Rapid Equilibrium Random complex

Figure 2.12 Reaction pathway for a bi-bi rapid equilibrium, random sequential ternary complex reaction mechanism.

THE COMBINED EQUILIBRIUM AND STEADY-STATE TREATMENT. There are a number of reasons why a rate equation should be derived by the combined equilibrium and steady-state approach. First, the experimentally observed kinetic patterns necessitate such a treatment. For example, several enzymic reactions have been proposed to proceed by the rapid-equilibrium random mechanism in one direction, but by the ordered pathway in the other. Second, steady-state treatment of complex mechanisms often results in equations that contain many higher-order terms. It is at times necessary to simplify the equation to bring it down to a manageable size and to reveal the basic kinetic properties of the mechanism. 

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... 

Rapid Equilibrium Case. In the absence of significant amounts of product (i.e., initial rate conditions thus, [P] 0), the rate expression for the rapid equilibrium random Bi Uni mechanism is v = Uniax[A][B]/(i iai b + i b[A] + i a[B] + [A][B]) where is the dissociation constant for the EA complex, and T b are the dissociation constants for the EAB complex with regard to ligands A and B, respectively, and Umax = 9[Etotai] where kg is the forward unimolecular rate constant for the conversion of EAB to EP. Double-reciprocal plots (1/v v. 1/[A] at different constant concentrations of B and 1/v v. 1/[B] at different constant concentrations of A) will be intersecting lines. Slope and intercept replots will provide values for the kinetic parameters. 

Cleland (160), steady-state kinetics of a Theorell-Chance mechanism can generally apply also to a rapid-equilibrium random mechanism with two dead-end complexes. However, in view of the data obtained with site-specific inhibitors this latter mechanism is unlikely in the case of the transhydrogenase (70, 71). The proposed mechanism is also consistent with the observation of Fisher and Kaplan (118) that the breakage of the C-H bonds of the reduced nicotinamide nucleotides is not a rate-limiting step in the mitochondrial transhydrogenase reaction. 

Random mechanisms wiU not show substrate inhibition of exchanges unless the levels of reactants that can form an abortive complex are varied together. The relative rates of the two exchanges will show whether catalysis is totally rate limiting (a rapid equilibrium random mechanism), or whether release of a reactant is slower. For kinases that phosphorylate sugars, the usual pattern is for sugar release to be partly rate limiting, but for nucleotides to dissociate rapidly (15, 16). 

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. 

In the case of enzymes working via a ternary complex mechanism, we have two extreme cases. The easiest to comprehend is the rapid equilibrium random mechanism (Scheme 5.4) this is the mechanism where the chemistry is most likely to be rate determining and kinetic isotope effects or structure-reactivity correlations are likely to be mechanistically informative. Enzymes acting on their physiological substrates at optimal pH are likely to show a degree of preference for one or the other substrate binding first, but they can often be induced to revert to a rapid equilibrium random mechanism by the use of non-optimal substrates or pH. 

The initial rate equation derived by steady-state analysis is of the second degree in A and B (SO). It simplifies to the form of Eq. (1) if the rates of dissociation of substrates and products from the complexes are assumed to be fast compared with the rates of interconversion of the ternary complexes k, k )] thus, the steady-state concentrations of the complexes approximate to their equilibrium concentrations, as was first shown by Haldane (14)- The kinetic coefficients for this rapid equilibrium random mechanism (Table I), together with the thermodynamic relations KeaKeab — KebKeba and KepKepq — KeqKeqp, suffice for the calculation of k, k and all the dissociation constants Kea = k-i/ki, Keab = k-i/ki, etc. 

Q. This finding eliminates a truly rapid equilibrium random mechanism, for which k and k must be much smaller than fc 4, k-i, k, and k-2, since the two exchange rates must then be equal. In fact, the differences between the two exchange rates show that the dissociation of A and/or P from the ternary complexes must be slow compared with that of B and/or Q, and also slow relative to the interconversions of the ternary complexes (32). This means that in at least one direction of reaction the dissociation of products in the overall reaction is essentially ordered for all these enzymes, the coenzymes dissociating last, as in the preferred pathway mechanism (Section I,B,4). With malate, lactate, and liver alcohol dehydrogenases, the NAD/NADH exchange rate increased to a... 

This brings us to the final mechanism we need to consider for a 2-substrate reaction, namely a random-order mechanism. We have assumed that we would be alerted to the possibility of a steady-state random-order mechanism by non-linear primary or secondary plots, but it is possible to get linear kinetics with a random-order mechanism. If we make the assumption that the further reaction of the ternary complex EAB is much slower than the network of reactions connecting E to EAB via EA and EB, then there are only 4 kinetically significant complexes and their concentrations are related to one another by substrate concentrations and dissociation constants. This is the rapid-equilibrium random-order mechanism, and the assumption made is analogous to the Michaelis-Menten equilibrium assumption for a 1-substrate mechanism. 

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). 

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)). 

It is interesting to note that, in all Bi Bi mechanisms, the kinetic constant composition next to a given substrate concentration term is always identical in aU mechanisms. Rapid Equilibrium Random Bi Bi mechanisms with dead-end complexes make an exception. Thus, if the dead-end complexes EBQ, or EAR + EBQ form, the kinetic constant composition next to the AP and BQ concentration terms, respectively, are... 

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. 

Table 2 summarizes the isotope exchange rate expressions for selected common sequential mechanisms, in addition to the Ordered Bi Bi mechanism outlined above. Since there is no such mechanism as Rapid Equilibrium Random Bi Bi without a dead-end complex, at least a dead-end complex EBQ must form (mechanism 4). If both dead-end complexes, EBQ and EAP, are formed, an extra KrP/KspB term is added in the denominator of rate equation for mechanism 4 K p represents the dissociation constant of P from the EAP complex. 

Rapid Equilibrium Random Bi Bi with dead-end complex EBQ Any H H H HCD... 

Equal isotope effect on the two V/K values suggests one of several possibilities, including an Equilibrium Ordered mechanism with or without a dead-end EB complex, a Rapid Equilibrium Random mechanism, or a Steady-State Random mechanism in which the rates of release of A and B from the central complex are equal. 

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. 

An enzyme-catalyzed reaction scheme in which the two substrates (A and B) can bind in any order, resulting in the formation of a single product of the enzyme-catalyzed reaction (hence, this reaction is the reverse of the random Uni Bi mechanism). Usually the mechanism is distinguished as to being rapid equilibrium (/.c., the ratedetermining step is the central complex interconversion, EAB EP) or steady-state (in which the substrate addition and/or product release steps are rate-contributing). See Multisubstrate Mechanisms... 

If the interconversion of the central EAB complex to the EP complex is rate determining and all other reactions are in a rapid equilibrium , the following rate equation for the random bi-uni reaction may be derived according to the above method ... 

The rate equations for fully random and ordered mechanisms for three-substrate reactions are shown in Table II and can only be briefly discussed here. For the random mechanism, the rate equation derived by the rapid equilibrium assumption 43) contains all the terms of Eq. (2), and from experimental values for the eight kinetic coefficients for the reaction in each direction the dissociation constants for all the complexes may be calculated (c/. 43). 

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