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     

On-line measurement of culture fluorescence during cultivation of Methylomonas mucosa

Summary This paper reports on the measurement of NAD(P)H dependent culture fluorescence during the batch culture of Methylomonas mucosa, a potential polysaccharide producer. The results obtained reveal that there is a high correlation between the fluorescence level and the cell concentration. The logistic equation was also found to be applicable for representing the biomass concentration data during the course of fermentation. A model was then presented for predicting both the biomass concentration and the culture fluorescence with high fidelity.     

Experimental design for parameter estimation from batch culture

An experimental design to estimate the parameters in a Monod-type equation from batch culture data was examined. Consideration was given to the design of experiments to estimate accurate values of the parameters. Sequential experimental design with the information index was used for this purpose. With this approach the standard deviation of the parameter values was reduced using simulated batch culture data.     

Quantitative patterns of development of community of <Emphasis Type="Italic">Pseudomonas aeruginosa</Emphasis> dissociants

The stationary phase of batch culture of Pseudomonas aeruginosa dissociants has been described by a variational model of consumption and growth. The generalized entropy functional was used as the objective function. The model parameters include the requirements of the dissociants for the main nutrients: carbon, nitrogen, and phosphorus. The variational model was used to calculate the limiting regions and microbial community composition during stationary growth for different initial combinations of the resources as a function of the limiting resources. A correspondence between the experimental data and model calculations has been demonstrated. A possibility to control the community structure is discussed.     

Unstructured models for growth and lactic acid production during two-stage continuous cultures of Lactobacillus helveticus

During lactic acid fermentation, seed culture is usually carried out without pH control, while culture is carried out at pH controlled at the optimal value to overcome inhibitory effects. The Luedeking–Piret expression was therefore previously modified by introducing additional terms involving the undissociated form of the lactic acid, the main inhibitory species, in case of batch cultures without pH control or involving the residual lactose concentration to account for the carbon substrate limitation, responsible for cessation of production during batch cultures of Lactobacillus helveticus at controlled pH. Both expressions were also merged to deduce a generalized model. Both models, as well as the Luedeking–Piret model, were developed to describe continuous two-stage culture of L. helveticus. By considering the parameter values obtained from the fitting of batch culture data, both modified Luedeking–Piret models showed interesting predictive potential. Indeed, some rather reliable predictive calculated values were recorded in both stages; the residual standard deviations were 0.5 and less than 8.8 for the biomass and the product concentrations at steady-state in the culture stage (second stage). The optimization of the parameters for growth- and non-growth-associated parameters improved the fitting in the culture stage, leading to residual standard deviations below 2.6 for lactic acid concentrations at steady-state.     

The monod equation: A revisit and a generalization to product inhibition situations

This paper shows how to treat the substrate-limiting Monod equation in a straight forward manner for different types of fermentors (plug-flow, batch, and mixed-flow) using the general language of chemical reaction engineering. Straight-line plots are developed for directly finding the kinetic constants of the equation, and an example using Monod's original data illustrates the procedure. The Monod equation is then generalized to account for the effects of both substrate and inhibitory toxic wastes. Finally, for pure product inhibition performance, expressions are derived for various reactor types, and correlation graphs are developed for finding the kinetic constants of the reaction. An example from the recent literature shows that this equation form fits the data extremely well.     

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures was examined, using on-line off-gas analyses to measure the oxygen and carbon dioxide consumption rates continuously. A cell suspension from continuous cultures at steady state was used as the inoculum. It was observed that a dynamic phase occurred in the initial phase of the experiment. In this phase the bacterial ferrous iron oxidation and growth were uncoupled. After about 16 h the bacteria were adapted and achieved a pseudo-steady state, in which the specific growth rate and oxygen consumption rate were coupled and their relationship was described by the Pirt equation. In pseudo-steady state, the growth and oxidation kinetics were accurately described by the rate equation for competitive product inhibition. Bacterial substrate consumption is regarded as the primary process, which is described by the equation for competitive product inhibition. Subsequently the kinetic equation for the specific growth rate, μ, is derived by applying the Pirt equation for bacterial substrate consumption and growth. The maximum specific growth rate, μ _max, measured in the batch culture agrees with the dilution rate at which washout occurs in continuous cultures. The maximum oxygen consumption rate, q _O2,max, of the cell suspension in the batch culture was determined by respiration measurements in a biological oxygen monitor at excess ferrous iron, and showed changes of up to 20% during the course of the experiment. The kinetic constants determined in the batch culture slightly differ from those in continuous cultures, such that, at equal ferric to ferrous iron concentration ratios, biomass-specific rates are up to 1.3 times higher in continuous cultures. Received: 8 February 1999 / Accepted: 17 February 1999     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.     

Gamma-interferon production and quality in stoichiometric fed-batch cultures of Chinese hamster ovary (CHO) cells under serum-free conditions

The application of a stoichiometric medium design approach was studied in fed-batch cultivation of Chinese hamster ovary (CHO) cells. A serum-free medium containing a very low protein concentration (2 mg/L insulin) was developed. A supplemental medium was formulated according to the stoichiometric equation governing cell growth using cell composition obtained from hybridoma cells. Fed-batch culture was conducted in spinner flasks using the supplemental medium for feeding. Significant improvement in cell growth, by-product reduction, and Gamma-Interferon (IFN-gamma) production was achieved as compared to a typical batch culture. Results indicate that the stoichiometric approach, originally developed for hybridoma cultures, is a fast and effective method for cell culture process design and improvement. The glycosylation of IFN-gamma was monitored off-line during the culture process. The accumulative IFN-gamma glycosylation efficiency was slightly improved as compared to that of the batch culture, due to the nutritional control through the stoichiometric feeding. Periodic glucose starvation was observed during the fed-batch culture as a result of the manual feeding. Pulse-chase radiolabeling assay shows that glucose starvation leads to a deteriorated IFN-gamma glycosylation efficiency. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 577-582, 1997.     

Continuous hydrocarbon fermentation with colloidal emulsion feed. A kinetic model for two-liquid phase culture

Candida tropicalis was cultured in a chemostat-type fermentor with n-hexadecane, dispersed in water as submicron droplets, as the only carbon substrate. The emulsion as well as the aqueous medium were fed continuously into the fermentor. A Monod-type equation can correlate the specific group rate in the continuous fermentor with the concentration of submicron droplets. The same equation can also be fitted to the data for the conventional-type batch culture in the same fermentor in which an oil phase as well as an aqueous phase existed, if the hydrocarbon concentration in the aqueous phase excluding oil drops is employed as the substrate concentration. This demonstrates that Candida tropicalis takes up only submicron droplets of n-hexadecane as the carbon substrate.     

Fed-batch cultivation of animal cells using different medium design concepts and feeding strategies

In animal cell cultivation, cell density and product concentration are often low due to the accumulation of toxic end-products such as ammonia and lactate and/or the depletion of essential nutrients. A hybridoma cell line (CRL-1606) was cultivated in T-flasks using a newly devised medium feeding strategy. The goals were to decrease ammonia and lactate formation by the design of an initial medium which would provide a starting environment to achieve optimal cell growth. This was followed by using a stoichiometric equation governing animal cell growth and then designing a supplemental medium for feeding strategy used to control the nutritional environment. The relationship between the stoichiometric demands for glutamine and nonessential amino acids was also studied. Through stoichiometric feeding, nutrient concentrations were controlled reasonably well. Consequently, the specific production rate of lactate was decreased by fourfold compared with conventional fed-batch culture and by 26-fold compared with conventional batch culture. The specific production rate of ammonia was decreased by tenfold compared with conventional fed-batch culture and by 50-fold compared with conventional batch culture. Most importantly, total cell density and monoclonal antibody concentration were increased by five- and tenfold respectively, compared with conventional batch culture. (c) 1994 John Wiley & Sons, Inc.     

Kinetic analysis of batch and continuous culture of Streptococcus cremoris HP1.

The growth of Streptococcus cremoris on a semidefined medium was studied at initial lactose concentrations of 0.2-5.0% in batch culture, and in lactose-limited chemostat cultures at 0.5% lactose. Kinetic analysis of the batch data, using statisitcal techniques, indicated the importance of lactose limitation and lactic acid inhibition of the growth of S. cremoris. A model for the biomass production, lactose utilization, and lactic acid production in batch culture was proposed. In continuous culture, it was found that steady state populations were maintained at higher dilution rates (D = 0.6-0.7 h-1) than the maximum predicted by batch culture (0.56h-1). No evidence for a selection of fast growing mutants was obtained. Copious growth adhering to the walls of the fermentor (i.e. wall growth) occurred very rapidly at higher dilution rates and this undoubtedly affected steady-state growth and wash-out and, as a consequence, the apparent maximum dilution rate.     

A general model for aerobic yeast growth: batch growth

A general model for aerobic yeast growth in batch culture is presented. It is based on the concept that the aerobic metabolism of all yeasts is determined by the relative sizes of the transport rate of sugar into the cell and the transport rate of respiratory intermediates into the mitochondrion. If the rate of sugar uptake rate exceeds the rate of transport of respiratory intermediates into the mitochondrion (as in Saccharomyces cerevisiae, S. uvarum, and S. pombe), the metabolism exhibits the features of ethanol excretion and limited specific oxygen uptake rate. If the rate of transport of respiratory intermediates into the mitochondrion is of the same order as the transport of sugar into the cell (as in Candida utilis), the metabolism is characterized by little or no ethanol excretion and a much higher specific oxygen uptake rate. Batch data from an extensive range of yeast and carbon sources is used to illustrate the use of this model. The ability of this model to fit such an extensive range of experimental data suggests that it can be used as a generalized model for aerobic yeast growth.     

Mathematical model of a batch acetone-butanol fermentation

A mathematical model for the batch culture of Clostridium acetobutylicum was formulated using experimental data for anaerobic solvent production. The model summarizes biochemical as well as physiological aspects of growth and metabolite synthesis by the production strain. The key fermentation rates are expressed and evaluated with regard to substrate consumption and butanol end-product inhibitory effects. Parametric sensitivity analysis of the batch process model was carried out, indicating the importance of the key process parameters.     

Catabolic control of hybridoma cells by glucose and glutamine limited fed batch cultures

Substrate limited fed batch cultures were used to study growth and overflow metabolism in hybridoma cells. A glucose limited fed batch, a glutamine limited fed batch, and a combined glucose and glutamine limited red batch culture were compared with batch cultures. In all cultures mu reaches its maximum early during growth and decreases thereafter so that no exponential growth and decreases thereafter so that no exponential growth rate limiting, although the glutamine concentration (>0.085mM) was lower than reported K(s) vales and glucose was below 0.9mM; but some other nutrients (s) was the cause as verified by simulations. Slightly more cells and antibodies were produced in the combined fed batch compared with the batch culture. The specific rates for consumption of glucose and glutamine were dramatically influenced in fed batch cultures resulting in major metabolic changes. Glucose limitation decreased lactate formation, but increased glutamine consumption and ammonium formation. Glutamine limitation decreased ammonium and alanine formation of lactate, alanine, and ammonium was negligible in the dual-substrate limited fed batch culture. The efficiency of the energy metabolism increased, as judged by the increase in the cellular yield coefficient for glucose by 100% and for glutamine by 150% and by the change in the metabolic ratios lac/glc, ala/ln, and NH(x)/ln, in the combined fed culture. The data indicate that a larger proportion of consumed glutamine enters the TCA cycle through the glutamate dehydrogenase pathway, which releases more energy from glutamine than the transamination pathway. We suggest that the main reasons for these changes are decreased uptake rates of glucose and glutamine, which in turn lead to a reduction of the pyruvate pool and a restriction of the flux through glutaminase and lactate dehydrogenase. There appears to be potential for further cell growth in the dual-substrate-limited fed batch culture as judged by a comparison of mu in the different cultures. (c) 1994 John Wiley & Sons, Inc.     

模拟青霉素发酵过程中菌体生长动态的细胞自动机模型

在青霉素发酵生产机理及其动力学微分方程模型的基础上,建立了模拟青霉素分批发酵过程中菌体生长动态的细胞自动机模型(CABGM)。CABGM采用三维细胞自动机作为菌体生长空间,采用Moore型邻域作为细胞邻域,其演化规则根据青霉素分批发酵过程中菌体生长机理和动力学微分方程模型设计。CABGM中的每一个细胞既可代表单个的青霉素产生菌,又可代表特定数量的青霉素产生菌,它具有不同的状态。对CABGM进行了统计特性的理论分析和仿真实验,理论分析和仿真实验结果均证明了CABGM能一致地复现动力学微分方程模型所描述的青霉素分批发酵菌体生长过程。最后,对所建模型在实际生产过程中的应用问题进行了分析,指出了需要进一步研究的问题。     

Kinetic studies of mold α-galactosidase on PNPG hydrolysis

The kinetic properties of α-galactosidase of Mortierella vinacea were investigated in detail using PNPG (p-nitrophenyl-α-D -galactopyranoside) as a substrate. Consequently, the enzyme was markedly inhibited not only by the substrate, but also by the galactose hydrolized. The initial rate of reaction at sufficiently high substrate concentrations, however, did not fall to zero and did approach a finite value. Galactose behaved as a mixed inhibitor and was neither totally competitive nor totally noncompetitive. A rate equation was obtained from a generalized equation derived from a kinetic model which took both the inhibitions into consideration. The constants used in the equation were appropriately estimated. The calculated rate agreed fairly well with the observed initial rate. Moreover, the PNPG hydrolysis progressing in a batch system was found to be approximately representable by simple first order kinetics in which the rate constant was dependent on the initial substrate concentration.     

Computer programs for modeling mammalian cell batch and fed-batch cultures using logistic equations

A MATLAB^® toolbox was developed for applying the logistic modeling approach to mammalian cell batch and fed-batch cultures. The programs in the toolbox encompass sensitivity analyses and simulations of the logistic equations in addition to cell specific rate estimation. The toolbox was first used to generate time courses of the sensitivity equations for characterizing the relationship between the logistic variable and the model parameters. Subsequently, the toolbox was used to describe CHO cell data from batch and fed-batch mammalian cell cultures. Cell density, product, glucose, lactate, glutamine, and ammonia data were analyzed for the batch culture while fed-batch analysis included cell density and product concentration. In all instances, experimental data were well described by the logistic equations and the resulting specific rate profiles were representative of the underlying cell physiology. The 6-variable batch culture data set was also used to compare the logistic specific rates with those from polynomial fitting and discrete derivative methods. The polynomial specific rates grossly misrepresented cell behavior in the initial and final stages of culture while those based on discrete derivatives had high variability due to computational artifacts. The utility of logistic specific rates to guide process development activities was demonstrated using specific protein productivity versus growth rate trajectories for the 3 cultures examined in this study. Overall, the computer programs developed in this study enable rapid and robust analysis of data from mammalian cell batch and fed-batch cultures which can help process development and metabolic flux estimation.     

Resolution of batch variations in pyrolysis mass spectrometry of bacteria by the use of artificial neural network analysis

A simple, but stringent, three group model of bacterial interstrain identity (two cultures of the same strain ofEscherichia coli) and difference (a culture of a serologically distinct strain) was used in multiple serial weekly subcultures for five weeks to demonstrate the effect of both growth-related (phenotypic) and machine-related variation on pyrolysis mass spectra. An aliquot of serum from a single sample was included in each pyrolysis batch to distinguish machine drift from culture drift. Conventional principal component (PC) canonical variate (CV) analysis was successful within each pyrolysis batch but the variations between batches precluded the use of data from more than one batch in successful PCCV analysis. In contrast, artificial neural networks (ANNs) trained with data from one batch could be successfully used to identify groups in data from non-contemporaneous pyrolysis batches. Although the ANN method will require validation in more complex settings than this simple model, it is a promising approach to the problem of batch constraint in pyrolysis mass spectrometry.