Dynamic Metabolic Modeling of Denitrifying Bacterial Growth: The

Article pubs.acs.org/IECR

Dynamic Metabolic Modeling of Denitrifying Bacterial Growth: The Cybernetic Approach Hyun-Seob Song* and Chongxuan Liu Pacific Northwest National Laboratory, Richland, Washington 99352, United States

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S Supporting Information *

ABSTRACT: Denitrification is a multistage reduction process converting nitrate ultimately to nitrogen gas, carried out mostly by facultative bacteria. Modeling of the denitrification process is challenging due to the complex metabolic regulation that modulates sequential formation and consumption of a series of nitrogen oxide intermediates, which serve as the final electron acceptors for denitrifying bacteria. In this work, we examined the effectiveness and accuracy of the cybernetic modeling framework in simulating the growth dynamics of denitrifying bacteria in comparison with kinetic models. In four different case studies using the literature data, we successfully simulated diauxic and triauxic growth patterns observed in anoxic and aerobic conditions, only by tuning two or three parameters. In order to understand the regulatory structure of the cybernetic model, we systematically analyzed the effect of cybernetic control variables on simulation accuracy. The results showed that the consideration of both enzyme synthesis and activity control through u- and v-variables is necessary and relevant and that uvariables are of greater importance in comparison to v-variables. In contrast, simple kinetic models were unable to accurately capture dynamic metabolic shifts across alternative electron acceptors, unless an inhibition term was additionally incorporated. Therefore, the denitrification process represents a reasonable example highlighting the criticality of considering dynamic regulation for successful metabolic modeling.

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INTRODUCTION Regulatory machinery endows microorganisms with a great capability to survive in widely changing environmental conditions. Accounting for regulation is therefore the key to predictive modeling of cellular metabolism. A molecular-level description of regulation is, however, often hindered due to incomplete knowledge on detailed mechanisms. Alternatively, the cybernetic approach provides a rational description of regulation by viewing a living cell as an optimal control system that regulates its internal state and functions (such as enzyme levels and activities, and subsequently biochemical reaction rates) to achieve a certain metabolic objective (such as the maximization of growth rate or nutrient uptake rate). For more than three decades since Ramkrishna introduced the cybernetic modeling idea for simulating biological systems in the early 1980s,1 cybernetic modelers have reported a great number of success in diverse applications,2 which include prediction of complex growth patterns on mixed substrates based on simple3−10 and detailed networks,11−17 nonlinear analysis of metabolic systems,18−20 estimation of intracellular flux distributions,21 prediction of dynamic cellular response to genetic modifications,22−25 and simulation of interspecies interactions in simple microbial consortia.26 Recently, the usefulness of the cybernetic approach in simulating complex microbial communities (as a sole framework or as a component of integrative modeling) has been discussed.27 One idea is to model the dynamics of microbial communities like the growth of single organisms by treating the community as a supra-organism. In doing so, consideration of community-level regulation is essential because natural environments contain diverse electron acceptors as well as electron donors. Along this direction, the framework termed functional © XXXX American Chemical Society

gene-centric approach simulated microbial biogeochemical processes considering enzyme synthesis kinetics,28 which were primarily based on empirical formulations without a fundamental description on regulation. Motivated by the functional gene-centric approach addressed above, in this article, we examined how the cybernetic approach can be extended to simulate microbial growth in environments containing alternative electron acceptors. The main focus of the cybernetic approach has been hitherto bacterial growth on multiple electron donors, but applications to multiple electron acceptors are rarely found. Using the developed model, we also identified key regulatory variables essential for accurate metabolic modeling. The cybernetic model typically describes cellular regulation at two levels, i.e., the control of enzyme synthesis and activity, but no rigorous analysis has yet been performed to confirm their impacts on simulation accuracy. For this purpose, we took the denitrification process as a modeling example.

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PROCESS AND MODEL DESCRIPTION Biological denitrification is an attractive method to remove nitrate from the groundwater and surface water contaminated, e.g., by eutrophication.29 Denitrification is a sequential reduction from nitrate to nitrogen gas,30 i.e., NO3− → NO2− → NO → N2O → N2

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Special Issue: Doraiswami Ramkrishna Festschrift Received: April 30, 2015 Revised: June 25, 2015 Accepted: June 29, 2015

A

DOI: 10.1021/acs.iecr.5b01615 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

For the reaction network illustrated in Figure 1, eq 2 is then simplified to

Each of these conversion steps is catalyzed by distinct enzymes synthesized in facultative bacteria (Figure 1). Denitrifying

rj = vjejrelr jkin ,

j=1−5

(3)

where r jkin ≡ μjmax

aj aj + Kj

(4)


Note that the above equations drop the subscript k from eq 2, meaning that biomass production rates are governed by the assimilation rates of electron acceptors. Dynamic balance for the jth enzyme (ej) is given as an ordinary differential equation, i.e., dej dt

bacteria preferably use oxygen, if available, as the terminal electron acceptor; in the absence of oxygen (i.e., anoxic conditions), they use nitrate (NO3−) and nitrogen oxide intermediates (NO2−, NO, N2O) as alternative electron acceptors following the order in eq 1. Thus, denitrifying microorganisms exhibit complex growth dynamics due to sequential production and consumption of a series of denitrification intermediates. In the cybernetic approach, sequential reduction in denitrification is viewed as the outcome of the competition among multiple alternative pathways associated with different electron acceptors. That is, the cybernetic model controls the syntheses and activities of enzymes catalyzing alternative reactions pathways such that the prescribed metabolic objective (i.e., the growth rate in this work) is maximized. In a general environment with J electron acceptors and K electron donors, the biomass production rate (rj,k) [mg/(h L)] that assimilates the jth acceptor (aj) [mg/L] and kth donor (dk) [mg/L] can be represented as follows: aj dk , rj , k = vj , kejrel, kμjmax ,k aj + Ka , j dk + Kd , k μmax j,k

(5)

where the four terms on the right-hand side in eq 5 represent constitutive enzyme synthesis, inductive enzyme synthesis, degradation, and dilution by growth, respectively. In situations where only limited data are available for parameter identification, we further introduce three simplifying assumptions: α1 = ··· = α5 = α [1/h], β1 = ··· = β5 = β [1/h], and the same kin kin kinetics between rkin E,j [enzyme level/(h·gDW)] and rj , i.e., rE,j = kE,j(aj/(aj + Kj)). The identical values of α’s and β’s imply that the variations in constitutive enzyme synthesis rates and in enzyme degradation rates are minimal across enzyme sets involved in different metabolic pathways. For the sake of convenience, eq 5 is rewritten in terms of the relative enzyme level (erel j ), i.e.,

Figure 1. Competitive enzymatic reactions carried out by aerobic denitrifying bacteria: (a) a schematic representation of aerobic respiration and anoxic denitrification and (b) normalized stoichiometric balances of electron acceptors and biomass. Res: representative enzyme involved in respiration. Nar: membrane-bound nitrate reductase. Nir: nitrite reductase. Nor: nitric oxide reductase. Nos: nitrous oxide reductase. BIOM: biomass. YBIOM,j (j = 1−5): biomass yield from each electron acceptor. YNO2−, YNO, YN2O, YN2: NO2− yield (from NO3−), NO yield (from NO2−), N2O yield (from NO), N2 yield (from N2O).

j = 1, 2, ... J ; k = 1, 2, ... K

= αj + ujrE,kinj − βjej − μej

dejrel dt

⎛ kE, j kin⎞ μmax + β ⎜ rel ⎟ u α + j max r j ⎟ − (β + μ)ej α + kE, j ⎜⎝ μj ⎠

=

(6)

The two control variables (i.e., uj’s [−] and vj’s [−]) in the cybernetic model represent transcriptional and translational regulation, and post-translational allosteric regulation, respectively. That is, in silico cell drives reactions through dynamic adjustment of the values of uj and vj (and thus enzyme level as well) such that the total biomass growth rate is maximized over each time interval. Such optimal control actions are realized by so-called the matching and proportional laws, i.e., uj =

ejrelr jkin 5

; kin

∑k = 1 ekrelrk

vj =

ejrelr jkin max(ekrelrkkin) k

(7)

Eq 7 was derived heuristically earlier,6 but Young and Ramkrisha later provided rigorous theoretical backgrounds based on optimal control theory.31 The cybernetic approach, therefore, accounts for genetic and enzymatic regulations in such a general manner without requesting for detailed knowledge on regulatory mechanisms. Finally, material balances for electron acceptors and biomass are listed up as follows:

(2)

erel j,k

where [1/h] is the maximal specific growth rate, [−] (dimensionless), and vj,k[−] denote the relative level of the enzyme (normalized by its theoretical maximum) and its activity, and Ka,j [mg/L] and Kd,k [mg/L] are half-saturation constants. For simplicity, we confine ourselves to the condition where the electron donor is present in excess (i.e., dk ≫ Kd,k). B


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Table 1. Model Parameters Used for Simulations in Figures 2 and 3: Fixed Yields (Y’s) and Optimized Growth Rate Constants a (μmax j ) fixed parameters


optimized parameters

R2 b

parameters

Figure 2a

Figure 2b

Figure 2c

Figure 2d

Figure 3

YBIOM,1 YBIOM,2 YBIOM,3 YBIOM,4 YBIOM,5 YNO2−

N/A 0.87 1.40 N/A N/A 0.88

N/A 1.05 0.15 N/A N/A 0.018

N/A 1.05 c N/A N/A c

37 1.94 1.54 N/A N/A 0.92

10 5 4 3 2 1

YNO YN2O

N/A N/A

N/A N/A

N/A N/A

N/A N/A

1 0.5

YN2

N/A

N/A

N/A

N/A

1

μmax 1 μmax 2 μmax 3 μmax 4 μmax 5 O2 NO3− NO2− BIOM

N/A 0.210 0.083 N/A N/A N/A 0.999 0.996 0.995

N/A 0.081 0.020 N/A N/A N/A 0.934 0.952 0.959

N/A 0.039 c N/A N/A N/A 0.992 c 0.977

0.317 0.236 0.116 N/A N/A 0.999 0.996 0.990 0.968

0.1 0.05 0.032 0.018 0.008 N/A N/A N/A N/A

a The coefficient of determination (R2) denotes a goodness-of-fit measure. bCoefficient of determination R2 ≡ 1 − Σi(xi − x̂i)/Σi(xi − xi̅ ), where x̅ = (1/n)Σixi; xi and x̂i denote the ith data point and the corresponding model estimate; and n is the number of data. cNot sought due to difficulty in data reading of nearly zero values of NO2− from the original paper.

for the last one.33 The data sets from Vasiliadou et al. represent organism-specific dynamics of three denitrifiers growing under anoxic conditions. In contrast, the experiment in Kornaros and Lyberatos considered the aerobic growth where nitrate reduction takes place after the preferred electron acceptor (i.e., oxygen) is depleted. In all experiments, they provided carbon sources in excess with a focus on the sequential reduction of nitrate. Complex dynamics of denitrification process in batch reactors makes kinetic modeling challenging due to (1) a large number of kinetic parameters, many of which (particularly, half-saturation constants) may not be identifiable34 and (2) the lack of information on suitable kinetic forms. We avoided these problems by formulating the cybernetic model based on a standard Michaelis−Menten (MM) kinetics with no additional inhibition term. While half-saturation constants (K1−K5) in MM kinetics may vary among reactions, typical batch experimental data is insufficient for accurate determination of their distinct values.35 Considering such typical circumstances, we fixed half-saturation constants with a constant value, i.e.,

dxO2

= −(1/YBIOM,1)r1x BIOM dt dx NO3− = −(1/YBIOM,2)r2x BIOM dt dx NO2− = [(YNO2−/YBIOM,2)r2 − (1/YBIOM,3)r3]x BIOM dt dx NO = [(YNO/YBIOM,3)r3 − (1/YBIOM,4)r4]x BIOM dt dx N2O = [(YN2O/YBIOM,4)r4 − (1/YBIOM,5)r5]x BIOM dt dx N2 = (YN2 /YBIOM,5)r5x BIOM dt 5

dx BIOM = x BIOM ∑ rj dt j=1

(8)

where x’s [mg/L] denote the concentrations of electron acceptors and biomass, Y’s [-] denote the yields of metabolites and biomass from given electron acceptors as defined in Figure 1, and r1−r5 represent growth rates through individual reactions. Note that mass balances in eq 8 can alternatively be reformulated in terms of substrate consumption rates, which would be a preferred formulation in conditions where substrate consumption does not necessarily lead to biomass production.

K1 = K 2 = K3 = K4 = K5 = 0.5 [mg/L]

(9)

While arbitrary, the choice for half-saturation constants in eq 9 was made considering the actual range of their reported values in the literature.32 For constitutive enzyme synthesis rate (α) and enzyme degradation rate (β), we used typical values frequently considered in the cybernetic modeling, i.e.,

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α = 0.01 [1/h],

RESULTS AND DISCUSSION We simulated four separate cases of batch process mediated by different individual denitrifying bacteria growing in synthetic liquid media: Acinetobacter sp. strain, Acidovorax sp. strain, Paracoccus sp. strain, and Pseudomonas denitrificans. We used experimental data collected by Vasiliadou et al. (2006) for the first three organisms32 and by Kornaros and Lyberatos (1998)

β = 0.05 [1/h]

(10)

The variation of K, α, and β values within a certain range does not significantly affect simulation results.15 The parameter kE,j [enzyme level/(h gDW)] in rkin E,j was determined following the work of Kim et al.,12 i.e., kE, j = (μjmax + β − α)ejmax C

(11) DOI: 10.1021/acs.iecr.5b01615 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 2. Batch denitrification data and model fit: (a) Acinetobacter sp. strain, (b) Acidovorax sp. strain, (c) Paracoccus sp. strain, and (d) Pseudomonas denitrificans (data sources: (a−c) ref 32 and (d) ref 33).

Figure 3. Qualitative simulation of a full denitrification process using arbitrarily assumed parameters: (a) concentrations of electron acceptors, (b) relative enzyme levels, and (c) biomass concentration.

For simplicity, we scaled eq 11 so that emax would be 1. Yields of j metabolites and biomass in each organism were directly obtained using the corresponding experimental data (Table 1). Thus, we optimized only a subset of μmax values [1/h]. The j actual number of optimized parameters (i.e., μmax j ) depends on the scope of simulation and available data. Simulation of Growth Dynamics in Anoxic and Aerobic Conditions. We compared the simulation results using the cybernetic model with four separate data sets of denitrification experiments (Figure 2). Acinetobacter sp. strain shows a typical pattern of denitrification in an anoxic batch culture (Figure 2a). The strain grows by taking nitrate as the electron acceptor with a concomitant generation of nitrite, which is later assimilated as an alternative acceptor when the level of nitrate becomes very low. Consequently, this bacterium

shows a diauxic growth: the initial fast growth on nitrate, followed by relatively slow growth on nitrite. The lag phase observed during the switch from nitrate to nitrite can be ascribed to the cell’s adaptation from nitrate-rich to nitrite-rich environment by synthesizing a new set of enzymes. Metabolite concentration profiles are organism-specific: formation of nitrate is low in Acidovorax sp. strain (Figure 2b) and almost negligible in the Paracoccus sp. strain (Figure 2c). As a result, the growth curves in these two organisms are not clearly separated by the lag phase. A more complex growth pattern is observed in aerobic conditions. P. denitrificans preferably assimilates oxygen even though the actual concentration of oxygen is much lower than that of nitrate (Figure 2d). After the oxygen is depleted, P. denitrificans shows a similar pattern to Acinetobacter sp. strain. D


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Table 2. Optimized Parameter Values and the Coefficients of Determination of Seven Different Models (Models 1−7)a model 1 fixed variables

optimized parameters

R


a

2

model 2

model 3

×

model 4

model 5

model 6

model 7

× × × 0.149 0.090 0.018 N/A

× × × 0.224 0.187 0.048 0.070

×

× ×

0.298 0.210 0.097 N/A

0.264 0.153 0.069 N/A

0.248 0.140 0.049 N/A

× × 0..221 0.181 0.040 N/A

N/A

N/A

N/A

N/A

N/A

N/A

0.038

0.999 0.996 0.990 0.968

0.999 0.991 0.992 0.981

0.999 0.984 0.971 0.985

0.998 0.956 0.913 0.991

0.994 0.989 0.970 0.986

0.994 0.924 0.854 0.981

0.994 0.997 0.989 0.968

v u e μmax 1 μmax 2 μmax 3 Kinh,O2

0.319 0.236 0.116 N/A

Kinh,NO3− O2 NO3− NO2− BIOM

The × symbols on the top rows mean that the corresponding cybernetic variables are fixed to a constant value (i.e., 1) in simulation.

As shown in Figure 2, the quality of model fit using the cybernetic approach is reasonably good in all cases (see also R2 values in Table 1). The cybernetic model accurately captured the metabolic shift from one electron acceptor to another occurring in denitrification process in both anoxic and aerobic conditions. Simulation profiles did not exactly match the data in some cases, e.g., biomass and nitrate profiles in Acidovorax sp. strain (Figure 2b) and growth profile in P. denitrificans (Figure 2d). This deviation would be ascribable to certain factors that may affect the bacterial growth but was not incorporated into the model, such as the inhibition effect of some of their metabolic products produced during denitrification. While practically important, further improvement of model fit through empirical modification of kinetics is not a primary goal of the current work that focuses on the evaluation of the basic capability of the cybernetic regulation in describing the complex bacterial growth in denitrification process. In this regard, we highlight that we obtained the simulation profiles in Figure 2 by tuning only a few parameters: two maximal growth rate constants in Figures 2a−c and three in Figure 2d (see also Table 1). The ability to simulate and predict complex metabolic shift through tuning such a small number of parameters is an important advantage of the cybernetic model over purely kinetic-based approaches. We extended simulations to the case that accounts for a full spectrum of electron acceptors as depicted in Figure 1. Due to the lack of real experimental data, the performed simulation was based on hypothetical values of parameters (Table 1). Simulation results in Figure 3a show the competitive assimilation of oxygen and nitrate and the subsequent conversion to nitrogen gas through a series of intermediates (i.e., nitrite, nitric oxide, and nitrous oxide). The temporal change of enzyme levels associated with each reaction step is represented in Figure 3b. Consequently, the biomass curve shows five distinct growth phases as in Figure 3c. Albeit qualitatively, Figure 3 shows that the cybernetic model is able to capture the complex multistage growth patterns caused by the sequential shift among a given set of electron acceptors. These results serve as test beds that require future validation through comparison with real experimental data. Cybernetic versus Kinetic Models. We examined the impact of regulatory variables in the cybernetic model on the simulation accuracy for the comparison with kinetic models. Thus, we considered the following seven models, each of which is characterized with a different level of regulation: (1) the standard cybernetic model; cybernetic models with one of the

regulatory variables removed by setting (2) v = 1 (i.e., no activity control) or (3) u = 1 (i.e., no control of enzyme synthesis); models with two regulatory variables removed by setting (4) v = u = 1 (i.e., no control of enzyme syntheses and activities) or (5) u = erel = 1 (i.e., highest enzyme level, but variable activity); models with no cybernetic regulations (i.e., kinetic models) (6) based on the standard MM kinetics or (7) based on empirical inhibition kinetics (Table 2). As a test example, we chose the denitrification in aerobic conditions (i.e., P. denitrificans data in Figure 2d), the case that provides the most comprehensive and complex growth pattern among others. We made groupwise comparisons: the standard cybernetic model (i.e., Model 1 in Table 2) versus cybernetic models that neglect one control variable (Group 1), versus cybernetic models that neglect two variables (Group 2), or versus kinetic models (Group 3). The original cybernetic model outperformed all other groups (as indicated by R2 values in Table 2), while the difference from Group 1 was the least. The comparison of Models 2 and 3 in Group 1 showed that the enzyme synthesis control by u-variables was more important than the enzyme activity control by v-variables (Figure 4 and Table 2). The Supporting Information shows how v-variables (Figure S1), u-variables (Figure S2), and relative enzyme levels (Figure S3) dynamically change in Models 1, 2, and 3. Model 2 with no enzyme activity control did not display appreciable difference in u-variables and enzyme levels. Interestingly, uvariables in Model 3 showed a more sensitive response to the change of electron acceptors in order to compensate for the absence of v-variables. In the comparison of Models 4 and 5 in Group 2, the u−v set has a stronger impact than the v−erel set, when eliminated (Figure 5 and Table 2). Finally, in the analysis of Group 3 models, simple kinetic model based on the standard MM kinetics (Model 6) showed poor performance and the quality of fit became comparable to the cybernetic model when inhibition term was additionally incorporated (Model 6) (Figure 6 and Table 2).

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CONCLUSIONS Through the case study using denitrification as an example, we have examined (1) how the cybernetic approach can be extended to model the bacterial growth on multiple alternative electron acceptors and (2) which control variables in the cybernetic model are the key parameters for quantitative simulation. The results showed that the cybernetic model indeed provided high simulation accuracy that could not be E


Article



Figure 6. Comparison of the original cybernetic model (Model 1) with kinetic models that are based on the standard MM kinetics (Model 6) or based on empirical inhibition kinetics (Model 7): (a) oxygen concentration, (b) nitrate concentration, (c) nitrite concentration, and (d) biomass concentration.

Figure 4. Comparison of the original cybernetic formulation (Model 1) with incomplete formulations that eliminate v-variables (Model 2) or u-variables (Model 3): (a) oxygen concentration, (b) nitrate concentration, (c) nitrite concentration, and (d) biomass concentration.

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ASSOCIATED CONTENT

S Supporting Information *

Figure S1: Trajectories of the cybernetic variables v1, v2, and v3 in Models 1, 2, and 3 defined in Table 2, respectively. Figure S2: Trajectories of the cybernetic variables u1, u2, and u3 in Models 1, 2, and 3 defined in Table 2, respectively. Figure S3: rel rel Trajectories of relative enzyme levels erel 1 , e2 , and e3 in Models 1, 2, and 3 defined in Table 2, respectively. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b01615.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +1-509-375-4485. Fax: +1-509-371-6946. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the U.S. Department of Energy (DOE) Office of Biological and Environmental Research (BER), as part of Foundational Scientific Focus Area (SFA) and Subsurface Biogeochemistry Research Program’s SFA at the Pacific Northwest National Laboratory (PNNL). PNNL is operated for the DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830.

Figure 5. Comparison of the original cybernetic formulation (Model 1) with incomplete formulations that eliminate v- and u-variables (Model 4) or u- and e-variables (Model 5): (a) oxygen concentration, (b) nitrate concentration, (c) nitrite concentration, and (d) biomass concentration.

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achieved by simple kinetic models; enzyme synthesis control played a more important role in ensuring simulation accuracy than enzyme activity control. While we focused on simple metabolic reactions affected by electron acceptors only, our modeling platform is readily applicable to expanded reaction networks that contain multiple electron donors and acceptors together. Particularly, the developed denitrification model can be used as a basis for modeling metabolic activities of complex microbial communities in natural environments, e.g., microbefacilitated biogeochemical cycling. The current work marks an initial step in that direction.

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Dynamic Metabolic Modeling of Denitrifying Bacterial Growth: The

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