Download Mathematical Description of Microbial Biofilms

SIAM REVIEW
Vol. 52, No. 2, pp. 000–000
c 2010 Society for Industrial and Applied Mathematics
Mathematical Description of
Microbial Biofilms∗
Isaac Klapper†
Jack Dockery†
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.
Key words. biofilms, quorum sensing, biomechanics, antimicrobial tolerance
AMS subject classifications. AUTHOR MUST PROVIDE
DOI. 10.1137/080739720
1. Introduction.
1.1. Microbial Ecology. Compared to the plant and animal kingdoms, the diversity of microbial life is considerably less explored and less understood (even the
notion of microbial species is a current topic 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 cataloging 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 toward the ultimate goal of constructing
descriptions of full microbial ecosystems.
Prokaryotes (bacteria and archaea) are estimated to make up approximately half
of the 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 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, it may be more truthful to say that we live in the “Age of
Bacteria” [90]. The familiar view of microbes in their free (planktonic) state, however,
is not the norm; rather, it is believed 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, 210]; see Figures 1 through 4. (Microbes in
∗ Received by the editors November 2, 2008; accepted for publication (in revised form) June 15,
2009; published electronically May 6, 2010. This work was supported by grant NSF/DMS 0856741.
http://www.siam.org/journals/sirev/52-2/73972.html
† Department of Mathematical Sciences, Center for Biofilm Engineering, Montana State University,
Bozeman, MT 59717 ([email protected]).
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ISAAC KLAPPER AND JACK DOCKERY
Fig. 1 Mixed-species photosynthetic mat, Biscuit Basin, Yellowstone National Park. Reprinted by
permission of Macmillan Publishers Ltd: Nature Reviews Microbiology from [96].
Fig. 2 Microbial structures in a mixed-species photosynthetic mat, Mushroom Spring, Yellowstone
National Park.
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
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Fig. 3 Laboratory-grown Pseudomonas aeruginosa biofilm. Reprinted by permission of Blackwell
Publishing from [126].
Fig. 4 Aquificales streamers (pink), Octopus Spring, Yellowstone National Park.
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collective behave very differently from their planktonic state; even genetic expression
patterns change [110, 210].) These matrices anchor and protect 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 [199]) 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 [183] (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 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; see, e.g., Figure 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, 218], just to name a few. Medical relevance is dramatic as well [26]. Quoting
from the National Institutes of Health [178],
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, biofilmassociated nosocomial (hospital acquired) infection is by itself a leading cause of death
in the United States [241].
1.2. Biofilm Development and Structure. Biofilm development is often characterized as a multistage process [159] (see 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 micro-
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
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Fig. 5 Stages of the biofilm life cycle, courtesy of the Montana State University Center for Biofilm
Engineering, P. Dirckx.
bial 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 macroscale 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 a remarkably forward-looking overview written by a pioneer
of the subject of biofilm mechanics and modeling, Bill Characklis [32].
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 quasi-static biofilm, and then
advection-reaction-diffusion processes are also assumed to be quasi-steady 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. Notably missing from Figure 6
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, 214]. 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
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ISAAC KLAPPER AND JACK DOCKERY
Fig. 6 Various time scales for biofilm-related processes. Box 1 includes 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 [169]
with permission from John Wiley & Sons.
chemicals. However, 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. 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 signaling, 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 ecological niches [207, 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 [184]. 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.
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
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2. Quorum Sensing. It has been known for some time that bacteria can communicate with each other using chemical signaling molecules. As in higher organisms,
these signaling 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, 185, 193, 22]. While there have
been a number of articles on the connections of biofilm formation to quorum sensing
[51, 99, 161], the specific relationships are still a matter of debate [122, 162]. The experiments are apparently difficult; for example, Stoodley et al. [208] 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, e.g., [176]).
There are many different types of quorum sensing systems and circuits, and many
bacteria have multiple signaling systems. The receptors for some signaling molecules
are very specific, which implies that some types of signaling are designed for intraspecies communication. On the other hand, there are universal communication
signaling molecules produced by different bacterial species as well as eukaryotic cells
which promote interspecies communication. Some species of bacteria can manipulate
the signaling of, and interfere with, other species’ abilities to assess and respond correctly to changes in cell population density. Cell-to-cell signaling, 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 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% to 10% of all genes in P.
aeruginosa are controlled by AHL quorum sensing signaling systems [194, 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
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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 [150,
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 characterized 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, notably including 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 resistance, the intracellular 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 [195]. An interesting fact is that different
target genes can be activated at different concentrations [195]. Activation by the Rprotein 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 to the AHL signal molecules [192]. These include strains of E. 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 signaling molecules are small surfactant-like molecules which are freely
diffusable, it has been suggested recently that the secretion of small molecules could
also be considered a form of diffusion sensing or efficiency sensing [100, 180]. Microbes
could produce and detect autoinducers as a method of evaluating the benefits of
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
9
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 modeling of QS is fairly brief. For a good 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, 149, 88, 222] as well as stochasticity
[89, 142, 189].
2.2. The Model. The quorum-sensing system of P. aeruginosa is unusual because it has two somewhat redundant regulatory systems [166, 167] . 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 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, 163, 164, 167, 101]), but preferentially promotes lasI
activity ([195]). 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 [155]. 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 Figure 7 [221].
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 mRNA are much shorter lived than
LasR, LasI, RhlR and RsaL, respectively. If we assume that all messenger RNAs
are produced by DNA at rates that are Michaelis–Menten in type, then, assuming
the mRNAs are in quasi-steady state, it follows (see, e.g., [58]) that production of
LasR, RasL, and RhlR follows Michaelis–Menten kinetics. We will also assume that
the production of 3-oxo-C12-HSL and C4-HSL also follow Michaelis–Menten kinetics.
Finally, we assume that the autoinducers freely diffuse across the cell membrane.
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Fig. 7 Schematic diagram showing gene regulation for the las and rhl systems in P. aeruginosa.
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/C4HSL complex, and the RhlR/3-oxo-C12-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
(1)
(2)
(3)
dC1
= rC1 P1 A1 − dc1 C1 ,
dt
dC2
= rC2 P2 A2 − dc2 C2 ,
dt
dC3
= rC3 P2 A1 − dc3 C3 .
dt
The equations for production of the proteins under the assumptions described above
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
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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]
are given by
(4)
(5)
(6)
dP1
= −rC1 P1 A1 + dc1 C1 − dp1 P1
dt
C1
+ VP1
+ P10 ,
KP1 + C1
dP2
= −rC2 P2 A2 + dc2 C2 − dp2 P2
dt
C1
+ VP2
+ P20
KP2 + C1
− rC3 P2 A1 + dc3 C3 ,
dP3
C1
= −dP3 P3 + VP3
+ P30 .
dt
KP3 + C1
Similarly, for the autoinducers,
(7)
(8)
dA1
= −rC1 P1 A1 + dC1 C1 − rC3 P2 A1 + dC3 C3 − dA1 A1
dt
C1
+ VA1
+ A01 + δ1 (E1 − A1 ),
C1 + KA1 (1 + KPP33r )
dA2
C2
= −rC2 P2 A2 + dC2 C2 + VA2
+ A02 + δ2 (E2 − A2 ).
dt
C2 + KA2
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 [165]. For this model we assume that diffusion alone acts
to transport the autoinducer (as is apparently correct for the rhl system [165]).
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
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ISAAC KLAPPER AND JACK DOCKERY
Fig. 8 Bifurcation Diagram for (1) through (8) with respect to the density parameter ρ for the
parameter values [72]: rc1 = rc2 = rc3 = 0.17 (1/µM s), dc1 = dc2 = dc 3 = 0.25 (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 ).
Ei , is governed by the equation
dEi
+ kEi Ei = δi (Ai − Ei ) for i = 1, 2.
(9)
(1 − ρ)
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
(10)
C2
dA2
δ2
= −rC2 P2 A2 + dC2 C2 + VA2
+ A02 + (E2 − A2 ).
dt
C2 + KA2
ρ
A bifurcation diagram for the steady states for (1) through (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
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
13
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 downregulated 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 that degrades the AHL signal and another
that disrupts the production of the signal by diffusing through the cell membrane
and destabilizing one of the proteins necessary for 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.
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
exoenzyme, 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. [149] 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 the regulatory system has
bistable behavior but a threshold effect at the population level. The analysis shows
14
ISAAC KLAPPER AND JACK DOCKERY
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 spatiotemporal 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 spatiotemporal changes of the signal within the
biofilm. The modeling results of Chopp et al. [35, 36] show that for QS to be upregulated 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 downregulated state followed by a rapid switch to the up-regulated state throughout the
biofilm except at the leading edges. Traveling waves of quorum sensing behavior are
also investigated. It is shown that while there is a range of possible traveling 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 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 from the behavior
observed in the corresponding model for a spatially homogeneous population of cells
in batch culture [7], where it is shown that the critical concentrations grow linearly
with bacterial carrying capacity.
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-QS and antibiotic
treatments on EPS concentration, signal level, bacterial numbers, and biofilm growth
rate is investigated. Analysis of the associated traveling wave behavior 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
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
15
many ways. Delivery of nutrients to the active biomass from which the QS molecules
are made is one example (see, e.g., [198]) 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. [176] found that at relatively high
flow rates, flow velocity was a stronger determinant of P. aeruginosa biofilm structure
than quorum-sensing-required 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, by
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 affect quorum sensing in a P. 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 may not play a significant role in biofilm formation.
For a recent review of how other environmental conditions may effect QS and
biofilm formation see, e.g., [105].
3. Growth. Much of the earliest work in biofilm modeling was, and continues to be, directed toward predicting growth balance, often for practical engineering applications, such as waste remediation, the work of Wanner and Gujer being particularly influential [11, 33, 121, 148, 228, 229]. These are generally onedimensional (1D) models combining reaction-diffusion equations for nutrient and
other substrates (such as electron donors or acceptors) 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 automatabased models [102, 103, 137, 154, 168, 172, 173, 250] (work by Picioreanu [171] has
had great impact), and subsequently individual-based models [132, 133], to study
growth processes in two and three dimensions. 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 semidiscrete 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
16
ISAAC KLAPPER AND JACK DOCKERY
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 (see [126], however), 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 singlespecies biofilm occupies the domain (in the 1D case) 0 < z < h(t) with height function
h(t). The portion of the domain z > h(t) is the bulk fluid region. A reaction-limiting
substrate, e.g., oxygen, with concentration c(z, t) is present throughout z > 0. It is
assumed that the bulk fluid is itself divided into a well-mixed “free stream,” in which
a constant substrate concentration 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,
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 because biofilms are watery and
apparently fairly porous [206]. The diffusive time scale 2 /κ, where is a system
length scale ( = 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
(11)
(κ(z)cz )z = H(h − z)r(c),
0 < z < h + L,
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 in turn generates a pressure p(z, t) that drives deformation with velocity
u(z, t). We balance pressure stress with friction, obtaining
(12)
1
u = −pz ,
λ
0 < z < h,
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 one dimemsion that u is determined by
(13)
uz = g,
0 < z < h.
17
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
BULK FLUID
(WELL MIXED)
BOUNDARY LAYER
L
z
h
Fig. 9 Biofilm occupies the 1D layer 0 < z < h and limiting substrate diffuses in from z = h + L.
r(c)
Max
c
Half
Saturation
Fig. 10 Typical form of saturating rate function r(c) = kc(ch + c)−1 . The parameters k, ch are the
maximum and half-saturation, respectively.
Combining (12) with (13) we obtain
1
pzz = − g,
λ
(14)
0 < z < h.
At the wall, u = 0 so that pz |z=0 = 0. Note that the biofilm–bulk fluid interface
moves according to
h
g(c(z, t)) dz,
(15)
ht = u(h, t) = −λpz (h, t) =
0
so that the interface moves to exactly accommodate new growth or decay. Also
observe that p is linear in 1/λ and 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
(16)
czz =
r(c)
,
κ
cz |z=0 = 0, c|z=h + Lcz |z=h = c0 .
18
ISAAC KLAPPER AND JACK DOCKERY
b
a + Lb = c
0
c /L
0
a
c
0
Fig. 11 Phase plane representation of solutions of system (17) corresponding to boundary conditions
given in (16).
Introducing the variables a = c, b = cz , results in the system
(17)
ȧ = b,
ḃ = r(a)/κ,
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, (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.
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
ȧ = b,
ḃ = (r (0)/κ)a,
i.e., ä = (r (0)/κ)a. Thus solutions behave like exp((z − h)/) with decay length
= 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 ≈ 10 µm [231], a number
consistent with observations.
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
19
Fig. 12 Active layer (counterstained green) in a Pseudomonas aeruginosa biofilm (stained red).
Scale bar is 100 µm. Reproduced with permission of the American Society for Microbiology
from [257].
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 [113, 114, 119, 203, 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 [179]. 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 [182].
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 mushroomlike structures are frequently observed [21] (Figure 5) with important
consequences [53] that can be addressed only 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 [144], 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 affects 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 , sat-
20
ISAAC KLAPPER AND JACK DOCKERY
isfying 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 typically can be
neglected, leaving the quasistatic approximations
∇ · (κi ∇ci ) = −χB ri .
(18)
We retain the same biofilm stress balance as before, namely,
1
u = −∇p,
λ
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
(19)
(20)
bi,t + ∇ · (ubi ) = gi ,
i = 1, . . . , Nb , where gi are the growth functions. (Note that 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
gi
(21)
θi,t + ∇ · (uθi ) = .
ρ
Using the constraint
Nb
θi = 1
i=1
and summing (21) we obtain the incompressibility condition
b
1
gi ,
ρ i=1
N
(22)
∇·u=
which can be combined with the divergence of (19), resulting in
(23)
b
1
−λ∇ p =
gi .
ρ i=1
2
N
The biofilm–bulk fluid interface moves with normal velocity
(24)
γ̇ = −λ∇p · n,
where n is a unit normal pointing into the bulk fluid. As a remark, note that equations
(21) can be rewritten in the forms
(25)
θi,t − λ∇p · ∇θi =
Nb
θi gi
−
gi .
ρ
ρ i=1
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
21
Fig. 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.
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 right-hand side of (24) [201, 229, 255].
Growth models are popular predictive tools in engineering applications such as
waste and water treatment [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
setup described so far is generally not the convenient one in the laboratory. In par-
22
ISAAC KLAPPER AND JACK DOCKERY
ticular, the stress balance posited in equation (19) omits important mechanics that
can be significant in laboratory systems, which often include nonnegligible 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., would 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 = 0 and where x = (x, y) is the vector of horizontal coordinates. Here h(0) (t)
is the location of the unperturbed interface; see (15). The first order perturbation
of the surface normal n is orthogonal to the zeroth order normal n0 = ẑ to the zdirection and hence orthogonal to the zeroth order pressure gradient which is parallel
to ẑ. Thus the interface velocity is, to first order,
(26)
(0)
(t), t)]
ḣ = −n0 · ∇p|z=h = ḣ(0) + eik·x [h(1) (t)g (0) (h(0) (t), t) − p(1)
z (h
(see (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
(1)
to bumps 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 toward 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
(27)
2
(0)
− z)κ−1 r (0))c(1)
c(1)
zz = (k + H(h
with accompanying boundary conditions and interface (at z = h(0) ) conditions. For
(1)
z > h(0) , (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
(28)
2
−1 r (0))c(1)
c(1)
zz = (k + κ
on 0 < z < h(0) with computed boundary conditions that can also be obtained
explicitly (see [59]). Note
within the biofilm decays like
√ that substrate perturbation
exp(−α(h(0) − z)), α = k 2 + −2 (recall = κ/r (0) is a measure of active layer
depth), away from the interface. Relative to zeroth order quantities, short wave
lengths (k 2 −1 ) decay quickly while long wave lengths (k 2 −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
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
23
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
(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)
, t)
z (h
= −Ck +
h(0)
0
Gc(1) cosh[k(h(0) − s)]ds.
The first term, −Ck, is the expected suppressive term; increased growth at the top of
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 −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, the 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 long-time behavior of biofilm-environmental interactions would seemingly require
an adequate model of biofilm mechanics including constitutive behavior.
Observations of biofilm mechanics have mainly been of two types: measurements
of constitutive response (e.g., linear regime stress-strain relations) [34, 123, 130, 131,
197, 209, 219, 220] and measurement of material failure (e.g., detachment due to mechanical failure) [143, 146, 156, 157, 158, 175, 211]. 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 rate proportional to the square of
the local biofilm height or by a similar rule [201, 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, 186, 248]. Note
that mechanically determined detachment or erosion is not the only loss mechanism;
dispersal [177] 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, 170]. 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 polymer-solvent two-phase (or more) theory [61, 145, 217].
In the simplest version, the system is divided into two phases: a sticky, viscoelastic
24
ISAAC KLAPPER AND JACK DOCKERY
biomaterial fraction consisting mainly of cells and matrix and 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 Figure 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
κ
f (φb ) + |∇φb |2 dV,
(30)
E(φb ) =
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
Energy
Density
0
1
Biomaterial Volume Fraction
Fig. 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.
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
25
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 than 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 models
can be found in [174]. We also note that surfactants may have an important role in
cohesion [50, 160].
Using (d/dt)φb + ∇ · (φb ub ) = 0 (and neglecting change of φb due to growth), we
can calculate
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
(32)
0 = φb ∇ · Π − ζ(ub − us ) + φb ∇p,
(33)
0 = −ζ(us − ub ) + φs ∇p,
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
(34)
∂φb
+ ∇ · (φb u) = ∇ · [a(φb )f (φb )∇φb ] − κ∇ · [a(φb )∇∇2 φb ],
∂t
where we suppose ζ = ζ0 φb φs and set a(φb ) = ζ0−1 φb φs = ζ0−1 φb (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, 188, 197, 209, 220]; 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 for mechanical respects only, by 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
26
ISAAC KLAPPER AND JACK DOCKERY
Fig. 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.
with center-of-mass x and end-to-end displacement q. Under this framework,
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
κ
E(ψ) =
f (φb ) + fe (ψ) + |∇φb |2 dV
2
with, in a simple version,
(35)
fe (ψ) = νkT
ln(ψ/φb ) + q 2 ψ dq,
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,
(36)
Π = −[f (φb ) − κ∇2 φb + νkT (q 2 + ln(ψ/φb ))]I.
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
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 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
(37)
∂M
+ ∇ · (uM ) − ∇uTb M − M ∇ub = ∇ · ζqqψ∇ · Πq ,
∂t
where · denotes averaging over q. The average on the right-hand 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 fluidlike contribution.
More complicated laws can arise using more involved physics [136]. See [227] for
computational examples.
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
27
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 [147]. 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, 196]. Such
dosages are expensive and may 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 strain-specific 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 to a nonspecific modulated population tolerance to general chemical attack. That is, we consider a situation whereby the same population 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 freefloating 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,
140, 205]: (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 our attention to 1D 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 a 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 [206]. Nevertheless, for many antimicrobials, penetration is slowed (e.g., delayed penetration of
aminoglycoside-type antibiotics due to binding to matrix material [87, 152]) or even
effectively halted altogether (e.g., limited depth penetration of chlorine into P. aerug-
28
ISAAC KLAPPER AND JACK DOCKERY
inosa 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 [204] 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 [153], and ciprofloxacin [212]),
(ii) interaction with biofilm material (perhaps as a part of the antimicrobial process)
can lead to irreversible reaction through which process 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
(38)
Bt = κBzz + f,
where f describes the three reactive mechanisms listed above [153, 204]. 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.,
(39)
S = g(B),
where g is a monotonic, saturating function of the sort illustrated in Figure 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
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
29
sites (for antimicrobial) and satisfies the equation
(40)
Xb,t = −
R(B)
Xb ,
Yb
Xb (z, 0) = Xb0 ,
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 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
(41)
(1 + g (B))Bt = κBzz − R(B)Xb − C(B).
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 exist only for a finite time. Then
equation (41) reduces approximately (away from z = h) to
Bt = κ̄Bzz − αB,
(42)
where
κ̄ =
κ
,
1 + g (0)
α=
R1 Xb + C1
1 + g (0)
and R1 and C1 are first order rate constants. Note that κ̄ < 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 [204]. While transient, this delay may be sufficient to
allow other defensive responses; see below.
The balance of diffusion and reaction
in (42) results in a decay distance = 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 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 are occupied.
The decay distance will thus also increase in time up
to a maximum of = κ0 /C1 .
On longer times, (38) equilibrates to
C(B) = κBzz ,
Below the resulting reactive layer, where B is small and C(B) ≈C1 B, antimicrobial
concentration is depleted exponentially with length scale = κ/C1 [203]. 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).
30
ISAAC KLAPPER AND JACK DOCKERY
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, (42) would
be replaced by
(43)
Ct = κCzz − αC,
where C is, say, oxygen concentration. Equation (43) is supplemented by a killing
rate equation of the form
(44)
bt = −k(C, B, b),
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
hand, 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 proceeds, 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.
However, it seems that, in actuality, this strategy does not necessarily work
well [15, 191]. 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 up-regulating production of protective enzymes [69, 85] or of extracellular
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
31
polysaccharides [13, 190], 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 [215], 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 an intranecine microbe-microbe conflict billions of years old, not 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 a 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 an 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 ),
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.
32
ISAAC KLAPPER AND JACK DOCKERY
If we modify the interface velocity (15) to include an erosion term [95, 201], 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 nontrivial stable steady state solutions. Existence
of such steady states (not necessarily unique) for a similar system was shown in [216]
with the necessary assumption that e(h)/h → ∞ as h → ∞. An important result
shown in [216] 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 to kill nongrowing 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 survives sustained exposure to penicillin [20]. A signature of these supertolerant 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 [140], 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 that provide 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 sporelike state (spores are not observed) nor cells caught coincidentally in a special, quiescent phase of cell division (the proportion of persister cells
is too small) [14, 116, 140]. 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 [213].
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, 213].
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 slowgrowing and generally tolerant in some way [140]; cells are able to switch in and out
of the persister state, with a greater tendency toward persistence when environmental
conditions are unfavorable, e.g., in stationary state conditions. In some versions,
switching is triggered by presence of antimicrobial stress [40, 54] and in others there
33
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
7
6
Log p
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
TIME
Fig. 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.
is an everpresent stochastic jump rate [135, 187]. Mathematically, these mechanisms
are similar to the adaptive stress response discussed in section 5.3, one difference 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 [200]. 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.
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
(45)
bt (σ, t) + (v(σ, t)b(σ, t))σ = −rd (σ, C, B)b(σ, t), σ > 0, t > 0,
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 supposedly less susceptible to
antimicrobials than less senescent ones. For simplicity we neglect C-dependence and
then a simple form for rd would be
(46)
rd (σ, B) = kd + H(δB − σ)kK ,
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)
34
ISAAC KLAPPER AND JACK DOCKERY
(a)
7
10
6
6
10
10
5
5
10
4
population
population
10
10
3
10
2
3
10
10
1
1
10
10
all cells
persisters
0
0
4
10
2
10
10
(b)
7
10
20
40
time (hr)
60
80
all cells
persisters
0
10
0
10
20
30
40
time (hr)
50
60
70
Fig. 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].
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
∞
(47)
b(0, t) =
rb (σ, C)b(σ, t) dσ,
0
where rb is a birth rate function. Equation (45) and condition (47) are supplemented
by
∞
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 persistence would seem to provide a reasonable qualitative match to available data, but so does the phenotype-switching model. In fact, the
MATHEMATICAL DESCRIPTION OF MICROBIAL BIOFILMS
35
two are difficult to distinguish at the population-modeling level. The reason for this
is that a nonsenescent/senescent divide is rather similar to a dual phenotype compartmentalization. Nonsenescent cells form a compartment and senescent cells form a
compartment with conversion of a sort occurring between the two: nonsenescent cells
eventually become senescent, while senescent cells produce, albeit at a slow rate, new
(and nonsenescent) 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. New 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 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 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.
Acknowledgments. The authors would like to thank Erik Alpkvist, Eric Becraft,
Shane Nowack, Phil Stewart, Bruce Ayati, and Tianyu Zhang for their assistance and
contributions.
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