Bull Math BiolDOI 10.1007/s11538-017-0259-4

ORIGINAL ARTICLE

Three-Dimensional Numerical Simulations of BiofilmDynamics with Quorum Sensing in a Flow Cell

Jia Zhao1,2 · Qi Wang1,3,4

Received: 14 July 2016 / Accepted: 15 February 2017© Society for Mathematical Biology 2017

Abstract We develop a multiphasic hydrodynamic theory for biofilms taking intoaccount interactions among various bacterial phenotypes, extracellular polymeric sub-stance (EPS), quorum sensing (QS) molecules, solvent, and antibiotics. In the model,bacteria are classified into down-regulated QS, up-regulated QS, and non-QS cellsbased on their QS ability. The model is first benchmarked against an experimentyielding an excellent fit to experimental measurements on the concentration of QSmolecules and the cell density during biofilm development. It is then applied to studydevelopment of heterogeneous structures in biofilms due to interactions of QS regula-tion, hydrodynamics, and antimicrobial treatment. Our 3D numerical simulations haveconfirmed that (i). QS is beneficial for biofilm development in a long run by building arobust EPS population to protect the biofilm; (ii). biofilms located upstream can induceQS downstream when the colonies are close enough spatially; (iii). QS induction maynot be fully operational and can even be compromised in strong laminar flows; (v).the hydrodynamic stress alters the biofilm morphology. Through further numericalinvestigations, our model suggests that (i). QS-regulated EPS production contributesto the structural formation of heterogeneous biofilms; (ii) QS down-regulated cells

B Qi Wangqwang@csrc.ac.cn

Jia Zhaozhaojia@email.unc.edu

1 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

2 Department of Mathematics, University of North Carolina at Chapel Hill,Chapel Hill, NC 27599, USA

3 Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China

4 School of Materials Sciences and Engineering, Nankai University, Tianjin 300071,People’s Republic of China

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tend to grow at the surface of the biofilm while QS up-regulated ones tend to grow inthe bulk; (iii) when nutrient supply is sufficient, QS induction might be more effectiveupstream than downstream; (iv) QSmay be of little benefit in a short timescale in termof fighting against invading strain/species; (v) the material properties of biomass (bac-teria and EPS) have strong impact on the dilution of QS molecules under strong shearflow. In addition, with thismodeling framework, hydrodynamic details and rheologicalquantities associated with biofilm formation under QS regulation can be resolved.

Keywords Biofilms · Quorum sensing phase field · Hydrodynamics

1 Introduction

Bacteria are ubiquitous in nature as well as in our daily life. In general, bacteria usuallydo not exist as isolated cells but rather live in organized communities. These bacterialcommunities, where bacteria are less motile and are glued together by extracellu-lar polymeric substance (EPS), are called biofilms. It is commonly perceived by themedical community that biofilms are responsible for many diseases or ailments asso-ciated with chronic infections, supported by the survey data that biofilms are presenton the removed tissue of 80% of patients undergoing surgery for chronic sinusitis(Sanclement et al. 2005). Unlike a planktonic (free-swimming) bacterium, bacteria inbiofilms appear to be more adaptive to the stressful environment; for instance, theyare more tolerable to antimicrobial agents and hydrodynamic stress. As the result,biofilms are always hard to be eradicated by standard antimicrobial treatment (Lewis2010), which perhaps explains the tendency for relapse of chronic diseases or ailmentscaused by biofilms.

One of the main features of biofilms, which makes bacteria in biofilms functioningquite different from planktonic bacteria, is that bacteria living in biofilms are observedto communicate, and furthermore, cooperate with each other by secreting solution-soluble signaling molecules known as autoinducers (Miller and Bassler 2001). Forgram-negative bacteria, the particular molecules are acyl homoserine lactones (AHLs)(Whitehead et al. 2001). Experimental results have shown that bacteria can sense theconcentration of the signaling molecules secreted by bacteria in the surrounding andfunction cooperatively to accomplish certain tasks when the concentration of autoin-ducers reaches a threshold value (Waters and Basseler 2005). This phenomenon iscommonly known as quorum sensing (QS). The signaling molecule or the autoin-ducer is therefore called the quorum sensing molecule. This cell–cell communicationmechanism via quorum sensing has been observed in many bacterial systems, such asvibrio fischeri (Nealson et al. 1970), where quorum sensing was first observed, Pseu-domonas aeruginosa, Escherichia coli and so on. This mechanism allows unicellularmicroorganism to perform behavior like multicellular organism.

Quorum sensing has been shown to be responsible for mediating a variety ofsocial activities in biofilms, which include the secretion of diverse by-products,biofilm growth (Chopp et al. 2002, 2003; Garcia-Aljaro and Melado-Rovira 2012),swarming motility and virulence gene expressions (Quinones et al. 2005), biofilm dis-persion (Cárcamo-Oyarce et al. 2015; Solano et al. 2014), and antimicrobial resistance

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(Thompson et al. 2015). In particular, the phenomenon of quorum sensing regulatingEPS production during biofilm formation has been widely reported, for instance inMarketon et al. (2003), Vuong et al. (2003), Waters and Basseler (2005).

EPS can be treated as the so-called public good since its a shared resource that eachindividual bacterium can take advantage of, even if it is produced by other bacteria.However, how quorum sensing regulation affect the EPS production is not clearlyresolved. Besides, in some bacterial systems, like Vibrio cholerae, the EPS productionis suppressed when the density of autoinducer reaches its threshold value (HeithoffandMahan 2004), whereas inPseudomonas aeruginosa, the EPS production would befacilitated once the quorum sensing is activated (Sakuragi and Kolter 2007). In addi-tion, as the phenomenon involves transport of signaling molecules, the heterogenousstructure in biofilm colonies and the solvent–biomass interaction cannot be ignored,as they can affect the convection and diffusion rates of the signaling molecules trans-ported as well as the morphology of biofilm colonies. This motivates us to develop afull 3Dmathematical model to study how quorum sensing regulates biofilm formationand development as well as the pros and cons of quorum sensing in an aqueous envi-ronment where hydrodynamic interaction between the various biofilm componentsand the ambient fluid flow is important.

The induction of quorum sensing is usually observed at high bacterial cell vol-ume fractions, where the concentration of QS molecules is accumulated to a certainthreshold value. However, a high bacterial cell density does not necessarily inducequorum sensing since many other factors have been observed to contribute to thisinduction such as nutrient supplies, hydrodynamic shear stress (Kirisits et al. 2007,)and the PH environment. Intuitively, a hydrodynamic flow may have a diluting effecton the signaling molecules, which then lead to a delay or even failure of QS inductionby preventing QS molecules, like AHLs, from reaching a certain threshold value. Forinstance, in Kirisits et al. (2007), the authors claim that the amount of biomass requiredfor the full QS induction of the population increases as the flow rate increases. On theother hand, coming with the inflow solvent, more essential substances, such as nutri-ent which enhances the production of QS molecules by bacteria, would be fresh in;thus hydrodynamic flowmay lead to higher production of the signaling molecules andthereby facilitate QS induction. Little work has been done to analyze the correlations,not to mention taking the heterogenous structure factor of biofilms into account. Thisadequately serves as another motivation for us to develop a full 3D hydrodynamicbiofilm model to study the hydrodynamic effect on quorum sensing.

In the literature, related with quorum sensing, some other factors have also beensingled out for possible impact to biofilm development. For instance, recently, it hasbeen verified that, except for cooperation, there probably exist cheaters in the bacte-rial population which exploit the signaling molecules produced by others (Popat et al.2012). In Fauvart et al. (2012), the authors observed that quorum sensing coordinatedsecretion of rhamnolipid acts as a surfactant, which leads to finger pattern formationin swimming bacteria aggregates. With the ever-increasing knowledge about quorumsensing, researchers areworking on developing potential applications by taking advan-tage of this cell–cell communication mechanism. For an overview of applications ofquorum sensing in biotechnology, readers are referred to Choudhary and Schmidt-Dannert (2010). One promising application is that it is probably more efficient to use

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anti-quorum sensing treatment than antibiotic agents in treating some biofilm infec-tions, which is likely to pose a selective pressure for the development of resistantmutants (Hentzer and Givskov 2003).

Inspired by experimental findings, researchers have come up with several math-ematical models to study biofilm formation and function, as well as related issues.Concerning coupling biofilm growth with quorum sensing features, several mathe-matical models have been developed. Ward et al. (2001) proposed an ODE systemmodeling quorum sensing in bacteria and later extended by Koerber et al. (2002) intoa 1DPDE system in space and time. The dependence of quorum sensing on the depth ofa growing biofilm is discussed in Chopp et al. (2003). In Nadell et al. (2008), the evolu-tion of quorum sensing in bacterial biofilms is simulated by an individual-basedmodel.A delay model (Barbarossa et al. 2010) is proposed by assuming a (not yet detected)enzyme, which degrades the autoinducer into an active form. In Perez-Velazquez et al.(2015), quorum sensing regulation and its heterogeneity in Pseudomonas syringae onleaves is studied by treating it as a nonnegative stochastic process. Recently, Maturet al. (2015) proposed a single-cell spatial model for quorum sensing. The inhibitionof quorum sensing is modeled by Fozard et al. (2012) as a stochastic process on thelevel of individual cells, claiming the time at which treatment is initiated is crucialfor the effective prevention of quorum sensing. Jabbari et al. (2012a) models it byreactive dynamics; the study is further extended in Jabbari et al. (2012b) with spa-tial diffusion. The hydrodynamic effects on quorum sensing induction have also beeninvestigated theoretically. A 2D partial differential equation model coupled with theStokes flow is proposed in Frederick et al. (2010) and later the authors extended thismodel (Frederick et al. 2011) and proved its well posedness in Sonner et al. (2011).Another 2D continuum model with Stokes background flow is proposed in Vaughanet al. (2010), where the shape of biofilms at the onset of hydrodynamic shear is studied.Recently, authors in Uecke et al. (2014) model quorum sensing with a backgroundflow by treating each bacteria as an individual. In Emerenini et al. (2015), the authorsstudy the quorum-sensing-induced biofilm detachment and later the model is furtheranalyzed in Emerenini et al. (2017). In Langebrake et al. (2014), the authors investi-gate QS inside colonies by studying the propagation of autoinducer waves, and somespatial–temporal patterns of autoinducers are also studied in Dilanji et al. (2012).

Despite the advances documented in the literature, there is very little work on3D hydrodynamics of biofilms in the ambient fluid environment available, whichtakes into account the spatial–temporal heterogeneous structure of biofilms as wellas the hydrodynamic flow effects. This is perhaps mainly because of computationalchallenges in resolving biofilms which are a truly 3D microorganism with highlyheterogeneous spatial–temporal structures and complex dynamics.

In this paper, we develop a new, full 3D hydrodynamic model for the mixture sys-tem consisting of biofilms and the ambient fluid, extending our previous phase-fieldmodels for biofilms of two and three effective components (Lindley et al. 2011; Zhanget al. 2008a). This model couples hydrodynamics of the biofilm–solvent mixture tothe quorum sensing activity, as well as dynamics of nutrient and antimicrobial agents.Using this model, we aim to investigate biofilm formation and development regulatedby quorum sensing in an aqueous environment under hydrodynamical flows and possi-bly antimicrobial treatment. A second-order numerical scheme in both time and space

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based on finite difference methods is devised to solve the governing system of partialdifferential equations in the model. The numerical solver for the discretized equationsis then implemented on graphical processing units (GPUs) in a 3D space and time.

The 3D numerical simulation tool is employed to investigate the interaction amongQS molecules and various biomass components under a hydrodynamic environment.Given the complexity of the problem, we select a couple of important case studies topresent in this paper. First of all, we benchmark the reactive kinetics with experimen-tal results on the concentration of QS molecules and the cell density during biofilmdevelopment. Then, we apply the full 3D simulation tool to investigate the interac-tion between hydrodynamics and quorum sensing regulation. Finally, we examinethe impact of hydrodynamic stresses on the biofilm morphology and biofilm materialproperties on QS molecule dilution under strong shear. The model and the accompa-nying computation tool are general enough to allow a wide spectrum of case studiesfor biofilm related issues, beyond what we report in this paper.

The rest of this paper is organized into four sections. In the second section, wepresent the detail of the hydrodynamic phase-fieldmodel for biofilms. Then, we designa linear numerical solver for the governing partial differential equations in the modelusing semi-implicit finite difference schemes and briefly discuss the implementationissue in the third section. In the fourth section, the numerical results and discussionsare presented. In the last section, we give a concluding remark for the study.

2 3D Hydrodynamic Model for Biofilms

In this section, we extend a previously developed two phase biofilm model (Zhanget al. 2008a) to include the effect of quorum sensing among the various bacterialphenotypes classified based on their quorum sensing abilities. The biofilm, consistingof bacteria, EPS, and solvent, is coarse-grained into a single fluid model with multi-components, in which different types of bacteria, EPS, as well as solvent are modeledeffectively as separate fluid components. Quorum sensing molecules, nutrients, andantibiotic agents are treated as phantom materials which exhibit chemical reactiveeffects without their mass and volume being considered. Treating substances of suchsmall molecules as phantom materials is an effective approximation to the complexmixture system given the little contribution from these materials to the transport ofthe total mass and volume of the entire material system.

2.1 Basic Notations

We study biofilms in a cubic domain � = [0, Lx ] × [0, Ly] × [0, Lz], withLx , Ly, Lz the length in x, y, z directions, respectively. Following our previous nota-tional approaches (Zhang et al. 2008a), we denote the volume fraction of the effectivesolvent and biomass as φs and φn , respectively. In biomass, we further divide it intoEPS and bacteria, whose volume fractions are denoted by φp and φb, respectively, i.e.,φn = φp + φb. The incompressibility condition of the mixture system requires

φs + φp + φb = 1. (1)

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Let ρb, vb, ρp, vp and ρs, vs be the density and velocity of the bacteria, EPS andsolvent, respectively. Then the volume-averaged density and velocity for the mixtureare denoted as

ρ = φbρb + φpρp + φsρs,

v = φbvb + φpvp + φsvs . (2)

In the biofilmmodel, we classify the bacteria into four different phenotypes accord-ing to their responses to quorum sensing (QS) molecules (Frederick et al. 2011; Nadellet al. 2008): QS down-regulated bacteria (QS−), QS up-regulated bacteria (QS+), non-QSbacteria (non-QS), aswell as the dead bacteria, whose volume fractions are denotedbyφb1, φb2, φb3 andφb4, respectively. Hence, the volume fraction of the bacteria couldbe written as

φb = φb1 + φb2 + φb3 + φb4. (3)

We note that since the bacteria in biofilms are less motile (Guttenplan and Kearns2013), we treat all bacteria having the same kinematics. In biofilms, QS− are thebacteria, which can convert into QS+ once the concentration of QS molecules reachesa threshold value (Frederick et al. 2011; Nadell et al. 2008) whereas QS+ are thebacteria, which can convert into QS− when the concentration of QSmolecules is lowerthan the threshold. In another word, QS− bacteria belong to the wild phenotype of thebacteria and QS+ bacteria represent the phenotype which has bonded QS moleculesand certain activated gene transcription for enhanced EPS production and thereforereacts differently from its original phenotype.Non-QSbacteria represent those bacteriawhich are not affected by the concentration of quorum sensing molecules but may beregulated by other proteins in the biofilm. These non-quorum sensing bacteria can beregarded a phenotype, which do not prefer a social relation with other bacteria, or theycan also be treated as a different type of bacteria, such as invaders. Dead bacteria arethose which lose their vitality but are still attached in the biofilm.

In addition to the above bacterial phenotypes, we also keep track of the functionalcomponents whose volume are omitted in the model, but their chemical effects areretained as their mass fractions are much smaller than those of biomass and sol-vent. These include (i) nutrients, (ii) antimicrobial agents and (iii) quorum sensingmolecules, whose concentrations are denoted by c, d and A, respectively.

We make a simplifying assumption here that all bacterial cells and EPS that theyproduce interact with solvent equally in the context of thermodynamics (i.e., entropicmotion andmixing). Thus the effective free energy density functional of the biologicalsystem is proposed as follows using the extended Flory–Huggin’s mixing free energydensity with an extra conformational entropy term (Cahn and Hilliard 1958; Coganand James 2004; Cogan and Keener 2005; Zhang et al. 2008a; Zhao et al. 2016a, b)

f = γ1|∇φn|2 + γ2

[φn

Nlogφn + φs logφs + χφnφs

], (4)

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where N is an effective polymerization index for the non-solvent biomass compo-nents, γ1 parametrizes the strength of the conformational entropy, γ2 parameterizesthe strength of the bulk interaction potential, χ is the mixing coefficient. We remarkthe mixing coefficient χ may depend on φb and φp. Here we treat it as constant, untilfurther experiment data are accessible.

2.2 Transport Equations for Biomass Components

With the free energy, the transport equation for each bacterial type is assumed to be gov-erned by a reactive Cahn–Hilliard-type equation (Cahn andHilliard 1958; Cahn 1959)

∂φbj

∂t+ ∇ · (vφbj ) = ∇ · (λφbj∇μ j

) + gbj , j = 1, 2, 3, 4, (5)

where λ is the mobility parameter, μ j = δ fδφbj

is the chemical potential with respectto bacterial type j , and gbj is the reactive rate for cell type j.

The reactive kinetics of these cell types include regulated growth due to nutrient,the phenotype conversion between the QS down-regulated and QS up-regulated cells,and decay due to antimicrobial agents as well as natural causes (death). We considerboth mechanisms of logistic growth dynamics for the bacteria (Almeida et al. 2012;Perez-Velazquez et al. 2015) and the Monod growth kinetics due to nutrient supply(Frederick et al. 2011). The inclusion of the logistic growth in this model is based onour assumption that the bacterial growth slows downwhen the spacing among bacteriais squeezed, especially in a confined geometry. Mathematically, this assumption elim-inates the potentially unbounded growth locally without the mechanism. Should wecontrol the carrying capacity φ∗

b as a large number, it would not qualitatively changethe model prediction compared with the linear growth model in most cases. The QSdown-regulated cells will produce the same phenotype, while the QS up-regulatedcells will produce β fraction of QS down-regulated cells and (1 − β) fraction of QSup-regulated cells (Ward et al. 2001). For each phenotype, we consider their deathrates due to natural causes, in which the decay rate for the dead cells is interpreted asthe dissolving or conversion rate of the cell into solvent. Between the quorum sensingcells, there exists the up and down regulation coordinated by the concentration of thequorum sensing molecule AHL (Frederick et al. 2010, 2011; Nadell et al. 2008; Wardet al. 2001). The reactive kinetics for all cell types are given by the following

gb1 = (Cb1φb1 + βCb2φb2)c

kb + c

(1 − φb

φ∗b

)− r12An

τ n + Anφb1

+ r21τ n

An + τ nφb2 − r1φb1 − Cd1d

kd1 + dφb1,

gb2 = (1 − β)Cb2c

kb + cφb2

(1 − φb

φ∗b

)+ r12An

An + τ nφb1

− r21τ n

An + τ nφb2 − r2φb2 − Cd2d

kd2 + dφb2,

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gb3 = Cb3c

kb + cφb3

(1 − φb

φ∗b

)− r3φb3 − Cd3d

kd3 + dφb3,

gb4 =∑3

i=1riφbi +

∑3

i=1

Cdid

kdi + dφbi − r4φb4, (6)

where the Cbi are the growth rates, β represents the percentage of QS+ bacteria thatproduces QS− bacteria (Ward et al. 2001), Cdi is the decay rate due to the antimi-crobial treatment (Cogan 2012), ri is the natural death rate (Frederick et al. 2010),kbi , kdi are the half saturation constants in the respective Monod model for each bac-terial phenotype, respectively, where i = 1, 2, 3. Here r4 is the natural conversion ratefrom dead bacteria into solvent due to cell lysis (Bayles 2007). And φ∗

b is the carryingcapacity of the bacterial volume fraction. In addition, the conversion rate betweenQS+and QS− are modeled by r12An

τ n+An and r21τ n

τ n+An , with τ the threshold AHL concentrationlocally required for quorum sensing induction to occur (Frederick et al. 2011) andthe exponent n the degree of polymerization in the synthesis of AHL (Frederick et al.2011). While the concentration of AHL gets bigger, quorum sensing induction gradu-ally takes place to increase the up-regulated bacterial cell population and decrease thedown-regulated bacterial cell population. The non-QS bacterial cells are not affectedby the quorum sensing molecules by definition. In this paper, we assume β = 1 tohypothesize that QS-regulated cells are born all as QS down-regulated.

The transport equation for EPS (φp), which is the product of the live bacteria, isproposed as a reactive Cahn–Hilliard-type equation as well and is given by,

∂φp

∂t+ ∇ · (vφp) = ∇ · (λφp∇μp) + gp, (7)

where μp = δ fδφp

is the chemical potential with respect to φp, and gp is the reactiveterm proposed as follows

gp =3∑

i=1

Cpi c

kpi + cφbi

(1 − φp

φ∗p

)− rpφp − Cdpd

kdp + dφp. (8)

Here Cpj is the EPS production rate with respect to the jth type cell, j = 1, 2, 3,facilitated by the nutrient (Zhang et al. 2008a). φ∗

p gives the upper bound for the EPSproduction. We assume that some EPS can be dissolved into solvent naturally withrate rp due to cell lysis (Bayles 2007) and accelerated by drugs with the rate given byCdp.kpi and kdp are the half saturation constants for the Monod models used.

2.3 Transport Equations for the Functional Components

There are three functional components in our biofilm model, namely, nutrient, antimi-crobial agents, and quorum sensingmolecules, whose concentrations are denoted by c,d and A, respectively. Thesemolecules all have smallmolecular weight comparedwith

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the components of the biomass, instead of tracking their volumes, we track their con-centrations, which are governed by convection–diffusion–reaction equations. Thesemolecules are transported with solvent, as they are dissolved in it.

Specifically, the transport equation for the concentration of the quorum sensingmolecule AHL (A) is given by

∂(φs A)

∂t+ ∇ · (φsvA) = ∇ · (φs Da∇A) + ga, (9)

where Da is the diffusion coefficient and ga is the reactive term given by

ga = −ra(φb1 + φb2)A +(

2∑i=1

(rai + Cai An

τ n + An

)φbi

) (1 − A

A∗

)− Cad

kda + dA.

(10)

Here ra is the decay rate of the AHL concentration due to binding with QS bacterialcells (QS receptors) (Waters andBasseler 2005), ra1 and ra2 are the background growthrate due to the down-regulated and up-regulated quorum sensing cells, respectively.Ca1 and Ca2 are growth rates, taking into account the effect that the concentration ofAHL would affect the productivity of AHL by the bacteria positively (Fekete et al.2010). Here A∗ gives the upper bound of solubility for AHL (Abraham 2006). Ca

is the decay rate of AHL due to antimicrobial agents and kda is the half saturationconstant. The diffusion coefficient is given by following the Hinson model (Hinsonand Kocher 1996),

Da = Da01 − φb

1 + 12φb

φs

φs + φpDpr

, (11)

where the first term on the right-hand side represents the reduction effect due to thepresence of bacteria while the second term is an empirical fitting accounting for thereduction effect due to the presence of EPS. Here Dpr is a model parameter fittedexperimentally.

The transport equation for nutrient is given as follows

∂φsc

∂t+ ∇ · (φsvc) = ∇ · (Dcφs∇c) + gc, (12)

where Dc is the diffusion coefficient for nutrient transport and the reactive rate is givenby

gc = −3∑

i=1

rbc

kbi + cφbi , (13)

which represents the nutrient consumption rate due to bacterial metabolism (Fredericket al. 2010; Zhao et al. 2016a), representing the nutrient consumption due to bothbacterial cell duplications and EPS production. For the diffusion rate of the nutrient(Dc), which is primarily oxygen in this model, Stewart (Stewart 1996) suggests it to

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be around 75% than that in the pure solvent. Since the molecular weight of oxygen isrelatively small, it is supposed to penetrate EPS and the membrane of cells equally.So we set the diffusion rate of oxygen in biofilms as follows

Dc = Dc01 − (φb + φp)

1 + 12 (φb + φp)

, (14)

where the second term in the expression represents the reduction in diffusion due tothe presence of the biomass.

Finally, the transport equation for antibiotic agents is proposed as follows

∂(φsd)

∂t+ ∇ · (φsvd) = ∇ · (Ddφs∇d) + gd , (15)

where Dd is the diffusion rate of antimicrobial agents and

gd = − rdd

kd + d

(4∑

i=1

φbi + φp

)− rdd, (16)

is the reactive term. Antibiotics (or antimicrobial agents) are assumed to be consumedby all bacterial cells and EPS at the same rate. Here rd represents the natural decayrate of antimicrobial agents (Shen et al. 2016). For the diffusion rate Dd , we also adopta Hinson type model (Hinson and Kocher 1996)

Dd = Dd01 − φb

1 + 12φb

φs

φs + φpDpr

. (17)

We remark that, in the development of these transport equations, we try to includeas many biological and physical effects as possible, which result in a large set ofmodel parameters. Some of them can be measured or found in the literature, whileothers are obtained either using our best guesses or calibrated against available exper-iments. Nevertheless, all these effects included in the model are present in the biofilmsystem.

2.4 Momentum and Continuity Equation

In order to couple the biofilm components to hydrodynamics, we need the governingequation for the averaged velocity v. We assume the fluid mixture as solenoidal andimpose the balance of linear momentum as follows

ρ

(∂v∂t

+ v · ∇v)

= −∇ p + ∇ · τ − γ1∇ · (∇φn ⊗ ∇φn), (18)

∇ · v = 0, (19)

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Three-Dimensional Numerical Simulations. . .

where ρ is the averaged density, p is the hydrostatic pressure, τ is the extra stress, andthe last term is the interfacial force due to the inhomogeneity of biomass distribution,derived from a variational principle (Chen 2002; Zhao et al. 2016d). In this paper, wepropose the mixture as an extended Newtonian fluid, then the extra stress tensor isgiven by

τ = 2ηD, η = φbηb + φpηp + φsηs, (20)

where ηb, ηp and ηs are the viscosity of bacteria, EPS and solvent. and D = 12 (∇v +

∇vT ) is the rate of strain tensor. EPS is viscoelastic, whose relaxation time rangesfrom few seconds to over 10min. When we study biofilm growth, the timescale ismuch larger than the EPS relaxation time. Therefore, treating the biomass in biofilmsas an extended Newtonian is a reasonable approximation.

2.5 Boundary Conditions

In order to simulate biofilms in a fixed domain, we need to impose boundary conditionsfor the governing partial differential equations, which mimic common situations forbiofilms being cultured and observed in vitro. In this paper, we focus on simulatingbiofilm flows in two special geometries: in a petri dish or a flow cell in 3D space andtime.

To mimic the biofilm development in a petri dish, both x and z directions areassumed periodic. At boundaries in the y direction, no-flux boundary conditions areimposed for the biomass and functional components and no-slip boundary conditionsare imposed for the velocity,

(ivsφs − Diφs∇i) · n|y=0,Ly = 0, i = c, d, A,

∇φi · n|y=0,Ly = 0, i = b1, b2, b3, b4, p,(vφi − λφi∇ δ f

δφi

)· n|y=0,Ly = 0, i = b1, b2, b3, b4, p, (21)

where n is the unit outer normal.We also impose a nutrient feeding condition c|y=Ly =c0 in place of the zero-flux condition wherever there is a steady supply of nutrientthrough the boundary at y = Ly .

For the case of biofilms in a finitely long water channel, which we call it a flowcell, periodic boundary conditions are imposed in the z direction, and y direction isbounded by solid walls, as Eq. (21). The channel is finite in the x direction. So, thereexist the inlet–outlet boundary conditions. The inlet velocity at x = 0 is given by

v0 = (p0y(1 − y), 0, 0) , (22)

where p0 is a prescribed pressure gradient. Suppose that solvent has already reachedequilibrium when flowing out of the cell at x = Lx , we impose vx = 0 at the outletboundary. For the nutrient concentration c, we prescribe boundary condition

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J. Zhao, Q. Wang

c|x=0 = c0(y), cx |x=Lx = 0. (23)

For QS molecules A, we prescribe boundary condition

Ax |x=0,Lx = 0. (24)

Here we use the Dirichlet boundary condition for nutrient concentration c at the inflow,because fresh nutrient supply is provided with the inflow. For QS molecules, we didnot consider any situation such as injecting A into the channel. Thus, we propose theflux-free boundary condition for A at the inflow, assuming that there are no biofilmsupstream.

For the biomass components, we impose no-flux boundary conditions in the xdirection, which applies to the situation where no biomass is flown in and out of theflow cell,

(vφi − λφi∇ δ f

δφi

)· n|x=0,Lx = 0, j = b1, b2, b3, b4, p. (25)

2.6 Non-dimensionalization

We denote t0 as the reference timescale, h the reference length scale, and c0, d0 and A

the characteristic concentration of the nutrient, antimicrobial agents and quorum sens-ing molecules, respectively. Then, we non-dimensionalize the variables as follows,

t = t

t0, x = x

h, v = vt0

h, τ = τ t20

ρ0h2, p = pt20

ρ0h2, c = c

c0, d = d

d0, A = A

A .

The following dimensionless parameters then emerge,

� = λρ0

t0, �1 = γ1kT t20

ρ0h4, �2 = γ2kT t20

ρ0h2, Res = ρ0h2

ηs t0,

Rep = ρ0h2

ηpt0, Reb = ρ0h2

ηbt0, ρ = φs

ρs

ρ0+ φn

ρn

ρ0,

Cbi = Cbi t0, rbi = rbi t0, Cdi = Cdi t0, Cai = Cai t0,

ri = ri t0, rdi = rdi t0, i = 1, 2, 3,

˜kbi = kbic0

, kdi = kdid0

, kd = kdd0

, kai = kaiA0

, k pi = kpic0

, i = 1, 2, 3,

rcp = rcpt0, kdp = kdpd0

, kcp = kcpc0

, Di = Di t0h2

, i = c0, d0, a0.

123

Three-Dimensional Numerical Simulations. . .

For simplicity, we drop the symbol , and the non-dimentionalized equations in thehydrodynamic model for biofilms are summarized as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(

∂v∂t + v · ∇v

) = ∇ · (φbτb + φpτp + φsτs) − ∇ p − �1∇ · (∇φn∇φn),

∇ · v = 0,∂φbj∂t + ∇ · (vφbj ) = ∇ · (�φbj∇μ j ) + gbj , j = 1, 2, 3, 4,

∂φp∂t + ∇ · (vφp) = ∇ · (�φp∇μp) + gp,

∂φs c∂t + ∇ · (φsvc) = ∇ · (Dcφs∇c) + gc,

∂φs A∂t + ∇ · (φsvA) = ∇ · (Daφs∇A) + ga,

∂φsd∂t + ∇ · (φsvd) = ∇ · (Ddφs∇d) + gd ,

(26)

where the reactive terms are given by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

gb1 = (Cb1φb1 + βCb2φb2)c

kb+c

(1 − φb

φ∗b

)− r12A2

τ 2+An φb1 + r21τ n

An+τ nφb2

−(r1 + Cd1d

kd1+d

)φb1,

gb2 = (1 − β)Cb2ckb+cφb2

(1 − φb

φ∗b

)+ r12An

An+τ nφb1 − r21τ n

An+τ nφb2 −

(r2 + Cd2d

kd2+d

)φb2,

gb3 = Cb3ckb+cφb3

(1 − φb

φ∗b

)−

(r3 + Cd3d

kd3+d

)φb3,

gb4 = ∑3i=1

(ri + Cdi d

kdi+d

)φbi − r4φb4,

gp = ∑3i=1

Cpi ckpi+cφbi

(1 − φp

φ∗p

)−

(rp + Cdpd

kdp+d

)φp,

ga = −ra A + ∑2i=1

(rai + Cai An

τ n+An

)φbi

(1 − A

A∗) − Cad

kda+d A,

gc = −∑3i=1

rbckbi+cφbi ,

gd = − rd0dkd+d

(∑4i=1 φbi + φp

)− rdd.

(27)

2.7 Numerical Methods

For the coupled PDEs in the governing system of equations, we devise a second-ordernumerical method to solve them. We note that, in the following notation, any variablewith overline represents the second-order extrapolation. In the following, we denotethe extrapolated data using over-lines; for instance, vn+1 = 2vn − vn−1.

Given the initial condition (v0 = 0, s0 = p0 = 0, φ0bj , φ

0p, c

0, d0, A0), we first

compute (v1, s1, p1, φ1bj , φ

1p, c

1, d1, A1) by a first-order scheme. Having computed

(vn−1, sn−1, pn−1, φn−1bj , φn−1

p , cn−1, dn−1, An−1) and (vn, sn, pn, φnbj , φ

np, c

n,

dn .An),wheren ≥ 2,we calculate (vn+1, sn+1, pn+1, φn+1bj , φn+1

p , cn+1, dn+1, An+1)in the following steps,

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J. Zhao, Q. Wang

1. Prediction:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ρn+1[3un+1−4vn+vn−1

2δt

]+ ρn+1vn+1 · ∇vn+1 + 1

2 (∇ · (ρn+1vn+1))vn+1,

+ 1Res

∇sn + ∇ pn − 1Rea

∇2un+1 = Rn+1 − 1

Rea∇2vn+1,

un+1 · n|y=0,Ly = 0,

un+1|x=0 = v0, un+1x |x=Lx = 0,

2. Projection:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∇ ·(

1ρn+1 ∇ψn+1

)= ∇ · un+1,

∂ψn+1

∂n |y=0,Ly = 0,

∂ψn+1

∂n |x=0,Lx = 0,

3. Correction:

⎧⎪⎪⎨⎪⎪⎩vn+1 = un+1 + 1

ρn+1 ∇ψn+1,

sn+1 = sn − ∇ · un+1,

pn+1 = pn − 3ψn+1

2δt + 1Rea

sn+1,

with

Rn+1 = −�1∇2φ

n+1n ∇φ

n+1n + ∇ ·

(φn+1n τ n+1

n + φn+1s τ n+1

s

),

and

1

Rea= φb,max

Reb+ φp,max

Rep+ 1 − φp,max − φb,max

Res,

where Reb , Rep and Res are theReynolds numbers for bacteria, EPSand solvent,respectively.

4. Update biomass (φbj , φp)

⎧⎪⎨⎪⎩

3φn+1bj −4φn

bj+φn−1bj

2δt + ∇ · (φn+1bj vn+1) = ∇ ·

[�φ

n+1bj F

(∑4i=1 φn+1

bi + φn+1p

)]+ gn+1

bi ,

3φn+1p −4φn

p+φn−1p

2δt + ∇ · (φn+1p vn+1) = ∇ ·

[�φ

n+1p F

(∑4i=1 φn+1

bi + φn+1p

)]+ gn+1

p ,

(28)

123

Three-Dimensional Numerical Simulations. . .

where j = 1, 2, 3, 4 and

gn+1b1 =

[Cb1cn+1

kb + cn+1

(1 − φb

n+1

φb,max

)− r1 − r12(A

n+1)2

τ 2 + (An+1

)2− Cd1d

n+1

kd1 + dn+1

]φn+1b1

+ r21τ 2

τ 2 + (An+1

)2φb2

n+1 + βCb2cn+1

kb + cn+1

(1 − φb

n+1

φb,max

)φb2

n+1

gn+1b2 =

[(1 − β)Cb2cn+1

kb + cn+1

(1 − φb

n+1

φb,max

)− r2 − r21τ 2

τ 2 + (An+1

)2− Cd2d

n+1

kd2 + dn+1

]φn+1b2

+ r12(An+1

)2

τ 2 + (An+1

)2φb1

n+1,

gn+1b3 =

[Cb3cn+1

kb + cn+1

(1 − φb

n+1

φb,max

)− r3 − Cd3d

n+1

kd3 + dn+1

]φn+1b3 ,

gn+1b4 =

3∑i=1

[ri + Cdid

n+1

kd3 + dn+1

]φn+1bi − r4φ

n+1b4 ,

gn+1p =

3∑i=1

Cpi cn+1

kpi + cn+1 φn+1bi

(1 − φn+1

p

φp,max

)+ αr4φ

n+1b4 − rpφ

n+1p

− Cdpdn+1

kdp + dn+1 φn+1

p , (29)

where

F(φ) = �2

(1

N

1

φn+1n + ε

+ 1

1 − φn+1n

− 2χ

)∇φ − �1∇∇2φ.

5. Update functional components:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

3φn+1s cn+1−4φn

s cn+φn−1

s cn−1

2δt + vn+1 · ∇ (cn+1φn+1

s

) = ∇ · (Dn+1s φn+1

s ∇cn+1) + gn+1

c ,

3φn+1s dn+1−4φn

s dn+φn−1

s dn−1

2δt + vn+1 · ∇ (dn+1φn+1

s

) = ∇ · (Dn+1e φn+1

s ∇dn+1) + gn+1

d ,

3φn+1s An+1−4φn

s An+φn−1

s An−1

2δt + vn+1 · ∇ (An+1φn+1

s

) = ∇ · (Dn+1a φn+1

s ∇an+1) + gn+1

a .

(30)

where the discrete schemes for reactive terms are summarized below

gn+1a = −ra A

n+1 +(

2∑i=1

(rai + Cai An

kai + An

)φn+1bi

) (1 − A

n+1

Amax

)

− Cadn+1

kda + dn+1 A

n+1,

123

J. Zhao, Q. Wang

gn+1c = −

3∑i=1

rbi cn

kb + cnφn+1bi −

3∑i=1

rpi cn

kpi + cnφn+1bi ,

gn+1d = − dn

kd + dn

(4∑

i=1

rdiφn+1bi + rdpφ

n+1p

)− rdd

n+1. (31)

The schemes above are presented as semi-discrete. The spatial discretization iscarried out using central finite differences (Zhao et al. 2016c, e, 2017). The boundaryconditions are discretized using the biased differencing at the solid boundaries. Thisscheme is implementation on GPUs using CUDA. We use a preconditioned BiCGsolver (using the package CUSP 2012) to solve the sparse linear system generatedfrom the full discretization using finite difference methods, where the preconditioneris constructed as the inverse of a good approximation of the implicit operator byapproximating the variable coefficient with a constant coefficient. The preconditionercan then be computed using the fast Fourier transform. The data are saved as HDF5format, which can be visualized usingVisIt software (Childs et al. 2012). Currently, thesolver can deal with up to 2563 meshes in space, limited by the global memory size ofa single GPU. The scheme is theoretically second order in space and time. Numericalconvergence tests have been conducted and the convergence rate is confirmed.

3 Results and Discussion

The hydrodynamic model of biofilms developed is applied to study dynamic prob-lems involving biofilms immersed in a viscous solvent. In the following, we applyit to investigate a few specific cases related to quorum sensing in biofilm formationand development, which include the effect of the threshold for quorum sensing induc-tion and its influence on biofilm formation, and the hydrodynamical effect on thedistribution of autoinducers and its consequence to biofilm development.

As alluded to earlier, the model has many parameters, and some of them can becalibrated via experiments, while others are either obtained from the literature or fromour best guesses. For instance, the washout rates in the model, r1, r2, r3, r4, rp, ra andrd are normally small; so, we set them into zero in the following case studies. For themonod constants (the k’s), we choose a common value 3.5 × 10−4 kg/m3 as used inAlpkvist et al. (2006). All the other parameters used in this paper are summarized inTable 1, unless mentioned otherwise.

3.1 Reactive Kinetics

We begin with the study on reactive kinetics in this model to elucidate bulk reactionsfor each component on two important mechanisms during biofilm development. Byneglecting hydrodynamics, diffusive effects, and the drug effect in the model, andassuming nutrient is sufficiently supplied (setting c = 1), we arrive at the governingequations for reactive kinetics in biofilms,

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Three-Dimensional Numerical Simulations. . .

Table 1 Parameters and their dimensional values

Symbol Description Value Unit Source

h Characteristic length scale 1 × 10−3 m Picioreanu et al. (1998)

t0 Characteristic timescale 1000 s Zhang et al. (2008a)

c0, d0 Characteristic massconcentration

8.24 × 10−3 kg/m3 Picioreanu et al. (1998)

A AHL molar solubility 4 × 10−4 mol/m3 Quinones et al. (2004)

ρ0 Characteristic density 1.0 × 103 kg/m3 Zhang et al. (2008a)

γ1 Distortional energycoefficient

8 × 106 m−1 Zhang et al. (2008a)

γ2 Mixing free energycoefficient

3 × 1017 m−3 Zhang et al. (2008a)

χ Flory–Huggins parameter 0.55 Zhang et al. (2008a)

λ Mobility parameter 1 × 10−9 kg−1 m3 s Zhang et al. (2008b)

N Generalized polymerization 1 × 103 Zhang et al. (2008a)

Dpr Hinson model constant 0.02 Lindley et al. (2011)

ηp , ηb Dynamic viscosity of biomass 10 kgm−1 s−1 Cogan (2011)

ηs Dynamic viscosity of solvent 1.002 × 10−3 kgm−1 s−1 Zhang et al. (2008b)

Dc0, Dd0 Nutrient and drug diffusioncoefficient

2.3 × 10−9 m2/s Zhang et al. (2008a)

Da0 AHL diffusion coefficient 4.9 × 10−10 m2/s Stewart (2003)

K Monod constant 3.5 × 10−4 kg/m3 Alpkvist et al. (2006)

Cb1,Cb2,Cb3 Bacteria growth rate 4 × 10−4 s−1 Roberts and Stewart (2005)

Cd1,Cd2,Cd3 Death rate due to drug 4 × 10−2 s−1 Cogan (2006)

rb, rd0 Consumption rate 4 × 10−2 m3/(kg · s) Alpkvist et al. (2006)

r12, r21 Transfer rate between φb1and φb2

4 × 10−3 s−1 Assumed

Cp1,Cp3 EPS production rate of φb1and φb3

1 × 10−7 s−1 Assumed

Cp2 EPS production rate of φb2 4 × 10−4 s−1 Kommedal et al. (2001)

ra1, ra2 Production rate of AHL 3.22 × 10−8 mol/(m3· s) Fekete et al. (2010)†

Ca1,Ca2 Activated production rate ofAHL

3.22 × 10−7 mol/(m3· s) Fekete et al. (2010)†

τ Threshold for quorumsensing induction

7 × 10−5 mol/m3 Fekete et al. (2010)

n Degree of polymerization 2.5 Fekete et al. (2010)

φ∗b Carrying capacity for bacteria 0.15 †

φ∗p Carrying capacity for EPS 0.2 Donlan (2002)

β Bacteria growth rate ratio 1.0 Ward et al. (2001)

Parameterswith references for their order ofmagnitude aremarked; otherwise, they are fitted by our previouswork and experience†We assume each bacteria volume is 20 µm3, such that the carrying capacity is 5 × 1015/m3, whenφb = 0.15. EPS 50– 80% to total biomass

123

J. Zhao, Q. Wang

Time (hr)

AH

L d

etec

ted

(nM

)

0

50

100

150

200

250

300

350

400modelexperiment

(a)

Time (hr)0 10 20 30 40 0 10 20 30 40

Cel

l con

cent

ratio

n L

og (C

FU/m

l)

6

6.5

7

7.5

8

8.5

9

9.5

10modelexperiment

(b)

Fig. 1 Fitting of the model to an experiment (Quinones et al. 2004). This demonstrates that the reactivekinetic model (or homogeneous model) predicts the experimental data on the AHL concentration and cellnumber density very well. The initial condition of the homogeneous model is (φb1, φb2, φb3, φp, A) =(3× 10−5, 0, 0, 0, 0), which is equivalent to 106 CFU/ml QS− bacteria, according to Table 1. Here ra1 =ra2 = 3.1 × 10−8, Ca1 = Ca2 = 3.1 × 10−7 and Cb1 = Cb2 = 1.11 × 10−4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dφb1dt = (Cb1φb1+βCb2φb2)c

kb1+c

(1 − φb

φ∗b

)− r12An

τ n+An φb1 + r21τ n

An+τ nφb2,

dφb2dt = (1 − β)

Cb2ckb2+cφb2

(1 − φb

φ∗b

)+ r12An

An+τ nφb1 − r21τ n

An+τ nφb2,

dφb3dt = Cb3c

kb3+cφb3

(1 − φb

φ∗b

)− Cd3d

kd3+d φb3,

dφp

dt = ∑3i=1

Cpi ckpi+cφbi

(1 − φp

φ∗p

),

dAdt = ∑2

i=1

(rai + Cai An

τ n+An

)φbi

(1 − A

A∗).

(32)

We fit the model parameters to the experiment in Quinones et al. (2004). The result isshown in Fig. 1. It agrees very well with the experiment, demonstrating the usefulnessof the simplified kinetic model.

A complete parameter study of this spatially homogeneous reaction model perhapswould warrant another paper, which will not be pursued in this paper.

We next examine the full hydrodynamic model for two quorum sensing relatedmechanisms that affect heterogeneous biofilm formation and functioning.

3.2 Biofilm Formation Coordinated by Quorum Sensing

Biofilm formation is a process in which bacteria aggregate in space and secrete glue-like exopopolysacharrides (EPS) to form a polymer-network around the bacteria,which creates a barrier between the bacteria and the surrounding media. It has beenobserved that quorum sensing controls biofilm formation in vibrio cholerae (Hammer

123

Three-Dimensional Numerical Simulations. . .

and Bassler 2003) by regulating transcription of genes involved in expopolysachar-ride production. Other observations on quorum sensing regulated EPS production canalso be found in Pseudomonas aeruginos (Sakuragi and Kolter 2007), sinorhizobiummeliloti (Marketon et al. 2003), as well as in some other bacterial systems (Vuong et al.2003). And biofilms adapt their architecture in order to cope with the hydrodynamicconditions and nutrient availability in Teodosio et al. (2011).

However, there has not been a consensus on what is the role of quorum sensingduring biofilm formation. In this section, we will focus on simulating the mechanismof quorum-sensing-regulated EPS production using the model in a heterogeneousbiofilm colony and studying pros and cons of QS regulation during biofilm formationin 3 dimensional space.

3.2.1 Biofilm Formation

First of all, we investigate heterogeneous biofilm formation in an aqueous environmentwhere laminar flows are present outside the biofilm. The scenarios include (i) a culturepetri dish filled with water where nutrient is supplied from the top surface; or (ii) along water channel where nutrient is constantly supplied through the flowing water.For simplicity, we suppress the non-QS-regulated bacteria in the study of the first casesince they are not important in this case, i.e., we consider a biofilm system where onlyQS-regulated bacteria are present initially. We set the initial value of φb3 (the volumefaction of non-QS-regulated bacteria) to zero and assume that the dead bacteria areall dissolved in the solvent instantly. This way, we can examine how quorum sensinginduction takes place and how biofilm dynamics are coordinated by the distributionof QS molecules.

Figure 2 depicts a simulation using the boundary condition (21), where ini-tial conditions and solutions at t = 300 are shown. In this simulation, someQS down-regulated bacteria (QS−) are assumed to be attached to the substratesurface initially, and the bacterial cell reproduction ensues as soon as the sim-ulation begins. As a result, the QS molecule is produced and its concentrationkeeps growing. As the concentration of the quorum sensing molecules approachesthe threshold value τ , QS− start to bind with QS molecules and undergo aphenotype change into QS up-regulated bacteria (QS+), for which the produc-tion for EPS is switched on. The biofilm colony starts growing exponentially byreproducing bacteria as well as secreting EPS. With more QS molecules pro-duced, QS+ eventually outgrow QS− and populate the bulk part of the biofilmcolony.

Our 3D model predicts that QS-regulated EPS production contributes to the forma-tion of heterogeneous biofilm structures. As shown in Fig. 2, this mushroom-shapedbiofilm morphology qualitatively agrees with the published result on the morpholog-ical pattern in Pseudomonas aeruginosa biofilms (Parsek and Tolker-Nielsen 2008).The 3D results for volume fractions of total bacteria, QS down-regulated and QSup-regulated bacteria, EPS, and the total biomass are depicted in Fig. 2, respec-tively.

In addition, our model also suggests the heterogenous distribution of live bacteriaand QS molecules. As shown in Fig. 2b, the distribution of live bacteria is highly

123

J. Zhao, Q. Wang

Fig. 2 (Color figure online) Biofilm formation with EPS production coordinated by quorum sensing ina biofilm in which non-QS bacteria are neglected. This simulation shows several clusters of QS− in adeveloping heterogeneous biofilm colony, where EPS production during biofilm formation is regulated byquorum sensing. The initial profile is shown in (a). The characteristic timescale is t0 = 103 s and thesimulation is conducted from t = 0 to t = 300, which represents roughly 3.4 days in real time. b A timet = 300, the bacterial distribution in terms of volume fractions is shown, where the red represents QS− andthe blue represents QS+; they are mixed. In the result, more QS+ are observed in the bulk of the biofilmwhile more QS− show up near the biofilm surface. In c–f, three 2D slices of QS−, QS+, EPS and the totalbiomass at t = 300 are depicted, respectively, from which more interior details of the biofilm colony areon display

heterogeneous, QS− are mainly located at the surface of the biofilm, where they canaccess nutrientsmore easily, andQS+ are locatedmore in the bulkof the biofilmcolony.In order to show details of this process, plots of three slices at x = 0.16, y = 0.04, z =0.16 for different biofilm components are shown in Figs. 2 and 3, respectively. InFig. 3c, where the average velocity is small, the distribution of the QS molecules(AHLs) is dominated mainly by spatial diffusion. We observe that QS molecules aremainly distributed within the biofilm colony although small amount can diffuse intothe solvent, illustrated by a three-slice plot in Fig. 3b. The biomass flux at z = 0.5is plotted in Fig. 3d. It shows that it is mainly distributed at the top surface betweenthe biomass the surrounding solvent, where there is more nutrient available near thesurface than in other places so that the biomass tends to grow the fastest in theseplaces. A pressure profile is also plotted in a 2D slice, which shows that high pressureis observed in the interior while low one is present near the surface of the biofilm.

This numerical simulation explains the growth process of the biofilm due toQS-regulated bacteria and QS-regulated EPS production together with some hydro-dynamic signatures. Of course, the biomass can’t grow without a sufficient nutrientsupply. But, in this case study, we assume its role is secondary.

123

Three-Dimensional Numerical Simulations. . .

Fig. 3 (Color figure online) Velocity, pressure and functional components from the simulation in Fig. 2.a–b 2D slices for the distribution of nutrient and QSmolecules, AHL. c–e 2D slice at z = 0.5 of the averagevelocity v, the biomass flux and the hydrostatic pressure at t = 300, respectively. Low pressures occur inthe interfacial region where the biofilm tends to grow. d Depicts explicitly the biomass flux given by φnvn .This shows that QS molecules (AHL) are mainly concentrated in the biofilm colony, but small amount isdiffused into the surrounding aqueous environment

3.2.2 Benefits and Detriments of Quorum Sensing Regulation

With biofilm formation regulated by quorum sensing, a straight forward question iswhether quorum sensing is always beneficial? Experimental findings seem to suggestthe answer is ‘it depends.’ As the production of QS molecules is energetically costly,i.e., consuming ATP (Keller and Surette 2006), bacteria probably would switch on thequorum sensing mechanism to start producing QS molecules whenever they are in the‘right time’ and ‘right place.’ In this subsection, we would like to further investigatethe so-called right time and right place such that QS regulation for EPS productioncan be in favor of the biofilm development.

Our numerical study confirms (Frederick et al. 2011) that quorum sensing regula-tion is not beneficial to biofilm development in a short time, especially, in terms ofgaining biomass. To explore the feature, we conduct another numerical simulation,where the quorum sensing mechanism is switched off by setting r21 = r12 = 0 whilethe other model parameters are retained as in the previous numerical experiment. Itis shown in Fig. 4 that the volume fraction of the total live bacteria without quorumsensing regulation is higher than that with quorum sensing regulation switched on.However, the total biomass in the system without QS regulation, including EPS andbacteria, is much less than that with QS regulation. This indicates that QS regula-tion is not a way to reproduce bacteria, but rather a way to increase the amount ofEPS so as to expand biofilm colony, especially, in a short time period. The result

123

J. Zhao, Q. Wang

Fig. 4 (Color figure online) Comparison between bacterial volume fractions during biofilm developmentwith and without quorum sensing. Here we set r12 = r21 = 0 for simplicity. a Volume fractions of totalbiomass.bVolume fractions ofQS down-regulated bacteria. cVolume fractions ofQS up-regulated bacteria.d Volume fractions of total live bacteria. The initial conditions are the same as the one used in Fig. 2. Withquorum sensing, the total biomass is higher than without, but the total live bacteria is not necessarily higher

agrees qualitatively with the experimental observation reported in Xavier and Foster(2007).

In the long time period, our numerical studies confirm that (Frederick et al. 2011)QSregulation is of benefits to stabilizing biofilm colony and settling on the host, as well asprotecting bacteria from environmental stress. Given the protective nature of EPS, thebiofilm colony can becomemore stable and resistant to external stress while more EPSis produced. EPS can protect bacteria from being attacked by antimicrobial agents orimmune systems of the host. To illustrate the beneficial effect of quorum sensing, weconduct a numerical experiment with the drug/disinfectant effect included. The resultis shown in Fig. 5. In this simulation (see Fig. 5f), non-QS-regulated bacteria arekilled dramatically while QS-regulated bacteria are killed in a much slower pace. Ourmodel and simulation confirm that EPS acts as a barrier to protect bacteria from beingattacked by antimicrobial agents. In Fig. 5a, b, we notice that bacteria staying withinthe environment filled with EPS lives longer while being treated by antimicrobialagents. In Fig. 5e, we see that antimicrobial agents penetrate less in the place withmore EPS than in the place with less EPS. This numerical result agrees qualitativelywith the claim made in Frederick et al. (2011) that biofilms can benefit from quorumsensing induced EPS production if bacteria cells have the objective of acquiring athick, protective layer of EPS.

In addition, via numerical simulations, our model predicts that quorum sensingregulation is not beneficial to biofilm formation in a short time period when competingwith invading strains of bacteria. In other words, given a highly reproductive invasionstrain (like non-QS-regulated bacteria), QS-regulated bacteria can lose the competition

123

Three-Dimensional Numerical Simulations. . .

Fig. 5 (Color figure online) Biofilm development subject to both quorum sensing and antibiotic (drug)treatment. The initial biofilm profile is a grown biofilm at t = 252 from Fig. 6. Antimicrobial agents aredosed through the top boundary. Here the characteristic time is t0 = 10 s. a–c The distribution of the livebacteria, EPS, dead bacteria in their volume fractions at t = 94, respectively. d, e Three slices of deadbacteria and antibiotics at t = 94, respectively. f shows total volume fractions of QS bacteria and non-QSbacteria from t = 0 to t = 600, respectively. After a long period of antimicrobial treatment, the bacterialpopulation is reduced to a negligible level in this simulation

such that the non-QS-regulated bacteria can take over and eventually populate thewhole biofilm colony, which is mentioned in the case study of bulk reactive kinetics.To illustrate the point, let’s consider a biofilm system with three different phenotypesof QS-regulated bacteria: QS−, QS+ and non-QS. In this system, we suppose thatnon-QS-regulated bacteria have the same reproductive rate as that of QS−, but donot bind QS molecules and convert themselves into QS+. We conduct a numericalsimulation, taking into account all these three phenotypes to study the detail on howthe interaction between these bacteria can affect biofilm development via differentself-production and EPS production rates. In this simulation, initially some QS− andnon-QS-regulated bacteria with the same volume fraction are assumed to be attachedto the substrate surface.We assumeCb1 = Cb3 and the growth rate forQS up-regulatedbacteriaCb2 is smaller. A 3D prediction of the biofilm growth process is shown in Fig.6. In Fig. 6b, non-QS-regulated bacteria outgrow the QS-regulated bacteria and takeover the whole biofilm colony eventually. This is because the QS-regulated bacteriaundergo mutual conversion which slows down the reproduction of QS+ bacteria aswell as QS−. The total volume fraction for each type of bacteria is plotted in Fig.

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Fig. 6 (Color figure online) Biofilmdevelopmentwith bothQS bacteria andNon-QS bacteria in the biofilm.Randomly distributed biofilm colonies are given initially. Here, the characteristic time is t0 = 103 s. a Initialbacterial volume fraction. b Bacterial volume fractions at time t = 252, where purple represents non-QSbacteria and red represents QS bacteria. c The volume fractions for each bacterial phenotype in the entirebiofilm colony, where non-QS-regulated bacteria outgrow QS-regulated bacteria. d The volume fraction ofnon-QS bacteria. e The volume fraction of QS bacteria at t = 252. f–i 2D slices at z = 0.5 for non-QS,QS bacteria, hydrostatic pressure, and biomass flux φnvn at t = 252, respectively. The non-QS-regulatedbacteria outgrow the QS-regulated bacteria in most part of the biofilm except in regions near the bottom ofthe domain. The hydrostatic pressure is again small near the interface and large in the interior of the biofilm.Here Cb1 = Cb3 = 4.0 × 10−4 and Cb2 = 10−4

6c, where non-QS bacteria grow exponentially, while QS-regulated bacteria growsonly slightly. Since non-QS-regulated bacteria have a higher effective reproductionrate than those of QS-regulated ones combined, non-QS-regulated bacteria eventuallygrow faster to gain more access to nutrient. Hence, the density of non-QS-regulatedbacteria eventually surpasses that of QS-regulated bacteria in the majority part of thebiofilm colony due exclusively to the quorum sensing effect.

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We remark that these findings perhaps also explain why antimicrobial treatment ismore effective for young biofilms, where EPS layer tends to be thinner. When biofilmshave grown for weeks, the effect of antimicrobial agents drops significantly. Whenthe biofilm is young, there exists less EPS surrounding it; when the biofilm is “old,”however, the QS-induced EPS has already accumulated around bacteria to preventdrugs from penetrating deeper into the biofilm. This agrees qualitatively well withthe observation reported in Shen et al. (2011) that biofilms at an earlier age is moresusceptible to antimicrobial treatment than those in an older age.

Next,we turn to discuss the interactionbetweenhydrodynamics andquorumsensingduring the development of biofilms in an aqueous environment.

3.3 Role of Hydrodynamics on Quorum Sensing

3.3.1 Quorum Sensing in an Aqueous Environment

Many biofilms grow in an aqueous environment encompassing laminar flows, such asriver bank or pipes, or quiescent aqueous environment. It is observed that a flow envi-ronment can impact on QS induction during biofilm formation (Kirisits et al. 2007).In return, the hydrodynamically altered QS induction can affect biofilm structures andfunctions. Recently, in Kim et al. (2016), authors found flows assist long-distance QSby enhancing autoinducer accumulation.

Here, we use the hydrodynamic model along with the computational tool to inves-tigate the effect in detail. For simplicity, we consider the case where there are onlyQS-regulated bacteria in the biofilm system. The boundary conditions used in the sim-ulation are set up tomimic biofilm development in a flow cell, where the fluid (solvent)flows through the biofilm domain continuously. We conduct numerical simulations ofthe biofilm system subject to varying inlet fluid velocities and focus on biofilm devel-opment under hydrodynamic shear. The result is summarized in Fig. 7. Initially, someQS− bacteria are attached at the center of the flow cell. Solvent containing nutrientkeeps flowing through the flow cell while the bacteria grow into a biofilm colony. Asshown in Fig. 7c, a higher hydrodynamic shear can actually dilute the concentrationof QS molecules, which leads to a delay in QS induction. 2D slices (at z = 0.5) ofQS molecules and QS+ bacteria at time t = 25 are shown in Fig. 8. This numericalexperiment demonstrates that the maximum concentration of QS molecules and thevolume fraction of QS+ are higher in the flow cell when subject to a slower inlet speedand than subject to a higher inlet speed.

Our numerical studies confirm that hydrodynamics have a pretty strong influence onquorum sensing in that a higher inlet flow dilutes the concentration of the QS signalingmolecules (Frederick et al. 2010; Uecke et al. 2014; Vaughan et al. 2010), which thenrequires a higher cell density and delayed time for quorum sensing induction to takeplace (Kim et al. 2016; Vaughan et al. 2010). This means that QS may not be fullyoperational at high flow rates and thus indicates that the role of QS in biofilm formationcan be compromised in a flowing environment.

In addition, through numerical study, our model suggests, with the diluted QSmolecular concentration, the biofilm in the flow cell with a higher inlet velocity has

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Fig. 7 (Color figure online) Biofilm development subject to varying inlet flow velocities. This gives acomparative study on how some QS− bacteria develop into a biofilm colony under different inlet velocitiesv0 = p0y(1 − y), where p0 = 1, 10, 20. a The initial distribution of QS− bacteria in volume fractions.b 2D slice (z = 0.5) of the initial velocity profile. c The total volume of QS− bacteria, QS+ bacteria,total bacteria, as well as the concentration of AHL, at three selected inlet velocities. The volume fractionof QS− bacteria and the total bacteria scales with the inlet speed, i.e., the higher the inlet speed is, thelarger the volume fractions are, whereas the volume fraction of QS+ bacteria and the AHL concentrationscales inversely with the inlet speed, i.e., the larger the inlet speed is, the smaller the volume fraction andthe concentration are

less QS+ bacteria. Note however that a biofilm with a higher inlet velocity resultsin more total bacteria. This is because the nutrient supply at a higher inlet velocitycan bring in more nutrient and thus facilitates the growth of all bacteria. A robustbiofilm colony needs not only the growth of bacteria, but also the build-up of EPS.Since the EPS growth by QS+ is more than that by QS−, a reduced QS+ productionleads to the reduced EPS production in the biofilm. Even though the total bacteriais not reduced, the EPS production is reduced by the higher inlet velocity. We notethat how quickly QS molecules are produced, how quickly the QS molecules diffuse

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Fig. 8 (Color figure online) The volume fraction of QS+ and AHL concentration at t = 15 with respect tothree selected inlet speeds. This figure shows 2D slices at z = 0.5 for the simulation depicted in Fig. 7. a–cAHL distribution at t = 15 for inlet velocity v0 = p0y(1 − y) with p0 = 1, 10, 20, respectively. d–f 2Dslices of volume fraction of QS+ (φb2 ) at t = 15 for inlet velocity v0 = p0y(1− y) with p0 = 1, 10, 20,respectively

away from the biofilms and the critical concentration of QS molecules are three mainfactors affecting biofilm formation. When taking into account of the effect of nutrient,especially, when the growth due to nutrient dominates, the diffusion rate and theconsumption rate of nutrient can be essential for the biofilm growth as well. We hopethe theoretical result can be quantitatively confirmed by experiments in the future.

3.3.2 Effect of Nutrient

Compared with a quiescent aqueous environment, the salient difference of biofilmsdeveloped in a flow cell is their structural heterogeneity in the flow field, due to thehydrodynamic stress and inhomogeneous nutrient distribution. Mass transfer in thefluid mixture can be affected by hydrodynamics of the bulk fluid and the morphologyof the biofilm colony. In a flow, one interesting question is which part of biofilms,upstream or downstream, is benefited more from quorum sensing regulation.

In Vaughan et al. (2010), the author suggests that biofilms upstream can inducedownstream biofilms to quorum sense in the flow if they are close enough spatiallywithout considering the heterogeneity of biofilms. Here we conduct a numerical sim-ulation with a distribution of biofilm colonies in the flow channel. Our 3D numericalsimulations agrees with this prediction. In details, in Fig. 9, it confirms (or quali-tatively agreeable) with the findings in Vaughan et al. (2010) that quorum sensingtakes place at downstream first, for instance quorum sensing induction takes place atthe last four colonies instead of the first one located upstream. It also shows that thedownstream region tends to have a higher concentration of QS molecules than at thelocation upstream in Fig. 9g, which we believe is an amplification of the convectionand diffusion of QS molecules.

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Fig. 9 (Color figure online) Quorum sensing induction under the hydrodynamic stress with a weak nutrientsupply. This figure shows quorum sensing induction in a flow cell with an inlet flow velocity. In thissimulation, c0 = 0.001, representing a scenario with a weak nutrient supply. Initially some QS− bacteriaare attached to the substrate in the middle of the flow cell. a The initial profile of QS− bacteria. b AHLconcentration at x = 0.5, y = 0.0625 and t = 25. c–f 2D slices at z = 0.5 for AHL, φb1, φb2, aswell as nutrient at time t = 200, respectively. When nutrient supply is weak in the entire flow cell, QSinduction is stronger downstream than upstream. Consequently, there exist more QS+ bacteria downstreamthan upstream

However, this may not be always true, especially when nutrient supply is strong.A comparison with different nutrient supply rates is shown in Fig. 10. We observethat, at an early stage, bacteria downstream can be benefited since QS molecules aremore diluted upstream (see Fig. 10c). QS induction is more effective downstreamsince there exist more QS+ bacteria there (see Fig. 10b). But when time passes by andthe nutrient supply is kept at a sufficiently high level, it seems that quorum sensinginduction benefits from a higher flow velocity since it sustains a sufficient nutrientsupply that facilitates the growth of all bacteria aswell the production ofQSmolecules.As a result, the overall production of QS molecules upstream increases such that boththe concentration of QS molecules and the volume fraction of bacteria are higherupstream. In Fig. 10e, we see there are more QS+ bacteria upstream than downstreamin this case. Figure 10h depicts the EPS production in the flow cell, which is also theresult of the excessive accessibility to nutrient.

3.3.3 Biofilm Morphology Under a Strong Shear Flow

In previous discussions, we focus on interactions between hydrodynamics and quorumsensing induction under the assumption that biofilms grow under weak flow fields.When the flow is strong, however, biofilms can undergo significant morphologicalchanges and most of the biofilms can perhaps be flushed out of the flow cell. To better

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Fig. 10 (Color figure online) Quorum sensing induction in a flow with a strong nutrient supply. This figureshows quorum sensing induction in a flow cell with inlet velocity v0 = 10y(1 − y) and a strong nutrientsupply (c0 = 1.0). In this simulation, we use the same initial distribution of QS− as used in Fig. 9 andcharacteristic time t0 = 103 s. a–c 2D slices of the volume fraction of QS− bacteria, QS+ bacteria andthe concentration of AHL at z = 0.5 and t = 1.25, respectively. d–i Depict the volume fraction of QS−bacteria, QS+ bacteria, the concentration of AHL, the concentration of nutrient, the volume fraction of EPS,and the velocity field at z = 0.5 and t = 25, respectively. In this case, the strong nutrient supply dominatesthe growth of bacteria so that QS+ grows the fastest near the upstream location than the downstream one

illustrate the hydrodynamic impact on biofilm structures and QS induction in strongflows, we conduct several numerical simulations where a grown biofilm colony issubject to a strong shear in a flow cell.

Here, we place two biofilm colonies in a 4 × 1 × 1 domain, representing a flowcell with inlet and outlet boundaries in the x direction. The initial biofilm profile isshown in Fig. 11a and the reference timescale is t0 = 1. From 11b, we notice that thebiofilm undergoes a significant morphological change under the shear. The coloniesare tilted under the shear. The one near the inlet boundary is about to detach from thebottom in Fig. 11d. In addition, a fast flow can dilute QS molecules as they are readilyconvected out of the flow cell shown in Fig. 11c, e.

The advantage of our full 3D hydrodynamic model, compared with those simplifiedmodels (without considering hydrodynamics) , is its capability to resolve the hydro-dynamic detail as well as significant rheological functions such as the normal stressdifferences and shear stress components. Here, these stress components related to thesimulation above are depicted in Fig. 12. In Fig. 12, we only show the sign of φnτ , theeffective stress for the biomass component, where we use red to represent negative and

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Fig. 11 (Color figure online) Flow effects on biofilm morphology and the distribution of QS molecules.This figure shows two grown biofilm colonies under shear in a flow cell, where the characteristic timet0 = 1s. a The initial profile of the two biofilm colonies. b–e show the data at t = 31. b The two coloniesin the shear flow. c Distribution of QS molecules. d A 2D slice of the biofilm colonies at z = 0.35. e A 2Dslice of the AHL distribution at z = 0.35

blue for positive values of the stress. It is seen through Fig. 12e that the first normalstress difference is positive at the top of the biofilm region while negative around theneck facing the inlet boundary, which means that the normal stress τxx is larger thanτyy at the top of the biofilm, providing a drag force along the x direction. This isconfirmed by Fig. 12c, where the top of the biofilm is stretched toward the x direction.In addition, through Fig. 12d, the second normal stress difference is positive at theneck facing the inlet boundary, where τyy is larger than τzz resulting in a stretchingof the biofilm colony toward the y direction. This is supported by the fact that the

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Fig. 12 (Color figure online) An anatomy of the flow-induced effect on biofilm morphology. a–c showthe first biofilm colony in Fig. (11) at time t = 0, 18, 42, respectively. We denote the negative value byred and the positive one by blue in the rest of the subfigures. The data shown for d–h are the solutions att = 18. d The second normal stress difference N1 = φn(τyy − τzz). e The first normal stress differenceN2 = φn(τxx − τyy). f The shear stress in the xy plane φnτxy . g The shear stress in the xz plane φnτxz . hThe shear stress in the yz plane φnτyz . The first and second normal stress difference are of opposite signsin most part of the colony. The primary shear stress φnτxy is always positive while the two secondary shearstress components can take on either signs

neck becomes thinner due to this stretch, shown in Fig. 12c. In addition, the otherthree components of shear stress tensor are also depicted in Fig. 12f–g. On the planey = a (0 < a < 1), we see the shear stress τxy is positive, which forces the biofilm tomove toward the x direction (shown in the comparison of Fig. 12b, c.) The effect ofτxz and τyz can also be observed on a z-plane, i.e., z = a (0 < a < 1), shown in Fig.12g, where τxz provides a shear stress toward the x direction on the top of the biofilmmushroom-shaped colony and τyz provides a shear stress toward the y direction onthe top of the mushroom.

It is known that QS will be repressed under strong hydrodynamic flow as AHLwill be advected (Kim et al. 2016). Here, in our model, we carried out a qualitativestudy by tuning the material parameter of biofilms, i.e., the viscosity of biomass.Given the same initial condition and parameters as shown in Fig. 11, we compare thevolume-averaged AHL concentration by changing the biomass viscosity. Numerical

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Fig. 13 (Color figure online)Effect of biofilm viscosity on QSmolecules accumulation. Thisfigure shows lower biofilmviscosity would help to trapAHL in biofilms in short timeperiod. Here A, B, C representthe case: ηp = ηb = 102;ηp = ηb = 10;ηp = ηb = 1,respectively. Here the Y axis isthe total volume-averagedconcentration of QS moleculesin flow cell, i.e., 1

|�|∫� Adx

results are shown in Fig. 13. As we observe, at the beginning, the volume-averagedconcentration of QS molecules in the domain does not change, as they are still inthe flow cell even though they may be flushed out. As time goes by, under samehydrodynamic conditions, it turns out that biofilms with a smaller viscosity (softer)can better trap QSmolecules inside than biofilms with a larger viscosity (harder). Thiscan be explained since biofilms with lager viscosity resists deformation such that theAHLwould be flush away. But biofilms with a smaller viscosity tends to deformmore,such that, instead of being flushed away into solvent and out of flow cell, more AHLwould be trapped in the biofilm. We remark a more realistic model accounting EPSviscoelasticity is needed to better understand biofilm deformation since in such shorttimescale biofilms behave like viscoelastic materials.

4 Conclusion

In this paper, a multiphasic hydrodynamic model for biofilms is developed to studybiofilm formation and development regulated by quorum sensing, extending our pre-vious 3D biofilm models (Zhang et al. 2008a). By identifying appropriate bacterialtypes based on their function in quorum sensing in the biofilm and proposing the corre-sponding reactive kinetics, we are able to describe several bulk phenomena related toquorum sensing induction using the new model. We then develop a numerical schemeto solve the governing system of equations in the model and implement it in 3D spaceand time on GPUs. With the numerical solver, we are able to simulate and predict het-erogeneous spatial–temporal structures of a biofilm in terms of its volume fractionsfor each bacterial component, EPS and solvent. The solver is used to study a set ofselected mechanisms in quorum-sensing-regulated biofilm development.

Numerical simulations indicate that QS-regulated EPS production competes forboth physical space and nutrient since a higher EPS production rate can expand biofilmcolonies spatially without reaching a higher local density. In addition, numerical sim-ulations confirm the hypothesis that QS-regulated EPS production promotes biofilmdevelopment as well as protects biofilms from the environment stress, such as antimi-crobial treatment and hydrodynamical stress. However, quorum sensing regulationis not universally beneficial to biofilm formation and development in all situations

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evidenced by what we have shown that an invasion strain of bacteria with a higherproduction rate can outgrow QS-regulated bacteria, where QS induction retards theproliferation of bacteria and thereby the development of biofilms in a short timescale.Our 3D simulations show that QS down-regulated cells tend to grow at the surface ofthe biofilm while QS up-regulated ones tend to grow in the bulk.

By simulating biofilms in a flow cell, hydrodynamical effects on quorum sensingare investigated in detail. The model captures the competition between the dynamicdiluting effect and the nutrient supply associated with QS induction well. Our simula-tions predict that downstream bacteria can be benefited from higher inlet flow speed toinitiate QS induction earlier. However, the model also shows that this only takes placein a short timescale, when nutrient supply is weak. When nutrient supply is strong, itis a competition between the flow-induced dilution of QS molecules and the nutrientsupply for the production of QS molecules that ultimately determines which parts ofbiofilms in a flow cell would benefit more.

This 3D hydrodynamic model provides a powerful simulation and predicative toolto allow us to investigate quorum-sensing-regulated biofilm development, some fac-tors that correlate with quorum sensing and their interaction with hydrodynamics. Inaddition to the results that we numerically confirmed, we hope new results predictedhere can be verified in experiments in the future. In our subsequent research, otherquorum-sensing-regulated factors such as virulence, growth factors, and resistance toantimicrobial agents will be investigated.

Acknowledgements Qi Wang is partially supported by NIH and NSF through awards DMS-1200487,DMS-1517347, R01GM078994-05A1, and NSFC awards 11571032 and 91630207. Jia Zhao is partiallysupported by an ASPIRE grant from the Office of the Vice President for Research at the University of SouthCarolina.

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