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.     

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.     

The relationship between kinetics of substrate-limited transitions and steady-state growth in continuous cultures ofAquaspirillum autotrophicum limited by pyruvate

Heterotrophic growth at steady state and during transient states caused by the sudden change of the concentration of the limiting factor in the feed medium was investigated experimentally for continuous cultures ofAquaspirillum autotrophicum limited by pyruvate. A model for describing the growth at steady state was selected from three unstructured models after statistical tests of the data. This model postulates that the growth yield increases linearly with the growth rate. Growth during transitions where the substrate remained limiting at all times was fitted with first-order kinetics. Theoretical predictions of these kinetics were derived from the unstructured models used to describe steady state. The predicted rate coefficients of the transients were compared to the experimental coefficients. It appeared that the model which best described steady-state growth also provided the best predictions for growth during the transient state. It is a widespread opinion that unstructured models are adequate to describe growth under steady-state conditions but not to predict transitions in continuous culture. However, for the particular case studied here, no higher degree of complexity was required to describe transitions, provided the growth of the culture was always limited by the substrate.     

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.     

Real-time monitoring and control of microbial bioprocesses with focus on the specific growth rate: current state and perspectives

Understanding the growth characteristics of microorganisms is an essential step in bioprocessing, not only because product formation may be growth-associated but also because they might influence cell physiology and thereby product quality. The specific growth rate, a key variable of many bioprocesses, cannot be measured directly and relies on the estimation through other measurable variables such as biomass, substrate, or product concentrations. Techniques for real-time estimation of the specific growth rate in microbial fed-batch cultures are discussed in the present paper. The advantages and limitations of different models and various monitoring techniques are discussed, highlighting the importance of the specific growth rate in the development of fast, reliable, and robust processes for the production of high-value products such as recombinant proteins.     

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.     

Dynamics of Predator‐Prey Interactions in Continuous Cultures

This paper studies the behavior of a general unstructured kinetic model for continuous bioreactors involving interactions between predator, prey and a limiting substrate. The analysis carried out in this paper shows how closed analytical conditions for arbitrary growth rates can be derived that describe the conditions for the existence of the interacting species in an oscillatory behavior. It is also demonstrated how practical diagrams in terms of operating and kinetic parameters can be constructed that classify the different behavior predicted by the model. Applications of these general results to a number of experimentally validated models have revealed that the saturation model always predicts hard oscillations for a certain range of dilution rates, for any values of model parameters. Bifurcation diagrams in the operating parameter space permitted the delineation of regions of hard oscillations, regions of static coexistence, regions of predator washout and regions of total washout. The analysis of the double saturation model has proven its ability to predict two Hopf points. Hard oscillations are therefore expected within the dilution rates corresponding to the two Hopf points. Practical diagrams were also constructed to delineate the boundaries separating hard oscillations from static coexistence and washout conditions.     

Kinetic effects of simultaneous inhibition by substrate and product

The starting point for the present investigations was the finding that increasing influent concentrations from 10 to 380 mmol/L glucose decreased the attainable growth rate of an acidogenic population in continuous culture from 0.52 to 0.05 h(-1) To account for this phenomenon, a new kinetic model is developed that combines substrate and product inhibition. Both effects are connected through the product yield, giving rise to a complex dependency of the growth rate on the substrate concentration. As a main feature, the maximum attainable growth rate decreases almost hyperbolically above some optimal substrate concentration in the influent. Furthermore, under certain conditions the kinetic model predicts the existence of three steady states: a high-conversion and a low-conversion state that are both stable and a metastable intermediate state. The latter states from the multiple-steady-state region are to be avoided, and eventual transitions to these states may have important consequences for the stability and the operation of such reaction systems. Substrate as well as product inhibition is reported for Propionibacterium freundenreichii and recently could be demonstrated for the above-mentioned acidogenic population. The proposed model allows optimization of anaerobic wastewater treatment processes and is applicable also to other fermentations.     

Dynamic behavior of the chemostat subject to substrate inhibition

A theoretical study was made on the dynamic behavior of a single-vessel continuous fermentation subject to a growth inhibition at, high concentration of the rate limiting substrate. Phase plane plotting and stability analysis showed three steady states to exist; namely, a “washout”; state, a nontrivial stable state, and an unstable state. Whether the system attains a nontrivial steady state or is to be washed-out depends upon the initial values of the cell and/or substrate concentration(s). Since this property is a characteristic feature of the present system, an experimental procedure was suggested to insure a stable operation in practice.     

A structured kinetic modeling framework for the dynamics of hybridoma growth and monoclonal antibody production in continuous suspension cultures

A structured kinetic model is developed to describe the dynamics of hybridoma growth and the production of monoclonal antibodies and metabolic waste products in suspension culture. The crucial details of known metabolic processes in hybridoma cells are incorporated by dividing the cell mass into four intracellular metabolic pools. The model framework and structure allow the dynamic calculation of the instantaneous specific growth rate of a hybridoma culture. The steady state and dynamic simulations of the model equations exhibit excellent agreement with experimentally observed trends in substrate utilization and product formation. The model represents the first to include any degree of metabolic detail and structure in describing a hybridoma culture. In so doing, it provides the basic modeling framework for incorporating further details of metabolism and can be a useful tool to study various strategies for enhancing hybridoma growth as well as viability and the production of monoclonal antibodies in suspension cultures.     

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     

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.     

Kinetic model identification in mixed populations using continuous culture data

The kinetic behavior of heterogeneous microbial populations of sewage origin was studied in a single-stage, isothermal, continuous flow, completely mixed aeration tank. A series of experiments were carried out at various dilutions rates using glucose as the limiting substrate. The cell dry weight and substrate concentration in terms of chemical oxygen demand (COD) were continuously monitored. The results indicate that reproducible steady-state conditions can generally be obtained; however, multiple steady states were observed at dilution rates near washout. At low dilution rates (below about 0.1 hr^?1) the contribution of microorganism decay became appreciable. Using the multiresponse data of cell dry weight and COD, the parameter values in various existing growth models were estimated. The analyses of variance and residuals revealed that models proposed by Moser, Monod, and Contois, each with a decay term added, were significantly better than the other models which were tested.     

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.     

Respirometric evaluation and modeling of glucose utilization by Escherichia coli under aerobic and mesophilic cultivation conditions

The study presents a mechanistic model for the evaluation of glucose utilization by Escherichia coli under aerobic and mesophilic growth conditions. In the first step, the experimental data was derived from batch respirometric experiments conducted at 37 degrees C, using two different initial substrate to microorganism (S(0)/X(0)) ratios of 15.0 and 1.3 mgCOD/mgSS. Acetate generation, glycogen formation and oxygen uptake rate profile were monitored together with glucose uptake and biomass increase throughout the experiments. The oxygen uptake rate (OUR) exhibited a typical profile accounting for growth on glucose, acetate and glycogen. No acetate formation (overflow) was detected at low initial S(0)/X(0) ratio. In the second step, the effect of culture history developed under long-term growth limiting conditions on the kinetics of glucose utilization by the same culture was evaluated in a sequencing batch reactor (SBR). The system was operated at cyclic steady state with a constant mean cell residence time of 5 days. The kinetic response of E.coli culture was followed by similar measurements within a complete cycle. Model calibration for the SBR system showed that E. coli culture regulated its growth metabolism by decreasing the maximum growth rate (lower microH) together with an increase of substrate affinity (lower K(S)) as compared to uncontrolled growth conditions. The continuous low rate operation of SBR system induced a significant biochemical substrate storage capability as glycogen in parallel to growth, which persisted throughout the operation. The acetate overflow was observed again as an important mechanism to be accounted for in the evaluation of process kinetics.     

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Mathematical modelling of dynamics and control in metabolic networks. V. Static bifurcations in single biochemical control loops

Here we expand an earlier study of feedback activation in simple linear reaction sequences by searching the parameter space of biologically realistic rate laws for multiple stable steady states. The impetus for this work is to seek the origin of decision making strategies at the metabolic level, with particular emphasis on the switching between the operating conditions needed to meet changing substrate availability and organism requirements. The control loop considered herein is a linear reaction chain in which the end product of the reaction sequence feedback activates the first reaction in the sequence to produce feedback control. It has been found that the criteria for the existence of multiple steady state solutions in such loops involve only the kinetics of the regulatory enzyme controlling the first reaction and that of end product removal. The effects of these kinetics are examined here using two representative models for the regulatory enzyme: the lumped controller, based on Hill-type kinetics, and the symmetry model. The behavior of these two models is qualitatively similar, and both show the characteristics needed for switching between low and high substrate utilization. The removal rate is assumed to be of the Michaelis-Menten type. Judicious scaling of the governing equations permits separation of genetically determined kinetic parameters from concentration dependent ones. This allows us to conclude that, for a fixed set of kinetic parameters, the steady state flux through the loop can be switched between stable steady states by merely varying metabolite or enzyme concentrations. In particular, when the initial substrate exceeds a certain critical level, the loop can be "switched on" (by a discontinuous increase in the flux through the chain), and similarly, when it falls below a critical level, the pathway is shut down. Similar effects can be realized by varying the ratios of enzyme concentrations. It is proposed that by identifying these critical points one can gain significant insight into the objectives of decision making at the metabolic level.     

Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.