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Chapter 19 : Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts

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Abstract:

This chapter presents interesting examples of the use of mathematical models in microbial ecology, since comprehensive coverage is not possible. The examples were also chosen to introduce various types of mathematical models to provide some guidance on the choice of suitable types of models to address a particular problem. The common thread running through the variety of examples and models presented is how spatial structure can change the interactions of microbes with each other and the host. To this end, the author begins with a model of a well-mixed system, the chemostat, as a reference point and then moves to spatially structured systems, including chemostats with wall growth, plug flow reactors (PFRs), colonies on agar plates, and finally biofilms. The discussion of chemostat dynamics focuses on the steady state, although more interesting dynamics such as damped or sustained oscillations can occur even in single-substrate, single-species chemostats, e.g., due to the slow induction of a transporter causing a delayed response. The indigenous flora of the intestinal tract is quite diverse and stable, implying that many species have been able to colonize the gut and coexist for a long time, yet the resident flora is resistant to colonization by new invaders. The common explanation that microbes form biofilms because they provide better growth conditions than the bulk liquid neglects the intense competition for resources diffusing into the biofilm, allowing only the top layer of cells to grow.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19

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Image of FIGURE 1
FIGURE 1

Schematic drawing of the essentials of a chemostat. The (idealized) chemostat is characterized by continuous flowthrough of material and perfect mixing of material within the reaction vessel. A vessel of volume, is fed at a constant flow rate, from a reservoir with substrate concentration, , replenishing the substrate concentration in the chemostat, . The reactor volume remains constant because the bulk liquid containing bacteria, substrate, and metabolites is removed from the reactor at the same . This leads to dilution of all contents of the chemostat bulk liquid (bacteria, substrate, and metabolites) with dilution rate, determined as . When the state of this system fluctuates or oscillates, the term chemostat, suggesting static conditions, is better replaced by the more technical term “continuous-flow stirred-tank reactor” (CSTR).

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 2
FIGURE 2

(a) Graph of the Monod function, the standard model describing the dependence of specific growth rate, μ(), on the substrate concentration, . is the substrate concentration at which half the maximal specific growth rate, μ, is reached; it therefore depends on μ. The specific affinity sensu Button, = μ/, is the initial slope of the curve and independent of μ. (b) Graphs of the Monod functions of two competitors for the most interesting case where the curves cross at a particular substrate concentration. This requires that the specific affinity of species 1 is higher than that of species 2, while the maximal specific growth rate of species 1 is lower than that of species 2. In the steady state of the chemostat, the specific growth rate becomes equal to the dilution rate set by the experimenter, indicated by the horizontal line. The competitor that can achieve this specific growth rate at a lower steady-state substrate concentration, *, will win the competition and invade a chemostat system dominated by the other species, because it has a positive net rate of increase if is above its own *, which includes the * of the weaker competitor. In short, the competitor with the lower * will win, which is known as the * rule ( ). While microbiologists use the term substrate, ecologists use the more general term resource, and that is where the * rule originates. In this case, competitor 1 has the lower * if the dilution rate is below the crossover point, but above, competitor 2 has the lower *.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 3
FIGURE 3

Scheme of the minimal structured model that can produce all observed phenomena of mixed substrate growth: diauxie, simultaneous use, and bistability. The enzyme dynamics of autocatalytic synthesis of the inducible enzymes and inhibition by dilution of cellular contents are analogous to the population dynamics of autocatalytic growth and resource competition. The model specifies the kinetics of all the lumped enzymatic reactions shown. Components are in roman type, and processes are in italics. Adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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FIGURE 4

Steady-state values for the biomass density of the planktonic population, *, and substrate concentration, *, in the (idealized) chemostat as a function of dilution rate, . The colonization steady state, * = ( *) and * = /(μ), is stable below the critical dilution rate, = μ /( + ). Above, the washout steady state, * = 0 and * = , is the only existing steady state; it is therefore stable. Parameters used in this example were μ = 1, = 0.1, = 1, and = 0.5.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 5
FIGURE 5

The standard model of chemostat competition can generate oscillations if competitiveness is not transitive; there is no need for external fluctuations to maintain biodiversity. The example is based on the growth of algae, which is limited by a handful of abiotic resources. (a) Competition of three species on three resources can sustain periodic oscillations, which allow coexistence of more species than resources. There is a cyclic chasing of competitors in time. Parameters for this simulation were taken from the first three species of Fig. 1c of reference . (b) Chaotic oscillations can result for a range of parameter values so that each species is an intermediate competitor for the resources that most limit its growth rate. Here, five species compete for three resources. Parameters for this simulation correspond to Fig. 1 of reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 6
FIGURE 6

Dependence of ATP yield, , substrate flux, , and ATP flux, , on pathway length, according to the kinetic theory of optimal pathway length ( ). For the sake of simplicity, one assumes a linear pathway with linear, irreversible, and identical kinetics for all enzymes. Further, ATP yield is assumed to be proportional to pathway length. In this simplest case, analytical expressions for the fluxes can be derived. is the rate constant for all enzymes, is the substrate concentration, the total concentration of all enzymes is restricted to , and the total concentration of all intermediates is restricted to . For this example, all parameters were set to unity, apart from = 1/4. Since ATP flux is the product of substrate flux and ATP yield, an optimal pathway length exists if the substrate flux decreases more steeply than the ATP yield increases. The optimal length of a pathway is shorter at higher substrate concentration and at higher metabolic costs of intermediates (restricting to lower values). Qualitatively the same results are obtained for reversible and/or Michaelis-Menten kinetics ( ). Adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 7
FIGURE 7

Graph of the function () = (1 − )/(1.1 − ), which models the fraction of biofilm offspring that remains grounded or bound to the wall because a free attachment site in the vicinity was available, which is therefore a monotone decreasing function of the fraction of the wall occupied, = ∑ B/ , where is the maximum number of attachment sites on the wall.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 8
FIGURE 8

Competition between resident and invader for the same limiting nutrients and adhesion sites in a chemostat. The resident and invader are the same strain, so they are equally fit, i.e., have equal parameters. (a) With adhesion to the gut wall; (b) without adhesion to the gut wall. Redrawn and adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 9
FIGURE 9

Schematic drawing of a PFR. A “plug” or disk-shaped volume element of material enters the PFR tube on one end, being transported with constant volumetric flow rate along the length of the tube without axial mixing of the liquid (diffusive spread of the plug’s components in the axial direction might be included). On the other hand, radial mixing is assumed to be perfect, so diffusion gradients cannot form in the radial direction. The plug’s material will “age” during transport, e.g., microbes grow and nutrients deplete, so at a certain position along the length of the tube the plug’s material will have a constant age and composition at steady state. Axial profiles of substrate concentrations resulting from enzymatic breakdown (assuming Michaelis-Menten kinetics and uniform enzyme concentration) or autocatalytic fermentation (assuming Monod kinetics for growth of the biomass) are shown. Note that a PFR can be approximated by linking CSTRs in series, since this reduces the dilution effect of a single CSTR and localizes the mixing. Indeed, as the number of CSTRs in the series increases, the behavior of the system rapidly approaches that of plug flow.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 10
FIGURE 10

An example of the spatial dynamics of colicin-producing (C), -resistant (R), and -sensitive (S) “colonies on agar plates” as simulated with the cellular automaton model described in reference , using the parameters of Fig. 1a to d of that reference. White patches are not colonized. (a) Initial distribution of colonies (only the bottom strip is shown since the distribution is uniform in the vertical direction). (b) After 10 steps (epochs) of the simulation. The small size of the CA lattice and uniformity of the initial lineup was chosen to show the workings of the CA rules described in the text. Since these rules are stochastic, runs will differ; however, the general trend is as shown: the C strain chases the S strain, the R strain chases the C strain, and the S strain chases the R strain. The empty patches can be colonized by the neighbors, here the S and R strains.

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 11
FIGURE 11

Diffusion of substrate is driven by the substrate consumption in the biomass, which creates a concentration gradient from the bulk liquid (source) through the concentration boundary layer to the biomass (sink). Two equally sized biofilm clusters are shown as gray areas. The 20 contour lines indicate substrate concentration from 100% in the bulk liquid down to 5% near the cluster surfaces, in steps of 5%, and the arrows indicate substrate flux. Substrate concentration in the bulk liquid is assumed to be in steady state. The length scale is in micrometers. Note the steeper gradient towards the cluster with the higher rate of substrate consumption and that most of the substrate is consumed in a thin layer along the biofilm surface. Adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 12
FIGURE 12

Effect of different spreading mechanisms on spatial mixing in multispecies biofilms. Shown are nitrifying biofilms with ammonia-oxidizing bacteria (light gray) and nitrite oxidizing bacteria (darker gray) competing for oxygen. The contour lines show the oxygen concentration in milligrams per liter. (a) CA use stochastic rules for placing newly formed biomass blocks in the neighborhood. These sudden division events inside the biofilm trigger recursive random displacements of neighbors until the biofilm surface has been reached, resulting in stronger mixing than in IbM. (b) IbM uses a shoving algorithm where cells push each other away if they happen to come too close, and only as far as necessary. The stronger mixing evident in the CA model allows nitrite-oxidizing bacteria deep in the biofilm, where the oxygen concentration is so low that they hardly grow, to mix into the still-growing biomass of the ammonia-oxidizing competitors, so that they are carried along with the flow of the growing biomass upwards, where the oxygen comes from. Thus, spatial mixing can profoundly change competitive interactions. Adapted from Fig. 7c and d of reference . A movie of the time course can be found on my website (http://www.theobio.uni-bonn.de/people/jan_kreft/biofilms.html).

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 13
FIGURE 13

Competition between groups of rate (light) and yield (dark) strategists in a biofilm population. The surface was initially colonized by single cells at equal distances, alternating between rate and yield strategists. (a) 5 cells each; (b) 10 cells each; (c) 20 cells each. If cells were far enough apart initially (a), the rate strategists could not overgrow the yield strategists during the initial phase of unlimited growth, when they grow faster. When growth becomes limited by diffusion of substrate into the biofilm, the yield strategists convert the influx of substrate more efficiently into biomass and grow faster as a group. At high initial cell density (c), the rate strategists overgrow the yield strategists, but by chance some yield strategists are carried upwards, with the flow of the spreading biomass, and form a cluster of yield strategists at the biofilm surface. If these yield clusters at the surface are large enough, they can overgrow the neighboring rate strategists. The breadth of the domain was 200 μm. Movies of these simulations that show the spreading of the biomass can be seen on my website (http://www.theobio.uni-bonn.de/people/jan_kreft/altruism.html). Adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Image of FIGURE 14
FIGURE 14

A simple mathematical model of autoinducer production and diffusion shows the effect of spatial distribution of the same number of cells (same cell density) on the steady-state concentration of autoinducer obtained. (a) Random cell distribution; (b) clustered cell distribution; (c) same clustered distribution but without positive feedback in the production of autoinducer. The autoinducer concentration is indicated by contour lines in percentage of the threshold for up-regulation and also by increased darkness at increasing concentration. It is much higher in the clusters and their vicinity because cells in clusters become up-regulated (white inside), which due to positive feedback leads to higher rates of signal production in the clusters. The scale is in micrometers. Adapted from reference .

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
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Tables

Generic image for table
TABLE 1

Hamilton’s classification of social behavior according to the fitness effects of the behavior for actor and recipient(s)

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19
Generic image for table
TABLE 2

An example of Simpson’s paradox showing that the frequency of altruists can increase globally despite a decrease of the frequency of altruists within each group

Citation: Kreft J. 2009. Mathematical Modeling of Microbial Ecology: Spatial Dynamics of Interactions in Biofilms and Guts, p 347-377. In Jaykus L, Wang H, Schlesinger L (ed), Food-Borne Microbes. ASM Press, Washington, DC. doi: 10.1128/9781555815479.ch19

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