biofilm in a flow channel using the bio-clogging flow mechanism. Flow is from left to right; food is suppliedfrom the upstream (left) boundary. Biofilm colonies are coloured to in-dicate the fraction of the cells that are downregulated ( M0

M0+M1); from

completely down-regulated (red) to fully up-regulated (blue). AHL iscoloured by isolines from zero (black) to maximum AHL (white).

One measurement of these effects is the “switching time,” which wedefine as the time when the AHL concentration first reaches the thresh-old value for induction somewhere in the domain. We also investigate thespatial distribution of up- and down-regulated cells in the flow channel,and biomass growth under these various environmental conditions.


A typical computer simulation of biofilm growth in a long, narrowchannel with aspect ratio ε = H/L = 0.1 and slow flow (Re = 10−3) isshown in Figure 1. The bottom surface, or substratum, is initially inocu-lated with twenty randomly chosen small colonies of down-regulated cellswith density 0.2 < M0 < 0.4. The growth period begins with biomassin the inocculated colonies growing, but not expanding. As the totalbiomass (M0 + M1) locally approaches 1.0, the colonies expand. Theindividual colonies initially grow at the same rate, but in later timestepsthe downstream colonies will experience slower growth due to substratedepletion by the upstream colonies, which are first to receive and utilizethe incoming nutrients. In time, some neighbouring colonies begin tomerge.

Isolines of AHL concentration are distributed between 0 and the max-imum AHL value (reported in each subfigure); the maximum AHL con-centration changes in time as it accumulates in the system. Initially(Figures 1(a),(b)), AHL concentrations increase in the main flow direc-tion. The colonies produce AHL molecules, which diffuse into the liquidregion, and are then transported downstream by convection. In latertime steps (Figures 1(c),(d)), the maximum AHL concentration is foundin the middle of the channel. This is due to both the accumulation ofAHL that has been transported via convection from the upstream re-gion, and the increased AHL produced by up-regulated colonies in themid-channel.

In regards to the biofilm composition, we observe that the down-regulated biofilm grows in time (Figures 1(a),(b)). The smaller, down-stream colonies begin to upregulate, or “switch” before the larger up-stream colonies in Figures 1(c). The switch occurs rapidly, and haspassed in Figure 1(d), where upregulation is present in nearly the entirebiofilm. The observation that the smaller, downstream colonies are firstto undergo the switch to up-regulation demonstrates the effects of masstransfer. AHL molecules produced by the large, upstream colonies aretransported downstream, and once there, contribute to upregulation.Thus, inter-colony communication has been facilitated by convectivetransport. In the convection dominated system of [45], we also observeimportant inter-colony interactions resulting from convective processes.

The concentration of C decreases in the flow direction (left to right)due to consumption by biomass (carbon concentration isolines are notshown). As the biofilm grows, the demand for carbon increases.

To analyze effects averaged across the computational domain, we willcalculate the occupancy (the fraction of the domain occupied by the


biofilm):

Occupancy (t) :=1

LH

∫Ω2(t)

dx dy

which is a simple measure of biofilm size. The total down-regulated andup-regulated biomass in the system are M0total and M1total:

M0total(t) :=

∫Ω

M0(t, x, y) dx dy,

M1total(t) :=

∫Ω

M1(t, x, y) dx dy

Finally, the average AHL concentration in the system is given by:

AHLaverage(t) :=1

LH

∫Ω

AHL(t, x, y) dx dy.

The occupancy plot in Figure 2(a) for our simulation in Figure 1 con-firms that biomass growth is exponential initially in time. Later, whennutrient demand begins to exceed the inflow of bulk substrate, growthcontinues, but at a lower rate. Figure 2(b) is a biofilm compositionplot, showing the total down-regulated and up-regulated biomass in thesystem. We initially see only growth of down-regulated biomass, untilthe switch to upregulation occurs, after which the biofilm compositionquickly becomes dominated by up-regulated biomass. This switch cor-responds to the rapid increase in average AHL concentration, shown inFigure 2(c).

The bioclogging flow mechanism was implemented in this example.The decrease in flow rate in time due to biomass accumulation in thechannel can be seen in Figure 2(d).

Three simulation experiments were conducted to investigate the in-fluence of environmental conditions and biofilm kinetics on autoinducerinduction in the growing biofilm. In these three experiments, we varied(i) nutrient availability Cbulk , (ii) up-regulation rate κ5, and (iii) theflow velocity, specified in terms of the dimensionless Reynolds number(Re).

3.2 Nutrient availability The bulk substrate concentration Cbulk wasvaried over one order of magnitude: Cbulk = 100, 50, 25. In each simu-lation, Re = 10−3. For higher bulk substrate concentrations, a greateramount of biomass is produced, and substrate limitations are observed


0

0.1

0.2

0.3

0 2 4 6 8 10 12

Occ

upan

cy

Time

(a) Occupancy.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12

Bio

mas

s M

0, M

1

Time

M0M1

(b) Biomass.

0

2

4

6

8

0 2 4 6 8 10 12

Ave

rage

AH

L

Time

(c) Average AHL.

0

0.0005

0.001

0 2 4 6 8 10 12

Re

Time

(d) Flow rate.

FIGURE 2: For the typical simulation in Figure 1, (a) the fraction ofthe domain occupied by biofilm (occupancy), (b) biofilm composition(the total M0,M1 biomass), (c) average AHL in the domain, and (d)average flow velocity for the bioclogging mechanism.

later in time (data not shown). This direct relation between nutrientsupply and growth confirms our intuitive expectations, and is in accor-dance with findings from previous biofilm growth studies [18].

It was found that, as anticipated, AHL concentrations are increasedand induction occurs at an earlier time for higher bulk substrate concen-trations. In the cases of Cbulk = 50 and 25 with the regular flow mecha-nism, the nutrients are severely limited, and the biofilm remains so smallthat insufficient AHL is produced for accumulation to the threshold levelfor switching.

Spatial induction patterns were observed. Induction occurs first forthe medium-sized colonies in the mid-channel, and upstream colonies arelast to upregulate. This is the result of the balance between local AHLproduction and AHL received from upstream colonies via convective


transport for the chosen flow regime.

3.3 Influence of quorum sensing regulation rate κ5 Among theexperimentally determined quorum sensing parameters, the quorum sens-ing up-regulation rate (κ5) had the most uncertainty. We varied κ5

between 1.0 and 5.0 (results not shown), and confirmed our intuitiveexpectations—a higher quorum sensing up-regulation rate led to earlierswitching times.

3.4 Influence of hydrodynamics The Reynolds number Re was var-ied over several orders of magnitude, between 10−5 ≤ Re ≤ 10−1. Thebulk substrate concentration on inflow was kept constant throughout theexperiment, at an abundant Cbulk value. To estimate the relative con-tributions of convective and diffusive mass transport, we will calculatethe dimensionless effective Peclet number:

Pe∗ =convection

diffusion= ε Re Sc,

where ε is the channel aspect ratio ε = H/L = 0.1, and Sc is theSchmitt number (Sc = ν/D), the ratio of the fluid momentum diffusiv-ity and substrate mass diffusivity [18]. For carbon and AHL in water,ν = 0.0864 m2d−1, DC = 10−4 m2d−1, and DAHL = 7.76−5 m2d−1,yielding Sc ≈ 103 for both dissolved substrates, with Sc slightly largerfor AHL. The effective Peclet numbers for our simulations are thenPe∗ ≈ 10.0, 0.1, and 0.001 for Re = 10−1, 10−3, and 10−5 respectively.So, our system is convection dominated for Re = 10−1, mildly dif-fusion dominated for Re = 10−3, and clearly diffusion dominated forRe = 10−5.

The roles of flow in our model have counteracting effects on quorumsensing induction. Increased flow implies an increased convective supplyof nutrients, implying growth is increased, and thus quorum sensinginduction is accelerated. However, increased flow velocities also implyincreased convective transport of AHL in the aqueous phase. ConvectiveAHL transport can have three effects on the onset of quorum sensing:

(i) AHL molecules that are produced in a biofilm colony and diffuseinto the aqueous phase are transported downstream, delaying localupregulation.

(ii) Downstream colonies receive AHL molecules from upstream. Ifthe AHL concentration in the aqueous phase is higher than in thebiofilm colonies, AHL will diffuse into the colony and can promoteupregulation, even when local cell densities are relatively low.


(iii) AHL is washed out from the channel by the flow field, having noeffect on quorum sensing.

Because of these counteracting roles, we do not know a priori the neteffect flow will have on our system, and so this will be investigated inthe following simulations.

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12

Occ

upan

cy

Time

Re=10e-1Re=10e-3Re=10e-5

(a) Occupancy.

0

2

4

6

8

0 2 4 6 8 10 12M

axim

um A

HL

in s

yste

m

Time

Re=10e-1Re=10e-3Re=10e-5

(b) Maximum AHL.

FIGURE 3: (a) The fraction of the domain occupied by biofilm (occu-pancy) for the flow velocity experiments, using the regular flow mech-anism. Higher flow velocities result in greater occupancies. (b) Themaximum AHL concentration in the channel in time for different Reusing the regular flow mechanism. For lower flow velocities, convectivetransport of AHL is decreased, and so more AHL may accumulate inthe channel.

The occupancy curves for the flow velocity experiments are plotted inFigure 3(a), using the regular flow mechanism. For an initial time period,the growth curves fall on the same line, indicating that the biofilmsare all growing at the same exponential rate. When growth becomeslimited, the curves will deviate from Re = 10−1; this biofilm continuesto grow at the maximal rate. We find that growth becomes limitedearlier for the smallest Re (Re = 10−5), and that higher flow velocitiesresult in greater occupancies (Table 2). The differences in the total(M0 + M1) and upregulated (M1) occupancies are listed in Table 2, forthe various flow rates at t = 9.7, the approximate end of the simulation.Interestingly, despite the fact that the biofilms are smaller for the slowerflows, the proportion of up-regulated biomass is greater for Re = 10−3

and 10−5 than for Re = 10−1. When the flow velocity decreases, less


AHL is removed via convective transport, allowing a greater proportionof bacteria to become up-regulated.

Re Total occupancy Proportion of up-regulated biomass

Max AHL

10−1 0.4737 0.001451 0.151910−3 0.2493 0.1350 1.18310−5 0.1876 0.9647 5.896

TABLE 2: A comparison of the total occupancy (M0 +M1), proportionof up-regulated biomass (M1/(M0 + M1)), and maximum AHL concen-trations in the channel, for the regular flow mechanism at t = 9.7.

To study the effects of flow on quorum sensing behaviour in ourbiofilm system, we measured the maximum AHL concentration in thechannel in time for various flow velocities. The maximum AHL concen-trations in time are plotted in Figure 3(b), in which we observe the rapidincrease in concentrations corresponding to induction, representative ofthe positive feedback in a quorum sensing system. The AHL concentra-tion values at t = 9.7 are given in Table 2. For higher flow velocities,convective AHL washout is increased, and so less AHL accumulates inthe domain. The influence of flow on the maximum AHL increases withincreasing velocity - for the regular flow case, the maximum AHL is fivetimes greater for Re = 10−5 than for Re = 10−3, and ten times greaterfor Re = 10−3 than for Re = 10−1.

To determine the temporal effects of flow on quorum sensing be-haviour in our biofilm system, we measured the switching time in eachsimulation—the time at which AHL concentrations first reach the thresh-old concentration for induction at some point within the biofilm do-main. Figure 4 shows the switching time for six simulations in whichRe = 10−1, 10−3, 10−5, and both flow mechanisms were used. Note thatfor the convection dominated regime Re = 10−1, using the regular flowmechanism, switching does not occur in the simulation time. These re-sults coincide with [26], in that flow rate can be set so high that AHLis washed out of the channel to the point that it cannot accumulate tothe induction threshold concentration, even though biomass growth isgreatest for the high flow velocities. However, each of the remaining


-6

-4

-2

0

8 8.5 9 9.5 10

log 1

0(R

e)

Switching Time

BiocloggingRegular

FIGURE 4: Switching time: the time at which AHL concentrations firstreach the threshold concentration in the domain. The switching timeis shown for six simulations in which Re= 10−1, 10−3, 10−5, using bothflow mechanisms. Little variation for the different Re is exhibited.

simulations undergo initial switching at approximately t = 9.0, indicat-ing that for both flow mechanisms, there is a minor influence of the flowvelocity on the switching time of the biofilm.

We now consider the spatial impacts of flow on quorum sensing in-duction. In Figure 1(c), we observe that flow induced convective AHLtransport has a great local impact on induction—large upstream coloniesexperience a delay in induction, or no induction at all. In contrast,smaller downstream colonies experience an accelerated induction due toa supply of AHL from upstream (that is, intercolony communication inour patchy biofilm community). In all experiments conducted where in-duction occurs, upregulation is initiated at the base of the biofilm, andthe faster growing upstream colonies are upregulated last. However, thespatial distribution of upregulated biomass varies with flow. For thelower flow velocities, initial induction occurs further upstream than forthe higher flow velocities. At low flow velocities, the removal of AHLby convection is to some extent balanced by the AHL produced by localcolonies, allowing for the effect of initial switching to be reached furtherupstream.

Figure 5 depicts simulations of 5(a) a convection-dominated system(Re = 10−1) and 5(b,c) diffusion-dominated systems (Re = 10−3, 10−5),allowing for a visual comparison of the systems at t = 9.0, for the regularflow mechanism. t = 9.0 is the approximate switching time for all flowregimes with the current Cbulk value, but here induction is not observed


(a) Re= 10−1, Maximum AHL = 0.12

(b) Re= 10−3, Maximum AHL = 1.66

(c) Re= 10−5, Maximum AHL = 2.24

FIGURE 5: Simulations of a convection-dominated system (a) anddiffusion-dominated systems (b) and (c) at time t=9.0. In (b) and (c),the highest fraction of upregulated cells is found in the medium-sizedcolonies in the centre of the domain, demonstrating the balance betweenAHL production by local colonies and AHL received from the upstreamregion by convective transport.

for the convection-dominated case. The maximum AHL concentrationis shown in each subfigure. Though the biofilm is much larger in 5(a),the forces of convection (seen by the AHL isolines) wash out AHL, andit cannot accumulate to the threshold. In Figure 5(b,c), the highestfraction of upregulated cells is found in the medium-sized colonies in thecentre of the domain, demonstrating the balance between AHL produc-tion by local colonies and AHL received from the upstream region byconvective transport. These results contrast those produced by systemsthat are highly convection dominated (e.g., [45]), in which the high-est AHL concentrations are only found at the outflow boundary of thedomain.

From these experiments on the effects of flow velocity on biofilmgrowth and quorum sensing behaviour, we conclude that the flow veloc-ity alone cannot be used to predict the induction time for the biofilm.However, hydrodynamics did have significant spatial impacts: for highflow velocities, convective transport of AHL was the dominant process,leading to increased biofilm growth but also increased AHL washoutdownstream. For low flow velocities, convective and diffusive AHL trans-port were balanced, allowing initial induction to occur further upstream,and for a greater accumulation of upregulated biomass and AHL in the


mid-channel.

3.5 Influence of bioclogging flow mechanism The regular and bio-clogging flow mechanisms produced results that were qualitatively verysimilar, though slight differences were found in the biofilm occupancy,proportion of upregulated biomass, and total AHL between simulationsusing the two flow mechanisms (not shown).

The differences in flow rates between the two mechanisms emerge atapproximately t = 5 in each simulation. We note that the drop in flowvelocities for the bioclogging case occurs at a late stage of biofilm growth,in fact, past the period of maximal growth. The difference in velocityat the end of the simulation is approximately two orders of magnitudein each case.

The direct relationship between nutrient supply and growth held forthe bioclogging flow mechanism as well. The maximum AHL for bioclog-ging was greater than for regular flow, and the bioclogging flow mech-anism led to earlier switching times. Though the regular flow mecha-nism allowed for greater biomass growth, the proportion of up-regulatedbiomass was higher for the bioclogging case. In one nutrient simulation(Cbulk = 50), AHL did not reach the threshold for induction with regu-lar flow, but it did surpass the threshold for the bioclogging flow case -indicating that for some range of bulk substrate concentrations, the flowregime chosen could lead us to draw different conclusions about quorumsensing behaviour in our biofilm system.

In each case, the regular flow mechanism allowed for greater biomassgrowth (measured by the total occupancy) than the bioclogging mech-anism, although the difference between the final occupancies of the reg-ular and bioclogging flow simulations was less noticeable with smallerRe (i.e., in diffusion dominated systems). Greater biomass growth withregular flow is due to the fact that the regular flow mechanism deliversnutrients at a fixed rate throughout the simulation experiment, whereasthe delivery rate decreases in time with bioclogging flow.

Overall, the bioclogging flow mechanism had a greater impact on quo-rum sensing than on biofilm growth. Bioclogging flow led to increasedquorum sensing activity in the biofilm, that is, more upregulated cellsand higher concentrations of AHL, even though biofilms were smallerwith this mechanism.

4 Ecological Implications Flow in the liquid surrounding thebiofilm clearly influences autoinduction, which is in accordance with theresults of [30]. Our results indicate that flow has an impact on both


the spatial and temporal induction patterns. The flow velocity, in therange we have chosen, has a significant local impact—the exact areawhere upregulation is initiated depended on the specific values of flowvelocity. The other important factor analyzed in our experiments wasthe dissolved growth limiting nutrient. Interestingly, the combinationof increased flow velocity and nutrients resulted in an increased colonygrowth upstream, but also resulted in an accelerated induction down-stream. So under the chosen conditions, smaller downstream colonieswere earlier upregulated than the larger upstream colonies. This resultis in conflict with the idea of quorum sensing in the strict sense, inwhich cell density is monitored by bacteria to determine whether den-sity is sufficient for induction. In the idea of diffusion sensing, individualcells measure the autoinducer concentration produced by themselves, todetect mass transfer limitations, that is, the amount of their AHL pro-duced and lost due to both diffusion and convection [39]. However, wehave groups of cells producing AHL, and there are clearly interactionsbetween the colonies. Our results are therefore best interpreted usingthe efficiency sensing concept, in which colonies sense a combination ofspatial distribution, cell density, and mass transfer limitations [24]. Insummary, the nutrient supply and the presence of flow play a role indetermining when cells switch from a single cell to a more collective be-haviour. The combination of nutrients and flow (via direct impact onAHL transfer and indirect impact on AHL production) controls the in-tracolonial induction and intercolonial interactions. In contrast to plank-ton or biofilms in the absence of flow, biofilms under flow conditions acthighly heterogeneously and interactively. There is additional value forcells to participate in collective behaviour, i.e., benefits to living in abiofilm, and the environmental conditions influence when these benefitsare recognized and the collective behaviour emerges.

The question arises whether the influence of flow disturbs the autoin-ducer regulation of behaviour, or if flow is an integral part of the reg-ulation - ultimately, can flow be used to interrupt bacterial behaviour?Autoinducers are thought to generate collective, cooperate behaviour,for example, in the release of effectors such as exoenzymes and an-tibiotica [24]. With the presence of flow, downstream cells lose theireffectors, but obtain them from upstream cells. In contrast, large up-stream colonies enforce cooperative behaviour by promoting inductionof downstream cells, but themselves do not necessarily participate in theinduced cooperative behaviour, depending on the flow velocity, nutrientconditions, and growth stage. This seems to contravene the reliabilityof the autoinducer signalling process. In the absence of flow, enforce-


ment of cooperation is attained by the positive feedback mechanismof autoinducer signalling. Within the colonies, enforcement can stabi-lize cooperativity—so under flow conditions, high enforcement (positivefeedback) will ensure cooperation within the colony. With flow, this en-forcement is of only limited utility beyond the boundaries of each colony.However, the number of non-cooperative cells or entire colonies is lim-ited and decreases over time, and thus may not be very relevant for theecological function of the sensing mechanism. Furthermore, most cellsmay obtain a major fraction of sensed autoinducers from their neigh-bouring cells within the colony, where flow likely does not relevantlydisturb cooperation.

Significantly higher flow velocities may restrict induction to be causedby autoinducers produced within the colony itself, but high flow veloc-ities can also diminish nutrient limitations for downstream cells, con-sequently promoting growth and induction. Thus, although flow doesalter the autoinducer regulation system, the goal of manipulating bac-teria through quorum sensing control would be best reached throughutilization of flow to affect nutrient supply. Note that in our model,nutrients affected biofilm growth. Nutrient limitations have also beenknown to affect the quorum sensing system, particularly autoinducerproduction. If nutrient-dependent AHL production were taken into ac-count, this process would add an additional complexity to our model.

It should also be mentioned that microcolony, or “patchy” structures,are representative of the first stage of biofilm development [11, 36]. Ac-cording to experimental and theoretical considerations, microcoloniesare the structures where quorum sensing primarily occurs, so the inves-tigation of quorum sensing using this structure is realistic. However, intime, microcolonies grow and may eventually merge. In such a homoge-neous thick biofilm, flow would less disturb inter-colony communications,resulting in a more homogeneous response by the biofilm as a whole.

5 Conclusion The model of biofilm growth and quorum sensingthat was presented here is the first mathematical model for two-dimen-sional patchy biofilm colonies with flow and nutrient-dependent growthincluded. It allows for the study of how AHL produced in one colonycan have a non-local effect on neighbouring colonies, as a consequenceof the fact that AHL produced by the bacteria is transported by con-vection and diffusion in the aqueous phase. Two flow mechanisms weretested, regular flow, and bioclogging, in which flow velocities decreasedwith biomass accumulation in the channel. We focused on slow flow-


ing systems in narrow conduits, mimicking in particular the situationin pore spaces in soils and fractured media. Moreover, while faster flowconditions would imply a quick washout of AHL molecules, the flowrange chosen here is in the range where the effects of AHL productionand convective transport intricately interact. Environmental conditionssuch as substrate availability and mass transfer clearly show an effect onlocal autoinduction. The key findings from the numerical simulationsare:

• increased bulk substrate concentration directly resulted in increasedbiofilm growth and earlier induction times,

• inter-colony communication: production of AHL from upstream colo-nies caused induction of downstream colonies. The exact locationwhere initial induction occurred depended on the combined effect ofnutrients and flow conditions,

• mass transfer processes influence quorum sensing: induction does notnecessarily occur first for the largest colonies. Our induction patternsexemplify the efficiency sensing concept, in which colonies sense acombination of spatial distribution, cell density, and mass transferlimitations,

• increased flow velocities resulted in greater biofilm growth, but not innoteworthy induction time variation. At a sufficiently high flow rate,biofilm growth is enhanced, yet AHL concentrations are reduced tothe point that quorum sensing induction does not occur,

• hydrodynamics had significant spatial impacts: for high flow veloci-ties, convective transport of AHL was dominant over diffusion, leadingto increased biofilm growth but also increased AHL washout down-stream. For low flow velocities, convective and diffusive AHL trans-port were balanced, allowing initial induction to occur further up-stream, and a greater accumulation of upregulated biomass and AHLin the mid-channel,

• both flow mechanisms produced results that were qualitatively verysimilar.

Acknowledgements This research was supported by the AdvancedFoods and Materials Network (AFMNET), a Network of Centers of Ex-cellence. M.R.F. also acknowledges the support received from an OntarioGraduate Scholarship in Science and Technology, and the Natural Sci-ence and Engineering Research Council (NSERC) through an AlexanderGraham Bell Canada Graduate Scholarship. The computing equipmentused in this study was funded by the Canada Foundation for Innova-


tion (CFI) with a Leaders Opportunity Fund award made to H.J.E. Wethank SHARCNET for the operation and maintenance of this computer.

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Corresponding author: M. Frederick

Department of Mathematics and Statistics, University of Guelph,

50 Stone Rd E, Guelph ON Canada N1G 2W1.

E-mail address: [email protected]

Center of Mathematical Sciences, Technical University of Munich,

Boltzmannstrasse 3, 85748 Garching, Germany.

Institute of Biomathematics and Biometry, Helmholtz Center Munich,

Ingolstaedter Landstrasse 1, 85764 Neuherberg, Germany.

Center of Mathematical Sciences, Technical University of Munich,

Boltzmannstrasse 3, 85748 Garching, Germany.

Department of Mathematics and Statistics, University of Guelph,

50 Stone Rd E, Guelph ON Canada N1G 2W1.

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A MATHEMATICAL MODEL OF QUORUM SENSING IN PATCHY … · QUORUM SENSING IN BIOFILMS 269 [11, 52], a comprehensive understanding of quorum sensing systems is highly desirable. 1.3 Modelling