Bistability in the isocitrate dehydrogenase reaction: an experimentally based theoretical study.

The enzyme isocitrate dehydrogenase (IDH, EC 1.1.1.42) can exhibit activation by one of its products, NADPH. This activation is competitively inhibited by the substrate NADP+, whereas NADPH competes with NADP+ for the catalytic site. Experimental observations briefly presented here have shown that if IDH is coupled to another enzyme, diaphorase (EC 1.8.1.4), which transforms NADPH into NADP+, the system can attain either one of two stable states, corresponding to a low and a high NADPH concentration. The evolution toward either one of these stable states depends on the time of addition of diaphorase to the medium containing IDH and its substrate NADP+. We present a theoretical and numerical analysis of a model for the IDH-diaphorase bienzymatic system, based on the regulatory properties of IDH. The results confirm the occurrence of bistability for parameter values derived from the experiments. Depending on the total concentration of NADP+ plus NADPH and the concentration of IDH, the system can either admit a single steady state or display bistability. We obtain an expression for the critical time t*, before which diaphorase addition leads to the lower steady state and after which addition of the enzyme leads to the upper steady state of NADPH. The analysis is extended to the case where the second substrate of IDH, isocitrate, is consumed in the course of the reaction without being regenerated. Bistability occurs only as a transient phenomenon in these conditions.     

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     

Dissipation and maintenance of stable states in an enzymatic system: analysis and simulation

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. An essential assumption for constraint-based analysis is the formation of a stable steady state. This work investigates dissipation and maintenance of stable states in a simple reversible enzymatic reaction with substrate inhibition. Under mass-action kinetics, the conditions under which the reaction maintains a stable steady state are analytically derived and numerically confirmed. It is shown that, in order to maintain a steady state in the regulated reaction, maximal enzyme activity must be much higher than input rate. Moreover, it is revealed that requirements for large enzyme activity are due to substrate inhibition. It is suggested that high activities of enzymes may play a vital role in protecting a stable state from its catastrophic collapse, giving an additional explanation to an intriguing problem—why the activities of some enzymes greatly exceed the flux capacity of a pathway. In addition, dissipation of the enzymatic reaction is analysed. It is shown that the collapse of stable states is always associated with a point at which dissipation is the highest. Therefore, in order to maintain a stable state, dissipation of the reaction must be less than a critical value. Moreover, although external forcing may not change net mass flow, it may lead to collapse of stable states. Furthermore, when stable states collapse at a critical forcing amplitude and period, dissipation also reaches a highest value. It is concluded that collapse of stable steady state in the enzyme system with substrate inhibition always corresponds to critical points at which dissipation is highest, regardless if the reaction is forced or not. Therefore, for the substrate inhibited reaction, maintenance of stable states is intrinsically related to level of dissipation.     

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.     

Transient behavior of a continuous stirred tank biological reactor utilizing phenol as an inhibitory substrate

Transient experiments were conducted on a Pseudomomas utilizing phenol in a continuous culture by disturbing the influent substrate concentration and dilution rate. Two stable steady states existed for some ranges of the parameters. Highly damped oscillations were observed in approaching a new high conversion steady state or in returning to a new high conversion steady state following a small disturbance. When a large disturbance was applied there was a smooth (overdamped) approach to a new low conversion steady state. The observed oscillatory behavior for small disturbances was predicted by a modified Powell-Ierusalemskii bottleneck model, but could not be predicted by a Monod-Haldane model; neither model was accurate for predicting the effect of large disturbances. A constant wall growth factor was used to account for microbial film activity, and the existence of two stable states was directly due to the presence of the film.     

Dynamic analysis of lactic acid fermentation in membrane bioreactor

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. This novel theoretical framework could result in an objective criterion on how to control the substrate concentration in order to keep a sustainable and steady output of lactic acid. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincaré-Bendixson Theorem, we obtain the conditions for the globally asymptotical stability of the equilibrium. Secondly, impulsive input substrate is also considered. Using Floquet's theorem and small-amplitude perturbation, we obtain the biomass-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

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.     

Effect of wall growth on scale-up problems and dynamic operating characteristics of the biological reactor

If inhibitory substrates are being utilized in a well-stirred biological reactor, microbiological growth on the walls of the reactor can create a scale-up problem. A simple model is proposed which shows that without such growth, of the three existing steady states only one is stable and nontrivial, but with wall growth the trivial, stable, steady state (washout) is impossible. In addition, wall growth reduces the region over which three steady states are feasible and reduces the minimum residence time for which there is only one steady state that corresponds to a high conversion. Thus, a laboratory process with a high surface area to volume ratio can give an over optimistic prediction of both necessary residence; time and stability of the full scale process unless wall growth is accounted for.     

SOME THOUGHTS ON NUTRIENT LIMITATION IN ALGAE1

An empirical relation relating specific growth, rate in steady state systems to nutrient status with respect to more than one nutrient simultaneously is proposed, based on 3 experimentally verifiable postulates: (1) that uptake depends on the external substrate concentration; (2) that growth depends on the interval substrate concentration; and (3) in a steady state system specific rate of uptake (in the absence of significant, excretion) is necessarily the product of the specific growth rate and internal substrate concentration. The implications of this model are discussed in particular in respect to the concept of luxury consumption and Liebig's law of minimum. Some aspects of uptake in transient situations are also discussed.     

Evaluation of kinetic parameters of biochemical reaction in three-phase fluidized bed biofilm reactor for wastewater treatment

This study evaluates the kinetic parameters of biochemical reaction in three-phase fluidized bed biofilm reactor from the steady state values of the response of the system to step changes in inlet concentration. It was observed from the outlet biological oxygen demand (BOD(5)) plot of the response of the system that as the inlet BOD(5) was increased, the outlet BOD(5) also increased, reached a peak value and then decreased until it leveled to a new steady state value corresponding to the new inlet concentration level. The increase in BOD(5) was attributed to the accumulation of substrate within the reactor as well as the decrease in biofilm substrate consumption rate as the microorganisms adjusted to the new environment. Using the substrate balance at steady state and assuming Monod kinetics, an equation relating the substrate consumption rate to substrate concentration (BOD(5)) and total biofilm surface area had been established. Monod kinetic parameters were found to be K=2.20g/m(2)/day, K(m)=17.41g/m(3) and K/K(m)=0.13m/day. The ratio K/K(m) can be taken as the indicator for biofilm substrate degradation effectiveness at low substrate concentrations.     

A kinetic description of sequential,reversible, Michaelis-Menten reactions: practical application of theory to metabolic pathways

Equations are presented which describe a linear coupled system of reactions that utilize a single substrate and convert it to product by way of several intermediate enzyme catalysed steps. The present analysis extends previous results by assuming that the enzymes obey reversible Michaelis-Menten kinetics. In order for the system to reach steady state one must assume that the initial substrate concentration and the final product concentration are buffered to a constant value. Using the present analysis it can be shown that the system will not enter a steady state if the maximal velocity of any forward reaction is less than the steady state flux through the system. This condition represents a practical test for determining if a system will enter steady state but is valid only when the rate of the primary enzyme is not affected allosterically be intermediates in the pathway. The equations are used to analyse a portion of the rat liver glycogenic pathway that catalyses the conversion of glucose to fructose 1,6-bisphosphate.     

Mixed culture biooxidation of phenol. III. Existence of multiple steady states in continuous culture with wall growth

It is shown that two steady states exist in certain regions of operation of a 2-liter continuous stirred tank biological reactor. Transition was made from one steady state to another by applying shock loads of either phenol substrate which is inhibitory to the culture at high concentrations or by adding large additional amounts of concentrated organisms. The existence of the multiple steady states is ascribed to the existence of wall growth, and their position is determined by the amount of wall growth. Transient behavior of the system did not follow the predictions of the simple wall growth model but the culture appeared to undergo a lag period immediately after applying the shock load to the system. It is concluded that the stability of a continuous culture utilizing an inhibitory substrate is improved by increasing the degree of wall growth and decreasing the substrate feed concentration. It is also concluded that small scale experiments can usually not be interpreted correctly unless the effect of wall growth is taken into account.     

Enzyme reactions at the surface of living cells. II. Destabilization in the membranes and conduction of signals

The dynamics of a bound-enzyme reaction is studied when the diffusion of both the substrate and the product is coupled to their electric repulsion and to enzyme reaction. Contrary to what is occurring when substrate diffusion is uncoupled with electric repulsion and enzyme reaction, no hysteresis loop of the partition coefficient exists. The electric partition coefficient monotonically declines as substrate or product concentration is increased in the reservoir. The random perturbation of a steady state may generate a localized destabilization of substrate and product concentration. This destabilization must propagate in the membrane and may be viewed as the conduction of a signal. These conduction phenomena are entirely due to electric effects. In the absence of these effects, the system is homeostatic, that is it returns back to its initial steady state after a perturbation. Obviously under these conditions conduction of signals cannot occur. Increasing the ionic strength of the external milieu tends to stabilize the system and to suppress conduction effects in the membrane.     

Metabolic homeostasis in the human erythrocyte: in silico analysis

A detailed computer model of human erythrocyte metabolism was shown to predict three steady states, two stable and one unstable. The most extreme steady state is characterized by almost zero concentrations of all the phosphorylated intermediates. The "normal" steady state is remarkably robust in the face of large changes in the activity of most of the enzymes of glycolysis and the pentose phosphate pathway: this steady state can be viewed as an attractor towards which the system returns following a metabolic perturbation. Focus is given to three responses of the system: (1) the 'energy charge' that pertains to the concentration of ATP relative to all purine nucleotides; (2) redox power expressed as the ratio of reduced-to-total glutathione and (3) the concentration of 2,3-bisphosphoglycerate, that directly affects the oxygen affinity of haemoglobin thus affecting the main physiological function of the cell. The collapse of the normal steady state in what can be viewed topologically as a catastrophe is posited as one key element of erythrocyte senescence and it is particularly important for erythrocyte destruction in patients with an inborn enzyme deficiency.     

Kinetics and performance of a co-immobilised system of amyloglucosidase and Zymomonas mobilis.

High operational stability and productivity of co-immobilised systems are important aspects for their successful application in industrial processes. A dynamic model is required to describe artificially co-immobilised systems because the time needed to reach steady state normally exceeds the operational life span of these systems. Time dependent intraparticle concentration profiles and macroscopic conversion were modelled to study the operational stability and productivity of these systems theoretically. The model was used to describe experimental results of ethanol production from maltose by a co-immobilised system of amyloglucosidase and Zymomonas mobilis. Furthermore, the influence of the immobilisation procedure with glutaraldehyde and polyethyleneimine could also be studied with and incorporated in the model. From the model it could be derived that co-immobilised systems performing a consecutive reaction evolve towards a steady state, characterised by a constant concentration of the intermediate in the particle if product inhibition is neglected. Such a situation develops independently of the biomass concentration and the radial position, and has important consequences for co-immobilised systems. When the concentration of the intermediate in the bulk liquid is lower than this constant value in the biocatalyst particle, two regions may be distinguished in the particle: an inactive peripheral region without biomass and an active core with a biomass concentration depending on the substrate and immobilised enzyme concentration. Unlike immobilised single cell systems, it is possible to obtain a real steady state and therefore a stable situation for co-immobilised systems. However, a high operational life time could only be achieved at the expense of the productivity of the biocatalyst particle. A stability criterion is derived which agrees very well with the simulation results.     

Simplified velocity equations for characterizing the partial inhibition or nonessential activation of bireactant enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820 825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V'max, K'ia, K'mA, K'mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

Simplified Velocity Equations for Characterizing the Partial Inhibition or Nonessential Activation of Bireactant Enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820–825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V¹_max, K¹_ia, K¹_mA,K¹_mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

Steady-state properties of a model ternary substrate cycle: theoretical predictions.

Numerous ternary substrate cycles are metabolically operative in vivo. The relative concentrations of the interconverted substrates are generally correlated with different physiological states. These cycles often include reversible and/or substrate-inhibited enzymic steps. The switch between one steady state (metabolic state) and another may be the consequence of either the effect of an exogeneous metabolite or signal, or the alteration of a cycle internal parameter. The interpretation of results obtained with currently designed experiments on substrate cycles seldom take into account the very dynamic and regulatory properties inherent in the cyclic and often autocatalytic nature of the pathway. In the present report, the various dynamic properties of a model ternary substrate cycle, bounded by moiety conservation, are investigated. Three situations with increasing complexity are considered: (i) the three enzymes are michaelian and catalyse irreversible steps; (ii) one of the enzymic steps is reversible; and (iii) one step is subjected to a destabilizing factor, i.e. inhibition by excess of substrate. The behavior(s) of the whole cycle is mainly controlled by four parameters, that is, ST, the total concentration of the substrate pool, and the three enzyme maximal velocities, VMi (i = 1,2,3). As ST (= S1 + S2 + S3) is constant, the Si steady-state concentrations (stable or not) can be represented in barycentric coordinates in a triangle (simplex). This convenient representation allows us to predict the different states of the system when one enzyme maximal activity is varied. The steady-state concentration dependencies as a function of one or several parameters may be either monostable (possibility of zero-order ultrasensitivity) or bistable (with or without reversible transitions). The physiological and experimental relevances of these observations are emphasized.     

Emergent properties of coupled enzyme reaction systems 1. Switching and clustering behaviour

The dynamics of locally and globally coupled cells that convert a substrate to a product via an uncompetitive substrate-inhibition mechanism is studied. When the cell-cell coupling strength is below a threshold value, the coupled system exhibits a large number of steady states; however, all cells cluster to one state when coupling exceeds the threshold value. The coupled system also exhibits a buffering capacity that maintains low and almost constant intracellular and extracellular substrate levels; however, there exists a threshold value on the influx rate of extracellular substrate beyond which the system switches to higher substrate levels. This transition becomes sharper as the number of coupled cells increases. Propagation failure of concentration fronts between adjacent cells is also exhibited by the system.     

The dynamics of single-substrate continuous cultures: the role of transport enzymes

A chemostat limited by a single growth-limiting substrate displays a rich spectrum of dynamics. Depending on the flow rate and feed concentration, the chemostat settles into a steady state or executes sustained oscillations. The transients in response to abrupt increases in the flow rate or the feed concentration are also quite complex. For example, if the increase in the flow rate is small, there is no perceptible change in the substrate concentration. If the increase in the flow rate is large, there is a large increase in the substrate concentration lasting several hours or days before the culture adjusts to a new steady state. In the latter case, the substrate concentration and cell density frequently undergo damped oscillations during their approach to the steady state. In this work, we formulate a simple structured model containing the inducible transport enzyme as the key intracellular variable. The model displays the foregoing dynamics under conditions similar to those employed in the experiments. The model suggests that long recovery times (on the order of several hours to several days) can occur because the initial transport enzyme level is too small to cope with the increased substrate supply. The substrate concentration, therefore, increases until the enzyme level is built up to a sufficiently high level by the slow process of enzyme induction. Damped and sustained oscillations can occur because transport enzyme synthesis is autocatalytic, and hence, destabilizing. At low dilution rates, the response of stabilizing processes, such as enzyme dilution and substrate consumption, becomes very slow, leading to damped and sustained oscillations.