Use of an Uncertainty Analysis for Genome-Scale Models as a

Article pubs.acs.org/est

Use of an Uncertainty Analysis for Genome-Scale Models as a Prediction Tool for Microbial Growth Processes in Subsurface Environments Christine Klier* HelmholtzZentrum München, German Research Centre for Environmental Health, Institute of Groundwater Ecology, Ingolstädter Landstrasse 1, D-85764 Neuherberg, Germany ABSTRACT: The integration of genome-scale, constraint-based models of microbial cell function into simulations of contaminant transport and fate in complex groundwater systems is a promising approach to help characterize the metabolic activities of microorganisms in natural environments. In constraint-based modeling, the specific uptake flux rates of external metabolites are usually determined by Michaelis−Menten kinetic theory. However, extensive data sets based on experimentally measured values are not always available. In this study, a genome-scale model of Pseudomonas putida was used to study the key issue of uncertainty arising from the parametrization of the influx of two growth-limiting substrates: oxygen and toluene. The results showed that simulated growth rates are highly sensitive to substrate affinity constants and that uncertainties in specific substrate uptake rates have a significant influence on the variability of simulated microbial growth. Michaelis−Menten kinetic theory does not, therefore, seem to be appropriate for descriptions of substrate uptake processes in the genome-scale model of P. putida. Microbial growth rates of P. putida in subsurface environments can only be accurately predicted if the processes of complex substrate transport and microbial uptake regulation are sufficiently understood in natural environments and if data-driven uptake flux constraints can be applied.

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Monod kinetic models17−21 that do not always account for the known complexity of actual microbial functions. Better quantitative analysis techniques are needed to describe the individual metabolic activities of microorganisms in order to directly parametrize metabolic microbial processes. The ability to predictively model the physiological responses of environmentally relevant microorganisms under a wide range of environmental conditions is one of the major goals of environmental biotechnology.22 The coupling of genomescale metabolic models with reactive transport models is a promising approach to help characterize the changing metabolic activities of microorganisms in natural environments8 and has already been tested with regard to the microbial cell function of G. sulf urreducens and R. ferrireducens and the in situ bioremediation of uranium in groundwater.23,24 Several important conclusions have been obtained solely from in silico genome-scale models of the important bioremediation microorganisms G. sulfurreducens and P. putida for various applications under laboratory conditions, including the prediction of phenotypic behavior, substrate preference and optimal growth requirements, as well as various metabolic engineering topics.25−30 Compared to theory-based cell models, which are dependent on biophysical equations that require

INTRODUCTION Genome-scale models are an increasingly important tool in many systems biology approaches, and their use has already resulted in various publications in high-level scientific journals.1−9 Advances in knowledge and the creation of suitable databases have paved the way for the systematic reconstruction of metabolic models based on microbial genomes. In the webbased resource SEED Model, 130 genome-scale metabolic models of bacteria (22 of them validated) are already available.5 Within the past decade, high-throughput genome sequencing and annotation, as well as the use of systems biology approaches, have already enabled the construction of expertcurated, genome-scale models describing the complete metabolism of several microorganisms (e.g., Geobacter sulfurreducens, Pseudomonas putida, Rhodoferax ferrireducens, Geobacter metallireducens, and Shewanella oneidensis), which play an important role in the bioremediation of contaminated groundwater.10−14 Porous aquifers are an essential resource for drinking water but, nowadays, various kinds of contamination also have to be managed15 and the bioremediation of contaminated field sites is becoming the focus of much hydrogeological, geochemical and microbiological research. To help address the complex interplay of processes found in groundwater systems,16 the transport and degradation of contaminants can be described by numerical bioreactive transport models. However, the representation of microbial processes in these reactive transport models is often based on the assumption of simple reaction rates or double© 2012 American Chemical Society

Received: Revised: Accepted: Published: 2790

October 6, 2011 February 13, 2012 February 15, 2012 February 15, 2012 dx.doi.org/10.1021/es203461u | Environ. Sci. Technol. 2012, 46, 2790−2798

Environmental Science & Technology

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Table 1. Literature Values of Experimentally Determined Substrate Uptake Fluxes for P. putida strain

exper. system/Temperature

substrate

substrate uptake (mmol gDW−1 h−1)

μmax (h−1)

P.putida mt2 (pWWO) (Duetz et al.,68) P.putida KT 24420 (pWWO) (del Castillo and Ramos69)

chemostat 28 °C

toluene

14.1 ± 1.0

0.05 (dilution rate)

batch 30 °C

toluene

11.9 ± 0.5

0.72 ± 0.02

glucose glucose and toluene glucose

13.1 ± 0.1 5.7 ± 0.5 6.4 ± 0.2 6.3 ± 0.1

0.73 ± 0.03 0.74 ± 0.07 0.56 ± 0.01

oxygen glucose

18.2 ± 0.8 11.0 ± 0.5

0.82 ± 0.02 0.82 ± 0.02

P.putida KT 24420 (del Castillo et al.70) E. coli (Fischer et al.37)

batch 30 °C reactor

where v ⃗ (mmol gDW−1 h−1, where DW stands for dry weight) is the flux vector describing the activity of internal and external metabolic fluxes, constrained by lower (lb) and upper (ub) bounds. Within the field of constraint-based modeling, Flux Balance Analysis (FBA) has become a standard technique.6,34 In this study, the genome-scale model of P. putida was analyzed using the COBRA Toolbox2 in the Matlab environment (The MathWorks). Substrate Uptake in Constraint-Based Modeling. When using FBA, the fluxes of metabolites that can, in accordance with their flux constraints, enter or leave the particular network determine the substrate uptake of the system and thus significantly influence the predicted growth rates μ (h−1). Exchange flux rates at the edge of the network can be estimated experimentally by means of biodegradation kinetic studies. In simulations, most external metabolites are allowed to enter and exit the network freely, whereas the uptake and, therefore, the vlb of limiting substrates (at least one) are constrained to specific flux rates. These specific uptake flux rates vlb (mmol gDW−1 h−1) are usually determined by Michaelis−Menten kinetic theory:

many difficult-to-measure kinetic parameters, constraint-based modeling does not require extensive model parametrization.6,25 This study investigated whether use of the current genomescale model of P. putida is applicable for natural environments, as described by Mahadevan et al.8 with the G. sulfurreducens model. An uncertainty analysis was used to investigate the lack of knowledge and the potential errors associated with model inputs. Such an approach is essential in understanding the degree of confidence that a user can place in the results obtained from any model.31 The key issue studied was the uncertainty arising from the parametrization of substrate uptake fluxes. The influx of extracellular substrates is an important capacity constraint in genome-scale metabolic modeling and any uncertainties present can have a significant influence on predicted microbial growth rates. In genome-scale modeling, the specific uptake flux rates of substrates are usually determined by Michaelis−Menten kinetic theory. However, extensive data sets based on experimentally measured values are not always available and many biochemical or cellular processes deviate significantly from the law of mass action. An uncertainty analysis, using oxygen and toluene as growthlimiting substrates, was performed for the genome-scale model of P. putida,11 perhaps the best-characterized species of aromatic hydrocarbon-degrading bacteria.

v lb = vmax

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CS CS + K m −1

where vmax (mmol gDW h ) denotes the maximum substrate uptake flux, Cs (mg L−1) the substrate concentration, and Km (mg L−1) the substrate affinity. In various model applications, the maximum rate of substrate uptake or substrate affinity are set to physiologically realistic levels,23 often estimated from experimentally determined parameters obtained for other organisms (e.g., E. coli),24 although some uptake rates are not further specified at all.27 Since published vmax and Km values are scare and sometimes inconsistent, values for experimentally determined Monod parameters for the maximum microbial growth rate μmax and the Monod affinity constant Ks are often used instead. Although, this assumption is only correct in special cases, when cell growth is controlled by the rate of active transport of a substrate.35 However, even the Monod parameters reported in the literature are often inconsistent, even for specified combinations of a single organism and a single substrate.36 Uncertainty Domain of Substrate Uptake Parameters. Determination of vmax and Km values. To define the domain of the lower bound for specific uptake flux rates vlb published experimental values measured for the uptake flux rates of oxygen and toluene were examined. Table 1 lists these reported values in terms of the bacterial strain and organism

MATERIALS AND METHODS Constraint-Based, Genome-Scale Modeling. Metabolic Network Reconstruction and Analysis. The reconstruction of cellular metabolic networks has already been described and reviewed in various publications.4,7,32,33 When reconstructing an organism-specific network, the potential metabolic reactions of the organism are mapped, based on the genome sequence and other available (transcript-, prote-, metabol-, or flux-) “omics” data, resulting in precise reaction stoichiometries. Network topology and capability characteristics can be studied by transforming the stoichiometric coefficients of all reactions into one, organism-specific numerical matrix (S), which can then be conveniently analyzed mathematically. The most popular approach used for computational network analysis and pathway-oriented interpretation is constraint-based modeling, which involves defining the constraints under which the network operates. Assuming steady-state mass balance for all metabolites in the system, this results in S × v⃗ = 0

(1)

v lb ⃗ ≤ v ⃗ ≤ vub ⃗

(2)

(3) −1

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dx.doi.org/10.1021/es203461u | Environ. Sci. Technol. 2012, 46, 2790−2798


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Table 2. Michaelis−Menten Kinetic Parameters of Oxygen and Toluene and Their Coefficients of Variation (CV) and Simulated Substrate Concentration Range Cs substrate

vmax (mmol gDW−1 h−1)

CV

Km (mg L−1)

CV

Cs,low (mg L−1)

Cs,high (mg L−1)

oxygen toluene

22.27 11.90

0.74 0.74

0.32 0.74

0.53 0.53

0.5 0.1

8.5 20.0

0.1 mg L−1). Uncertainties in the parameters vmax and Km were also included, as shown in Table 2, and compared with the sensitivities shown by the growth rates, with regard to the Km values. Latin Hypercube Sampling (LHS). Probabilistic modeling approaches can produce an estimate of the uncertainty range of modeling results, with respect to the domain of possible model inputs. This is achieved by repeatedly running a deterministic model many times over for a large number of different input values.40 However, to allow the number of model runs to be kept to a minimum in the present study, LHS from distributions of model input parameters was used. Based on the subdivision of the probability distribution of each input parameter in N disjunctive equiprobable intervals,41 random sampling of one value in each interval was performed and N samples were obtained for each parameter. Random sampling into statistical distributions was accomplished using the Random function of Mathematica. The sampled values of the first parameter were then randomly paired with the sampled values of the second parameter, and also randomly paired with further combinations, resulting in N combinations of p parameters. This set of p-tuples is known as the Latin Hypercube sample and, according to Janssen et al.,42 the choice of N > 4/3p usually gives satisfactory results. A modeling exercise was carried out where the probability distribution of vmax,oxygen, vmax,toluene, Km,oxygen, and Km,toluene was analyzed and a value of N = 25 was chosen. Probability distributions of these variables were specified by the means and the standard deviation, assuming normal distributions. The use of a normal distribution was justified here due to the lack of sufficient data for a distribution assessment. The CV values of the parameters (Table 2) were set in accordance with the propagated experimental error in measured values, as discussed previously. The uncertainty analysis was performed for subsets of the concentration ranges, as given in Table 2. One substrate concentration was fixed to either the maximum or minimum value, respectively, while the other substrate concentration was varied in the concentration intervals used for the reference simulation. The results of the LHS study were assessed using a one-way ANOVA and two different posthoc tests (the Bonferroni test and the Tukey test) from the Mathematica software package (version 7.0, Wolfram Research; cp. Table 4). In both these tests, the significance level criteria were set to α = 0.05. The Bonferroni test is a mean comparison test based on the Student t distribution, with modified α based on the number of groups. The Tukey test is a mean comparison test based on the Studentized range distribution.

used, experimental system, and substrate composition. Since no published values for the oxygen uptake rates of P. putida could be found, Fick’s first law was used to calculate the diffusive substrate flux in a spherical cell, using the formula: 4πrDa (Cmax − C0) (4) DW where r (m) denotes the radius of a spherical cell, Da (7.31 × 10−2 m2 h−1 in air at 20 °C) the substrate diffusion coefficient, Cmax (8.71 × 10−6 mmol m−3 in air) the maximum substrate concentration, C0 (set to 0 mmol m−3) the substrate concentration within the cell, and DW (g) describes the dry weight of the spherical cell. This results in vmax,oxygen = 22.27 mmol h−1 gDW−1 (Table 2), which is similar to the value reported by Fischer et al.37 for E. coli in a reactor. A value of 0.74 mg L−1 as determined by Duetz et al.38 in a chemostat experiment, was used as the Km of toluene for P. putida. Since no published values could be found for the Km of oxygen for P. putida, a parameter sensitivity analysis was performed instead. Determination of coefficients of variation (CV). To define the uncertainty domain of vmax, the results of LofererKrössbacher et al.39 were considered. They documented E. coli DSM 613 at different growth phases with a range of cell volumes from 0.1 up to 3.5 μm3 (n = 678) and examined natural bacterial assemblages in two lakes, ranging from 0.003 up to 3.5 μm3 (n = 465). They also described the following relationship between cell dry weight and the volume of a bacterial cell: vmax =

DW = 435Vc 0.86

(5)

Combining Fick’s first law with the results of LofererKrössbacher et al.,39 the coefficient of variation for substrate uptake in a spherical bacterial cell can be estimated according to the range of bacterial cell volumes measured. Assuming possible cell volumes between 0.1 and 3.5 μm3, a CV value of 74% was calculated for vmax. To define the uncertainty domain of the Km values, a CV value of 53% was used, based on chemostat data for E. coli grown on glucose (as summarized in Table 2 from Kovarova-Kovar and Egli35). Since the estimated CV values were rather high, an uncertainty analysis was also performed in which the CVs for all parameters were set at 30%. In Table 2, parameter values are summarized and the estimated ranges of substrate concentrations, Cs, in environmental groundwater systems are listed. For reference simulation purposes, the concentration ranges in Table 2 were divided into equal concentration intervals of 1.0 mg L−1 for Cs,oxygen and into exponentially distributed intervals from 0.1 to 6.0 mg L−1 for Cs,toluene. Sensitivity Analysis. Using the estimated concentration range of the substrates listed in Table 2, a sensitivity analysis was performed for Km values of 0.1, 0.01, and 0.001 mM (for a better representation here given in mM; 0.1 mM = Km,oxygen 3.20 mg L−1 = Km,toluene 9.21 mg L−1) at “high” and “low” levels of oxygen (Cs,high oxygen = 8.5 mg L−1, Cs,low oxygen= 0.5 mg L−1) and of toluene supply (Cs,high toluene = 20.0 mg L−1, Cs,low toluene =

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RESULTS Sensitivity Analysis of Km Values. Because of the lack of consistent Km values cited in the literature for the estimation of specific uptake flux rates vlb and the wide variability range (several orders of magnitude) for reported values (e.g., see ref 43), a sensitivity analysis was performed prior to the uncertainty analysis. The effects of Km values on simulated 2792



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microbial growth rates μ were investigated using the genomescale model of Nogales et al.11 At “high substrate supply” Cs,high for oxygen and toluene, the mean simulated growth rates of P. putida were sensitive to changes in the substrate affinity Km for oxygen, but not for toluene (Table 3b). Even changes in 2

were relatively insensitive to changes in the effective Km value. As expected, the lower the Km values the higher the variability in the simulated growth rates over the substrate concentration ranges studied. Overall, the sensitivity analysis showed that a Km,toluene of 0.74 mg L−1 for P. putida, as used by Duetz et al.,38 and a Km,oxygen of 0.32 mg L−1 seemed to be appropriate for the simulated concentration ranges used in the uncertainty analysis. However, the comparison of the sensitivity of the growth rates to the Km values and the possible uncertainty range for the kinetic uptake parameters also showed that, in the absence of reliable measured substrate affinity constants, the Michaelis− Menten approach is not appropriate for the estimation of substrate uptake rates. Uncertainty Analysis for Substrate Uptake Fluxes. First, reference growth rates for P. putida were simulated using the model of Nogales et al.11 for substrate concentrations of oxygen and toluene − based on the assumption that these are representative of many contaminated groundwater environments (Table 2; Cs,low to Cs,high), as shown in Figure 1. Up to a

Table 3. Results of Sensitivity Analysis of Simulated Microbial Growth Rates, with Substrate Affinities of Oxygen Km,oxygen and Toluene Km,toluene at “High Substrate Supply” (Cs,high oxygen = 8.5 mg L−1, Cs,high toluene = 20.0 mg L−1) and at “Low Substrate Supply” (Cs,low oxygen = 0.5 mg L−1, Cs,low toluene = 0.1 mg L−1)

Figure 1. Mean simulated P. putida growth rates for a range of possible substrate concentrations (Cs) of toluene and oxygen in environmental groundwater systems.

orders of magnitude in Km,oxygen resulted in only small changes of the growth rates (Δμmean = 0.137 h−1). At “low substrate supply” Cs,low the mean simulated growth rates were sensitive to both affinity constants. Changes in 2 orders of magnitude in Km values resulted in large changes in the simulated growth rates (Δμmean values up to 0.461 h−1 for toluene and Δμmean values up to 0.408 h−1 for oxygen), with slightly increased sensitivity for Km,toluene. The influence of estimated uncertainty in the parameters vmax and Km was also included and the effect on simulated microbial growth rate noted. The inclusion of the maximum (mean vmax value plus 74%, mean Km value plus 54%) and the minimum (mean vmax value minus 74%, mean Km value minus 54%) uncertainty range in the measured Michaelis− Menten kinetics parameters is shown (Table 3a and c) at “high substrate supply”. The sensitivity of the simulated growth rates to Km,oxygen was insignificant compared to the simulated variability range for growth rates brought about by the inclusion of an uncertainty range for the measured parameters. At “low substrate supply” when substrate affinity constants were low the uncertainties in the measured parameters had only a slightly higher effect on simulated growth rates (Δμ = 0.670) than the sensitivity shown by these rates to Km,oxygen and Km,toluene. Even if the initial substrate concentrations (Cs,high) were slightly higher than the affinity constants (Km,oxygen = 3.20 mg L−1 and Km,toluene = 9.21 mg L−1) the simulated growth rates

concentration of 0.3 mg L−1, toluene was the major growthlimiting substrate in the model simulations while, at higher toluene concentrations, oxygen became the major growthlimiting substrate. According to the simulated growth rates shown in Figure 1 for the subsequent uncertainty analysis carried out using LHS, one substrate concentration was fixed to the maximum (Figure 2) or minimum (Table 4) concentration, respectively, while the other substrate concentration was varied (in keeping with the particular concentration intervals adopted). A modeling exercise was then carried out in which the effect of the probability distribution of four model input parameters was analyzed. Statistical assessment (Table 4) showed that a coefficient of variation for the Michaelis−Menten substrate uptake parameters of CVvmax = 74% and CVKm = 53%, using the LHS approach (N = 25), only produced a significantly different growth rate at different substrate concentrations in a single test case (Figure 2). At maximum oxygen supply and varying toluene concentrations, there was a significant difference in simulated growth rate between the highest concentrations (Cs,toluene = 20.0−9.8 mg L−1) and the lowest concentration (Cs,toluene = 0.1 mg L−1). Reducing the CVs to 30% produced significant differences in growth rate between the various concentration groups. At maximum oxygen supply and varying toluene concentrations, significant differ2793



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Figure 2. Simulated growth rates for an uncertainty analysis performed using Latin Hypercube Sampling (number of simulation runs =25) at maximum oxygen supply (Cs,oxygen = 8.5 mg L−1) and variable toluene concentrations (a, b), and at maximum toluene supply (Cs,toluene = 20.0 mg L−1) and variable oxygen concentrations (c, d) with different coefficient of variation (CV) values for Michaelis−Menten kinetic parameters vmax and Km.

Table 4. ANOVA Post-Hoc Test Results of Simulated Substrate Concentration Groups substrate

post hoc tests

groupsa significantly different at the 5% level

toluene at high oxygen suppply CV 30%

Bonferroni Tukey Bonf. and Tukey

toluene at low oxygen supply CV 30%

Bonf. and Tukey Bonf. and Tukey

oxygen at high toluene supply CV 30% oxygen at low toluene supply CV 30%

Bonf. Bonf. Bonf. Bonf.

{{1, 15}, {2, 15}, {3, 15}} {{1, 15}, {2, 15}, {3, 15}, {4, 15}, {5, 15}, {6, 15}} {{1, 14}, {2, 14}, {3, 14}, {4, 14}, {5, 14}, {6, 14}, {7, 14}, {8, 14}, {9, 14}, {10, 14}, {11, 14}, {1, 15}, {2, 15}, {3, 15}, {4, 15}, {5, 15}, {6, 15}, {7, 15}, {8, 15}, {9, 15}, {10, 15}, {11, 15}, {12, 15}, {13, 15}} {} {{1, 15}, {2, 15}, {3, 15}, {4, 15}, {5, 15}, {6, 15}, {7, 15}, {8, 15}, {9, 15}, {10, 15}, {11, 15}, {12, 15}, {13, 15}} {} {{1, 9}, {2, 9}, {3, 9}, {4, 9}, {5, 9}, {6, 9}} {} {}

and and and and

Tukey Tukey Tukey Tukey

Toluene: Group 1 to group 15 decreasing toluene concentrations Cs,toluene 20.0−0.1 mg L−1. Oxygen: Group 1 to group 9 decreasing oxygen concentrations Cs,oxygen 8.5−0.5 mg L−1.

a

between C s,oxygen = 8.5−3.5 mg L −1 and the lowest concentration (Cs,oxygen = 0.5 mg L−1). As expected from the results shown in Figure 1, however, at minimum toluene supply and varying oxygen concentrations (Table 4) simulated growth rates were not significantly different. The results of Figure 2 and Table 4 show that the inclusion of uncertainty measures in the substrate uptake parameters reduced the amount of difference between the simulated growth rates produced by

ences in simulated growth rates were found between Cs,toluene = 20.0−0.6 mg L−1 and the lowest concentrations (Cs,toluene = 0.2−0.1 mg L−1) (Figure 2b). At minimum oxygen supply and varying toluene concentrations (Table 4) simulated growth rates were significantly different between Cs,toluene = 20.0−0.3 mg L−1 and the lowest concentration (Cs,toluene = 0.1 mg L−1). At maximum toluene supply and varying oxygen concentrations, simulated growth rates were significantly different 2794



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substrate-limited and metabolic stress conditions.50 While primary transport systems (e.g., ABC transporters) are intrinsically unidirectional, secondary transport systems mediate fluxes in the uptake or efflux direction in accordance with the prevailing electrochemical gradients.51 E. coli, for example, possesses four phosphorus transport systems which can be classified on the basis of substrate specificity, as well as bioenergetic and structural criteria. Outer membrane (OM) and periplasm transport is also controlled in quite a complex manner in gram-negative bacterial cells. Extracellular electron transfer onto insoluble Fe(III) oxides in G. sulf urreducens is thought to require proteins that must be exported to the outer surface of the cell. Voordeckers et al.52 pointed out that G. sulf urreducens can adapt to the loss of most outer surface cytochromes in order to reduce soluble, chelated Fe(III), but accessing insoluble electron acceptors is likely to be more difficult and may require specific arrangements of outer surface molecules. The use of a single vmax and Km value (e.g., for acetate uptake under different electron acceptor conditions in G. sulf urreducens) has not yet been verified.53 Specific transport systems are also included in some genomescale metabolic models, but their full complexity is not always represented. In addition, maximum uptake fluxes and substrate affinity values differ between these single transport systems and also depend on environmental conditions (e.g., pH, temperature, presence of metal ions). Reay et al.54 studied the temperature dependence of inorganic nitrogen uptake by determining specific affinities based on estimates of kinetic parameters obtained from chemostat experiments. They hypothesized that active substrate transport decreases as the temperature decreases below the optimum for growth, while passive uptake is less affected by temperature effects. The nonspecificity of induced enzymes allows P. putida to simultaneously utilize several similar substrates without an excess of redundant genetic coding for enzyme induction convergence.43,55 Surprisingly, benzene and phenol produce exactly the same intermediate, but yield very different biodegradation kinetics. It is, therefore, reasonable to conclude that the factors responsible are probably associated with either the rates of the reactions that produce the intermediate (catechol) or the transport of the substrate into the cytoplasm.43 Izallalen et al.,26 seeking to increase the rate of extracellular electron transport, detected several changes in gene expression levels involving processes not included in the G. sulf urreducens genome-scale model. Thus, Mahadevan et al.56 pointed out that the incorporation of data-driven flux constraints is an important consideration when trying to understand the regulation mechanism for the metabolism of G. sulf urreducens in nonoptimal environments, specifically the down-regulation of acetate permease genes during substrate limitation. The kinetic properties exhibited by a microbial cell for a particular substrate should be intimately linked to the expression levels of the enzymes involved in the metabolic pathway of that substrate, as the steady-state extracellular concentration of the growthcontrolling substrate and the content of the enzymes involved in transport and catabolism influence each other.35 The network-level interplay between the regulation and metabolism of S. cerevisiae was studied by Herrgard et al.57 The results showed that a combined approach, using an integrated model of regulation and metabolism, helped improve the prediction of growth phenotypes. This was confirmed by Ishii et al.,58 who found that the intracellular metabolic network of E. coli is

the genome-scale model, as shown in Figure 1. At CVvmax = 74% and CVKm = 53%, as estimated from the Michaelis− Menten parameters reported in the literature, significant differences in simulated growth rates can only be distinguished between the highest (mean μ = 0.516−0.514 h−1) and lowest (mean μ = 0.197 h−1) toluene concentrations when the oxygen supply is not restricted. Standard deviations in the simulated growth rates produced by LHS in the single-substrate concentration groups were high, overall, and only when input parameter CVs were small (30%), could significant differences in growth rate be distinguished between very high and very low substrate concentrations. The uncertainties in simulated growth rates caused by the CVs of the input parameters were, in most cases, higher than the simulated growth variability associated with the substrate concentration ranges. As there were no experimentally measured data available in the literature for vmax and Km values under changing environmental conditions, the validity of Michaelis−Menten kinetics when describing substrate uptake processes in genome-scale modeling was extremely restricted.

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DISCUSSION When genome-scale metabolic models are coupled to reactive transport models and applied to environmental conditions, their predictive power (reducing the need for empirical parameter calibration) is often emphasized.8,23,24 However, the results of this study highlighted an important issue (also pointed out by King et al.25) in that the FBA approach for genome-scale models requires a priori knowledge of uptake fluxes, restricting its use to settings at which they are constrained by the availability of experimental data. The microbial growth rates of P. putida predicted using the genomescale approach were highly sensitive to the vmax and Km parameters selected for the substrate uptake flux (Table 3) and it must be assumed that the parameter values determined in batch and chemostat experiments will differ significantly from actual field parameters. Thus, Scheibe et al.23 had to reduce the uptake rates of all relevant substrates (acetate, iron, and ammonium) when using the FBA approach by 1 order of magnitude when carrying out their field-scale simulation. One important point highlighted in the present work is that Michaelis−Menten kinetic theory is generally unsuitable for the description of substrate uptake flux rates in environmental systems where different rate limiting steps operate. Due to the presence of microscopic mass-transfer processes, biodegradation in environmental systems often does not conform to Michaelis−Menten theory, which was developed to describe the metabolic processes occurring in solutions in which microorganisms and their substrates are well mixed.44−46 Evidently, increased microbial conversion capacities do not lead to higher biotransformation rates when mass-transfer is the limiting factor and diffusion-limited substrate uptake is, in most cases, the critical factor influencing microbial growth and substrate bioremediation.47,48 Rate-limiting steps may also include the uptake of a substrate by the cell, transport of the substrate within the cell, or binding of the substrate to an enzyme complex.49 Another important point to consider when bacterial substrate transport systems are being described using the Michaelis− Menten approach is whether uncertainties in the measurement of specific uptake flux rates can be minimized by using an improved experimental setup. Bacterial substrate transport systems are complex and have a significant effect under 2795



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better understanding of microbial substrate uptake processes in natural environments is needed.

remarkably stable in the face of environmental changes, and that E. coli can actively respond to changes in the concentration of growth-limiting substrates by regulating the level of enzyme expression in order to maximize its growth rate, even though this strategy might be costly in terms of energy requirements. Implications for further work must be that aerobic and anaerobic degrading microorganisms possess actively respond systems to changes in substrate concentrations and environmental conditions, and that the resulting uncertainties in measured specific uptake flux rates are often not fully understood because of their complexity. Uncertainties in measured uptake rates may be minimized somewhat by better experimental setup systems. The results of the toluene-based uncertainty analysis carried out in this study are also likely to be applicable to other growth substrates used by P. putida and similar uncertainty analyses could perhaps be usefully conducted for other genome-scale models e.g., for the anaerobic degrading bacterium G. sulf urreducens. Additional uncertainties in genome-scale models that originate from the model reconstruction process itself,7,59 or from uncertainties in mathematical model analysis,60,61 have not been discussed in the present study, but should be addressed in future work. For example, the utilization of less energetic and more kinetically efficient processes in natural systems may provide a competitive advantage in terms of flux per unit enzyme when substrate concentrations are low.14 FBA cannot predict the behavior of cells that do not metabolize substrates in the most efficient manner.62 Nor can microbial switching from a high-rate regime (maximizing pathway flux) to a high-yield regime63 be predicted accurately by FBA analysis using the optimality principle of yield maximization. Steadystate flux approximation is the underlying principle on which FBA is based, but it becomes undefined at the “edges” of metabolic activity, where stochasticity, enzyme kinetics, spatial distributions, and varying levels of metabolic regulation become dominant forces in cell activity.64 Dynamic models would be an important complement to the analysis process, but they typically involve a large number of unmeasured parameters.59,65 Yizhak et al.66 recently published a novel approach for integrating quantitative proteomic and metabolomic data with a genome-scale metabolic network model in order to predict flux alterations under different perturbations. Recently, Henry et al.67 have presented a method for integrating metabolomic data with a thermodynamic-based metabolic flux analysis in which constraints on reaction directionality are derived from metabolic concentration data based on thermodynamic principles. In conclusion, the results of the study show that uncertainties in specific substrate uptake rates have a significant influence on simulated growth variability and that Michaelis−Menten kinetic theory does not seem to be appropriate to describe substrate uptake processes in the genome-scale model of P. putida. Microbial growth rates of P. putida in subsurface environments can only be accurately predicted by a genome-scale model if the complex substrate transport and microbial uptake regulation processes are fully understood and data-driven uptake flux constraints can be incorporated. The integration of genomescale, constraint-based models of microbial cell function into simulations of contaminant transport and fate in complex groundwater systems is a promising approach to help improve the characterization of metabolic activities of microorganisms but, as shown by the model of P. putida adopted in this study, a

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 89 3187 2916. Fax: +49 89 3187 3361. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the Helmholtz Zentrum München, German Research Centre for Environmental Health. The author would like to thank Jan Krumsiek for comments and assistance with computations, and Rainer Meckenstock for helpful discussions.

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