Kinetics bisubstrate reactions

In this chapter we described the thermodynamics of enzyme-inhibitor interactions and defined three potential modes of reversible binding of inhibitors to enzyme molecules. Competitive inhibitors bind to the free enzyme form in direct competition with substrate molecules. Noncompetitive inhibitors bind to both the free enzyme and to the ES complex or subsequent enzyme forms that are populated during catalysis. Uncompetitive inhibitors bind exclusively to the ES complex or to subsequent enzyme forms. We saw that one can distinguish among these inhibition modes by their effects on the apparent values of the steady state kinetic parameters Umax, Km, and VmdX/KM. We further saw that for bisubstrate reactions, the inhibition modality depends on the reaction mechanism used by the enzyme. Finally, we described how one may use the dissociation constant for inhibition (Kh o.K or both) to best evaluate the relative affinity of different inhibitors for ones target enzyme, and thus drive compound optimization through medicinal chemistry efforts. 

The determination of bisubstrate reaction mechanism is based on a combination of steady state and, possibly, pre-steady state kinetic studies. This can include determination of apparent substrate cooperativity, as described in Chapter 2, study of product and dead-end inhibiton patterns (Chapter 2), and attempts to identify... 

We have introduced kinetics as the primary method for studying the steps in an enzymatic reaction, and we have also outlined the limitations of the most common kinetic parameters in providing such information. The two most important experimental parameters obtained from steady-state kinetics are kcat and kcat/Km. Variation in kcat and kcat/Km with changes in pH or temperature can provide additional information about steps in a reaction pathway. In the case of bisubstrate reactions, steady-state kinetics can help determine whether a ternary complex is formed during the reaction (Fig. 6-14). A more complete picture generally requires more sophisticated kinetic methods that go beyond the scope of an introductory text. Here, we briefly introduce one of the most important kinetic approaches for studying reaction mechanisms, pre-steady state kinetics. 

FIGURE 6-14 Steady-state kinetic analysis of bisubstrate reactions. 

In practice, uncompetitive and mixed inhibition are observed only for enzymes with two or more substrates—say, Sj and S2—and are very important in the experimental analysis of such enzymes. If an inhibitor binds to the site normally occupied by it may act as a competitive inhibitor in experiments in which [SJ is varied. If an inhibitor binds to the site normally occupied by S2, it may act as a mixed or uncompetitive inhibitor of Si. The actual inhibition patterns observed depend on whether the and S2-binding events are ordered or random, and thus the order in which substrates bind and products leave the active site can be determined. Use of one of the reaction products as an inhibitor is often particularly informative. If only one of two reaction products is present, no reverse reaction can take place. However, a product generally binds to some part of the active site, thus serving as an inhibitor. Enzymologists can use elaborate kinetic studies involving different combinations and amounts of products and inhibitors to develop a detailed picture of the mechanism of a bisubstrate reaction. 

The general rule for writing the rate equation according to the quasi-equilibrium treatment of enzyme kinetics can be exemplified for the random bisubstrate reaction with substrates A and B forming products P and Q (Figure 7.1), where KaKab = KbKba and KpKpq = KqKqp. 

While almost all synthetically useful catalytic reactions involve two reactants it will be best to first illustrate this graphical approach using a unimolecular process and the way it can be affected by inhibitors. After this some different types of bisubstrate reactions will be discussed. More detailed analyses of the kinetics of multisubstrate systems have been published in texts 13 and... 

Typically, the kinetics of bisubstrate reactions are studied by measuring the initial reaction rates over a range of concentrations of one substrate. A, while holding the concentration of the second substrate, B, constant and doing this for several fixed values of [B]. If specific concentrations of A are used for all of the reaction series, these same rates can also be used to examine changes in reaction rate when [B] is varied at fixed concentrations of A for several values of [A]. 

When one of the substrates is water (i.e., when the process is one of hydrolysis), with the reaction taking place in aqueous solution, only a fraction of the total number of water molecules present participates in the reaction. The small change in the concentration of water has no effect on the rate of reaction and these pseudo-one substrate reactions are described by one-substi ate kinetics. More generally the concentrations of both substrates may be variable, and both may affect the rate of reaction. Among the bisubstrate reactions important in clinical enzymology are the reactions catalyzed by dehydrogenases, in which the second substrate is a specific coenzyme, such as the oxidized or reduced forms of nicotinamide adenine dinucleotide, (NADH), or nicotinamide adenine dinucleotide phosphate, (NADPH), and the amino-group transfers catalyzed by the aminotransferases. 

Most enzymatic reactions involving two-substrate reactions show more complex kinetics than do one-substrate reactions. Examples are catalyzed by dehydrogenases and aminotransferases. Hydrolytic reactions are bisubstrate reactions in which water is one of the substrates. The change in water concentration is negligible and has no effect on the rate of reaction. A two-suhstrate reaction can be written as... 

Previously, while discussing the general theory of complex reactions, we have considered some other mechanisms with linear steps, such as one given by eq. (4.107) corresponding to three-step sequence or eq. (4.116). In a similar way kinetic expressions could be derived for more complicated reaction networks, as presented for instance in Chapter 5 (see equations 5.76 for 4 step sequence, eq. 5.84 for 6 steps eq. 5.88 and 5.89 for a mechanism with 8 linear steps and the general form for n-step mechanism eq. 5.94). Ordered sequential bisubstrate reactions can be expressed by eq. 5.76 for the 4 step sequence (Figure 6.11)... 

TABLE 11.5 Cleland nomenclature for bisubstrate reactions exemplified. Three common kinetic mechanisms for bisubstrate enzymatic reactions are exemplified. The forward rate equations for the order bi bi and ping pong bi hi are derived according to the steady-state assumption, whereas that of the random bi bi is based on the quasi-equilibrium assumption. These rate equations are first order in both A and B, and their double reciprocal plots (1A versus 1/A or 1/B) are linear. They are convergent for the order bi bi and random bi bi but parallel for the ping pong bi bi due to the absence of the constant term (KiaKb) in the denominator. These three kinetic mechanisms can be further differentiated by their product inhibition patterns (Cleland, 1963b)... 

In Chapters 5 and 6, we have examined different types of inhibition, both in monosubstrate and in bisubstrate reactions. All these types of inhibition can be classified into several kinetic forms. The linear types of inhibition, described in Chapter 5, are always complete inhibitions and their rate equations can be presented in a generd form ... 

Kinetic measurements with bisubstrate reactions are performed by measuring the initial reaction rates in the presence of increasing concentrations of substrate A, keeping the substrate B constant and repeating the experiment at several fixed concentrations of substrate B thus, A represents a variable substrate and B a constant substrate. In the double reciprocal plot, the experimental data present a family of straight hnes, with a common intersection point which is found on ordinate, on abscissa, in the HI or in the IV quadrant. One can use the same set... 

In the literature, literally dozens of kinetic mechanisms have been proposed for bisubstrate enzymes (Alberty, 1958 Alberty Hammes, 1958 Teller Alberty, 1959 Wong Hanes, 1962 Fromm, 1967 Dalziel, 1969 Hurst, 1969 Rudolph Fromm, 1969, 1971, 1973). However, only those pathways that are either weU documented, or seem to be a logical extension of established mechanisms, will be presented in this and the following chapters. Thus, we shall divide the rapid equilibrium bisubstrate reactions into the following major types, according to the type and number of enzyme-substrate or enzyme-product complexes that can form (Alberty, 1953 Cleland, 1970, 1977 Fromm, 1979 Engel, 1996 Purich Allison, 2000) ... 

The velocity equations for bisubstrate reactions are usually formidable if expressed in terms of individual rate constants. The resulting equations are almost useless until the rate constants are grouped into relatively simple kinetic constants that can be experimentally determined. Various methods for grouping the individual rate constants have been developed by Alberty (1953), Ddziel (1957), Bloomfield etal. (1962), Wong Hanes (1962), Cleland (1963), Mahler Coides (1966), and others. 

Table 5. The physical significance of kinetic coefficients for bisubstrate reactions ...

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