The Michaelis-Menten Approach to Enzyme Kinetics

A particularly useful model for the kinetics of enzyme-catalyzed reactions was devised in 1913 by Leonor Michaelis and Maud Menten. It is stiU the basic model for nonallosteric enzymes and is widely used, even though it has undergone many modifications. 

A typical reaction might be the conversion of some substrate, S, to a product, P. The stoichiometric equation for the reaction is 

The mechanism for an enzyme-catalyzed reaction can be summarized in the form 

Note the assumption that the product is not converted to substrate to any appreciable extent. In this equation, is the rate constant for the formation of the enzyme—substrate complex, ES, from the enzyme, E, and the substrate, S k is the rate constant for the reverse reaction, dissociation of the ES complex to free enzyme and substrate and Ag is the rate constant for the conversion of the ES complex to product P and the subsequent release of product from the enzyme. The enzyme appears explicitly in the mechanism, and the concentrations of both free enzyme, E, and enzyme—substrate complex, ES, therefore, appear in the rate equations. Catalysts characteristically are regenerated at the end of the reaction, and this is true of enzymes. 

The substrate concentration at which the reaction proceeds at one-half its maximum velocity has a special significance. It is given the symbol K, which can be considered an inverse measure of the affinity of the enzyme for the substrate. The lower the K, the higher the affinity. 

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The kinetic parameters for a free enzyme in solution are readily derived using the Michaelis-Menten approach describing pseudo-steady-state conversions. Consider Equation (31.1) representing the conversion of a substrate S into a product P, catalyzed by an enzyme E. The rate of formation of an enzyme/substrate complex, ES, is denoted as ku the reverse reaction by and the rate of subsequent conversion to the free product by k2. 

Until relatively recently this was the only method that could be used conveniently to fit data by regression. This is the reason why so many classical approaches for evaluating biochemical data depended on linearising data, sometimes by quite complex transformations. The best known examples are the use of the Lineweaver-Burk transformation of the Michaelis-Menten model to derive enzyme kinetic data, and of the Scatchard plot to analyse ligand binding equilibria. These linearisation procedures are generally no longer recommended, or necessary. 

The quantitative treatment of kinetic data is based on the pseudophase separation approach, i.e. the assumption that the aggregate constitutes a (pseudo)phase separated from the bulk solution where it is dispersed. Some of the equations below are reminiscent of the well-known Michaelis- Menten equation of enzyme kinetics [101]. This formal similarity has led many authors to draw a parallel between micelle and enzyme catalysis. However, the analogy is limited because most enzymatic reactions are studied with the substrate in a large excess over the enzyme. Even for systems showing a real catalytic behavior of micelles and/or vesicles, the above assumption of the aggregate as a pseudophase does not allow operation with excess substrate. The condition... 

The Eadie-Hofstee plot does a betterjob than the Line-weaver-Burke plot in evenly distributing the data points over the entire substrate concentration range, and can be a useful visual technique for ascertaining whether enzyme kinetics are typical (as shown) or atypical (see Figure 8.18, B and G). The Michaelis-Menten approach basically assumes that enzymes present a single binding site to each substrate. Estimates of V x of drug... 

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

This equation is fundamental to all aspects of the kinetics of enzyme action. The Michaelis-Menten constant, KM, is defined as the concentration of the substrate at which a given enzyme yields one-half of its maximum velocity. is the maximum velocity, which is the rate approached at infinitely high substrate concentration. The Michaelis-Menten equation is the rate equation for a one-substrate enzyme-catalyzed reaction. It provides the quantitative calculation of enzyme characteristics and the analysis for a specific substrate under defined conditions of pH and temperature. KM is a direct measure of the strength of the binding between the enzyme and the substrate. For example, chymotrypsin has a Ku value of 108 mM when glycyltyrosinylglycine is used as its substrate, while the Km value is 2.5 mM when N-20 benzoyltyrosineamide is used as a substrate... 

It has been found experimentally that in most cases v is directly proportional to the concentration of enzyme [.E0] and that v generally follows saturation kinetics with respect to the concentration of substrate [limiting value called Vmax. This is expressed quantitatively in the Michaelis-Menten equation originally proposed by Michaelis and Menten. Km can be seen as an apparent dissociation constant for the enzyme-substrate complex ES. The maximal velocity Vmax = kcat E0. ... 

There are methods used Lo study enzymes other than those of chemical instrumental analysis, such as chromatography, that have already been mentioned. Many enzymes can be crystallized, and their structure investigated by x-ray or electron diffraction methods. Studies of the kinetics of enzyme-catalyzed reactions often yield useful data, much of this work being based on the Michaelis-Menten treatment. Basic to this approach is the concept (hat the action of enzymes depends upon the formation by the enzyme and substrate molecules of a complex, which has a definite, though transient, existence, and then decomposes into the products, of the reaction. Note that this point of view was the basis of the discussion of the specilicity of the active sites discussed abuve. 

Most enzymes react with two or more substrates. For this reason, the Michaelis-Menten equation is inadequate for a full kinetic analysis of these enzyme reactions. Nonetheless, the same general approach can be used to derive appropriate equations for two or more substrates. For example, most enzymes that react with two substrates, A and B, are found to obey one of two equations if initial velocity measurements are made as a function of the concentration of both A and B (with product concentrations equal to zero). These are... 

Many substances interact with enzymes to lower their activity that is, to inhibit them. Valuable information about the mechanism of action of the inhibitor can frequently be obtained through a kinetic analysis of its effects. To illustrate, let us consider a case of competitive inhibition, in which an inhibitor molecule, I, combines only with the free enzyme, E, but cannot combine with the enzyme to which the substrate is attached, ES. Such a competitive inhibitor often has a chemical structure similar to the substrate, but is not acted on by the enzyme. For example, malonate (-OOCCH2COO-) is a competitive inhibitor of succinate (-OOCCH2CH2COO-) dehydrogenase. If we use the same approach that was used in deriving the Michaelis-Menten equation together with the additional equilibrium that defines a new constant, an inhibitor constant, A),... 

Interestingly, a fully appropriate model was developed at the same time as the Langmuir model using a similar basic approach. This is the Michaelis-Menten equation which has proved to be so useful in the interpretation of enzyme kinetics and, thereby, understanding the mechanisms of enzyme reactions. Another advantage in using this model is the fact that a graphical presentation of the data is commonly used to obtain the reaction kinetic parameters. Some basic concepts and applications will be presented here but a more complete discussion can be found in a number of texts. ... 

The concentration of the substrate should be, wherever possible, sufficient for saturation of the enzyme, so that the kinetics in the standard assay approach zero order with regard to substrate. Should this be impossible because of the poor solubility of the substrate or at a low affinity of the enzyme for the substrate, it would be advisable to determine the Michaelis-Menten constant so that the observed values of the reaction fate can be converted into those that would be obtained on saturation with the substrate. 

Early applications of the transform-both-sides approach generally were done to transform a nonlinear problem into a linear one. One of the most common examples is found in enzyme kinetics. Given the Michae-lis-Menten model of enzyme kinetics... 

The Michaelis-Menten equation developed in 1913 ushered in the era of enzyme kinetics and mechanism (chapter 2). Experimentally, its application involves graphing rates (velocities) of reaction (v) against trial concentrations of substrate ([S]). A "saturation" curve is usually observed in which there is a leveling off of v, so as to approach the maximum rate (V a ) as [S] reaches saturation concentration. In practice, it is difficult to accurately determine the onset of saturation and this led to considerable uncertainty in the values of Vmax as well as the enzyme-substrate binding constants (K in chapter 2). 

The derivation mathematics are detailed in many publications dealing with enzyme kinetics. The Michaelis-Menten constant is, however, due to the individual approximation used, not always the same. The simplest values result from the implementation of the equilibrium approximation in which represents the inverse equilibrium constant (eqn (4.2(a))). A more common method is the steady-state approach for which Briggs and Haldane assumed that a steady state would be reached in which the concentration of the intermediate was constant (eqn (4.2(b))). The last important approach, which should be mentioned, is the assumption of an irreversible formation of the substrate complex [k--y = 0) (eqn (4.2(c))), which is of course very unlikely. In real enzyme reactions and even in modelled oxo-transfer reactions, this seems not to be the case. 

The concept of an enzyme-substrate complex is fundamental to the appreciation of enzyme reactions and was initially developed in 1913 by Michaelis and Menten, who derived an equation that is crucial to enzyme studies. Subsequent to Michaelis and Menten several other workers approached the problem from different viewpoints and although their work is particularly useful in advanced kinetic and mechanistic studies, they confirmed the basic concepts of Michaelis and Menten. 

Analyses of enzyme reaction rates continued to support the formulations of Henri and Michaelis-Menten and the idea of an enzyme-substrate complex, although the kinetics would still be consistent with adsorption catalysis. Direct evidence for the participation of the enzyme in the catalyzed reaction came from a number of approaches. From the 1930s analysis of the mode of inhibition of thiol enzymes—especially glyceraldehyde-phosphate dehydrogenase—by iodoacetate and heavy metals established that cysteinyl groups within the enzyme were essential for its catalytic function. The mechanism by which the SH group participated in the reaction was finally shown when sufficient quantities of purified G-3-PDH became available (Chapter 4). 

For an enzyme with typical Michaelis-Menten kinetics, the value of e ranges from about 1 at substrate concentrations far below Km to near 0 as Vmax is approached. Allosteric enzymes can have elasticities greater than 1.0, but not larger than their Hill coefficients (p. 167). 

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