Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.     

Run-to-run fed-batch optimization for protein production using recombinant Escherichia coli

A two-phase design approach is introduced to determine the optimal feed rate, fed glucose concentration and fermentation time to maximize protein productivity using recombinant Escherichia coli BL21 (pBAW2) strain. The first phase is applied to determine a primary S-system kinetic model using batch time-series data. Two runs were carried out in the second phase to achieve the maximum protein productivity for the fed-batch fermentation process. The computational results using the S-system kinetic model obtained from the second run are in better agreement with the experiments than those using the kinetic model obtained from batch time-series data. For cross-validation, two extra fed-batch experiments with different feed strategies were carried out for comparison with the optimal fed-batch result. From the experimental results, this approach could improve productivity by at least 3%.     

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%.     

Use of 16S rRNA gene terminal restriction fragment analysis to assess the impact of solids retention time on the bacterial diversity of activated sludge

Terminal restriction fragment length polymorphism (T-RFLP) analysis of 16S rRNA genes was used to investigate the reproducibility and stability in the bacterial community structure of laboratory-scale sequencing batch bioreactors (SBR) and to assess the impact of solids retention time (SRT) on bacterial diversity. Two experiments were performed. In each experiment two sets of replicate SBRs were operated for a periods of three times the SRT. One set was operated at an SRT of 2 days and another set was operated at an SRT of 8 days. Samples for T-RFLP analysis were collected from the two sets of replicate reactors. HhaI, MspI, and RsaI T-RFLP profiles were analyzed using cluster analysis and diversity statistics. Cluster analysis with Ward's method using Jaccard distance and Hellinger distance showed that the bacterial community structure in both sets of reactors from both experimental runs was dynamic and that replicate reactors were clustered together and evolved similarly from startup. Richness (S), evenness (E), the Shannon-Weaver index (H), and the reciprocal of Simpson's index (1/D) were calculated, and the values were compared between the two sets of reactors. Evenness values were higher for reactors operated at an SRT of 2 days. Statistically significant differences in diversity (H and D) between the two sets of reactors were tested using a randomization procedure, and the results showed that reactors from both experimental runs that were operated at an SRT of 2 days had higher diversity (H and D) at the 5% level. T-RFLP analysis with diversity indices proved to be a powerful tool to analyze changes in the bacterial community diversity in response to changes in the operational parameters of activated-sludge systems.     

Use of 16S rRNA Gene Terminal Restriction Fragment Analysis To Assess the Impact of Solids Retention Time on the Bacterial Diversity of Activated Sludge

Terminal restriction fragment length polymorphism (T-RFLP) analysis of 16S rRNA genes was used to investigate the reproducibility and stability in the bacterial community structure of laboratory-scale sequencing batch bioreactors (SBR) and to assess the impact of solids retention time (SRT) on bacterial diversity. Two experiments were performed. In each experiment two sets of replicate SBRs were operated for a periods of three times the SRT. One set was operated at an SRT of 2 days and another set was operated at an SRT of 8 days. Samples for T-RFLP analysis were collected from the two sets of replicate reactors. HhaI, MspI, and RsaI T-RFLP profiles were analyzed using cluster analysis and diversity statistics. Cluster analysis with Ward's method using Jaccard distance and Hellinger distance showed that the bacterial community structure in both sets of reactors from both experimental runs was dynamic and that replicate reactors were clustered together and evolved similarly from startup. Richness (S), evenness (E), the Shannon-Weaver index (H), and the reciprocal of Simpson's index (1/D) were calculated, and the values were compared between the two sets of reactors. Evenness values were higher for reactors operated at an SRT of 2 days. Statistically significant differences in diversity (H and D) between the two sets of reactors were tested using a randomization procedure, and the results showed that reactors from both experimental runs that were operated at an SRT of 2 days had higher diversity (H and D) at the 5% level. T-RFLP analysis with diversity indices proved to be a powerful tool to analyze changes in the bacterial community diversity in response to changes in the operational parameters of activated-sludge systems.     

Biodesulphurization of mengen lignite by Rhodoccocus rhodochrous in a batch stirred and aerated tank fermenter

The biodesulphurization of Mengen lignite by a mesophilic bacterium, Rhodococcus rhodochrus ATCC 53968, was investigated in a batch stirred and aerated reactor. The experiments were carried out at 28°C with an inoculum percentage, initial pH, initial sodium acetate and lignite concentration of the biodesulphurization medium of 8% [v/v], 6.5 mM, 20 mM and 20 g/l, respectively. Variations in the sulphur contents of the lignite relative to the biodesulphurization period were monitored. The effects of the stirring and aeration rates on the removal of different sulphur forms from coal were investigated in the ranges 450–1,200 rpm and 0.1–0.53 vvm and the optimum values were found to be 500 rpm and 0.18 vvm, respectively. An increase in the total sulphur reduction with increasing biodesulphurization time was observed. The maximum total sulphur removal percentage was found to be 15.2% at 1,200 rpm after four days of incubation. The highest total sulphur removal rate was calculated on the second day of microbial desulphurization for each run. The total and organic sulphur contents of the coal after biodesulphurization were correlated with the stirring and aeration rates by using the non-linear least squares regression method. In the experimental runs lasting 8 days, the highest organic sulphur reducing percentage of 10.1% was obtained at a stirring rate of 500 rpm and an aeration rate of 0.40 vvm.     

An integrated framework for the inference of viral population history from reconstructed genealogies

We describe a unified set of methods for the inference of demographic history using genealogies reconstructed from gene sequence data. We introduce the skyline plot, a graphical, nonparametric estimate of demographic history. We discuss both maximum-likelihood parameter estimation and demographic hypothesis testing. Simulations are carried out to investigate the statistical properties of maximum-likelihood estimates of demographic parameters. The simulations reveal that (i) the performance of exponential growth model estimates is determined by a simple function of the true parameter values and (ii) under some conditions, estimates from reconstructed trees perform as well as estimates from perfect trees. We apply our methods to HIV-1 sequence data and find strong evidence that subtypes A and B have different demographic histories. We also provide the first (albeit tentative) genetic evidence for a recent decrease in the growth rate of subtype B.     

Model Testing in Radioligand/Receptor Interaction by Monte Carlo Simulation

Abstract

Monte Carlo simulations have been applied for evaluating the reliability of parameter estimates as well as for testing models in radioligand saturation binding experiments. Scatchard analysis was compared to the nonlinear least-square curve fitting method for one-site saturation binding curves. It was found that linear regression analysis from the transformed data in the Scatchard plot yielded generally less accurate parameter estimates than nonlinear regression analysis of untransformed data. The advantage of the nonlinear least-squares curve fitting method was especially pronounced in cases where the scatter and number of data points, as well as the radioligand concentration range, were chosen similar to less optimal experimental conditions. Under such circumstances, several K_D and B_max values derived by Scatchard analysis led to physically impossible negative values whereas the same data analyzed by nonlinear regression yielded reasonable parameter estimates. Furthermore, it was found that for both means of analysis, K_D and B_max correlated positively. In another set of Monte Carlo experiments, saturation binding curves involving two receptor sites were generated and subsequently analyzed according to both a one-site and a two-site model. The confidence with which one is able to distinguish the two-site model from nonlinear least-squares curve fitting was then estimated for optimal, as well as for, less ideal experimental condigions.     

2,3-Butanediol production by Enterobacter aerogenes: selection of the optimal conditions and application to food industry residues

Optimum values of temperature, pH, and starting substrate concentration are experimentally determined for 2,3-butanediol production by Enterobacter aerogenes through three set of batch fermentations of synthetic glucose solutions. The results of tests carried out at variable temperature show an optimum of 39 °C and are used to estimate, for both fermentation and thermal inactivation, the activation enthalpies (7.19 and 23.6 kJ mol^у) and the related entropies (т.32 and т.27 kJ mol^у K^у). An optimum pH value of 6.0 is evidenced from batch runs at variable pH, whose results are also used to make reasonable hypotheses on the reaction controlling the metabolic pathway which leads to butanediol. The fermentability of different food industry wastes, namely starch hydrolysate, both raw and decoloured molasses, and whey, is finally checked.     

The effect of carbon source on microbial community structure and Cr(VI) reduction rate

In the present work, the effect of the carbon source on microbial community structure in batch cultures derived from industrial sludge and hexavalent chromium reduction was studied. Experiments in aerobic batch reactors were carried out by amending industrial sludge with two different carbon sources: sodium acetate and sucrose. In each of the experiments performed, four different initial Cr(VI) concentrations of: 6, 13, 30 and 115 mg/L were tested. The change of carbon source in the batch reactor from sodium acetate to sucrose led to a 1.3–2.1 fold increase in chromium reduction rate and to a 5‐ to 9.5‐fold increase in biomass. Analysis of the microbial structure in the batch reactor showed that the dominant communities were bacterial species (Acinetobacter lwoffii, Defluvibacter lusatiensis, Pseudoxanthomonas japonensis, Mesorhizium chacoense, and Flavobacterium suncheonense) when sodium acetate was used as carbon source and fungal strains (Trichoderma viride and Pichia jadinii), when sodium acetate was replaced by sucrose. These results indicate that the carbon source is a key parameter for microbial dynamics and enhanced chromium reduction and should be taken into account for efficient bioreactor design. Biotechnol. Bioeng. 2010;107: 478–487. © 2010 Wiley Periodicals, Inc.     

Modeling kinetics of a large‐scale fed‐batch CHO cell culture by Markov chain Monte Carlo method

Markov chain Monte Carlo (MCMC) method was applied to model kinetics of a fed‐batch Chinese hamster ovary cell culture process in 5,000‐L bioreactors. The kinetic model consists of six differential equations, which describe dynamics of viable cell density and concentrations of glucose, glutamine, ammonia, lactate, and the antibody fusion protein B1 (B1). The kinetic model has 18 parameters, six of which were calculated from the cell culture data, whereas the other 12 were estimated from a training data set that comprised of seven cell culture runs using a MCMC method. The model was confirmed in two validation data sets that represented a perturbation of the cell culture condition. The agreement between the predicted and measured values of both validation data sets may indicate high reliability of the model estimates. The kinetic model uniquely incorporated the ammonia removal and the exponential function of B1 protein concentration. The model indicated that ammonia and lactate play critical roles in cell growth and that low concentrations of glucose (0.17 mM) and glutamine (0.09 mM) in the cell culture medium may help reduce ammonia and lactate production. The model demonstrated that 83% of the glucose consumed was used for cell maintenance during the late phase of the cell cultures, whereas the maintenance coefficient for glutamine was negligible. Finally, the kinetic model suggests that it is critical for B1 production to sustain a high number of viable cells. The MCMC methodology may be a useful tool for modeling kinetics of a fed‐batch mammalian cell culture process. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2010     

Robust parameter estimation during logistic modeling of batch and fed‐batch culture kinetics

Methods for robust logistic modeling of batch and fed‐batch mammalian cell cultures are presented in this study. Linearized forms of the logistic growth, logistic decline, and generalized logistic equation were derived to obtain initial estimates of the parameters by linear least squares. These initial estimates facilitated subsequent determination of refined values by nonlinear optimization using three different algorithms. Data from BHK, CHO, and hybridoma cells in batch or fed‐batch cultures at volumes ranging from 100 mL–300 L were tested with the above approach and solution convergence was obtained for all three nonlinear optimization approaches for all data sets. This result, despite the sensitivity of logistic equations to parameter variation because of their exponential nature, demonstrated that robust estimation of logistic parameters was possible by this combination of linearization followed by nonlinear optimization. The approach is relatively simple and can be implemented in a spreadsheet to robustly model mammalian cell culture batch or fed‐batch data. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature

Optimal experiment design for parameter estimation (OED/PE) has become a popular tool for efficient and accurate estimation of kinetic model parameters. When the kinetic model under study encloses multiple parameters, different optimization strategies can be constructed. The most straightforward approach is to estimate all parameters simultaneously from one optimal experiment (single OED/PE strategy). However, due to the complexity of the optimization problem or the stringent limitations on the system's dynamics, the experimental information can be limited and parameter estimation convergence problems can arise. As an alternative, we propose to reduce the optimization problem to a series of two-parameter estimation problems, i.e., an optimal experiment is designed for a combination of two parameters while presuming the other parameters known. Two different approaches can be followed: (i) all two-parameter optimal experiments are designed based on identical initial parameter estimates and parameters are estimated simultaneously from all resulting experimental data (global OED/PE strategy), and (ii) optimal experiments are calculated and implemented sequentially whereby the parameter values are updated intermediately (sequential OED/PE strategy).This work exploits OED/PE for the identification of the Cardinal Temperature Model with Inflection (CTMI) (Rosso et al., 1993). This kinetic model describes the effect of temperature on the microbial growth rate and encloses four parameters. The three OED/PE strategies are considered and the impact of the OED/PE design strategy on the accuracy of the CTMI parameter estimation is evaluated. Based on a simulation study, it is observed that the parameter values derived from the sequential approach deviate more from the true parameters than the single and global strategy estimates. The single and global OED/PE strategies are further compared based on experimental data obtained from design implementation in a bioreactor. Comparable estimates are obtained, but global OED/PE estimates are, in general, more accurate and reliable.     

Mathematical description of bikaverin production in a fluidized bed bioreactor

Summary  Growth of Gibberella fujikuroi in submerged cultures occurs as micelles or filamentous hyphae dispersed in fluid and pellets or stable, spherical agglomerations. Gibberella fujikuroi growth, substrate consumption and bikaverin production kinetics obtained from submerged batch fermentation were fitted to three different sigmoid models: two and three-parameter Gompertz models and one Logistic model. Growth fitting was used to compare between models and select the best one by means of an F test. The best model for describing growth was the two-parameter Gompertz model and was used for glucose consumption and bikaverin production fitting. Data from eight different schemes of fermentations were analysed and parameter estimation was carried out by means of minimization of residual sum of squares. Some characteristic values obtained with the two-parameter Gompertz model fit are: μ=0.028 h⁻¹, Y_x/s=0.1089 g substrate/g biomass, α =0.1384 g product/g biomass.     

Kinetic behavior of mixed populations of activated sludge

The kinetic behavior of heterogeneous microbial populations of sewage origin was studied in a single-stage isothermal continuous flow completely mixed aeration tank. A series of experiments were carried out at various dilution rates using glucose as the growth limiting substrate. The steady-state behavior of the system was observed at each dilution rate and the results were found to fit fairly well with the steady-state equation bayed on the Monod model with an endogenous respiration term included, i.e., μ = μ_mS/(K_s + S) ? K_d. The growth kinetics of cells harvested at steady state for each dilution rate were studied using batch experiments. The multiple response data of the system as functions of time were used to estimate the parameter values in the above kinetic model. It was found that values of the growth parameters changed significantly and systematically with cell population. For example, values of μ_m were high at high dilution rates and low at low dilution rates. It was also found that only those batch growth parameters from cells obtained at fairly high dilution rates are comparable with those estimated by the results of steady-state operations. The results of this investigation suggest that (1) different cell populations pre dominated at different steady-state dilution rates, with high dilution rates resulting in predominantly fast-growing organisms and low dilution rates resulting in predominantly slow-growing cells, and (2) risk exists in any randomly picked batch experiment to predict the steady-state behavior of the system when heterogeneous microbial populations must be used.     

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.     

Testing for Independence of Binary Responses

In the context of experiments involving visual inspection of random dot patterns the problem of testing the null hypothesis of independence of binary responses is considered. A flexible model for dependence between binary responses is proposed. Two tests, optimal under different versions of the model, are derived. These two tests turn out to involve the same computations as the Wilcoxon two sample test and the runs test respectively.     

Process model comparison and transferability across bioreactor scales and modes of operation for a mammalian cell bioprocess

A Monod kinetic model, logistic equation model, and statistical regression model were developed for a Chinese hamster ovary cell bioprocess operated under three different modes of operation (batch, bolus fed‐batch, and continuous fed‐batch) and grown on two different bioreactor scales (3 L bench‐top and 15 L pilot‐scale). The Monod kinetic model was developed for all modes of operation under study and predicted cell density, glucose glutamine, lactate, and ammonia concentrations well for the bioprocess. However, it was computationally demanding due to the large number of parameters necessary to produce a good model fit. The transferability of the Monod kinetic model structure and parameter set across bioreactor scales and modes of operation was investigated and a parameter sensitivity analysis performed. The experimentally determined parameters had the greatest influence on model performance. They changed with scale and mode of operation, but were easily calculated. The remaining parameters, which were fitted using a differential evolutionary algorithm, were not as crucial. Logistic equation and statistical regression models were investigated as alternatives to the Monod kinetic model. They were less computationally intensive to develop due to the absence of a large parameter set. However, modeling of the nutrient and metabolite concentrations proved to be troublesome due to the logistic equation model structure and the inability of both models to incorporate a feed. The complexity, computational load, and effort required for model development has to be balanced with the necessary level of model sophistication when choosing which model type to develop for a particular application. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2013     

Bias and Sampling Error of the Estimated Proportion of Genotypic Variance Explained by Quantitative Trait Loci Determined From Experimental Data in Maize Using Cross Validation and Validation With Independent Samples

Cross validation (CV) was used to analyze the effects of different environments and different genotypic samples on estimates of the proportion of genotypic variance explained by QTL (p). Testcrosses of 344 F(3) maize lines grown in four environments were evaluated for a number of agronomic traits. In each of 200 replicated CV runs, this data set was subdivided into an estimation set (ES) and various test sets (TS). ES were used to map QTL and estimate p for each run (p(ES)) and its median (p(ES)) across all runs. The bias of these estimates was assessed by comparison with the median (p(TS.ES)) obtained from TS. We also used two independent validation samples derived from the same cross for further comparison. The median p(ES) showed a large upward bias compared to p(TS.ES). Environmental sampling generally had a smaller effect on the bias of p(ES) than genotypic sampling or both factors simultaneously. In independent validation, p(TS.ES) was on average only 50% of p(ES). A wide range among p(ES) reflected a large sampling error of these estimates. QTL frequency distributions and comparison of estimated QTL effects indicated a low precision of QTL localization and an upward bias in the absolute values of estimated QTL effects from ES. CV with data from three QTL studies reported in the literature yielded similar results as those obtained with maize testcrosses. We therefore recommend CV for obtaining asymptotically unbiased estimates of p and consequently a realistic assessment of the prospects of MAS.