Ordered Bisubstrate Reactions

The equilibria shown in Eqn. 7.53 describes the system in which the reactants A and B combine with the catalyst in the specific order, A adsorbing before B.23 Such cases can arise when the adsorption of A causes the electronic changes in the active site which are necessary for B to adsorb. Without this modification of the electronic nature of the site B could not be adsorbed. 

The reciprocal rate expression for reactions run with varying itrations of B at a constant concentration of A is given by Eqn. 7.54. 

The replot of slope versus 1/[A] has a slope of a y-axis intercept at 

In contrast to what is observed with a random bisubstrate reaction, with an ordered bisubstrate process the rate data obtained on varying the concentration of A at constant concentrations of B do not give reciprocal plots which are symmetrical to Fig. 7.12. Comparing Eqn. 7.54 with Eqn. 7.58, the double reciprocal equation for reactions in which [A] is varied at constant [B], shows the differences between the two systems. 

These differences are further illustrated by comparing Fig. 7.12 with Fig. 7.13, the plot of 1/v versus 1/lA). In Fig. 7.13 the lines intersect at a point above the x-axis but not on the y-axis as in Fig. 7.12. The intercept correlates on the y-axis to 1/V,j ax and on the x-axis to -1/Ka. When the reaction is run under zero order conditions in B, the slope of the plot is zero and the y-intercept is 1/Vn,ax- 

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Fig. 7.12. Plots of 1/v versus 1/[B] for an ordered bisubstrate reaction where A adsorbs before B.

As a mle, an uncompetitive inhibition occurs only if there are more than one substrate or product (Huang, 1990). For example, an uncompetitive inhibition will take place in a Rapid Equilibrium Order bisubstrate reaction, when an inhibitor competes with B while A is the variable substrate. Thus, the equilibria shown below describe an ordered bisubstrate system in which an inhibitor competes with B but does not bind to free enzyme. 

Figuer 3. Ordered bisubstrate reaction. Data points are drawn according to numerical values in Table 1. 

Figure 4. Ordered bisubstrate reaction. The initial rate data from Table 1 are presented in the linear form, according to Line weaver-Burk (top) or Hanes (bottom).

FIGURE 6-13 Common mechanisms for enzyme-catalyzed bisubstrate reactions, (a) The enzyme and both substrates come together to form a ternary complex. In ordered binding, substrate 1 must bind before substrate 2 can bind productively. In random binding, the substrates can bind in either order. 

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. 

Sequential Reactions. In sequential reactions, all substrates must bind to the enzyme before any product is released. Consequently, in a bisubstrate reaction, a ternary complex of the enzyme and both substrates forms. Sequential mechanisms are of two types ordered, in which the substrates bind the enzyme in a defined sequence, and random. 

Figure. 6.8. Sequential ordered mechanism for bisubstrate reactions.

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)... 

This nomenclature has been introduced by Cleland (1963), but other descriptions of bisubstrate mechanisms are also found in the biochemical literature. For example, a sequential addition in bisubstrate reactions, an Ordered Bi Bi mechanism is also called a compulsory-order ternary-complex mechanism whereas a Random Bi Bi mechanism is called a random-order ternary-complex... 

This example clearly shows that completely randomized steady-state bisubstrate reactions wiU produce extremely complex rate equations which are, in most cases, unmanageable and almost useless for practical purposes. Thus, for example, the rate equation for an Ordered Bi Bi mechanism has 12 terms in the denominator (compare Eq. (9.8)). A completely Random Bi Bi mechanism yields an even more comphcated rate equation with 37 new terms in the denominator. Eor this reason, and in such cases, we shah usuahy revert to simplifying assumptions, usually introducing the rapid equilibrium segments in the mechanism in order to reduce the rate equations to manageable forms. 

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. 

Distribution equations for bisubstrate reactions in the steady state are often very complex expressions (Chapter 9). However, in the chemical equilibrium, the distribution equations for all enzyme forms are usually less complex. Consider an Ordered Bi Bi mechanism in reaction (16.12) with a single central complex ... 

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. 

An experimental protocol in Fig. 8 is shown for an ordered bisubstrate mechanism. A small volume of enzyme is incubated with sufficient labeled substrate. A, to convert most or all of the enzyme into a binary complex, EA. This solution is then diluted into a large volume containing the unlabeled substrate. A, plus variable amounts of cosubstrate, B. After several seconds, add is added to stop the enzymatic reaction, and labeled product, Q, is determined analytically. A blank is then run with the labeled reactant already diluted in the large solution, plus only the enzyme present in the small volume. The experiment is then repeated at different levels of the second substrate B, and a reciprocal plot is made of the amount of labeled product, Q, as a function of the reciprocal of the substrate B concentration thus, by extrapolation, the maximum amount of labeled product formed, Q a. is obtained. 

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... 

Bisubstrate reactions (Chapters 8 and 9). In bisubstrate reactions, a frequent case is a need to distinguish between the Steady-State Ordered, Ping Pong, and Equihbrium Ordered mechanism the rate equations involved are... 

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