Substrate influx can modulate the persistence of the active state in hysteretic enzymes: A theoretical analysis

A simple kinetic model is proposed for a hysteretic enzyme with an inflow of substrate (or transported ion). There can be two steady states of activity, with an abrupt transition to the lower level. The duration of the first, activated state rather than its height is determined by the effective substrate (ion) concentration, while oscillatory switchover between states is an intrinsic property of hysteretic enzymes. With plasma membrane Ca-ATPase as an example, it is shown that the magnitude of the input calcium signal can be translated into the time span for which the enzyme persists in the activated state.     

Interaction of mannose-6-phosphate with the hysteretic transition in glucose-6-phosphate hydrolysis in intact liver microsomes.

We showed previously that glucose-6-phosphatase activity was characterised in intact liver microsomes by a hysteretic transition between a rapid and a slower catalytic form of the enzyme. We have now further investigated the substrate specificity of these two kinetic forms. It was found that the pre-incubation of intact microsomes with mannose-6-phosphate or glucose-6-phosphate (50 microM for 30 s) suppressed the burst in glucose-6-phosphatase activity, that the hysteretic transition was reversible and that mannose-6-phosphate inhibited glucose-6-phosphate hydrolysis during the first seconds of incubation, but not anymore after the burst. Our results indicate (i) that mannose-6-phosphate is recognised by the enzyme and can promote the hysteretic transition and (ii) that the transient phase is part of the catalytic mechanism itself.     

Time-dependent kinetic complexities in cholinesterase-catalyzed reactions

Cholinesterases (ChEs) display a hysteretic behavior with certain substrates and inhibitors. Kinetic cooperativity in hysteresis of ChE-catalyzed reactions is characterized by a lag or burst phase in the approach to steady state. With some substrates damped oscillations are shown to superimpose on hysteretic lags. These time dependent peculiarities are observed for both butyrylcholinesterase and acetylcholinesterase from different sources. Hysteresis in ChE-catalyzed reactions can be interpreted in terms of slow transitions between two enzyme conformers E and E′. Substrate can bind to E and/or E′, both Michaelian complexes ES and E’s can be catalytically competent, or only one of them can make products. The formal reaction pathway depends on both the chemical structure of the substrate and the type of enzyme. In particular, damped oscillations develop when substrate exists in different, slowly interconvertible, conformational, and/or micellar forms, of which only the minor form is capable of binding and reacting with the enzyme. Biphasic pseudo-first-order progressive inhibition of ChEs by certain carbamates and organophosphates also fits with a slow equilibrium between two reactive enzyme forms. Hysteresis can be modulated by medium parameters (pH, chaotropic and kosmotropic salts, organic solvents, temperature, osmotic pressure, and hydrostatic pressure). These studies showed that water structure plays a role in hysteretic behavior of ChEs. Attempts to provide a molecular mechanism for ChE hysteresis from mutagenesis studies or crystallographic studies failed so far. In fact, several lines of evidence suggest that hysteresis is controlled by the conformation of His438, a key residue in the catalytic triad of cholinesterases. Induction time may depend on the probability of His438 to adopt the operative conformation in the catalytic triad. The functional significance of ChE hysteresis is puzzling. However, the accepted view that proteins are in equilibrium between preexisting functional and non-functional conformers, and that binding of a ligand to the functional form shifts equilibrium towards the functional conformation, suggests that slow equilibrium between two conformational states of these enzymes may have a regulatory function in damping out the response to certain ligands and irreversible inhibitors. This is particularly true for immobilized (membrane bound) enzymes where the local substrate and/or inhibitor concentrations depend on influx in crowded organellar systems, e.g. cholinergic synaptic clefts. Therefore, physiological or toxicological relevance of the hysteretic behavior and damped oscillations in ChE-catalyzed reactions and inhibition cannot be ruled out.     

A generalized theory of the transition time for sequential enzyme reactions.

In a sequence of coupled enzyme reactions the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. Provided that a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaelis-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Application of the theory to coupled enzyme assays allows a definition of the minimum requirements for successful operation of the assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration.     

The malignant transformation as a metabolic steady state transition: The possible significance of the phasing of enzyme synthesis and related aspects

It has been suggested by Sel'kov and by the author that the malignant transformation may be due to a transition between alternative steady states at the metabolic level without necessarily involving a genetic defect. It had previously been indicated by the author that the properties of a cell can be expected to reflect its pattern of temporal organisation. These two aspects are now considered in relation to one another by examining the behaviour of a coupled enzymic system (involving substrate inhibition characteristics) which is capable of exhibiting multiple steady states. It is shown that the phasing of the synthesis of the enzymes concerned during the cell cycle (normal or disturbed as a result of the action of agents) can determine whether a transition occurs or not and also if it is stable or not. It is also pointed out that the phasing between the fluctuations in the levels of inhibitors and activators of the enzymes involved and in their isozyme patterns can be equally significant in these respects. Hence a transition can be effected by a variety of agents acting at diverse sites within or without the cell. The transition is discussed as an example of a process that may be involved in the malignant transformation. It is emphasized that once a transition has occurred in a cell, there is no reason to presuppose that the new state cannot be inherited by the progeny of that cell, despite the removal of the causative agent: reversal is possible, however, and is discussed as an explanation for abortive oncogenic transformation. Other aspects briefly discussed in relation to metabolic steady state transitions include - the random nature and distribution of transformation in culture; the heterogeneity of transformed cells; resistance to transformation; the significance of the timing and duration of action of an oncogenic agent; the effect of gene duplication on the ‘fixation’ of the transformation; cell death. Finally it is pointed out that the same considerations are likely to apply to other systems exhibiting multiple steady states as a result of the existence of the phenomenon of hysteresis and hence probably to genetic switch systems also.     

A dynamical model for a co-operative enzyme.

Proteins partially immersed in the hydrophobic portion of a lipid bilayer interact by means of London-van der Waals non-bonding dispersion forces. Moreover, in certain organelles, enzymes are structured in a lattice or ordered matrix. These conditions may facilitate the establishment of long-range correlations between proteins. We studied the dynamical properties of a model for an enzyme endowed with a highly co-operative conformational transition between two reactive states. Two cases were considered, a closed system and an open system. In the closed system for different degrees of interaction among the proteins, it was found that for a substrate concentration greater than a certain threshold an abrupt change of enzymatic activity occurs. This biphasic behavior has been observed in the enzymatic activity of crystalline mitochondrial aspartate aminotransferase and for some other crystalline enzymes. In the analysis of the open system, for a specific input rate of the substrate, two different dynamics were found depending on the selected degree of interaction. For a certain value of a parameter phi, representing the degree of interaction among the reacting units, three steady states co-exist. This multiplicity confers excitable properties to the model. For larger values of phi, limit cycle type solutions were obtained. Thus, a sustained oscillatory product formation of the enzymatic reaction is observed. These results are compared with experimental observations of enzyme extracts detected by NMR.     

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.     

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.     

Kinetic studies of the lipid-activated pyruvate oxidase flavoprotein of Escherichia coli

Pyruvate oxidase is a flavoprotein dehydrogenase isolated from Escherichia coli which catalyzes the oxidative decarboxylation of pyruvate to acetate and CO2. In vivo, the enzyme can bind to the bacterial membrane and reduce ubiquinone-8, feeding electrons into the respiratory chain. The purified enzyme has been shown previously to bind to phospholipids and detergents and, upon doing so, is activated. The turnover with ferricyanide as an electron acceptor increases 20- to 30-fold upon lipid binding. In this work, initial velocity and stop-flow kinetics are used to investigate the activation of this enzyme. It is shown that the unactivated form of the enzyme is markedly hysteretic. Progress curves at low substrate concentrations show an initial acceleration in enzyme turnover. This is consistent with the results of stop-flow experiments. Rates obtained for either the reduction of the unactivated flavoprotein by pyruvate or its reoxidation by ferricyanide in single turnover experiments are much slower than the rates predicted by observed turnover in initial velocity studies, in some cases by more than 2 orders of magnitude. The data are best explained by the slow interconversion between two forms of the enzyme, one with low turnover and one which rapidly turns over. As isolated, the enzyme is highly unreactive, as revealed by the stop-flow experiments. During turnover, even in the absence of lipid activators, some of the enzyme converts to the rapid-turnover form. This slow interconversion is shown by kinetic simulation to preclude a steady state from being established. Lipid activators appear to shift the equilibrium to favor the rapid-turnover form of the enzyme. Once the enzyme is "locked" into an activated conformation, the hysteresis is no longer observed, and the stop-flow results are in agreement with data obtained from initial velocity experiments. Activation appears to result in both increased rates of electron transfer into and out of the flavin.     

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.     

Functional changes associated with the sequential transformation of L'4 into L4 pyruvate kinase.

The functional changes, associated with the sequential transformation of L'4 into L4 pyruvate kinase (ATP:pyruvate 2-O-phosphotransferase, EC 2.7.1.40) were studied. L'4 enzyme from human erythrocytes shows strong hysteretic behaviour: the initial rate of the enzyme preincubated with an unsaturating concentration of phosphoenolpyruvate is much higher than of the enzyme preincubated with ADP, at the same phosphoenolpyruvate concentration, although the "final activity" (the activity of the linear part of the reaction progress curve) was the same in both cases. This phenomenon was observed both in the presence and absence of fructose 1,6-diphosphate. High concentrations of both Mg2+free and MgATP2- diminish the difference in initial rate, between the ADP and phosphoenolpyruvate preincubated enzymes: Mg2+free by stabilizing the phosphoenolpyruvate-induced form; ATPMg2- by stabilizing the ADP-induced form. The magnitude of the difference in initial rates of the ADP-or phosphoenolpyruvate-preincubated enzyme is a function of both substrates. L4 pyruvate kinase (either from human liver or trypsin treated L'4 enzyme) does not, or to a very slight extent, show such behaviour. L'2L2 pyruvate kinase shows behaviour intermediate between L'4 and L4 enzymes. A model is proposed to describe the kinetic behaviour of L'4 and L4 enzymes.     

Integration of temporal analysis and control analysis of metabolic systems.

A theory is developed that integrates approaches to the analysis of pathway transient response and metabolic control analysis. A Temporal Control Coefficient is defined that is a measure of the system's transient response to modulation of enzyme activity or concentration. The approach allows for the analysis of the establishment of a steady state from rest, of the system's 'agility' of response to minor perturbations of a pre-existing steady state and of the macroscopic transition between steady states. In the last-mentioned case it is shown that, like the transient time itself, the control of transient response retains the property of independence from the mechanism of the transition. In consequence, the Temporal Control Coefficient can be defined in terms of the control properties of the initial and final states alone without reference to the mechanism of transition. A summation property is shown to apply to the Temporal Control Coefficients in each case. Connectivity relationships between elasticities and Temporal Control Coefficients are also established.     

Influence of Ca<Superscript>2+</Superscript> Oscillatory Influx on Membrane Ca<Superscript>2+</Superscript>-ATPase Activity: a Kinetic Model

A kinetic model for the membrane Ca²⁺-ATPase is considered. The catalytic cycle in the model is extended by enzyme auto-inhibition and by oscillatory calcium influx. It is shown that the conductive enzyme activity can be registered as damped or sustained Ca²⁺ pulses similar to observed experimentally. It is shown that frequency variations in Ca²⁺ oscillatory influx induce changes of pulsating enzyme activity. Encoding is observed for the signal frequency into a number of fixed levels of sustained pulses in the enzyme activity. At certain calcium signal frequencies, the calculated Ca²⁺-ATPase conductivity demonstrates chaotic multi-level pulses, similar to those observed experimentally.__________Translated from Biokhimiya, Vol. 70, No. 4, 2005, pp. 539–544.Original Russian Text Copyright © 2005 by Goldstein, Mayevsky, Zakrjevskaya.     

Hysteretic behaviour and GSSG substrate inhibition shown by glutathione reductase from Phycomyces blakesleeanus

Phycomyces blakesleeanus glutathione reductase shows hysteretic behaviour under experimental conditions, when GSSG substrate inhibition is observed. The progress curves for the reaction show an acceleration phase. The degree of hysteresis varied inversely as the enzyme concentration. It increased when GSSG or NADPH concentration increased, whereas the addition of GSH or NADP+ to the initial reaction mixture prevented it from occurring. In addition, hysteresis was dependent on pH, ionic strength and temperature, decreasing as any of these parameters increased. The parallel effects of pH and ionic strength on the GSSG substrate inhibition and hysteretic behaviour suggest a relationship between these two mechanisms. From the overall results reported in this paper, we propose that the hysteretic behaviour shown by Phycomyces glutathione reductase could be due to a process of time-dependent accumulation of reaction products rather than to a slow conformational change.     

Self-regulation of phytoalexin production: a non-biosynthetic enzyme controls alkaloid biosynthesis in cultured cells of Eschscholzia californica

Benzophenanthridine alkaloids are strong antimicrobials of Papaveraceae and attractive lead compounds for drug development. The cytotoxicity of these compounds requires the producing plant to limit the pathogen-triggered burst of biosynthesis. Cells of Eschscholzia californica excrete early benzophenanthridines to the cell wall, followed by re-uptake and reduction in the cytoplasm by the detoxifying enzyme sanguinarine reductase. We now discovered that this enzyme is a core component of self-control in alkaloid production. RNAi-based silencing of sanguinarine reductase gave rise to mutants that either show a complete stop of elicitor-triggered alkaloid production or a burst of biosynthesis that severalfold surpasses the wild type level. These unexpected phenotypes reflect impacts of substrate or product of sanguinarine reductase: the substrate, sanguinarine, inhibits phospholipase A2 at the plasma membrane, an initial component of the signal path towards expression of biosynthetic enzymes. The product, dihydrosanguinarine, inhibits enzymes of early biosynthesis, prior to reticuline formation. By tuning these steady states, sanguinarine reductase adjusts the capacity of alkaloid biosynthesis: a minimum activity is sufficient to prevent the blockade of the induction pathway by sanguinarine, while the full activity of the same enzyme causes a limitation of the biosynthetic flow via dihydrosanguinarine.     

Ligand-induced interconversions of maize homoserine dehydrogenase among different states

The threonine-sensitive homoserine dehydrogenase has been isolated and extensively purified from shoots of Zea mays L. var. earliking. This enzyme is shown to be hysteretic under certain conditions. Progress curves of the NAD-dependent reaction catalyzed by the maize enzyme can be characterized by distinct lags prior to achievement of steady state velocities, reflecting transitions from less active species to a more active steady state form of the enzyme. Incubation of the enzyme for 1 min at 25 degrees C prior to initiation of the reaction profoundly influences the properties of the less active enzyme and the nature of the subsequent slow transitions during assay. When the feedback modifier, L-threonine, or KCl is included in the preincubation mixture, the transitions involve biomolecular association reactions. In the absence of either ligand, or in the presence of an appropriate mixture of both, a unimolecular transition occurs during assay. Three unique preincubation states of the enzyme have been identified on the basis of their response to substrates and effectors; whereas, the kinetic and regulatory properties of the steady state form of the enzyme are independent of preincubation conditions. Steady state can thus be achieved by three different transitions. Each transition is retarded by threonine and favored by substrates and potassium, although the effects of these compounds differ quantitatively. Under the conditions tested, monovalent cations have no effect on the steady state velocity of the enzyme. A model describing the relationships among the four unique states of the enzyme which is consistent with the present results and supported by previous observations is proposed.     

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.     

Stochastic ensembles, conformationally adaptive teamwork, and enzymatic detoxification

It has been appreciated for a long time that enzymes exist as conformational ensembles throughout multiple stages of the reactions they catalyze, but there is renewed interest in the functional implications. The energy landscape that results from conformationlly diverse poteins is a complex surface with an energetic topography in multiple dimensions, even at the transition state(s) leading to product formation, and this represents a new paradigm. At the same time there has been renewed interest in conformational ensembles, a new paradigm concerning enzyme function has emerged, wherein catalytic promiscuity has clear biological advantages in some cases. "Useful", or biologically functional, promiscuity or the related behavior of "multifunctionality" can be found in the immune system, enzymatic detoxification, signal transduction, and the evolution of new function from an existing pool of folded protein scaffolds. Experimental evidence supports the widely held assumption that conformational heterogeneity promotes functional promiscuity. The common link between these coevolving paradigms is the inherent structural plasticity and conformational dynamics of proteins that, on one hand, lead to complex but evolutionarily selected energy landscapes and, on the other hand, promote functional promiscuity. Here we consider a logical extension of the overlap between these two nascent paradigms: functionally promiscuous and multifunctional enzymes such as detoxification enzymes are expected to have an ensemble landscape with more states accessible on multiple time scales than substrate specific enzymes. Two attributes of detoxification enzymes become important in the context of conformational ensembles: these enzymes metabolize multiple substrates, often in substrate mixtures, and they can form multiple products from a single substrate. These properties, combined with complex conformational landscapes, lead to the possibility of interesting time-dependent, or emergent, properties. Here we demonstrate these properties with kinetic simulations of nonequilibrium steady state (NESS) behavior resulting from energy landscapes expected for detoxification enzymes. Analogous scenarios with other promiscuous enzymes may be worthy of consideration.     

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.