A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates

A mathematical model is presented for both batch and continuous cultures of microorganisms utilizing inhibitory substrates. The key feature of the model is the use of a inhibition function to relate substrate concentration and specific growth rate. Simulation studies show that the primary result of inhibition by substrate in a batch culture is an increase in the lag time whereas in continuous culture inhibition by substrate may result in process instability. The model should be of value in investigations of the stability of biological processes used for the treatment of certain industrial wastes such as those containing phenols, thiocyanates, nitrates, ammonia, volatile acids, etc., which are known to be inhibitory to many of the organisms metabolizing them.     

Scale-up from shake flasks to fermenters in batch and continuous mode with Corynebacterium glutamicum on lactic acid based on oxygen transfer and pH

Scale-up from shake flasks to fermenters has been hampered by the lack of knowledge concerning the influence of operating conditions on mass transfer, hydromechanics, and power input. However, in recent years the properties of shake flasks have been described with empirical models. A practical scale-up strategy for everyday use is introduced for the scale-up of aerobic cultures from shake flasks to fermenters in batch and continuous mode. The strategy is based on empirical correlations of the volumetric mass transfer coefficient (k(L) a) and the pH. The accuracy of the empirical k(L) a correlations and the assumptions required to use these correlations for an arbitrary biological medium are discussed. To determine the optimal pH of the culture medium a simple laboratory method based on titration curves of the medium and a mechanistic pH model, which is solely based on the medium composition, is applied. The effectiveness of the scale-up strategy is demonstrated by comparing the behavior of Corynebacterium glutamicum on lactic acid in shake flasks and fermenters in batch and continuous mode. The maximum growth rate (micro(max) = 0.32 h(-1)) and the oxygen substrate coefficient (Y O2 /S= 0.0174 mol/l) of C. glutamicum on lactic acid were equal for shake flask, fermenter, batch, and continuous cultures. The biomass substrate yield was independent of the scale, but was lower in batch cultures (Y(X/S) = 0.36 g/g) than in continuous cultures (Y(X/S) = 0.45 g/g). The experimental data (biomass, respiration, pH) could be described with a simple biological model combined with a mechanistic pH model.     

Approximation of continuous fermentation by semicontinuous cultures

Semicontinuous fermentations, in which a fraction of a culture is replaced with fresh media at regular intervals, have been previously used as a means of approximating continuous growth. In most cases deviations from continuous operation were erroneously estimated using Fencl's model, which is only valid when the specific growth rate is independent of the substrate concentration. An approach to modeling Semicontinuous growth that incorporates the same kinetics followed in batch and continuous growth was developed and tested for Monod's expression for the specific growth rate. A dimensionless form of the model was used to simulate Semicontinuous fermentations for comparison to continuous growth. Differences between Semicontinuous and continuous growth were found to depend on three dimensionless variables: feed concentration, replacement rate, and time between replacements. For given values of the dimensionless feed concentration and time between replacements, a range of dimensionless replacement rates can be determined over which semi-continuous cultures are approximately continuous.     

A dynamic fed batch strategy for a Pichia pastoris mixed feed system to increase process understanding

Mixed substrate feeding strategies are frequently investigated to enhance the productivity of recombinant Pichia pastoris processes. For this purpose, numerous fed batch experiments or time-consuming continuous cultivations are required to optimize control parameters such as the substrate mixing ratio and the applied methanol concentration. In this study, we decoupled the feeding of methanol and glycerol in a mixed substrate fed batch environment to gain process understanding for a recombinant P. pastoris Muts strain producing the model enzyme horseradish peroxidase. Specific substrate uptake rates (qs) were controlled separately, and a stepwise increased qGly-control scheme was applied to investigate the effect of various substrate fluxes on the culture. The qs-controlled strategy allowed a parallel characterization of the metabolism and the recombinant protein expression in a fed batch environment. A critical-specific glycerol uptake rate was determined, where a decline of the specific productivity occurred, and a time-dependent acceleration of protein expression was characterized with the dynamic fed batch approach. Based on the observations on recombinant protein expression, propositions for an optimal feeding design to target maximal productivities were stated. Thus, the dynamic fed batch strategy was found to be a valuable tool for both process understanding and optimization of product formation for P. pastoris in a mixed substrate environment.     

Kinetics of Streptomyces baarnensis growth and antibiotic synthesis in batch and dialysis cultures

The kinetics of biomass and antibiotic formation in batch and dialysis culture of Streptomyces baarnensis at various initial concentrations limiting the substrate growth (glucose) has been studied. The antibiotic substances were synthesized by actively growing culture, its concentration in the cultural media was maximum in the log-phase. In continuous dialysis culture on the background of biomass lianer growth in the course of time the constant antibiotic concentration in the media proportional to the glucose input concentration has been established. The inactivation (decomposition) of antibiotic was immediately initiated after discontinuation of substrate supply and followed first kinetics order. Observed features were used for construction of kinetical model of antibiotic biosynthesis. A conclusion has been made that the dialysis culture gives opportunity for more effective antibiotic synthesis as compared with the batch one.     

A model for diauxic growth.

A model is developed to predict batch and continuous culture behavior of fermentations on two different carbon and energy sources. The basic assumption of the model is that the permease for the favored substrate is constitutive, whereas the premease for the second substrate is subject to induction and catabolite repression. Simulations of the model show features of diauxic growth described in the literature.     

Growth models of cultures with two lipid phases. I. Substrate dissolved in dispersed phase

Mathematical models which can be used to describe batch growth in fermentations with two liquid phases are developed for systems in which the growth limiting substrate is dissolved the dispersed liquid phase. In the models, the possibilities of growth occurring at the surface of the dispersed phase and in the continuous phase are considered. It is assumed that the composition of the dispersed phase is such that substrate utilization from it causes little of no change, in interfacial area. Three special cases are examined. In the first, it is assumed that all growth occurs at the surface of the dispersed phase. In the second and third, it is assumed that growth occurs both at the interface and in the continuous phase. The second case assumes that substrate equilibrium is continuously established between the two phases while the third assumes substrate consumption in the continuous phase is limited by rate of transport of substrate to that phase. Comparison of the first model with available experimental data shows good agreement between model and data.     

An energetic model for oxygen-limited metabolism

Microbial production of 2,3-butanediol by Klebsiella oxytoca occurs under conditions of an oxygen limitation. The extent to which substrate is oxidized to 2,3-butanediol and its coproducts, (acetic acid, acetoin, and ethanol) and the relative flow rates of substrate to energetic and biosynthetic pathways are controlled by the degree of oxygen limitation. Two energetic relationships which describe the response to an oxygen limitation have been derived. The first relationship describes the coupling between growth and energy production observed under oxygen-limited conditions. This allows calculation of energetic parameters and modeling of the cell mass and substrate profiles in terms of the degree of oxygen limitation only. The second relationship describes the average degree of oxidation and the rate of the end-product flow. The model has been tested with both batch and continuous culture. During these kinetic studies, two phases of growth have been observed: energy-coupled growth, which was described above; and, energy-uncoupled growth, which arises when the degree of oxygen limitation reaches a critical value. Optimal culture performance with respect to 2,3-butanediol productivity occurs during energy-coupled growth. (c) 1993 John Wiley & Sons, Inc.     

The influence of high substrate concentrations on microbial kinetics

High substrate concentrations inhibit growth and may distort the metabolism of microorganisms. Mechanisms causing substrate inhibition are discussed and used to derive several mathematical models representative of the entire concentration range, including stimulation of growth by low substrate concentrations. These kinetic models are tested with a variety of batch culture measurements of specific growth rate and respiration rate at widely-ranging substrate concentrations. Using one of the kinetic models, equations are developed for batch, continuous, and exponential-feed reactors. Comparison of results obtained in continuous culture with results from exponential-feed culture systems is shown to offer a novel experimental method for evaluating the effect of the cell age distribution on the properties and metabolic activity of a culture.     

Continuos IBE fermentation by immobilized growing Clostridium beijerinckii cells in a stirred-tank fermentor

The potential of continuous isopropanol-butanol-ethanol (IBE) fermentation by Ca-alginate-immobilized Clostridium beijerinckii cells in a continuous stirred-tank reactor is investigated. A mathematical model is presented to describe steady-state reactor performance. It appeared to be possible to use the biocatalyst particles repeatedly for successive fermentations (at least three times for a total duration of two months). Reactor productivity was 6-16 times higher than that of a batch fermentation (free cells), while the solvents yield was also increased. Measurements of substrate, product, and biomass concentrations were only partially in agreement with the model; however, a solid basis for further technological developement of the process has been laid.     

Dynamic modeling and simulation of continuous airlift bioreactors

For dynamic behaviors of continuous airlift bioreactors, a mathematical model based on a tanks-in-series model with backflow has been developed. The equations describing the dynamics of airlift bioreactors are material balances for micro-organism, substrate, dissolved oxygen and oxygen in gas-phase and heat balances. Non-ideal mixing of liquid and gas phases is taken into account using a tanks-in-series model with backflow. The batch operation, startup operation and the consequence of plant failure were simulated and the effects of design and operating parameters for an airlift bioreactor on its dynamic behaviors were discussed. The concentration profiles of micro-organism, substrate, dissolved oxygen and oxygen in gas-phase and the temperature profile in an airlift bioreactors and their dynamics were obtained. The computational results indicate that the transients of a chemostat in the case of bubble column bioreactor are slower compared with those in the case of airlift bioreactor. The proposed simulator is more precise as compared with models published previously in the literature and therefore provides more reliable and rational examination of continuous airlift bioreactor performance.     

Kinetic studies of the lactic acid fermentation in batch and continuous cultures

The fermentation kinetics of the homofermentative organism Lactobacillus delbrueckii in a glucose-yeast extract medium is studied in both batch and continuous culture under conditions of controlled pH. From a graphical analysis of the batch data, a mathematical model of the process is derived which relates bacterial growth, glucose utilization, and lactic acid formation. The parameters in the model represent the activity of the organism and are a function of pH, having a maximum value at about 5.90. In a continuous stirred tank fermentor (CSTF), the effect of pH, feed concentration, and residence time is observed. The feed medium is a constant ratio of two parts glucose to one part yeast extract plus added mineral salts. An approximate prediction of the steady-state behavior of the CSTF can be made using a method based on the kinetic model derived for the batch case. In making step changes from one steady state to another, the transient response is observed. Using the kinetic model to simulate the transient period, the calculated behavior qualitatively predicts the observed response.     

Gene Regulation in Continuous Cultures: A Unified Theory for Bacteria and Yeasts

During batch growth on mixtures of two growth-limiting substrates, microbes consume the substrates either sequentially (diauxie) or simultaneously. The ubiquity of these growth patterns suggests that they may be driven by a universal mechanism common to all microbial species. Recently, we showed that a minimal model accounting only for enzyme induction and dilution, the two processes that occur in all microbes, explains the phenotypes observed in batch cultures of various wild-type and mutant/recombinant cells (Narang and Pilyugin in J. Theor. Biol. 244:326–348, 2007). Here, we examine the extension of the minimal model to continuous cultures. We show that: (1) Several enzymatic trends, attributed entirely to cross-regulatory mechanisms, such as catabolite repression and inducer exclusion, can be quantitatively explained by enzyme dilution. (2) The bifurcation diagram of the minimal model for continuous cultures, which classifies the substrate consumption pattern at any given dilution rate and feed concentrations, provides a precise explanation for the empirically observed correlations between the growth patterns in batch and continuous cultures. (3) Numerical simulations of the model are in excellent agreement with the data. The model captures the variation of the steady state substrate concentrations, cell densities, and enzyme levels during the single- and mixed-substrate growth of bacteria and yeasts at various dilution rates and feed concentrations. This variation is well approximated by simple analytical expressions that furnish deep physical insights. (4) Since the minimal model describes the behavior of the cells in the absence of cross-regulatory mechanisms, it provides a rigorous framework for quantifying the effect of these mechanisms. We illustrate this by analyzing several data sets from the literature.     

Growth models of cultures with two liquid phases. III. Continuous cultures

Some mathematical models, which have been used to describe batch growth in fermentations with two liquid phases present, are used to predict the behavior of continuous fermentations in a chemostat. Two types of dispersed systems are considered in this investigation. In the first, type, it is assumed that the composition of the dispersed phase is such that, increased substrate utilization results in a decreased substrate concentration with no change in the interfacial area. In the second type of system, the dispersed phase is assumed to be pure substrate; therefore, the substrate concentration in the dispersed phase remains constant but the interfacial area is affected by changes in dilution rate. Three special cases are examined for each type of system in order to examine the effect of the interfacial area, the phase equilibrium constant, and the mass transfer coefficient on system performance. Comparison of two of the models with available experimental data shows fair agreement, between model and data.     

Application of a competition model to the growth of Streptococcus mutans and Streptococcus sanguis in binary continuous culture.

Streptococcus mutans 6715-15 and Streptococcus sanguis 10558 were grown together in continuous culture with glucose as the limiting carbon source. The relationship of growth rate to substrate concentration was determined for pure cultures of each organism in continuous and batch cultures. A model based on competition for a growth-limiting substrate (glucose) was used to predict the proportions of each organism when grown in binary cultures. The results indicate that interactions other than competition for glucose carbon exist between S. mutans and S. sanguis grown under these conditions.     

Application of a competition model to the growth of Streptococcus mutans and Streptococcus sanguis in binary continuous culture

Streptococcus mutans 6715-15 and Streptococcus sanguis 10558 were grown together in continuous culture with glucose as the limiting carbon source. The relationship of growth rate to substrate concentration was determined for pure cultures of each organism in continuous and batch cultures. A model based on competition for a growth-limiting substrate (glucose) was used to predict the proportions of each organism when grown in binary cultures. The results indicate that interactions other than competition for glucose carbon exist between S. mutans and S. sanguis grown under these conditions.     

Enzymatic synthesis of a series of alkyl esters using novozyme 435 in a packed-bed,miniaturized, continuous flow reactor

Using lipase catalysed, enzymatic esterification as a model reaction, we successfully demonstrate the use of miniaturized technology for biocatalytic reactions. Benchmarked against batch reactions, nine alkyl esters have been synthesized effectively, using Novozyme 435 in a pressure driven, packed-bed, miniaturized, continuous flow reactor. In some cases close to 100% ester conversions were obtained. The paper also demonstrates the ability to screen the enzyme for substrate specificity.     

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     

Fermentation of Cellulosic Substrates in Batch and Continuous Culture by Clostridium thermocellum

Fermentation of dilute-acid-pretreated mixed hardwood and Avicel by Clostridium thermocellum was compared in batch and continuous cultures. Maximum specific growth rates per hour obtained on cellulosic substrates were 0.1 in batch culture and >0.13 in continuous culture. Cell yields (grams of cells per gram of substrate) in batch culture were 0.17 for pretreated wood and 0.15 for Avicel. Ethanol and acetate were the main products observed under all conditions. Ethanol:acetate ratios (in grams) were approximately 1.8:1 in batch culture and generally slightly less than 1:1 in continuous culture. Utilization of cellulosic substrates was essentially complete in batch culture. A prolonged lag phase was initially observed in batch culture on pretreated wood; the length of the lag phase could be shortened by addition of cell-free spent medium. In continuous culture with ~5 g of glucose equivalent per liter in the feed, substrate conversion relative to theoretical ranged from 0.86 at a dilution rate (D) of 0.05/h to 0.48 at a D of 0.167/h for Avicel and from 0.75 at a D of 0.05/h to 0.43 at a D of 0.11/h for pretreated wood. At feed concentrations of <4.5 g of glucose equivalent per liter, conversion of pretreated wood was 80 to 90% at D = 0.083/h. Lower conversion was obtained at higher feed substrate concentrations, consistent with a limiting factor other than cellulose. Free Avicelase activities of 12 to 84 mU/ml were observed, with activity increasing in this order: batch cellobiose, batch pretreated wood < batch Avicel, continuous pretreated wood < continuous Avicel. Free cellulase activity was higher at increasing extents of substrate utilization for both pretreated wood and Avicel under all conditions tested. The results indicate that fermentation parameters, with the exception of free cellulase activity, are essentially the same for pretreated mixed hardwood and Avicel under a variety of conditions. Hydrolysis yields obtained with C. thermocellum cellulase acting either in vitro or in vivo were comparable to those previously reported for Trichoderma reesei on the same substrates.