Modeling Antibiotic Tolerance in Biofilms by Accounting for Nutrient Limitation

A mathematical model of biofilm dynamics was used to investigatethe protection from antibiotic killing that can be affordedto microorganisms in biofilms based on a mechanism of localizednutrient limitation and slow growth. The model assumed thatthe rate of killing by the antibiotic was directly proportionalto the local growth rate. Growth rates in the biofilm were calculatedby using the local concentration of a single growth-limitingsubstrate with Monod kinetics. The concentration profile ofthis metabolic substrate was calculated by solving a reaction-diffusionproblem. The model predicted the following features: stratifiedpatterns of growth with zones of no growth in the biofilm interior,slow killing of biofilm microorganisms that was further retardedas the initial biofilm thickness increased, nonuniform spatialpatterns of killing inside the biofilm, biofilm killing ratesthat decrease in a nonlinear way as the concentration of thegrowth-limiting substrate feeding the biofilm is decreased,and heightened tolerance when external mass transfer resistanceis manifested. This modeling study also provides motivationfor further investigation of a hypothetical cell state in whichdamaged cells score as nonviable but continue to consume substrate.The existence of such a cell state can further retard biofilmkilling, according to the simulations. The results support theimportant contributions of nutrient limitation and slow growthto the antibiotic tolerance of microorganisms in biofilms.

INTRODUCTION

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Introduction

Bacteria and yeast that grow in biofilms are responsible fordiverse persistent infections (2, 4). The tenacity of such infectionsis attributed, at least in part, to the reduced susceptibilitiesof microorganisms growing in biofilms to antimicrobial chemotherapy(12). One of the long-standing hypotheses to explain the poorkilling of biofilm cells by antibiotics is that the biofilmcontains slowly growing or nongrowing microorganisms that areprotected by virtue of their inactivity (1, 18). Nutrient limitationand slow growth are known to be common features of the biofilmmode of growth. Experimental tests of the slow-growth mechanismof biofilm protection are generally consistent with the ideathat this mechanism provides at least a partial explanationfor the recalcitrance of biofilms to chemotherapy (5).

There is a reason to be skeptical of a biofilm defense basedon nutrient limitation and slow growth. One would expect thatas the growing cells in a biofilm are killed, nutrients shouldpenetrate the biofilm, which would then feed the more deeplyembedded cells and render them susceptible. The protection affordedby this mechanism would be only transient by this reasoning.This paradox is the motivation for the modeling study reportedin this article. The interaction of microbial growth, nutrientutilization, nutrient diffusion, and antibiotic killing is complexand nonintuitive. The purpose of this study was to investigate,using a computer model of biofilm dynamics, the degree of protectionthat can be anticipated in a biofilm exposed to a growth-dependentantibiotic.

MATERIALS AND METHODS

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Materials and Methods

The basic mathematical model used in this study has been describedin detail elsewhere (11, 14). The model was based on the conceptualand mathematical formulation derived by Wanner and Reichert(16). The model described the growth of a uniformly thick biofilmin a continuous-flow stirred tank reactor, i.e., a chemostatwith wall growth. Biologically, the system was conceptualizedas a single species whose growth rate was determined by theconcentration of a single substrate, according to Monod kinetics.Some of the processes integrated in this model included bulkflow into and out of the reactor, transport of solutes intothe biofilm by Fickian diffusion, substrate consumption by themicroorganism, microbial growth, transport of cells within thebiofilm by advective displacement, detachment of biomass fromthe surface of the film, and killing of microorganisms in thepresence of an antibiotic. Macroscopic material balances aroundthe entire reactor vessel were coupled to one-dimensional differentialmaterial balances that described processes occurring withinthe biofilm at the microscale.

In the base case model, two cell states were simulated: liveand dead (Fig. 1A). Both cell types occupied the same volume,but only live cells consumed substrate and were capable of growth.Live cells could be converted to dead cells upon exposure tothe antibiotic. The rate of killing was taken to be directlyproportional to the concentration of live cells, the concentrationof antibiotic, and the local specific growth rate of the livecells. In a special version of the model, a third cell statewas introduced (Fig. 1B). This state, termed "damaged," wasa hypothetical intermediate between live and dead. Damaged cellscould still consume substrate, but they did not grow and werenot capable of recovering to the live cell state. In actuality,cells might occupy a spectrum of states from viable and fullyactive to inactive and dead. The use of a single intermediatestate is a convenient simplification. The damaged cell stateis supported by some experimental evidence (8, 13).

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FIG. 1. Two conceptual models of antibiotic action against bacteria. In the first model (A), there are only two cell states: live and dead. Live cells are converted to dead cells upon exposure to antibiotic. The rate of this transformation is characterized by the coefficient k₁ and is proportional to the growth rate of the cell. In the second conceptual model (B), three cell states are proposed and are labeled live, damaged, and dead, respectively. Live cells are converted to damaged cells upon exposure to antibiotic. The rate of this transformation is characterized by the coefficient k₁ and is also proportional to the growth rate of the cell. Damaged cells are converted to dead cells at a fixed rate, independent of additional exposure to antibiotic; the rate of this process is characterized by the coefficient k₂. The second step is assumed to be independent of growth rate. Live cells and damaged cells are capable of using substrate, while dead cells exhibit no metabolic activity.

Base case parameter values are summarized in Table 1. For thisstudy, the dilution rate was set to a very large value (417h^-1, which is 1,000 times the maximum specific growth rate of0.42 h^-1). The effect of this was that the bulk fluid concentrationswere essentially equal to the influent settings. All of thegradients that occurred in the system were inside the biofilmor in the concentration boundary layer immediately adjacentto the biofilm.

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TABLE 1. Parameter values for biofilm modeling

RESULTS AND DISCUSSION

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Results and Discussion

Once a biofilm is thick enough, it invariably experiences gradientsin the concentrations of metabolic substrates. This phenomenonis illustrated by the results in Fig. 2A, which show the concentrationprofiles of the limiting substrate as predicted in our simulations.These profiles indicate that the biofilm becomes substrate limitedif it is thicker than approximately 50 µm. The microscalevariation in the availability of this substrate results in gradientsin the local growth rate (Fig. 2B). These predictions are consistent,at least in qualitative terms, with experimental measurementsof the concentration gradients in key substrates, for example,dissolved oxygen (3, 9). They are also consistent with the fewexperimental measurements of the spatial patterns of growthand metabolic activity within biofilms (10, 15, 17, 19). Theseexperimental measurements indicate zones of metabolic and proteinsynthetic activity of approximately 10 to 50 µm.

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FIG. 2. Predicted substrate concentration (A) and growth rate (B) profiles within biofilms of various thicknesses. Biofilms thicker than approximately 50 µm are predicted to experience nonuniform spatial distributions of substrate concentration and microbial growth. The normalized distance is the distance from the substratum divided by the biofilm thickness.

The predicted time course of biofilm killing is shown in Fig.3. Figure 3 illustrates two main points. First, biofilm bacteriaare killed more slowly than growing free-floating cells. Second,the rate of killing in the biofilm decreases as the biofilmthickness increases. It is natural to wonder whether the secondresult reflects inadequate penetration of the antibiotic intothe biofilm. However, the antibiotic is predicted to permeatethroughout the biofilm within 20 min after dosing, even forthe thickest biofilm. The antibiotic concentration in the biofilmremains at the applied concentration throughout the treatmentperiod. Slow antibiotic penetration is therefore not an importantfactor in protection of the biofilm in these simulations.

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FIG. 3. Biofilm killing (solid lines) versus time as a function of initial biofilm thickness. The dashed line indicates the predicted killing of planktonic cells. The protection is predicted to be greater the thicker the biofilm is.

A bit more insight into the time course of killing can be gleanedby examining the profiles of live and dead cell concentrationswithin the biofilm (Fig. 4). Bacteria are killed first nearthe biofilm-bulk fluid interface. With time, the killing frontprogresses inward toward the substratum. The movement of thekilling front (defined as the location in the biofilm wherethe live and dead cell concentrations are equal) is chartedin Fig. 5. These results indicate that the killing front ispredicted to move at a velocity ranging from about 20 to 50µm h^-1. The rate at which the killing front advances appearsto be determined by the rate at which the substrate penetratesinto the biofilm. It is important that even though dead bacteriaare unable to consume substrate in this version of the model,their physical presence imposes a resistance to the transportof substrate into the depths of the biofilm. In other words,dead cells shield their more deeply embedded and living neighborsfrom substrate, and this shielding retards the advance of thekilling effects of the antibiotic.

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FIG. 4. Temporal evolution of live (solid line) and dead (dotted line) cell spatial distributions within an antibiotic-treated biofilm. Prior to antibiotic treatment, all of the cells were live cells. Each solid and dashed line pair represents a different stage in the time course of antibiotic treatment, as labeled. This simulation was conducted for a biofilm that was initially 300 µm thick.

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FIG. 5. Movement of killing front into a biofilm during antibiotic treatment. The position of the killing front, defined as the location where the fractions of live and dead cells are equal, is measured as the distance from the substratum.

If substrate delivery controls the rate of killing, then increasingthe substrate concentration in the system should improve theaction of the antibiotic against the biofilm. The results ofthe simulations of this experiment are reported in Fig. 6. Thetime required to achieve a certain extent of killing (a 6-logreduction) is predicted to be a strong and nonlinear functionof the substrate concentration in the bulk fluid. Increasingthe substrate concentration decreases the time required to killthe biofilm, and decreasing the substrate concentration significantlyprolongs the time required to kill the biofilm. For example,the killing time (i.e., the time required to achieve a 6-logreduction) is predicted to be 13 h in the presence of the basecase substrate concentration of 8 mg/liter, but it lengthensto 66 h (2.7 days) when the substrate concentration bathingthe biofilm is reduced to 1 mg/liter. In light of this prediction,it would be interesting to see experimental measurements ofbiofilm susceptibility as a function of the nutrient concentration.

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FIG. 6. Time required for a 6-log reduction in biofilm viable cell numbers as a function of bulk fluid substrate concentration (solid line). The dotted line indicates the base case oxygen concentration of 8 mg/liter, and the dashed line indicates the minimal killing time attained at very high oxygen concentrations. The killing time is strongly dependent on the substrate concentration, especially for low substrate concentrations. This simulation was conducted for a biofilm that was initially 300 µm thick.

All of the preceding results are based on a model in which antibiotic-treatedbacteria are assumed to lose all ability to consume substratecoincident with their loss of viability. This is probably notrealistic. Bacteria that have been damaged by an antibioticmay continue to respire and even make new proteins for hoursafter they have lost the ability to form a colony on a plate.To account for this possibility, we developed a version of ourmodel that allowed for three possible cell states, as shownin Fig. 1B. An intermediate cell state, which we have termed"damaged," represents cells that continue to consume substrate,even though they would not score as viable in a conventionalexperimental measurement. When this intermediate cell stateis included in the model, antibiotic killing of biofilm cellsis retarded (Fig. 7). The degree of retardation ranges from1 to more than 5 for a 300-µm-thick biofilm, dependingon the relative rates of the transformation of live cells todamaged cells and damaged cells to dead cells. These relativerates are captured in the ratio k₂/k₁. Large values of k₂/k₁indicate that damaged cells die more rapidly than they are formed,and therefore, damaged cells will be few and are expected tohave little impact on the survival of the biofilm. Small valuesof k₂/k₁ indicate that damaged cells die slowly and that theywill occupy a significant fraction of the biofilm and can affectthe time course of killing. The rates that we have used arepurely hypothetical, but these simulations illustrate that theexistence of a nonviable, yet respiring cell state will addto the protection in the biofilm state. This intermediate cellstate would not be expected to protect bacteria in the planktonicstate because free-floating cells are all exposed to the samesubstrate concentration. The model predicts that damaged cellsare found throughout the biofilm but that their numbers aregreatest near the killing front.

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FIG. 7. Retardation of time required to achieve a 6-log reduction in biofilm viable cell numbers as a function of the ratio of damaged cell and live cell death rates (k₂/k₁). The kill time retardation ratio is the killing time computed for a biofilm with damaged cells to that for one in which no damaged cells are present.

Another process that would act to retard biofilm killing ismaintenance utilization of substrate. Even bacteria that arenot growing, if they consume substrate for maintenance purposes,would act to keep more deeply embedded cells in a state of nutrientdeprivation.

Implicit in all of the preceding simulations is the assumptionthat the slowly moving fluid adjacent to the biofilm imposesminimal resistance to delivery of the substrate to the biofilmsurface. More precisely, an external mass transfer film thicknessof 10 µm was assumed. This might be an acceptable approximationfor a biofilm system in which there is vigorous mixing or rapidflow resulting in turbulent conditions. Under laminar flow conditions,the external mass transfer resistance to substrate transportis likely to be larger. This has been simulated by increasingthe external liquid film layer thickness in the model from 10to 300 µm. This range of external film thicknesses isconsistent with limited experimental measurements of mass transfercoefficients (6, 7). The result is significant retardation ofbiofilm killing (Fig. 8). Slower killing results because thesubstrate flux to the biofilm is reduced. This set of simulationsshows that external mass transfer resistance exacerbates nutrientlimitation and further protects biofilm bacteria from antibiotickilling.

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FIG. 8. Effect of external mass transfer resistance on biofilm killing. External mass transfer resistance is determined by the thickness of the external fluid film, L_L. The base case external fluid film thickness used in all other simulations was 10 µm.

The combined effects of nutrient limitation, utilization ofsubstrate by damaged, nonviable cells, and external mass transferresistance have the potential to account for substantially reducedsusceptibility in the biofilm state. For example, a 300-µm-thickbiofilm that is bounded by a 300-µm-thick external fluidfilm and in which the value of the k₁ rate coefficient is threetimes the value of k₂ would be highly protected. These conditionslead to the prediction that the biofilm would be killed (6-logreduction) only after 52 h of continuous treatment. For comparison,planktonic cells would be killed to the same extent after only2.7 h of antibiotic treatment.

These simulations collectively demonstrate that nutrient limitationand slow growth do constitute a plausible protective mechanismin biofilms when the antibiotic depends on substrate availabilityor growth for its killing action. These results do not provethat this is a sufficient explanation for antibiotic tolerancein biofilms. Our simulations suggest that nutrient-limited growthcan retard killing in biofilms but cannot explain indefiniteprotection. Nutrient limitation and slow growth probably contributeto reduced biofilm susceptibility and operate in concert withother protective mechanisms to achieve the full degree of recalcitrancethat is observed in vitro and in vivo.

We wish to emphasize that the variations in growth rate withinthe biofilm manifested in this model were predicted by calculatingthe simultaneous interaction of reaction and diffusion fromfirst principles. This model serves as a basis for the designof experiments to test the effects of the substrate concentrationon biofilm susceptibility and to investigate the possibilityof substrate utilization by antibiotic-damaged cells.

ACKNOWLEDGMENTS

This work was supported by an award from the W. M. Keck Foundationand by NIH award GM67245-01 to P.S.S.

Peg Dirckx drew Fig. 1.

FOOTNOTES

* Corresponding author. Mailing address: Center for Biofilm Engineering and Department of Chemical Engineering, Montana State University—Bozeman, Bozeman, Montana 59717-3980. Phone: (406) 994-2890. Fax: (406) 994-6098. E-mail: phil_s{at}erc.montana.edu.

REFERENCES

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