Klapper2009 Mathematical Description Of Microbial Biofilms Review4

Mathematical Description of Microbial Biofilms Isaac 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 some of their important aspects, including quorum sensing, growth, mechanics, and antimicrobial tolerance mechanisms.

1 1.1

Introduction 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 of debate [41, 235]) in part because classification and characterization of microbes have been comparatively difficult tasks. Recently, though, new efficient technologies for genetic and genomic cataloguing of microbial communities are changing the landscape of prokaryotic and archael ecology, uncovering surprising diversity in the process. Such information is important as microorganisms are closely connected with geochemical cycling as well as with production and degradation of organic materials. Thus a set of guiding principles is needed to understand the distribution and diversity of microorganisms in the context of their micro- and macroenvironments as a step towards the ultimate goal of constructing descriptions of full microbial ecosystems. 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 estimated 100 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 coli inhabiting a given person will easily exceed the number of people who ever lived: to paraphrase Stephen Jay Gould, though we may think we are living in a human dominated world, but in actuality it may be more truthful to say that we live in the “Age of Bacteria” [90]. The familar view of microbes in their free (planktonic) state is however not the norm; rather it is believed 1

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 biofilms and microbial mats, each consisting of large numbers of organisms living within a self-secreted matrix constructed of polymers and other molecules [1, 47, 46, 49, 96, 211], see Figures 14. (Microbes in collective behave very differently from their planktonic state; even genetic expression patterns change [110, 211].) These matrices serve the purposes of anchoring and protecting their communities in favorable locations [76], generally wet or damp but not always so [237], while providing a framework in which structured populations can differentiate and self-organize. In this way, as a result of internal spatial variation, microbial community ecology is 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 stresses and accompanying physical and chemical material properties. One can expect that, in both directions, the connection between physics and ecology in microbial communities is profound at 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 that biofilm formation is observed today in the Aquificales [184] (Figure 4), the lowest known branch of bacteria (on the phylogenetic tree of life), and, likewise, Korarcheota [111], the corresponding lowest known branch of archaea. Prokaryotic mats may once have been a dominant life form 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

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

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

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 consist of communities of photoautotrophs and attending heterotrophs, though communities based on chemoautotrophy 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 almost any 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 waste treatment and remediation [42, 219], just to name a few. Medical relevance is dramatic as well [26]. Quoting from the National Institutes of Health [179]: “Biofilms are clinically important, 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 and peritoneal dialysis catheters, in-dwelling catheters for hemodialysis and for chronic administration of chemotherapeutic agents (Hickman catheters); cardiac implants such as pacemakers, 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 associated nosocomial (hospital acquired) infection is by itself a leading cause of death in the United States [241].

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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 first stage, isolated planktonic-state microbes attach to a surface or interface. Once attached to a surface a phenotypic transformation (change of gene expression pattern) into a biofilm state occurs. At this point cells begin manufacturing and excreting extracellular matrix material. The exact nature of this material can vary greatly depending on microbial strain and, even within a single strain, on environmental conditions and history though its function shares commonality. With formation of an anchoring matrix, the nascent biofilm begins to grow. Secondary colonizing strains might arrive and join the new community. But as the biofilm matures, meso- and macro-scale physical realities like diffusion-reaction limitations and mechanical stress distribution become more important, probably to the extent that they significantly influence structure and ecology. Finally, the fully mature biofilm reaches a quasi-steady state where growth is balanced by loss through erosion and detachment due to mechanical stress, self-induced dispersal from starved subregions, and predation and loss to grazers and viruses. We recommend reference [32] to the reader, a remarkably forward-looking overview written by a pioneer of the subject of biofilm mechanics and modeling, Bill Characklis. On close inspection, we note a wide variation of relevant time scales, see Figure 6, ranging from fast time scales for bulk fluid dynamics and reaction-diffusion chemical subprocesses to slow time scales of biological development. The roughly ten orders of magnitude variation in range is a challenge to modelers of course, one that necessitates compromises. The usual practice is to introduce equilibrium in the fast processes: bulk fluid flow, when considered, is assumed steady over a quasistatic biofilm, and then advection-reaction-diffusion processes are also assumed quasisteady relative to the given fluid velocity field. Given those fluid velocity and chemical profiles, the modeler can take advantage of scale separation and focus on the slow growth and loss processes that are typically the ones of interest anyway. Notably missing from

5

Figure 6: Various time scales for biofilm related processes. Box 1 include processes related to bulk fluid (measured with respect to a biofilm length scale). Box 2 includes processes related to chemical processes (again, measured with respect to a biofilm length scale). Box 3 includes biological processes (essentially independent of biofilm length scale). Reprinted from [170] Fig. 6 though are time scales related to viscoelastic mechanics within the biofilm itself that may 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 corrosive products, etc.) which performs many roles [78, 215]. First and foremost it serves as an anchoring structure for its inhabitants, keeping them in place and protected from external mechanical forces as well as hostile predators and dangerous chemicals. Though one should be mindful that, in the words of H.-C. Flemming, “a biofilm is not a prison”, meaning that it is necessary for biofilm inhabitants to be able to escape in order to form new colonies elsewhere. These topics will be further pursued later. But for now we note that the matrix is thought to accomplish a number of other purposes as well; it provides a food source when needed and it protects from dehydration; by keeping microbes in proximity to each other, the matrix plays a role in community ecology, facilitating signalling, ecological interactions, and exchange of genetic material [134]; conversely, it has been suggested that production of matrix material can serve as a sort of exclusion zone by pushing competing microorganisms away, thus improving access to resources [256]. The physical makeup of a biofilm creates a heterogeneous environment, in part due to reaction-diffusion limitations on both small and large scales, resulting in a rich structure of

6

ecological niches [208, 260]. As a simple but important example, oxygen consumption in a layer at the top of a biofilm can create a low oxygen environment in the sublayer below that is favorable for anaerobic organisms (those that function without oxygen). One such instance of this 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 microbial fuel cells [185]. Even single-species biofilms, though, can present phenotypic heterogeneity in their ecology by changing genetic expression patterns in response to spatially varying environmental conditions within the biofilm. Regarding the biofilm as a whole organism, spatial heterogeneity-induced niche structure confers considerable flexibility and may be one of the reasons 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 chemical signalling molecules. As in higher organisms, these signalling processes allow bacteria to synchronize activities and behave as multicellular microorganisms. Many bacteria produce, release, and respond to small signal molecules in order to monitor their own population density and control the expression of specific genes in response to change in population density. This type of gene regulation is known as quorum sensing (QS) [238]. Quorum sensing systems have been shown to regulate the expression of genes involved in plasmid transfer, population mobility, biofilm formation, biofilm maturation, dissolution of biofilms and virulence [246, 186, 194, 22]. While there have been a number of articles on the connections of biofilm formation to quorum sensing [51, 99, 162], the specific relationships are 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 a QS mutant were visually indistinguishable from wild type biofilms. It was found later through a more refined analysis that there are discernable differences (see eg. [177]). There are many different types of quorum sensing systems and circuits and many bacteria have multiple signalling systems. The receptors for some signalling molecules are very specific which implies that some types of signalling are designed for intraspecies communication. On the other hand, there are universal communication signalling molecules produced by different bacterial species as well as eukaryotic cells which promote interspecies communication. Some species of bacteria can manipulate the signalling of, and interfere with, other species’ abilities to assess and respond correctly to changes in cell population density. Cell to cell signalling, and interference 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 various types of quorum sensing systems see [109, 246, 247]. One type of quorum sensing signaling in Gram-negative bacteria is mediated by the secretion of low-molecular-weight compounds called acyl homoserine lactones (AHL). Because these water soluble molecules are small and can permeate cell membranes, they cannot accumulate to critical concentrations within the cells unless the bacteria are at a sufficiently large cell density or there is some type of restriction on chemical diffusion. AHL signals are used to trigger expression of 7

specific genes when the bacteria have reached a critical density. This “census-taking” enables the 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 unproductive 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 influence their host’s function to allow the pathogen to thrive. It would not be beneficial for a small population to produce these factors if there were not sufficient numbers to mount a viable attack on the host. This cell-to-cell communication tactic appears to be extremely important in the pathogenic bacterium Pseudomonas aeruginosa. It has been estimated that 6-10% of all 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 luminescence in the symbiotic marine bacterium Vibrio fischeri, which colonizes light organs of the Hawaiian squid 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 bacteria were present [91]. The original thought was that the growth media itself contained an inhibitor which was degraded by the bacteria so that when large numbers were present, bioluminescence was induced [115]. Later it was shown that luminescence was initiated not by removal of an 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 to a sufficiently high concentration. Bacteria are able to sense their cell density by monitoring autoinducer concentration and do not up regulate genes required for bioluminescence until the population size is sufficient. The squid uses the light produced by bacteria to camouflage itself on moonlit nights by masking its shadow and the bacteria benefit because the light organ is rich 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 the genes involved in QS in V. fischeri was first carried out by Engebrecht et al. [70]. This led to the basic model for quorum sensing in V. fischeri that is now a paradigm for other similar QS systems. 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 systems are composed of the autoinducer, which is synthesized by an autoinducer synthase, and a transcriptional regulatory protein (R protein). The production of autoinducer is regulated by a specific 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 target genes, enhancing their transcription including notably genes that regulate the autoinducer synthesis proteins. At low cell density, the autoinducer is synthesized at basal levels and diffuses into the surrounding medium, where it is diluted. With increasing cell density or increasing diffusional resistence, the intra-cellular concentration of the autoinducer builds until it reaches a threshold concentration for which the R-protein/autoinducer complex is effective in up-regulating a number of genes [196]. An interesting fact is that different target genes can 8

be activated at different concentrations [196]. Activation by the R-protein of the gene that produces the autoinducer synthase creates a positive feedback loop. The result is a dramatic increase in autoinducer concentration. Autoinducer concentration provides a rough correlation with 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 bacterial–host relationships, both symbiotic and pathogenic in other ways as well. Some organisms compete with bacteria by disrupting quorum signal production or the signal receptor or by degrading the signal molecules themselves. Many of the behaviors regulated by QS appear to be cooperative, for example, producing public goods such as exoenzymes, biosurfactants, antibiotics, and exopolysaccharides. In this view, secreting shared molecules is an inherently cooperative behavior, and as such is exploitable by cheaters, e.g. bacteria that are unable to synthesize signal molecules but are able to detect and respond the AHL signal molecules [193]. These include 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 expense of producing signals themselves. Since most signalling molecules are small surfactant-like molecules which are freely diffusable, it has been recently suggested that the secretion of small molecules could also be considered a form of diffusion sensing or efficiency sensing [100, 181]. Microbes could produce and detect autoinducers as a method of evaluating the benefits of secreting more expensive molecules. A bacterium could use the signal to estimate mass-transfer properties of the media and spatial distribution of cells as well as cell densities. The thought is that it would be an inefficient use of resources to produce enzymes if they are going to be rapidly transported away by diffusion or some other means like advection. Note that this need not be a cooperative strategy since a 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 microbiology story 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 molecular level was started by the almost simultaneous publications of James et al. [108] and Dockery and Keener [58]. These models indicate that QS works as a switch via a positive feedback loop which gives rise to bistability for sufficient cell density, an up regulated and down regulated state. There have been several extensions of these types of deterministic models including much more regulatory network detail [73, 74, 150, 88, 222] as well as stochasticity [89, 143, 190].

2.2

The Model

The quorum-sensing system of P. aeruginosa is unusual because it has two somewhat redundant regulatory systems [167, 168] . The first system was shown to regulate expression of the elastase LasB and was therefore named the las system. The two enzymes, LasB elastase and LasA 9

Vfr + lasR

+

3-oxo-C12-HSL LasR

+

LasI

+ lasI

+

-

LasR + GacA

rsaL

+ rhlR

RsaL

C4-HSL RhlI

RhlR

+

RhrR

rhlI

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 las system is composed of lasI, the autoinducer synthase gene responsible for synthesis of the autoinducer 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 of genes ([251, 164, 165, 168, 101]), but preferentially promotes lasI activity ([196]). The las system is positively controlled by both GacA and Vfr, which are needed for transcription of lasR. 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 of its ability to control the production of rhamnolipid [156]. Rhamnolipid has a detergent-like structure and is responsible for the degradation of lung surfactant and inhibits the mucociliary transport 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 transcriptional activator 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 is no shortage of substrate for autoinducer production then there is no need to explicitly model the biosynthesis of 3-oxo-C12-HSL and C4-HSL by LasI and RhlI. Second, there is evidence that 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

Table 1: Variables used to identify concentrations. Variable A1 A2 P1 P2 P3 C1 C2 C3 E1 E2

concentration [3-oxo-C12-HSL] [C4-HSL] [LasR] [RhlR] [RsaL] [LasR/3-oxo-C12-HSL] [RhlR/C4-HSL] [RhlR/3-oxo-C12-HSL] (External) [3-oxo-C12-HSL] (External) [C4-HSL]

mRNA are much shorter lived than LasR, LasI, RhlR and RsaL, respectively. If we assume that all messenger RNA’s are produced by DNA at rates that are Michaelis-Menten in type, then, assuming the mRNA’s 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 of 3-oxo-C12-HSL and C4-HSL also follow Michaelis-Menten kinetics. Lastly we assume that the autoinducers freely diffuse across the cell membrane. Following [72] the intercellular dynamics is described by a system of eight coupled differential equations 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-oxoC12-HSL complex. We introduce variables for all the concentrations (shown in Table 1) as well as 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 at rate rCi and dissociate at the rate dCi , i = 1, 2, 3, then dC1 = rC1 P1 A1 − dc1 C1 , dt dC2 = rC2 P2 A2 − dc2 C2 , dt dC3 = rC3 P2 A1 − dc3 C3 . dt

(1) (2) (3)

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

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by dP1 = −rC1 P1 A1 + dc1 C1 − dp1 P1 dt C1 +VP1 + P10 , KP 1 + C 1 dP2 = −rC2 P2 A2 + dc2 C2 − dp2 P2 dt C1 +VP2 + P20 KP 2 + C 1 −rC3 P2 A1 + dc3 C3 C1 dP3 = −dP3 P3 + VP3 + P30 . dt KP 3 + C 1

(4)

(5)

(6)

Similarly, for the autoinducers, dA1 = −rC1 P1 A1 + dC1 C1 − rC3 P2 A1 + dC3 C3 − dA1 A1 dt C1 + A01 + δ1 (E1 − A1 ), +VA1 C1 + KA1 (1 + KPP33r ) C2 dA2 + A02 + δ2 (E2 − A2 ). = −rC2 P2 A2 + dC2 C2 + VA2 dt C2 + KA2

(7)

(8)

Next, we need to determine how the density of organisms controls the activity of this network. We assume that autoinducer Ai (i = 1, 2) diffuses across the cell membrane, and that the local volume fraction of cells is ρ. Then, by assumption, the local volume fraction of extracellular space is 1 − ρ. As indicated above, the extracellular autoinducer is assumed to diffuse freely across the cell membrane with conductance δi and to naturally degrade at rate kEi . We remark that there is some evidence that the transport of the autoinducer may involve both passive diffusion and a cotransport mechanism [166]. For this model we assume that diffusion alone 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 the equation   dEi (1 − ρ) + kEi Ei = δi (Ai − Ei ), for i = 1, 2 (9) dt Here we emphasize that the factor 1 − ρ must be included to scale for the difference between concentration in the extracellular space and concentration viewed as amount per unit total volume. Similarly, the governing equation for intracellular autoinducer Ai must be modified to account for the cell density. For example, (8) becomes dA2 δ2 C2 + A02 + (E2 − A2 ). = −rC2 P2 A2 + dC2 C2 + VA2 dt C 2 + KA 2 ρ 12

(10)

A bifurcation diagram for the steady states for (1)-(8) is shown in Figure 8. The parameter ρ provides a switch between the two stable steady solutions. For small values of ρ, there is a unique stable steady state with small values of A1 and A2 . As ρ increases, two more solutions appear (a saddle node bifurcation), and as ρ increases yet further, the small solution disappears (another saddle-node) leaving a unique solution, thus initiating a “switch”. For intermediate values of ρ, there are three steady solutions. The small and large values are stable steady solutions while the intermediate solution is unstable, a saddle point. It is easy to arrange parameter values that have this switching behavior, and the switch can be adjusted to occur at any desired density level. One can give the following verbal explanation of how quorum sensing works. The quantity A1 , the autoinducer, is produced by cells at some nominal rate. However, each cell must dump its production of A1 or else the autocatalytic reaction would turn on. As the density of cells increases, the dumping process becomes less effective, and so the autocatalytic reaction is enabled. It is interesting that, to the best of the authors’ knowledge, bistability has not yet been observed experimentally. This may be due to difficulties such as the requirement for a reporter gene product with an extremely short half-life. Haseltine and Arnold [98] have shown experimentally in an artificial quorum-sensing network that simple changes in network architecture can change the steady-state behavior of a QS network. In particular, it is possible to obtain graded threshold and bistable responses via plasmid manipulations. They also have shown that threshold 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 genetic circuits by using the approach described by Angeli et al. [10]. The first population level models were developed by Ward et al. [233] including population growth. 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 of increased QS molecule concentration as the population size increases and the induced increase of the up-regulated subpopulation. In a relatively short time (compared to the time scale for growth) there is a rapid increase of the QS molecule concentration. Ward et al. [234] extended these models to include a negative feedback mechanism known to be involved in QS. A similar modeling approach has been developed to investigate QS in a wound [127]. Since the virulence of P. aeruginosa is controlled by QS, there is the possibility that agents designed to block cell-to-cell communication can act as novel antibacterials. One advantage of this 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 degrades the AHL signal and another that disrupts the production of the signal by diffusing through the cell membrane and destabilising one of the proteins necessary for the signaling molecule production. A conventional antibiotic is also investigated. Through numerical simulations and mathematical analysis it is shown that while it is possible to reduce the QS signal (and hence virulence) to a negligible level, the qualitative response to treatment is sensitive to parameter values. 13

8

7

6

[A1] (µ M)

5

4

3

2

1

0

0

0.1

0.2

0.3

0.4

0.5

ρ

0.6

0.7

0.8

0.9

1

Figure 8: Bifurcation Diagram for (1)-(8) with respect to the density parameter ρ for the parameter values [72]: rc1 = rc2 = rc3 = .17(1/µMs), dc1 = dc2 = dc3 = .25(1/s, VP1 = Vp2 = 0.6(µM/s), KP1 = k(µM), KP2 = 1.2(µM), VP3 = 0.9(µM/s), KP3 = 1(µM/S), VA1 = 1.2(µM/s), VA2 = 1(µM/s), KA1 = 0.4(µM), KA2 = 1.0(µM), dP1 = dP2 = 0.2(1/s), dP3 = 0.25(1/s), δ1 = 0.2(1/s), δ2 = 0.4(1/s), KE1 = KE2 = 0.2(µM), dA1 = 0.2(1/s), dA2 = 0.4(1/s), P10 = P20 = P30 = 0.004(µM), KP 3r = 2.5(µM), A01 = 0.4(µM), A02 = 0.004(µM)

14

Koerber et al. have published an interesting application of deterministic population level models [128]. They use a deterministic model for QS to help develop a stochastic model for the escape of Staphylococcus aureus from the endosome. The deterministic model includes the QS molecule, 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 stochastic model of a single cell. Muller et al. [150] have developed a deterministic model that includes the regulatory network for 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 cell size is studied and the model is validated with experimental data. It is interesting that the submodel for regulatory system has bistable behavior but at the population level a threshold effect. The analysis shows that a single cell is mainly influenced by its own signal and thus is computing diffusion sensing. However, interaction with other cells does provide a higher order effect. 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 et al. [232] model a biofilm spreading over a surface using a thin film approach. Both approaches allow one to study the distribution of up-regulated cells as well as the spatio-temporal changes of the signal within the biofilm. The modeling results of Chopp et al. [35, 36] show that for QS to be up-regulated near the substratum, cells in oxygen-deficient regions of the biofilm must still be synthesizing the signaling compound. As one would expect, the induction of QS is related to biofilm depth; once the biofilm grows to a critical depth, QS is up-regulated. The critical biofilm depth varies with the pH of the surrounding fluid. One very interesting prediction is that 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 surface and include the QS model developed in [233]. The analysis shows that there is a phase of biofilm maturation during which the cells remain in the down-regulated state followed by a rapid switch to the up-regulated state throughout the biofilm except at the leading edges. Travelling waves of quorum sensing behaviour are also investigated. It is shown that while there is a range of possible 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 critical biofilm 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 that in the thick-biofilm limit, the critical concentration of each drug increases exponentially with the biofilm thickness; this is different than the behaviour observed in the corresponding model for spatially homogeneous population of cells in batch culture, [7], where it is shown that the critical concentrations grow linearly with bacterial carrying capacity. 15

The work of [8] is further extended in [9] to study a mature biofilm and the effects of QS and anti-QS drugs on EPS production. In [9] EPS synthesis is assumed to be regulated by the signal molecule. Furthermore EPS, QS signal production, and bacterial growth, are assumed to be limited by a single substrate, thus factoring in the nutrient poor conditions within the biofilm. The effect of anti-quorum sensing and antibiotic treatments on EPS concentration, signal level, bacterial numbers and biofilm growth rate is investigated. Analysis of the associated travellingwave behaviour is also investigated. For further investigations into anti-QS drugs on biofilm development see [236]. Here a comparison is made between batch cultures and biofilms. In both cases it is shown that an early application of an anti-QS drug will delay or prevent the onset of mass up-regulation. There is a hysteresis affect though; once up-regulation takes place a considerable amount of drug is needed to force down-regulation of the cells.

2.3

Hydrodynamics and QS

Mass transfer refers to the physical process by which substances are transported in a system: it has the potential to affect QS in many ways. Delivery of nutrients to the active biomass from which the QS molecules are made is one example, see eg. [199], that could affect the production rates of the signal. Mass transfer can be affected by the hydrodynamics of the bulk fluid and the geometry of the biofilm community. These two factors affect each other because a biofilm community both shapes and is shaped by its external environment. Few studies have directly addressed the effect of hydrodynamics on quorum sensing in structured communities. Purevdorj et al. [177] found that at relatively high flow rates, flow velocity was a stronger determinant of P. aeruginosa biofilm structure than quorum-sensingrequired functions. Of course it may be the case that at these flow rates, the flow had a diluting effect on signaling molecules. In the absence of QS-inducing concentrations, it may be expected that there would be little difference between the wild type (WT) and QS mutant biofilms. Yarwood et al. [258] grew WT and QS mutant biofilms of S. aureus by batch culture under static conditions, batch culture on spinning disks, and in flow cells. The QS mutation had the greatest effect under static conditions, resulting in an increase in biofilm formation, consistent with the hypotheses that flow can have a diluting effect on the signaling molecules. The only modeling results known to the authors along these lines are due to Kirisits et al. [120]. They provide experimental and modeling evidence that the hydrodynamic environment can impact quorum sensing (QS) in a Pseudomonas aeruginosa biofilm. As one would expect, the amount of biofilm biomass required for full QS induction of the population increased as the flow rate increased. The experimental data and model result suggest that signal washout might explain why QS fails to fully induce at the highest flow rate, although one cannot rule out mass transfer effects on nutrient gradients. The data also suggest that QS may not be fully operative at high flow rates and thus not play a significant role in biofilm formation. For a recent review of how other environmental conditions may effect QS and biofilm formation see eg. [105].

16

3

Growth

Much of the earliest work in biofilm modeling was, and continues to be, directed towards predicting 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 and other substrates (such as electron donors or acceptors) together with some sort of growthgenerated velocity and a moving boundary, see below. Close examination, confirmed by confocal microscopy, however revealed that 1D approximations could be suspect [21, 86]. First efforts at 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 individualbased models [132, 133], to study growth processes in 2D and 3D. In these systems, new-growth biomaterial 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, unoccupied location could be found. Substrate was treated differently; in contrast to discrete processing of biomass, in most cases substrate transport and consumption was handled as a continuum reaction-diffusion process. While appealing in their simplicity, these semi-discrete models suffer a certain degree of arbitrariness in their choice of rules. Hence, more recently, fully continuum models based on standard continuum mechanics have been introduced; this is where we place our attention here. The idea is to treat the biofilm as a viscous or viscoelastic material that expands in response to growth-induced pressure, much like avascular tumor models [92, 93]. We note that other continuum models based on chemotactic and diffusive mechanisms have also been 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 continuum models to the cellular scale has been considered in [253, 254], but, lacking many details of the microscale, 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 single species 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 c0 is maintained, and a diffusive boundary layer of thickness L adjacent to the biofilm in which substrate freely diffuses [114]. Within the biofilm, substrate diffuses and is consumed. Overall, c obeys ct = (κ(z)cz )z − H(h − z)r(c), 0 < z < h + L,

17

BULK FLUID (WELL MIXED) BOUNDARY LAYER

L

z

h

Figure 9: biofilm occupies the 1D layer 0 < z < h and limiting substrate diffuses in from z = h + L. r(c) Max

c Half Saturation

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 function r(c) = kc + m (with k or m possibly zero) is adequate. The diffusivity κ(z) is discontinuous across the biofilm–bulk fluid interface (and may vary within the biofilm as well), but in practice the variation is not that large, especially for smaller molecules such as oxygen as biofilms are watery and apparently fairly porous [207]. The diffusive time scale ℓ2 /κ, where ℓ is a system length scale (ℓ = h say in the present setup), is usually very small compared to other relevant time scales, see Figure 6, in which case the quasi-steady approximation (κ(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 substrate concentration in the well-mixed zone, and cz (0, t) = 0, i.e., no flux of substrate through the wall. Substrate consumption rate r(c) drives biofilm growth at rate g(r(c)) (typically a form g = (r(c) − b)Y is used, where Y is a yield coefficient and b is a base subsistence level), which 18

in turn generates a pressure p(z, t) that drives deformation with velocity u(z, t). We balance pressure stress with friction, obtaining 1 u = −pz , λ

0 < z < h,

(12)

where λ 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 consequence in 1D that u is determined by uz = g,

0 < z < h.

(13)

Combining (12) with (13) we obtain 1 pzz = − g, λ

0 < z < h.

(14)

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

ht = u(h, t) = −λpz (h, t) =

g(c(z, t)) dz,

(15)

0

so that the interface moves so as to exactly accomodate new growth or decay. Also observe that p is linear in 1/λ so, for this same reason, that the interface speed ht is independent of the friction parameter λ. 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 at z = h. In order to evaluate the right-hand side of (15), we need to solve (11) for the substrate concentration 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 r(c) czz = , cz |z=0 = 0, c|z=h + Lcz |z=h = c0 (16) κ Introducing the variables a = c, b = cz , results in the system a˙ = b b˙ = r(a)/κ

(17)

and phase-plane analysis reveals, see Figure 11, that a solution of (16) corresponds to an orbit in the positive quadrant connecting the a axis to the line a + Lb = c0 over “time” z = 0 to z = h. Under mild smoothness conditions on r, we observe that the orbit starting from the origin takes infinite “time” to reach the target line a + Lb = c0 (corresponding to an infinitely deep layer) and the orbit starting from a = c0 takes 0 “time” (corresponding to a degenerate layer), with continuous variation of orbit “time”, i.e., depth, between. Thus for a layer of given depth, equation (11) has a solution. This solution is unique if r is monotone but otherwise may not be. Note that if h is large, then the solution spends most of its “time” near the (a,b)-plane origin, i.e., c(z) is close to zero except near z = h. 19

b a + Lb = c 0 c /L 0

a c 0

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 of sharp 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 to the origin, the so-called substrate limited regime. In this limit, r(c) ≈ r ′ (0)c and (17) can be approximated by a˙ = b b˙ = (r ′ (0)/κ)a, p i.e., a ¨ = (r ′ (0)/κ)a. Thus solutions behave like exp((z − h)/ℓ) with decay length ℓ = κ/r′ (0). In terms of the biofilm, this indicates exponential decay of substrate concentration and hence a sharp transition between high and low biological activity. Typical parameter values used in models amount to ℓ ≈ 10 µm [231], a number consistent with observations. The existence of active and inactive layers (see Figure 12) is a central component of biofilm behavior, as will be discussed later, and has been well studied especially since the introduction of microsensor and CLSM technology, e.g. [113, 114, 119, 204, 249, 242, 243, 257, 259]. Appearance of other types of microenvironments demarcated by sharp localized chemical transitions is also a frequently observed phenomenon and an indicator of ecological diversity [180]. Sharp boundary layers are often seen for example at electron acceptor transitions, usually separating important niche regions. These boundaries can even move in time, for example due to the day-night cycle and its control of photosynthetic oxygenesis [183].

20

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

3.3

Multidimensional, multispecies, multisubstrate models

The simple model of the previous section presents a useful picture of the role of diffusive transport; we will return to it later. However biofilms are often not 1D systems; rather channels and 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 usually composed of many species, upwards of 500 in dental biofilms for example [145], including both autotrophs (users of inorganic materials and external energy) and heterotrophs (users of organic products of autotrophs) cycling and throughputting numerous products and byproducts. Even within individual species, considerable behavioral variation exists (indeed, as already mentioned even the notion of microbial species is unsettled). Inactive and inert material also impacts spatial distribution and hence influences ecology [121]. So we introduce Nb different “biophases” [229, 230] – material phases that could be species or other quantities like phenotypic variants, extracellular material, inert biomass, free water, etc. – each with concentration bi (x, t), i = 1, . . . , Nb . Similarly, confronted with the ecology of an interactive microflora, simplifying to a single limiting substrate is also likely insufficient. So we generalize to a set of substrates with concentrations ci (x, t), i = 1, . . . , Nc , satisfying mass balance equations ci,t = ∇ · (κi ∇ci ) + χB ri where χB (x, t) is the characteristic function of the biofilm-occupied region and ri are the multispecies usage rate functions. As before, the time derivatives can typically be neglected, leaving the quasistatic approximations ∇ · (κi ∇ci ) = −χB ri . (18) 21

We retain the same biofilm stress balance as before, namely 1 u = −∇p λ

(19)

and continue to consider only growth-induced pressure balanced by friction. Note the implicit introduction of the assumption in (19) that friction is strong enough so that all phases are locked together, i.e., all phases are advected with the same velocity u (this assumption will be relaxed later). The result is that we obtain biophase transport equations bi,t + ∇ · (ubi ) = gi ,

(20)

i = 1, . . . , Nb , where gi are the growth functions. (Remark: we neglect the possibility of diffusion as is common practice. Diffusion of biomaterial is generally considered to be insignificant though we are not aware of measurements of diffusion coefficients.) Defining θi (x, t) and ρi (x, t) to be respectively the volume fraction and specific density of biophase i, we have bi = ρi θi . The various biophases are generally mostly composed of water and hence can be regarded as incompressible, i.e., ρi (x, t) is a constant ρ. Then (20) reduces to θi,t + ∇ · (uθi ) =

gi . ρ

(21)

Using the constraint Nb X

θi = 1

i=1

and summing (21) we obtain the incompressibility condition N

b 1X gi ∇·u= ρ i=1

(22)

which can be combined with the divergence of (19), resulting in N

−λ∇2 p =

b 1X gi . ρ i=1

(23)

The biofilm–bulk fluid interface moves with normal velocity γ˙ = −λ∇p · n

(24)

where n is a unit normal pointing into the bulk fluid. As a remark, note that equations (21) can be rewritten in the forms Nb gi θi X θi,t − λ∇p · ∇θi = − gi . ρ ρ i=1

22

(25)

From (23), observe as in the 1D single phase, single substrate case, that p is proportional to 1/λ and hence the interface velocity and biophase volume fractions are independent of λ, In fact, excepting a scaling factor of p, the friction coefficient λ is arbitrary and, as before, can be set equal to 1. This simplification is a consequence of the choice of momentum balance (19). Equations (18), (23), (24), and (25) together with boundary conditions and choices for the utilization rate functions ri and growth rate functions gi constitute a model system. See [231] for typical rate function choices (often multilinear or multilinear with saturation forms). Boundary conditions are ∇c · n = ∇p · n = 0 on the solid boundary (where n is a normal vector), p = 0 at the biofilm–bulk fluid interface, and ci = constanti at a boundary above the biofilm. This boundary is sometimes placed at a fixed location and is sometimes allowed to move to a fixed distance above the highest point of the biofilm (imitating the top of a fluid diffusive boundary layer). We have delayed mention of one important component of these types of growth models, namely biomaterial loss. Without some loss mechanism, the biofilm region can grow without bound so, in order to study the long-time behavior, some method of material removal is necessary. Material loss is often taken into account by allowing the growth rate functions gi(x, t) to be negative in sum or by including an erosion term at the biofilm–bulk fluid interface through addition of an inward directed term to the righthand side of (24) [202, 229, 255]. Growth models are popular predictive tools in engineering applications such as waste and water treatment, see [231]. See Figure 13 for an example computation of the sort reported in [5]. These sorts of models can produce pictures that “look” good, with indeed some work on quantitative verification, but there is not a good deal of experimental data appropriate for comparison because the computationally convenient set-up described so far is generally not the convenient one in the laboratory. In particular, the stress balance posited in equation (19) omits important mechanics that can be significant in laboratory systems which often include non-negligible fluid shear stress.

3.4

Linear Analysis

Nevertheless, these models seem to do a reasonable job at describing some important zeroth order quantities like approximate net production of biomaterial and net kinetics. We wish however to explore another direction, namely creation of structured biofilm (as opposed to flat ones). One might expect that 1D solutions would be susceptible to a Mullins-Sekerka type instability, i.e., be unstable to perturbation of a surface protrusion that extends into richer substrate and hence tends to be amplified. In order to study stability, then, we consider linear perturbations to the upper biofilm surface (the biofilm-bulk fluid interface) of the form h(x, t) = h(0) (t) + h(1) (t)eik·x with k 6= 0 and where x = (x, y) is the vector of horizontal coordinates. Here h(0) (t) is the location of the unperturbed interface, see equation (15). The first order perturbation of the surface normal n is orthogonal to the zeroth order normal n0 = zˆ to the z-direction and hence 23

Figure 13: Simulation of a two component system (active and inert biomass) with a single limiting 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 ˆz. Thus the interface velocity is, to first order, (0) h˙ = −n0 · ∇p|z=h = h˙ (0) + eik·x [h(1) (t)g (0) (h(0) (t), t) − p(1) z (h (t), t)]

(26)

(see equation (15)) where superscript (i) refers to the ith order quantity. The first term in the perturbed velocity, eik·x h(1) (t)g (0) (h(0) (t), t), indicates perturbation enhancement due to bumps (1) extending into richer substrate. The second term, −eik·x pz (h(0) (t), t), is considered below. Note that this pressure gradient term can be expected to oppose instability since the perturbed pressure 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 so surface protrusions receive little advantage. Specializing then as previously to the substrate limitation case where r(c) ∼ = r ′ (0)c, equation (11) linearizes, with κ constant, to 2 (0) c(1) − z)κ−1 r ′ (0))c(1) zz = (k + H(h

(27)

with accompanying boundary conditions and interface (at z = h(0) ) conditions. For z > h(0) , (1) (27) reduces to czz − k 2 c(1) = 0, which can be solved explicitly, and thus the interface problem (27) in turn reduces to a boundary value problem 2 −1 ′ (1) c(1) zz = (k + κ r (0))c

(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(−α(h(0) − z)), 24

p √ α = k 2 + ℓ−2 (recall ℓ = κ/r′ (0) is a measure of active layer depth), away from the interface. Relative to zeroth order quantities, short wave lengths (k 2 ≫ ℓ−1 ) decay quickly while long wave lengths (k 2 ≪ ℓ−1 ) decay on a roughly k-independent depth scale determined instead by the active layer depth. Given the perturbed substrate concentration c(1) , we can then determine the perturbation to the pressure field p by linearizing the multidimensional generalization of (14) (setting λ = 1): 2 (1) p(1) − G(c(0) )c(1) zz = k p

(29)

where G(c(0) ) = g ′ (r(c(0) ))r ′(c(0) ) ∼ = g ′ (0)r ′ (0) > 0. Equation (29) has solutions of the form Z z (1) k(z−h(0) ) −1 p (z) = Ce −k Gc(1) sinh[k(z − s)]ds, 0

on 0 < z < h(0) with C > 0, and so (0) −p(1) z (h , t)

= −Ck +

Z

h(0)

Gc(1) cosh[k(h(0) − s)]ds.

0

The first term, −Ck, is the expected suppressive term; increased growth at the top of bumps induces increased pressure and hence a positive perturbation to the pressure gradient. This effect dominates for large k. The integral term mitigates (due to reduction of extra growth because of substrate usage and diffusion), but mitigation is small for k 2 ≫ ℓ−1 where c(1) decays quickly. Hence, putting everything together in (26), we see a long-wave instability regularized at large k by pressure – in the absence of mechanical forcing other than growth-induced expansive pressure, finger formation is predicted with a preferred wavelength.

4

Biofilm Mechanics

The simple mechanical environment assumed so far is not always the reality. Rather, presence of exterior driving forces is probably common. Though composed of microorganisms, a biofilm is a macroscale structure that interacts with its environment as a macroscale material. Indeed it may well be that a principal fitness benefit of extracellular matrix formation (which is, after all, 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 longtime 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 constitutive 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 use of an erosion mechanism where material is lost locally from the biofilm-bulk fluid interface at a 25

rate proportional to the square of the local biofilm height or by a similar rule [202, 229, 255]. It is clear that balance of new growth by more or less concurrent biomaterial elimination is a key element of long time quasi-stationarity in biofilms and it seems that detachment and hence general 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 sections through use of a mixture model [37, 39, 124, 261, 262], for example following ideas for polymersolvent two (or more) phase theory [61, 146, 218]. In the simplest version, the system is divided into two phases: a sticky, viscoelastic biomaterial fraction consisting mainly of cells and matrix and also a solvent fraction consisting mainly of free water. We denote the biomaterial and solvent volume fractions by φb (x, t) and φs (x, t) with φb + φs = 1. The corresponding specific densities with respect to volume fraction are denoted ρb (x, t) and ρs (x, t). As the biomaterial is to a large extent also water, we assume both specific densities are constant. We define ub to be the velocity of the biofilm fraction and us to be the velocity of the solvent fraction, and set average velocity u = φb ub +φs us . Again, as biofilm is mostly water, we impose incompressibility on u, i.e., ∇ · u = 0. The aim then is to find (and solve) equations describing the time course of 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 important question 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 since flocs retain many of the advantages of biofilms [81]. It is believed that the biofilm matrix is sticky in the sense that it includes a mesh of polymers with a tendency to form physical links and entanglements [79]. We can construct a model of this tendency by introducing a so-called cohesion energy E(φb ) of the form Z   κ E(φb ) = f (φb ) + |∇φb |2 dV (30) 2 where f (φb ) is an energy density of mixing, Figure 14, with a minimum at a volume fraction φb = φb,0 where attractive forces are balanced by repulsive, packing interactions [124]. If attractive forces are weak at long distances, as is the case for hydrogen bonding for example, then f flattens as φb → 0. The term (κ/2)|∇φb |2 is a distortional energy which penalizes formation of small-scale structure (necessary if for no other reason then for regularization) and also introduces a surface energy penalty. We remark that we neglect mixing entropy effects in (30), but that they can also be included, in which case E can be characterized as a free energy [37]. An alternative approach to modeling biomaterial adhesion in biofilms using Potts 26

Energy Density

0

1

Biomaterial Volume Fraction

Figure 14: Representative mixing energy density. Note that biomaterial volume fraction φb is confined between 0 and 1. As φb 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 in cohesion [50, 161], Using (d/dt)φb +∇·(φb ub ) = 0 (and neglecting change of φb due to growth), we can calculate Z dE = − (φb ∇ · Π) · ub dV (31) dt where Π = −[f ′ (φb ) − κ∇2 φb ]I is a cohesive stress tensor. Equation (31) takes the form of a work integral and thus indicates that biomaterial motion as dictated by biomaterial velocity ub does work. Hence the term φb ∇·Π must 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 ζ(ub − us ), we obtain force balance equations 0 = φb ∇ · Π − ζ(ub − us ) + φb ∇p 0 = −ζ(us − ub ) + φs ∇p

(32) (33)

where p is hydrostatic pressure. Equations (32) and (33) are stripped down force balances: we neglect inertial forces and assume for the moment that viscous and viscoelastic forces are small compared to friction. With some manipulation, we obtain from (32) and (33) that ∂φb + ∇ · (φb u) = ∇ · [a(φb )f ′′ (φb )∇φb ] − κ∇ · [a(φb )∇∇2 φb ] ∂t 27

(34)

Figure 15: Phase separation – low volume fraction biomaterial (left) is unstable to spinodal decomposition, resulting in phase separation (right) with one phase of high biomaterial volume fraction (dark) and the other with no biomaterial. where we suppose ζ = ζ0 φb φs and set a(φb ) = ζ0−1 φb φs = ζ0−1φb (1 − φb ). In the 1D case, incompressibility and boundary conditions dictate that u = 0, leaving us with a modified Cahn-Hilliard equation. For sufficiently small φb , f ′′ < 0 resulting in instability, the so-called spinodal decomposition [28]. As a consequence, low volume fractions of biomaterial will tend to condense 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 the effect 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 on their characterization as viscoelastic fluids [106, 123, 130, 189, 198, 210, 221]; biofilms respond elastically 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 need to 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 of cells, polymers, and other materials, for ease we will simplify the dynamical description by, for mechanical respects only, supposing it to be made up of uniform polymers. Following [60, 227] we introduce a function ψ(x, q, t) tracking a weighting of polymers with center-of-mass x and end-to-end displacement q. Under this framework, Z φb (x, t) = ψ(x, q, t) dq. Note that ψ/φb , with ψ/φb defined to be 0 when φb = 0, is a probability density in q. We can now extend the energy (30) to include elastic contributions as Z   κ E(ψ) = f (φb) + fe (ψ) + |∇φb|2 dV 2 with, in a simple version, fe (ψ) = νkT

Z


ln(ψ/φb ) + q 2 ψ dq 28

(35)

where ν is a polymer matrix density parameter. The log term is an entropy contribution that opposes, for example, polymer extension in any one particular direction. The quadratic term is a contribution of a standard linear spring potential. Proceeding as before, we calculate dE/dt with respect to the advective influence of a biomaterial velocity ub , obtaining the same form (31) with, in this more general setting, Π = −[f ′ (φb ) − κ∇2 φb + νkT (q 2 + ln(ψ/φb ))]I.

(36)

Assuming that biomaterial is deformed by the biomaterial velocity ub , then ψ can be supposed to obey a Smoluchowski equation [60] of the form ∂ψ + ∇ · (ψu) = ∇ · (ζψ∇ · Π) − ∇q · (ψ∇ub q) ∂t

(37)

where ζ is a mobility coefficient. The first term on the left is due to chemical mixing stress (as in the previous section) and the second is due to viscoelastic stress. Note that averaging over q reproduces equation (34) for φb . Classical linear R molecular models indicate that the viscoelastic stress tensor is approximated by M = 2kT λ qqψ dq where λ is an elasticity parameter [136]. To obtain M, we can multiply (37) by qq and average over q, resulting in the equation ∂M + ∇ · (uM) − ∇uTb M − M∇ub = ∇· < ζqqψ∇ · Π >q ∂t where < · > denotes averaging over q. The average on the righthand side reproduces the chemical forcing terms from earlier as well as new viscoelastic terms (from the third term in Π above, equation (36)). In the case of the simple elastic energy density contribution (35), the averaging process produces a Maxwell fluid-like contribution. More complicated laws can arise using 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 attack as compared to free-floating, planktonic populations [148]. Many studies have noted that bacterial cells within a biofilm require higher minimum antimicrobial concentrations (MIC) (or minimum biofilm eliminating concentration), even by a factor of 103 or more for effective control 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 therapeutically safe levels. However, sub-MIC application can result for example in an infection whose symptoms are only temporarily subdued by even an extended treatment regime but which recur following treatment cessation. It is important to emphasize that we are not referring to strainspecific genetic mechanisms whereby individual cells from a special, resistant strain in some manner acquire altered genes able to transform the specific target or access to that target by a particular class of antimicrobials [240], but rather a non-specific modulated population tolerance to general chemical attack. That is, we consider a situation whereby the same population 29

of cells that would be susceptible in their free swimming form becomes much less so within the protection of the biofilm phenotype and vice-versa. Indeed, it has even been suggested that competitive disadvantage (for nutrients and space) would preclude resistant genetic variant strains from surviving long, in the absence of antimicrobial challenge, in environmental biofilms where competitive pressures are high [84]. Identical cells have the same genetic potential whether situated in a biofilm or free-floating in a planktonic state. So an effective defensive strategy available in biofilms but not planktonic states must either rely on special regulatory networks that operate only in the biofilm state or else on the special environmental conditions created in the biofilm. Lacking, presently, evidence for the former [245], we concentrate on the latter. This is not in any way to say that regulation will not be found to be important, but at least it seems that one key factor is not genetic at all but instead physical: spatially varied environments created by reaction-diffusion processes. The consequence is spatial variation in antimicrobial concentration as well as in microbial physiology and activity. We will argue below that physically modulated processes can either defeat antimicrobial attack or at least slow it sufficiently that biological responses such as altered gene expression patterns can occur. We concentrate on discussion and mathematical description of four particular tolerance mechanisms, all related in varying extents to reaction-diffusion barriers [30, 141, 206]: (1) diffusive and reactive penetration barriers, (2) adaptive stress response below a reactive layer, (3) nutrient limitation and slow growth below the active layer, and (4) presence of persister cells. For illustrative purposes it will be sufficient to confine attention to one dimensional descriptions, i.e., flat biofilms with variation in the height direction only, though higher dimensional effects may have consequence as well, due for example to fluid transport influences [38, 67].

5.1

Penetration Barriers

It is not surprising that biofilm matrix would act as a protective barrier against large-scale attack, for instance by host-defense phagocyte cells [223] – its microbe-matrix structure may simply provide a physical obstruction to engulfment (although the detailed mode of defense may be more subtle, see e.g. [112]). However, down at the molecular scale, penetration into and through the matrix by diffusion can be rather free, even for relatively large molecules [207]. Nevertheless, for many antimicrobials, penetration is slowed (e.g. delayed penetration of aminoglycoside-type antibiotics due to binding to matrix material [87, 153]) or even effectively halted altogether (e.g. limited depth penetration of chlorine into Pseudomonas aeruginosa and Klebsiella pneumoniae biofilms [52]), so somehow a matrix barrier seems to provide shelter even against small invaders. Within a biofilm, the main transport mechanism is diffusion, as advective processes are generally extremely slow comparatively. One should not overlook the fact that there are also reactive processes present that might influence transport rates significantly however. Following Stewart [205] we list three possibilities: (i) interaction with charged matrix material can lead to reversible adsorption (as suggested for example in studies of antibiotics vancomycin [48], tobramycin [154], and ciprofloxacin [213]), (ii) interaction with biofilm material (perhaps as a part of the antimicrobial process) can lead to irreversible reaction through which process

30

both antimicrobial and reactant are neutralized, for example reduction of oxidizing agents like chlorine [52], and (iii) reaction with a catalytic agent in which antimicrobial is neutralized but the reacting agent is unaltered by the process. An example of the latter is enzymatic activity of extracellular 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, the confined and concentrated biofilm environment (relative to the planktonic environment) makes these defense mechanisms much more effective. For purposes of describing these possible mechanisms mathematically, we consider as before a 1D biofilm domain 0 < z < h with the substratum at z = 0 and the bulk fluid interface at z = h, and set B(z, t) to be the concentration of free antimicrobial. Height h is assumed constant, i.e., we suppose that no significant growth occurs over the time-scale of antimicrobial application and so do not consider here extended disinfection schedules. B satisfies a no-flux condition Bz |z=0 = 0 at the substratum and a Dirichlet condition B|z=h = B0 at the bulk fluid interface. 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 conditions B(z, 0) = 0, 0 < z < h, i.e., application beginning at t = 0 of antimicrobial to an initially antimicrobial-free biofilm. The aim is to describe the spatial profile of B as a function of time in order to model the effects of penetration barriers. The concentration B satisfies an equation of the from Bt = κBzz + 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 reversible component 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, the Monod form g(B) = kS B/(B + KB ). We also assume that S, being matrix associated, does not diffuse. (ii) Xb is a density of binding sites, and R is a reaction rate function, also monotonic and saturating (with respect to B only – we assume that biofilms do not choose to produce so many reaction binding sites such that saturation of f in Xb would occur), measuring removal rate of antimicrobial due to irreversible adsorption. The function Xb (z, t), 0 < z < h, is the concentration of free binding sites (for antimicrobial) and satisfies the equation Xb,t = −

R(B) Xb , Yb

Xb (z, 0) = Xb0 ,

(40)

where Yb is a yield coefficient. (iii) C is a catalysis rate function, again monotonic and saturating with respect to its argument, measuring removal rate of antimicrobial due to catalytic 31

destruction. Note that g, R, and C may all depend on other factors such as microbial activity as one example, but these extra complications are neglected for purposes of clarity. Using (39), equation (38) can be rewritten as (1 + g ′(B))Bt = κBzz − R(B)Xb − C(B).

(41)

At the top of the biofilm, if the biocide dosage is saturating, then (41) simplifies to, approximately, Bt = κBzz − C0 , where the constant C0 is the saturation level of C(B), so that antimicrobial shows a quadratically decreasing profile. We focus attention on an antimicrobial depletion zone in the deeper parts of the biofilm where B is small although this zone may only exist for a finite time. Then equation (41) reduces approximately (away from z = h) to Bt = κ ¯ Bzz − αB where κ ¯=

κ , 1 + g ′ (0)

α=

(42)

R1 Xb + C1 1 + g ′ (0)

and R1 and C1 are first order rate constants. Note that κ ¯ < κ. i.e., that the effective diffusivity is smaller than the true diffusivity as a consequence of mechanism (i), reversible sorption. Based on available data, Stewart estimated the decrease to be approximately by a factor of 10 leading to a penetration delay for typical biofilms of about 1 to 100 minutes [205]. While transient, this delay may be sufficient to allow other defensive responses, p see below. The balance of diffusion and reaction in (42) results in a decay distance ℓ = κ0 /(R1 Xb0 + C1 ), similar to that seen in Section 3.2 in the context of substrate limitation and active layer formation, with resulting antimicrobial limitation below a reactive layer located at the top of the biofilm if ℓ is smaller than h. Note that reversible sorption plays no role. A part of the reactive term though, namely −R1 Xb B/(1 + g ′ (0)), corresponds to irreversible sorption and is temporary within any reactive layer; Xb decays on time scale Yb /R1 B0 as binding sites p are occupied. The decay distance ℓ will thus also increase in time up to a maximum of ℓ = κ0 /C1 . On longer times, (38) equilibrates to C(B) = κBzz , Below the resulting reactive layer, where B is small p and C(B) ≈ C1 B, antimicrobial concentration is depleted exponentially with length scale ℓ = κ/C1 [204]. Thus for those antimicrobials like hydrogen peroxide that are susceptible to enzymatic deactivation, a permanent reactive penetration layer results (at least, permanent on the time-scales considered here).

32

5.2

Altered Microenvironment

But many antimicrobials do not appear susceptible to a defensive mechanism like enzymatic degradation 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 at levels above the MIC, significant killing of microbes in the deeper regions of the biofilm is often not observed, see as an example [48]. This lack of efficacy seems related to decreased potency against inactive microbes; in most cases antimicrobial-induced killing proceeds in the active layer where metabolic activity is high, while the same antimicrobial agent may not be as effective against less active organisms [25, 71, 226] such as those which predominate below the active layer; microbes that are not “eating” will not ingest poison as easily, so to speak. Other microenvironmental 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 antimicrobial penetration barrier except with a focus on a limiting substrate concentration, e.g. oxygen concentration, replacing antimicrobial concentration. That is, equation (42) would be replaced by Ct = κCzz − αC (43) where C is, say, oxygen concentration. Equation (43) is supplemented by a killing rate equation of 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 as C, B, or b approach 0. Even assuming that the antimicrobial is able to penetrate the entire biofilm in quantity without significant resistance, it would still be the case that below the active layer, where C is exponentially small, killing is limited. Note however that, in (43), substrate usage generally depends on presence of live organisms, i.e., α = α(b) with α approaching 0 as b does. Thus this model predicts that substrate penetration depth would gradually increase as disinfection extinguishes live microbes in upper parts of the biofilm. Hence, it can generally be expected that, barring other defense, eventually the biofilm will be uniformly penetrated by substrate and protected environmental conditions (absent antimicrobial) that favor inactivity may disappear in time.

5.3

Adaptive Stress Response

Thus one method for overcoming diffusive and reactive barriers could be application of a sustained 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 sustained attack, and on the other, reactive resistance, while perhaps not transient, cannot prevent disinfection of an upper layer of biofilm which over time will be shed in the process exposing previously protected microorganisms. Further, as disinfection procedes, organisms near the surface are killed and cease substrate usage so that the active layer may extend farther into the biofilm, thus improving susceptibility. So a patient antimicrobial course might be expected to be eventually rewarded with effective disinfection. 33

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 effective disinfection method than low, long-duration control attempts [94]. Why does this happen? In at 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 of extracellular polysaccharides [13, 191] among other means [245]. This tolerance is specific to the applied antimicrobial [16] unlike the general tolerance exhibited by persisters, see below. It is worth emphasizing that these adaptive or acclimitizing defenses are individually modulated and hence available to organisms in both biofilm and free-floating planktonic state; the difference is that, even at super-MIC antimicrobial levels, sufficiently thick biofilms can result in a “safe” zone 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 sufficient time to mount a genetically controlled defense [216], a strategy not available to planktonic communities. As an aside, we remark that it is perhaps not so surprising to discover that microorganisms have the know-how to mount a responsive defense to a wide spectrum of known antimicrobials. The natural world is full of antimicrobial substances deployed in a literally billions of years old intranecine microbe-microbe conflict, not even to mention multicellular organisms and their responses, providing a rich source for discovery and subsequent use for our own advantage. However, those same billions of years have provided plenty of opportunity for microbial responses to be constructed. And biofilms may be likely place for those defenses to operate effectively for reasons already discussed. To describe adaptive response mathematically within a simple 1D biofilm model, we can return to the multispecies, multisubstrate setup (18)-(24) particularizing to three species and two substrates, namely the three species θu = live unadapted cell volume fraction, θa = adapted cell volume fraction, θd = dead (or more generally inactive) cell volume fraction, and the two substrates cs = limiting substrate concentration, cB = antimicrobial concentration (previously denoted as B). Live cells grow in the presence of limiting substrate; dead cells do not obviously but nevertheless need to be tracked since they take up space (although one might allow dead cells to lyse and then disappear, possibly releasing substrate in the process). Cells can die both from natural causes and from disinfection by antimicrobial. Unadapted and adapted cells interconvert with rates depending on the antimicrobial concentration. The equations for the cell densities are θu,t + (uθu )z = −ϕ(cB )θu − λ(cB )θu + αu (cs )θu + γ(cB )θa θa,t + (uθa )z = λ(cB )θu − ψ(cB )θa + αa (cs )θa − γ(cB )θa θd,t + (uθd )z = φ(cB )θu + ψ(cB )θa , on the domain z ∈ [0, h(t)]. Here ϕ and ψ are death rates (combining natural and disinfection fatality), αu and αa are growth rates, λ is the adaption rate, and γ is the reversion rate. Quantities cs and cB satisfy Ds cs,zz = rs (θu , θa , θd ) DB cB,zz = rB (θu , θa , θd ) 34

where rs and rB are reactive terms, Ds and DB are diffusivities, and, due to the relative rapidity of diffusive equilibration, we have neglected advective transport as well as the explicit time variation. The incompressibility condition θu + θa + θd = 1 translates into the equation uz = ρ−1 (αu (cs )θu + αa (cs )θa ) for velocity u where ρ is the specific density, see (22). Boundary conditions are as described in Section 3.3. If we modify the interface velocity (15) to include an erosion term [95, 202], as is commonly done in 1D models, then h(t) satisfies ht = u(h, t) − e(h, θu , θa , θd ). Here e is an erosion function (usually chosen to be e(h) = αh2 ) that allows, through balance of growth 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 assumption that e(h)/h → ∞ as h → ∞. An important result shown in [217] is that if adapted cells are not 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 of adaption rate to reversion rate is crucial in designing a dosage scheme. Not surprisingly, this ratio 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 of antibiotic challenge a small number of surviving cells are able to repopulate and return the biofilm to viability. These special cells, called persisters, are in fact not particular in their tolerance to fluoroquinolones or to any one organism, and have been consistently observed in antimicrobially dosed planktonic bacterial cultures since at least the 1940s when a study observed that about 1 out of 106 cells of staphylococci cultures survive sustained exposure to 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(−k1 t) + C2 exp(−k2 t) with k1 > k2 and C1 ≫ C2 , see Figure 16, with C1 being the initial population of normal cells and C2 the initial population of tolerant cells, see e.g. [14]. Persister cells are not specific to biofilms; indeed, most studies of them have focused on planktonic populations. However, as recently realized by Lewis [141], persistence may be most dangerous in biofilms where surviving cells, though few in number, can hide from predators and have access to dead, lysed cells providing a nutrient-rich environment. In contrast, in a planktonic culture, isolated surviving cells are more vulnerable to predators or immune defenses. Characterization of persisters has been uncertain. Persisters would seem not to be genetic variants (they seed regrowth of populations indistinguishable from the original) nor cells in a spore-like state (spores are not observed) nor cells caught coincidentally in a special, quiescent 35

7

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0.5

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TIME

Figure 16: Biphasic killing curve. In the initial phase, a large susceptible population is rapidly reduced, 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]. Unlike adaptive responders, they show cross-tolerance to antimicrobials – survivors of a challenge from one 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, the percentage 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, there is still no definitive answer to this question. A common hypothesis is that a persister cell is a special, protected phenotype of the main population that is slow-growing and generally tolerant in some way [141]; cells are able to switch in and out of the persister state, with a greater tendency towards persistence when environmental conditions are unfavorable, e.g. in stationary state conditions. In some versions, switching is triggered by presence of antimicrobial stress [40, 140] and in others there is an everpresent stochastic jump rate [135, 188]. Mathematically, these mechanisms are similar to the adaptive stress response discussed in Section 5.3, one difference though being that the growth rate αa would be supposed to be very small for persisters. (Recall also that adaptive response is antimicrobial-specific.) We thus consider here instead an alternative – persister cells are senescent cells [12, 125] – based on the observation that bacterial cells can demonstrate aging effects [201]. The idea is that senescent cells are slow-growing, and slow-growing cells have increased tolerance to antimicrobials, so perhaps persister cells are defined by their senescence. 36

In order to consider implications of a persister-senescence connection we construct a model of senescing microbial populations, necessitating introduction of structured population densities [239] as follows. Set b(σ, t) to be the bacterial population at time t of senescence level σ in a well-mixed batch culture. Then b satisfies bt (σ, t) + (v(σ, t)b(σ, t))σ = −rd (σ, C, B)b(σ, t), σ > 0, t > 0

(45)

where v is a senescing rate function (we will set v = 1 here, i.e., senescence = chronological age, for simplicity), and rd is a death rate function which may depend on senescence level σ, C is available limiting substrate concentration, and B is antimicrobial concentration. More senescent 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 (σ, B) = kd + H(δB − σ)kK

(46)

where kd is a natural death rate and kK is a disinfection rate. The parameter δ is a tolerance coefficient. The form (46) indicates that sufficiently senesced cells (σ > δB) are entirely tolerant. This is a rather extreme choice, but the exact form of tolerance relation does not effect observations to follow very much. Equation (45) is a linear wave equation in the region σ > 0, t > 0, with characteristics emanating from both the σ- and t-axes, and thus requires initial conditions b(σ, 0) = b0 (σ) and also a birth boundary condition at σ = 0 of the form Z ∞ b(0, t) = rb (σ, C)b(σ, t) dσ, (47) 0

where rb is a birth rate function. Equation (45) and condition (47) are supplemented by Z ∞ Ct = − rC (σ, C)b(σ, t) dσ, 0

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 subject to antimicrobial challenge at hour 17 and then recultured at hour 27. The dashed line indicates the number of those sufficiently aged cells as to qualify as persisters. Note the biphasic killing during dosage, as well as the relative scarcity of persisters during growth periods as opposed to during stationary, especially late stationary phases. This latter difference is due to abundance of young cells during exponential growth as well as the time lag for young cells to age their way into senescence. The senescence model of persistance would seem to provide a reasonable qualitative match to available data, but so does the phenotype-switching model. In fact, the two are difficult to distinguish at the population-modeling level. The reason for this is that a non-senescent/senescent divide is rather similar to a dual phenotype compartmentalization. Non-senescent cells form a compartment and senescent cells form a compartment with conversion of a sort occuring 37

(a)

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population

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all cells persisters

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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 removed at 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 which time the surviving cells are recultured at the t=0 h nutrient level. Solid lines, all cells; dashed lines, persisters. Reprinted by permission of the Society for General Microbiology from [125]. between the two: non-senescent cells eventually become senescent, while senescent cells produce, albeit at a slow rate, new (and non-senescent) offspring. So resolution awaits further observational data. And of course, it would not be a great surprise if neither explanation were sufficient.

6

Conclusion

The physiology and ecology of microbes is a fast growing field of study. Advancing technologies in microsensing and molecular analysis, and accompanying rapidly lowering barriers for their use, are transforming understanding and appreciation of these organisms in vivo and of the central roles they play in geo- and bio-processes. Newly and rapidly accumulating knowledge will be important not only in advances in medical and industrial fields, but also in the context of key geochemical cycles like the carbon and nitrogen cycles. But microbial systems are complicated 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 acting on and within a given community have been treated separately. This dichotomy is unfortunate as, in many microbial systems, community structure and function are intimately connected with chemical and energetic processes and with protection from the physical environment. Of 38

course, embedding all the complications of a living, reacting ecosystem in a moving interface, multiphase material system raises many difficult problems; physics, chemistry, and biology all combine at a variety of time- and length-scales. We have attempted to convey some of those 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 gene transfer, applications such as bioremediation, etc. All of these are additional ripe topics for mathematical 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 assistance and contributions.

39

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