Klapper2009 Mathematical Description of Microbial Biofilms Review4

8/14/2019 Klapper2009 Mathematical Description of Microbial Biofilms Review4

1/58

Mathematical Description of Microbial BiofilmsIsaac Klapper, Jack Dockery

Department of Mathematical Sciences &Center for Biofilm Engineering,

Montana State University,Bozeman, Montana 59717,

email: [email protected]

Abstract. We describe microbial communities denoted biofilms and efforts to model someof their important aspects, including quorum sensing, growth, mechanics, and antimicrobialtolerance mechanisms.

1 Introduction

1.1 Microbial Ecology

Compared to the plant and animal kingdoms, the diversity of microbial life is considerably

less explored and less understood (even the notion of microbial species is a current topic ofdebate [41, 235]) in part because classification and characterization of microbes have been com-paratively difficult tasks. Recently, though, new efficient technologies for genetic and genomiccataloguing of microbial communities are changing the landscape of prokaryotic and archaelecology, uncovering surprising diversity in the process. Such information is important as mi-croorganisms are closely connected with geochemical cycling as well as with production anddegradation of organic materials. Thus a set of guiding principles is needed to understandthe distribution and diversity of microorganisms in the context of their micro- and macro-environments as a step towards the ultimate goal of constructing descriptions of full microbialecosystems.

Prokaryotes (bacteria and archaea) are estimated to make up approximately half of extant

biomass [244]. Their numbers are staggering; for example each human harbors an estimated100 trillion microbes (bacteria and archaea), ten times more microbes than human cells [19],and, as another example, during the course of a normal lifetime, the number of Escherichia coliinhabiting a given person will easily exceed the number of people who ever lived: to paraphraseStephen Jay Gould, though we may think we are living in a human dominated world, but inactuality it may be more truthful to say that we live in the Age of Bacteria [90]. The familarview of microbes in their free (planktonic) state is however not the norm; rather it is believed

1


2/58

Figure 1: Mixed species photosynthetic mat, Biscuit Basin, Yellowstone National Park.Reprinted by permission of Macmillan Publishers Ltd: Nature Reviews Microbiology from [96]

that much of the microbial biomass is located in close-knit communities, designated biofilmsand microbial mats, each consisting of large numbers of organisms living within a self-secretedmatrix constructed of polymers and other molecules [1, 47, 46, 49, 96, 211], see Figures 1-

4. (Microbes in collective behave very differently from their planktonic state; even geneticexpression patterns change [110, 211].) These matrices serve the purposes of anchoring andprotecting their communities in favorable locations [76], generally wet or damp but not alwaysso [237], while providing a framework in which structured populations can differentiate andself-organize. In this way, as a result of internal spatial variation, microbial community ecologyis determined not only by classical ecological interactions such as competition and cooperation(as for example in chemostat communities [200]) but also by local physical and chemical stressesand accompanying physical and chemical material properties. One can expect that, in bothdirections, the connection between physics and ecology in microbial communities is profoundat multiple scales and that these ecosystems are not merely the sum of their parts.

Microbial mats and biofilms have a long lineage it is interesting and suggestive to note thatbiofilm formation is observed today in the Aquificales [184] (Figure 4), the lowest known branchof bacteria (on the phylogenetic tree of life), and, likewise, Korarcheota [111], the correspond-ing lowest known branch of archaea. Prokaryotic mats may once have been a dominant lifeform on earth and may have played an ecologically crucial role in precambrian earth chemistry.Within those billions of years of dominance, enormous generation of genetic diversity occurred;today, prokaryotes are estimated to carry 100 times as many genes as eukaryotes [244]. These

2


3/58

Figure 2: Microbial structures in a mixed species photosynthetic mat, Mushroom Spring, Yel-lowstone National Park.

Figure 3: Laboratory grown Pseudomonas aeruginosa biofilm. Reprinted by permission ofBlackwell Publishing from [126]

.

3


4/58

Figure 4: Aquificales streamers (pink), Octopus Spring, Yellowstone National Park.

days, possibly due to the existence of grazers, mats are mostly found in extreme environments(e.g. high or low temperature, high salinity, deep ocean vents) where they typically consistof communities of photoautotrophs and attending heterotrophs, though communities based onchemoautotrophy also exist, e.g. Fig. 4. However, while microbial mats are now relatively rare,their thinner cousins, microbial biofilms, are ubiquitous. One can and will find them in almostany damp or wet environment, and they are often key players in problems such as human and

animal infections [43, 82], fouling of industrial equipment and water systems [18], and wastetreatment and remediation [42, 219], just to name a few. Medical relevance is dramatic aswell [26]. Quoting from the National Institutes of Health [179]: Biofilms are clinically im-portant, accounting for over 80 percent of microbial infections in the body. Examples include:infections of the oral soft tissues, teeth and dental implants; middle ear; gastrointestinal tract;urogenital tract; airway/lung tissue; eye; urinary tract prostheses; peritoneal membrane andperitoneal dialysis catheters, in-dwelling catheters for hemodialysis and for chronic adminis-tration of chemotherapeutic agents (Hickman catheters); cardiac implants such as pacemak-ers, prosthetic heart valves, ventricular assist devices, and synthetic vascular grafts and stents;prostheses, internal fixation devices, percutaneous sutures; and tracheal and ventilator tubing.Biofilms can be dangerous and opportunistic pathogenic structures: in fact, biofilm associatednosocomial (hospital acquired) infection is by itself a leading cause of death in the UnitedStates [241].

4


5/58

Figure 5: Stages of the biofilm life cycle, courtesy of the Montana State University Center for

Biofilm Engineering, P. Dirckx.

1.2 Biofilm Development and Structure

Biofilm development is often characterized as a multistage process [160], Figure 5. In the firststage, isolated planktonic-state microbes attach to a surface or interface. Once attached to asurface a phenotypic transformation (change of gene expression pattern) into a biofilm state oc-curs. At this point cells begin manufacturing and excreting extracellular matrix material. Theexact nature of this material can vary greatly depending on microbial strain and, even within asingle strain, on environmental conditions and history though its function shares commonality.With formation of an anchoring matrix, the nascent biofilm begins to grow. Secondary coloniz-

ing strains might arrive and join the new community. But as the biofilm matures, meso- andmacro-scale physical realities like diffusion-reaction limitations and mechanical stress distribu-tion become more important, probably to the extent that they significantly influence structureand ecology. Finally, the fully mature biofilm reaches a quasi-steady state where growth isbalanced by loss through erosion and detachment due to mechanical stress, self-induced dis-persal from starved subregions, and predation and loss to grazers and viruses. We recommendreference [32] to the reader, a remarkably forward-looking overview written by a pioneer of thesubject of biofilm mechanics and modeling, Bill Characklis.

On close inspection, we note a wide variation of relevant time scales, see Figure 6, rangingfrom fast time scales for bulk fluid dynamics and reaction-diffusion chemical subprocesses toslow time scales of biological development. The roughly ten orders of magnitude variationin range is a challenge to modelers of course, one that necessitates compromises. The usualpractice is to introduce equilibrium in the fast processes: bulk fluid flow, when considered, isassumed steady over a quasistatic biofilm, and then advection-reaction-diffusion processes arealso assumed quasisteady relative to the given fluid velocity field. Given those fluid velocityand chemical profiles, the modeler can take advantage of scale separation and focus on the slowgrowth and loss processes that are typically the ones of interest anyway. Notably missing from

5


6/58

Figure 6: Various time scales for biofilm related processes. Box 1 include processes related tobulk fluid (measured with respect to a biofilm length scale). Box 2 includes processes relatedto chemical processes (again, measured with respect to a biofilm length scale). Box 3 includesbiological processes (essentially independent of biofilm length scale). Reprinted from [170]

Fig. 6 though are time scales related to viscoelastic mechanics within the biofilm itself thatmay inconveniently straddle several time scales. Also missing are external forcing time-scales,

e.g. the day-night cycle, which can also significantly effect biofilm community structure [55]Biofilms are housed in a self-secreted matrix composed of polysaccharides, proteins, DNA,

and other materials (detritus from lysed cells, abiotic material like precipitates and corrosiveproducts, etc.) which performs many roles [78, 215]. First and foremost it serves as an anchoringstructure for its inhabitants, keeping them in place and protected from external mechanicalforces as well as hostile predators and dangerous chemicals. Though one should be mindfulthat, in the words of H.-C. Flemming, a biofilm is not a prison, meaning that it is necessaryfor biofilm inhabitants to be able to escape in order to form new colonies elsewhere. These topicswill be further pursued later. But for now we note that the matrix is thought to accomplisha number of other purposes as well; it provides a food source when needed and it protectsfrom dehydration; by keeping microbes in proximity to each other, the matrix plays a rolein community ecology, facilitating signalling, ecological interactions, and exchange of geneticmaterial [134]; conversely, it has been suggested that production of matrix material can serveas a sort of exclusion zone by pushing competing microorganisms away, thus improving accessto resources [256].

The physical makeup of a biofilm creates a heterogeneous environment, in part due toreaction-diffusion limitations on both small and large scales, resulting in a rich structure of

6


7/58

ecological niches [208, 260]. As a simple but important example, oxygen consumption in alayer at the top of a biofilm can create a low oxygen environment in the sublayer below that isfavorable for anaerobic organisms (those that function without oxygen). One such instance ofthis commensalism is surface corrosion by anaerobic sulfate reducing bacteria protected in the

depths of a biofilm [139, 224], a phenomenon that might be exploitable in the form of microbialfuel cells [185]. Even single-species biofilms, though, can present phenotypic heterogeneity intheir ecology by changing genetic expression patterns in response to spatially varying envi-ronmental conditions within the biofilm. Regarding the biofilm as a whole organism, spatialheterogeneity-induced niche structure confers considerable flexibility and may be one of thereasons that bacteria have such a broad role in geocycling.

2 Quorum Sensing

It has been known for some time that bacteria can communicate with each other using chem-

ical signalling molecules. As in higher organisms, these signalling processes allow bacteria tosynchronize activities and behave as multicellular microorganisms. Many bacteria produce,release, and respond to small signal molecules in order to monitor their own population densityand control the expression of specific genes in response to change in population density. Thistype of gene regulation is known as quorum sensing (QS) [238].

Quorum sensing systems have been shown to regulate the expression of genes involved inplasmid transfer, population mobility, biofilm formation, biofilm maturation, dissolution ofbiofilms and virulence [246, 186, 194, 22]. While there have been a number of articles onthe connections of biofilm formation to quorum sensing [51, 99, 162], the specific relationshipsare still a matter of debate [122, 163]. The experiments are apparently difficult; for example,

Stoodley et al. [209] initially reported that under high shear conditions biofilms formed by aQS mutant were visually indistinguishable from wild type biofilms. It was found later througha more refined analysis that there are discernable differences (see eg. [177]).

There are many different types of quorum sensing systems and circuits and many bacteriahave multiple signalling systems. The receptors for some signalling molecules are very specificwhich implies that some types of signalling are designed for intraspecies communication. Onthe other hand, there are universal communication signalling molecules produced by differentbacterial species as well as eukaryotic cells which promote interspecies communication. Somespecies of bacteria can manipulate the signalling of, and interfere with, other species abilitiesto assess and respond correctly to changes in cell population density. Cell to cell signalling, andinterference with it, could have important ramifications for eukaryotes in the maintenance of

normal microflora and in protection from pathogenic bacteria. For recent reviews of the varioustypes of quorum sensing systems see [109, 246, 247].

One type of quorum sensing signaling in Gram-negative bacteria is mediated by the secretionof low-molecular-weight compounds called acyl homoserine lactones (AHL). Because these watersoluble molecules are small and can permeate cell membranes, they cannot accumulate to criticalconcentrations within the cells unless the bacteria are at a sufficiently large cell density or thereis some type of restriction on chemical diffusion. AHL signals are used to trigger expression of

7


8/58

specific genes when the bacteria have reached a critical density. This census-taking enablesthe group as a whole to express specific genes only at large population densities.

Most often the processes regulated by quorum sensing are ones that would be unproduc-tive when undertaken by an individual bacterium but become effective when initiated by the

collective. For example, virulence factors are molecules produced by a pathogen that influencetheir hosts function to allow the pathogen to thrive. It would not be beneficial for a smallpopulation to produce these factors if there were not sufficient numbers to mount a viableattack on the host. This cell-to-cell communication tactic appears to be extremely importantin the pathogenic bacterium Pseudomonas aeruginosa. It has been estimated that 6-10% ofall genes in Pseudomonas aeruginosa are controlled by AHL quorum sensing signaling systems[195, 225].

Perhaps the best characterized example of AHL QS is in the autoinduction of luminescencein the symbiotic marine bacterium Vibrio fischeri, which colonizes light organs of the Hawaiiansquid Euprymna scolopes [80]. When the bioluminescent bacteria were grown in liquid cultures

it was noted that the cultures produced light only when sufficiently large numbers of the bacteriawere present [91]. The original thought was that the growth media itself contained an inhibitorwhich was degraded by the bacteria so that when large numbers were present, bioluminescencewas induced [115]. Later it was shown that luminescence was initiated not by removal ofan inhibitor but instead by accumulation of an activator molecule, the autoinducer [151, 64].This molecule is made by the bacteria and activates luminescence when it has accumulated toa sufficiently high concentration. Bacteria are able to sense their cell density by monitoringautoinducer concentration and do not up regulate genes required for bioluminescence until thepopulation size is sufficient. The squid uses the light produced by bacteria to camouflage itselfon moonlit nights by masking its shadow and the bacteria benefit because the light organ isrich in nutrients allowing for local cell densities which are not attainable in seawater. The QS

molecule produced by V. fischeri, first isolated and characterised in 1981 by Eberhard et al.[65], was identified as N-(3-oxohexanoyl)-homoserine lactone (3-oxo-C6-HSL). Analysis of thegenes involved in QS in V. fischeri was first carried out by Engebrecht et al. [70]. This led tothe basic model for quorum sensing in V. fischeri that is now a paradigm for other similar QSsystems.

The fundamental mechanism of AHL based quorum-sensing is the same across systems,although the biochemical details are often very different. These cell-to-cell signaling systemsare composed of the autoinducer, which is synthesized by an autoinducer synthase, and atranscriptional regulatory protein (R protein). The production of autoinducer is regulated by aspecific R-protein. The R-protein by itself is not active without the corresponding autoinducer,

but the R-protein/autoinducer complex binds to specific DNA sequences upstream of targetgenes, enhancing their transcription including notably genes that regulate the autoinducersynthesis proteins. At low cell density, the autoinducer is synthesized at basal levels anddiffuses into the surrounding medium, where it is diluted. With increasing cell density orincreasing diffusional resistence, the intra-cellular concentration of the autoinducer builds untilit reaches a threshold concentration for which the R-protein/autoinducer complex is effectivein up-regulating a number of genes [196]. An interesting fact is that different target genes can

8


9/58

be activated at different concentrations [196]. Activation by the R-protein of the gene thatproduces the autoinducer synthase creates a positive feedback loop. The result is a dramaticincrease in autoinducer concentration. Autoinducer concentration provides a rough correlationwith the bacteria population density and together with transcriptional activators allows the

group to express specific genes at specific population levels.Quorum-sensing controlled processes are often crucial for successful bacterialhost relation-

ships, both symbiotic and pathogenic in other ways as well. Some organisms compete withbacteria by disrupting quorum signal production or the signal receptor or by degrading thesignal molecules themselves. Many of the behaviors regulated by QS appear to be cooper-ative, for example, producing public goods such as exoenzymes, biosurfactants, antibiotics,and exopolysaccharides. In this view, secreting shared molecules is an inherently cooperativebehavior, and as such is exploitable by cheaters, e.g. bacteria that are unable to synthesizesignal molecules but are able to detect and respond the AHL signal molecules [193]. Theseinclude strains of Escherichia coli. This type of cheating allows bacteria to sense the presence

of other AHL-producing bacteria and respond to the AHL signal molecules without the expenseof producing signals themselves.Since most signalling molecules are small surfactant-like molecules which are freely diffus-

able, it has been recently suggested that the secretion of small molecules could also be considereda form of diffusion sensing or efficiency sensing [100, 181]. Microbes could produce and detectautoinducers as a method of evaluating the benefits of secreting more expensive molecules. Abacterium could use the signal to estimate mass-transfer properties of the media and spatialdistribution of cells as well as cell densities. The thought is that it would be an inefficient useof resources to produce enzymes if they are going to be rapidly transported away by diffusionor some other means like advection. Note that this need not be a cooperative strategy sincea single bacterium might find itself in a confined region where the autoinducer could reach

threshold without any other cells present.

2.1 Modeling of QS

While quorum sensing has been described as the most consequential molecular microbiologystory of the last decade [27, 252], the history of mathematical modelling of QS is fairly brief.For a very nice survey up to about 2005 see Ward [236]. Modeling of QS at the molecularlevel was started by the almost simultaneous publications of James et al. [108] and Dockeryand Keener [58]. These models indicate that QS works as a switch via a positive feedback loopwhich gives rise to bistability for sufficient cell density, an up regulated and down regulatedstate. There have been several extensions of these types of deterministic models including muchmore regulatory network detail [73, 74, 150, 88, 222] as well as stochasticity [89, 143, 190].

2.2 The Model

The quorum-sensing system ofP. aeruginosais unusual because it has two somewhat redundantregulatory systems [167, 168] . The first system was shown to regulate expression of the elastaseLasB and was therefore named the las system. The two enzymes, LasB elastase and LasA

9


10/58

lasR

lasI

rhlR

rhlI

+

+

+

+

+

+-

+

+

-

Vfr

LasR

LasR

LasI

RsaLGacA

RhlR

RhrR

RhlI

C4-HSL

3-oxo-C12-HSL

rsaL

Figure 7: Schematic diagram showing the gene regulation for the las and rhl systems in P.aeruginosa.

elastase, are responsible for elastolytic activity which destroys elastin-containing human lung

tissue and causes pulmonary hemorrhages associated with P. aeruginosa infections. The lassystem is composed of lasI, the autoinducer synthase gene responsible for synthesis of theautoinducer 3-oxo-C12-HSL, and the lasR gene that codes for transcriptional activator protein.The LasR/3-oxo-C12-HSL dimer, which is the activated form of LasR, activates a variety ofgenes ([251, 164, 165, 168, 101]), but preferentially promotes lasI activity ([196]). The lassystem is positively controlled by both GacA and Vfr, which are needed for transcription oflasR. The transcription of lasI is also repressed by the inhibitor RsaL.

The second quorum-sensing system in P. aeruginosa is named the rhl system because ofits ability to control the production of rhamnolipid [156]. Rhamnolipid has a detergent-likestructure and is responsible for the degradation of lung surfactant and inhibits the mucociliarytransport and ciliary function of human respiratory epithelium. This system is composed of rhlI,

the synthase gene for the autoinducer C4-HSL, and the rhlR gene encoding a transcriptionalactivator protein. A diagram depicting these two systems is shown in Fig. 7 [54].

We make several simplifying assumptions as in [58, 72]. First if we assume that there isno shortage of substrate for autoinducer production then there is no need to explicitly modelthe biosynthesis of 3-oxo-C12-HSL and C4-HSL by LasI and RhlI. Second, there is evidencethat many proteins are more stable than the mRNA that code for them (see, for example,[6, 29, 68]). Assuming this is the case then LasR mRNA, lasI mRNA, rhlI mRNA, and rsaL

10


11/58

Table 1: Variables used to identify concentrations.

Variable concentration

A1 [3-oxo-C12-HSL]A2 [C4-HSL]P1 [LasR]P2 [RhlR]P3 [RsaL]C1 [LasR/3-oxo-C12-HSL]C2 [RhlR/C4-HSL]C3 [RhlR/3-oxo-C12-HSL]E1 (External) [3-oxo-C12-HSL]E2 (External) [C4-HSL]

mRNA are much shorter lived than LasR, LasI, RhlR and RsaL, respectively. If we assume thatall messenger RNAs are produced by DNA at rates that are Michaelis-Menten in type, then,assuming the mRNAs are in quasi-steady state, it follows (see eg. [58]) that production of LasR,RasL and RhlR follow Michaelis-Menten kinetics. We will also assume that the production of3-oxo-C12-HSL and C4-HSL also follow Michaelis-Menten kinetics. Lastly we assume that theautoinducers freely diffuse across the cell membrane.

Following [72] the intercellular dynamics is described by a system of eight coupled differentialequations that model the intracellular concentrations of LasR, RhlR, RasL, 3-oxo-C12-HSL,C4-HSL, the LasR/3-oxo-C12-HSL complex, the RhlR/C4-HSL complex and the RhlR/3-oxo-

C12-HSL complex. We introduce variables for all the concentrations (shown in Table 1) as wellas the external concentrations of the autoinducers, 3-oxo-C12-HSL and C4-HSL.

If we assume that all the dimers C1, C2 and C3 are formed via the law of mass action atrate rCi and dissociate at the rate dCi , i = 1, 2, 3, then

dC1dt

= rC1P1A1 dc1C1, (1)dC2dt

= rC2P2A2 dc2C2, (2)dC3dt

= rC3P2A1 dc3C3. (3)

The equations for production of the proteins under the assumptions described above are given

11


12/58

by

dP1dt

= rC1P1A1 + dc1C1 dp1P1 (4)

+VP1 C1

KP1 + C1+ P01 ,

dP2dt

= rC2P2A2 + dc2C2 dp2P2 (5)

+VP2C1

KP2 + C1+ P02

rC3P2A1 + dc3C3dP3dt

= dP3P3 + VP3C1

KP3 + C1+ P03 . (6)

Similarly, for the autoinducers,

dA1dt

= rC1P1A1 + dC1C1 rC3P2A1 + dC3C3 dA1A1 (7)

+VA1C1

C1 + KA1(1 +P3

KP3r)

+ A01 + 1(E1 A1),

dA2dt

= rC2P2A2 + dC2C2 + VA2C2

C2 + KA2+ A02 + 2(E2 A2). (8)

Next, we need to determine how the density of organisms controls the activity of thisnetwork. We assume that autoinducer Ai (i = 1, 2) diffuses across the cell membrane, and

that the local volume fraction of cells is . Then, by assumption, the local volume fraction ofextracellular space is 1 . As indicated above, the extracellular autoinducer is assumed todiffuse freely across the cell membrane with conductance i and to naturally degrade at rate kEi.We remark that there is some evidence that the transport of the autoinducer may involve bothpassive diffusion and a cotransport mechanism [166]. For this model we assume that diffusionalone acts to transport the autoinducer (as is apparently correct for the rhl system [166]).

If we suppose that the density of cells is uniform and the extracellular space is well mixed,then the concentration of autoinducer in the extracellular space, denoted Ei, is governed by theequation

(1 )

dEidt

+ kEiEi

= i(Ai Ei), for i = 1, 2 (9)

Here we emphasize that the factor 1 must be included to scale for the difference betweenconcentration in the extracellular space and concentration viewed as amount per unit totalvolume. Similarly, the governing equation for intracellular autoinducer Ai must be modified toaccount for the cell density. For example, (8) becomes

dA2dt

= rC2P2A2 + dC2C2 + VA2C2

C2 + KA2+ A02 +

2

(E2 A2). (10)

12


13/58

A bifurcation diagram for the steady states for (1)-(8) is shown in Figure 8. The parameter provides a switch between the two stable steady solutions. For small values of , there is aunique stable steady state with small values of A1 and A2. As increases, two more solutionsappear (a saddle node bifurcation), and as increases yet further, the small solution disappears

(another saddle-node) leaving a unique solution, thus initiating a switch. For intermediatevalues of , there are three steady solutions. The small and large values are stable steadysolutions while the intermediate solution is unstable, a saddle point. It is easy to arrangeparameter values that have this switching behavior, and the switch can be adjusted to occurat any desired density level.

One can give the following verbal explanation of how quorum sensing works. The quantityA1, the autoinducer, is produced by cells at some nominal rate. However, each cell mustdump its production of A1 or else the autocatalytic reaction would turn on. As the density ofcells increases, the dumping process becomes less effective, and so the autocatalytic reaction isenabled.

It is interesting that, to the best of the authors knowledge, bistability has not yet beenobserved experimentally. This may be due to difficulties such as the requirement for a reportergene product with an extremely short half-life. Haseltine and Arnold [98] have shown exper-imentally in an artificial quorum-sensing network that simple changes in network architecturecan change the steady-state behavior of a QS network. In particular, it is possible to obtaingraded threshold and bistable responses via plasmid manipulations. They also have shown thatthreshold response is more consistent with the wild type lux operon than the bistable response.It would seem possible to conduct experiments to explore the multistability in synthetic geneticcircuits by using the approach described by Angeli et al. [10].

The first population level models were developed by Ward et al. [233] including populationgrowth. This system models subpopulations of up-regulated and down-regulated cells and a

single QS molecule species, predicting that QS occurs due to the combination of the effects ofincreased QS molecule concentration as the population size increases and the induced increaseof the up-regulated subpopulation. In a relatively short time (compared to the time scale forgrowth) there is a rapid increase of the QS molecule concentration. Ward et al. [234] extendedthese models to include a negative feedback mechanism known to be involved in QS. A similarmodeling approach has been developed to investigate QS in a wound [127].

Since the virulence ofP. aeruginosa is controlled by QS, there is the possibility that agentsdesigned to block cell-to-cell communication can act as novel antibacterials. One advantage ofthis approach is that it is nonlethal and hence less selective for resistance than lethal methods.Anguige et al. [7] have extended the basic model developed in [233] to investigate therapies that

are designed to disrupt QS. Two types of QS blockers were investigated, one which degradesthe AHL signal and another that disrupts the production of the signal by diffusing throughthe cell membrane and destabilising one of the proteins necessary for the signaling moleculeproduction. A conventional antibiotic is also investigated. Through numerical simulations andmathematical analysis it is shown that while it is possible to reduce the QS signal (and hencevirulence) to a negligible level, the qualitative response to treatment is sensitive to parametervalues.

13


14/58


15/58

Koerber et al. have published an interesting application of deterministic population levelmodels [128]. They use a deterministic model for QS to help develop a stochastic model for theescape of Staphylococcus aureus from the endosome. The deterministic model includes the QSmolecule, up and down-regulated cell types, an exo-enzyme, and the thickness of the endosome.

The deterministic model is used to validate the escape time asymptotically for the stochasticmodel of a single cell.

Muller et al. [150] have developed a deterministic model that includes the regulatory networkfor QS in a single cell and have incorporated this single cell model into a population level model.The population model is a spatially structured model. The scaling behavior with regard to cellsize is studied and the model is validated with experimental data. It is interesting that thesubmodel for regulatory system has bistable behavior but at the population level a thresholdeffect. The analysis shows that a single cell is mainly influenced by its own signal and thus iscomputing diffusion sensing. However, interaction with other cells does provide a higher ordereffect. Note that it is assumed that distances between cells are large compared to cell sizes.

The first biofilm models incorporating QS were due to Chopp et al. [35, 36] and Ward et al.[232]. Both groups modeled the spatio-temporal evolution of a growing one-dimensional biofilm.Chopp et al. [35, 36] consider the growth of a biofilm normal to a surface whereas Ward etal. [232] model a biofilm spreading over a surface using a thin film approach. Both approachesallow one to study the distribution of up-regulated cells as well as the spatio-temporal changesof the signal within the biofilm. The modeling results of Chopp et al. [35, 36] show that for QSto be up-regulated near the substratum, cells in oxygen-deficient regions of the biofilm must stillbe synthesizing the signaling compound. As one would expect, the induction of QS is relatedto biofilm depth; once the biofilm grows to a critical depth, QS is up-regulated. The criticalbiofilm depth varies with the pH of the surrounding fluid. One very interesting prediction isthat of a critical pH threshold above which quorum sensing is not possible at any depth. The

results indicate the importance of the relationships among metabolic activity of the bacterium,signal synthesis, and the chemistry of the surrounding environment.

Ward et al. [232] consider the growth of a densely packed biofilm spreading out on a surfaceand include the QS model developed in [233]. The analysis shows that there is a phase of biofilmmaturation during which the cells remain in the down-regulated state followed by a rapid switchto the up-regulated state throughout the biofilm except at the leading edges. Travelling wavesof quorum sensing behaviour are also investigated. It is shown that while there is a range ofpossible travelling wave speeds, numerical simulations suggest that the minimum wave speed,determined by linearization, is realized for a wide class of initial conditions.

Anguige et al. [8] extended the results of [7] to the biofilm setting by modifying the model

[232] to include anti-QS drugs. Using numerical methods, it is shown that there is a criticalbiofilm depth such that treatment is successful until this depth is reached, but fails thereafter(see Section 5 below for discussion of biofilm response to antimicrobials). It is interesting thatin the thick-biofilm limit, the critical concentration of each drug increases exponentially withthe biofilm thickness; this is different than the behaviour observed in the corresponding modelfor spatially homogeneous population of cells in batch culture, [7], where it is shown that thecritical concentrations grow linearly with bacterial carrying capacity.

15


16/58


17/58

3 Growth

Much of the earliest work in biofilm modeling was, and continues to be, directed towardspredicting growth balance, often with practical engineering applications in mind such as waste

remediation, the work of Wanner & Gujer being particularly influential [11, 33, 121, 149, 228,229]. These are generally 1D models combining reaction-diffusion equations for nutrient andother substrates (such as electron donors or acceptors) together with some sort of growth-generated velocity and a moving boundary, see below. Close examination, confirmed by confocalmicroscopy, however revealed that 1D approximations could be suspect [21, 86]. First effortsat multidimensionality made use of cellular automata-based models [102, 103, 137, 155, 169,173, 174, 250] (work by Picioreanu [172] has had great impact), and subsequently individual-based models [132, 133], to study growth processes in 2D and 3D. In these systems, new-growthbiomaterial was introduced by pushing old material out of the way according to given rules for example, newly created biomaterial displaced biomaterial in an occupied neighboring cell,which in turn displaced biomaterial in a next neighboring cell, etc., until an open, unoccupiedlocation could be found. Substrate was treated differently; in contrast to discrete processingof biomass, in most cases substrate transport and consumption was handled as a continuumreaction-diffusion process. While appealing in their simplicity, these semi-discrete models suffera certain degree of arbitrariness in their choice of rules. Hence, more recently, fully continuummodels based on standard continuum mechanics have been introduced; this is where we placeour attention here. The idea is to treat the biofilm as a viscous or viscoelastic material thatexpands in response to growth-induced pressure, much like avascular tumor models [92, 93]. Wenote that other continuum models based on chemotactic and diffusive mechanisms have alsobeen considered [2, 66, 117], and continuum and individual-based methods were combined in [3].In the absence of observational evidence distinguishing these approaches (though see [126]), and

while recognizing that different methodologies may have the advantage in different situations,we focus on the classical continuum mechanics platform. We remark that connecting continuummodels to the cellular scale has been considered in [253, 254], but, lacking many details of themicroscale, this line of research is currently difficult.

3.1 Example: 1D, single species, single limiting substrate model

We consider the half-space z > 0 in R3, Figure 9, with a wall placed at z = 0. A singlespecies biofilm occupies the domain (in the 1D case) 0 < z < h(t) with height function h(t).The portion of the domain z > h(t) is the bulk fluid region. A reaction limiting substrate, e.g.oxygen, with concentration c(z, t) is present throughout z > 0. It is assumed that the bulk fluid

is itself divided into a well-mixed free-stream in which a constant substrate concentration c0is maintained, and a diffusive boundary layer of thickness L adjacent to the biofilm in whichsubstrate freely diffuses [114]. Within the biofilm, substrate diffuses and is consumed. Overall,c obeys

ct = ((z)cz)z H(h z)r(c), 0 < z < h + L,

17


18/58

L

h

BOUNDARY LAYER

BULK FLUID

(WELL MIXED)

z

Figure 9: biofilm occupies the 1D layer 0 < z < h and limiting substrate diffuses in fromz = h + L.

r(c)

c

Saturation

Half

Max

Figure 10: Typical form of saturating rate function r(c) = kc(ch + c)1. The parameters k, ch

are the maximum and half-saturation respectively.

where H is the heaviside function and r is a consumption rate function. We assume r(0) = 0,and r(c) > 0 for c > 0. Generally, we can expect a saturating form for r such as in Figure 10,e.g. the Monod form r(c) = kc(ch+c)

1 [231], although in some circumstances a linear functionr(c) = kc + m (with k or m possibly zero) is adequate. The diffusivity (z) is discontinuousacross the biofilmbulk fluid interface (and may vary within the biofilm as well), but in practicethe variation is not that large, especially for smaller molecules such as oxygen as biofilms arewatery and apparently fairly porous [207]. The diffusive time scale 2/, where is a systemlength scale ( = h say in the present setup), is usually very small compared to other relevanttime scales, see Figure 6, in which case the quasi-steady approximation

((z)cz)z = H(h z)r(c), 0 < z < h + L, (11)is generally appropriate. Boundary conditions are c(h + L, t) = c0, i.e., constant substrateconcentration in the well-mixed zone, and cz(0, t) = 0, i.e., no flux of substrate through thewall.

Substrate consumption rate r(c) drives biofilm growth at rate g(r(c)) (typically a formg = (r(c) b)Y is used, where Y is a yield coefficient and b is a base subsistence level), which

18


19/58

in turn generates a pressure p(z, t) that drives deformation with velocity u(z, t). We balancepressure stress with friction, obtaining

1

u =

pz, 0 < z < h, (12)

where is a friction coefficient [59]. Alternatively, a pressure-viscosity balance was used in [37].More elaborate stress balances will be described later.

Biofilm, being mostly water [44], can be considered to be incompressible, with the conse-quence in 1D that u is determined by

uz = g, 0 < z < h. (13)

Combining (12) with (13) we obtain

pzz =

1

g, 0 < z < h. (14)

At the wall, u = 0 so that pz|z=0 = 0. Note that the biofilm-bulk fluid interface moves accordingto

ht = u(h, t) = pz(h, t) =h0

g(c(z, t)) dz, (15)

so that the interface moves so as to exactly accomodate new growth or decay. Also observethat p is linear in 1/ so, for this same reason, that the interface speed ht is independent ofthe friction parameter . Completing the system description, a second boundary condition for

p is provided by the assumption that pressure is constant in the bulk fluid, so we set p = 0 atz = h.

In order to evaluate the right-hand side of (15), we need to solve (11) for the substrateconcentration c(z, t), 0 < z < h. For z > h, c is linear in z and a short calculation reduces (11)on 0 < z < h to

czz =r(c)

, cz|z=0 = 0, c|z=h + Lcz|z=h = c0 (16)

Introducing the variables a = c, b = cz, results in the system

a = b

b = r(a)/(17)

and phase-plane analysis reveals, see Figure 11, that a solution of (16) corresponds to an orbitin the positive quadrant connecting the a axis to the line a + Lb = c0 over time z = 0 toz = h. Under mild smoothness conditions on r, we observe that the orbit starting from theorigin takes infinite time to reach the target line a + Lb = c0 (corresponding to an infinitelydeep layer) and the orbit starting from a = c0 takes 0 time (corresponding to a degeneratelayer), with continuous variation of orbit time, i.e., depth, between. Thus for a layer of givendepth, equation (11) has a solution. This solution is unique ifr is monotone but otherwise maynot be. Note that ifh is large, then the solution spends most of its time near the (a,b)-planeorigin, i.e., c(z) is close to zero except near z = h.

19


20/58

c0

c0

/L

c0

a + Lb =

a

b

Figure 11: Phase plane representation of solutions of system (17) corresponding to boundary

conditions given in (16).

3.2 Active Layers and Microenvironments

System (17) illustrates an important physical phenomenon of biofilms, namely the existence ofsharp transitions resulting in what are sometimes called microenvironments [45]. As just noted,if layer height h is sufficiently large, then solutions of (17) spend most of their time close tothe origin, the so-called substrate limited regime. In this limit, r(c) r(0)c and (17) can beapproximated by

a = b

b = (r(0)/)a,

i.e., a = (r(0)/)a. Thus solutions behave like exp((z h)/) with decay length =

/r(0).In terms of the biofilm, this indicates exponential decay of substrate concentration and hencea sharp transition between high and low biological activity. Typical parameter values used inmodels amount to 10 m [231], a number consistent with observations.

The existence of active and inactive layers (see Figure 12) is a central component of biofilmbehavior, as will be discussed later, and has been well studied especially since the introduction ofmicrosensor and CLSM technology, e.g. [113, 114, 119, 204, 249, 242, 243, 257, 259]. Appearanceof other types of microenvironments demarcated by sharp localized chemical transitions is also a

frequently observed phenomenon and an indicator of ecological diversity [180]. Sharp boundarylayers are often seen for example at electron acceptor transitions, usually separating importantniche regions. These boundaries can even move in time, for example due to the day-night cycleand its control of photosynthetic oxygenesis [183].

20


21/58

Figure 12: Active layer (counter-stained green) in a Pseudomonas aeruginosa biofilm (stainedred). Scale bar is 100 m. Reproduced with permission of the American Society for Microbi-ology from [257]

.

3.3 Multidimensional, multispecies, multisubstrate models

The simple model of the previous section presents a useful picture of the role of diffusivetransport; we will return to it later. However biofilms are often not 1D systems; rather channelsand mushroom-like structures are frequently observed [21] (Fig. 5) with important consequences,e.g. [53], that can only be addressed by models allowing multidimensional structures [37, 59,62]. Further, we have neglected much important ecology in reality, biofilms are usuallycomposed of many species, upwards of 500 in dental biofilms for example [145], includingboth autotrophs (users of inorganic materials and external energy) and heterotrophs (users oforganic products of autotrophs) cycling and throughputting numerous products and byproducts.Even within individual species, considerable behavioral variation exists (indeed, as alreadymentioned even the notion of microbial species is unsettled). Inactive and inert material alsoimpacts spatial distribution and hence influences ecology [121]. So we introduce Nb differentbiophases [229, 230] material phases that could be species or other quantities like phenotypicvariants, extracellular material, inert biomass, free water, etc. each with concentration bi(x, t),i = 1, . . . , N b. Similarly, confronted with the ecology of an interactive microflora, simplifyingto a single limiting substrate is also likely insufficient. So we generalize to a set of substrates

with concentrations ci(x, t), i = 1, . . . , N c, satisfying mass balance equations

ci,t = (ici) + Briwhere B(x, t) is the characteristic function of the biofilm-occupied region and ri are the multi-species usage rate functions. As before, the time derivatives can typically be neglected, leavingthe quasistatic approximations

(ici) = Bri. (18)

21


22/58


23/58


24/58

Figure 13: Simulation of a two component system (active and inert biomass) with a singlelimiting substrate (oxygen). Left: molecular oxygen concentration (kg/m3). Right: active(green) and inert (blue) biomass.

orthogonal to the zeroth order pressure gradient which is parallel to z. Thus the interfacevelocity is, to first order,

h = n0 p|z=h = h(0) + eikx[h(1)(t)g(0)(h(0)(t), t) p(1)z (h(0)(t), t)] (26)(see equation (15)) where superscript (i) refers to the ith order quantity. The first term in theperturbed velocity, eikxh(1)(t)g(0)(h(0)(t), t), indicates perturbation enhancement due to bumps

extending into richer substrate. The second term, eikxp(1)z (h(0)(t), t), is considered below.Note that this pressure gradient term can be expected to oppose instability since the perturbedpressure increases towards the top of bumps where perturbed growth expansion is strongest.

Perturbation effects might be expected to be most noticeable in the substrate limited case under growth limitation (i.e, substrate saturation), substrate variation has little effect sosurface protrusions receive little advantage. Specializing then as previously to the substratelimitation case where r(c) = r(0)c, equation (11) linearizes, with constant, to

c(1)zz = (k2 + H(h(0) z)1r(0))c(1) (27)

with accompanying boundary conditions and interface (at z = h(0)) conditions. For z > h(0),(27) reduces to c

(1)zz k2c(1) = 0, which can be solved explicitly, and thus the interface prob-

lem (27) in turn reduces to a boundary value problem

c(1)zz = (k2 + 1r(0))c(1) (28)

on 0 < z < h(0) with computed boundary conditions that can also be obtained explicitly(see [59]). Note that substrate perturbation within the biofilm decays like exp((h(0) z)),

24


25/58

=

k2 + 2 (recall =

/r(0) is a measure of active layer depth), away from the interface.Relative to zeroth order quantities, short wave lengths (k2 1) decay quickly while long wavelengths (k2 1) decay on a roughly k-independent depth scale determined instead by theactive layer depth.

Given the perturbed substrate concentration c(1), we can then determine the perturbationto the pressure field p by linearizing the multidimensional generalization of (14) (setting = 1):

p(1)zz = k2p(1) G(c(0))c(1) (29)

where G(c(0)) = g(r(c(0)))r(c(0)) = g(0)r(0) > 0. Equation (29) has solutions of the form

p(1)(z) = Cek(zh(0)) k1

z0

Gc(1) sinh[k(z s)]ds,

on 0 < z < h(0) with C > 0, and so

p(1)z (h(0), t) = Ck +h(0)0

Gc(1) cosh[k(h(0) s)]ds.

The first term, Ck, is the expected suppressive term; increased growth at the top of bumpsinduces increased pressure and hence a positive perturbation to the pressure gradient. Thiseffect dominates for large k. The integral term mitigates (due to reduction of extra growthbecause of substrate usage and diffusion), but mitigation is small for k2 1 where c(1) decaysquickly. Hence, putting everything together in (26), we see a long-wave instability regularized atlarge k by pressure in the absence of mechanical forcing other than growth-induced expansivepressure, finger formation is predicted with a preferred wavelength.

4 Biofilm Mechanics

The simple mechanical environment assumed so far is not always the reality. Rather, presenceof exterior driving forces is probably common. Though composed of microorganisms, a biofilmis a macroscale structure that interacts with its environment as a macroscale material. Indeedit may well be that a principal fitness benefit of extracellular matrix formation (which is, afterall, a process with a cost) is the ability of biofilms to withstand external mechanical stress.Large-scale physics and small-scale biology are thus intertwined, and description of the long-time behavior of biofilm-environmental interactions would seemingly require an adequate model

of biofilm mechanics including constitutive behavior.Observations of biofilm mechanics have mainly been of two types: measurements of consti-

tutive response (e.g. linear regime stress-strain relations) [34, 123, 130, 131, 198, 210, 220, 221],and measurement of material failure (e.g. detachment due to mechanical failure) [144, 147,157, 158, 159, 176, 212]. The latter has not been addressed in great detail in modeling efforts detachment tends to be treated in an ad-hoc manner, the most popular technique being useof an erosion mechanism where material is lost locally from the biofilm-bulk fluid interface at a

25


26/58

rate proportional to the square of the local biofilm height or by a similar rule [202, 229, 255]. Itis clear that balance of new growth by more or less concurrent biomaterial elimination is a keyelement of long time quasi-stationarity in biofilms and it seems that detachment and hence gen-eral aspects of constitutive response play an important role in loss [31, 104, 187, 248]. Note that

mechanically determined detachment or erosion is not the only loss mechanism; dispersal [178]and predation [107, 138] can also be significant.

There have been some attempts to use mechanical principles in models in simplified ways [4,23, 56, 57, 63, 171]. We describe here one method for generalizing setups of the previous sectionsthrough use of a mixture model [37, 39, 124, 261, 262], for example following ideas for polymer-solvent two (or more) phase theory [61, 146, 218]. In the simplest version, the system is dividedinto two phases: a sticky, viscoelastic biomaterial fraction consisting mainly of cells and matrixand also a solvent fraction consisting mainly of free water. We denote the biomaterial andsolvent volume fractions by b(x, t) and s(x, t) with b + s = 1. The corresponding specificdensities with respect to volume fraction are denoted b(x, t) and s(x, t). As the biomaterial

is to a large extent also water, we assume both specific densities are constant. We define ub tobe the velocity of the biofilm fraction and us to be the velocity of the solvent fraction, and setaverage velocity u = bub+sus. Again, as biofilm is mostly water, we impose incompressibilityon u, i.e., u = 0. The aim then is to find (and solve) equations describing the time courseof the volume fractions which incorporate the relevant physical and biological influences.

4.1 Cohesion

We begin by considering forces of cohesion what is it that keeps a biofilm together, or,conversely, how would one pull a biofilm apart [77]? This is an interesting and importantquestion because biofilms need to balance their need for structural integrity with a capability for

effective dispersal. Indeed measurements suggest that floc (i.e., clump, see Fig. 5) detachment,as opposed to erosion of single or small numbers of cells, is significant, which is notable sinceflocs retain many of the advantages of biofilms [81]. It is believed that the biofilm matrix issticky in the sense that it includes a mesh of polymers with a tendency to form physical linksand entanglements [79]. We can construct a model of this tendency by introducing a so-calledcohesion energy E(b) of the form

E(b) =

f(b) +

2|b|2

dV (30)

where f(b) is an energy density of mixing, Figure 14, with a minimum at a volume fraction

b = b,0 where attractive forces are balanced by repulsive, packing interactions [124]. Ifattractive forces are weak at long distances, as is the case for hydrogen bonding for example,then f flattens as b 0. The term (/2)|b|2 is a distortional energy which penalizesformation of small-scale structure (necessary if for no other reason then for regularization) andalso introduces a surface energy penalty. We remark that we neglect mixing entropy effectsin (30), but that they can also be included, in which case E can be characterized as a freeenergy [37]. An alternative approach to modeling biomaterial adhesion in biofilms using Potts

26


27/58

Density

Energy

Biomaterial Volume Fraction

0 1

Figure 14: Representative mixing energy density. Note that biomaterial volume fraction b isconfined between 0 and 1. As b approaches 1, energy density may become large or infinite.

models can be found in [175]. We also note that surfactants may have an important role incohesion [50, 161],

Using (d/dt)b+(bub) = 0 (and neglecting change ofb due to growth), we can calculatedE

dt=

(b ) ubdV (31)

where = [f(b) 2b]I

is a cohesive stress tensor. Equation (31) takes the form of a work integral and thus indicatesthat biomaterial motion as dictated by biomaterial velocity ub does work. Hence the term bmust be present in the biomaterial force balance equation in order to maintain consistency.Introducing a frictional coupling between biomaterial and solvent fractions of the form (ub us), we obtain force balance equations

0 = b (ub us) + bp (32)0 =

(us

ub) + s

p (33)

where p is hydrostatic pressure. Equations (32) and (33) are stripped down force balances: weneglect inertial forces and assume for the moment that viscous and viscoelastic forces are smallcompared to friction. With some manipulation, we obtain from (32) and (33) that

bt

+ (bu) = [a(b)f(b)b] [a(b)2b] (34)

27


28/58

Figure 15: Phase separation low volume fraction biomaterial (left) is unstable to spinodaldecomposition, resulting in phase separation (right) with one phase of high biomaterial volumefraction (dark) and the other with no biomaterial.

where we suppose = 0bs and set a(b) = 10 bs =

10 b(1 b). In the 1D case,

incompressibility and boundary conditions dictate that u = 0, leaving us with a modifiedCahn-Hilliard equation. For sufficiently small b, f

< 0 resulting in instability, the so-calledspinodal decomposition [28]. As a consequence, low volume fractions of biomaterial will tend tocondense and phase-separate into a higher volume fraction concentrates, i.e., cohesive biofilms,expelling solvent in the process, Figure 15. That is, introduction of a cohesion energy has theeffect of transforming our model biofilm into a true, coherent material phase.

4.2 Viscoelasticity

There is by now a fairly extensive literature on viscoelasticity of biofilms, in particular ontheir characterization as viscoelastic fluids [106, 123, 130, 189, 198, 210, 221]; biofilms respondelastically to mechanical stress on short time scales and as a viscous fluid on long time scales.In order to accurately describe biofilm mechanics then, at least over short time scales, we needto replace the frictional balance described previously with a suitable viscoelastic dynamics [75].

Although the biomaterial phase consists of a complicated collection of a variety of types ofcells, polymers, and other materials, for ease we will simplify the dynamical description by, formechanical respects only, supposing it to be made up of uniform polymers. Following [60, 227]we introduce a function (x, q, t) tracking a weighting of polymers with center-of-mass x andend-to-end displacement q. Under this framework,

b(x, t) =

(x, q, t) dq.

Note that /b, with /b defined to be 0 when b = 0, is a probability density in q. We cannow extend the energy (30) to include elastic contributions as

E() =

f(b) + fe() +2|b|2

dV

with, in a simple version,

fe() = kT

ln(/b) + q

2

dq (35)

28


29/58

where is a polymer matrix density parameter. The log term is an entropy contribution thatopposes, for example, polymer extension in any one particular direction. The quadratic termis a contribution of a standard linear spring potential.

Proceeding as before, we calculate dE/dt with respect to the advective influence of a bio-

material velocity ub, obtaining the same form (31) with, in this more general setting,

= [f(b) 2b + kT(q2 + ln(/b))]I. (36)Assuming that biomaterial is deformed by the biomaterial velocity ub, then can be supposedto obey a Smoluchowski equation [60] of the form

t+ (u) = ( ) q (ubq) (37)

where is a mobility coefficient. The first term on the left is due to chemical mixing stress (asin the previous section) and the second is due to viscoelastic stress. Note that averaging over

q reproduces equation (34) for b.Classical linear molecular models indicate that the viscoelastic stress tensor is approximated

by M = 2kT

qq dq where is an elasticity parameter [136]. To obtain M, we can multi-ply (37) by qq and average over q, resulting in the equation

M

t+ (uM) uTb M Mub = < qq >q

where < > denotes averaging over q. The average on the righthand side reproduces thechemical forcing terms from earlier as well as new viscoelastic terms (from the third term in above, equation (36)). In the case of the simple elastic energy density contribution (35), theaveraging process produces a Maxwell fluid-like contribution. More complicated laws can ariseusing more involved physics [136]. See [227] for computational examples.

5 Tolerance and Disinfection

One of the singular aspects of bacterial biofilms is their relative tolerance to chemical attackas compared to free-floating, planktonic populations [148]. Many studies have noted that bac-terial cells within a biofilm require higher minimum antimicrobial concentrations (MIC) (orminimum biofilm eliminating concentration), even by a factor of 103 or more for effective con-trol as compared to the MIC for those same bacteria in their planktonic state, e.g. [118, 197].Such dosages are expensive and may in fact translate into dosages that exceed therapeuticallysafe levels. However, sub-MIC application can result for example in an infection whose symp-toms are only temporarily subdued by even an extended treatment regime but which recurfollowing treatment cessation. It is important to emphasize that we are not referring to strain-specific genetic mechanisms whereby individual cells from a special, resistant strain in somemanner acquire altered genes able to transform the specific target or access to that target bya particular class of antimicrobials [240], but rather a non-specific modulated population toler-ance to general chemical attack. That is, we consider a situation whereby the same population

29


30/58


31/58

both antimicrobial and reactant are neutralized, for example reduction of oxidizing agents likechlorine [52], and (iii) reaction with a catalytic agent in which antimicrobial is neutralized butthe reacting agent is unaltered by the process. An example of the latter is enzymatic activity ofextracellular catalase in protecting against hydrogen peroxide [69]. In each of the three cases,

the biofilm context is important even if matrix material is not directly involved in defense, theconfined and concentrated biofilm environment (relative to the planktonic environment) makesthese defense mechanisms much more effective.

For purposes of describing these possible mechanisms mathematically, we consider as beforea 1D biofilm domain 0 < z < h with the substratum at z = 0 and the bulk fluid interfaceat z = h, and set B(z, t) to be the concentration of free antimicrobial. Height h is assumedconstant, i.e., we suppose that no significant growth occurs over the time-scale of antimicrobialapplication and so do not consider here extended disinfection schedules. B satisfies a no-fluxcondition Bz|z=0 = 0 at the substratum and a Dirichlet condition B|z=h = B0 at the bulk fluidinterface. A diffusive boundary layer above the interface z = h could be included to allow for

mass transfer resistance, but does not qualitatively affect results. We suppose initial conditionsB(z, 0) = 0, 0 < z < h, i.e., application beginning at t = 0 of antimicrobial to an initiallyantimicrobial-free biofilm.

The aim is to describe the spatial profile of B as a function of time in order to model theeffects of penetration barriers. The concentration B satisfies an equation of the from

Bt = Bzz + f (38)

where f describes the three reactive mechanisms listed above [154, 205]. We set

f(B,S,Xb) = St R(B)Xb C(B).

Here (i) S is the sorbed antimicrobial concentration. We assume in fact that this reversiblecomponent of sorption equilibrates quickly relative to other relevant processes, i.e.,

S = g(B) (39)

where g is a monotonic, saturating function of the sort illustrated in Fig. 10, e.g., say, theMonod form g(B) = kSB/(B + KB). We also assume that S, being matrix associated, does notdiffuse. (ii) Xb is a density of binding sites, and R is a reaction rate function, also monotonicand saturating (with respect to B only we assume that biofilms do not choose to produce somany reaction binding sites such that saturation of f in Xb would occur), measuring removalrate of antimicrobial due to irreversible adsorption. The function Xb(z, t), 0 < z < h, is theconcentration of free binding sites (for antimicrobial) and satisfies the equation

Xb,t = R(B)Yb

Xb, Xb(z, 0) = Xb0, (40)

where Yb is a yield coefficient. (iii) C is a catalysis rate function, again monotonic and satu-rating with respect to its argument, measuring removal rate of antimicrobial due to catalytic

31


32/58

destruction. Note that g, R, and C may all depend on other factors such as microbial activityas one example, but these extra complications are neglected for purposes of clarity.

Using (39), equation (38) can be rewritten as

(1 + g(B))Bt = Bzz R(B)Xb C(B). (41)At the top of the biofilm, if the biocide dosage is saturating, then (41) simplifies to, approxi-mately,

Bt = Bzz C0,where the constant C0 is the saturation level of C(B), so that antimicrobial shows a quadrati-cally decreasing profile.

We focus attention on an antimicrobial depletion zone in the deeper parts of the biofilmwhere B is small although this zone may only exist for a finite time. Then equation (41) reducesapproximately (away from z = h) to

Bt = Bzz B (42)

where

=

1 + g(0), =

R1Xb + C11 + g(0)

and R1 and C1 are first order rate constants. Note that < . i.e., that the effective diffusivityis smaller than the true diffusivity as a consequence of mechanism (i), reversible sorption. Basedon available data, Stewart estimated the decrease to be approximately by a factor of 10 leadingto a penetration delay for typical biofilms of about 1 to 100 minutes [205]. While transient, thisdelay may be sufficient to allow other defensive responses, see below. The balance of diffusion

and reaction in (42) results in a decay distance =

0/(R1Xb0 + C1), similar to that seenin Section 3.2 in the context of substrate limitation and active layer formation, with resultingantimicrobial limitation below a reactive layer located at the top of the biofilm if is smallerthan h. Note that reversible sorption plays no role. A part of the reactive term though, namelyR1XbB/(1 + g(0)), corresponds to irreversible sorption and is temporary within any reactivelayer; Xb decays on time scale Yb/R1B0 as binding sites are occupied. The decay distance willthus also increase in time up to a maximum of =

0/C1.

On longer times, (38) equilibrates to

C(B) = Bzz ,

Below the resulting reactive layer, where B is small and C(B) C1B, antimicrobial concentra-tion is depleted exponentially with length scale =

/C1 [204]. Thus for those antimicrobials

like hydrogen peroxide that are susceptible to enzymatic deactivation, a permanent reactivepenetration layer results (at least, permanent on the time-scales considered here).

32


33/58

5.2 Altered Microenvironment

But many antimicrobials do not appear susceptible to a defensive mechanism like enzymaticdegradation and in fact are able to fully penetrate the biofilm at concentrations roughly equal

to the externally applied concentration. Nevertheless, even when full penetration occurs atlevels above the MIC, significant killing of microbes in the deeper regions of the biofilm isoften not observed, see as an example [48]. This lack of efficacy seems related to decreasedpotency against inactive microbes; in most cases antimicrobial-induced killing proceeds in theactive layer where metabolic activity is high, while the same antimicrobial agent may not be aseffective against less active organisms [25, 71, 226] such as those which predominate below theactive layer; microbes that are not eating will not ingest poison as easily, so to speak. Othermicroenvironmental parameters that vary with depth, such as pH, may also be relevant.

A basic model of this defense would look much like the previous one for antimicrobialpenetration barrier except with a focus on a limiting substrate concentration, e.g. oxygenconcentration, replacing antimicrobial concentration. That is, equation (42) would be replaced

byCt = Czz C (43)

where C is, say, oxygen concentration. Equation (43) is supplemented by a killing rate equationof the form

bt = k(C,B,b) (44)where b(z, t) is the concentration of microbes and k is a killing function that approaches 0 asC, B, or b approach 0. Even assuming that the antimicrobial is able to penetrate the entirebiofilm in quantity without significant resistance, it would still be the case that below the activelayer, where C is exponentially small, killing is limited. Note however that, in (43), substrate

usage generally depends on presence of live organisms, i.e., = (b) with approaching 0 asb does. Thus this model predicts that substrate penetration depth would gradually increase asdisinfection extinguishes live microbes in upper parts of the biofilm. Hence, it can generallybe expected that, barring other defense, eventually the biofilm will be uniformly penetrated bysubstrate and protected environmental conditions (absent antimicrobial) that favor inactivitymay disappear in time.

5.3 Adaptive Stress Response

Thus one method for overcoming diffusive and reactive barriers could be application of a sus-tained low-level dosage of antimicrobial agent. The motivation is just this: on the one hand,

diffusive resistance is temporary for many antimicrobials and can be overcome by sustainedattack, and on the other, reactive resistance, while perhaps not transient, cannot prevent dis-infection of an upper layer of biofilm which over time will be shed in the process exposingpreviously protected microorganisms. Further, as disinfection procedes, organisms near thesurface are killed and cease substrate usage so that the active layer may extend farther into thebiofilm, thus improving susceptibility. So a patient antimicrobial course might be expected tobe eventually rewarded with effective disinfection.

33


34/58

However, it seems that, in actuality, this strategy does not necessarily work well [15, 192].Indeed, available evidence suggests that high, short-duration dosing can be a more effectivedisinfection method than low, long-duration control attempts [94]. Why does this happen? Inat least some cases, microbes apparently respond to sublethal antimicrobial dosage by acquiring

increased tolerance, for example by upregulating production of protective enzymes [69, 85] or ofextracellular polysaccharides [13, 191] among other means [245]. This tolerance is specific to theapplied antimicrobial [16] unlike the general tolerance exhibited by persisters, see below. It isworth emphasizing that these adaptive or acclimitizing defenses are individually modulated andhence available to organisms in both biofilm and free-floating planktonic state; the differenceis that, even at super-MIC antimicrobial levels, sufficiently thick biofilms can result in a safezone where either antimicrobial does not penetrate in strength or else metabolic activity is low.Hence biofilms can contain microbes which are locally protected and thus allowed sufficienttime to mount a genetically controlled defense [216], a strategy not available to planktoniccommunities.

As an aside, we remark that it is perhaps not so surprising to discover that microorganismshave the know-how to mount a responsive defense to a wide spectrum of known antimicro-bials. The natural world is full of antimicrobial substances deployed in a literally billions ofyears old intranecine microbe-microbe conflict, not even to mention multicellular organisms andtheir responses, providing a rich source for discovery and subsequent use for our own advan-tage. However, those same billions of years have provided plenty of opportunity for microbialresponses to be constructed. And biofilms may be likely place for those defenses to operateeffectively for reasons already discussed.

To describe adaptive response mathematically within a simple 1D biofilm model, we canreturn to the multispecies, multisubstrate setup (18)-(24) particularizing to three species andtwo substrates, namely the three species u = live unadapted cell volume fraction, a = adapted

cell volume fraction, d = dead (or more generally inactive) cell volume fraction, and the twosubstrates cs = limiting substrate concentration, cB = antimicrobial concentration (previouslydenoted as B). Live cells grow in the presence of limiting substrate; dead cells do not obviouslybut nevertheless need to be tracked since they take up space (although one might allow deadcells to lyse and then disappear, possibly releasing substrate in the process). Cells can dieboth from natural causes and from disinfection by antimicrobial. Unadapted and adapted cellsinterconvert with rates depending on the antimicrobial concentration.

The equations for the cell densities are

u,t + (uu)z = (cB)u (cB)u + u(cs)u + (cB)aa,t + (ua)z = (cB)u

(cB)a + a(cs)a

(cB)a

d,t + (ud)z = (cB)u + (cB)a,

on the domain z [0, h(t)]. Here and are death rates (combining natural and disinfectionfatality), u and a are growth rates, is the adaption rate, and is the reversion rate.Quantities cs and cB satisfy

Dscs,zz = rs(u, a, d)

DBcB,zz = rB(u, a, d)

34


35/58

where rs and rB are reactive terms, Ds and DB are diffusivities, and, due to the relativerapidity of diffusive equilibration, we have neglected advective transport as well as the explicittime variation. The incompressibility condition u + a + d = 1 translates into the equation

uz =

1(u(cs)u + a(cs)a)

for velocity u where is the specific density, see (22). Boundary conditions are as described inSection 3.3.

If we modify the interface velocity (15) to include an erosion term [95, 202], as is commonlydone in 1D models, then h(t) satisfies

ht = u(h, t) e(h, u, a, d).Here e is an erosion function (usually chosen to be e(h) = h2) that allows, through balance ofgrowth and erosion, for non-trivial stable steady state solutions. Existence of such steady states

(not necessarily unique) for a similar system was shown in [217] with the necessary assumptionthat e(h)/h as h . An important result shown in [217] is that if adapted cells arenot subject to disinfection and if they grow faster than they revert (to the unadapted state),then biofilm eradication is not possible at any level of disinfectant dosage. Thus the ratio ofadaption rate to reversion rate is crucial in designing a dosage scheme. Not surprisingly, thisratio seems to be relatively large [17].

5.4 Persisters

There are certain antibiotics that are able to defeat all of the above defenses; as an example,fluoroquinolones are able to rapidly penetrate biofilms and also able to kill non-growing bacteria,

albeit less effectively than growing ones [24]. Nevertheless, even in this case, upon removal ofantibiotic challenge a small number of surviving cells are able to repopulate and return thebiofilm to viability. These special cells, called persisters, are in fact not particular in theirtolerance to fluoroquinolones or to any one organism, and have been consistently observedin antimicrobially dosed planktonic bacterial cultures since at least the 1940s when a studyobserved that about 1 out of 106 cells of staphylococci cultures survive sustained exposureto penicillin [20]. A signature of these super-tolerant cells is a biphasic killing curve, i.e.,a population versus disinfection time of the form p(t) = C1 exp(k1t) + C2 exp(k2t) withk1 > k2 and C1 C2, see Figure 16, with C1 being the initial population of normal cellsand C2 the initial population of tolerant cells, see e.g. [14]. Persister cells are not specific tobiofilms; indeed, most studies of them have focused on planktonic populations. However, asrecently realized by Lewis [141], persistence may be most dangerous in biofilms where survivingcells, though few in number, can hide from predators and have access to dead, lysed cellsproviding a nutrient-rich environment. In contrast, in a planktonic culture, isolated survivingcells are more vulnerable to predators or immune defenses.

Characterization of persisters has been uncertain. Persisters would seem not to be geneticvariants (they seed regrowth of populations indistinguishable from the original) nor cells in aspore-like state (spores are not observed) nor cells caught coincidentally in a special, quiescent

35


36/58

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

TIME

Logp

Figure 16: Biphasic killing curve. In the initial phase, a large susceptible population is rapidlyreduced, leaving, in the final phase, a small (initially 1/10000 of the size of the susceptible group)but tolerant population. For illustration purposes the total population p(t) = 107 exp(5t) +103 exp(0.5t) is plotted.

phase of cell division (the proportion of persister cells is too small) [14, 116, 141]. Unlikeadaptive responders, they show cross-tolerance to antimicrobials survivors of a challenge fromone type of antimicrobial can also tolerate challenges from representatives of widely different

families of antimicrobials [214]. Persister cells appear to be slowly growing cells, and in fact, thepercentage of persisters is larger in more slowly growing cultures (so-called stationary phase)as compared to faster growing cultures (so-called log phase) [116, 214].

So what are these mysterious persister cells? Despite many years of observations, thereis still no definitive answer to this question. A common hypothesis is that a persister cellis a special, protected phenotype of the main population that is slow-growing and generallytolerant in some way [141]; cells are able to switch in and out of the persister state, with agreater tendency towards persistence when environmental conditions are unfavorable, e.g. instationary state conditions. In some versions, switching is triggered by presence of antimicrobialstress [40, 140] and in others there is an everpresent stochastic jump rate [135, 188]. Mathe-

matically, these mechanisms are similar to the adaptive stress response discussed in Section 5.3,one difference though being that the growth rate a would be supposed to be very small forpersisters. (Recall also that adaptive response is antimicrobial-specific.) We thus consider hereinstead an alternative persister cells are senescent cells [12, 125] based on the observationthat bacterial cells can demonstrate aging effects [201]. The idea is that senescent cells areslow-growing, and slow-growing cells have increased tolerance to antimicrobials, so perhapspersister cells are defined by their senescence.

36


37/58

In order to consider implications of a persister-senescence connection we construct a modelof senescing microbial populations, necessitating introduction of structured population densi-ties [239] as follows. Set b(, t) to be the bacterial population at time t of senescence level ina well-mixed batch culture. Then b satisfies

bt(, t) + (v(, t)b(, t)) = rd(,C,B)b(, t), > 0, t > 0 (45)where v is a senescing rate function (we will set v = 1 here, i.e., senescence = chronologicalage, for simplicity), and rd is a death rate function which may depend on senescence level ,C is available limiting substrate concentration, and B is antimicrobial concentration. Moresenescent organisms are supposed less susceptible to antimicrobials than less senescent ones.For simplicity we neglect C-dependence and then a simple form for rd would be

rd(, B) = kd + H(B )kK (46)

where kd is a natural death rate and kK is a disinfection rate. The parameter is a tolerancecoefficient. The form (46) indicates that sufficiently senesced cells ( > B) are entirely toler-ant. This is a rather extreme choice, but the exact form of tolerance relation does not effectobservations to follow very much.

Equation (45) is a linear wave equation in the region > 0, t > 0, with characteristicsemanating from both the - and t-axes, and thus requires initial conditions b(, 0) = b0() andalso a birth boundary condition at = 0 of the form

b(0, t) =

0

rb(, C)b(, t) d, (47)

where rb is a birth rate function. Equation (45) and condition (47) are supplemented by

Ct =

0

rC(, C)b(, t) d,

an equation for substrate consumption, where rC is a substrate usage function.A typical computational experiment (from [125]) using parameter estimates from the general

literature is reprinted in Figure 17. The solid lines follow populations of cultures of cells subjectto antimicrobial challenge at hour 17 and then recultured at hour 27. The dashed line indicatesthe number of those sufficiently aged cells as to qualify as persisters. Note the biphasic killingduring dosage, as well as the relative scarcity of persisters during growth periods as opposed toduring stationary, especially late stationary phases. This latter difference is due to abundanceof young cells during exponential growth as well as the time lag for young cells to age their wayinto senescence.

The senescence model of persistance would seem to provide a reasonable qualitative match toavailable data, but so does the phenotype-switching model. In fact, the two are difficult to dis-tinguish at the population-modeling level. The reason for this is that a non-senescent/senescentdivide is rather similar to a dual phenotype compartmentalization. Non-senescent cells forma compartment and senescent cells form a compartment with conversion of a sort occuring

37


38/58

0 20 40 60 8010

0

101

102

103

104

105

106

107

(a)

time (hr)

population

all cells

persisters

0 10 20 30 40 50 60 7010

0

101

102

103

104

105

106

107

(b)

time (hr)

population

all cells

persisters

Figure 17: Exposure of a batch culture to an antimicrobial with senescence determined by

chronological age. (a) Antimicrobial applied during stationary phase at t=17 h and removedat t=27 h, at which time the surviving cells are recultured at the t=0 h nutrient level. (b)Antimicrobial applied during exponential phase at t=5 h and removed at t=15 h, at whichtime the surviving cells are recultured at the t=0 h nutrient level. Solid lines, all cells; dashedlines, persisters. Reprinted by permission of the Society for General Microbiology from [125].

between the two: non-senescent cells eventually become senescent, while senescent cells pro-duce, albeit at a slow rate, new (and non-senescent) offspring. So resolution awaits furtherobservational data. And of course, it would not be a great surprise if neither explanation weresufficient.

6 Conclusion

The physiology and ecology of microbes is a fast growing field of study. Advancing technologiesin microsensing and molecular analysis, and accompanying rapidly lowering barriers for theiruse, are transforming understanding and appreciation of these organisms in vivo and of thecentral roles they play in geo- and bio-processes. Newly and rapidly accumulating knowledgewill be important not only in advances in medical and industrial fields, but also in the contextof key geochemical cycles like the carbon and nitrogen cycles. But microbial systems arecomplicated enough that effective modeling may be essential for fully exploiting this knowledge.

The applied mathematics community would in our opinion be well-advised to take note.In this review we have aimed to raise questions of how macroscale physical factors might

influence composition, structure, and function of ecosystems within microbial communities.Historically, as in many biological contexts, ecological phenomena and physical forces actingon and within a given community have been treated separately. This dichotomy is unfortunateas, in many microbial systems, community structure and function are intimately connectedwith chemical and energetic processes and with protection from the physical environment. Of

38


39/58

course, embedding all the complications of a living, reacting ecosystem in a moving interface,multiphase material system raises many difficult problems; physics, chemistry, and biologyall combine at a variety of time- and length-scales. We have attempted to convey some ofthose difficulties here. That said, there are nevertheless many important topics that have been

omitted, especially a detailed discussion of microbial ecology and its role in biofilms and mats.Also on the omitted list are the role of viruses, mechanisms and influence of horizontal genetransfer, applications such as bioremediation, etc. All of these are additional ripe topics formathematical modeling.

7 Acknowledgements

The authors would like to note support through NSF/DMS 0856741 and thank Erik Alpkvist,Eric Becraft, Shane Nowack, Phil Stewart, Bruce Ayati, and Tianyu Zhang for their assistanceand contributions.

39


40/58

References

[1] D.G. Allison, The biofilm matrix, Biofouling 19, 139-150 (2003).

[2] E. Alpkvist, N. Overgaard, S. Gustafsson, A. Heyden, A new mathematical model forchemotactic bacterial colony growth. Wat. Sci. Tech. 49, 187-192 (2004)

[3] E. Alpkvist, C. Picioreanu, M.C.M. van Loosdrecht, A. Heyden, Three-dimensional biofilmmodel with individual cells and continuum EPS matrix, Biotech. Bioeng. 94, 961-979(2006).

[4] E. Alpkvist, I.Klapper, Description of Mechanical Response Including Detachment Usinga Novel Particle Method of Biofilm/Flow Interaction, Wat. Sci. Tech. 55, 265-273 (2007)

[5] E. Alpkvist and I. Klapper. A multidimensional multispecies continuum model for het-erogenous biofilm. Bull. Math. Biol. 69, 765-789 (2007)

[6] J. B. Anderson, C. Sternberg, L. K. Poulsen, M. Givskov, and S. Molin. New unstablevariants of green fluorescent protein for studies of transient gene expression in bacteria.Appl. Environ. Microbiol, 64:22402246, 1998.

[7] K. Anguige, J.R. King, J.P. Ward and P. Williams. Mathematical modelling of therapiestargeted at bacterial quorum sensing. Math. Biosci. 192, 39-83 (2004)

[8] K. Anguige, J.R. King and J.P. Ward. Modelling antibiotic- and anti-quorum sensingtreatment of a spatially-structured Pseudomonas aeruginosa population. J. Math. Biol.51, 557-594 (2005).

[9] K. Anguige, J.R. King and J.P. Ward. A multi-phase mathematical model of quorumsensing in a maturing Pseudomonas aeruginosa biofilm. Math. Biosci. 203, 240-276 (2006).

[10] D. Angeli, J. E. Ferrell, Jr., and E. D. Sontag Detect

Klapper2009 Mathematical Description of Microbial Biofilms Review4

Documents

Transcript of Klapper2009 Mathematical Description of Microbial Biofilms Review4