Exponential growth kinetics for Polyporus versicolor and Pleurotus ostreatus in submerged culture.

Simple mathematical models for a batch culture of pellet-forming fungi in submerged culture were tested on growth data for Polyporus versicolor (ATCC 12679) and Pleurotus ostreatus (ATCC 9415). A kinetic model based on a growth rate proportional to the two-thirds power of the cell mass was shown to be satisfactory. A model based on a growth rate directly proportional to the cell mass fitted the data equally well, however, and may be preferable because of mathematical simplicity.     

Mathematical model for aerobic culture of a recombinant yeast

A mathematical model was formulated to simulate cell growth, plasmid loss and recombinant protein production during the aerobic culture of a recombinant yeast S. cerevisiae. Model development was based on three simplified metabolic events in the yeast: glucose fermentation, glucose oxidation and ethanol oxidation. Cell growth was expressed as a composite of these metabolic events. Their contributions to the total specific growth rate depended on the activities of the pacemaker enzyme pools of the individual pathways. The pacemaker enzyme pools were regulated by the specific glucose uptake rate. The effect of substrate concentrations on the specific growth rate was described by a modified Monod equation. It was assumed that recombinant protein formation is only associated with oxidative pathways. Plasmid loss kinetics was formulated based on segregational instability during cell division by assuming constant probability of plasmid loss. Experiments on batch fermentation of recombinant S. cerevisiae C468/pGAC9 (ATCC 20690), which expresses Aspergillus awamori glucoamylase gene and secretes glucoamylase into the extracellular medium, were carried out in an airlift bioreactor in order to evaluate the proposed model. The model successfully predicted the dynamics of cell growth, glucose consumption, ethanol metabolism, glucoamylase production and plasmid instability. Excellent agreement between model simulations and our experimental data was achieved. Using published experimental data, model agreement was also found for other recombinant yeast strains. In general, the proposed model appears to be useful for the design, scale-up, control and optimization of recombinant yeast bioprocesses.     

Dynamic analysis of a microbial process: A systems engineering approach

A general mathematical model of the chemostat system is developed in order to define an experimental program of dynamic testing. A glucose-limited culture ofSaccharomyces cerevisiae was grown in a chemostat using chemically defined medium. The chemostat was perturbed from an initial steady state by changes in input glucose concentration, dilution rate, pH, and temperature. Dynamic responses of cell mass, glucose, cell number, RNA, and protein concentrations were measured. A number of simulation techniques were used in developing a dynamic mathematical model and in comparing the developed model with experimental data as well as the Monod model. The resulting model was found to be quantitatively accurate and superior to the Monod model. The developed model was interpreted in the light of cell physiology. Adjustment of intracellular RNA fraction was found to be rate limiting in acceleration of cell specific growth rate.     

Regulation of mammalian cell growth by autocrine growth factors: analysis of consequences for inoculum cell density effects

Effects of inoculum cell density on mammalian cell growth in culture have been observed in a variety of experimental systems. Although these effects have been attributed generally to medium conditioning by the cells, there has previously been no quantitative theory proposed for this phenomenon based on developments in molecular and cell biology. In this article, we offer such a theory founded on the regulatory action of autocrine growth factors. A particularly relevant example of these is platelet- derived growth factor (PDGF), which is produced by fibroblastic cells in response to stimulation by transforming growth factor beta (TGFbeta), a common serum constituent, and provides a mitogenic signal for the same cells. A simple mathematical model for the production, diffusive transport, and binding of autocrine growth factors to cell surface receptors, coupled to a model for the dependence of cell proliferation on growth factor receptor binding allows prediction of initial cell population growth rate as a function of inoculum cell density. We focus on situations involving anchorage-dependent cell growth, in which the cells are attached to a surface. A number of clear results are obtained, most notably the following: 1) for cells cultured on spherical microcarrier bead surfaces, the inoculum cell density needed to produce a given growth rate is linearly proportional to the bead radius; and 2) all other factors being equal, the inoculum cell density on a unit surface area basis needed to produce a given growth rate is greater for spherical microcarrier surfaces than for flat culture dish surfaces. These two results are consistent with the experimental observations of Hu and coworkers(1,2) for fibroblast growth in minimal medium plus serum. The model also allows elucidation of the influence of other system parameters, both biological and physical, on initial cell proliferation rate and the inoculum cell density dependence.     

Cell cycle model for growth rate and death rate in continuous suspension hybridoma cultures

As a result of recent advances in flow cytometry, renewed interest is shown in modeling the kinetic behavior of cells in culture on the basis of cell cycle parameters. An important but often overlooked kinetic variable in hybridoma cultures is the cell death rate. Not only the overall cell growth but also the kinetics of nutrient metabolism and monoclonal antibody production have been shown to depend on the cell death rate in continuous suspension hybridoma cultures. The present study shows that the death rate in hybridoma cultures is proportional to the fraction of cells arrested in the G(1) phase of the cell cycle. The steady-state cell age distributions in the various phases of the division cycle have been calculated analytically. A simple mathematical model has been used to produce the profiles of the cycling and arrested cell fractions with respect to the dilution rate. The calculated steady-state growth rate, death rate, and viability profiles are shown to be in agreement with recently published experimental data from continuous suspension hybridoma cultures. (c) 1992 John Wiley & Sons, Inc.     

Kinetic model for nitrogen-limited wine fermentations.

A physical and mathematical model for wine fermentation kinetics has been developed to predict sugar utilization curves based on experimental data from wine fermentations with various initial nitrogen and sugar concentrations in the juice. The model is based on: (1) yeast cell growth limited by nitrogen; (2) sugar utilization rates and ethanol production rates proportional solely to the number of viable cells; and (3) a death rate for cells proportional to alcohol content. All but one parameter in the model can be estimated from existing data. However, experiments to find this final parameter, a constant describing cell death, indicate that cell death may not be the critical factor in determining fermentation kinetics as cell viability remains significant until sugar utilization has ceased. The model, nevertheless, predicts a transition from normal to sluggish to stuck fermentations as initial nitrogen levels decrease. It also predicts that fermentations with high initial Brix levels may go to completion when supplemented with nitrogen in the form of ammonia. Therefore, we hypothesize that the model is valid but that ethanol causes the yeast cells to become inactive while remaining viable. Experimental verification of the model has been performed using flask-scale experiments. The model has also been used to evaluate the possibility of using nitrogen or viable cell additions to avoid or correct problem (i.e., sluggish or stuck) fermentations.     

Mathematical model of transitional processes in the chemostat culture of microorganisms

A mathematical model of the growth of the cell culture was developed. The model takes into account changes of the levels of the enzymes which define the metabolism rate, transport of the substrate into the cell, regeneration of the donors of energy. The model is based on the proposition that the rate of overall protein synthesis in the cell is defined by the concentration of a few aminoacids limiting the growth. The chemostat culture of the methanol-assimilating yeast was used as the object of modelling. The model allows to explain the experimental kinetics of alterations in cell number (biomass) and other measurable characteristics of the culture during the transient process when the dilution rate was changed.     

Generalization of monod kinetics for analysis of growth data with substrate inhibition

The inhibitory effect of butanol on yeast growth has been studied for the strain Candida utilis ATCC 8205 growing aerobically on butanol under batch conditions. A mathematical expression was then proposed to fit the kinetic pattern of butanol inhibition on the specific growth rate: \documentclass{article}\pagestyle{empty}\begin{document}$$ \mu = \frac{{\mu _m S}}{{K_s + S}}\left[{1 - \frac{S}{{S_m }}} \right];n $$\end{document}The maximum allowable butanol concentration above which cells do not grow was predicted to be 9.16g/L. The proposed model appears to accurately represent the experimental data obtained in this study and the literature data developed for a variety of batch culture systems at widely ranging substrate concentrations.     

Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications

Competition between two microbial populations for a single pollutant (phenol) was studied in a sequencing fed-batch reactor (SFBR). A mathematical model describing this system was developed and tested experimentally. It is based on specific growth rate expressions revealed from pure culture batch experiments. The species employed were Pseudomonas putida (ATCC 17514) and Pseudomonas resinovorans (ATCC 14235). It was found that both species biodegrade phenol following inhibitory kinetics which can be described by Andrews' expression. The model predicts that the dynamics of a SFBR, and the kinetics of biodegradation, result in a complex set of operating regimes in which neither species, only one species, or both species can survive at steady cycle. The model also predicts the existence of multiple outcomes, achievable from different start-up conditions, in some domains of the operating parameter space. Experimental results confirmed the model predictions. There was excellent agreement between predicted and measured concentrations of phenol, total biomass, and the biomass of each individual species. This study shows how serious discrepancies can arise in scale-up of biodegradation data if population dynamics are not taken into account. It also further confirms experimentally the theory of microbial competition in periodically forced bioreactors. (c) 1993 John Wiley & Sons, Inc.     

Kinetics of microbial growth on mixtures of pentachlorophenol and chlorinated aromatic compounds

Batch experiments were conducted to examine the effects of several substrate analogs on the degradation of pentachlorophenol by an enrichment culture of pentachlorophenol-utilizing bacteria. The presence of substrate analogs which were unable to serve as a carbon source for growth of the culture (e.g., 3,5,6,-trichloro-2-pyridinol, 2,4-dichlorophenoxyacetic acid) decreased the rate of pentachlorophenol degradation. The presence of a utilizable substrate analog (e.g., phenol, 2,4,5-trichlorophenol) also inhibited the initial rate of pentachlorophenol degradation; however, the overall removal rate was accelerated due to an increase in cell mass concentration as a result of simultaneous growth on both substrates. These effects were shown to be predicted by a mathematical model based on a modified Monod equation. Kinetic parameters obtained from the results of laboratory studies can be used for further process analysis to define the optimal conditions for the biological treatment of complex mixtures of phenolic compounds.     

红豆杉细胞悬浮培养结构化数学模型的探讨

用１０Ｌ机械搅拌式生物反应器悬浮培养红豆杉细胞,得到细胞生长、基质消耗和紫杉醇合成动力学曲线。经过代谢动力学分析建立了结构化数学模型。并将模型值与实验值进行比较,结果表明模型预测值与实验值较吻合。     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

A model of xylitol production by the yeast Candida mogii

Production of xylitol from xylose in batch fermentations of Candida mogii ATCC 18364 is discussed in the presence of glucose as the cosubstrate. Various initial ratios of glucose and xylose concentrations are assessed for their impact on yield and rate of production of xylitol. Supplementation with glucose at the beginning of the fermentation increased the specific growth rate, biomass yield and volumetric productivity of xylitol compared with fermentation that used xylose as the sole carbon source. A mathematical model is developed for eventual use in predicting the product formation rate and yield. The model parameters were estimated from experimental observations, using a genetic algorithm. Batch fermentations, which were carried out with xylose alone and a mixture of xylose and glucose, were used to validate the model. The model fitted well with the experimental data of cell growth, substrate consumption and xylitol production.     

In vitro ovine articular chondrocyte proliferation: experiments and modelling

This study focuses on analysis of in vitro cultures of chondrocytes from ovine articular cartilage. Isolated cells were seeded in Petri dishes, then expanded to confluence and phenotypically characterized by flow cytometry. The sigmoidal temporal profile of total counts was obtained by classic haemocytometry and corresponding cell size distributions were measured electronically using a Coulter Counter. A mathematical model recently proposed ( 1 ) was adopted for quantitative interpretation of these experimental data. The model is based on a 1‐D (that is, mass‐structured), single‐staged population balance approach capable of taking into account contact inhibition at confluence. The model’s parameters were determined by fitting measured total cell counts and size distributions. Model reliability was verified by predicting cell proliferation counts and corresponding size distributions at culture times longer than those used when tuning the model’s parameters. It was found that adoption of cell mass as the intrinsic characteristic of a growing chondrocyte population enables sigmoidal temporal profiles of total counts in the Petri dish, as well as cell size distributions at ‘balanced growth’, to be adequately predicted.     

Cell growth and catecholase production for Polyporus versicolor in submerged culture.

Cell growth and catecholase production for Polyporus veriscolor (ATCC 12679) were studied in mechanically agitated submerged culture, as functions of temperature. The exponential-phase growth rate exhibited a maximum at 28 degrees C. Over the range of 20 degrees C to approximately 30 degrees C, both cell mass and enzyme yield factors were constant. At higher temperatures (30 to 40 degrees C) cell mass yield factor decreased and enzyme yield factor increased. Specific respiration rate of P. versicolor was determined. Thermal deactivation of catecholase was investigated between 30 and 50 degrees C, and deactivation rates were fit to an Arrhenius rate expression.     

A comparison of mathematical model predictions to experimental measurements for growth and recombinant protein production in induced cultures of Escherichia coli

Recombinant cell growth and protein synthesis by a recombinant Escherichia coli under various inducing conditions are compared to the predictions of a mathematical model. The mathematical model used was a combination of two literature models: (1) an empirical kinetic model for recombinant growth and product formation and (2) a genetically structured model of the lac promoter-operator on a multicopy plasmid. The experimental system utilized was recombinant E. coli CSH22 bearing the temperature-sensitive plasmid pVH106/172, which codes for the synthesis of beta-galactosidase and the other lac operon genes under the control of a lac promoter. Mathematical model predictions for recombinant beta-galactosidase yield and specific growth rate were compared with fermentation measurements of these same quantities for conditions of chemical induction with cyclic AMP and IPTG, copy number amplification (by shifting culture temperature), and combined chemical induction and copy number amplification. The model successfully predicted experimental product yields for most cases of chemical induction even though the product yields varied from 0.34 x 10(3) to 1500 x 10(3) units/g cell mass. The kinetic model also correctly predicted a decline in the specific growth rate with increasing levels of plasmid and recombinant protein. The model was less successful at predicting product amplification at high copy numbers. A comparison of model predictions and experimental results was also used to investigate some of the assumptions used in constructing the mathematical models.     

Mathematical modeling for mixed culture growth of two bacterial populations with opposite substrate preferences

An unstructured mathematical model is proposed for mixed culture growth of two different bacterial species that exhibit "opposite" substrate preferences in response to the "same" environmental conditions. The model incorporates enzymatic control mechanisms such as induction, repression, and inhibition in the microorganisms as manifested in their preferential utilization of substrates and microbial interactions such as amensalism and competition. The model predicts cell mass, substrate concentrations, dissolved oxygen tension, as well as key enzyme levels. The predictions of the model are compared with experimental data for pure culture growth and for mixed culture growth on two substrates, glucose and citrate, in a batch reactor.     

A mathematical model for fluid shear-sensitive 3D tissue construct development

This research studies dynamic culture for 3D tissue construct development with computational fluid dynamics. It proposes a mathematical model to evaluate the impact of flow rates and flow shear stress on cell growth in 3D constructs under perfusion. The modeling results show that dynamic flow, even at flow rate as low as 0.002 cm/s, can support much better mass exchange, higher cell number, and more even cell and nutrient distribution compared to static culture. Higher flow rate can further improve nutrient supply and mass exchange in the construct, promoting better nutritious environment and cell proliferation compared to lower flow rate. In addition, consideration of flow shear stress predicts much higher cell number in the construct compared to that without shear consideration. While the nutrient can dominate shear stress in influencing cell proliferation, the shear effect increases with flow rate. The proposed model helps tissue engineers better understand the cell-flow relationship at the molecular level during dynamic culture.     

Determination of dissolved CO(2) concentration and CO(2) production rate of mammalian cell suspension culture based on off-gas measurement

The determination of dissolved CO(2) and HCO(3)(-) concentrations as well as the carbon dioxide production rate in mammalian cell suspension culture is attracting more and more attention since the effects on major cell properties, such as cell growth rate, product quality/production rate, intracellular pH and apoptosis, have been revealed. But the determination of these parameters by gas analysis is complicated by the solution/dissolution of carbon dioxide in the culture medium. This means that the carbon dioxide transfer rate (CTR; which can easily be calculated from off-gas measurement) is not necessarily equal to carbon dioxide production rate (CPR). In this paper, a mathematical method to utilize off-gas measurement and culture pH for cell suspension culture is presented. The method takes pH changes, buffer and medium characteristics that effect CO(2) mass transfer into account. These calculations, based on a profound set of equations, allow the determination of the respiratory activity of the cells, as well as the determination of dissolved CO(2), HCO(3)(-) and total dissolved carbonate. The method is illustrated by application to experimental data. The calculated dissolved CO(2) concentrations are compared with measurements from an electrochemical CO(2) probe.