Unveiling steady-state multiplicity in hybridoma cultures: the cybernetic approach

Mammalian cells grown in suspension produce waste metabolites such as lactate, alanine, and ammonia, which reduce the yield of cell mass and the desired product on the nutrients supplied. Previous studies (Cruz et al., 1999; Europa et al., 2000; Follstad et al., 1999) have shown that the cells can be made to alter their metabolism by starving them on their nutrients in continuous cultures at low dilution rates or starting the culture as a fed-batch. This leads to multiple steady states in continuous reactors, with some states being more favorable than others. Mathematical models that take into account the metabolic regulation that leads to these multiple steady states are invaluable tools for bioreactor control. In this article we present a cybernetic modeling strategy in which Metabolic Flux Analysis (MFA) is used to guide the cybernetic formulation. The hybridoma model presented as a result of this strategy considers the partially substitutable, partially complementary nature of glucose and glutamine. The choice of competitions within the network is guided by MFA and the model is successful in explaining the three multiple steady states observed. The cybernetic model though identified for the hybridoma experiments of Hu and others (Europa et al., 2000) seem generally applicable to mammalian systems as it captures the pathways that are common to mammalian cells grown in suspension. The model presented here could be used for start-up strategies for continuous reactors and model-based feedback control for maintaining high productivity of the reactor.     

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.     

Growth of mixed cultures on mixtures of substitutable substrates: the operating diagram for a structured model

The growth of mixed microbial cultures on mixtures of substrates is a problem of fundamental biological interest. In the last two decades, several unstructured models of mixed-substrate growth have been studied. It is well known, however, that the growth patterns in mixed-substrate environments are dictated by the enzymes that catalyse the transport of substrates into the cell. We have shown previously that a model taking due account of transport enzymes captures and explains all the observed patterns of growth of a single species on two substitutable substrates (J. Theor. Biol. 190 (1998) 241). Here, we extend the model to study the steady states of growth of two species on two substitutable substrates. The model is analysed to determine the conditions for existence and stability of the various steady states. Simulations are performed to determine the flow rates and feed concentrations at which both species coexist. We show that if the interaction between the two species is purely competitive, then at any given flow rate, coexistence is possible only if the ratio of the two feed concentrations lies within a certain interval; excessive supply of either one of the two substrates leads to annihilation of one of the species. This result simplifies the construction of the operating diagram for purely competing species. This is because the two-dimensional surface that bounds the flow rates and feed concentrations at which both species coexist has a particularly simple geometry: It is completely determined by only two coordinates, the flow rate and the ratio of the two feed concentrations. We also study commensalistic interactions between the two species by assuming that one of the species excretes a product that can support the growth of the other species. We show that such interactions enhance the coexistence region.     

Chemostat-cultivated Escherichia coli at high dilution rate: multiple steady states and drift

The representation of metabolic network reaction kinetics in a scaled, polynomial form can allow for the prediction of multiple steady states. The polynomial formalism is used to study chemostat-cultured Escherichia coli which has been observed to exhibit two multiple steady states under ammonium ion-limited growth conditions: a high cell density-low ammonium ion concentration steady state and a low cell density-high ammonium ion concentration steady state. Additionally, the low-cell-density steady state has been observed to drift to the high-cell-density steady state. Inspection of the steady-state rate expressions for the ammonium ion transport/assimilation network (in polynomial form) suggests that at low ammonium ion concentrations, two steady states are possible. One corresponds to heavy use of the glutamine synthetase-glutamate synthase (GLNS-GS) branch and the second to heavy use of the glutamate dehydrogenase (GDH) branch. Realization of the predicted intracellular steady states is also found to be dependent on the parameters of the transport process. Moreover, the two steady states differ in where their energy intensity lies. To explain the drift, GLNS, which is inducible under low ammonium ion concentrations, is suggested to be a "memory element." A chemostat-based model is developed to illustrate that perturbations in dilution rate can lead to drift between the two steady states provided that the disturbance in dilution rate is sufficiently large and/or long in duration.     

Nonlinear dynamics of eucaryotic pyruvate dehydrogenase multienzyme complex: decarboxylation rate,oscillations, and multiplicity

Pyruvate conversion to acetyl-CoA by the pyruvate dehydrogenase (PDH) multienzyme complex is known as a key node in affecting the metabolic fluxes of animal cell culture. However, its possible role in causing possible nonlinear dynamic behavior such as oscillations and multiplicity of animal cells has received little attention. In this work, the kinetic and dynamic behavior of PDH of eucaryotic cells has been analyzed by using both in vitro and simplified in vivo models. With the in vitro model the overall reaction rate (nu(1)) of PDH is shown to be a nonlinear function of pyruvate concentration, leading to oscillations under certain conditions. All enzyme components affect nu(1) and the nonlinearity of PDH significantly, the protein X and the core enzyme dihydrolipoamide acyltransferase (E2) being mostly predominant. By considering the synthesis rates of pyruvate and PDH components the in vitro model is expanded to emulate in vivo conditions. Analysis using the in vivo model reveals another interesting kinetic feature of the PDH system, namely, multiple steady states. Depending on the pyruvate and enzyme levels or the operation mode, either a steady state with high pyruvate decarboxylation rate or a steady state with significantly lower decarboxylation rate can be achieved under otherwise identical conditions. In general, the more efficient steady state is associated with a lower pyruvate concentration. A possible time delay in the substrate supply and enzyme synthesis can also affect the steady state to be achieved and leads to oscillations under certain conditions. Overall, the predictions of multiplicity for the PDH system agree qualitatively well with recent experimental observations in animal cell cultures. The model analysis gives some hints for improving pyruvate metabolism in animal cell culture.     

Predicting stability of mixed microbial cultures from single species experiments: 2. Physiological model

In this paper, we study the equilibria of a physiological model describing the continuous culture in which two microbial populations compete for two substitutable resources. This work is an extension of the stability analysis of the phenomenological model of mixed microbial growth [M.M. Ballyk, G.S.K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci. 118 (1993) 127-180; S.S. Pilyugin, G.T. Reeves, A. Narang, Predicting stability of mixed microbial cultures from single species experiments: 2. Phenomenological model]. Here, we investigate the influence of the peripheral enzymes that catabolize the substrate uptake on the stability of the mixed culture. We show that, under steady state conditions, an increase in the concentration of one substrate inhibits the uptake of the other substrate(s). We present the criteria for existence, uniqueness, and stability of various types of equilibria. We formulate these criteria in terms of growth isoclines and consumption curves for each of the competing species. Since both types of curves can be obtained from a single species experiment, our approach provides a direct connection between theory and experiment and allows one to infer the dynamics of mixed cultures from the dynamics of single species cultures. By expressing the stability criteria in terms of intracellular properties, the model establishes a link between ecology and molecular biology.     

Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture: IV. Enzymes and fluxes of pyruvate metabolism

The activities of pyruvate kinase (PK), pyruvate: formate-lyase (PFL), pyruvate dehydrogenase (PDH), and citrate synthase (CS) involved in the anaerobic glycerol conversion by Klebsiella pneumoniae were studied in continuous culture under conditions of steady states and sustained oscillations. Both the in vitro and in vivo activities of PK, PFL, and PDH are strongly affected by the substrate concentration and its uptake rate, as is the in vitro activity of CS. The flux from phosphoenolpyruvate to pyruvate is found to be mainly regulated on a genetic level by the synthesis rate of PK, particularly at low substrate concentration and low growth rate. In contrast, the conversion of pyruvate to acetyl-CoA is mainly regulated on a metabolic level by the in vivo activities of PFL and PDH. The ratio of in vitro to in vivo activities is in the range of 1 to 1.5 for PK, 5 to 17 for PFL and 5 to 80 for PDH under the experimental conditions. The regulation of in vivo activity and synthesis of these enzymes is sensitive to fluctuations of culture conditions, leading to oscillations of both the in vitro and in vivo activities. In particular, PFL is strongly affected during oscillations; its average in vitro activity is only about half of its corresponding steady-state value under similar environmental conditions. The average in vitro activities of PDH and PK under oscillations are close to their corresponding steady-state values. In contrast to all other enzymes measured for the glycerol metabolism by K. pneumoniae PFL and PDH are more effectively in vivo utilized under oscillations than under steady state, underlining the peculiar role of pyruvate metabolism in the dynamic responses of the culture.     

Competition between two species for two complementary or substitutable resources

The classical models of interspecific competition are phenomenological, that is, they purport to describe the trajectories followed by the abundances of both competitors, without specifying either mechanisms or the dynamics of what the competition is for, i.e. resources. Yet the conditions for the different outcomes of competition inferred from these models are most often interpreted in terms of resources. Here it is contended that the dynamics of resource supply and exploitation must be explicitly taken into account if these conditions are to be obtained in any accurate form. First, the distinction between perfectly substitutable, imperfectly substitutable and perfectly complementary resources is examined. This leads to an array of models intended to mirror interspecific exploitative competition for two resources for each category considered. Then, the conditions for the coexistence of the two competitors in each case are derived and presented with direct reference to the modes of resource use.     

Structured kinetic model to represent the utilization of multiple substrates in complex media during rifamycin B fermentation

Industrial fermentations typically use media that are balanced with multiple substitutable substrates including complex carbon and nitrogen source. Yet, much of the modeling effort to date has mainly focused on defined media. Here, we present a structured model that accounts for growth and product formation kinetics of rifamycin B fermentation in a multi-substrate complex medium. The phenomenological model considers the organism to be an optimal strategist with an in-built mechanism that regulates the sequential and simultaneous uptake of the substrate combinations. This regulatory process is modeled by assuming that the uptake of a substrate depends on the level of a key enzyme or a set of enzymes, which may be inducible. Further, the fraction of flux through a given metabolic branch is estimated using a simple multi-variable constrained optimization. The model has the typical form of Monod equation with terms incorporating multiple limiting substrates and substrate inhibition. Several batch runs were set up with varying initial substrate concentrations to estimate the kinetic parameters for the rifamycin overproducer strain Amycolatopsis mediterranei S699. Glucose and ammonium sulfate (AMS) demonstrated significant substrate inhibition toward growth as well as product formation. The model correctly predicts the experimentally observed regulated simultaneous uptake of the substitutable substrate combinations under different fermentation conditions. The modeling results may have applications in the optimization and control of rifamycin B fermentation while the modeling strategy presented here would be applicable to other industrially important fermentations.     

Coexistence and displacement in consumer-resource systems with local and shared resources

Competition for local and shared resources is widespread. For example, colonial waterbirds consume local prey in the immediate vicinity of their colony, as well as shared prey across multiple colonies. However, there is little understanding of conditions facilitating coexistence vs. displacement in such systems. Extending traditional models based on type I and type II functional responses, we simulate consumer-resource systems in which resources are “substitutable,” “essential,” or “complementary.” It is shown that when resources are complementary or essential, a small increase in carrying capacity or decrease in handling time of a local resource may displace a spatially separate consumer species, even when the effect on shared resources is small. This work underscores the importance of determining both the nature of resource competition (substitutable, essential, or complementary) and appropriate scale-dependencies when studying metacommunities. We discuss model applicability to complex systems, e.g., urban wildlife that consume natural and anthropogenic resources which may displace rural competitors by depleting shared prey.     

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.     

Bacterial consortium proteomics under 4‐chlorosalicylate carbon‐limiting conditions

In this study, the stable consortium composed by Pseudomonas reinekei strain MT1 and Achromobacter xylosoxidans strain MT3 (cell numbers in proportion 9:1) was under investigation to reveal bacterial interactions that take place under severe nutrient‐limiting conditions. The analysis of steady states in continuous cultures was carried out at the proteome, metabolic profile, and population dynamic levels. Carbon‐limiting studies showed a higher metabolic versatility in the community through upregulation of parallel catabolic enzymes (salicylate 5‐hydroxylase and 17‐fold on 2‐keto‐4‐pentenoate hydratase) indicating a possible alternative carbon routing in the upper degradation pathway highlighting the effect of minor proportions of strain MT3 over the major consortia component strain MT1 with a significant change in the expression levels of the enzymes of the mainly induced biodegradation pathway such as salicylate 1‐hydroxylase and catechol 1,2‐dioxygenase together with important changes in the outer membrane composition of P. reinekei MT1 under different culture conditions. The study has demonstrated the importance of the outer membrane as a sensing/response protective barrier caused by interspecies interactions highlighting the role of the major outer membrane proteins OprF and porin D in P. reinekei sp. MT1 under the culture conditions tested.     

An analysis of extended and exponentially-fed-batch cultures.

Under certain conditions it is shown that an extended culture is equivalent to an exponentially-fed-batch culture, that an exponentially-fed-batch culture (and an extended culture) can be maintained at a steady state and that an exponentially-fed-batch culture may be mimicked by a continuous-flow culture with a constant dilution rate. Operational conditions required to maintain steady states are specified.     

Application of hybrid cybernetic model in simulating myeloma cell culture co-consuming glucose and glutamine with mixed consumption patterns

A dynamic model called hybrid cybernetic model (HCM) based on structured metabolic network is established for simulating mammalian cell metabolism featured with partially substitutable and partially complementary consumption patterns of two substrates, glucose and glutamine. Benefiting from the application of elementary mode analysis (EMA), the complicated metabolic network is decomposed into elementary modes (EMs) facilitating the employment of the hybrid cybernetic framework to investigate the external and internal flux distribution and the regulation mechanism among them. According to different substrate combination, two groups of EMs are obtained, i.e., EMs associated with glucose uptake and simultaneous uptake of glucose and glutamine. Uptake fluxes through various EMs are coupled together via cybernetic variables to maximize substrate uptake. External fluxes and internal fluxes could be calculated and estimated respectively, by the combination of the stoichiometrics of metabolic networks and fluxes through regulated EMs. The model performance is well validated via three sets of experimental data. Through parameter identification of limited number of experimental data, other external metabolites are precisely predicted. The obtained kinetic parameters of three experimental cultures have similar values, which indicates the robustness of the model. Furthermore, the prediction performance of the model is successfully validated based on identified parameters.     

A metabolic model of the biological phosphorus removal process: II. Validation during start-up conditions

A metabolic model of the biological phosphorus removal process has been developed and validated previously for complex conversions during the process under anaerobic and aerobic conditions at different growth rates in sequencing batch reactors in steady state. For additional validation of the metabolic model, the model was applied to the dynamic conditions which occur during the start-up phase of the biological P removal in the presence and absence of non-polyP heterotrophic microorganisms. In a laboratory scale sequencing batch reactor, experiments were performed to examine the enrichment of the population with polyphosphate organisms during the start-up and the subsequent shift from non-polyP, heterotrophic organisms to polyP organisms in the sludge. The effect of different influent loading patterns for acetate and phosphate was studied. In these experiments, the maximal growth rate of the polyP organisms and the behavior of the internal storage compounds could be derived. The metabolic model was capable of describing the experimental results, without the need to adjust the kinetic or stoichiometric parameters obtained under steady state conditions. (c) 1995 John Wiley & Sons, Inc.     

On microbial states of growth†

It is crucial to the reproducibility of results and their proper interpretation that the conditions under which experiments are carried out be defined with rigour and consistency, in this review we attempt to clarify the differences and interrelationships among steady, balanced and exponential states of culture growth. Basic thermodynamic concepts are used to introduce the idea of steady-state growth in open, biological systems. The classical, sometimes conflicting, definitions of steady-state and balanced growth are presented, and a consistent terminology is proposed. The conditions under which a culture in balanced growth is also in exponential growth and in steady-state growth are indicated. It is pointed out that steady-state growth always implies both balanced and exponential growth, and examples in which the converse does not hold are described. More complex situations are then characterized and the terminology extended accordingly. This leads to the notion of normal growth and growth that can be synchronous or otherwise unbalanced but still reproducible, and to the condition of approximate steady state manifested by growth in batch culture and by asymmetrically dividing cells, which is analysed in some detail.     

Analysis of CHO cells metabolic redistribution in a glutamate-based defined medium in continuous culture.

The effect of glutamine replacement by glutamate and the balance between glutamate and glucose metabolism on the redistribution of t-PA-producing recombinant CHO cells metabolism is studied in a series of glucose shift down and shift up experiments in continuous culture. These experiments reveal the existence of multiple steady states, and experimental data are used to perform metabolic flux analysis to gain a better insight into cellular metabolism and its redistribution. Regulation of glucose feed rate promotes a higher efficiency of glucose and nitrogen source utilization, with lower production of metabolic byproducts, but this reduces t-PA specific production rate. This reduction under glucose limitation can be attributed to the fact that the cells are forced to efficiently utilize the carbon and energy source for growth, impairing the production of dispensable metabolites. It is, therefore, the combination of growth rate and carbon and energy source availability that determines the level of t-PA production in continuous culture.     

Continuous culture growth of photoautotrophic cell suspensions from Chenopodium rubrum

The construction and operation of a continuous culture system for the propagation of cell suspensions from Chenopodium rubrum under photoautotrophic conditions has been described. A dilution rate of 0.16/day gave an equilibrium culture density of 1,100,000 cells/ml and a mean doubling time of 150 hours. During continuous culture steady state conditions with respect to nutrient uptake, cell protein and chlorophyll content, starch accumulation, in vitro activities of enzymes related to different metabolic pathways could be established. Photosynthetic CO₂ assimilation of steady state cells was about 100 mol CO₂/mg chlorophyll x hour. Dark CO₂ fixation was 3% of the light values.