Rapid Equilibrium Random System

Rapid Equilibrium Random system, in the absence of products, occurs if the substrates A and B bind randomly. If the binding of one substrate changes the dissociation constant for the other substrate by the same factor ( = Xa/Xa = Xb/Xb), the system can be described by the equilibria shown below  

In the random case, in order to obtain the rapid equihbrium conditions, only the off rate constants of A and B from their binary complexes have to be much faster then fceat- If Vi = kcaiEo, the velocity equation for reaction scheme (8.6) is 

The same equation is obtained if one assumes that A and B dissociate from the EAB complex with rates similar or smaller than fccat- In such cases, the MichaeMs constants, Xa and Xb, are no longer dissociation constants, but are more complex expressions XiA and Xb are stiU true dissociation constants of respective binary complexes. 

any substrate that can add last can bind either in rapid equilibrium or in steady-state fashion without changing the form of the rate equation. Comparison of Eqs. (8.2) and (8.7), however, shows that a substrate that cannot add last will change Ae rate equation if it adds in rapid equilibrium. 

Equations (8.9) and (8.10) are completely symmetrical and, therefore, their graphical presentations are also symmetrical. For this reason, the graphical presentation of Eq. (8.10) is omitted, because it is not necessary to perform two separate experiments to constmct the reciprocal plots for two varied substrates. One primary plot, for example, i/vo versus 1/A, contains all the information necessary to calculate all kinetic constants. 

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Fi re 4-42 (a) Plot of 1/u versus 1/[A] at different fixed concentrations of B for a rapid equilibrium random system where or, the... 

The Haldane relationship is identical for all rapid equilibrium random systems (Haldane, 1930 Cleland, 1982). Thus, from Eq. (8.37), one also obtains... 

In rapid equilibrium random systems, one substrate may have an appreciable affinity for the other substrate s binding site, particularly when the two substrates are chemically similar. In such cases, a substrate inhibition by one of the substrates may take place. Let us examine the case when the binding of substrate B to the A site does not prevent B from binding to its own site, so that two deadend complexes form, BE and BEB. 

Let us examine several ordered and random-ordered rapid equilibrium Ter Ter systems, in addition to the aforementioned total rapid equilibrium random system ... 

The procedure to be described here was originally developed by Cha. The basic principle of his approach is to treat the rapid-equilibrium segment as though it were a single enzyme species at steady state with the other species. Let us consider the hybrid Rapid-Equilibrium Random-Ordered Bi Bi system ... 

A potential limitation encountered when one seeks to characterize the kinetic binding order of certain rapid equilibrium enzyme-catalyzed reactions containing specific abortive complexes. Frieden pointed out that initial rate kinetics alone were limited in the ability to distinguish a rapid equilibrium random Bi Bi mechanism from a rapid equilibrium ordered Bi Bi mechanism if the ordered mechanism could also form the EB and EP abortive complexes. Isotope exchange at equilibrium experiments would also be ineffective. However, such a dilemma would be a problem only for those rapid equilibrium enzymes having fccat values less than 30-50 sec h For those rapid equilibrium systems in which kcat is small, Frieden s dilemma necessitates the use of procedures other than standard initial rate kinetics. 

Fromm and Rudolph have discussed the practical limitations on interpreting product inhibition experiments. The table below illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict nonhnearity in product inhibition studies. This nonlin-earity might not be obvious under standard initial rate protocols, but products that would be competitive in rapid equilibrium systems might appear to be noncompetitive in steady-state random schemes , depending on the relative magnitude of those squared terms. See Abortive Complex... 

A binds to free E with a dissociation constant Ka (also called Ku, in the Cleland nomenclature). B binds to free E with a dissociation constant -Kb (or Kn). The binding of one substrate may alter the affinity of the enzyme for the other. Thus, A binds to EB with a dissociation constant ctKa. Since the overall equilibrium constant between A and E must be the same regardless of the path taken, B binds to EA with a dissociation constant aKs. o Ka is the same as Km (the K for A at saturating [B]). ocKb is the same as (the for B at saturating [A]). If the rate-limidng step is the slow conversion of EAB to EPQ, we can derive the velocity equation for the forward reaction in the absence of P and Q in the usual manner. In fact, the only difference between the rapid equilibrium random bireactant system and noncompetitive or linear mixed-type inhibition is that now the ternary complex (EAB) is catalyticaUy active, while ESI was not. 

Hexokinase does not yield parallel reciprocal plots, so the Ping Pong mechanism can be discarded. However, initial velocity studies alone will noi discriminate between the rapid equilibrium random and steady-state ordered mechanisms. Both yield ihe same velocity equation and families of intersecting reciprocal plots. Other diagnostic procedures must be used (e.g., product inhibition, dead-end inhibition, equilibrium substrate binding, and isotope exchange studies). These procedures are described in detail in the author s Enzyme Kinetics behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley-Interscience (1975),... 

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 2. Noncompetitive inhibition. Rapid Equilibrium Random bisubstrate system with an inhibitor noncompetitive with B. Graphical presentation of Eq. (S-io), with A as a constant and B as a variable substrate.

Consider a hypothetical Rapid Equilibrium Random Bi Bi system. Both products are present in the reaction mixture, A and B or P and Q can occupy the enzyme active site simultaneously, and all four Mgands can bind by themselves however, the dead-end complexes EAP and EBQ are not formed. 

RAPID EQUILIBRIUM RANDOM BIBI SYSTEM WITH A DEAD-END EBQ COMPLEX... 

Product inhibition in the Rapid Equilibrium Random Bi Bi system with a dead-end EBQ complex takes place if, in addition to the presence of substrates A and B, also the product P, or tematively the product Q, is also present. 

In the general case, the Rapid Equilibrium Random Bi Bi system with both dead-end complexes ElAP and EBQ (Reaction (8.35)), the denominator of the rate equation has nine terms, each for one form of enzyme the unity represents the free enzyme (Eq. (8.37)). From the general rate Eq. (8.37), one can write down directly the rate equations for all other possible combinations of the Rapid Equilibrium Random Bi Bi system. 

For example, the rate equation for the Rapid Equilibrium Random Bi Bi system with a dead-end EBQ con ilex (Reaction (8.23)) is obtained by omitting from Eq. (8.38) the term that contains AP, because the EAR complex does not form, thus affording directly the corresponding rate Eq. (8.25). Similarly, from the general rate Eq. (8.38), one can also delete denominator terms corresponding to B, P, BQ and AP, and obtain directly the general rale equation for the Rapid EquiKbrium Ordered Bi Bi system (Eq. (8.12)). 

Note that Eqs. (9.15) and (9.16) are identical with the corresponding rate equations for the Rapid Equilibrium Random bisubstrate system (Chapter 8 Eqs. (8.7) and (8.8)). 

Substrate Inhibition in a Rapid Equilibrium Random Bisubstrate System... 

If A, B, and C are the three substrates that yield three products, P, Q, and R, and aU forms of enzyme are in a rapid equilibrium, the system can be designated as a Rapid Equilibrium Random Ter Ter, and the equilibria shown as... 

Ordered and Random-Ordered Rapid Equilibrium Trisubstrate Systems... 

An enzyme-catalyzed reaction involving two substrates and one product. There are two basic Bi Uni mechanisms (not considering reactions containing abortive complexes or those catagorized as Iso mechanisms). These mechanisms are the ordered Bi Uni scheme, in which the two substrates bind in a specific order, and the random Bi Uni mechanism, in which either substrate can bind first. Each of these mechanisms can be either rapid equilibrium or steady-state systems. 

The primary graphs alone, the double reciprocal plots in the absence of products, can be quite valuable in the diagnostics of rapid equilibrium mechanisms. Thus, one can easily distinguish the ordered from the random rapid equilibrium system, even from the primary graphs in the absence of the products of reaction (Table 1). 

Isotope exchange is the only way to determine whether a random system is a rapid equilibrium one. If the conversion of the central complexes is the sole rate-limiting step, then all exchanges in a given direction must proceed at the same rate at any given set of reactant concentrations. Therefore, i>a-q = a-p = >b-p. and i>Q A = Vp-A = Up-B The equality of exchange rates holds even if dead-end complexes form. 

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