Article pubs.acs.org/est
Evaluating Alternate Biokinetic Models for Trace Pollutant Cometabolism Li Liu,† Philip J. Binning,† and Barth F. Smets*,† †
Department of Environmental Engineering, Technical, University of Denmark, Bygningstorvet 115, 2800 Kgs.Lyngby, Denmark S Supporting Information *
ABSTRACT: Mathematical models of cometabolic biodegradation kinetics can improve our understanding of the relevant microbial reactions and allow us to design in situ or in-reactor applications of cometabolic bioremediation. A variety of models are available, but their ability to describe experimental data has not been systematically evaluated for a variety of operational/ experimental conditions. Here five different models were considered: firstorder; Michaelis−Menten; reductant; competition; and combined models. The models were assessed on their ability to fit data from simulated batch experiments covering a realistic range of experimental conditions. The simulated observations were generated by using the most complex model structure and parameters based on the literature, with added experimental error. Three criteria were used to evaluate model fit: ability to fit the simulated experimental data, identifiability of parameters using a colinearity analysis, and suitability of the model size and complexity using the Bayesian and Akaike Information criteria. Results show that no single model fits data well for a range of experimental conditions. The reductant model achieved best results, but required very different parameter sets to simulate each experiment. Parameter nonuniqueness was likely to be due to the parameter correlation. These results suggest that the cometabolic models must be further developed if they are to reliably simulate experimental and operational data.
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INTRODUCTION The unabated use of anthropogenic chemicals will result in their continued presence in our water bodies. Our ability to remove such chemicals, via biological or chemical processes, is critical as we must be able to exploit such impaired waters for various consumptive uses. Cometabolic microbiological reactions are one possible treatment option for removal of trace contaminants in potable water production. Cometabolic reactions are microbiological processes that are mediated by enzymes, but without immediate benefit to the host cell or organism.1,2 Cometabolic transformation of trace chemicals has been observed or implemented under both aerobic3−6 and anaerobic conditions7−10 for a wide variety of chemical classes in different environments. The carbon or energy necessary for growth is typically supplied by other substances, the primary substrates.11 Many anthropogenic chemicals are known to be susceptible to cometabolic transformation by various microbes, methanogens,12 methanotrophs,13 Pseudomonas,14 and nitrifiers.3 While oxygenases and reductases are implicated enzymes under aerobic and anaerobic conditions, respectively, the actual enzymes mediating the cometabolic transformation are often unknown. Micropollutant cometabolic transformation processes are influenced by a variety of factors including sorption, mass transfer, biodegradation, and cosolute effects.15,16 Among these processes, the biodegradation kinetics can be of major practical importance for application of in situ remediation. The © 2014 American Chemical Society
likelihood for biodegradation kinetics, in contrast to mass transfer rates, to control the overall contaminant removal rate is greater for cometabolic than for metabolic-based treatment scenarios.17 Hence, this study does not capture all the cometabolic transformation processes as it focuses on the biokinetic processes.Cometabolic tranformation is typically inferred from observations of the trace chemicals (the cometabolic substrate) and primary substrates in batch kinetic experiments.11 These experiments provide data that can be used to understand how biodegradation kinetics are controlled by environmental conditions. In particular, data on the interaction between primary substrate and cometabolic substrate degradation is required in order to control in situ, or in reactor conditions, to optimize trace pollutant removal.17 Biokinetic behavior can be captured in mathematical models, which describe how transformation rates are dependent on concentrations of relevant compounds and properties of the mediating catalysts. Early cometabolic models only considered the cometabolic substrate, using simple first-order18 or mixedorder Michaelis−Menten kinetic expressions.7,19−23 More recent models consider both the cometabolic and primary substrate; and capture phenomena such as direct competitive Received: Revised: Accepted: Published: 2230
July 21, 2014 December 19, 2014 December 29, 2014 December 29, 2014 DOI: 10.1021/es5035393 Environ. Sci. Technol. 2015, 49, 2230−2236
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Environmental Science & Technology Table 1. Summary of Growth-Supported Cometabolic Biokinetic Models name
differential equations for substrate and biomass growth substrate (Sg)
M 1: first-order model
dSg
M 2: Michaelis− Menten (MM) model
dSg
M 3: reductant model
dSg
M 4: competition model
dSg
M 5: combined model
dt
dt
dt
dt
dSg dt
=−
=−
=−
=−
=−
kgXSg K sg + Sg
kgXSg K sg + Sg kgXSg K sg + Sg
kgXSg K sg + Sg
kgXSg K sg + Sg
cometabolic substrate (Sc)
biomass (X)
ref
dSc = − kcXSc dt
kgXSg k XS dX =− c c +Y − bX T K sg + Sg dt
18
dSc k XS =− c c dt K sc + Sc
kgXSg dX 1 kcXSc =− +Y − bX T K sc + Sc K sg + Sg dt
7, 19, 20
⎛ k XS ⎞⎛ Sg ⎞ dSc ⎟⎟ = − ⎜ c c ⎟⎜⎜ dt ⎝ K sc + Sc ⎠⎝ K sg + Sg ⎠
kgXSg dX 1 ⎛ k XS ⎞⎛ Sg ⎞⎟ = − ⎜ c c ⎟⎜⎜ +Y − bX T ⎝ K sc + Sc ⎠⎝ K sg + Sg ⎟⎠ K sg + Sg dt
3, 6
dSc kcXSc =− K dt K sc + Sc + Sg K sc
⎛ ⎞ kgXSg kcXSc ⎟ dX 1⎜ =− ⎜ − bX ⎟+Y dt T ⎜ K sc + Sc + Sg Kc ⎟ K sg + Sg Kg ⎠ ⎝
5, 20, 25, 26
sg
dSc =− dt
⎛ Sg ⎞ ⎜⎜ ⎟ Sg ⎞ ⎛ K + Sg ⎟⎠ K sc⎜1 + K ⎟ + Sc ⎝ sg ⎝ sg ⎠ kcXSc
interaction between the substrates,20,24−27 indirect interaction by the primary substrates supplying energy,3,5,6 and direct interactions exerted by toxicity of cometabolic transformation products.28−30 A variety of cometabolic models have been calibrated against experimental data to yield kinetic parameter estimates. However, these calibrations have so far been rudimentary, using goodness of fit as the key criterion, with limited consideration of parameter sensitivity and identifiability, or model size and complexity.25 Models are becoming more complex as we seek to capture more mechanistic details.25 This increased complexity must be weighed against the need for parameter identifiability: Can accurate and precise model parameters be estimated from experimental data?31 Chambon et al.,9 for example, showed that model parameters describing dehalorespiration kinetics vary widely. Such wide ranges can be due to weak parameter identifiability associated with poor experimental design.32 A critical assessment of cometabolic degradation models is needed. This is evident from the lack of consensus in the literature on the best modeling approach. For example, the reductant model was selected to represent THM (Trihalomethanes) cometabolism degradation by Wahman et al.,33 as it best fit the experimental data. But Simkins et al.34 demonstrated that the first-order model provided a good fit to the disapearance of benzoate for low initial concentration zones. These seemingly contradictory results indicate that we need criteria other than measures of goodness of fit to evaluate these models. Model assessment consists of two interrelated tasks: parameter estimation and model discrimination (also called model structure selection).35 Various methods are available, for example Smets et al.36 addressed parameter identifiability for biofilm growth biokinetics by adding a Monte Carlo parameter retrievability analysis on top of the typical residual error response surface analysis. More advanced methods include the evaluation of the Fisher information matrix (FIM), or of the linear dependency of sensitivity functions.37−39 Brun et al.,37 for
dX 1 =− dt T
⎛ Sg ⎞ k XS ⎜⎜ ⎟⎟ + Y g g − bX Sg ⎞ ⎛ + K S K ⎝ ⎠ sg g sg + Sg K sc⎜1 + K ⎟ + Sc ⎝ sg ⎠ kcXSc
33
example, quantified the degree of linear dependency of local sensitivity function (Sij) in a colinearity index (γκ), an estimate of the capacity of individual parameters to correct for each other in calibration, with smaller colinearity indices being preferable. To evaluate model parsimony, goodness of fit can be weighed against the number of model parameters, with extra parameters imposing a penalty:40 tools employing this approach include the Akaike Information Criterion (AIC),41,42 and the Bayesian Information Criterion (BIC). The objective of this research was to use available model assessment tools to compare different cometabolic kinetic models for their ability to simulate and fit observations across a range of conditions representing typical batch degradation assays. Because real experimental data obtained by a consistent methodological approach were not available for this purpose, we generated synthetic data sets. Model structure deficiencies become more apparent as the quality and temporal resolution of data increases. This is particularly important when experiments are used to gain information on lumped parameters, which are then used for process assessment or design. To illustrate this, the performance of each model was tested using the generated data sets and an objective statistical analysis. The statistical tools explored the whole space of possible model outcomes, an outcome that cannot be achieved with real data. Batch depletion profiles of growth substrate and a cometabolically transformed nongrowth substrate were generated using the most complex model structure available, with parameters informed by the literature, and with noise added to simulate experimental error. Different models were then evaluated for their ability to fit the synthetic observations and return reasonable parameter estimates. Three criteria were chosen to evaluate the cometabolic models: goodness of fit, parameters identifiability, and model size and complexity. Discussions encompassed comparison of simulation results for each model under different condition and their mathematical implementation. 2231
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MATERIALS AND METHODS Cometabolism is defined as transformation of a nongrowth substrate by growing cells in the presence of a growth substrate, by resting cells in the absence of a growth substrate, or by resting cells in the presence of an energy substrate.21 Here, five cometabolic model structures17,21,33 were evaluated, specifically focusing on growth-supported cometabolism. Each model structure consisted of three kinetic expressions, respectively describing changes in the concentration of cometabolic substrate (Sg), primary substrate (Sc), and microbial biomass (X). Changes in the biomass concentration were modeled by positive and negative stoichiometric links to primary and cometabolic substrate removal, respectively, augmented by a first order endogenous decay process. The negative link to cometabolic substrate removal captures inactivation associated with accumulation of cometabolite transformation products.29,43 Recovery after inactivation was not considered.29 Primary substrate removal was modeled identically in all cases by Michaelis−Menten (MM) dependency of specific removal rates on substrate concentrations. The main difference between the model structures was in the kinetic expressions describing cometabolic substrate removal. In the simplest case, the kinetics were only a function of the cometabolic substrate (Sg) (M1: the first-order model, M2: the MM model), in the more complex cases the kinetics were also governed by the primary substrate (Sc), as a colimiting substrate (M3: reductant model), as a competitive substrate (competitive inhibition of cometabolic substrate by primary substrate) (M4: competition model), or both (M5: combined model, Table 1). Reciprocal competitive inhibition of primary substrate removal by cometabolic substrate (competitive inhibition of primary substrate removal by cometabolic substrate) was not considered, as such is rarely relevant.33 This assumption may not be appropriate when growth substrates are at low relative concentrations compared to cometabolic substrates; then mutual inhibition must be considered. Although the models are progressively more complex, the number of kinetic and stoichiometric parameters remains 5 (kc, kg, Kc, Kg, b) and 2 (Y, T) for all models, except for the first order model, which has only 4 kinetic parameters (kc, kg, Kg, b). Complete model descriptions are provided in Table 1, and relevant parameters in Table 2. Models that reflect true observations with high certainty should have the following attributes: a high goodness of fit,
randomly distributed residuals, parameter estimates close to true values (if known), and simple model structures.44 We selected three criteria to discriminate between models: the Chisquare (χ 2 ) statistic, representing goodness of fit to experimental data; the colinearity index γκ measuring parameter correlation; and the Akaike information criterion (AIC) weighing model complexity. These terms are fully defined in the Supporting Information (SI) and in Table S1. Model discrimination can be based on comparing different model structures against a consistent set of experimental data. However, dense and comprehensive data sets for cometabolic biodegradation, which include at least primary and cometabolic substrate profiles, are sparse. Hence, our analysis relied on generated sets of synthetic data to which artificial error (of same order of magnitude as experimental error) was added.45,46 This approach allowed us to compare models over a wider range of experimental and initial conditions. Hence, batch biodegradation experiments were simulated where the concentration of both cometabolic and primary substrate were monitored. Initial concentration of both substrates as well as of the microbial biomass were assumed known. All five mentioned models were then tested for their ability to fit the synthetic observations and return reasonable parameter estimates. The overall approach involved five sequential steps. In the first step simulated data were generated using Model 5. A wide range of relative model parameter values was obtained by varying Ksg/Ksc from 102 to 10−2, and maintaining kg, kc, b, Tc and Y at an arbitrary value of 0.1 (in their respective units, within the reported range of each parameter17,25,26).Initial conditions were kept constant for Xo (100 mg L−1) and Sgo (1 mg L−1), but Sco was varied from 1 to 106 ng L−1. The ratio of half-saturation coefficient Ksg/Ksc was set to vary from 102 to 10−2 and initial conditions Sgo/Sco was varied from 1 to 106, we believe this could cover a large range of cometabolic kinetic conditions of different enzyme system and different contaminants/substrates.17,23 This resulted in 35 different scenarios (SI Table S2, with 7 Group and each Group has 5 subgroup). Synthetic data sets were generated from the simulated Sg and Sc profiles by adding error (randomly sampled from normal distributions with relative deviation of 0.05). Then, each Sg and Sc profile was sampled to yield 10 data points chosen at time intervals adjusted to cover the complete concentration profile. In step 2, Models 1 through 5 were fit to the synthetic data (Sg and Sc) by minimization of sum of squared relative residuals (SI) to yield best-fit and covariance estimates of the parameters {kg, kc, Ksg, Ksc, b, T, Y}. In step 3, the AIC and BIC were calculated based on residuals analysis. In step 4, the colinearity index γκ of parameter subset {kg, kc, Ksg, b, T, Y} was calculated based on the parameter uncertainty analysis (SI Table S1). Ksc was omitted from the colinearity analysis for consistency, as this parameter does not occur in Model 1 (Table 1). All calculations were made in Matlab, with parameter estimation using the Levenberg−Marquardt algorithm. A range of initial conditions (expressed as the S0/KS ratio) was simulated, as changing initial conditions can be used in experiments to determine the degree of correlation between the k and KS parameters.47 Here, a very wide range of conditions was employed, with and Sco/KSc and Sgo/KSg varying from 10−6 to 1 and 0.01 to 100, respectively. The simulated experiments were primarily aimed at estimating the kinetic parameters, and not the stoichiometric parameters b, T, and Y, whose estimation requires observation of significant changes in the biomass
Table 2. Summary of the Parameters used in the Cometabolic Model Structures symbol
description
b kc
first order decay coefficient maximum specific utilization rate of cometabolic substrate or first order utilization rate of cometabolic substrate
Ksc
half-saturation coefficient for cometabolic substrate maximum specific utilization rate of growth substrate half-saturation coefficient for growth substrate transformation capacity for biomass for the cometabolic substrate
kg Ksg T Y
biomass yield on growth substrate
units −1
d mg nongrowth substrate (mg cells)−1 d−1 or (mg cells)−1 d−1 mg L−1 mg growth substrate (mg cells)−1 d−1 mg L−1 mg cells (mg cometabolic substrate)−1 mg cells (mg growth substrate)−1 2232
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Environmental Science & Technology concentration. The effect of varying the ratios of kg/kc and kgY/ T was not considered in specified systems.
smallest AIC value. The procedure illustrated in Figure 1 is then repeated for 34 more combinations of initial conditions and results are summarized in the following. Goodness of Fit. For each combination of initial conditions/parameters, the goodness of fit of each model to each synthetic data set was determined in terms of χ2 (SI Table S6). Initial conditions affected the χ2 value differently for an individual model. For example, for high Sg values, the χ2 values of the first-order model were notably high because metabolic transformation rates were higher than could be predicted by the best fit first-order kinetic model. Results can be used to evaluate which of the 5 models gives the best fit to the data for the range of initial conditions/ parameters tested. The model with the minimum χ2 value for each point of the parameter/initial substrate concentration matrix is shown in Figure 2, where the models with minimum
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RESULTS AND DISCUSSION Figure 1 shows an example of the procedure with initial conditions Sgo = 1 mg L−1, Sco = 0.1 mg L−1, Ksg = 10 mg L−1,
Figure 2. Cometabolic models with lowest χ2 value for each parameter/initial substrate concentration combination.
χ2 value is marked using a symbol. In the figure, the vertical axis shows the value of Sg/Ksg (from 10−6 − 1), and the lateral axis the value of Sc/Ksc (from 0.01 to 100). In Figure 2 we shaded the background with the color symbolizing the reductant model to indicate that the part of the parameter space where the reductant model provides the best possible description of the data. Results show that the reductant model is the model that best fits the data for the largest range of initial conditions/ parameters. In order to test the robustness of this conclusion the fitting process was repeated with several new sets of synthetic data, where the data sets differed in the amount of random error added to each data point. Results (not shown), showed that the conclusions were not affected by the data set. The χ2 metric is only one possible measure of goodness of fit. A second metric of goodness-of-fit, the scaled root mean squared error (SRMSE, eq 3, SI), also produced similar results to those using the χ2 metric (SI Table S3 compares the χ2 and SRMSE for models fit to new sets of synthetic data of Group1.1 and 1.2). Parameter Identifiability. The previous section showed that the reductant model fits the data best when compared with other models. However, while the reduction model fits each data set well in the parameter/initial substrate concentration domain, it achieves these fits with a widely varying set of parameters, see SI Table S4. It is important that a single parameter set can describe a variety of experimental conditions so additional measures of model fit are required to evaluate
Figure 1. Synthetic (dashed lines) and best-fit (continuous lines) (Sg,t) and (Sc,t) profiles for scenario Group 2.2 (SI Table S2). Fits by each individual model are shown. The χ2, colinearity, and AIC value associated with these analyses for each model are summarized in the table.
Ksc = 1 mg L−1. The values of the parameters kg, kc, b, Tc and Y are all set at 0.1 in their respective units. Error free profiles of Sg and Sc are generated with Model 5 (dashedlines). Discrete data points are then sampled to which a random error is added. These data points are then fitted by each of the other individual models (continuous lines). Results show that the reductant model fits the data best, with the smallest χ2 value (red numbers in the table below Figure 1) of 0.11; the colinearity index γκ value shows that the first-order model has the smallest parameter correlation; while the reductant model has the 2233
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of goodness of fit with a penalty for more complex models with a larger parameter space. While commonly used to compare linear regression models,49,50 this approach has rarely been applied to discriminate between cometabolic biodegradation models. Both the AIC and BIC were calculated for all models for all combinations of initial conditions/parameters. AIC and BIC analysis resulted in nearly identical rankings across the calculation domain (SI Figure S2). The best model structures, based on the AIC, are shown in Figure 4. The figure shows that
model fit. One possible cause of parameter variability is parameter correlation, which means that multiple combinations of parameters can be found with very similar fitting results. Such correlation can be evaluated using a colinearity analysis. When a set of parameters have linearly independent sensitivity functions, their colinearity index approaches unity; higher values indicate higher degree of linear dependence, and poorer identificability.37 For a set of parameters to be simultaneously identifiable, critical values for their colinearity index have been proposed ranging from 5 to 20.37 Parameter subsets having colinearity index higher than this range are not well determined.48 The colinearity indices γκ of the 6 parameter subset {kg, kc, Ksg, b, T, Y} for each model are summarized in SI Figure S1 and Table S5. The colinearity index γκ of the first-order model was lowest for small values of Sc/Ksc, and for high Sc/Ksc values the combined model, which was also used to generate the synthetic data, was superior. For some initial values/parameters, the Fisher Information Matrix was singular or almost singular and could not be inverted so that γκ could not be calculated. The colinearity values of the six parameter set examined above, range from 11 (first-order model) to 7.7 × 107 (MM model), indicating poor identifiability−even for the best scenarios. This is not surprising, and typically the 3 parameters, b, Y, and Tc would be estimated separately from other dedicated experimental data.23 Therefore, we repeated our analysis for the 3-parameter subset Ksg, kg, kc (values were summarized in SI Table S7). Results are compared in Figure 3.
Figure 4. Cometabolic models with lowest AIC values for each parameter/initial substrate concentration combination.
the first-order model provides the best compromise between model fit and model complexity. The AIC and BIC of reductant model were much better than MM model, competition model and combined model, but inferior to the First-order model because of the additional parameter. Implications of This Work. We have established a general method for cometabolic biokinetic model selection and comparison, which combines an evaluation of model fit, parameter identifiability and model parsimony. The method was tested by comparing five different biokinetic models for growth-supported cometabolism across the entire space of possible model outcomes of different substrate initial concentration and parameter conditions. The χ2 value of the reductant model was consistently lower than the other three models for a range of initial conditions, but with a set of fitted parameters that varied greatly between experimental data sets. When the Sc concentration is very low, the first-order model and MM could be considered. The colinearity index γκ of the first-order model was small for low values of Sc/Ksc indicating low parameter correlation, while for high Sc/Ksc values, the combined model had lower parameter correlation than the other models. AIC and BIC results show that the first-order model may be preferable to the other models because it fits data well and employs one less parameter; and the reductant model was generally the best of the remaining models. Overall across the tested regimes, we could not generally conclude that one particular type of model is intrinsically superior over another. Nevertheless, this work clearly and comprehensively assesses the parameter identification and model structure of each model. The general method presented can offer insight
Figure 3. Cometabolic models with the highest identifiability for the 3parameter subsets {kg, kc, Ksg} for each parameter/initial substrate concentration combination.
Results show that the first-order model and the reductant model provide the most identifiable three parameter subsets in the upper left and lower right region. On average, the parameter identifiability of the first-order model was higher than the other models especially in regions of lower relative initial cometabolic substrate concentration (SI Figure S1 and Figure 3). Model Size and Complexity. The suitability of alternate models can be evaluated using the Akaike and Bayesian Information Criterion (AIC, BIC). These model evaluation measures determine model parsimony by combining a measure 2234
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into model selection and may be practically extended to more sophisticated model systems.
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ASSOCIATED CONTENT
* Supporting Information S
Five tables and three supplemental figures are provided. Mathematical models to describe cometabolism, Chi-square (χ2) test statistic, parameter uncertainty analysis and Akaike information criterion (AIC) are also included in Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Fax: +45 45 93 28 50; e-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the EU-FP7 project BIOTREAT (GA No. 266039).
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REFERENCES
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DOI: 10.1021/es5035393 Environ. Sci. Technol. 2015, 49, 2230−2236
