Random Bisubstrate Reactions

If two reactants. A and B, bind to the catalytically active site randomly and if the binding of one changes the dissociation constant of the other by a factor, a, the system can be described by Eqn. 7.44.22 jhjs reaction sequence is similar to the non-competitive inhibition described by Eqn. 7.32 but has the MAB complex as the active species rather than the non-productive entity present during inhibition. 

Since the rate of the reaction depends on the concentrations of both A and B the rate equation can be written in two forms to show the rate dependency on each. Eqn. 7.45 is the reciprocal rate expression when [B] is held constant and [A] varies. The rate expression for those reactions run with [A] constant and [B] changing is symmetrical to Eqn. 7.45. This is also true for all of the ofter equations and plots described below. 

In Fig. 7.11 are shown the plots of 1/v versus 1/[BJ at constant [A] for several values of [A]. These plots intersect at a point above the x-axis and with slopes having Eqn. 7.46. 

The plot in Fig. 7.11 for the reaction run under zero order conditions in A, [A] is saturating, has a y-axis intercept at and an x-axis intercept at 

This is the case when a is less than one, that is, when the binding of one reactant increases the binding affinity of the other. When a equals one the binding of one reactant has no effect on the other and the lines intersect on the x-axis at -l/Kg. If the binding of one reactant decreases the binding of the other, a is greater than one and the lines of these plots will intersect below the x-axis. The intersection still corresponds on the x-axis to -l/Kg. 

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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. 

Fig. 7.11. Double reciprocal plots for reactions run at different fixed concentrations of A for a random bisubstrate reaction with a< 1.

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. 

As with the random bisubstrate reactions, the plots and replots for data obtained when [B] is varied and [A] is constant are symmetrical to those obtained when [A] is varied and [B] is held constant. All of the pertinent information can... 

As a mle, a noncompetitive inhibition occurs only if there are more than one substrate or product (Todhunter, 1979 Fromm, 1995). For example, a noncompetitive inhibition will take place in a random bisubstrate reaction, when an inhibitor competes with one substrate while the other substrate is varied. Thus, the equilibria shown below describe a Rapid Equilibrium Random bisubstrate system in which an inhibitor competes with A but allows B to bind. 

Figure i. Substrate inhibition in a Rapid Equilibrium Random bisubstrate reaction. Rate equation (u.6) was drawn assuming that Vi = rA=. iA =. e — i = i< Dotted line represents the uninhibited reaction ( =< ). 

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. 

There are three general patterns observed with bisubstrate reactions random, sequential and ping-pong. 

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. 6.9. Random sequential mechanism for bisubstrate reactions.

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)... 

Why product inhibition occurs. The products of reaction are formed at the active site of enzyme and are the substrates for the reverse reaction. Consequently, a product may act as an inhibitor by occupying the same site as the substrate from which it is derived. In the Rapid Equilibrium Random bisubstrate mechanism, most ligand dissociations are very rapid compared to the interconversion of EAB and EPQ. Thus, the levels of EP and EQ are essmtiaUy zero in the absence of added P and Q. In the presence of only one of the products, the reverse reaction can be neglected, as the concentration of the other product is essentially zero during the early part of the reaction. Nevertheless, the forward reaction will be inhibited because finite P (or Q) ties up some of the enzyme. The type of this product inhibition depends on the number and type of enzyme-product complexes that can form. Consequently, product inhibition studies can be very valuable in the diagnostics of kinetic mechanisms (Rudolph, 1979). 

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... 

If the breakdown of the central complex in bisubstrate reactions is not the sole rate-limiting step, than the rate equation becomes quite complex. For example, consider the Steady-State Random Bi Uni system shown below ... 

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. 

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 ... 

The first synthetic bisubstrate analog inhibitor 18 was successfully designed for al,2-FucT II by Palcic et al., based on the proposed ion-pair mechanism shown in Scheme 8 [22]. Analog 18, where the Gal unit is attached to the terminal phosfor of GDP through a flexible ethylene linkage, was found to be a competitive inhibitor with respect to both donor and acceptor substrates with K values of 16 and 2.3 pM, respectively. Inhibition studies with the bisubstrate analog also helped establish the kinetic mechanism of the enzyme reaction. These dual competitive inhibition patterns are only consistent with a random kinetic mechanism where either substrate can bind to free enzyme. 

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