STEADY STATE GROWTH OF PHYTOPLANKTON IN CONTINUOUS CULTURE: COMPARISON OF INTERNAL AND EXTERNAL NUTRIENT EQUATIONS1,2

Previously, there have been conflicts over whether external or internal nutrient concentrations control phytoplankton growth rates at steady state in continuous culture. To experimentally demonstrate that both equations equally describe steady state growth, continuous culture studies with phosphorus-limited growth of the chrysophyte Monochrysis lutheri Droop were carried out over the entire growth rate region up to biomass washout. Data were examined using both the Monod and Droop equations, and, even though there were significant variations in the yield coefficient with growth rate, the data fit both equations reasonably well. Because of their relative simplicity, the Droop equation and an equation combining both the Monod and Droop equations are better suited for expressing kinetic data than the Monod equation. It is crucial, though, that the criteria necessary to achieve steady state be fulfilled.     

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     

The application of growth kinetics to the control of Eichhornia crassipes (Mart) solms. Through nutrient removal by mechanical harvesting

The growth of Eichhornia crassipes (water hyacinth) under nitrate-nitrogen limiting conditions can be explained by the Monod rectangular model. The kinetic constants maximum specific growth rate, half saturation constant and yield coefficient were determined under nitrate-nitrogen limiting conditions in nutrient culture at an air temperature of 25°C. The practical application of these constants for the possible control of water hyacinth through nutrient limitation is illustrated.     

Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

嗜酸氧化亚铁硫杆菌生长动力学

在确定二价铁离子为A.f生长过程中惟一限制性底物条件下,通过考察初始亚铁离子浓度、初始pH值两种影响亚铁离子氧化代谢的主要因素来研究细菌的生长特性,得到以限制性底物亚铁离子浓度为表征的细菌生长曲线。利用基于Monod方程建立的细菌生长动力学方程模型,采用Matlab软件中的Gauss-Newton算法确定了在不同条件下细菌生长动力学参数,包括最大比生长速率μm、Monod常数K及Ro,推导出了不同条件下A.f对数期以底物Fe(Ⅱ)浓度为表征的生长动力学方程。     

Kinetics and modelling of cell growth and substrate uptake inCentella asiatica cell culture

In this study, we have conducted kinetics and modelling studies ofCentella asiatica cell growth and substrate uptake, in an attempt to evaluate cell growth for a better understanding and control of the process. In our bioreactor cultivation experiment, we observed a growth rate of 0.18/day, a value only 20% higher than was seen in the shake flask cultivation trial. However, the observed maximum cell dry weight in the shake flask, 10.5 g/L, was 14% higher than was achieved in the bioreactor. Ninety seven percentage confidence was achieved via the fitting of three unstructured growth models; the Monod, Logistic, and Gompertz equations, to the cell growth data. The Monod equation adequately described cell growth in both cultures. The specific growth rate, however, was not effectively predicted with the Logistic and Gompertz equations, which resulted in deviations of up to 73 and 393%, respectively. These deviations in the Logistic and Gompertz models may be attributable to the fact that these models were developed for substrate-independent growth and fungi growth, respectively.     

Determination of monod kinetic coefficients for volatile hydrophobic organic compounds

A new procedure is presented to determine Monod kinetic coefficients and the microbial yield coefficient for volatile hydrophobic compounds such as phenanthrene. Batch experiments were conducted with a mixed culture capable of degrading phenanthrene. The phenanthrene disappearance and carbon dioxide production were monitored with time. A maximum likelihood estimator was formulated to fit the set of equations that describe the system to the measured data. The model takes into account a number of processes such as partition onto the apparatus, volatilization, and partition onto the biomass. The parameters required to describe these processes were obtained by independent experiments. The yield coefficient could be determined within a small range. However, the specific growth rate and the half-saturation constant were found to vary widely, with pairs of them describing the system adequately. It was shown that partition and volatilization processes can significantly affect the determination of the yield and Monod kinetic coefficients and need to be taken into account. (c) 1996 John Wiley & Sons, Inc.     

Kinetics of lactose fermentation using a recombinant Saccharomyces cerevisiae strain

This work presents a multi-route, non-structural kinetic model for interpretation of ethanol fermentation of lactose using a recombinant flocculent Saccharomyces cerevisiae strain expressing both the LAC4 (coding for beta-galactosidase) and LAC12 (coding for lactose permease) genes of Kluyveromyces lactis. In this model, the values of different metabolic pathways are calculated applying a modified Monod equation rate in which the growth rate is proportional to the concentration of a key enzyme controlling the single metabolic pathway. In this study, three main metabolic routes for S. cerevisiae are considered: oxidation of lactose, reduction of lactose (producing ethanol), and oxidation of ethanol. The main bioprocess variables determined experimentally were lactose, ethanol, biomass, and dissolved oxygen concentrations. Parameters of the proposed kinetic model were established by fitting the experimental data obtained in a small lab-scale fermentor with the initial lactose concentrations ranging from 5 g/dm3 to 50 g/dm3. A very good agreement between experimental data and simulated profiles of the main variables (lactose, ethanol, biomass, and dissolved oxygen concentrations) was achieved.     

Intraparticle diffusional effects in immobilized cell particles

Summary Immobilized cell technology frequently relies on the entrapment of the biomass in a gel particle, and it is generally observed that mass transfer limitations within the gel particle lead to nonuniform cell distribution. This note addresses the consequence of maintaining a very high cell mass density within a biopolymer particle. We illustrate that conventional effectiveness factor calculations can be used to determine particle sizes which would avoid nonuniform cell growth. The analysis is based on simple Monod kinetics. Special attention is given to near zero order kinetic systems in which the effectiveness factor remains high although the limiting nutrient may be depleted near the center of the particle. Mina Dalili is at the Department of Chemical Engineering, North Carolina State University     

Modeling of Xanthophyllomyces dendrorhous growth on glucose and overflow metabolism in batch and fed-batch cultures for astaxanthin production

An astaxanthin-producing yeast Xanthophyllomyces dendrorhous ENM5 was cultivated in a liquid medium containing 50 g/L glucose as the major carbon source in stirred fermentors (1.5-L working volume) in fully aerobic conditions. Ethanol was produced during the exponential growth phase as a result of overflow metabolism or fermentative catabolism of glucose by yeast cells. After accumulating to a peak of 3.5 g/L, the ethanol was consumed by yeast cells as a carbon source when glucose in the culture was nearly exhausted. High initial glucose concentrations and ethanol accumulation in the culture had inhibitory effects on cell growth. Astaxanthin production was partially associated with cell growth. Based on these culture characteristics, we constructed a modified Monod kinetic model incorporating substrate (glucose) and product (ethanol) inhibition to describe the relationship of cell growth rate with glucose and ethanol concentrations. This kinetic model, coupled with the Luedeking-Piret equation for the astaxanthin production, gave satisfactory prediction of the biomass production, glucose consumption, ethanol formation and consumption, and astaxanthin production in batch cultures over 25-75 g/L glucose concentration ranges. The model was also applied to fed-batch cultures to predict the optimum feeding scheme (feeding glucose and corn steep liquor) for astaxanthin production, leading to a high volumetric yield (28.6 mg/L) and a high productivity (5.36 mg/L/day).     

Effect of Nutrient Concentration on the Growth of Escherichia coli

The relationship between specific growth rate of Escherichia coli and the concentration of limiting nutrient (glucose or phosphate or tryptophan) has been determined for populations in a steady state. At high concentrations the specific growth rate is independent of the concentration of nutrient, but at low concentrations the specific growth rate is a strong function of the nutrient concentration. Such a relationship was predicted by Monod; however, Monod's equation does not predict the relationship over the entire range of nutrient concentration. If parameters of the equation are estimated from the results obtained at low concentrations, then at high concentrations of nutrient, the specific growth rate is significantly higher than that predicted by Monod's equation. These results were interpreted on the basis that the rate of growth is controlled by at least two parallel reactions and that the affinities of the enzymes catalyzing these reactions are different. The relationship between specific growth rate and mean cell volume was also measured, and the results indicate that mean cell volume depends not only on the specific growth rate but also on the nature of the limiting nutrient. There are different mean cell volumes at the same specific growth rate established by different limiting nutrients. Therefore, the mean cell volume is not uniquely determined by the specific growth rate.     

Extended monod kinetics for substrate, product, and cell inhibition

A generalized form of Monod kinetics is proposed to account for all kinds of product, cell, and substrate inhibition. This model assumes that there exists a critical inhibitor concentration above which cells cannot grow, and that the constants of the Monod equation are functions of this limiting inhibitor concentration. Methods for evaluating the constants of this rate form are presented. Finally the proposed kinetic form is compared with the available data in the literature, which unfortunately is very sparse. In all cases, this equation form fitted the data very well.     

Stoichiometric food quality and herbivore dynamics

Herbivores may grow with nutrient or energy limitation, depending on food abundance and the chemical composition of their food. We present a model that describes herbivore growth as a continuous function of two limiting factors. This function uses the synthesizing unit concept, has the hyperbolic Monod model as a limiting case, and has the same number of parameters as the Monod model coupled to Liebig's discontinuous minimum rule. We use the model to explore nutrient-limited herbivore growth in a closed system with algae, Daphnia and phosphorus as the limiting nutrient. Phosphorus in algae may substantially influence Daphnia growth. This influence changes over time and is most pronounced when algae and Daphnia populations fluctuate strongly. Relative to classic models that only consider food quantity as a determinant of Daphnia growth, our model shows richer dynamical behaviour. In addition to the standard positive equilibrium, which may be stable or unstable depending on nutrient availability, a new positive equilibrium may arise in our model when mortality rates are relatively high. This equilibrium is unstable and reduces the likelihood of long-term persistence of Daphnia in the system.     

A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

Modelling the simultaneous processes occurring during the BOD5 test using synthetic waters as a pattern

Synthetic waters imitating BOD₅ test conditions have been studied to model the progress curves of cell growth, organic matter consumption and oxygen demand. Glucose and glutamic acid were used as organic matter and Bacillus subtilis, Escherichia coli and Pseudomonas putida as microorganisms. The concentrations of all the products were measured simultaneously over time and were fitted to the Logistic and Monod models. Following this, the goodness of fit was analysed. Discrimination between models allowed us to conclude that the Logistic model has acceptable accuracy to describe the kinetic behaviour inside a BOD₅ bottle, while the extra parameter in the Monod model would not be statistically justified. Thus, the logistic parameters were determined under different conditions by non-linear regression and their biological meaning was interpreted.     

Kinetics of biotransformation of 1,1,1-trichloroethane by Clostridium sp. strain TCAIIB.

Batch experiments were conducted to examine the effects of high concentrations of 1,1,1-trichloroethane (TCA) on the biotransformation of TCA by Clostridium sp. strain TCAIIB. The biotic dehalogenation of TCA to 1,1-dichloroethane by nongrowing cells was measured at 35 degrees C, and the data were used to obtain the kinetic parameters of the Monod relationship half-velocity coefficient Ks (31 microM) and the coefficient of maximum rate of TCA biotransformation (kTCA; 0.28 mumol per mg per day). The yield of biomass decreased with an increase in the TCA concentration, although TCA concentrations up to 750 microM did not completely inhibit bacterial growth. Also, kTCA was higher in the presence of high concentrations of TCA. A mathematical model based on a modified Monod equation was used to describe the biotransformation of TCA. The abiotic transformation of TCA to 1,1-dichloroethene was measured at 35 degrees C, and the first-order formation rate coefficient for 1,1-dichloroethene (ke) was determined to be 0.86 per year.     

A kinetic model describing Shewanella oneidensis MR-1 growth, substrate consumption, and product secretion

Aerobic growth of Shewanella oneidensis MR-1 in minimal lactate medium was studied in batch cultivation. Acetate production was observed in the middle of the exponential growth phase and was enhanced when the dissolved oxygen (DO) concentration was low. Once the lactate was nearly exhausted, S. oneidensis MR-1 used the acetate produced during growth on lactate with a similar biomass yield as lactate. A two-substrate Monod model, with competitive and uncompetitive substrate inhibition, was devised to describe the dependence of biomass growth on lactate, acetate, and oxygen and the acetate growth inhibition across a broad range of concentrations. The parameters estimated for this model indicate interesting growth kinetics: lactate is converted to acetate stoichiometrically regardless of the DO concentration; cells grow well even at low DO levels, presumably due to a very low K(m) for oxygen; cells metabolize acetate (maximum specific growth rate, micro(max,A) of 0.28 h(-1)) as a single carbon source slower than they metabolize lactate (micro(max,L) of 0.47 h(-1)); and growth on acetate is self-inhibiting at a concentration greater than 10 mM. After estimating model parameters to describe growth and metabolism under six different nutrient conditions, the model was able to successfully estimate growth, oxygen and lactate consumption, and acetate production and consumption under entirely different growth conditions.     

A simple kinetic model for myeloma cell culture with consideration of lysine limitation

A simple kinetic model is developed to describe the dynamic behavior of myeloma cell growth and cell metabolism. Glucose, glutamine as well as lysine are considered as growth limiting substrates. The cell growth was restricted as soon as the extracellular lysine is exhausted and then intracellular lysine becomes a growth limiting substrate. In addition, a metabolic regulator model together with the Monod model is used to deal with the growth lag phase after inoculation or feeding. By using these models, concentrations of substrates and metabolites, as well as densities of viable and dead cells are quantitatively described. One batch cultivation and two fed-batch cultivations with pulse feeding of nutrients are used to validate the model.