Michaelis-Menten formalism

Figure 8 Plot of the initial rate of the enzyme-catalyzed oxidation of 1-phenylpropanol as a function of % ee. The solid line represents a fit of the data to the Michaelis-Menten formalism for competitive inhibition where [S] = [ -(60)] and [ ] = [ -(60)]. The total alcohol concentration was maintained constant at lOmM.100...

The decrease in absorption at 339 nm was measured for 5 min. From the experimental data, the reciprocal plot (l/F versus 1/[NADH]) was obtained. In this case, pyruvate was present at saturating concentration. Therefore, data were interpreted in terms of the basic Michaelis-Menten formalism. 

These properties are likely to have an important influence on the behavior of intact biochemical systems, e.g., within the living cell, enzymes do not function in dilute homogeneous conditions isolated from one another. The postulates of the Michaelis-Menten formalism are violated in these processes and other formalisms must be considered for the analysis of kinetics in situ. The intracellular environment is very heterogeneous indeed. Many enzymes are now known to be localized within 2-dimensional membranes or quasi 1-dimensional channels, and studies of enzyme organization in situ [26] have shown that essentially all enzymes are found in highly organized states. The mechanisms are more complex, but they are still composed of elementary steps governed by fractal kinetics. 

Identifying Relevant Interactions Obtaining Appropriate Rate Laws Critique of the Michaelis-Menten Formalism Conclusions... 

Many of the subsequent developments in enzyme kinetics share the same basic postulates of Michaelis-Menten kinetics. Although the mechanisms and equations may be different in detail, they all lead to rate laws that are linear functions of enzyme concentration and rational functions of the reactant and modifier concentrations. Hence, all these developments are based upon the same underlying formalism, which I shall refer to as the Michaelis-Menten Formalism. 

We shall conelude this section by emphasizing some general properties of the rate law in the Michaelis-Menten Formalism (Wong and Hanes, 1962 Savageau, 1969a). 

Rate laws based on the Michaelis-Menten Formalism are functions only of kinetic parameters and concentration variables. 

Because rate laws based on the Michaelis-Menten Formalism have this particular mathematical form, they can be analyzed with a general method developed for such functions by Bode (1945). For details see Savageau (1976). 

The Michaelis-Menten Formalism provides the mathematical framework within which most of the kinetic theory of biochemical reactions has developed. The common steps in the application of this approach are ... 

A number of practical prescriptions have been developed over the years that are designed in large part to ensure that the postulates of the Michaelis-Menten Formalism are satisfied. 

The Michaelis-Menten Formalism has been remarkably successful in elucidating the mechanisms of isolated reactions in the test tube. There are numerous treatments of this use of kinetics, and many of these provide a thoughtful critique of the potential pit falls. In short, reliable results can be obtained with steady-state methods if one is careful to follow the canons and if one remembers that several mechanisms may yield the same kinetic behavior. Isotope exchange, pre-steady state, and other transient or relaxation kinetic techniques, as well as various chemical and physical methods, also have been applied in conjunction with steady-state kinetic methods to dissect the elementary reactions within an enzyme-catalyzed reaction and to distinguish between various models (e.g., see Cleland, 1970 Kirschner, 1971 Segel, 1975 Hammes, 1982 Fersht, 1985). 

Finally, it should be noted that while the Michaelis-Menten Formalism may be appropriate for many isolated enzymes in vitro, this does not imply that the resulting rate law for the reaction will be the classical Michaelis-Menten rate law [Eqn. (22)]. Hill et al. (1977) have made a careful assessment of this issue and, on the basis of their results, have come to question whether the simple Michaelis-Menten rate law fits any enzyme that is examined with sufficient care. The tendency to ignore inconsistencies, and continue to treat rate laws as if they were the classical case, indicates that the grip of the conventional Michaelis-Menten paradigm is very strong. We shall examine this point from another perspective in the following section. 

Bode analysis (Bode, 1945) in principle provides a general method for estimating the parameter values in rational functions of the type that characterize rate laws based on the Michaelis-Menten Formalism. The usefulness of this method, which... 

Given the limitations described above, it is easy to understand why full rate laws for all the enzymes of an intact biochemical system are unlikely to become available in the near future. Current practice typically involves the use of approximate rate laws obtained for other purposes and under experimental conditions in vitro that tend to differ somewhat for each enzyme. Since the appropriate physiological conditions are seldom known, adjustments are often made in the rate law for each enzyme to reflect a common set of conditions that is thought to exist in vivo. This involves a number of assumptions and ad hoc adjustments, but this is often the only course of action if one is to utilize existing information on the rate laws for the enzymes of the system. It has to be recognized that the practice of using rate laws obtained under conditions in vitro is potentially a serious source of error that must be properly evaluated. Hence, let us turn to a critical re-examination of the Michaelis-Menten Formalism with an emphasis on its appropriateness for kinetics in situ. 

The Michaelis-Menten Formalism did not anticipate the type of enzyme-enzyme organization described above. One of its fundamental assumptions has been that complexes do not occur between different forms of an enzyme or between different enzymes (Segal, 1959 Webb, 1963 Cleland, 1970 Segel, 1975 Wong, 1975). From the derivation of the classical Michaelis-Menten rate law, it can be seen that such complexes must be excluded or they will destroy the linear structure of the kinetic equations. 

Although in recent years specific cases of enzyme-enzyme interaction have been approximated in various ways and treated by various modifications of the Michaelis-Menten Formalism, no general method for dealing with this class of mechanisms has developed from this approach. In any case, the rate laws that result from such interactions can be quite different in mathematical form from the rational functions characteristic of the Michaelis-Menten Formalism. 

From the results presented in this section, we conclude that the postulates of the Michaelis-Menten Formalism and the canons of good enzymological practice in vitro are not appropriate for characterizing the behavior of integrated biochemical systems. The very conditions that may have made it possible to identify important qualitative features of an enzymatic mechanism and produce a rate law in vitro tend to make the quantitative characterization of the reaction rate in vivo by this rate law invalid. 

This is not the case for the other two formalisms commonly used in biochemistry—the Linear Formalism and the Michaelis-Menten Formalism. The Linear Formalism implies linear relationships among the constituents of a system in quasi-steady state, which is inconsistent with the wealth of experimental evidence showing that these relationships are nonlinear in most cases. The case of the Michaelis-Menten Formalism is more problematic. An arbitrary system of reactions described by rational functions of the type associated with the Michaelis-Menten Formalism has no known solution in terms of elementary mathematical functions, so it is difficult to determine whether or not this formalism is consistent with the experimentally observed data. It is possible to deduce the systemic behavior of simple specific systems involving a few rational functions and find examples in which the elements do not exhibit allometric relationships. So, in... 

The Mass-Action representation is clearly a special case of the GMA representation in which all exponents are positive integers. The Michaelis-Menten representation is, in turn, a special case of the traditional Mass-Action representation in which two important restrictions have been imposed (Savageau, 1992). First, it is assumed that the mechanism is in quasi-steady state. The derivatives of the dependent state variables in the Mass-Action Formalism can then be set to zero, thereby reducing the description from differential equations to algebraic equations. Second, it is assumed that complexes do not occur between different forms of an enzyme or between different enzymes. The algebraic equations will then be linear in the concentrations of the various enzyme forms, and one can derive the rational function that is the representation of the rate law within the Michaelis-Menten Formalism. 

The use of kinetics to characterize the behavior of integrated biochemical systems is a more recent and less developed practice. One of the more important issues in this integrative context is the selection of an appropriate formal representation. The most common approach is simply to adopt the Michaelis-Menten Formalism that has served so well for the elucidation of isolated reaction mechanisms. However, as the discussion above showed, there are difficulties in estimating the parameters of this formalism in general, and there is a combinatorial explosion in the amount of data required to characterize the rate law by kinetic means. Thus, even if the Michaelis-Menten Formalism were appropriate in principle, there... 

Fig. 4.14 Plot of the results of a calculation of the steady-state concentration of frnctose 6-phosphate for the system shown in fig. 4.13. The enzyme models are either based on Michaelis-Menten formalisms or modifications of multiple allosteric effector equations. The gate exhibits a function with both AND and OR properties. At low concentrations of both inpnts, the mechanism functions similarly to an OR gate, while at simultaneously high concentrations of the inpnt species (citrate and cAMP), the output behavior more closely resembles a fuzzy logic AND gate. The mechanism satisfies the requirements for a fuzzy aggregation function. (From [7].)...

In all instances, the activity of BCA towards its physiological substrates has been found to fully conform to the Michaelis-Menten formalism. Under such analysis, BCA exhibits an extremely high turnover number, the magnitude of which is sufficient to render diffusion limitations on protons and substrate a significant constraint on any proposed mechanism. 

Plots of V versus V/[s] were sensibly linear for both substrates as shown in Figure 1, indicating that for all practical purposes this system can be treated by the Michaelis-Menten formalism. For both substrates, increase with an increase in buffer... 

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