Kinetics ordered 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. 

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 some steady-state mechanisms, such as an Ordered Bi Bi mechanism, all or some of the rate constants can be calculated directly from the kinetic constants. Quantities like and /Cm wiU not necessarily show the normal temperature behavior, since they are usually combinations of several rate constants, but if certain rate constants predominate, normal temperature behavior maybe obeyed. This is often the case with bimolecular rate constants /Ccat/ A and fccat/AB in bisubstrate reactions. 

The kinetic expression for observed isotope effects is the ratio of both entire rate equations describing the disappearance of hydrogen and deuterium substrates. The isotopically sensitive step appears in multiple terms and cannot be factored out. In order to achieve factoring and subsequent simplification to useful kinetic equations, it is necessary to examine the Umits of rate equations at low and high substrate concentrations, where enzyme reactions approach first-order and zero-oider kinetics, respectively. To understand this, we must consider how isotope effects in bisubstrate reactions are measured. 

Tlie problems of the statistical analysis of kinetic data are most easily understood in relation to a specific example. Therefore, let us consider a bisubstrate reaction that proceeds in an ordered fashion the leading substrate A is added first, followed by substrate B, and the initial velocity equation is... 

The order of addition of substrates in the Bi Bi mechanisms, with a central ternary complex, can be strictly ordered, completely random or partially random. We can employ reaction (10.7) in order to analyze most kinetic mechanisms that occur in bisubstrate systems ... 

As already pointed out in Chapter 9, the steady-state expressions for the catalytic constant, Vu and for the specificity constant, Vi/Ksi for bisubstrate mechanisms are rather complex. For the ordered mechanism in reaction (17.18), as we have already pointed out in Section 9.2.2, even if we leave out the isomerizations, that is, the complexes EAB and EPQ, expressions for VJK and V, are rather complex. If we include the isomerization complexes, IiAB and EPQ, the rate equations for the catalytic constant, V, and for the specificity constant, VJKji, appear quite formidable compared to equations for the monosubstrate reaction (Eqs. (17.13) and (17.14)). Further, if we remember that the kinetic expression for isotope efects is the ratio of both entire rate equations describing the disappearance of hydrogen and deuterium substrates (or other isotopes), than the rate equations for isotope effects may appear awesome. 

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