Practical identifiability of biokinetic parameters of a model describing two-step nitrification in biofilms

Parameter estimation and model calibration are key problems in the application of biofilm models in engineering practice, where a large number of model parameters need to be determined usually based on experimental data with only limited information content. In this article, identifiability of biokinetic parameters of a biofilm model describing two-step nitrification was evaluated based solely on bulk phase measurements of ammonium, nitrite, and nitrate. In addition to evaluating the impact of experimental conditions and available measurements, the influence of mass transport limitation within the biofilm and the initial parameter values on identifiability of biokinetic parameters was evaluated. Selection of parameters for identifiability analysis was based on global mean sensitivities while parameter identifiability was analyzed using local sensitivity functions. At most, four of the six most sensitive biokinetic parameters were identifiable from results of batch experiments at bulk phase dissolved oxygen concentrations of 0.8 or 5 mg O(2)/L. High linear dependences between the parameters of the subsets (KO2,AOB,muAOB) and (KO2,NOB,muNOB) resulted in reduced identifiability. Mass transport limitation within the biofilm did not influence the number of identifiable parameters but, in fact, decreased collinearity between parameters, especially for parameters that are otherwise correlated (e.g., muAOB) and KO2,AOB, or muNOB and KO2,NOB). The choice of the initial parameter values had a significant impact on the identifiability of two parameter subsets, both including the parameters muAOB and KO2,AOB. Parameter subsets that did not include the subsets muAOB and KO2,AOB or muNOB and KO2,NOB were clearly identifiable independently of the choice of the initial parameter values.     

Kinetics and modeling of autotrophic thiocyanate biodegradation

The biokinetic parameters for autotrophic systems are difficult to obtain and are often mistakenly determined because the size of the autotrophic population in mixed (i.e., heterotrophic and autotrophic) cultures cannot be accurately estimated. This article presents a systematic approach, combining bioenergetic calculations and experimental data, to obtain values of the biokinetic parameters pertinent to the aerobic, autotrophic biodegradation of thiocyanate. Nonlinear regression techniques were employed using both initial thiocyanate utilization rate data and single thiocyanate depletion curves. Both types of data were necessary to overcome the problems arising from the linear nature of the substrate depletion curves and the high correlation of the biokinetic model parameters inherent in nonlinear regression analysis. The aerobic biodegradation of thiocyanate followed a substrate inhibition pattern that was successfully described by the Haldane-Andrews model. Although regression analysis did not yield unique biokinetic parameter estimates, the following parameter value ranges were obtained: maximum specific substrate utilization rate (k), 0.26 to 0.44 mg SCN-/mg biomass h; half-saturation coefficient (Ks), 2.3 to 7.1 mg SCN-/L; and inhibition coefficient (Ki), 28 to 109 mg SCN-/L. Based on the estimated biokinetic parameter values, a design and operation diagram was constructed that depicts the steady-state thiocyanate concentration as a function of solids retention time for a completely mixed, continuous-flow reactor.     

Kinetic analysis of simultaneous 2,4-dinitrotoluene (DNT) and 2, 6-DNT biodegradation in an aerobic fluidized-bed biofilm reactor.

We previously reported on the mineralization of 2,4-dinitrotoluene (2,4-DNT) and 2,6-dinitrotoluene (2,6-DNT) in an aerobic fluidized-bed bioreactor (FBBR) (Lendenmann et al. 1998 Environ Sci Technol 32:82-87). The current study examines the kinetics of 2, 4-DNT and 2,6-DNT mineralization at increasing loading rates in the FBBR with the goal of obtaining system-independent kinetic parameters. At each steady state, the FBBR was subjected to a set of transient load experiments in which substrate flux in the biofilm and bulk substrate concentrations were measured. The pseudo-steady-state data were used to estimate the biokinetic parameters for 2,4-DNT and 2,6-DNT removal using a mechanistic mathematical biofilm model and a routine that minimized the sum of the squared residuals (RSS). Estimated kinetic parameters varied slightly for each steady-state; retrieved parameters for qm were 0. 83 to 0.98 g DNT/g XCOD d for 2,4-DNT removal and 0.14 to 0.33 g DNT/g XCOD d for 2,6-DNT removal. Ks values for 2,4-DNT removal (0. 029 to 0.36 g DNT/m3) were consistently lower than Ks values for 2, 6-DNT removal (0.21 to 0.84 g DNT/m3). A new approach was introduced to estimate the fundamental biofilm kinetic parameter S*b,min from steady-state performance information. Values of S*b,min indicated that the FBBR performance was limited by growth potential. Adequate performance of the examined FBBR technology at higher loading rates will depend on an improvement in the growth potential. The obtained kinetic parameters, qm, Ks, and S*b,min, can be used to aid in the design of aerobic FBBRs treating waters containing DNT mixtures.     

A critical comparison of extant batch respirometric and substrate depletion assays for estimation of nitrification biokinetics

Estimation of nitrification biokinetics has been conducted using different batch techniques via measurement of nitrogen species or surrogates such as oxygen (respirometry). However, there are no reports that specifically compare kinetic parameters estimated from respirometry with those from direct nitrogen species measurements. In this study, we evaluated the ability of parameter estimates from isolated and optimally designed complete extant respirometric assays to describe concurrently obtained ammonia and nitrite depletion profiles. Additionally, we mapped the different parameter sets to steady-state bioreactor performance. Using multivariate analysis of variance, we found that estimates from respirometric and substrate depletion assays were predominantly statistically different at the 95% confidence level. The sensitivity of predicted stead-state nitrifying reactor performance to differences in parameter estimates was highest close to the limiting solids retention time (SRT). However, at characteristic nitrifying SRTs the predicted reactor performance using parameter estimates from respirometry and substrate depletion assays were in very close correspondence. Therefore, parameters estimated from extant respirometric assays can be used to adequately predict nitrifying reactor performance.     

Stable adaptive algorithm for simultaneous estimation of time-varying parameters and state variables in aerobic bioprocesses

The application of modern model based control algorithms in the bioprocesses is hampered by the lack of accurate and cheap on-line sensors, capable of providing on-line measurements of the main process variables and parameters. In this paper, a new approach for estimation of immeasurable time-varying parameters and state variable is presented for a class of aerobic bioprocesses using only on-line measurements of the oxygen uptake rate. The approach consists in the design of a new parameter estimator of biomass growth rate and yield coefficient for oxygen consumption on the basis of the theory of adaptive estimation. The dynamical equation of the measurable reaction rate, oxygen uptake rate, is presented as a linear one with respect to the biomass growth rate and the yield coefficient for oxygen consumption. In this way, the structure of the proposed estimator becomes linear time-varying one. After some mathematical transformations, that structure is presented in a form, allowing to be derived the stability conditions using some theoretical results concerning the stability of adaptive observers. The estimates of the yield coefficient for oxygen consumption, the biomass concentration and specific growth rate are obtained then on the basis of the generated estimates using well known kinetic models of bioprocesses. With respect to previous similar approaches, the new estimation algorithm gives stable estimates not only of immeasurable state variable and reaction rates but likewise of an yield coefficient. The behavior of the proposed estimator is studied under inexact initial conditions, step changes of dilution rate and in the presence of measurement noise by simulations using a process model, which belongs to the investigated class of bioprocesses.     

Dynamic investigation and modeling of the nitrogen cometabolism in Methylococcus capsulatus ( Bath)

The methanotrophic bacterium Methylococcus capsulatus is capable of assimilating methane and oxygen into protein-rich biomass, however, the diverse metabolism of the microorganism also allows for several undesired cometabolic side-reactions to occur. In this study, the ammonia cometabolism in Methylococcus capsulatus is investigated using pulse experiments. Surprisingly Methylococcus capsulatus oxidizes ammonia to nitrate through a yet unknown mechanism and fixes molecular nitrogen even at a high dissolved oxygen tension. The observed phenomena can be modeled using 14 ordinary differential equations and 18 kinetic parameters, of which 6 were revealed by Morris screening to be identifiable from the experimental data. Monte Carlo simulations showed that the model was robust and accurate even with uncertainty in the parameter values as confirmed by statistical error analysis.     

Modeling particle transport by bubbles for performance guidelines in airlift fermentors

A calculation method has been developed to model the statistical transport of biological particles in bubble-driven flows, with special reference to the biokinetics of environmental excursions experienced by individual cells, aggregated cells, or immobilization beads in airlift bioreactors. Interim developments on modeling the transport of such particles in concentric tube devices are reported. The calculation is driven by user-prescribed global parameters for the bioreactor geometry, bulk air flow rate, and particle parameters (size and slip speed). The algorithm calls on empirical data correlations for void fraction, bulk liquid flow rate, and bubble sizes and slip speeds, optimally selected from a large bibliographic database. The Monte Carlo algorithm concentrates on simulating particle transport in the bubbly riser flows.The packaged family of correlations and calculations represents, in effect, an expert system augmented by a transport simulation suited to characterizing the biokinetic response of cells cultured in airlift bioreactors.     

An efficient approach to automate the manual trial and error calibration of activated sludge models

An efficient approach is introduced to help automate the rather tedious manual trial and error way of model calibration currently used in activated sludge modeling practice. To this end, we have evaluated a Monte Carlo based calibration approach consisting of four steps: (i) parameter subset selection, (ii) defining parameter space, (iii) parameter sampling for Monte Carlo simulations and (iv) selecting the best Monte Carlo simulation thereby providing the calibrated parameter values. The approach was evaluated on a formerly calibrated full-scale ASM2d model for a domestic plant (located in The Netherlands), using in total 3 months of dynamic oxygen, ammonia and nitrate sensor data. The Monte Carlo calibrated model was validated successfully using ammonia, oxygen and nitrate data collected at high measurement frequency. Statistical analysis of the residuals using mean absolute error (MAE), root mean square error (RMSE) and Janus coefficient showed that the calibrated model was able to provide statistically accurate and valid predictions for ammonium, oxygen and nitrate. This shows that this pragmatic approach can perform the task of model calibration and therefore be used in practice to save the valuable time of modelers spent on this step of activated sludge modeling. The high computational demand is a downside of this approach but this can be overcome by using distributed computing. Overall we expect that the use of such systems analysis tools in the application of activated sludge models will improve the quality of model predictions and their use in decision making.     

Biofilm model calibration and microbial diversity study using Monte Carlo simulations

Mathematical models are useful tools for studying and exploring biological conversion processes as well as microbial competition in biological treatment processes. A single‐species biofilm model was used to describe biofilm reactor operation at three different hydraulic retention times (HRT). The single‐species biofilm model was calibrated with sparse experimental data using the Monte Carlo filtering method. This calibrated single‐species biofilm model was then extended to a multi‐species model considering 10 different heterotrophic bacteria. The aim was to study microbial diversity in bulk phase biomass and biofilm, as well as the competition between suspended and attached biomass. At steady state and independently of the HRT, Monte Carlo simulations resulted only in one unique dominating bacterial species for suspended and attached biomass. The dominating bacterial species was determined by the highest specific substrate affinity (ratio of µ/K_S). At a short HRT of 20 min, the structure of the microbial community in the bulk liquid was determined by biomass detached from the biofilm. At a long HRT of 8 h, both biofilm detachment and microbial growth in the bulk liquid influenced the microbial community distribution. Biotechnol. Bioeng. 2013; 110: 1323–1332. © 2012 Wiley Periodicals, Inc.     

Adaptive dissolved oxygen control through the glycerol feeding in a recombinant Pichia pastoris cultivation in conditions of oxygen transfer limitation

In high cell density cultivation processes the productivity is frequently constrained by the bioreactor maximum oxygen transfer capacity. The productivity can often be increased by operating the process at low dissolved oxygen concentrations close to the limitation level. This may be accomplished with a closed-loop controller that regulates the dissolved oxygen concentration by manipulating the dominant carbon source feeding rate. In this work we study this control problem in a pilot 50l bioreactor with a high cell density recombinant P. pastoris cultivation in complex media. The study focuses on the design of accurate stable adaptive controllers, with guaranteed exponential convergence and its relation with the calibration of controller parameters. Two adaptive control strategies were tested in the pilot bioreactor: a model reference adaptive controller with a linear reference model and an integral feedback controller with adaptive gain. The latter alternative proved to be more robust to errors in the measurements of the off-gas composition. Concerning the instrumentation, algorithms were derived assuming that both the dissolved oxygen tension and off-gas composition are measured on-line, but also the case of only dissolved oxygen being measured is addressed. It was verified that the measurement of off-gas composition might not improve the controller performance due to measurement and process time delays.     

Identification of insulin receptor systems: assessing the impact of model selection and measurement error on precision of parameter estimates using Monte Carlo study

An extensive Monte Carlo study has been carried out in order to study the effect of measurement error on the precision of parameter estimates of an insulin binding system. Hypothetical radioimmunoassay experiments were generated for insulin binding to erythrocytes. The design of experiments followed strictly the protocol of real experiments. Randomly generated error was added to the synthetic data. The standard technique, a weighted non-linear regression analysis, was employed to re-estimate parameters of a model of two receptor sites and a model of negative co-operativity. As the original parameter values were known, the differences between original and estimated values was studied for (a) measurement error in the range from 0-17%, (b) random initial estimates and (c) error-free non-specific binding. In addition, analytical estimates of parameter precision were compared with the true between-experiment variation of parameter estimates. At the measurement error of 12%, a one site model is recommended to estimate the high affinity population of the two sites model. Plausible results can be expected in 90% of experiments, the between-experiment variation being approximately 30%. The model of two receptor sites gives approximately two thirds of plausible results. The high affinity population can be estimated with the between-experiment variation of 40%, the low affinity population is virtually unidentifiable with the between-experiment variation of approximately 100% and parameter estimates biased to higher values. Only half of the results obtained from the model of negative co-operativity are plausible, the variation in parameter estimates ranges from 90-150% and estimates are biased to higher values. At the level of 12% measurement error, random initial estimates do not significantly affect the estimation process, provided initial estimates are selected from a feasible range. At the same measurement error, the error-free non-specific binding does not improve the results, indicating that the mean of six replicates may be taken as a reliable estimate of non-specific binding. The analytical estimates of the coefficient of variation systematically underestimates the true between-experiments coefficient of variation, the difference has been found to be about 50%.     

单木生物量模型估计区域尺度生物量的不确定性

采用系统抽样体系江西省固定样地杉木连续观测数据和生物量数据,通过Monte Carlo法反复模拟由单木生物量模型推算区域尺度地上生物量的过程,估计了江西省杉木地上总生物量。基于不同水平建模样本量n及不同决定系数R~2的设计,分别研究了单木生物量模型参数变异性及模型残差变异性对区域尺度生物量估计不确定性的影响。研究结果表明:2009年江西省杉木地上生物量估计值为(19.84±1.27)t/hm~2,不确定性占生物量估计值约6.41%。生物量估计值和不确定性值达到平稳状态所需的运算时间随建模样本量及决定系数R~2的增大而减小;相对于模型参数变异性,残差变异性对不确定性的影响更小。     

Low-dose-rate extrapolation using the multistage model

The distribution of the maximum likelihood estimates of virtually safe levels of exposure to environmental chemicals is derived by using large-sample theory and Monte Carlo simulation according to the Armitage-Doll multistage model. Using historical dose-response we develop a set of 33 two-stage models upon which we base our conclusions. The large-sample distributions of the virtually safe dose are normal for cases in which the multistage-model parameters have nonzero expectation, and are skewed in other cases. The large-sample theory does not provide a good approximation of the distribution observed for small bioassays when Monte Carlo simulation is used. The constrained nature of the multistage-model parameters leads to bimodal distributions for small bioassays. The two modes are the direct result of estimating the linear parameter in the multistage model; the lower mode results from estimating this parameter to be nonzero, and the upper mode from estimating it to be zero. The results of this research emphasize the need for incorporation of the biological theory in the model-selection process.     

Parameter Estimation Methods for Chaotic Intercellular Networks

We have investigated simulation-based techniques for parameter estimation in chaotic intercellular networks. The proposed methodology combines a synchronization–based framework for parameter estimation in coupled chaotic systems with some state–of–the–art computational inference methods borrowed from the field of computational statistics. The first method is a stochastic optimization algorithm, known as accelerated random search method, and the other two techniques are based on approximate Bayesian computation. The latter is a general methodology for non–parametric inference that can be applied to practically any system of interest. The first method based on approximate Bayesian computation is a Markov Chain Monte Carlo scheme that generates a series of random parameter realizations for which a low synchronization error is guaranteed. We show that accurate parameter estimates can be obtained by averaging over these realizations. The second ABC–based technique is a Sequential Monte Carlo scheme. The algorithm generates a sequence of “populations”, i.e., sets of randomly generated parameter values, where the members of a certain population attain a synchronization error that is lesser than the error attained by members of the previous population. Again, we show that accurate estimates can be obtained by averaging over the parameter values in the last population of the sequence. We have analysed how effective these methods are from a computational perspective. For the numerical simulations we have considered a network that consists of two modified repressilators with identical parameters, coupled by the fast diffusion of the autoinducer across the cell membranes.     

Uncertainty analysis of forest carbon sink forecast with varying measurement errors: a data assimilation approach

Aims Accurate forecast of ecosystem states is critical for improving natural resource management and climate change mitigation. Assimilating observed data into models is an effective way to reduce uncertainties in ecological forecasting. However, influences of measurement errors on parameter estimation and forecasted state changes have not been carefully examined. This study analyzed the parameter identifiability of a process-based ecosystem carbon cycle model, the sensitivity of parameter estimates and model forecasts to the magnitudes of measurement errors and the information contributions of the assimilated data to model forecasts with a data assimilation approach.Methods We applied a Markov Chain Monte Carlo method to assimilate eight biometric data sets into the Terrestrial ECOsystem model. The data were the observations of foliage biomass, wood biomass, fine root biomass, microbial biomass, litter fall, litter, soil carbon and soil respiration, collected at the Duke Forest free-air CO₂ enrichment facilities from 1996 to 2005. Three levels of measurement errors were assigned to these data sets by halving and doubling their original standard deviations.Important findings Results showed that only less than half of the 30 parameters could be constrained, though the observations were extensive and the model was relatively simple. Higher measurement errors led to higher uncertainties in parameters estimates and forecasted carbon (C) pool sizes. The long-term predictions of the slow turnover pools were affected less by the measurement errors than those of fast turnover pools. Assimilated data contributed less information for the pools with long residence times in long-term forecasts. These results indicate the residence times of C pools played a key role in regulating propagation of errors from measurements to model forecasts in a data assimilation system. Improving the estimation of parameters of slow turnover C pools is the key to better forecast long-term ecosystem C dynamics.     

Sensitivity analysis of a biofilm model describing a one-stage completely autotrophic nitrogen removal (CANON) process.

A mathematical model for nitrification and anaerobic ammonium oxidation (ANAMMOX) processes in a single biofilm reactor (CANON) was developed. This model describes completely autotrophic conversion of ammonium to dinitrogen gas. Aerobic ammonium and nitrite oxidation were modeled together with ANAMMOX. The sensitivity of kinetic constants and biofilm and process parameters to the process performance was evaluated, and the total effluent concentrations were, in general, found to be insensitive to affinity constants. Increasing the amount of biomass by either increasing biofilm thickness and density or decreasing porosity had no significant influence on the total effluent concentrations, provided that a minimum total biomass was present in the reactor. The ANAMMOX process always occurred in the depth of the biofilm provided that the oxygen concentration was limiting. The optimal dissolved oxygen concentration level at which the maximum nitrogen removal occurred is related to a certain ammonium surface load on the biofilm. An ammonium surface load of 2 g N/m2. d, associated with a dissolved oxygen concentration level of 1.3 g O2/m3 in the bulk liquid and with a minimum biofilm depth of 1 mm seems a proper design condition for the one-stage ammonium removal process. Under this condition, the ammonium removal efficiency is 94% (82% for the total nitrogen removal efficiency) (30 degrees C). Better ammonium removal could be achieved with an increase in the dissolved oxygen concentration level, but this would strongly limit the ANAMMOX process and decrease total nitrogen removal. It can be concluded that a one-stage process is probably not optimal if a good nitrogen effluent is required. A two-stage process like the combined SHARON and ANAMMOX process would be advised for complete nitrogen removal.     

Degradation of toluene by a mixed population of archetypal aerobes, microaerophiles, and denitrifiers: laboratory sand column experiment and multispecies biofilm model formulation

An experiment was conducted in a saturated sand column with three bacterial strains that have different growth characteristics on toluene, Pseudomonas putida F1 which degrades toluene only under aerobic conditions, Thauera aromatica T1 which degrades toluene only under denitrifying conditions, and Ralstonia pickettii PKO1 has a facultative nature and can perform nitrate-enhanced biodegradation of toluene under hypoxic conditions (DO <2 mg/L). Steady-state concentration profiles showed that oxygen and nitrate appeared to be utilized simultaneously, regardless of the dissolved oxygen concentration and the results from fluorescent in-situ hybridization (FISH) indicated that PKO1 maintained stable cells numbers throughout the column, even when the pore water oxygen concentration was high. Since PKO1's growth rate under aerobic condition is much lower than that of F1, except under hypoxic conditions, these observations were not anticipated. Therefore these observations require a mechanistic explanation that can account for localized low oxygen concentrations under aerobic conditions. To simulate the observed dynamics, a multispecies biofilm model was implemented. This model formulation assumes the formation of a thin biofilm that is composed of the three bacterial strains. The individual strains grow in response to the substrate and electron acceptor flux from bulk fluid into the biofilm. The model was implemented such that internal changes in bacterial composition and substrate concentration can be simulated over time and space. The model simulations from oxic to denitrifying conditions compared well to the experimental profiles of the chemical species and the bacterial strains, indicating the importance of accounting for the biological activity of individual strains in biofilms that span different redox conditions.     

Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.