Single-molecule enzyme kinetics in the presence of inhibitors

Recent studies in single-molecule enzyme kinetics reveal that the turnover statistics of a single enzyme is governed by the waiting time distribution that decays as mono-exponential at low substrate concentration and multi-exponential at high substrate concentration. The multi-exponentiality arises due to protein conformational fluctuations, which act on the time scale longer than or comparable to the catalytic reaction step, thereby inducing temporal fluctuations in the catalytic rate resulting in dynamic disorder. In this work, we study the turnover statistics of a single enzyme in the presence of inhibitors to show that the multi-exponentiality in the waiting time distribution can arise even when protein conformational fluctuations do not influence the catalytic rate. From the Michaelis-Menten mechanism of inhibited enzymes, we derive exact expressions for the waiting time distribution for competitive, uncompetitive, and mixed inhibitions to quantitatively show that the presence of inhibitors can induce dynamic disorder in all three modes of inhibitions resulting in temporal fluctuations in the reaction rate. In the presence of inhibitors, dynamic disorder arises due to transitions between active and inhibited states of enzymes, which occur on time scale longer than or comparable to the catalytic step. In this limit, the randomness parameter (dimensionless variance) is greater than unity indicating the presence of dynamic disorder in all three modes of inhibitions. In the opposite limit, when the time scale of the catalytic step is longer than the time scale of transitions between active and inhibited enzymatic states, the randomness parameter is unity, implying no dynamic disorder in the reaction pathway.     

When does the Michaelis-Menten equation hold for fluctuating enzymes?

Enzymes are dynamic entities: both their conformation and catalytic activity fluctuate over time. When such fluctuations are relatively fast, it is not surprising that the classical Michaelis-Menten (MM) relationship between the steady-state enzymatic velocity and the substrate concentration still holds. However, recent single-molecule experiments have shown that this is the case even for an enzyme whose catalytic activity fluctuates on the 10(-4)-10 s range. The purpose of this paper is to examine various scenarios in which slowly fluctuating enzymes would still obey the MM relationship. Specifically, we consider (1) the quasi-static condition (e.g., the conformational fluctuation of the enzyme-substrate complex is much slower than binding, catalysis, and the conformational fluctuations of the free enzyme), (2) the quasi-equilibrium condition (when the substrate dissociation is much faster than catalysis, irrespective of the time scales or amplitudes of conformational fluctuations), and (3) the conformational-equilibrium condition (when the dissociation and catalytic rates depend on the conformational coordinate in the same way). For each of these scenarios, the physical meaning of the apparent Michaelis constant and catalytic rate constant is provided. Finally, as an example, the theoretical analysis of a recent single-molecule enzyme assay is considered in light of the perspectives presented in this paper.     

紫外光谱法实时测定腈水合酶的催化动力学

采用紫外光谱法研究了腈水合酶催化丙烯腈水合的过程,在不同丙烯腈初始浓度下,测定了催化过程中275nm紫外吸光度的变化,计算出丙烯酰胺的生成速率.用Michaelis-Menten方程对不同丙烯腈浓度下的Nocardiasp.腈水合酶催化速率进行了拟合,得到该酶以丙烯腈为底物的米氏常数(K_m)为8.46mmol/L,单位质量腈水合酶的催化速率常数(k_cat)为2398μmol/(min·mg).     

Michaelis-Menten relations for complex enzymatic networks

Most biological processes are controlled by complex systems of enzymatic chemical reactions. Although the majority of enzymatic networks have very elaborate structures, there are many experimental observations indicating that some turnover rates still follow a simple Michaelis-Menten relation with a hyperbolic dependence on a substrate concentration. The original Michaelis-Menten mechanism has been derived as a steady-state approximation for a single-pathway enzymatic chain. The validity of this mechanism for many complex enzymatic systems is surprising. To determine general conditions when this relation might be observed in experiments, enzymatic networks consisting of coupled parallel pathways are investigated theoretically. It is found that the Michaelis-Menten equation is satisfied for specific relations between chemical rates, and it also corresponds to a situation with no fluxes between parallel pathways. Our results are illustrated for a simple model. The importance of the Michaelis-Menten relationship and derived criteria for single-molecule experimental studies of enzymatic processes are discussed.     

On the simple Michaelis-Menten mechanism for chemical reactions

Several mathematical properties associated with the simple Michaelis-Menten mechanism for enzymatic reactions are proven. In particular it is shown that the usual interpretation of the slope of the experimental Michaelis-Menten rate law in terms of the reaction constants of the mechanism can be obtained, in the approximation in which the total concentration of the enzyme is small compared with the Michaelis-Menten constant, independently of the ratio between the total initial concentrations of the enzyme and substrate. Furthermore, the ratio of the total concentration of the enzyme to the Michaelis-Menten constant allows for the elimination of a fast variable in a singular perturbation method, yielding the Michaelis-Menten rate law as a first order approximation.     

Effect of AOT on enzymatic activity of the organic solvent resistant tyrosinase from Streptomyces sp. REN-21 in aqueous solutions and water-in-oil microemulsions

The effect of AOT (sodium-bis(2-ethylhexyl sulfosuccinate)) on enzymatic activity of the organic solvent resistant tyrosinase (OSRT) in aqueous phosphate buffer solutions and in water-in-oil microemulsions of the water/AOT/isooctane system has been investigated. In contrast to mushroom tyrosinase, AOT does not activate OSRT in aqueous solutions, altering its activity very little at concentrations lower than 2 mM. Increasing contents of AOT in isooctane reduce the observed initial reaction rates of oxidation of t-butylcatechol (tBC) and 4-methylcatechol (4-MC). Similarly to mushroom tyrosinase, the effect has been described using an equation based on preferential binding of the substrates by surfactant interface layers. The apparent Michaelis-Menten substrate binding constants increase linearly with AOT concentration (with slopes of 0.12+/-0.02 and 0.051+/-0.006 for tBC and 4-MC, respectively), and the effective enzyme turnover number in the microemulsions remains practically constant.     

Steering the enzymatic activity of proteins by ionic liquids. A case study of the enzyme kinetics of yeast alcohol dehydrogenase

We explore ion-specific effects exerted by ionic liquids (ILs) on the enzyme kinetics of yeast alcohol dehydrogenase. The Michaelis-Menten reaction scheme is used to parameterize the observed kinetics in terms of the apparent dissociation constant of the substrate (Michaelis-Menten constant) K(M), the turnover number k(cat), which reflects the number of product molecules per enzyme molecule per second, and the enzymatic efficiency k(cat)/K(M) of the reaction. Results for fifteen salts are used to deduce Hofmeister anion and cation series. The ion rankings derived from K(M), k(cat) and k(cat)/K(M) differ markedly. Only the results for the enzymatic efficiency correspond to expectations from other phenomena, such as the thermal stability of native proteins. Anion variation has a significantly larger effect on the enzymatic efficiency than cation variation. All ILs decrease k(cat) relative to its value for the IL-free solution, thus driving enzyme deactivation. Enhancements of the enzymatic efficiency by some ions are founded in their effects on the Michaelis-Menten constant. The observed Hofmeister anion and cation series point toward hydrophobic interactions as an important factor controlling ion-specific effects on the enzymatic activity.     

基于分子机制的蛋白质酶解反应经验修饰动力学模型

蛋白质酶解反应动力学行为的复杂性在于体系中底物与产物的多样性,以及由此决定的动力学常数的可变性.基于此,以酪蛋白(case in)—胰蛋白酶(trypsin)为模式体系,拟合求得动力学常数(Km和kcat)随水解度(DH)值变化的函数表达式,其规律为:随DH值增加,Km增大,kcat减小,kcat/Km减小,证明:酶与底物的亲和力随肽链缩短而减小,即高分子量多肽为蛋白酶的适宜底物,而酶解效率与酶解专一性随反应进行逐渐降低.进一步,根据分子水平的蛋白质酶解作用机制,关联水解实验数据,得到case in-trypsin酶解反应的经验修饰动力学方程(模型平均相对误差<5%),为定量表征复杂酶解历程以及高效制备活性多肽提供了理论基础.     

Stationary Kinetics of Light Olefin Oxidation by Hydrogen Peroxide in Aqueous Alkali Solutions in the Presence of Fe(III) Oxide

The steady-state kinetics of ethylene and propylene oxidation by hydrogen peroxide in the presence of Fe(III) oxide in aqueous solutions with the permanent adding of H₂O₂ to the reaction medium was studied. The use of an original method for the study of the steady-state reaction kinetics with gas chromatographic detection of substrate consumption from the gas phase made it possible to estimate the apparent rate constants of ethylene oxidation, the ratio of the rate constants of propylene and ethylene oxidation, the reaction orders with respect to the substrate and oxidant concentration, the dependence of the apparent rate constant of ethylene oxidation on the catalyst weight and on the pH of solution, and the apparent activation energy of the process under condition of substrate distribution between the gas and liquid phases. It was found that the kinetic isotope effect in ethylene oxidation is almost absent when completely deuterated ethylene is used.     

On-line characterization of the activity and reaction kinetics of immobilized enzyme by high-performance frontal analysis

A microreactor by immobilized trypsin on the activated glycidyl methacrylate-modified cellulose membrane packed column was constructed. Immobilized trypsin mirrored the properties of the free enzyme and showed high stability. A novel method to characterize the activity and reaction kinetics of the immobilized enzyme has been developed based on the frontal analysis of enzymatic reaction products, which was performed by the on-line monitoring of the absorption at 410 nm of p-nitroaniline from the hydrolysis of N-alpha-benzoyl-DL-arginine-p-nitroanilide (BAPNA). The hydrolytic activity of the immobilized enzyme was 55.6% of free trypsin. The apparent Michaelis-Menten kinetics constant (Km) and Vmax values measured by the frontal analysis method were, respectively, 0.12 mM and 0.079 mM min(-1) mg enzyme(-1). The former is very close to that observed by the static and off-line detection methods, but the latter is about 15% higher than that of the static method. Inhibition of the immobilized trypsin by addition of benzamidine into substrate solution has been studied by the frontal analysis method. The apparent Michaelis-Menten constant of BAPNA (Km), the inhibition constant of benzamidine (Ki) and Vmax were determined. It was indicated that the interaction of BAPNA and benzamidine with trypsin is competitive, the Km value was affected but the Vmax was unaffected by the benzamidine concentration.     

Off-line form of the Michaelis-Menten equation for studying the reaction kinetics in a polymer microchip integrated with enzyme microreactor

We firstly transformed the traditional Michaelis-Menten equation into an off-line form which can be used for evaluating the Michaelis-Menten constant after the enzymatic reaction. For experimental estimation of the kinetics of enzymatic reactions, we have developed a facile and effective method by integrating an enzyme microreactor into direct-printing polymer microchips. Strong nonspecific adsorption of proteins was utilized to effectively immobilize enzymes onto the microchannel wall, forming the integrated on-column enzyme microreactor in a microchip. The properties of the integrated enzyme microreactor were evaluated by using the enzymatic reaction of glucose oxidase (GOx) with its substrate glucose as a model system. The reaction product, hydrogen peroxide, was electrochemically (EC) analyzed using a Pt microelectrode. The data for enzyme kinetics using our off-line form of the Michaelis-Menten equation was obtained (K(m) = 2.64 mM), which is much smaller than that reported in solution (K(m) = 6.0 mM). Due to the hydrophobic property and the native mesoscopic structure of the poly(ethylene terephthalate) film, the immobilized enzyme in the microreactor shows good stability and bioactivity under the flowing conditions.     

Surface enzyme kinetics for biopolymer microarrays: a combination of Langmuir and Michaelis-Menten concepts

Real-time surface plasmon resonance (SPR) imaging measurements of surface enzymatic reactions on DNA microarrays are analyzed using a kinetics model that couples the contributions of both enzyme adsorption and surface enzyme reaction kinetics. For the case of a 1:1 binding of an enzyme molecule (E) to a surface-immobilized substrate (S), the overall enzymatic reaction can be described in terms of classical Langmuir adsorption and Michaelis-Menten concepts and three rate constants: enzyme adsorption (k(a)), enzyme desorption (k(d)) and enzyme catalysis (k(cat)). In contrast to solution enzyme kinetics, the amount of enzyme in solution is in excess as compared to the amount of substrate on the surface. Moreover, the surface concentration of the intermediary enzyme-substrate complex (ES) is not constant with time, but goes to zero as the reaction is completed. However, kinetic simulations show that the fractional surface coverage of ES on the remaining unreacted sites does reach a steady-state value throughout the course of the surface reaction. This steady-state value approaches the Langmuir equilibrium value for cases where k(a)[E] > k(cat). Experiments using the 3' --> 5' exodeoxyribonuclease activity of Exonuclease III on double-stranded DNA microarrays as a function of temperature and enzyme concentration are used to demonstrate how this model can be applied to quantitatively analyze the SPR imaging data.     

Understanding dynamic disorder fluctuations in single-molecule enzymatic reactions

Single-molecule studies of enzymatic reactions reveal fluctuations in the reaction rate, which cannot be explained by classic Markovian dynamics. This dynamic disorder is attributed to slow transitions in enzyme conformations that take place over timescales longer than reaction cycle times. In this review we summarize current theoretical models for reaction kinetics in fluctuating, single enzyme systems. Also examined are some of the implications of single-molecule fluctuations for reaction rates in systems such as cells or biosensors that contain a moderate number of molecular copies. We conclude that the dynamic disorder in single-molecule enzyme systems is well-described by available models. However, more work is required to study the effect of single-molecule fluctuations on finite systems over limited periods of time.     

Measurement of Catalytic Reaction Kinetics for Adsorbed Enzyme Monolayers

We present a new assay based on total internal reflection fluorescence (TIRF) to quantify the catalytic activity of adsorbed enzyme monolayers on macroscopically flat surfaces. The need for such an assay derives from a general shortage of assay methods that are sufficiently sensitive to measure reaction kinetics for just a single monolayer of enzymes. The assay is based on the enzymatic conversion of a soluble, nonfluorescent fluorogenic substrate reagent to a soluble, highly fluorescent product. The reaction occurs at the solid-liquid interface where the enzymes are adsorbed. Fluorogenic substrates are introduced to the adsorbed layer by convective diffusion from solutions undergoing steady laminar slit flow. The exponentially decaying evanescent wave that is produced by total internal reflection serves as a "spectroscopic ruler" to resolve the spatial concentration profile of fluorescent products in solution near the interface. By measuring the steady-state fluorescence signal as a function of the Peclet number that characterizes mass transfer conditions in the experiment, it is possible to determine the enzymatic reaction rate. Here we present the development of the method and its application to a test system of beta-galactosidase adsorbed to methylated silica surfaces. Compared to the enzymatic rate constants for this enzyme in free solution, adsorption decreased the Michaelis-Menten rate constant kcat by a factor of 10 and increased the equilibrium binding constant Km by a factor of 4.5. Thus the intrinsic activity of the enzyme, as represented by the ratio kcat/Km, decreased 45-fold due to adsorption. Copyright 1999 Academic Press.     

Kinetics of oxidation of o-dianisidine by hydrogen peroxide in the presence of antibody complexes of iron(III) coproporphyrin

The complex of iron(III) coproporphyrinl (FeCPI) with antibody D5E3 was studied as an artificial peroxidase, usingo-dianisidine as a substrate. At saturation with respect to antibody, the initial rates ofo-dianisidine oxidation are practically the same for free and bound FeCPI at a concentration 5 × 10^-9M, but the catalytic rate constant (k_c) for bound FeCPI exceed (k_c) for free FeCPI by two-to threefold. This difference can be explained by a real enhancement of (k_c) at the antibody-active site. The dependence of initial rates of the reaction on substrate concentrations obeyed Michaelis-Menten kinetics and revealed substrate activation at high concentrations ofo-dianisidine. A comparison of the Stern-Volmer constants foro-dianisidineinduced quenching of the porphyrin fluorescence proves that antibody-bound coproporphyrin is equivalently accessible to the substrate as protoporphyrin bound to apoperoxidase from horseradish peroxidase (HRP). Based on analysis of the (k_c) dependence on H₂O₂ concentrations in the FeCPI-antibody system, we suggest that interaction with hydrogen peroxide is the rate-limiting step for the oxidation reaction.     

Is there anything left to say on enzyme kinetic constants and quasi-steady state approximation?

In this paper we re-examine the commonly accepted meaning of the two kinetic constants characterizing any enzymatic reaction, according to Michaelis-Menten kinetics. Expanding in terms of exponentials the solutions of the ODEs governing the reaction, we determine a new constant, which corrects some misinterpretations of current biochemical literature.     

Enzymatic Hydrolysis of N-Acylated 1-Aminophosphonic Acids

Abstract

Penicillin acylase from E. coli (FC 3.5.1.11) was found to hydrolyse N-phenylacetylated 1-aminoalkylphosphonic acids and their esters. Enzyme preferentially converts the R-form of the substrates: the ratios of the bimolecular rate constants of penicillin acylase-catalysed hydrolysis of R-and S- forms of 1-(N-phenylacetaminol-ethylphosphonic acid and its dimethyl- and diisopropyl- esters are 58000, 2600, 1800; these derivatives were shown to have the greatest values of the catalytic constants for enzymatic hydrolysis of all known substrates of penicillin acylase: 237, 148, and 134 s; corresponding values of Michaelis constants are 3.7×10^?5, 6.8×10^?4, and 6.2×10^?4 M. The kinetics of the enzymatic hydrolysis of 1-(N-phenylacetaminol-ethylphosphonic acid was investigated up to high degrees of conversion. The inhibition of penicillin acylase by high concentrations of the R-form of the substrate (with substrate inhibition constant 0.07 Ml and competitive inhibition by the reaction product phenylacetic acid (K_i=3.5×10_?5 M) was observed. Penicillin acylase was shown to possess quite broad substrate specificity among N-acylated 1-aminoalkylphosphonic acids and was found to be capable of hydrolysing 1-(N-phenylacetaminol-substituted 2-phenylethyl-, 1-phenylmethyl- and 3-methylbutylphosphonic acids with high efficiency and enantioselectivity.     

Flaws in foldamers: conformational uniformity and signal decay in achiral helical peptide oligomers

Although foldamers, by definition, are extended molecular structures with a well-defined conformation, minor conformers must be populated at least to some extent in solution. We present a quantitative analysis of these minor conformers for a series of helical oligomers built from achiral but helicogenic α-amino acids. By measuring the chain length dependence or chain position dependence of NMR or CD quantities that measure screw-sense preference in a helical oligomer, we quantify values for the decay constant of a conformational signal as it passes through the molecular structure. This conformational signal is a perturbation of the racemic mixture of M and P helices that such oligomers typically adopt by the inclusion of an N or C terminal chiral inducer. We show that decay constants may be very low (<1% signal loss per residue) in non-polar solvents, and we evaluate the increase in decay constant that results in polar solvents, at higher temperatures, and with more conformationally flexible residues such as Gly. Decay constants are independent of whether the signal originates from the N or the C terminus. By interpreting the decay constant in terms of the probability with which conformations containing a screw-sense reversal are populated, we quantify the populations of these alternative minor conformers within the overall ensemble of secondary structures adopted by the foldamer. We deduce helical persistence lengths for Aib polymers that allow us to show that in a non-polar solvent a peptide helix, even in the absence of chiral residues, may continue with the same screw sense for approximately 200 residues.     

Interpreting the catalytic voltammetry of an adsorbed enzyme by considering substrate mass transfer, enzyme turnover, and interfacial electron transport

Redox active enzymes can be adsorbed onto electrode surfaces to catalyze the interconversion of oxidized and reduced substrates in solution, driven by the supply or removal of electrons by the electrode. The catalytic current is directly proportional to the rate of enzyme turnover, and its dependence on the electrode potential can be exploited to define both the kinetics and thermodynamics of the enzyme's catalytic cycle. However, observed electrocatalytic voltammograms are often complex because the identity of the rate limiting step changes with the electrode potential and under different experimental conditions. Consequently, extracting mechanistic information requires that accurate models be constructed to deconvolute and analyze the observed behavior. Here, a basic model for catalysis by an adsorbed enzyme is described. It incorporates substrate mass transport, enzyme kinetics, and interfacial electron transport, and it accurately reproduces experimentally recorded voltammograms from the oxidation of NADH by subcomplex Ilambda (the hydrophilic subcomplex of NADH:ubiquinone oxidoreductase), under a range of conditions. Mass transport is imposed by a rotating disk electrode and described by the Levich equation. Interfacial electron transport is controlled by the electrode potential and characterized by a dispersion of rate constants, according to the model of Léger and co-workers. Here, the Michaelis-Menten equation is used for the enzyme kinetics, but our methodology can also be readily applied to derive and apply analogous equations relating to alternative enzyme mechanisms. Therefore, our results are highly relevant to the interpretation of electrocatalytic voltammograms for adsorbed enzymes in general.