Coexistence of S. cerevisiae and E. coli in chemostat under substrate competition and product inhibition

A mixed culture of Saccharomyces cerevisiae and Escherichia coli was established in a stable coexistence steady state in a chemostat under constant operating conditions. The species competed for glucose, the growth-limiting resource, and produced acetate and ethanol. The acetic acid was shown to be very inhibitory to E. coli in pure culture at pH 5 while ethanol inhibition was only marginal. No significant inhibition of S. cerevisiae growth was observed by either acetate or ethanol. Pure culture parameters were measured and used in the analysis. Linearized stability analysis for the case when both organisms produce the inhibitor showed that a transition through three stable outcomes was possible as the feed concentration is lowered. Experimental studies verified these predictions, and successive transitions from a yeast growth steady state, to a coexistence steady state, and to an E. coli growth steady state were obtained by lowering the glucose concentration in the feed from 10 to 5 to 2.5 g/L, respectively. This dynamic behavior is distinct from the outcomes of other competition-inhibition combinations and experimentally demonstrates for the first time that coexistence is possible due to substrate competition and product inhibition.     

Modeling of an aerobic biofilm reactor with double-limiting substrate kinetics: bifurcational and dynamical analysis

A mathematical model of an aerobic biofilm reactor is presented to investigate the bifurcational patterns and the dynamical behavior of the reactor as a function of different key operating parameters. Suspended cells and biofilm are assumed to grow according to double limiting kinetics with phenol inhibition (carbon source) and oxygen limitation. The model presented by Russo et al. is extended to embody key features of the phenomenology of the granular‐supported biofilm: biofilm growth and detachment, gas–liquid oxygen transport, phenol, and oxygen uptake by both suspended and immobilized cells, and substrate diffusion into the biofilm. Steady‐state conditions and stability, and local dynamic behavior have been characterized. The multiplicity of steady states and their stability depend on key operating parameter values (dilution rate, gas–liquid mass transfer coefficient, biofilm detachment rate, and inlet substrate concentration). Small changes in the operating conditions may be coupled with a drastic change of the steady‐state scenario with transcritical and saddle‐node bifurcations. The relevance of concentration profiles establishing within the biofilm is also addressed. When the oxygen level in the liquid phase is <10% of the saturation level, the biofilm undergoes oxygen starvation and the active biofilm fraction becomes independent of the dilution rate. © 2011 American Institute of Chemical Engineers Biotechnol. Prog., 2011     

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.     

On the mechanistic origin of damped oscillations in biochemical reaction systems

A generalized reaction scheme for the kinetic interaction of two reactants in a metabolic pathway has been examined in order to establish what minimal mechanistic patterns are required to support a damped oscillatory transient-state kinetic behaviour of such a two-component system when operating near a steady state. All potentially oscillating sub-systems inherent in this scheme are listed and briefly characterized. The list includes several mechanistic patterns that may be frequently encountered in biological system (e.g. involving feedback inhibition, feed-forward activation, substrate inhibition or product activation), but also draw attention to some hitherto unforeseen mechanisms by which the kinetic interaction of two metabolites may trigger damped oscillations. The results can be used to identify possible sources of oscillations in metabolic pathways without detailed knowledge about the explicit rate equations that apply.     

Exacting predictions by cybernetic model confirmed experimentally: Steady state multiplicity in the chemostat

We demonstrate strong experimental support for the cybernetic model based on maximizing carbon uptake rate in describing the microorganism's regulatory behavior by verifying exacting predictions of steady state multiplicity in a chemostat. Experiments with a feed mixture of glucose and pyruvate show multiple steady state behavior as predicted by the cybernetic model. When multiplicity occurs at a dilution (growth) rate, it results in hysteretic behavior following switches in dilution rate from above and below. This phenomenon is caused by transient paths leading to different steady states through dynamic maximization of the carbon uptake rate. Thus steady state multiplicity is a manifestation of the nonlinearity arising from cybernetic mechanisms rather than of the nonlinear kinetics. The predicted metabolic multiplicity would extend to intracellular states such as enzyme levels and fluxes to be verified in future experiments. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Multiple stable states and hysteresis in continuous, oscillating cultures of budding yeast

The conditions that precede the onset of autonomous oscillations in continuous yeast cultures were studied in three different types of experiments. It was found that the final state of the culture depended on the protocol used to start up the reactor. Batch cultures, switched to continuous operation at different stages of the batch growth curve, all exhibited similar dynamics-ethanol depletion followed by autonomous oscillations. Small perturbations of the distribution of states in the reactor, achieved by addition of externally grown cells, were able to quench the oscillatory dynamics. Reaching the desired operating point by slow dilution rate changes gave rise to different final states, two oscillatory states and one steady state, depending on the rate of change in dilution rate. The multiplicity of stable states at a single operating point is not explained by any current distributed model and points toward a segregated mechanism of these oscillations.     

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.     

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.     

Qualitative analysis of a mathematical model of a open enzyme reaction

A general case of the set of two differential equations, describing an open reaction v1 leads to S v reversible E P v2 leads to, has been considered. The requirements to the character of the functions v1([S]), v2([P]) and v([S], [P]) were formulated for the case of existence and absence of alternative steady states and sustained oscillations. The formulae were derived to determine the slope of the unstable portion of the quasi-steady state characteristic. The generalized model of Monod, Wyman and Changeux has been considered as an example of v([S], [P]). It has been shown that with monotonically decreasing v1 and monotonically increasing v2, the alternative steady states and oscillations are possible only in the presence of substrate inhibition or product activation. However, under the joint action of substrate inhibition and product activation, the system will exhibit bistability rather than an oscillatory behavior. In the case of an irreversible two-substrate reaction which can be described by a similar mathematical model, inhibition by the first and second substrate is equivalent to substrate inhibition and product activation.     

Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture: II. Analysis of metabolic rates and pathways under oscillation and steady-state conditions

The oscillation phenomena reported in the preceding article for the anaerobic continuous fermentation of glycerol by Klebsiella pneumoniae are analyzed in terms of metabolic fluxes (metabolic rates and yields) and stoichiometry of pathways. Significant oscillations in the fluxes of CO(2), H(2), formic acid, ethanol, and reducing equivalents are observed which show obvious relationships to each other. Changes in the consumption or production rates of glycerol, acetic acid, 1,3-propanediol, and ATP are irregular and have relatively small amplitudes compared with their absolute values. By comparing the metabolic fluxes under oscillation and steady state that have nearly the same environmental conditions it could be shown that pyruvate metabolism is the main step affected under oscillation conditions. The specific formation rates of all the products originating from pyruvate metabolism (CO(2), H(2), formic acid, ethanol, acetic acid, lactic acid, and 2,3-butanediol) show significant differences under conditions of oscillation and steady state. In contrast, the specific rates of substrate uptake, ATP generation, and formation of products deriving either directly from glycerol (1,3-propanediol) or from the upstream of pyruvate metabolism (e.g., succinic acid) are not, or at least not significantly, affected during oscillation. Stoichiometric analysis of metabolic pathways confirms that other enzyme systems, in addition to pyruvate: formate-lyase, must be simultaneously involved in the pyruvate decarboxylation under both oscillation and steady-state conditions. The results strongly suggest oscillations of activities of these enzymes under oscillation conditions. It appears that the reason for the occurrence of oscillation and hysteresis lies in an unstable regulation of pyruvate metabolism of different enzymes triggered by substrate excess and drastic change(s) of environmental conditions. (c) 1996 John Wiley & Sons, Inc.     

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     

Dynamics of commensalistic systems with self- and cross-inhibition

Up to three stable steady states are possible in a simple commensalistic system, taking place in an open-loop mixed reactor when the growth rates of the two species are inhibited by the substrates they prey on (Self-inhibition). Two stable states are possible in a system with noncompetitive inhibition of the species by the substrate they are not preying on (cross-inhibition). A large number of steady states as well as oscillatory states are possible when both self- and cross-inhibition are strong. Multiplicity of steady states is also possible in a reactor with biomas recirculation for these kinetics. Yet, the latter is more stable than the open-loop reactor in the sense that the domain of steady-state multiplicity is narrower. The stability of steady states and the dynamics of the systems for each of the investigated kinetics are summarized in a qualitative phase plane. The importance of the analysis for improving the selectively and yield of the system and for predicting the response of the system to changes in the operating conditions, is discussed.     

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.     

Overflow metabolism during anaerobic growth of Klebsiella aerogenes NCTC 418 on glycerol and dihydroxyacetone in chemostat culture

Klebsiella aerogenes NCTC 418 was grown anaerobically in chemostat culture with glycerol as source of carbon and energy. Glycerol-limited cultures did not ferment the carbon source with maximal efficiency but produced considerable amounts of 1,3-propanediol. The fraction of glycerol converted to this product depended on the growth rate and on the limitation: faster growing cells produced relatively more of this compound. Under glycerol excess conditions the energetic efficiency of fermentation was decreased due to the high 1,3-propanediol excretion rate. Evidence is presented that 1,3-propanediol accumulation exerts a profound effect on the cells' metabolic behaviour.When steady state glycerol-limited cultures were instantaneously relieved of the growth limitation a vastly enhanced glycerol uptake rate was observed, accompanied by a shift in the fermentation pattern towards 1,3-propanediol and acetate. This observation was consistent with the extremely high glycerol dehydrogenase activity that was measured in vitro. Some mechanisms that could be responsible for the energy dissipation during this response are discussed.     

Modelling effects of high product and substrate inhibition on oscillatory behavior in continuous bioreactors

In this study we consider a model for continuous bioreactors which incorporates the effects of high product and substrate inhibition on the kinetics as well as biomass and product yields. We theoretically investigate the possibility of various dynamic behaviors in the bioreactor over different ranges of operating parameters to determine the delineating process conditions which may lead to oscillatory behavior. Application of the singular perturbation technique allows us to derive explicit conditions on the system parameters which specifically ascertain the existence of limit cycles composed of concatenations of catastrophic transitions occurring at different speeds. We discover further that the interactions between the limiting substrate and the growing microorganisms can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low-frequency cycle. Such a study not only can describe more fully the kinetics in a fermentor but also assist in formulating optimum fermentor operating conditions and in developing control strategy for maintaining optimum productivity.     

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.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

Steady state growth space study of <Emphasis Type="Italic">Lactococcus lactis</Emphasis> in D-stat cultures

Growth space of Lactococcus lactis subsp. lactis IL1403 was studied at constant growth rate using D-stat cultivation technique. Starting from steady state conditions in a chemostat culture (μ = 0.2 h⁻¹), the pH and/or temperature were continuously changed in the range of 5.4–6.4 and 26–34°C, respectively, followed by the return to the initial environmental conditions. Based on substrate consumption and product formation yields and expression changes of 1,920 genes, it was shown that changes of physiological state were not dependent on the direction of movement (from pH 6.3 to 5.4 or from 5.4 to 6.3), showing that quasi steady state values in D-stat corresponded to the steady state values in chemostats. Relative standard deviation of growth characteristics in triplicate D-stat experiments was below 10%. Continuing the experiment and reestablishing initial growth conditions revealed in average 7% difference (hysteresis) in growth characteristics when comparing chemostat steady state cultures prior and after the change of environmental conditions. Similarly, shifts were also seen at gene expression levels. The large amount of quantitatively reliable data obtained in this study provided a new insight into dynamic properties of bacterial physiology, and can be used for describing the growth space of microorganisms by modeling cell metabolism.     

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.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.