Enzymes (Second Edition)

Table 11.1 General nomenclature for enzymatic reactions

bi, ter, and so on to refer to one, two, three, and more chemical entities. For example, a reaction that utilizes two substrates to produce two products is referred to as a bi bi reaction, a reaction using three substrates to form two products is as a ter bi reaction, and so on (Table 11.1).

Let us consider in some detail a group transfer reaction that proceeds as a bi bi reaction:

E AX B E A BX

The reaction scheme as written leaves several important questions unanswered. Does one substrate bind and leave before the second substrate can bind? Is the order in which the substrates bind random, or must binding occur in a specific sequence? Does group X transfer directly from A to B when both are bound at the active site of the enzyme, or does the reaction proceed by transfer of the group from the donor molecule, A, to a site on the enzyme, whereupon there is a second transfer of the group from the enzyme site to the acceptor molecule

B(i.e., a reaction that proceeds through formation of an E—X intermediate)? These questions raise the potential for at least three distinct mechanisms for

the generalized scheme; these are referred to as random ordered, compulsory ordered, and double-displacement or ‘‘Ping-Pong’’ bi bi mechanisms. Often a major goal of steady state kinetic measurements is to differentiate between these varied mechanisms. We shall therefore present a description of each and describe graphical methods for distinguishing among them.

In the treatments that follow we shall use the general steady state rate equations of Alberty (1953), which cast multisubstrate reactions in terms of the equilibrium constants that are familiar from our discussions of the Henri— Michaelis—Menten equation. This approach works well for enzymes that utilize one or two substrates and produce one or two products. For more complex reaction schemes, it is often more informative to view the enzymatic reactions instead in terms of the rate constants for individual steps (Dalziel, 1975). At the end of this chapter we shall briefly introduce the method of King and Altman (1956) by which relevant rate constants for complex reaction schemes can be determined diagrammatically.

352 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

11.2 Bi Bi REACTION MECHANISMS

11.2.1 Random Ordered Bi Bi Reactions

In the random ordered bi bi mechanism, either substrate can bind first to the enzyme, and either product can leave first. Regardless of which substrate binds first, the reaction goes through an intermediate ternary complex (E · AX · B), as illustrated:

Here the binding of AX to the free enzyme (E) is described by the dissociation constant K , and the binding of B to E is likewise described by K . Note that the binding of one substrate may very well affect the affinity of the enzyme for the second substrate. Hence, we may find that the binding of AX to the preformed E · B complex is described by the constant K . Likewise, since the overall equilibrium between E · AX · B and E must be path independent, the binding of B to the preformed E · AX complex is described by K . When B is saturating, the value of K is equal to the Michaelis constant for AX (i.e., K ). Likewise, when AX is saturating, K K . The velocity of such an enzymatic reaction is given by Equation 11.1:

If we express the concentrations of the various species in terms of the free enzyme concentration [E], we obtain:

If we fix the concentration of one of the substrates, we can rearrange and simplify Equation 11.2 significantly. For example, when [B] is fixed and [AX] varies, we obtain:

Figure 11.1 Double-reciprocal plot for a random ordered bi bi enzymatic reaction.

At high, fixed concentrations of B, the terms K /[B] and K /[B] go to zero. Thus, at saturating concentrations of B we find:

If we measure the reaction velocity over a range of AX concentrations at several, fixed concentrations of B, the reciprocal plots will display a nest of lines that converge to the left of the y axis, as illustrated in Figure 11.1. The data from Figure 11.1 can be replotted as the slopes of the lines as a function of 1/[B], and the y intercepts (i.e., 1/V ) as a function of 1/[B] (Figure 11.2). The y intercept of the plot of slope versus 1/[B] yields an estimate of

and 1/ K , respectively. Thus from the data contained in the two replots, one can calculate the values of K , K , and V simultaneously.

354 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

Figure 11.2 (A) Slope and (B) y-intercept replots of the data from Figure 11.1, illustrating the graphical determination of K , K , and V for a random ordered bi bi enzymatic reaction.

11.2.2 Compulsory Ordered Bi Bi Reactions

In compulsory ordered bi bi reactions, one substrate, say AX, must bind to the enzyme before the other substrate (B) can bind. As with random ordered reactions, the mechanism proceeds through formation of a ternary intermedi-

356 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

Figure 11.3 Double-reciprocal plot for a double-displacement (Ping-Pong) bi bi enzymatic reaction.

Using steady state assumptions, the velocity equation for a double-displace- ment reaction can be obtained:

If we fix the value of [B], then Equation 11.8 for variable [AX] becomes:

Reciprocal plots of a reaction that conforms to the double-displacement mechanism for varying concentrations of AX at several fixed concentrations of B will yield a nest of parallel lines, as seen in Figure 11.3. For each concentration of substrate B, the values of 1/V and 1/K can be determined from the y and x intercepts, respectively, of the double-reciprocal

Figure 11.4 Replots of the data from Figure 11.3 as (A) 1/Vmaxapp versus 1/[B] and (B) 1/K AX,appm versus 1/[B], illustrating the graphical determination of K AXm , K Bm, and V max for a doubledisplacement (Ping-Pong) bi bi enzymatic reaction.

11.3 DISTINGUISHING BETWEEN RANDOM AND COMPULSORY ORDERED MECHANISMS BY INHIBITION PATTERN

It should be clear from Figures 11.1 and 11.3, and the foregoing discussion, that the qualitative form of the double-reciprocal plots makes it easy to distinguish between a double-displacement mechanism and a mechanism

C, competitive; N, noncompetitive; U, uncompetitive; —, no inhibition.

0 for the scheme in Figure 8.1) or product inhibitor, and the mechanism of substrate interaction with the enzyme. For a bi bi reaction, one observes specific inhibitor patterns for the different mechanisms we have discussed when a competitive dead-end inhibitor or a product of the reaction is used as the inhibitor. The patterns for both dead-end and product inhibition additionally depend on whether the fixed substrate is at a saturating or nonsaturating (typically at [S] K ) concentration with respect to its apparent K .

The relationships leading to these differing patterns of dead-end and product inhibition for bi bi reactions have been derived elsewhere (see, e.g., Segel, 1975). Rather than rederiving these relationships, we present them as diagnostic tools for determining the mechanism of reaction. The patterns are summarized in Tables 11.2 and 11.3 for dead-end and product inhibition, respectively. By measuring the initial velocity of the reaction in the presence of several concentrations of inhibitor, and varying separately the concentrations of AX and B, one can identify the reaction mechanism from the pattern of doublereciprocal plots and reference to these tables.