A century of enzyme kinetic analysis, 1913 to 2013

This review traces the history and logical progression of methods for quantitative analysis of enzyme kinetics from the 1913 Michaelis and Menten paper to the application of modern computational methods today. Following a brief review of methods for fitting steady state kinetic data, modern methods are highlighted for fitting full progress curve kinetics based upon numerical integration of rate equations, including a re-analysis of the original Michaelis–Menten full time course kinetic data. Finally, several illustrations of modern transient state kinetic methods of analysis are shown which enable the elucidation of reactions occurring at the active sites of enzymes in order to relate structure and function.     

On true and apparent Michaelis constants in enzymology. II. Is the equation k(m)app = k(s) + k)cat)/k(1) true for enzyme-catalysed reactions with activator participation?

The article is dedicated to analysis of equation which expresses apparent Michaelis constant K(m)app) of enzyme-catalysed reactions with activator participation by means of the substrate constant K(s) and rate constant of enzyme-substrate complex decomposition k(cat). It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model and to derive equation of apparent Michaelis constant as K(m(app) = K(s) + k(cat)/k(1), but this approach cannot be used for investigation of all reactions with activator participation. The equation mentioned above is not obeyed in the general case, it may be true for some mechanisms only or under certain ratio of kinetic parameters of enzyme-catalysed reactions.     

Formate as an Intermediate in the Bovine Rumen Fermentation

An average of 11 (range, 2 to 47) mumoles of formate per g per hr was produced and used in whole bovine rumen contents incubated in vitro, as calculated from the product of the specific turnover rate constant, k, times the concentration of intercellular formate. The latter varied between 5 and 26 (average, 12) nmoles/g. The concentration of formate in the total rumen contents was as much as 1,000 times greater, presumably owing to formate within the microbial cells. The concentration of formate in rumen contents minus most of the plant solids was varied, and from the rates of methanogenesis the Michaelis constant, K(m), for formate conversion to CH(4) was estimated at 30 nmoles/g. Also, the dissolved H(2) was measured in relation to methane production, and a K(m) of 1 nmole/g was obtained. A pure culture of Methanobacterium ruminantium showed a K(m) of 1 nmole of H(2)/g, but the K(m) for formate was much higher than the 30 nmoles for the rumen contents. It is concluded that nonmethanogenic microbes metabolize intercellular formate in the rumen. CO(2) and H(2) are the principal substrates for rumen methanogenesis. Eighteen per cent of the rumen methane is derived from formate, as calculated from the intercellular concentration of hydrogen and formate in the rumen, the Michaelis constants for conversion of these substrates by rumen liquid, and the relative capacities of whole rumen contents to ferment these substrates.     

Hydrolysis of liquefied corn starch in a membrane reactor

The objective of this study was to develop a continuous hydrolysis process for the enzymatic saccharification of liquefied corn starch using a membrane reactor. A residence time distribution study confirmed that the membrane reactor could be modeled as a simple continuous stirred tank reactor (CSTR). Kinetic studies indicated that the continuous reactor operated in the first-order region with respect to substrate concentration at substrate concentrations greater than 200 g/L. At a residence time of 1 h and an enzyme concentration of 1 g/L, the maximum reaction velocity (V(m)) was 3.86 g glucose/L min and the apparent Michaelis constant (K(m) (')) was 562 g/L. The K(m) (') value for the continuous reactor was 2-7 times greater than that obtained in a batch reactor.Kinetic data were fit to a model based on the Michaelis-Menten rate expression and the design equation for a CSTR. Application of the model at low reactor space times was successful. At space times of 6 min or less, the model predicted the reactor's performance reasonably well. Additional work on the detection and quantitation of reversion products formed by glucoamylase is required. Isolation, detection, and quantitation of reversion products by HPLC was difficult. Detailed analysis on the formation of these reversion products could lead to better reactor designs in the future.     

Removal of fiber from vines by solid state fermentation/enzymatic degradation: a comparison of flax and kudzu retting

Kinetic data describing the decomposition of the outer sheath of kudzu vines (undergoing a solid fermentation process in a glass beaker of soil) have been analyzed to determine the two constants, K(m)/S(o) and V/S(o), where S(o) is the initial substrate concentration, K(m) the Michaelis constant, and V the maximum product rate. The kinetic data are expressed by a simple time-varying desheathing index, obtained from the number of spatula scrapings required to reach the desired hard cellulosic fibers (vascular bundles) of the plant. A simple relationship between the desheathing index, D.I. and the dimensionless product concentration, P/S(o), is proposed to relate the D.I. data and enzyme kinetic concentration data. Thus, the Michaelis-Menten enzyme kinetic parameters can be estimated from easily obtained physical (non-chemical data; the D.I.(t) measurements). This low energy process for processing vines into valuable fibers is similar to the traditional microbial method for recovering flax fibers for linen cloth, by retting of the flax plant vines, except there is no unbound liquid water present in the soil medium.     

The integrated Michaelis-Menten rate equation: déjà vu or vu jàdé?

A recent article of Johnson and Goody (Biochemistry, 2011;50:8264–8269) described the almost-100-years-old paper of Michaelis and Menten. Johnson and Goody translated this classic article and presented the historical perspective to one of incipient enzyme-reaction data analysis, including a pioneering global fit of the integrated rate equation in its implicit form to the experimental time-course data. They reanalyzed these data, although only numerical techniques were used to solve the model equations. However, there is also the still little known algebraic rate-integration equation in a closed form that enables direct fitting of the data. Therefore, in this commentary, I briefly present the integral solution of the Michaelis-Menten rate equation, which has been largely overlooked for three decades. This solution is expressed in terms of the Lambert W function, and I demonstrate here its use for global nonlinear regression curve fitting, as carried out with the original time-course dataset of Michaelis and Menten.     

Nutrient affinity,half‐saturation constants and the cost of toxin production in dinoflagellates

The two parameters of the Michaelis–Menten model, the maximum uptake rate and the half‐saturation constant, are not stochastically independent, and the half‐saturation constant is not a measure of nutrient affinity, as commonly assumed. Failure to realise their interdependence and mechanistic interpretation may lead to the emergence of false trade‐offs.     

Accurate kinetic parameter estimation during progress curve analysis of systems with endogenous substrate production

We have identified an error in the published integral form of the modified Michaelis–Menten equation that accounts for endogenous substrate production. The correct solution is presented and the error in both the substrate concentration, S, and the kinetic parameters V_m, K_m, and R resulting from the incorrect solution was characterized. The incorrect integral form resulted in substrate concentration errors as high as 50% resulting in 7–50% error in kinetic parameter estimates. To better reflect experimental scenarios, noise containing substrate depletion data were analyzed by both the incorrect and correct integral equations. While both equations resulted in identical fits to substrate depletion data, the final estimates of V_m, K_m, and R were different and K_m and R estimates from the incorrect integral equation deviated substantially from the actual values. Another observation was that at R = 0, the incorrect integral equation reduced to the correct form of the Michaelis–Menten equation. We believe this combination of excellent fits to experimental data, albeit with incorrect kinetic parameter estimates, and the reduction to the Michaelis–Menten equation at R = 0 is primarily responsible for the incorrectness to go unnoticed. However, the resulting error in kinetic parameter estimates will lead to incorrect biological interpretation and we urge the use of the correct integral form presented in this study. Biotechnol. Bioeng. 2011;108: 2499–2503. © 2011 Wiley Periodicals, Inc.     

Predicting the performance of immobilized enzyme reactors using reversible Michaelis–Menten kinetics

A general mathematical model is developed in the present work for predicting the steady state performance of immobilized enzyme reactor performing reversible Michaelis - Menten kinetics. The model takes into account the effect of external diffusional limitations, the axial dispersion and the equilibrium constant on reactor performance quantified as relative substrate conversion and yield. The performance of reactor is characterized using the dimensionless parameters of Damkohler number, Stanton number, Peclet number, the equilibrium constant and the dimensionless input substrate concentration. The reactor performance is described for the two extreme cases of plug flow reactor (PFR) and continuous stirred tank reactor (CSTR) in addition to the intermediate case of dispersed plug flow reactor (DPFR). The performance of reactor is compared for the two cases of zero order and reversible first order kinetics.     

Nonfixed relationship of the Michaelis constant and maximum velocity with their corresponding rate constants

The Michaelis constant (K(m)) and V(mas) (E0k(cat)) values for two mutant sets of enzymes were studied from the viewpoint of their definition in a rapid equilibrium reaction model and in a steady state reaction model. The "AMP set enzyme" had a mutation at the AMP-binding site (Y95F, V67I, and V67I/L76V), and the "ATP set enzyme" had a mutation at a possible ATP-binding region (Y32F, Y34F, and Y32A/Y34A). Reaction rate constants obtained using steady state model analysis explained discrepancies found by the rapid equilibrium model analysis. (i) The unchanged number of bound AMPs for Y95F and the wild type despite the markedly increased K(m) values for AMP of the AMP set of enzymes was explained by alteration of the rate constants of the AMP step (k(+2), k(-2)) to retain the ratio k(+2)/k(-2). (ii) A 100 times weakened selectivity of ATP for Y34F in contrast to no marked changes in K(m) values for both ATP and AMP for the ATP set of enzymes was explained by the alteration of the rate constants of the ATP steps. A similar alteration of the K(m) and k(cat) values of these enzymes resulted from distinctive alterations of their rate constants. The pattern of alteration was highly suggestive. The most interesting finding was that the rate constants that decided the K(m) and k(cat) values were replaced by the mutation, and the simple relationships between K(m), k(cat), and the rate constants of K(m)1 = k(+1)/k(-1) and k(cat) = k(f) were not valid. The nature of the K(m) and k(cat) alterations was discussed.     

Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

A Kinetic Analysis of Coupled (or Auxiliary) Enzyme Reactions

As a case study, we consider a coupled (or auxiliary) enzyme assay of two reactions obeying the Michaelis–Menten mechanism. The coupled reaction consists of a single-substrate, single-enzyme non-observable reaction followed by another single-substrate, single-enzyme observable reaction (indicator reaction). In this assay, the product of the non-observable reaction is the substrate of the indicator reaction. A mathematical analysis of the reaction kinetics is performed, and it is found that after an initial fast transient, the coupled reaction is described by a pair of interacting Michaelis–Menten equations. Moreover, we show that when the indicator reaction is fast, the quasi-steady-state dynamics are governed by three fast variables and one slow variable. Timescales that approximate the respective lengths of the indicator and non-observable reactions, as well as conditions for the validity of the Michaelis–Menten equations, are derived. The theory can be extended to deal with more complex sequences of enzyme-catalyzed reactions.     

You wrote it; you own it!

Authors of papers published in Rockefeller University Press journals (The Journal of Cell Biology, The Journal of Experimental Medicine, or The Journal of General Physiology) now retain copyright to their published work. This permits authors to reuse their own work in any way, as long as they attribute it to the original publication. Third parties may use our published materials under a Creative Commons license, six months after publication.     

You wrote it; you own it!

Authors of papers published in Rockefeller University Press journals (The Journal of Cell Biology, The Journal of Experimental Medicine, or The Journal of General Physiology) now retain copyright to their published work. This permits authors to reuse their own work in any way, as long as they attribute it to the original publication. Third parties may use our published materials under a Creative Commons license, six months after publication.     

Regulation of potassium absorption in barley roots: an allosteric model

Plasmalemma influx isotherms for K(+) were measured in the system I concentration range (0.01-0.32 mm), for barley (Hordeum vulgare L.) roots of varying internal K(+) concentration, and Km values for influx calculated. In plants grown for several days in CaSO(4) or in CaSO(4) plus KCl solutions, as well as in plants grown in CaSO(4) for several days and then rapidly loaded with KCl during a pretreatment period, Michaelis constant values were positively correlated with internal K(+) concentrations. Influx of K(+) is shown to be sigmoidally related to internal K(+) concentration and Hill plots of influx data give linear transformations with n = 4. This information is taken as support for an allosteric model for the regulation of K(+) influx in which the "carrier" is envisaged as possessing a single external binding site for K(+) as well as four internal sites for allosteric control of influx.     

Immobilized enzyme cellulose hollow fibers: II. Kinetic analysis

Immobilized enzyme hollow fibers may be useful in the purification or treatment of whole blood under clinical conditions. In this study, catalytically pure heparinase was immobilized to cellulose to analyze the feasibility for the removal of heparin's anticoagulant activity from whole blood. The kinetics of catalytically pure heparinase immobilized to regenerated cellulose hollow fibers were quantified with respect to mass transfer coefficient and enzyme loading. The kinetic analysis showed that increases in the mass transfer coefficient of heparin in the fiber lumen decreased the apparent Michaelis constant while increases in enzyme activity immobilized to the fiber lumen increased the apparent Michaelis constant. The apparent Michaelis constant was an order of magnitude greater than the intrinsic K(m) value for the system. The intrinsic K(m) value for heparinase-cellulose is 0.4 +/- 0.3 mg/mL (N = 6) and it is the same order of magnitude as the K(m) value for soluble heparinase.     

Adsorption of Trichoderma reesei cellulase on cellulose: experimental data and their analysis by different equations

The adsorption of cellulase from Trichoderma reesei MCG 77 on Avicel was measured at varying cellulase (2-8 g/L) and Avicel (10-200 g/L) concentrations at pH 4.8 and 50 degrees C. Different mathematical equations were derived for the evaluation of the experimental data. The fraction of cellulase protein that can maximally be adsorbed is 0.96, and 1 g Avicel can bind maximally 0.092 g cellulase protein. The Michaelis constant for the adsorption equilibrium [cellulase] + [Avicel] right harpoon over left harpoon [cellulase Avicel] complex is between 2.0 and 2.3 . 10(-5) mol/L. This value is based on the assumption that cellulase has an average molecular weight of 48.000. The apparent molecular weight of Avicel, i.e., that amount in grams that can bind 1 mol cellulase, is 520,000. Under maximum binding the enzyme covers on Avicel a surface of 42 m(2)/g, and the occupied volume is 0.186 cm(3)/g Avicel.     

Patterns of sulfur isotope fractionation during microbial sulfate reduction

Studies of microbial sulfate reduction have suggested that the magnitude of sulfur isotope fractionation varies with sulfate concentration. Small apparent sulfur isotope fractionations preserved in Archean rocks have been interpreted as suggesting Archean sulfate concentrations of <200 μm , while larger fractionations thereafter have been interpreted to require higher concentrations. In this work, we demonstrate that fractionation imposed by sulfate reduction can be a function of concentration over a millimolar range, but that nature of this relationship depends on the organism studied. Two sulfate‐reducing bacteria grown in continuous culture with sulfate concentrations ranging from 0.1 to 6 mm showed markedly different relationships between sulfate concentration and isotope fractionation. Desulfovibrio vulgaris str. Hildenborough showed a large and relatively constant isotope fractionation (³⁴ε_SO4‐H2S ? 25‰), while fractionation by Desulfovibrio alaskensis G20 strongly correlated with sulfate concentration over the same range. Both data sets can be modeled as Michaelis–Menten (MM)‐type relationships but with very different MM constants, suggesting that the fractionations imposed by these organisms are highly dependent on strain‐specific factors. These data reveal complexity in the sulfate concentration–fractionation relationship. Fractionation during MSR relates to sulfate concentration but also to strain‐specific physiological parameters such as the affinity for sulfate and electron donors. Previous studies have suggested that the sulfate concentration–fractionation relationship is best described with a MM fit. We present a simple model in which the MM fit with sulfate concentration and hyperbolic fit with growth rate emerge from simple physiological assumptions. As both environmental and biological factors influence the fractionation recorded in geological samples, understanding their relationship is critical to interpreting the sulfur isotope record. As the uptake machinery for both sulfate and electrons has been subject to selective pressure over Earth history, its evolution may complicate efforts to uniquely reconstruct ambient sulfate concentrations from a single sulfur isotopic composition.     

New insights into activation and substrate recognition of polyhydroxyalkanoate synthase from Ralstonia eutropha

The polyhydroxyalkanoate synthase of Ralstonia eutropha (PhaC_Re) shows a lag time for the start of its polymerization reaction, which complicates kinetic analysis of PhaC_Re. In this study, we found that the lag can be virtually eliminated by addition of 50 mg/L TritonX-100 detergent into the reaction mixture, as well as addition of 2.5 g/L Hecameg detergent as previously reported by Gerngross and Martin (Proc Natl Sci USA 92: 6279–6283, 1995). TritonX-100 is an effective lag eliminator working at much lower concentration than Hecameg. Kinetic analysis of PhaC_Re was conducted in the presence of TritonX-100, and PhaC_Re obeyed Michaelis–Menten kinetics for (R)-3-hydroxybutyryl-CoA substrate. In inhibitory assays using various compounds such as adenosine derivatives and CoA derivatives, CoA free acid showed competitive inhibition but other compounds including 3′-dephospho CoA had no inhibitory effect. Furthermore, PhaC_Re showed a considerably reduced reaction rate for 3′-dephospho (R)-3-hydroxybutyryl CoA substrate and did not follow typical Michaelis–Menten kinetics. These results suggest that the 3′-phosphate group of CoA plays a critical role in substrate recognition by PhaC_Re.     

Ground-state destabilization in orotate phosphoribosyltransferases by binding isotope effects

Orotate phosphoribosyltransferases (OPRTs) form and break the N-ribosidic bond to pyrimidines by way of ribocation-like transition states (TSs) and therefore exhibit large α-secondary 1'-(3)H k(cat)/K(m) kinetic isotope effects (KIEs) [Zhang, Y., and Schramm, V. L. (2010) J. Am. Chem. Soc. 132, 8787-8794]. Substrate binding isotope effects (BIEs) with OPRTs report on the degree of ground-state destabilization for these complexes and permit resolution of binding and transition-state effects from the k(cat)/K(m) KIEs. The BIEs for interactions of [1'-(3)H]orotidine 5'-monophosphate (OMP) with the catalytic sites of Plasmodium falciparum and human OPRTs are 1.104 and 1.108, respectively. These large BIEs establish altered sp(3) bond hybridization of C1' toward the sp(2) geometry of the transition states upon OMP binding. Thus, the complexes of these OPRTs distort OMP part of the way toward the transition state. As the [1'-(3)H]OMP k(cat)/K(m) KIEs are approximately 1.20, half of the intrinsic k(cat)/K(m) KIEs originate from BIEs. Orotidine, a slow substrate for these enzymes, binds to the catalytic site with no significant [1'-(3)H]orotidine BIEs. Thus, OPRTs are unable to initiate ground-state destabilization of orotidine by altered C1' hybridization because of the missing 5'-phosphate. However the k(cat)/K(m) KIEs for [1'-(3)H]orotidine are also approximately 1.20. The C1' distortion for OMP happens in two steps, half upon binding and half on going from the Michaelis complex to the TS. With orotidine as the substrate, there is no ground-state destabilization in the Michaelis complexes, but the C1' distortion at the TS is equal to that of OMP. The large single barrier for TS formation with orotidine slows the rate of barrier crossing.