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.     

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.     

Denaturable immobilized enzymatic reaction in a packed-bed reactor with and without substrate inhibitions

Summary An immobilized enzymatic reaction in a packed-bed reactor is investigated in this paper. The thermal denaturation of immobilized enzyme caused by excessive reacting temperature rise is considered. An unsteady state dispersion model is employed to examine the dynamic behaviors of the substrate concentration, temperature and enzyme activity along the reactor. Also included in the present paper is the effect of substrate inhibition which occurs rather frequently in many enzymatic reactions. Comparison of results of the immobilized enzymatic reactions with and without substrate inhibitions are made to show the extent the substrate inhibition affects the enzymatic reaction. Furthermore, the effects of heat reaction and the Peclet number which characterize the reaction and flow behaviors, respectively, on the system considered are analyzed in detail.     

Effectiveness factors for substrate and product inhibition

The transformation technique of Na and Na (Math. Biosci., 6 , 25, 1970) is extended to convert boundary-value problems involving the steady-state diffusion equation for spherical immobilized enzyme particles exhibiting substrate and product inhibition to initial-value problems. This allows a study of the influence of external mass transfer resistances on the effectiveness factors. It also considerably reduces the number of calculations required to investigate the effect of changes in the kinetic parameters on the overall rate of reaction. The existence of multiple steady states for substrate inhibition kinetics in spherical catalyst particles is illustrated and a criterion for uniqueness of steady states is developed. Effectiveness factors for competitive and noncompetitive product inhibition increase with increasing value of the Sherwood number for the substrate and increasing value of the ratio of substrate to product effective diffusivities within the particle.     

Microscopic diffusion-reaction coupling in steady-state enzyme kinetics

The theory of diffusion-controlled processes is applied to describe the steady state of a reversible enzymatic reaction with special emphasis on the effects of enzyme saturation. A standard macroscopic steady-state treatment requires only that the average diffusional influx of substrate equals the net reaction flux as well as the average diffusional efflux of product. In contrast, the microscopic diffusion-reaction coupling used here takes properly into account the conditional concentration distributions of substrate and product: Only when the enzyme is unoccupied will there be a diffusional association flux; when the enzyme is occupied, the concentration distributions will relax towards their homogeneous bulk values. In this way the relaxation effects of the non-steady state will be constantly reoccurring as the enzyme shifts between unoccupied and occupied states. Thus, one is forced to describe the steady state as the weighted sum of properly time-averaged non-stationary conditional distributions. The consequences of the theory for an appropriate assessment of the parameters obtained in Lineweaver-Burk plots are discussed. In general, our results serve to justify the simpler macroscopic coupling scheme. However, considerable deviations between the standard treatment and our analysis can occur for fast enzymes with an essentially irreversible product release.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.     

Subunit interactions in enzyme transition states--antagonism between substrate binding and reaction rate

The principles of structural kinetics as applied to polymeric enzymes have been reinvestigated in order to take account of the probable existence of subunit interactions in the enzyme transition states. On the basis of simple and plausible postulates, structural rate equations have been derived for dimeric enzymes and compared to substrate binding isotherms. It then becomes possible to understand how subunit interactions affect substrate affinity and enzyme reaction rate. There exists an antagonism between substrate binding to the enzyme and the steady state rate of product appearance. If subunit interactions increase the rate of product appearance, they decrease the fractional saturation of the enzyme by the substrate. Alternatively, if they decrease the reaction velocity they increase the fractional saturation. This seemingly paradoxical effect is the direct consequence of subunit interactions occurring in both the ground and the transition states.     

Bistability and the ordered bimolecular mechanism.

An equation describing the instantaneous velocity of an ordered bimolecular enzymatic reaction that exhibits inhibition by substrate and product was derived. Using kinetic constant values for horse liver alcohol dehydrogenase, the velocity expression was applied to an open-reaction system. The calculated steady-state surfaces displayed regions of bistability, which further substantiates the link between substrate inhibition and multiple steady states. This general computational approach may be applied to any system that can be described by an instantaneous velocity equation.     

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.     

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.     

Hormone regulation of cellular metabolism: Interplay of membrane transport and consecutive enzymic reaction

The regulatory effect of hormones on a steady-state process that consists of the mediated transport of a substrate across a membrane and a consecutive enzymic reaction is examined theoretically. The regulation of such a process depends not only on the effect of the hormone on the transport system, but also on the kinetic parameters of the reaction. The rate of metabolism can both increase and decrease due to a hormonal increase in the transport capacity, the affinity of the carrier to the substrate or the amount of energy involved, although the rate of mediated transport per se is always enhanced by such effects. Substrate inhibition of the enzyme can lead to multiple steady states and, thus, amplify hormone action. The quantitative results are believed to give insight into the hormonal regulation of both cellular uptake and intracellular metabolism.     

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.     

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.     

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     

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

Kinetic consequences of a slow substrate binding step in group transferases. Interpretation of product inhibition experiments.

The steady state kinetic properties of a simple model for an enzyme catalyzed group transfer reaction between two substrates have been calculated. One substrate is assumed to bind slowly and the other rapidly to the enzyme. Apparent substrate inhibition or substrate activation by the rapidly binding substrate may result if the slowly binding substrate binds at unequal rates to the free enzyme and to the complex between the enzyme and the rapidly binding substrate. Competitive inhibition by each product with respect to its structurally analogous substrate is to be expected if both substrates are in rapid equilibrium with their enzyme-substrate complexes. This product inhibition pattern, however, may also be observed when one substrate binds slowly. Noncompetitive inhibition with respect to the rapidly binding substrate by its structurally analogous product may result if the slowly binding substrate binds more slowly to the enzyme-product complex than to the free enzyme. Inhibition by substrate analogs which are not products should follow the same rules as inhibition by products. Thus substrate analog inhibition experiments are not particularly informative. The form of inhibition by "transition state analog" inhibitors should reveal which substrate binds slowly. There is no sharp conceptual distinction between ordered and random "kinetic mechanisms". I therefore suggest that the use of these concepts should be abandoned.     

Steady state kinetics of an enzyme reaction with one substrate and one modifier

This study examines the steady state kinetics of a reaction involving an enzyme, a substrate and a modifier when the reaction follows Michaelis-Menten kinetics. Conditions for Michaelis-Menten kinetics are deduced, and it is shown that an analogue of detailed balance determines the complexity of the rate equations in these cases. A scheme to distinguish many cases of Michaelis-Menten kinetics is presented. It is shown that steady state kinetics are, in general, insufficient to specify the mechanism of a reaction, since different effects of a modifier can give identical steady state kinetic data.     

Kinetic constraints for formation of steady states in biochemical networks

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. This work introduces the concept of kinetic constraints. It is shown that maximal reaction rates are appropriate constraints only for isolated enzymatic reactions. For biochemical networks, it is revealed that constraints for formation of a steady state require specific relationships between maximal reaction rates of all enzymes. The constraints for a branched network are significantly different from those for a cyclic network. Moreover, the constraints do not require Michaelis-Menten constants for most enzymes, and they only require the constants for the enzymes at the branching or cyclic point. Reversibility of reactions at system boundary or branching point may significantly impact on kinetic constraints. When enzymes are regulated, regulations may impose severe kinetic constraints for the formation of steady states. As the complexity of a network increases, kinetic constraints become more severe. In addition, it is demonstrated that kinetic constraints for networks with co-regulation can be analyzed using the approach. In general, co-regulation enhances the constraints and therefore larger fluctuations in fluxes can be accommodated in the networks with co-regulation. As a first example of the application, we derive the kinetic constraints for an actual network that describes sucrose accumulation in the sugar cane culm, and confirm their validity using numerical simulations.     

Formation of C-C bonds by mandelonitrile lyase in organic solvents

Mandelonitrile lyase (EC 4.1.2.10) catalyzes the formation of D-mandelonitrile from HCN and benzaldehyde. Mandelonitrile lyase was immobilized by adsorption to support materials, for example, Celite. The enzyme preparations were used in diisopropyl ether for production of D-mandelonitrile. In order to obtain optically pure D-mandelonitrile it was necessary to use reaction conditions which favor the enzymatic reaction and suppress the competing spontaneous reaction, which yields a racemic mixture of D, L-mandelonitrile. The effects of substrate concentrations, water content, and support materials on both the spontaneous and enzymatic reactions were studied. The enzymatic reaction was carried out under conditions where the importance of the spontaneous reaction was negligible and high enantiomeric purity of D-mandelonitrile was achieved (at least 98% enantiomeric excess). The operational stability of the enzyme preparations was studied in batch as well as in continuous systems. It was vital to control the water content in the system to maintain an active preparation. In a packed bed reactor the enzyme preparations were shown to be active and stable. The reactors were run for 50 h with only a small decrease in product yield.