Validity of quasi-steady-state and transfer-function representations for input-output relation in a Michaelis-Menten reaction

In relation to the input-output characteristics of enzymatic reactions in the cellular metabolism and biochemical reactors, the validity of the quasi-steady-state and transfer-function representations of reaction velocity has been examined for a basic Michaelis-Menten reaction employing computer simulation, that is, numerical integration of the rate equation. The well-known S-v relationship (relationship between substrate concentration and reaction velocity)derived on the quasi-steady-state assumption is found to be in general a good approximation to the actual velocity throughout the temporal progress of the reaction. The validity of the approximation depends on a ratio of the Michaelis constant to the total enzyme concentration in the reaction system rather than on the individual rate constants. A transfer-function representation is derived on assuming an exponential change in the reaction velocity for the indicial response to the substrate influx rate. The representation has a wider valid region with a decrease in influx rate than with an increase in the influx rate. The validity is most dependent on a ratio of total enzyme concentration to the steady-state concentration of the substrate. The analysis of the linear sensitivity of the reaction velocity to rate constants reveals that the characteristics of these valid representations in systems analysis change according to the phase of the reaction.     

Michaelis-Menten Kinetics for Determining Enzymatic Activity of Lysostaphin

The rate of lysostaphin-catalyzed lysis of staphylococci follows the Michaelis-Menten equation at [E](0) < [S](0), i.e., the activity of the enzyme is proportional to its concentration. This equation is proposed for determining the specific activity of lysostaphin. The apparent activation energy of hydrolysis of pentaglycine bridges in Staphylococcus peptidoglycan is 77.9 kJ/mol.     

Kinetic analysis of urea-inactivation of beta-galactosidase in the presence of galactose

The effect of galactose on the inactivation of purified beta-galactosidase from the black bean, Kestingiella geocarpa, in 5 M urea at 50 degrees C and at pH 4.5, was determined. Lineweaver-Burk plots of initial velocity data in the presence and absence of urea and galactose were used to determine the relevant K(m) and V(max) values, with p-nitrophenyl beta-D-galactopyranoside (PNPG) as substrate, S. The inactivation data were analysed using the Tsou equation and plots. Plots of ln([P](infinity) - [P](t) ) against time in the presence of urea yielded the inactivation rate constant, A. Plots of A vs [S] at different galactose concentrations were zero order showing that A was independent of [S]. Plots of [P](infinity) vs [S] were used to determine the mode of inhibition of the enzyme by galactose, and slopes and intercepts of the 1/[P](infinity) vs. 1/[S] yielded k(+0) and k '(+0), the microscopic rate constants for the free enzyme and the enzyme-substrate complex, respectively. Plots of k(+0) and k '(+0) vs. galactose concentrations showed that galactose protected the free enzyme and not the enzyme-substrate complex against urea inactivation via a noncompetitive mechanism at low galactose concentrations and a competitive pattern of inhibition at high galactose concentrations. The implication of the different modes of inhibition in protecting the free enzyme was discussed.     

Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis-Menten constant

In vitro enzyme reactions are traditionally conducted under conditions of pronounced substrate excess since this guarantees that the bound enzyme is at quasi-steady-state (QSS) with respect to the free substrate, thereby justifying the Briggs-Haldane approximation (BHA). In contrast, intracellular reactions, amplification assays, allergen digestion assays and industrial applications span a range of enzyme-to-substrate ratios for which the BHA is invalid, including the extreme of enzyme excess. The quasi-equilibrium approximation (QEA) is valid for a subset of enzyme excess states. Previously, we showed that the total QSSA (tQSSA) overlaps and extends the validity of the BHA and the QEA, and that it is at least roughly valid for any total substrate and enzyme concentrations. The analysis of the tQSSA is hampered by square root nonlinearity. Previous simplifications of the tQSSA rate law are valid in a parameter domain that overlaps the validity domains of the BHA and the QEA and only slightly extends them. We now integrate the tQSSA rate equation in closed form, without resorting to further approximations. Moreover, we introduce a complimentary simplification of the tQSSA rate law that is valid in states of enzyme excess when the absolute difference between total enzyme and substrate concentrations greatly exceeds the Michaelis-Menten constant. This includes a wide range of enzyme and substrate concentrations where both the BHA and the QEA are invalid and allows us to define precisely the conditions for zero-order and first-order product formation. Remarkably, analytical approximations provided by the tQSSA closely match the expected stochastic kinetics for as few as 15 reactant molecules, suggesting that the conditions for the validity of the tQSSA and for its various simplifications are also of relevance at low molecule numbers.     

Steady-state kinetics of yeast cytochrome c peroxidase catalyzed oxidation of inorganic reductants by hydrogen peroxide

Rates of yeast cytochrome c peroxidase (ferrocytochrome c:hydrogen-peroxide oxidoreductase, EC 1.11.1.5) catalyzed oxidation of bis(tripyridine)cobalt(II) ion, penta(amine)pyridineruthenium(II) ion and ferrocyanide ion by hydrogen peroxide have been found to obey the empirical equation: (formula; see text) in the pH range 5 to 8, and at saturating H2O2 concentrations. [( S] and [CcP] are the concentrations of the reductant and the enzyme, respectively.) Values of k2 were found to be independent of the reductant. The term k0[S] is only significant with the cobalt and ruthenium complexes at high pH. The mechanism proposed to account for this rate equation differs significantly from previous mechanistic proposals. In particular, the rate data require the assignment of the rate-limiting step at high substrate concentrations to a slow electron-transfer within the enzyme, and not, as previously suggested, to saturation of substrate binding to the enzyme. Also, the term k0[S] implies that the reactive substrates, including the natural substrate (yeast cytochrome c), react with the hydrogen peroxide-heme complex and not with the radical species formed by reaction with hydrogen peroxide in the absence of reductants.     

Generalized rate equation for single-substrate enzyme catalyzed reactions

The most widely used rate expression for single-substrate enzyme catalyzed reactions, namely the Michaelis-Menten kinetics is based upon the assumption that enzyme concentration is in excess of the substrate in the medium and the rate is mainly limited by the substrate concentration according to saturation kinetics. However, this is only a special case and the actual rate expression varies depending on the initial enzyme/substrate ratio (E₀/S₀). When the substrate concentration exceeds the enzyme concentration the limitation is due to low enzyme concentration and the rate increases with the enzyme concentration according to saturation kinetics. The maximum rate is obtained when the initial concentrations of the enzyme and the substrate are equal. A generalized rate equation was developed in this study and special cases were discussed for enzyme catalyzed reactions.     

Iron-sulfur cluster interconversions in biotin synthase: dissociation and reassociation of iron during conversion of [2Fe-2S] to [4Fe-4S] clusters

Biotin synthase catalyzes the insertion of a sulfur atom into the saturated C6 and C9 carbons of dethiobiotin. This reaction has long been presumed to occur through radical chemistry, and recent experimental results suggest that biotin synthase belongs to a family of enzymes that contain an iron-sulfur cluster and reductively cleave S-adenosylmethionine, forming an enzyme or substrate radical, 5'-deoxyadenosine, and methionine. Biotin synthase (BioB) is aerobically purified as a dimer of 38 kDa monomers that contains two [2Fe-2S](2+) clusters per dimer. Maximal in vitro biotin synthesis requires incubation of BioB with dethiobiotin, AdoMet, reductants, exogenous iron, and crude bacterial protein extracts. It has previously been shown that reduction of BioB with dithionite in 60% ethylene glycol produces one [4Fe-4S](2+/1+) cluster per dimer. In the present work, we use UV/visible and electron paramagnetic resonance spectroscopy to show that [2Fe-2S] to [4Fe-4S] cluster conversion occurs through rapid dissociation of iron from the protein followed by rate-limiting reassociation. While in 60% ethylene glycol the product of dithionite reduction is one [4Fe-4S](2+) cluster per dimer, the product in water is one [4Fe-4S](1+) cluster per dimer. Further, incubation with excess iron, sulfide, and dithiothreitol produces protein that contains two [4Fe-4S](2+) clusters per dimer; subsequent reduction with dithionite produces two [4Fe-4S](1+) clusters per BioB dimer. BioB that contains two [4Fe-4S](2+/1+) clusters per dimer is rapidly and reversibly reduced and oxidized, suggesting that this is the redox-active form of the iron-sulfur cluster in the anaerobic enzyme.     

Further theoretical refinement on the internal thermodynamics of perfect enzymes.

By solving simultaneously the equation for ''uniform binding'' [Albery & Knowles (1976) Biochemistry 15, 5631-5640] and the equation for ''differential binding'' [Chin (1983) J. Am. Chem. Soc. 105, 6502-6503], I derived the following simple equation for perfect enzymes (with single substrate and single product) under irreversible conditions: K2 = beta(1 + Rs)/1-beta(1 + Rs) where K2 is the internal equilibrium constant and beta is the Brönsted coefficient of the elementary catalytic step, and Rs is defined as [S]0/Ks, with [S]0 being the physiological substrate concentration and Ks being the substrate dissociation constant. The equation suggests that the perfect enzyme can have different internal thermodynamic properties depending on physiological conditions.     

A linearization method for low catalytic activity enzyme kinetic analysis

A kinetic analysis was made and a linear plot based on the general rate equation derived by Laidler [Can. J. Chem. 33, 1614-1624] is proposed. This linearization method allows determining the kinetic parameters (K(m), k(cat)) and [E](0) for enzymes with low catalytic activity. The method was applied to chloroperoxidase from Caldariomyces fumago [EC 1.11.1.10], whose kinetic parameters K(m)(app), k(cat)(app), and [E](0) with monochlorodimedone as substrate, were obtained by using the linearization plot and the V(max) value (calculated by Eadie-Hofstee plot). This plot could also be useful to the study of abenzyme kinetics provided the concentration of the latter is either higher or equal than K(m) value.     

Fluorescence monitoring of the conformational change in alpha 2-macroglobulin induced by trypsin under second-order conditions: the macroglobulin acts both as a substrate and a competitive inhibitor of the protease

The reaction of bovine pancreatic trypsin with human plasma alpha(2)-macroglobulin (alpha(2)M) was studied at 25 degrees C, using equimolar mixtures of E and I in 50 mM potassium phosphate buffer, pH 7. The conformational change in alpha(2)M was monitored through the increase in protein fluorescence at 320 nm (exc lambda, 280 nm). At [alpha(2)M](0) =[E](0) =11.5-200 nM, the fluorescence change data fit the integrated second-order rate equation, (F(infinity) -F(0) )/(F(infinity) -F(t) )=1+k(i,obsd) [alpha(2)M](0) t, indicating that cleavage of the bait region in alpha(2)M was the rate-determining step. The apparent rate constant (k(i,obsd)) was found to be inversely related to reactant concentration. The kinetic behavior of the system was compatible with a model involving reversible, nonbait region binding of E to alpha(2)M, competitively limiting the concentration of E available for bait region cleavage. The intrinsic value of k(i) was (1.7+/-0.24) x 10(7) M(-1) s(-1).K(p), the inhibitory constant associated with peripheral binding, was estimated to be in the submicromolar range. The results of the present study point to a potential problem in interpreting kinetic data relating to protease-induced structural changes in macromolecular substrates. If there is nonproductive binding, as in the case of trypsin and alpha(2)M, and the reactions are monitored under pseudo first-order conditions ([S](0) >[E](0) ), an intrinsically second-order process (such as the rate-limiting bait region cleavage in alpha(2)M) may become kinetically indistinguishable from an intrinsically first-order process (e.g. rate-limiting conformational change). Hence an excess of one component over the other should be avoided in kinetic studies addressing such systems.     

Iron-sulfur clusters of biotin synthase in vivo: a Mössbauer study

Biotin synthase, the enzyme that catalyzes the last step of the biosynthesis of biotin, contains only [2Fe-2S](2+) clusters when isolated under aerobic conditions. Previous results showed that reconstitution with an excess of FeCl(3) and Na(2)S under reducing and anaerobic conditions leads to either [4Fe-4S](2+), [4Fe-4S](+), or a mixture of [4Fe-4S](2+) and [2Fe-2S](2+) clusters. To determine whether any of these possibilities or other different cluster configuration could correspond to the physiological in vivo state, we have used (57)Fe M?ssbauer spectroscopy to investigate the clusters of biotin synthase in whole cells. The results show that, in aerobically grown cells, biotin synthase contains a mixture of [4Fe-4S](2+) and [2Fe-2S](2+) clusters. A mixed [4Fe-4S](2+):[2Fe-2S](2+) cluster form has already been observed under certain in vitro conditions, and it has been proposed that both clusters might each play a significant role in the mechanism of biotin synthase. Their presence in vivo is now another argument in favor of this mixed cluster form.     

Biotin synthase contains two distinct iron-sulfur cluster binding sites: chemical and spectroelectrochemical analysis of iron-sulfur cluster interconversions.

Biotin synthase is an iron-sulfur protein that utilizes AdoMet to catalyze the presumed radical-mediated insertion of a sulfur atom between the saturated C6 and C9 carbons of dethiobiotin. Biotin synthase (BioB) is aerobically purified as a dimer that contains [2Fe-2S](2+) clusters and is inactive in the absence of additional iron and reductants, and anaerobic reduction of BioB with sodium dithionite results in conversion to enzyme containing [4Fe-4S](2+) and/or [4Fe-4S](+) clusters. To establish the predominant cluster forms present in biotin synthase in anaerobic assays, and by inference in Escherichia coli, we have accurately determined the extinction coefficient and cluster content of the enzyme under oxidized and reduced conditions and have examined the equilibrium reduction potentials at which cluster reductions and conversions occur as monitored by UV/visible and EPR spectroscopy. In contrast to previous reports, we find that aerobically purified BioB contains ca. 1.2-1.5 [2Fe-2S](2+) clusters per monomer with epsilon(452) = 8400 M(-)(1) cm(-)(1) per monomer. Upon reduction, the [2Fe-2S](2+) clusters are converted to [4Fe-4S] clusters with two widely separate reduction potentials of -140 and -430 mV. BioB reconstituted with excess iron and sulfide in 60% ethylene glycol was found to contain two [4Fe-4S](2+) clusters per monomer with epsilon(400) = 30 000 M(-)(1) cm(-)(1) per monomer and is reduced with lower midpoint potentials of -440 and -505 mV, respectively. Finally, as predicted by the measured redox potentials, enzyme incubated under typical anaerobic assay conditions is repurified containing one [2Fe-2S](2+) cluster and one [4Fe-4S](2+) cluster per monomer. These results indicate that the dominant stable cluster state for biotin synthase is a dimer containing two [2Fe-2S](2+) and two [4Fe-4S](2+) clusters.     

Mechanism of action of Moloney murine leukemia virus RNase H III.

The mechanism of action of Moloney murine leukemia virus RNase H III was studied, utilizing the model substrate (A)n. (dT)n and polyacrylamide gel electrophoresis to assay enzyme activity. Examination by electrophoresis on 15% polyacrylamide gels in 7 M urea and on DEAE-cellulose paper in 7 M urea revealed that, early in a reaction with [3H](A)n. (dT)n as substrate, RNase H III generated products ranging in length from 80 to 90 nucleotides to less than 10 nucleotides and that after extended incubation the limit digest products generated were 3 to 15 nucleotides long. Product oligomers were of the following configuration: [5'-P, 3'-OH](A)n. RNase H III was shown to be an exonuclease requiring free ends in its substrate for activity by the inability to degrade RNA inserted in Escherichia coli ColE1 plasmid DNA. The enzyme was capable of attacking RNA in RNA-DNA hybrids in the 5' to 3' and 3' to 5' directions as demonstrated by the use of [3H, 5'-32P](A)600. (dT)n and cellulose-[3H](A)n. (dT)n. Rnase H III was random in its mode of action because addition of excess unlabeled (A)n. (dT)n to an ongoing reaction with [3H](A)n. (dT)n as substrate resulted in immediate inhibition of enzyme activity.     

Inhibition and activation of enzymes. The effect of a modifier on the reaction rate and on kinetic parameters

A combined analysis of enzyme inhibition and activation is presented, based on a rapid equilibrium model assumption in which one molecule of enzyme binds one molecule of substrate (S) and/or one molecule of a modifier X. The modifier acts as activator (essential or non-essential), as inhibitor (total or partial), or has no effect on the reaction rate (v), depending on the values of the equilibrium constants, the rate constants of the limiting velocity steps, and the concentration of substrate ([S]). Different possibilities have been analyzed from an equation written to emphasize that v = f([X]) is, in general and at a fixed [S], a hyperbolic function. Formulas for Su (the value of [S], different from zero, at which v is unaffected by the modifier) and v(su) (v at that particular [S]) were deduced. In Lineweaver-Burk plots, the straight lines related to different [X] generally cross in a point (P) with coordinates (Su, v(su)). In certain cases, point P is located in the first quadrant which implies that X acts as activator, as inhibitor, or has no effect, depending on [S]. Furthermore, we discuss: (1) the apparent Vmax and Km displayed by the enzyme in different situations; (2) the degree of effect (inhibition or activation) observed at different concentrations of substrate and modifier; (3) the concept of Ke, a parameter that depends on the concentration of substrate and helps to evaluate the effect of the modifier: it equals the value of [X] at which the increase or decrease in the reaction rate is half of that achieved at saturating [X]. Equations were deduced for the general case and for particular situations, and used to obtain computer-drawn graphs that are presented and discussed. Formulas for apparent Vmax, Km and Ke have been written in a way making it evident that these parameters can be expressed as pondered means.     

On true and apparent Michaelis constants in enzymology. III. Is it linear dependence between apparent michaelis constant and limiting rate and is it possible to determine the substrate constant value using this dependence?

The Slater-Bonner method which is used for graphic determination of substrate constant (Ks) by linear dependence of apparent Michaelis constant (Km(app)) on the limiting rate (V(app)) of enzyme-catalysed reactions with activator participation has been critically analysed. It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model which allow to find correlation Km(app) and V(app) as equation Km(app) = Ks + V(app)/k1[E]0 ([E]0 is a total enzyme concentration, k1 is a rate constant of enzyme-substrate complex formation from free enzyme and substrate) in order to calculate Ks and individual rate constants (k1, k(-1)), but this approach for investigation of all reactions with activator participation ought not to be used. The above equation is not obeyed in general, it may be true for some mechanisms only or under certain ratios of kinetic parameters of enzyme-catalysed reactions.     

Substrate inhibition of xanthine oxidase and its influence on superoxide radical production.

The influence of substrate inhibition on xanthine oxidase-intramolecular electron transport was studied by steady-state kinetic analysis. Experiments with hypoxanthine and xanthine up to 900 microM indicated an inhibition pattern which fitted an equation of the general form nu 0 = nu max . [S]/(Km + a[S] + b[S]2/Ki). Univalent electron flux to oxygen was favored at substrate concentrations above 50 microM. This augmentation of univalent flux percentage that appeared at a high substrate concentration was greater for hypoxanthine that xanthine and at pH 8.3 than at 9.5. Our results support a mechanism of inhibition in which a substrate-reduced enzyme, non-productive Michaelis complex was formed. It is possible that this non-productive complex favored the univalent pathway of enzyme reoxidation (superoxide production) by increasing the midpoint redox potential of the molybdenum active site.     

The time course of hydrolysis of a beta-lactam antibiotic by intact gram-negative bacteria possessing a periplasmic beta-lactamase.

An equation is derived from first principles for describing the change in concentration with time of a beta-lactam antibiotic in the presence of intact Gram-negative bacteria possessing a beta-lactamase located in the periplasmic space. The equation predicts a first-order decline in beta-lactam concentration of the form [S] = [Si]e lambda t, where [S] is the exogenous concentration of beta-lactam, [Si] is the value of [S] at time zero, t is the time from mixing of cells and antibiotic and lambda (less than 0) is the decay constant. The value of lambda is exactly described by the theory in terms of experimentally measurable quantities. Quantitative data concerning cephaloridine hydrolysis by intact cells of Haemophilus influenzae agreed well with the theory, as did data concerning the uptake of 2-nitrophenyl galactoside by intact cells of Escherichia coli. Cephalosporin C hydrolysis by intact cells of Pseudomonas aeruginosa did not progress as predicted by the theory. The theory is applicable to any substrate which is acted on by an enzyme that is located solely in the periplasmic space and that obeys the Michaelis-Menten equation of enzyme kinetics.     

A semi-integrated method for the determination of enzyme kinetic parameters and graphical representation of the Michaelis-Menten equation

A semi-integrated method for the determination of the enzyme kinetics parameters (Km and V) and graphical representation of the Michaelis-Menten equation is proposed as a variation of determination of initial reaction rate (v) as a function of initial substrate concentration ([S]0). The method is based on the determination of the time required to exhaust half of the initial substrate concentration as a function of the initial substrate concentration. The advantages and limitations of this method are discussed.     

Increased intracellular [dATP] enhances cardiac contraction in embryonic chick cardiomyocytes

Although ATP is the physiological substrate for cardiac contraction, cardiac contractility is significantly enhanced in vitro when only 10% of ATP substrate is replaced with 2'-deoxy-ATP (dATP). To determine the functional effects of increased intracellular [dATP] ([dATP](i)) within living cardiac cells, we used hypertonic loading with varying exogenous dATP/ATP ratios, but constant total nucleotide concentration, to elevate [dATP](i) in contractile monolayers of embryonic chick cardiomyocytes. The increase in [dATP](i) was estimated from dilution of dye added in parallel with dATP. Cell viability, average contractile amplitude, rates of contraction/relaxation, spontaneous beat frequency, and Ca2+ transient amplitude and kinetics were examined. At total [dATP](i) above approximately 70 microM, spontaneous contractions ceased, and above approximately 100 microM [dATP](i), membrane blebbing was also observed, consistent with apoptosis. Interestingly, [dATP](i) of approximately 60 microM ( approximately 40% increase over basal [dATP](i) levels) enhanced both amplitude of contraction and the rates of contraction and relaxation without affecting beat frequency. With total [dATP](i) of approximately 60 microM or less, we found no significant change in Ca2+ transients. These data indicate that there is an "optimal" concentration of exogenously loaded [dATP](i) that under controlled conditions can enhance contractility in living cardiomyocytes without affecting beat frequency or Ca2+ transients.