Rapid Equilibrium Random 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. 

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... 

Multisubstrate or multiproduct enzyme-catalyzed reaction mechanisms in which one or more substrates and/ or products bind and/or are released in a random fashion. Note that this definition does not imply that there has to be an equal preference for any particular binding sequence. The flux through the different binding sequences could very easily be different. However, in rapid equilibrium random mechanisms, the flux rates are equivalent. See Multisubstrate Mechanisms... 

ATP + L-arginine = ADP + N-phospho-L-arginine (<7> rapid equilibrium random mechanism [6])... 

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). 

Christensen and coworkers suggested that D-glucitol dehydrogenase follows a rapid-equilibrium, random mechanism.426... 

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. 

Scheme 5.4 Rapid equilibrium random mechanism for a two-substrate enzyme, illustrated for a glycosyl transfer.

Rapid Equilibrium Random Mechanism The random mechanism is EA... 

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... 

If /cs = 0 (ordered mechanism), = kglk at all times, and one sees the full (VIK >) isotope effect regardless of A level. This is also the isotope effect when B is varied at low A (see above). In a random mechanism, Cf. varies from kgl(k4 + ks) at low A to kg/k4 at very high A. Thus, in a rapid equilibrium random mechanism ( 4, ks kg), one sees kg as the isotope effect regardless of which substrate is varied or labeled. When there is some stickiness to one or both substrates, however, the above analysis will demonstrate it. 

The Ordered and TheoreU-Chance mechanisms have certain constraints placed on their intersection points, but the Rapid Equilibrium Random mechanism has none, since only the equilibrium constant relates the forward and reverse reactions. 

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. 

Based on isotope effects only, it is not possible to distinguish the Rapid Equilibrium Ordered from the Rapid Equilibrium Random mechanism. However, the first mechanism gives a distinctive initial velocity pattern that intersects on the ordinate with B as the varied substrate. To teU the difference between the Rapid Equilibrium Random and the Steady-State Random mechanism will require other methods, such as the isotope trapping method (Rose et al, 1974), or isotopic exchange. 

In an ordered bisubstrate mechanism one must vary the second substrate, B, and determine V/JSTb, regardless of whether the label is in A or B, since ly AwiU not show an isotope effect. For a random mechanism one must vary both A and B, since one may see different isotope effects on VfK and V/Aa, a distinction that may help to characterize the mechanism. The effects onV/K and V/Aa shouldbe different when one or both substrates are sticky, that is, dissociate more slowly from the enzyme than they react to give products. The substrate with the lower V7A is the sticky one. Larger effects on Vthan on either V7A show that both substrates are sticky smaller ones show that a slow step follows release of the first product. A rapid equilibrium random mechanism will show equal isotope effects on V, V/Ab, and F/Aa, aU larger than unity. 

Detailed studies with agmatine as the acceptor in the presence and absence of the product inhibitor nicotinamide (data not shown) were in agreement with the sequential rapid equilibrium, random mechanism shown below (kinetic parameters are summarized in Table 1). 

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