pH modulation of transient state kinetics of enzymes. I. Simple theoretical models

The effect of proton concentration on pre-steady-state kinetics has been investigated theoretically for enzyme reactions involving the breaking of one substrate into two products. Even for the simple double-intermediate mechanism the approach to the steady state may exhibit a rather complex kinetics, which is pH-dependent. This process may even exhibit damped oscillations. A change of pH may completely change this transient kinetics and even suppresses the oscillatory regime. A simple method is presented which allows estimation of the values of the rate and ionization constants. This procedure allows one to distinguish the simple double-intermediate mechanism from a more complex process where the 'fast' binding of the substrate induces a 'slow' conformation change of the enzyme.     

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.     

Oscillatory enzyme reactions and Michaelis–Menten kinetics

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis–Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis–Menten phosphorylation–dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.     

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.     

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.     

Oscillatory reaction of chondroitin sulfate induced by gradual introduction of calcium ion

Rhythm is an important dynamic behavior in biological systems. We have been studying oscillatory reactions of enzymes induced by gradual entry of substances through semipermeable membrane. Not only enzymes but also a few species of substance of living system have been elucidated to cause oscillatory reaction. Here we present the oscillatory reaction by chondroitin sulfate in a system of gradual entry of calcium ion. Introducing calcium ion through dialysis membrane into chondroitin sulfate solution induces an oscillation of free calcium ion concentration in chondroitin sulfate solution. Simultaneously, it is elucidated that oscillation of conformation occurs with permeation of calcium ion. In both measurements, oscillations with 25h period are obtained. The phases of oscillation, however, differ slightly from each other. From these results, it is suggested that autocatalysis exerts in the contraction of chondroitin sulfate conformation. These phenomena are very intriguing for elucidating oscillation in living system.     

Oscillatory reaction of alcohol dehydrogenase in an oil/water system

With the use of an oil/water system, oscillatory reactions of an enzyme have been demonstrated. This reaction system has been conceived as an example of the metabolic oscillations of living cells. When a substrate (ethanol) in the oil phase of toluene or chloroform slowly migrated into the aqueous phase containing alcohol dehydrogenase and NAD+, oscillations were observed in the concentration of NADH produced. The gradual entry of substrate into the aqueous phase was essential for the oscillatory reactions to occur. A possible mechanism to account for the appearance of oscillatory reactions of enzymes is proposed, which differs from that presented previously.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

A general solution for the steady-state kinetics of immobilized enzyme systems

A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.     

Non-michaelian behavior of 5 alpha-reductase in human prostate

An in-depth analysis of the kinetics of 5 alpha-reductase in human prostatic tissue gave findings inconsistent with the claim that the enzyme is michaelian. In both hyperplastic and malignant tissue, the time-course of the conversion of testosterone (T) into dihydrotestosterone (DHT) was non-linear under conditions ensuring less than 15% conversion of substrate and cofactor. An initial rapid phase of conversion was followed by a long steady-state phase. This time-dependent change in conversion rate was not due to enzyme denaturation, fast inhibition by substrate or product effects. It resulted from a true slow transient kinetic process induced in the reactive enzyme by the substrates. Under our experimental conditions at pH 5.5, 5 alpha-reductase appeared to undergo a conformational change from an initially highly reactive form to a less reactive form. Since this "hysteretic" behavior was correlated with apparently negative cooperativity in enzyme kinetics, we postulate that, as previously described for other key metabolic enzymes, regulation of 5 alpha-reductase activity in the prostate depends on the molecular flexibility of the enzyme and on changes in the cooperativity of different enzyme forms over time. This original non-michaelian behavior may explain the conflicting kinetics reported so far in the literature for this enzyme. The clinical implications of 5 alpha-reductase hysteresis and its involvement in the damping of DHT production within the prostate are discussed.     

Kinetic mechanism of phosphofructokinase-2 from Escherichia coli. A mutant enzyme with a different mechanism

The kinetic mechanisms of Escherichia coli phosphofructokinase-2 (Pfk-2) and of the mutant enzyme Pfk-2 were investigated. Initial velocity studies showed that both enzymes have a sequential kinetic mechanism, indicating that both substrates must bind to the enzyme before any products are released. For Pfk-2, the product inhibition kinetics was as follows: fructose-1,6-P2 was a competitive inhibitor versus fructose-6-P at two ATP concentrations (0.1 and 0.4 mM), and noncompetitive versus ATP. The other product inhibition patterns, ADP versus either ATP or fructose-6-P were noncompetitive. Dead-end inhibition studies with an ATP analogue, adenylyl imidodiphosphate, showed uncompetitive inhibition when fructose-6-P was the varied substrate. For Pfk-2, the product inhibition studies revealed that ADP was a competitive inhibitor versus ATP at two fructose-6-P concentrations (0.05 and 0.5 mM), and noncompetitive versus fructose-6-P. The other product, fructose-1, 6-P2, showed noncompetitive inhibition versus both substrates, ATP and fructose-6-P. Sorbitol-6-P, a dead-end inhibitor, exhibited competitive inhibition versus fructose-6-P and uncompetitive versus ATP. These results are in accordance with an Ordered Bi Bi reaction mechanism for both enzymes. In the case of Pfk-2, fructose-6-P would be the first substrate to bind to the enzyme, and fructose-1,6-P2 the last product to be released. For Pfk-2, ATP would be the first substrate to bind to the enzyme, and APD the last product to be released.     

A mechanistic model for enzymatic saccharification of cellulose using continuous distribution kinetics II: cooperative enzyme action, solution kinetics, and product inhibition

The projected cost for the enzymatic hydrolysis of cellulosic biomass continues to be a barrier for the commercial production of liquid transportation fuels from renewable feedstocks. Predictive models for the kinetics of the enzymatic reactions will enable an improved understanding of current limitations, such as the slow-down of the overall conversion rate, and may point the way for more efficient utilization of the enzymes in order to achieve higher conversion yields. A mechanistically based kinetic model for the enzymatic hydrolysis of cellulose was recently reported in Griggs et al. (2011) (Part I). In this article (Part II), the enzyme system is expanded to include solution-phase kinetics, particularly cellobiose-to-glucose conversion by β-glucosidase (βG), and novel adsorption and product inhibition schemes have been incorporated, based on current structural knowledge of the component enzymes. Model results show cases of cooperative and non-cooperative hydrolysis for an enzyme system consisting of EG(I) and CBH(I). The model is used to explore various potential rate-limiting phenomena, such as substrate accessibility, product inhibition, sterically hindered enzyme adsorption, and the molecular weight of the cellulose substrate.     

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.     

Nucleotide regulation of Escherichia coli glycerol kinase: initial-velocity and substrate binding studies

Substrate binding to Escherichia coli glycerol kinase (EC 2.7.1.30; ATP-glycerol 3-phosphotransferase) was investigated by using both kinetics and binding methods. Initial-velocity studies in both reaction directions show a sequential kinetic mechanism with apparent substrate activation by ATP and substrate inhibition by ADP. In addition, the Michaelis constants differ greatly from the substrate dissociation constants. Results of product inhibition studies and dead-end inhibition studies using 5'-adenylyl imidodiphosphate show the enzyme has a random kinetic mechanism, which is consistent with the observed formation of binary complexes with all the substrates and the glycerol-independent MgATPase activity of the enzyme. Dissociation constants for substrate binding determined by using ligand protection from inactivation by N-ethylmaleimide agree with those estimated from the initial-velocity studies. Determinations of substrate binding stoichiometry by equilibrium dialysis show half-of-the-sites binding for ATP, ADP, and glycerol. Thus, the regulation by nucleotides does not appear to reflect binding at a separate regulatory site. The random kinetic mechanism obviates the need to postulate such a site to explain the formation of binary complexes with the nucleotides. The observed stoichiometry is consistent with a model for the nucleotide regulatory behavior in which the dimer is the enzyme form present in the assay and its subunits display different substrate binding affinities. Several properties of the enzyme are consistent with negative cooperativity as the basis for the difference in affinities. The possible physiological importance of the regulatory behavior with respect to ATP is considered.     

Damped oscillatory hysteretic behaviour of butyrylcholinesterase with benzoylcholine as substrate.

Steady-state kinetics for the hydrolysis of benzoylcholine (BzCh) and benzoylthiocholine (BzSCh) by wild-type human butyrylcholinesterase (BuChE) and by the peripheral anionic site mutant D70G were compared. kcat/Km for the hydrolysis of BzSCh was 17-fold and 32-fold lower than that for hydrolysis of BzCh by wild-type and D70G, respectively. The rate-limiting step for hydrolysis of BzCh was deacylation, whereas acylation was rate-limiting for hydrolysis of BzSCh. Wild-type enzyme and the D70G mutant were found to reach steady-state velocity slowly with BzCh as the substrate. At pH 6, the approach to steady-state for both enzymes consisted of a mono-exponential acceleration upon which a set of damped oscillations was superimposed. From pH 7 to 8.5, the approach to steady-state consisted of a simple exponential acceleration. The damped oscillations were analyzed by both a numerical approximation and simulation based on a theoretical model. BuChE-catalyzed hydrolysis of the thiocholine analogue of BzCh showed neither lags nor oscillations, under the same conditions. The frequency and amplitude of the damped oscillations decreased as the BzCh concentration increased. The apparent induction time for the exponential portion of the lag was calculated from the envelope of the damped oscillations or from the smooth lag. Wild-type BuChE showed a hyperbolic increase in induction time as the BzCh concentration increased (tau max = 210 s at pH 6.0). However, the induction time for D70G was constant over the whole range of BzCh concentrations (tau max = 60 s at pH 6.0). Thus, the induction time does not conform to a simple hysteretic model in which there is a slow conformational transition of the enzyme from an inactive form E to an active form E'. No pH-dependence of the induction time was found between pH 6.0 and 8.5 in sodium phosphate buffers of various concentrations (from 1 mm to 1 m). However, increasing the pH tended to abolish the oscillations (increase the damping factor). This effect was more pronounced for D70G than for wild-type. Although the lyotropic properties of phosphate change from chaotropic at pH 6.0 to kosmotropic at pH > 8.0, no effect of phosphate concentration on the oscillations was noticed at the different pH values, suggesting that the oscillations are not related to a pH-dependent Hofmeister effect of phosphate ions. Simulation and theoretical analysis of the oscillatory behaviour of the approach to the steady-state for BuChE led us to propose a model for the hysteresis of BuChE with BzCh. In this model, the substrate-free enzyme is present as an equilibrium mixture of two forms, E and E'. Substrate binds to E and E', but only Epsilon'S makes products. It is proposed that oscillations originate from a time-dependent change in the local concentration, solvation and/or conformation of substrate in the bulk solution. 1H-NMR measurements provided evidence for a slow equilibrium between two BzCh conformers. Binding of the conformationally preferred substrate conformer leads to products.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Enzyme deactivation

Enzyme deactivation kinetics is often first-order. Different examples of first-order deactivation kinetics exhibited by different enzymes under a wide variety of conditions are presented. Examples of both soluble and immobilized enzymes are presented. The influence of different parameters, chemical modification of specific residues, inhibitors, inactivators, protecting agents, induced conformational changes by external agents, enzyme concentration, and different substrates on the first-order inactivation kinetics of different enzymes is analyzed. The different examples presented from a variety of different areas provides a judicious framework and collection demonstrating the wide applicability of first-order deactivation kinetics. Examples of reversible first-order deactivation kinetics and deactivation-disguise kinetics are also presented.Different mechanisms are also presented to model complex enzyme deactivations. The non-series type mechanisms are emphasized and these involve the substrate and chemical modifiers. Substrate-dependent deactivation rate expressions that are of "separable" and "non-separable" type are presented. Rate expressions involving time-dependent rate constants along with their corresponding mechanisms are presented. Examples of enzymes that exhibit a deactivation-free grace period are also given. An interesting case of enzyme inactivation is the loss of activity in the presence of an auto-decaying reagent. The method is presented by which the intrinsic inactivation rate constants may be obtained. Examples of pH-dependent enzyme inactivation are presented that may be modelled by a five-step (or a simplified two-step) mechanism, and also by a single-step mechanism involving residual activity for the final state. Appropriate examples of enzyme inactivation are presented in each case to highlight the different mechanisms involved.     

Theory of the kinetics of reactions catalyzed by enzymes attached to membranes

A theoretical treatment has been worked out for the kinetics of solid-supported enzyme systems, with diffusive and electrostatic effects taken into account. A utilization factor, defined as the ratio of the actual reaction rate to the rate of substrate consumption in the outer solution, is defined, and equations to evaluate the utilization factor are given for five kinetic conditions: (a) Michaelis-Menten behavior, (b) substrate inhibition, (c) product inhibition (competitive), (d) product inhibition (noncompetitive), and (e) product inhibition (anticompetitive). When the solid-supported enzymes obey a Michaelis-Menten relationship, an equation for the apparent Michaelis constant is given and a criterion for insignificant diffusion effects is shown. A substrate-inhibited enzyme reaction may display multiple steady-state behavior, and a criterion for uniqueness is presented. In the case of product-inhibited enzyme reactions, the utilization factor is always less than that which corresponds to a Michaelis-Menten relationship. Equations to evaluate the apparent Michaelis and inhibition constants are given.     

Conservation of glycolytic oscillations in Saccharomyces cerevisiae and human pancreatic beta-cells: a study of metabolic robustness

The present study compares two computer models of the first part of glucose catabolism in different organisms in search of evolutionarily conserved characteristics of the glycolysis cycle and proposes the main parameters that define the stable steady-state or oscillatory behavior of the glycolytic system. It is suggested that in both human pancreatic beta-cells and Saccharomyces cerevisiae there are oscillations that, despite differences in wave form and period of oscillation, share the same robustness strategy: the oscillation is not controlled by only one but by at least two parameters that will have more or less control over the pathway flux depending on the initial state of the system as well as on extra-cellular conditions. This observation leads to two important interpretations: the first is that in both S. cerevisiae and human beta-cells, despite differences in enzyme kinetics and mechanism of feedback control, evolution seems to have kept an oscillatory behavior coupled to the glucose concentration outside the cytoplasm, and the second is that the development of drugs to regulate metabolic dysfunctions in more complex systems may require further study, not only determining which enzyme is controlling the flux of the system but also under which conditions and how its control is maintained by the enzyme or transferred to other enzymes in the pathway as the drug starts acting.     

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.