Self-organization and dynamics of an open futile cycle

The dynamic behaviour of an open futile cycle composed of two enzymes has been investigated in the vicinity of a steady-state. A necessary condition required for damped or sustained oscillations of the system is that enzyme E₂, which controls recycling of the substrate S₂, be inhibited by an excess of this substrate. In order for the system to be neutrally stable and therefore to exhibit sustained oscillations, it is not necessary for antagonist enzyme E₁ to be activated by its product S₂. If it is enzyme E₁ which is inhibited by an excess of its substrate S₁, the system has a saddle point. Other conditions for stability or instability of the system have been determined. If the enzyme E₁, which is not inhibited by the substrate, exhibits a slow conformational transition of the mnemonical type, this transition dramatically alters the stability behavior of the system. If the mnemonical enzyme E₁ were exhibiting a positive kinetic co-operativity, decreasing the rate of the conformational transition of the mnemonical enzyme will increase the stability of the whole system and will tend to damp the oscillations in the vicinity of the steady-state. If conversely the mnemonical enzyme E₁ were exhibiting a negative kinetic co-operativity, decreasing the rate of the enzyme conformational transition will decrease the stability of the system and will tend to create or amplify oscillations of the system taken as a whole. If these results may be extended to more complex metabolic cycles, involving more than two enzymes, it may be tentatively considered that positive co-operativity associated with slow transition has emerged in the course of evolution in order to limit temporal instabilities of metabolic cycles. Alternatively one may speculate that the “biological function” of negative co-operativity is to create or amplify these temporal instabilities.     

Ionic control of immobilized enzymes. Kinetics of acid phosphatase bound to plant cell walls.

When an enzyme is bound to an insoluble polyelectrolyte it may acquire novel kinetic properties generated by Donnan effects. It the enzyme is homogeneously distributed within the matrix, a variation of the electrostatic partition coefficient, when substrate concentration is varied, mimics either positive or negative co-operativity. This type of non-hyperbolic behaviour may be distinguished from true co-operativity by an analysis of the Hill plots. If the enzyme is heterogeneously distributed within the polyelectrolyte matrix, an apparent negative co-operativity occurs, even if the electrostatic partition coefficient does not vary when substrate concentration is varied in the bulk phase. If the partition coefficient varies, mixed positive and negative co-operativities may occur. All these effects must be suppressed by raising the ionic strength in the bulk phase. Attraction of cations by fixed negative charges of the polyanionic matrix may be associated with a significant decrease of the local pH. The magnitude of this effect is controlled by the pK of the fixed charges groups of the Donnan phase. The local pH cannot be much lower than the value of this pK. This effect may be considered as a regulatory device of the local pH. Acid phosphatase of sycamore (Acer pseudoplatanus) cell walls is a monomeric enzyme that displays classical Michaelis-Menten kinetics in free solution. However, when bound to small cell-wall fragments or to intact cells, it has an apparent negative co-operativity at low ionic strength. Moreover a slight increase of ionic strength apparently activates the bound enzymes and tends to suppress the apparent co-operativity. At I0.1, or higher, the bound enzyme has a kinetic behavior indistinguishable from that of the purified enzyme in free solution. These results are interpreted in the light of the Donnan theory. Owing to the repulsion of the substrate by the negative charges of cell-wall polygalacturonates, the local substrate concentration in the vicinity of the bound enzyme is smaller than the corresponding concentration in bulk solution. The kinetic results obtained are consistent with the view that there exist at least three populations of bound enzyme with different ionic environments: a first population with enzyme molecules not submitted to electrostatic effects, and two other populations with molecules differently submitted to these effects. The theory allows one to estimate the proportions of enzyme belonging to these populations, as well as the local pH values and the partition coefficients within the cell walls.     

Kinetic implications of the occurrence of several relaxations in the conformational transition of mnemonical enzymes

If the conformational transition involved in enzyme memory occurs in several elementary steps, the time constant of the overall 'slow' relaxation is mostly determined by the individual values of the rate constants pertaining to the overall transconformation. The extent of kinetic co-operativity of the enzyme reaction, however, is mostly controlled by the degree of reversibility of the elementary steps of the conformational transition. There is then no simple relation between the time scale of the 'slow' transition and the extent of kinetic co-operativity of the enzyme reaction. A slow transition of about 10(-3) s-1 is therefore perfectly compatible with a strong positive or negative co-operativity and in particular with the negative co-operativity observed with wheat germ hexokinase LI. The relationship that has been established recently [Pettersson, G. (1986) Eur. J. Biochem. 154, 167-170] between the 'slow' enzyme relaxation and the extent of kinetic co-operativity holds only in the specific case where the transconformation occurs in one step. Owing to the possible occurrence of a multistep conformation change, the lack of this relationship means nothing as to the validity, or the invalidity, of the concept of mnemonical transition. More informative than the time scale of the 'slow' transition is its dependence with respect to glucose and glucose 6-phosphate, which both react with the enzyme. The effect of reaction products on the modulation of kinetic co-operativity is also of cardinal importance in the diagnosis of enzyme memory. Since an alternative model has been recently proposed by Pettersson (cited above) to explain the mechanistic origin of kinetic co-operativity of monomeric enzymes, the effect of products on the kinetic co-operativity predicted by this alternative model has been studied theoretically, in order to determine whether it is consistent with the experimental results obtained with wheat germ hexokinase LI. This analysis shows that the predictions of this model are in total disagreement with both the predictions of the mnemonical model and the experimental results obtained with wheat germ hexokinase LI, as well as with other enzymes. This alternative model cannot therefore be considered as a sensible explanation of the mechanistic origin of co-operativity of monomeric enzymes. It is therefore concluded that the mnemonical model which rests on numerous experimental results, obtained by different research groups, on different enzymes is the simplest and most likely explanation of the kinetic subtleties displayed by some monomeric enzymes, and in particular wheat germ hexokinase LI.     

Kinetics of hexokinase D (''glucokinase'') with inosine triphosphate as phosphate donor. Loss of kinetic co-operativity with respect to glucose.

When ATP, the normal phosphate donor for hexokinase D ('glucokinase'), is replaced by ITP, the positive co-operativity with respect to glucose disappears. This may be rationalized in relation to kinetic models for hexokinase D co-operativity, which assume that with the normal substrates the chemical reaction and subsequent release of products occur so rapidly that binding of substrates cannot approach equilibrium and is therefore not constrained by the thermodynamic requirement that the Hill coefficient for substrate binding cannot exceed the number of binding sites. ITP is a much poorer substrate than ATP, however: its Km value at high glucose concentrations is 24 times the value for ATP, whereas the value of the limiting rate V is decreased about 8-fold. Consequently it is no longer possible for the ternary complex to be converted into products rapidly enough to generate kinetic co-operativity. The negative co-operativity with respect to glucose observed in 2H2O with ATP as phosphate donor also disappears when ITP is used instead of ATP.     

Inhibition of co-operative enzymes by substrate-analogues: Possible implications for the physiological significance of negative co-operativity illustrated by phenylalanine metabolism in higher plants

The “Hill” equation for co-operative binding-systems has been extended to describe the effect of substrate-analogue on the binding of substrate to an oligomeric protein. It is demonstrated that the more negatively co-operative the binding-system, the more sensitive is the binding of substrate to inhibition by increases in the relative concentration of substrate-analogue. It is proposed that the physiological significance of negative co-operativity for enzymes may be complementary to the physiological significance of positive co-operativity. The effect of negative co-operativity is to make substrate binding more sensitive to inhibition by relative increases in the concentration of substrate-analogue (e.g. for many enzymes product of the reaction) at the expense of decreased sensitivity of substrate binding to relative changes in substrate concentration compared to a system with equivalent, independent substrate binding sites. In contrast, the effect of positive co-operativity is to make the enzyme more sensitive to relative changes in substrate concentration at the expense of decreased sensitivity to inhibition by relative increases in product concentration, compared to an enzyme without co-operative binding.     

Generalized microscopic reversibility, kinetic co-operativity of enzymes and evolution.

Generalized microscopic reversibility implies that the apparent rate of any catalytic process in a complex mechanism is paralleled by substrate desorption in such a way that this ratio is held constant within the reaction mechanism [Whitehead (1976) Biochem. J. 159, 449--456]. The physical and evolutionary significances of this concept, for both polymeric and monomeric enzymes, are discussed. For polymeric enzymes, generalized microscopic reversibility of necessity occurs if, within the same reaction sequence, the substrate stabilizes one type of conformation of the active site only. Generalized microscopic reversibility suppresses the kinetic co-operativity of the slow transition model [Ainslie, Shill & Neet (1972) J. Biol. Chem. 247, 7088--7096]. This situation is obtained if the free-energy difference between the corresponding transition states of the two enzyme forms is held constant along the reaction co-ordinate. This situation implies that the 'extra costs' of energy (required to pass each energy barrier) that are not covered by the corresponding binding energies of the transition states vary in a similar way along the two reaction co-ordinates. The regulatory behaviour of monomeric enzymes is discussed in the light of the concept of 'catalytic perfection' proposed by Albery & Knowles [(1976) Biochemistry 15, 5631--5640]. These authors claim that an enzyme will be catalytically 'perfect' when its catalytic efficiency is maximum. If this situation occurs for a monomeric enzyme obeying either the slow transition or the mnemonical model, it can be shown that the kinetic co-operativity disappears. In other words, kinetic co-operativity of a monomeric enzyme is 'paid for' at the expense of catalytic efficiency, and the monomeric enzyme cannot be simultaneously co-operative and catalytically very efficient. This is precisely what has been found experimentally in a number of cases.     

Influence of pH on the allosteric properties of lactate dehydrogenase activity of Phycomyces blakesleeanus.

1. Lactate dehydrogenase from mycelium of Phycomyces blakesleeanus showed positive homotropic interactions with NADH at all pH values studied (pH 5.0-7.7). The calculated values for the first and last intrinsic association constants remained unaltered with pH, in contrast with the Hill coefficient value, which varied significantly, reaching its maximum values at pH 6.0 and 7.7. This suggests the hypothesis that pH regulates these homotropic effects by changes in the value of the intermediate intrinsic association constants. 2. From pH 7.2 to 7.7 lactate dehydrogenase exhibited, likewise, positive homotropic interactions with pyruvate. There were practically no changes in the first and last intrinsic association constants and in Hill coefficient values with pH. At pH values below 7.2 (pH 5.0-6.8) the enzyme showed high substrate inhibition, which was highly dependent on pH, NADH concentration and temperature. By way of substrate inhibition pH regulates, primarily, lactate dehydrogenase activity towards pyruvate, since the homotropic effects appear not to be dependent on pH. 3. Fructose 1,6-bisphosphate is a true allosteric effector of lactate dehydrogenase of Phycomyces blakesleeanus. it decreases positive co-operativity with NADH, and on the other hand pyruvate co-operativity turns into mixed co-operativity. In addition, the effector decreases the inhibitory effect caused by pyruvate.     

Regulation of enzyme activity in adsorptive enzyme systems

The kinetic behaviour of adsorptive enzyme systems with free and adsorbed enzyme forms in rapid equilibrium has been analysed. It has been shown that the dependences of enzymic reaction rate on substrate or “adsorptive effector” concentrations reveal the deviations from simple kinetic laws of Michaelis-Menten type (positive or negative kinetic co-operativity). Such kinetic anomalies should be observed when adsorption of the enzyme results in the changing catalytic properties and when the state of the equilibrium between free and bound enzyme forms depends on the presence of low molecular substances (substrates, coenzymes and various cellular metabolites). The physiological significance of adsorption-desorption processes for the enzyme activity regulation has been emphasized.     

Subunit coupling and kinetic co-operativity of polymeric enzymes. Amplification, attenuation and inversion effects

The principles of structural kinetics, as applied to dimeric enzymes, allow us to understand how the strength of subunit coupling controls both substrate-binding co-operativity, under equilibrium conditions, and kinetic co-operativity, under steady state conditions. When subunits are loosely coupled, positive substrate-binding co-operativity may result in either an inhibition by excess substrate or a positive kinetic co-operativity. Alternatively, negative substrate-binding co-operativity is of necessity accompanied by negative kinetic co-operativity. Whereas the extent of negative kinetic co-operativity is attenuated with respect to the corresponding substrate-binding co-operativity, the positive kinetic co-operativity is amplified with respect to that of the substrate-binding co-operativity. Strong kinetic co-operativity cannot be generated by a loose coupling of subunits. If subunit is propagated to the other, the dimeric enzyme may display apparently surprising co-operativity effects. If the strain of the active sites generated by subunit coupling is relieved in the non-liganded and fully-liganded states, both substrate-binding co-operativity and kinetic co-operativity cannot be negative. If the strain of the active sites however, is not relieved in these states, negative substrate-binding co-operativity is accompanied by either a positive or a negative co-operativity. The possible occurrence of a reversal of kinetic co-operativity, with respect to substrate-binding co-operativity, is the direct consequence of quaternary constraints in the dimeric enzyme. Moreover, tight coupling between subunits may generate a positive kinetic co-operativity which is not associated with any substrate-binding co-operativity. In other words a dimeric enzyme may well bind the substrate in a non co-operative fashion and display a positive kinetic co-operativity generated by the strain of the active sites.     

Investigation on the kinetic mechanism of octopine dehydrogenase. A regulatory behavior.

The kinetic scheme of octopine dehydrogenase of Pecten maximus L., a monomeric enzyme obeying a bi-ter sequential mechanism, was completed, essentially in the forward reaction, by steady-state studies over a wide range of substrate concentration at pH 7.0. Deviation from the Michaelis-Menten behavior with respect to NAD+ and other significant kinetic data led us to ascribe for octopine dehydrogenase mechanism the mnemonical enzyme concept. In addition, another regulatory behavior can be envisaged involving the formation of two dead-end complexes enzyme.NADH.D-octopine and enzyme.NAD+.pyruvate.L-arginine.     

Kinetic study of an enzyme-catalysed reaction in the presence of novel irreversible-type inhibitors that react with the product of enzymatic catalysis

In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.     

Partial purification and properties of purine nucleoside phosphorylase from rabbit erythrocytes.

1. The partial purification of purine nucleoside phosphorylase from rabbit erythrocytes is described. 2. Analytical and preparative isoelectric focusing gave a pI value for the enzyme of 4.65. 3. Gel-chromatography and sucrose-density-gradient-centrifugation techniques gave estimates of the molecular weight in the range 75000-83000. 4. Lineweaver-Burk plots of kinetic data were non-linear at high inosine concentrations. Extrapolation of the linear part of such plots yielded a Km value for inosine of about 70 micrometer for the rabbit erythrocyte and liver enzymes. 5. A Hill interaction coefficient of 0.75 was obtained, suggesting negative co-operativity with respect to the binding of inosine. 6. Treatment of the enzyme with 5,5'-dithiobis-(2-nitrobenzoic acid) caused partial inactivation, and subsequent Lineweaver-Burk plots with inosine as substrate displayed complete linearity, with an increase in Km value for inosine to 200 micrometer. 7. Starch-gel electrophoresis did not reveal the presence of secondary isoenzymes; all tissue extracts examined gave electrophoretic patterns similar to those obtained with the partially purified enzyme from erythrocytes. 8. Results of hybridization studies with nucleoside phosphorylase from human foetal liver suggest that the rabbit enzyme is also a trimer.     

pH-induced co-operative effects in hysteretic enzymes. 1. A theoretical model of a new type of co-operative behaviour controlled by pH

A new model which provides an explanation for pH-induced co-operativity of hysteretic enzymes is proposed. The essence of the model is that a region, or a domain, of the enzyme undergoes a spontaneous 'slow' conformational change which does not affect the geometry of the active site. The region which undergoes this spontaneous conformational transition bears an ionizable group. Kinetic co-operativity occurs if the pK of this ionizable group changes upon this conformational transition. Thus co-operativity does not arise from a distortion of the active site. An interesting prediction of the model is that at 'extreme' pH values co-operativity must be suppressed. Although the kinetic equation pertaining to the model is of the 2:2 type, co-operativity is not maximum or minimum at half-saturation of the enzyme by the substrate, as occurs with 2:2 binding isotherms. A new index of maximum or minimum kinetic co-operativity, whether this extreme occurs at half-saturation or not, has been proposed which allows the change of kinetic co-operativity to be followed as a function of pH. It is believed that this model will be useful in explaining the behaviour of enzymes attached to biological polyelectrolytes, such as membranes or cell envelopes.     

Regulation of glycolysis in sea bass liver: phosphofructokinase isozymes

Sea bass (Dicentrarchus labrax L.) liver phosphofructokinase (PFK) presents biphasic kinetics with respect to fructose-6-phosphate (F-6-P) in experiments carried out with crude extract. After the enzyme had been purified, two isozymes have been detected after chromatographic treatment. The two isozymes present different kinetic behaviour PFK-L1, the first eluted phosphofructokinase activity shows positive cooperativity with respect to fructose-6-phosphate and PFK-L2, the second activity fraction, has a Hill coefficient of 0.38 (negative cooperativity). The first isozyme shows less affinity for fructose-6-phosphate than that shown by PFK-L2. The joint kinetics of both isozymes produces a biphasic kinetics with respect to fructose-6-phosphate, similar to that observed in crude extracts.     

The binding of nucleotides and bivalent cations to the calcium-and-magnesium ion-dependent adenosine triphosphatase from rabbit muscle sarcoplasmic reticulum.

The binding of MgATP to purified Ca2+Mg2+-dependent adenosine triphosphatase from rabbit muscle sarcoplasmic reticulum was studied by using a flow-dialysis method. Phosphoryl-enzyme formation and catalytic activity were also measured, and all three processes demonstrated negative co-operativity, with half-saturation of all three parameters at a MgATP concentration of 40-50muM, and a Hill coefficient (h) of 0.8. The variation of the binding constant with with pH was measured and showed tighter binding of MgATP with increasing pH over the range 6.8-8.5. Binding parameters for ATP analogues were also measured. The binding of Ca2+ in the presence and absence of ATP analogues gave half saturation at a Ca2+ concentration of 1.2-1.3muM. Hill plots of Ca2+-binding data gave a slope of 0.8. These results show that the binding of MgATP and Ca2+ can occur in a random manner, with neither substrate influencing the affinity of the enzyme for the other.     

Letter: Kinetic negative co-operativity in the allosteric model of Monod, Wyman and Changeux

The co-operativity of homotropic interactions between substrate molecules in oligomeric enzymes is analyzed in the frame of the concerted transition theory of Monod et al. (1965). A discussion of the Hill coefficient n_H allows determination of the conditions for negative co-operativity (n_H < 1). This phenonomenon, usually taken as indicative of a sequential mechanism (Koshland et al., 1966), can be accounted for by the concerted model when the enzyme represents a K-V or V system, i.e. when the two protomer conformational states postulated in the theory differ in their catalytic activity. However, only negative co-operativity for catalysis can be explained by the concerted model, not negative co-operativity of binding.     

Observation of a kinetic slow transition in monomeric glucokinase

Rat liver glucokinase (EC 2.7.1.2) is a monomeric enzyme with positive cooperativity for glucose phosphorylation for which several kinetic mechanisms have been proposed. We have observed a slow kinetic transition when the enzyme is assayed in the presence of 30% glycerol. When the enzyme had been preincubated or stored in 50 mM glucose, the initially rapid activity decayed, via a first-order process, to a new steady-state velocity. The glucose-induced process is reversible since if the enzyme is preincubated without glucose, an initially low activity accelerates over minutes to the same steady-state velocity. This final velocity is independent of the preincubation conditions and is determined solely by the glucose and ATP concentrations in the assay. Possible artifacts which might cause nonlinear progress curves have been ruled out. The transition has a half-time of 2-10 min depending on glucose and ATP concentrations and temperature. In the steady-state kinetics, positive cooperativity occurs with glucose with a Hill coefficient (nH) = 1.3 at high ATP concentrations, approaching unity as the ATP concentration decreases. This pattern is similar to that seen in the linear velocities in the absence of glycerol. Similarly, negative cooperativity with MgATP is seen in the steady-state velocities at low glucose concentrations with the Hill coefficient approaching 1 as the glucose concentrations approach saturation. The initial velocity for enzyme preincubated in high glucose concentration was either Michaelis-Menten as a function of glucose at high MgATP concentration or heterogeneous (nH less than 1, negatively cooperative) at low MgATP concentration.(ABSTRACT TRUNCATED AT 250 WORDS)     

Mechanistic origin of the sigmoidal rate behaviour of rat liver hexokinase D (''glucokinase'').

Two recent proposals to account for the kinetic co-operativity of hexokinase D ('glucokinase') from rat liver are examined. A model in which the deviations from Michaelis-Menten kinetics result from a random order of binding of the substrates [Pettersson (1986) Biochem. J. 233, 347-350] accounts satisfactorily for the behaviour as a function of glucose concentrations, but it also predicts observable substrate inhibition by MgATP, which is in fact not observed. An alternative proposal in which the deviations arise from recycling of an enzyme-MgADP complex [Pettersson (1986) Eur. J. Biochem. 154, 167-170] also accounts satisfactorily for some of the data, but the required enzyme-MgADP complex could not be detected in isotope-exchange measurements. Thus the mnemonical mechanism proposed originally [Storer & Cornish-Bowden (1977) Biochem. J. 165, 61-69], which explains the deviations in terms of a relatively slow interconversion between two forms of free enzyme, remains the most parsimonious explanation of the behavior of hexokinase D.     

Enzyme reactions at the surface of living cells. I. Electric repulsion of charged ligands and recognition of signals from the external milieu

The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics.     

The regulatory design of an allosteric feedback loop: the effect of saturation by pathway substrate

Aspects of metabolic regulation can be fruitfully studied with a combination of generic modelling, control analysis and graphical analysis using rate characteristics. This paper analyses a prototypical supply-demand system consisting of a biosynthetic subsystem subject to allosteric inhibition by its product and a demand process that consumes this product. The effect of changes in affinity of the committing supply enzyme for the pathway substrate on the regulatory properties of the supply subsystem is compared for the Monod-Wyman-Changeux and the reversible Hill allosteric enzyme models. We found that the Hill model has a distinct advantage in that the steady-state concentration at which it maintains the product is set by the half-saturating product concentration and is independent of changes in the degree of saturation for substrate. In contrast, with the Monod-Wyman-Changeux model this set point varies with affinity for substrate. Explicitly incorporating reversibility in all rate equations made it possible to distinguish between kinetic and thermodynamic aspects of regulation. Combining the supply and demand rate characteristics allows us to explore both the control distribution at steady state and the regulatory performance of the system over a wide range of demand activities.