Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

On the reliable estimation of sequential Monod kinetic parameters

While dozens of studies have attempted to estimate the Monod kinetic parameters of microbial reductive dechlorination, published values in the literature vary by 2-6 orders of magnitude. This lack of consensus can be attributed in part to limitations of both experimental design and parameter estimation techniques. To address these issues, Hamiltonian Monte Carlo was used to produce more than one million sets of realistic simulated microcosm data under a variety of experimental conditions. These data were then employed in model fitting experiments using a number of parameter estimation algorithms for determining Monod kinetic parameters. Analysis of data from conventional triplicate microcosms yielded parameter estimates characterized by high collinearity, resulting in poor estimation accuracy and precision. Additionally, confidence intervals computed by commonly used classical regression analysis techniques contained true parameter values much less frequently than their nominal confidence levels. Use of an alternative experimental design, requiring the same number of analyses as conventional experiments but comprised of microcosms with varying initial chlorinated ethene concentrations, is shown to result in order-of-magnitude decreases in parameter uncertainty. A Metropolis algorithm which can be run on a typical personal computer is demonstrated to return more reliable parameter interval estimates.

Dosage concentration and pulsing frequency affect the degradation efficiency in simulated bacterial polycyclic aromatic hydrocarbon-degrading cultures

A major source of anthropogenic polycyclic aromatic hydrocarbon (PAH) inputs into marine environments are diffuse emissions which result in low PAH concentrations in the ocean water, posing a potential threat for the affected ecosystems. However, the remediation of low-dosage PAH contaminations through microbial processes remains largely unknown. Here, we developed a process-based numerical model to simulate batch cultures receiving repeated low-dosage naphthalene pulses compared to the conventionally used one-time high-dosage. Pulsing frequency as well as dosage concentration had a large impact on the degradation efficiency. After 10 days, 99.7%, 97.2%, 86.6%, or 83.5% of the 145 mg L^-1 naphthalene was degraded when given as a one-time high-dosage or in 2, 5, or 10 repeated low-concentration dosages equally spaced throughout the experiment, respectively. If the simulation was altered, giving the system that received 10 pulses time to recover to 99.7%, pulsing patterns affected the degradation of naphthalene. When pulsing 10 days at once per day, naphthalene accumulated following each pulse and if the degradation was allowed to continue until the recovered state was reached, the incubation time was prolonged to 17 days with a generation time of 3.81 days. If a full recovery was conditional before the next pulse was added, the scenario elongated to 55 days and generation time increased to 14.15 days. This indicates that dissolution kinetics dominate biodegradation kinetics, and the biomass concentration of PAH-degrading bacteria alone is not a sufficient indicator for quantifying active biodegradation. Applying those findings to the environment, a one-time input of a high dosage is potentially degraded faster than repeated low-dosage PAH pollution and repeated low-dosage input could lead to PAH accumulation in vulnerable pristine environments. Further research on the overlooked field of chronic low-dosage PAH contamination is necessary.

Stoichiometric ratios for biotics and xenobiotics capture effective metabolic coupling to re(de)fine biodegradation

Preserving human and environmental health requires anthropogenic pollutants to be biologically degradable. Depending on concentration, both nutrients and pollutants induce and activate metabolic capacity in the endemic bacterial consortium, which in turn aids their degradation. Knowledge on such 'acclimation' is rarely implemented in risk assessment cost-effectively. As a result, an accurate description of the mechanisms and kinetics of biodegradation remains problematic. In this study, we defined a yield 'effectivity', comprising the effectiveness at which a pollutant (substrate) enhances its own degradation by inducing (biomass) cofactors involved therein. Our architecture for calculation represents the interplay between concentration and metabolism via both stoichiometric and thermodynamic concepts. The calculus for yield 'effectivity' is biochemically intuitive, implicitly embeds co-metabolism and distinguishes 'endogenic' from 'exogenic' substances' reflecting various phenomena in biodegradation and bio-transformation studies. We combined data on half-lives of pollutants/nutrients in wastewater and surface water with transition-state rate theory to obtain also experimental values for effective yields. These quantify the state of acclimation: the portion of biodegradation kinetics attributable to (contributed by) 'natural metabolism', in view of similarity to natural substances. Calculated and experimental values showed statistically significant correspondence. Particularly, carbohydrate metabolism and nucleic acid metabolism appeared relevant for acclimation (R² = 0.11-0.42), affecting rates up to 10^4.9(±0.7) times: under steady-state acclimation, a compound stoichiometrically identical to carbohydrates or nucleic acids, is 10^3.2 to 10^4.9 times faster aerobically degraded than a compound marginally similar. Our new method, simulating (contribution by) the state of acclimation, supplements existing structure-biodegradation and kinetic models for predicting biodegradation in wastewater and surface water. The accuracy of prediction may increase when characterizing nutrients/co-metabolites in terms of, e.g., elemental analysis. We discuss strengths and limitations of our approach by comparison to empirical and mechanism-based methods.

Quantifying Impacts of Microcosm Mass Loss on Kinetic Constant Estimation

Microcosm experiments to assess microbial reductive dechlorination of chlorinated aliphatic hydrocarbons typically experience 5-50% mass loss due to frequent sampling events and diffusion through septa. A literature review, however, reveals that models fit to such experiments for kinetic constant estimation have generally failed to account for experimental mass loss. To investigate possible resultant bias in best-fit parameters, a series of numerical experiments was conducted in which Monod kinetic models with and without mass loss were fit to more than 1300 synthetic data sets, generated using published microcosm data. Models that failed to account for mass loss resulted in significant fitted parameter bias. Bias ranged from 5 to 45% of the parameter magnitude for Monte Carlo simulations with low (approximately 10%) mass loss to 20-120% for simulations with high (approximately 40%) mass loss. In addition, for high mass loss simulations, best-fit values consistently fell along the bounds of the optimization range. These results suggest that failure to properly account for mass loss in microcosms may lead to inaccurate estimation of kinetic constants and may explain some of the literature-reported variability in these parameters. A model is presented that provides a method for including sampling and diffusional mass losses to improve kinetic constant estimation accuracy.

Accurate and reliable estimation of kinetic parameters for environmental engineering applications: A global, multi objective, Bayesian optimization approach

Accurate and reliable predictions of bacterial growth and metabolism from unstructured kinetic models are critical to the proper operation and design of engineered biological treatment and remediation systems. As such, parameter estimation has progressed into a routine challenge in the field of Environmental Engineering. Among the main issues identified with parameter estimation, the model-data calibration approach is a crucial, yet an often overlooked and difficult optimization problem. Here, a novel and rigorous global, multi objective, and fully Bayesian optimization approach that overcomes challenges associated with multi-variate, sparse and noisy data, as well as highly non-linear model structures commonly encountered in Environmental Engineering practice is presented. This optimization approach allows an improved definition and targeting of the compromise solution space for all multivariate problems, allowing efficient convergence, and a Bayesian component to thoroughly explore parameter and model prediction uncertainty. This global optimization approach outperformed, in terms of parameter accuracy and precision, standard, local non-linear regression routines and overcomes issues associated with premature convergence and addresses overfitting of different variables in the calibration process. •A sequential single, multi-objective, and Bayesian optimization workflow was developed to accurately and reliably estimate unstructured kinetic model parameters.•The global, single objective approach defines the global optimum (the best compromise solution) and "extreme" parameter solutions for each variable, while the global, multi-objective approach confirms the "best" compromise solution space for the Bayesian search to target and convergence is assessed using the single objective results.•The Approximate Bayesian Computational approach fully explores parameter and model prediction uncertainty targeting the compromise solution space previously identified.

Application of unstructured kinetic models to predict microcystin biodegradation: Towards a practical approach for drinking water treatment

Biological drinking water treatment technologies offer a cost-effective and sustainable approach to mitigate microcystin (MC) toxins from harmful algal blooms. To effectively engineer these systems, an improved predictive understanding of the bacteria degrading these toxins is required. This study reports an initial comparison of several unstructured kinetic models to describe MC microbial metabolism by isolated degrading populations. Experimental data was acquired from the literature describing both MC removal and cell growth kinetics when MC was utilized as the primary carbon and energy source. A novel model-data calibration approach melding global single-objective, multi-objective, and Bayesian optimization in addition to a fully Bayesian approach to model selection and hypothesis testing were applied to identify and compare parameter and predictive uncertainties associated with each model structure. The results indicated that models incorporating mechanisms of enzyme-MC saturation, affinity, and cooperative binding interactions of a theoretical single, rate limiting reaction accurately and reliably predicted MC degradation and bacterial growth kinetics. Diverse growth characteristics were observed among MC degraders, including moderate to high maximum specific growth rates, very low to substantial affinities for MC, high yield of new biomass, and varying degrees of cooperative enzyme-MC binding. Model predictions suggest that low specific growth rates and MC removal rates of degraders are expected in practice, as MC concentrations in the environment are well below saturating levels for optimal growth. Overall, this study represents an initial step towards the development of a practical and comprehensive kinetic model to describe MC biodegradation in the environment.

Kinetic Studies on the Conversion of Levoglucosan to Glucose in Water Using Brønsted Acids as the Catalysts

Fast pyrolysis is as a promising and versatile technology to depolymerize and concentrate sugars from lignocellulosic biomass. The pyrolysis liquids produced contain considerable amounts of levoglucosan (1,6-anhydro-β-d-glucopyranose), which is an interesting source for glucose (GLC). Here, we report a kinetic study on the conversion of levoglucosan (LG) to GLC in water using sulfuric and acetic acid as the catalysts under a wide range of conditions in a batch setup. The effects of the initial LG loading (0.1-1 M), sulfuric and acetic acid concentrations (0.05-0.5 M and 0.5-1 M, respectively), and reaction temperatures (80-200 °C) were determined. Highest GLC yields were obtained using sulfuric acid (98 mol %), whereas the yields were lower for acetic acid (maximum 90 mol %) due to the formation of byproducts such as insoluble polymers (humins). The experimental data were modeled using MATLAB software, and relevant kinetic parameters were determined. Good agreement between experimental and model was obtained when assuming that the reaction is first order with respect to LG. The activation energies were 123.4 kJ mol^-1 and 120.9 kJ mol^-1 for sulfuric and acetic acid, respectively.

The role of organic matter and clay content in sediments for bioavailability of pyrene

Evaluation of the bioavailable fractions of organic contaminants such as polycyclic aromatic hydrocarbons (PAHs) is extremely important for assessing their risk to the environment. This available fraction, which can be solubilised and/or easily extracted, is believed to be the most accessible for bioaccumulation, biosorption and/or transformation. Sediment organic matter (OM) and clay play an important role in the biodegradation and bioavailability of PAHs. The strong association of PAHs with OM and clay in sediments has a great influence not only on their distribution but also on their long-term environmental impact. This paper investigates correlations between bioavailability and the clay and OM contents in sediments. The results show that OM is a better sorbent for pyrene (chosen as a model PAH) and that increasing the OM content reduces the bioavailable fraction. A mathematical model was used to predict the kinetic desorption, and these results showed that the sediment with the lowest content of OM had an F_fast value of 24%, whereas sediment with 20% OM gave a value of 9%. In the experiments with sediments with different clay contents, no clear dependence between clay and rate constants of the fast desorbing fractions was observed, which can be explained by the numerous possible interactions at the molecular level.

Experimental and Kinetic Modeling Studies on the Conversion of Sucrose to Levulinic Acid and 5-Hydroxymethylfurfural Using Sulfuric Acid in Water

We here report experimental and kinetic modeling studies on the conversion of sucrose to levulinic acid (LA) and 5-hydroxymethylfurfural (HMF) in water using sulfuric acid as the catalyst. Both compounds are versatile building blocks for the synthesis of various biobased (bulk) chemicals. A total of 24 experiments were performed in a temperature window of 80-180 °C, a sulfuric acid concentration between 0.005 and 0.5 M, and an initial sucrose concentration between 0.05 and 0.5 M. Glucose, fructose, and HMF were detected as the intermediate products. The maximum LA yield was 61 mol %, obtained at 160 °C, an initial sucrose concentration of 0.05 M, and an acid concentration of 0.2 M. The maximum HMF yield (22 mol %) was found for an acid concentration of 0.05 M, an initial sucrose concentration of 0.05 M, and a temperature of 140 °C. The experimental data were modeled using a number of possible reaction networks. The best model was obtained when using a first order approach in substrates (except for the reversion of glucose) and agreement between experiment and model was satisfactorily. The implication of the model regarding batch optimization is also discussed.

Biobased Furanics: Kinetic Studies on the Acid Catalyzed Decomposition of 2-Hydroxyacetyl Furan in Water Using Brönsted Acid Catalysts

Biobased furanics like 5-hydroxymethylfurfural (5-HMF) are interesting platform chemicals for the synthesis of biofuel additives and polymer precursors. 5-HMF is typically prepared from C6 ketoses like fructose, psicose, sorbose and tagatose. A known byproduct is 2-hydroxyacetylfuran (2-HAF), particularly when using sorbose and psicose as the reactants. We here report an experimental and kinetic modeling study on the rate of decomposition of 2-HAF in a typical reaction medium for 5-HMF synthesis (water, Brönsted acid), with the incentive to gain insights in the stability of 2-HAF. A total of 12 experiments were performed (batch setup) in water with sulfuric acid as the catalyst (100-170 °C, C_H2SO4 ranging between 0.033 and 1.37 M and an initial 2-HAF concentration between 0.04 and 0.26 M). Analysis of the reaction mixtures showed a multitude of products, of which levulinic acid (LA) and formic acid (FA) were the most prominent (Y_max,FA = 24 mol %, Y_max,LA = 10 mol %) when using HCl. In contrast, both LA and FA were formed in minor amounts when using H₂SO₄ as the catalyst. The decomposition reaction of 2-HAF using sulfuric acid was successfully modeled (R² = 0.9957) using a first-order approach in 2-HAF and acid. The activation energy was found to be 98.7 (±2.2) kJ mol^-1.

Modeling sorption and biodegradation of phenanthrene in mangrove sediment slurry

A mathematical model, combining both sorption and biodegradation process, was developed to predict the biodegradation of phenanthrene by Sphingomonas sp. in different sediment slurries. The model includes two sorption parameters, α (the partition coefficient) and 1/K (the diffusion resistance); a kinetic parameter k (the first order rate constant); and a sediment parameter, A(V) (the specific sediment surface area in unit volume of slurry). These parameters were evaluated and verified in three types of sediment slurry systems (namely sandy clay loam Ho Chung sediment with fastest degradation, sandy Kei Ling Ha sediment with medium degradation, and clay Mai Po sediment with slowest degradation) at different initial phenanthrene concentrations. High R(2) values, ranging from 0.935 to 0.969, were obtained. Based on this integrated sorption-biodegradation model, the phenanthrene biodegradation in any sediment slurry could be predicted as long as the parameters of the specific sediment surface area in unit volume of slurry, total organic carbon and clay content were measured.

Log transformation benefits parameter estimation in microwave tomographic imaging

Microwave tomographic imaging falls under a broad category of nonlinear parameter estimation methods when a Gauss-Newton iterative reconstruction technique is used. A fundamental requirement in using these approaches is evaluating the appropriateness of the regression model. While there have been numerous investigations of regularization techniques to improve overall image quality, few, if any, studies have explored the underlying statistical properties of the model itself. The ordinary least squares (OLS) approach is used most often, but there are other options such as the weighted least squares (WLS), maximum likelihood (ML), and maximum a posteriori (MAP) that may be more appropriate. In addition, a number of variance stabilizing transformations can be applied to make the inversion intrinsically more linear. In this paper, a statistical analysis is performed of the properties of the residual errors from the reconstructed images utilizing actual measured data and it is demonstrated that the OLS algorithm with a log transformation (OLSlog) is clearly advantageous relative to the more commonly used OLS approach by itself. In addition, several high contrast imaging experiments are performed, which demonstrate that different subsets of data are emphasized in each method and may contribute to the overall image quality differences.

Measurement of biodegradability parameters for single unsubstituted and methylated polycyclic aromatic hydrocarbons in liquid bacterial suspensions

Substrate depletion experiments were conducted to characterize aerobic biodegradation of 20 single polycyclic aromatic hydrocarbons (PAHs) by induced Sphingomonas paucimobilis strain EPA505 in liquid suspensions. PAHs consisted of low molecular weight, unsubstituted, and methyl-substituted homologs. A material balance equation containing the Andrews kinetic model, an extension of the Monod model accounting for substrate inhibition, was numerically fitted to batch depletion data to estimate extant kinetic parameters including the maximal specific uptake rates, q(max), the affinity coefficients, K(S), and the substrate inhibition coefficients, K(I). Strain EPA505 degraded all PAHs tested. Applied kinetic models adequately simulated experimental data. A cell proliferation assay involving reduction of the tetrazolium dye WST-1 was used to evaluate the ability of strain EPA505 to utilize individual PAHs as sole energy and carbon sources. Of the 22 PAHs tested, 9 supported bacterial growth. Evaluation of the biokinetic data showed that q(max) correlated highly with transmembrane flux as theoretically estimated by a diffusion model, pointing to transmembrane transport as a potential rate-determining process. The biodegradability data generated in this study is essential for the development of quantitative structure-activity relationships (QSARs) for biodegradability and for modeling biodegradation of simple PAH mixtures.

A process-based mathematical model on methane production with emission indices for control

In this paper, a process-based mathematical model is developed for the production of methane through biodegradation. It is a three-dimensional model given by ordinary differential equations. The results of the analysis of the model are interpreted through three emission indices, which are introduced for the first time. The estimation of either one or all of them can interpret the feasibility of the equilibrium and the long-term emission tendency of methane. The vulnerability of the methane production process with respect to soil temperature effects in methanogenic phase has been discussed and a feasible condition within a specified temperature range has defined for the nonvulnerability of the methane production process and also it has shown that under the same condition, zero-emission process of methane will be nonvulnerable with respect to the soil temperature effects in methanogenic phase. Lastly, condition for zero emission of methane is also obtained and it is interpreted through the emission indices.

Kinetic analysis of the inhibitory effect of trichloroethylene (TCE) on nitrification in cometabolic degradation

In this study, the inhibitory effect of TCE on nitrification process was investigated with an enriched nitrifier culture. TCE was found to be a competitive inhibitor of ammonia oxidation and the inhibition constant (K(I)) was determined as 666-802 microg/l. The TCE affinity for the AMO enzyme was significantly higher than ammonium. The effect of TCE on ammonium utilization was evaluated with linearized plots of Monod equation (e.g., Lineweaver-Burk, Hanes-Woolf and Eadie-Hofstee plots) and non-linear least square regression (NLSR). No significant differences were found among these data evaluation methods in terms of kinetic parameters obtained.

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.

Effects of carbonate species on the kinetics of dechlorination of 1,1,1-trichloroethane by zero-valent iron

The effect of precipitates on the reactivity of iron metal (Fe0) with 1,1,1-trichloroethane (TCA) was studied in batch systems designed to model groundwaters that contain dissolved carbonate species (i.e., C(IV)). At representative concentrations for high-C(IV) groundwaters (approximately 10(-2) M), the pH in batch reactors containing Fe0 was effectively buffered until most of the aqueous C(IV) precipitated. The precipitate was mainly FeCO3 (siderite) but may also have included some carbonate green rust. Exposure of the Fe0 to dissolved C(IV) accelerated reduction of TCA, and the products formed under these conditions consisted mainly of ethane and ethene, with minor amounts of several butenes. The kinetics of TCA reduction were first-order when C(IV)-enhanced corrosion predominated but showed mixed-order kinetics (zero- and first-order) in experiments performed with passivated Fe0 (i.e., before the onset of pitting corrosion and after repassivation by precipitation of FeCO3). All these data were described by fitting a Michaelis-Menten-type kinetic model and approximating the first-order rate constant as the ratio of the maximum reaction rate (Vm) and the concentration of TCA at half of the maximum rate (K(1/2)). The decrease in Vm/K(1/2) with increasing C(IV) exposure time was fit to a heuristic model assuming proportionality between changes in TCA reduction rate and changes in surface coverage with FeCO3.

Biokinetics in acidogenesis of highly suspended organic wastewater by adenosine 5' triphosphate analysis

In this paper, we pointed out the problems of using conventional volatile suspended solids (VSS) and chemical oxygen demand (COD) to evaluate biokinetic coefficients, especially for the treatment of highly suspended organic wastewater. We also introduced a novel approach to evaluate biokinetic coefficients by measurement of adenosine 5'-triphosphate (ATP) of microorganisms. The concept of using ATP analysis in biokinetic evaluations with highly suspended wastewater was shown to be effective. This study also showed that the conventional VSS and COD methods were strongly affected by incoming suspended organics in the wastewater and by biokinetics of microorganisms. A cheese-processing wastewater was used in evaluating the biokinetics of mesophilic acidogens. The concentration of COD and total suspended solids in the wastewater was 63.3 g/L and 12.4 g/L, respectively. The TSS was 23.6% of total solids concentration. A high ratio of VSS to total suspended solids of 96.7% indicated that most of the suspended particles were organic materials. Lactose and protein were the major organic components contributing COD in the wastewater, and a total of 94.2% of the COD in the wastewater was due to the presence of lactose and protein. Two different physiological conditions where the maximum rates of acetate and butyrate production occurred were tested. These were pH 7 (condition A for acetate production) and pH 7.3 (condition B for butyrate production) at 36.2C, respectively. Based on the molecular structures of the major organic substances and microbial ATP analysis, the residual substrate and microbial concentrations were stoichiometrically converted to substrate COD (SuCOD) and microbial VSS (MVSS), respectively, using correlation coefficients reported previously. These SuCOD and MVSS were simultaneously used to evaluate the biokinetic coefficients using Monod-based mathematical equations. The nonlinear least squares method with 95% confidence interval was used to evaluate biokinetic coefficients. The maximum microbial growth rate, mu(max) and half saturation coefficient, K(s), for conditions A and B were determined to be 9.9 +/- 0.3 and 9.3 +/- 1.0 day(-1) and 134.0 +/- 58.3 and 482.5 +/- 156.5 mg SuCOD/L, respectively. The microbial yield coefficient, Y, and microbial decay rate coefficient, k(d) for conditions A and B were determined to be 0.29 +/- 0.03 and 0.20 +/- 0.05 mg MVSS/mg SuCOD, and 0.14 +/- 0.05 and 0.25 +/- 0.05 day(-1), respectively. Specific substrate utilization rate at condition B was 43.8 +/- 20.6 mg SuCOD/mg MVSS/day, which was 31% higher than that at condition A.

Maximization of acetic acid production in partial acidogenesis of swine wastewater

Swine wastewater was biologically treated to produce short-chain volatile organic acids (VOAs) in laboratory-scale continuously stirred tank reactors. The maximum production rates of acetic and butyric acids associated with simultaneous changes in pH and hydraulic retention time (HRT) were investigated, in which the degree of acidification of swine wastewater to the short-chain VOAs was <25% of influent chemical oxygen demand (COD) concentration. A constant inoculum system was used to minimize the experimental error due to the use of inconsistent inoculum. The inoculum system was operated with synthetic wastewater at 6000 mg soluble chemical oxygen demand per liter (pH 6.0) and 35 degrees C at 0.5 day hydraulic retention time. Response surface methodology was applied successfully to determine the optimum physiological condition for which the maximum rate of acetic acid production occurred, which was pH 5.90 and 0.88 day hydraulic retention time at 35 degrees C. The partial acidification process to manage swine waste should be operated in the optimum condition for acetic acid production because the optimum operating condition for butyric acid production approached the washout point.