A simple model to describe transient differences between cell number and biomass growth rates of Escherichia coli

Information on the response of a microbial culture to dynamic environmental conditions is necessary for the design of transient operation processes. However, most attempts at modelling culture response have been directed at describing the steady-state behavior. Thus, there is a need for adequate dynamic models for process design. Simulations of nutrient shifts were completed using a "single-cell" model for Escherichia coli. It was discovered that the specific mass growth rate and the specific number of cells growth rate were different under transient conditions, whereas at steady state (balanced growth) these rates are equivalent. Using these observations, a simple delay model to describe the transient behavior of the two growth rates is formulated and tested. The model contains as state variables only the readily measurable macroscopic quantities (biomass, cell number, and limiting nutrient). This model agreed well with the predictions of the single-cell model.     

Operability of Continuous Bioprocesses: Static Behavior of a Large Class of Unstructured Models

The complete static behavior of a large class of unstructured models of continuous bioprocesses is classified using elementary concepts of the singularity theory and continuation techniques. The class consists of models for which the cell growth rate is proportional to the rate of utilization of limiting substrate while the kinetics of cell growth, utilization of limiting substrate and synthesis of the desired non-biomass product are allowed to assume general forms of substrate and product. This class of models was used extensively in the literature to model fermentation processes. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of wash-out conditions with non-trivial steady states and multistability resulting from hysteresis. These general results are applied to a number of experimentally validated models of fermentation processes, and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models. Practical criteria are also derived for the safe operation of the bioprocesses.     

A simple structured model describing the growth of Streptomyces lividans

The growth of Streptomyces lividans in defined media was modeled using a simple structured growth model. Conventional unstructured models like Monod kinetics, substrate inhibition kinetics, and the logistic equation were also used in an attempt to fit the data, but the results were all unsatisfactory. The main reason for failure in applying simple unstructured models is that they cannot describe the long lag phases sometimes observed during growth of S. lividans. The simple structured growth model was derived along similar principles to cybernetic growth models. This model quite accurately describes the growth of S. lividans. It assumes that the rate of assimilation of a substrate depends on the concentration of a specific key enzyme. This key enzyme is only produced in the presence of the substrate, and it is broken down at a steady rate. An enzyme synthesis allocation variable, w, similar to the cybernetic variable, u, described in cybernetic growth models, is proposed to control enzyme synthesis. Until the key enzyme concentration approaches its maximum level, very little substrate is consumed. And consequently, the lag phase is sustained.     

On describing microbial growth kinetics from continuous culture data: Some general considerations,observations, and concepts

Analysis of continuous culture methodology suggests that this potentially powerful tool for kinetic analysis can be improved by minimizing several inherent shortcomings. Medium background substrates — organic carbon, phosphate, and manganese — were shown to dominate kinetic observations at concentrations below chemical detection methods. Reactor wall growth, culture size distribution changes, sample removal-induced steady state perturbations, and limiting substrate leakage from organisms are treated in terms of kinetic measurement errors. Large variations in maximal growth rates and substrate uptake rates found are attributed to experimental protocol-induced transient states. Relationships are presented for correcting limiting substrate concentrations for lability during sampling, contamination with unreacted medium, and background substrate effects. Analytical procedures are discussed for improved measurement of limiting substrate kinetics involving enzymes, isotopes, and material balance manipulation. Relaxation methods as applied to continuous culture are introduced as a means for isolating separate rate constants describing net substrate transport and for evaluating cellular metabolite leakage. Low velocity growth, multiple substrate metabolism, and endogenous metabolism are discussed along with measurements showing that 1-month generation times for aquatic microorganisms can be quite normal and that the kinetics are compatible withμg/liter limiting substrate concentrations. The concept of regarding growth kinetics as the sum of several net accumulation processes is suggested.     

Classification of static behavior of a class of unstructured models of continuous bioprocesses

The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.     

Linear growth and poly(beta-hydroxybutyrate) synthesis in response to pulse-wise addition of the growth-limiting substrate to steady-state heterotrophic continuous cultures of Aquaspirillum autotrophicum.

Heterotrophic pyruvate-limited steady-state continuous cultures of the bacterium Aquaspirillum autotrophicum were perturbed with a pulse injection of a small volume of concentrated pyruvate solution. These cultures exhibited an instantaneous change in the growth dynamics, turning from steady state to apparently linear growth. These transient growth-responses had no lag phase and were clearly distinct from unlimited exponential growth according to the initial rates of increase of biomass and substrate disappearance kinetics. A linear accumulation with time of poly(beta-hydroxybutyrate) was observed within the cells. Slopes of these linear responses were negatively correlated with the dilution rate. Physiological bases of linear growth are discussed in the light of the models of H. E. Kubitschek. Poly(beta-hydroxybutyrate) synthesis in the absence of exogenous limitation may serve to protect the cells against a transient metabolic overflow.     

Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

Inhibition Kinetics of Phenol Degradation by Pseudomonas aeruginosa from Continuous Culture and Washout Data

Steady states of a continuous culture with an inhibitory substrate were used to estimate kinetic parameters under substrate limitation (chemostat operation). Pure cultures of an indigenous Pseudomonas aeruginosa were grown in continuous culture on phenol, the sole source of carbon and energy, at dilution rates of 0.010 to 0.20 h^{- 1}. Using different dilution rates, several steady states were investigated and the specific phenol consumption rates were calculated. In addition, phenol degradation was investigated by increasing the dilution rate above the critical dilution rate (washout cultivation). The results showed that the specific phenol consumption rate increased with increased dilution rate at steady state and that the degradation by Pseudomonas aeruginosa can be described by simple substrate inhibition kinetics under substrate limitation but cannot be described by simple substrate inhibition kinetics under washout cultivation. Fitting of the steady-state data from continuous cultivation to various inhibition models resulted in the best fit for the Yano and Koga kinetic inhibition model. The r_{s max} value of 0.278 mg/mg/h obtained from the Yano and Koga equation was comparable to the experimentally calculated r_{s max} value of 0.283 mg/mg/h obtained under washout cultivation.     

Inhibition kinetics of phenol degradation from unstable steady-state data

Multiplicity of steady states of a continuous culture with an inhibitory substrate was used to estimate kinetic parameters under steady-state conditions. A continuous culture of Pseudomonas cepacia G4, using phenol as the sole source of carbon and energy, was overloaded by increasing the dilution rate above the critical dilution rate. The culture was then stabilized in the inhibitory branch by a proportional controller using the carbon dioxide concentration in the reactor exhaust gas as the controlled variable and the dilution rate as the manipulated variable. By variation of the set point, several unstable steady states in the inhibitory branch were investigated and the specific phenol conversion rates calculated. In addition, phenol degradation was investigated under substrate limitation (chemostat operation).The results show that the phenol degradation by P. cepacia can be described by the same set of inhibition parameters under substrate limitation and under high substrate concentrations in the inhibitory branch. Biomass yield and maintenance coefficients were identical. Fitting of the data to various inhibition models resulted in the best fit for the Yano and Koga equation. The well-known Haldane model, which is most often used to describe substrate inhibition by phenol, gave the poorest fit. The described method allows a precise data estimation under steady-state conditions from the maximum of the biological reaction rate up to high substrate concentrations in the inhibitory branch. Inhibition parameter estimation by controlling unstable steady states may thus be useful in avoiding discrepancies between data generated by batch runs and their application to continuous cultures which have been often described in the literature. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 54: 567-576, 1997.     

Effect of carbon:nitrogen ratio on kinetics of phenol biodegradation byAcinetobacter johnsonii in saturated sand

In polluted soil or ground water, inorganic nutrients such as nitrogen may be limiting, so that Monod kinetics for carbon limitation may not describe microbial growth and contaminant biodegradation rates. To test this hypothesis we measured¹⁴CO₂ evolved by a pure culture ofAcinetobacter johnsonii degrading 120 µg¹⁴C-phenol per ml in saturated sand with molar carbon:nitrogen (CN) ratios ranging from 1.5 to 560. We fit kinetics models to the data using non-linear least squares regression. Phenol disappearance and population growth were also measured at CN1.5 and CN560.After a 5- to 10-hour lag period, most of the¹⁴CO₂ evolution curves at all CN ratios displayed a sigmoidal shape, suggesting that the microbial populations grew. As CN ratio increased, the initial rate of¹⁴CO₂ evolution decreased. Cell growth and phenol consumption occurred at both CN1.5 and CN560, and showed the same trends as the¹⁴CO₂ data. A kinetics model assuming population growth limited by a single substrate best fit the¹⁴CO₂ evolution data for CN1.5. At intermediate to high CN ratios, the data were best fit by a model originally formulated to describe no-growth metabolism of one substrate coupled with microbial growth on a second substrate. We suggest that this dual-substrate model describes linear growth on phenol while nitrogen is available and first-order metabolism of phenol without growth after nitrogen is depleted.     

A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria

When the growth of bacteria in a chemostat is controlled by limiting the supply of a single essential nutrient, the growth rate is affected both by the concentration of this nutrient in the culture medium and by the amount of time that it takes for the chemical and physiological processes that result in the production of new biomass. Thus, although the uptake of nutrient by cells is an essentially instantaneous process, the addition of new biomass is delayed by the amount of time that it takes to metabolize the nutrient. Mathematical models that incorporate this "delayed growth response" (DGR) phenomenon have been developed and analysed. However, because they are formulated in terms of parameters that are difficult to measure directly, these models are of limited value to experimentalists. In this paper, we introduce a DGR model that is formulated in terms of measurable parameters. In addition, we provide for this model a complete set of criteria for determining persistence versus extinction of the bacterial culture in the chemostat. Specifically, we show that DGR plays a role in determining persistence versus extinction only under certain ranges of chemostat operating parameters. It is also shown, however, that DGR plays a role in determining the steady-state nutrient and bacteria concentrations in all instances of persistence. The steady state and transient behavior of solutions of our model is found to be in agreement with data that we obtained in growing Escherichia coli 23716 in a chemostat with glucose as a limiting nutrient. One of the theoretical predictions of our model that does not occur in other DGR models is that under certain conditions a large delay in growth response might actually have a positive effect on the bacteria's ability to persist.     

Der substratlimitierte pH-Auxostat – Eine neue Methode zur kontinuierlichen Kultur

A method for continuous cultivation of microorganisms is demonstrated, the substrate limited pH-auxostat. The limiting substrate only is added with constant velocity. In this culture the cells grow with high utilization of the limiting substrate and with the highest specific growth rate possible at the given conditions. Yield coefficients and dilution rates of stable K⁺-limited steady states in yeast cultures with different pH-values and biomass concentrations were measured.     

Continuous cultures limited by a gaseous substrate: Development of a simple, unstructured mathematical model and experimental verification with Methanobacterium thermoautotrophicum

This article presents a simple, unstructured mathematical model describing microbial growth in continuous culture limited by a gaseous substrate. The model predicts constant gas conversion rates and a decreasing biomass concentration with increasing dilution rate. It has been found that the parameters influencing growth are primarily the gas transfer rate and the dilution rate. Furthermore, it is shown that, for correct simulation of growth, the influence of gaseous substrate consumption on the effective gas flow through the system has to be taken into account.Continuous cultures of Methanobacterium thermoautotrophicum were performed at three different gassing rates. In addition to the measurement of the rates of biomass production, product formation, and substrate consumption, microbial heat dissipation was assessed using a reaction calorimeter. For the on-line measurement of the concentration of the growth-limiting substrate, H(2), a specially developed probe has been used. Experimental data from continuous cultures were in good agreement with the model simulations. An increase in gassing rate enhanced gaseous substrate consumption and methane production rates. However, the biomass yield as well as the specific conversion rates remained constant, irrespective of the gassing rate. It was found that growth performance in continuous culture limited by a gaseous substrate is substantially different from "classic" continuous culture in which the limiting substrate is provided by the liquid feed. In this report, the differences between both continuous culture systems are discussed.     

Acetic acid production by Acetogenium kivui in continuous culture — kinetic studies and computer simulations

Summary This work considers the continuous production of acetic acid by the homoacetogenic and thermophilic bacterium Acetogenium kivui. A mathematical model for the growth kinetics has been developed. The unstructured model for growth and product formation includes product and substrate inhibition as well as maintenance energy effects. The associated model parameters have been identified by non-linear optimization and evidenced experimentally in continuous culture as steady-state data. By using a mineral medium with glucose as the energy and carbon source for the bacteria proper carbon balances are available. The model permits good predictions of steady-state concentrations. Offprint requests to: J. von Eysmondt     

Cybernetic modeling and regulation of metabolic pathways in multiple steady states of hybridoma cells

Hybridoma cells utilize a pair of complementary and partially substitutable substrates, glucose and glutamine, for growth. It has been shown that cellular metabolism shifts under different culture conditions. When those cultures at different metabolic states are switched to a continuous mode, they reach different steady states under the same operating conditions. A cybernetic model was constructed to describe the complementary and partial substitutable nature of substrate utilization. The model successfully predicted the metabolic shift and multiple steady-state behavior. The results are consistent with the experimental observation that the history of the culture affects the resulting steady state.     

Dynamics and stability analysis of the growth and astaxanthin production system of Haematococcus pluvialis

This paper investigated high cell density cultivation of Haematococcus pluvialis for astaxanthin production in 3.7-L bioreactors. A biomass concentration of 2.74 g L⁻¹and an astaxanthin yield of 64.4 mg L⁻¹ were obtained. Based on the experimental results, a new and simple dynamic model is proposed, differing from Monod kinetics, to describe cell growth, product formation and substrate consumption. Good agreement was found between the model predictions and experimental data. The model revealed that there was cell growth inhibition on product formation and product feedback compensation for substrate consumption, but no substrate inhibition or product inhibition of cell growth. Stability analysis demonstrated that no multiplicity of steady states was observed; the unique positive steady state was locally asymptotically stable; and the effect of dilution rate on steady states was greater than that of the initial substrate concentration. Received 23 February 1999/ Accepted in revised form 08 June 1999     

Growth kinetics of a yeast grown on glucose or hexadecane

The kinetics of growth of a Torulopsis sp. was investigated in a continuous culture with glucose or hexadecane as the carbon source; growth was limited by either carbon or nitrogen. The relationship between the concentration of the limited substrate and the steady-state growth rate of the organism was examined and tested against various models of growth. No existing model was found to describe the growth accurately and a new model has been proposed: It is postulated that this behavior would result from a simple first order reaction between the reactants of the rate-limiting enzymic reaction of the organism's metabolism.     

Transient kinetics of yeast growth in batch and continous culture with an inhibitory carbon and energy source

Growth and acetate metabolism by Candida utilis (ATCC 9226) is reported for both acetate- and zinc-limited cultures in defined media. Acetate concentrations were varied from suboptimal to inhibitory levels in both types of media in differential shake flask culture and in batch and continuous cultures in stirred fermentors. Transient responses of steady-state cultures to small or large additions of concentrated sodium acetate, or to shifts in dilution rate or inlet acetate concentration are compared with one another and with simple mathematical models of growth and acetate metabolism. Exponential growth was observed during unrestricted growth (differential shake flask and batch cultures) with both types of media. Addition of acetate during unrestricted growth always caused lags and for larger pulses, lower specific growth rates were observed after exponential growth resumed. Inhibition by high acetate concentrations was much greater in acetate–limited than in zinc–limited cultures. During restricted growth (steady-state, continuous cultures), high acetate concentrations again consistently caused growth lags but stimulated, inhibited, or temporarily stopped acetage uptake. Qualitative agreement between the predictions of a simple mathematical model of acetate inhibition fitted to differential shake flask data and the observed transient data was surprisingly good.