Rapid Equilibrium Random

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

Along these fines, Liebermeister and Klipp [161] suggested the use of a rapid-equilibrium random-order binding scheme as a generic mechanism for all enzymes, independent of the actual reaction stoichiometry. While there will be deviations from the (unknown) actual kinetics, such a choice, still outperforms power-law or lin-log approximations [161]. 

This muscle phosphotransferase (EC 2.132) catalyzes the reversible rephosphorylation of ADP to form ATP (/.c., T eq = [ATP][Creatine]/([Creatine phosphate] [ADP]) = 30). In resting muscle, creatine phosphate is synthesized at the expense of abundant stores of ATP intracellular creatine phosphate stores often reach 50-60 mM. If ATP is suddenly depleted by muscle contraction, its product ADP is immediately converted back into ATP by the reverse of the creatine kinase reaction. Depending on the pH at which the enzyme is studied, the kinetic reaction can be either rapid equilibrium random or rapid equilibrium ordered. A-Ethylglycocyamine can also act as a substrate. 

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. 

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 ... 

The area enclosed by dashed lines in Scheme 6a is the rapid-equilibrium random segment. The segment is called A, and Scheme 6a is reduced to a basic figure of X and EQ connected by two pathways (Scheme 6b) ... 

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. 

Alberty first proposed the use of Haldane relations to distinguish among the ordered Bi Bi, the ordered Bi Bi Theorell-Chance, and the rapid equilibrium random Bi Bi mechanisms. Nordlie and Fromm used Haldane relationships to rule out certain mechanisms for ribitol dehydrogenase. 

Haldane is also valid for the ordered Bi Bi Theorell-Chance mechanism and the rapid equilibrium random Bi Bi mechanism. The reverse reaction of the yeast enzyme is easily studied an observation not true for the brain enzyme, even though both enzymes catalyze the exact same reaction. A crucial difference between the two enzymes is the dissociation constant (i iq) for Q (in this case, glucose 6-phosphate). For the yeast enzyme, this value is about 5 mM whereas for the brain enzyme the value is 1 tM. Hence, in order for Keq to remain constant (and assuming Kp, and are all approximately the same for both enzymes) the Hmax,f/f max,r ratio for the brain enzyme must be considerably larger than the corresponding ratio for the yeast enzyme. In fact, the differences between the two ratios is more than a thousandfold. Hence, the Haldane relationship helps to explain how one enzyme appears to be more kmeticaUy reversible than another catalyzing the same reaction. 

A procedure that assists in the characterization of binding mechanisms for sequential (/.e., non-ping pong) reactions . The same general initial rate expression applies to the steady-state ordered Bi Bi reaction, the rapid-equilibrium random Bi Bi reaction, and the Theorell-... 

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. 

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])... 

When taken collectively, the overall evidence indicates that hexokinase can both add and release substrates and products in a random mechanism. However, the mechanism cannot be described as rapid equilibrium random. The evidence also indicates that the preferred pathway is the ordered addition of glucose followed by ATP, then the release of ADP followed by glucose-6-P. Danenberg and Cleland have recently attempted to assign relative rate constants to a general random mechanism for hexokinase as shown in Fig. 14 (30). 

Earlier studies on the properties of phosphorylases isolated from various sources have indicated that their subunits are similar in size with about 100,000 daltons.15-17 The reaction proceeds in a rapid equilibrium random Bi-Bi mechanism as has been shown by kinetic studies with rabbit skeletal muscle phosphorylases a18-20 and b,21,22 rabbit liver enzyme,23 potato tuber enzyme,24 and the enzyme from E. coli.25) In contrast, the substrate specificities for various glucans differ considerably depending on the enzyme sources. The rabbit muscle enzyme has high affinity for branched glucans such as glycogen and amylopectin but low affinity for amylose and maltodextrin.26,27 The potato tuber enzyme can act on amylose, amylopectin, and maltodextrin but only poorly on glycogen,28,29 while the E. coli enzyme shows high affinity for maltodextrin.10 ... 

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. 

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... 

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),... 

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.

Figure 5.4 Patterns of double reciprocal plots for two substrate reactions, (a) Ping-pong. The double reciprocal plots for one substrate in the presence of fixed concentrations of the other are parallel, no matter which substrate reacts first, (b) Rapid equilibrium random, (c) Rapid equilibrium ordered, first binding substrate only.

Fig. 7-19 and Fig. 7-20 show a number of effects on the activity of aminoacylase, which cannot be described by a standard kinetic model alone. To derive a kinetic model for the aminoacylase and for describing all influences, a reaction scheme based on the rapid equilibrium random uni-bi model was used (compare with Eq. (40)). This model was supplemented, according to the above measured influences of substrate inhibition by acetate and inhibition by D-alanine. Consequently, the reaction scheme Eq. (40) had to be extended to yield equation (46) and the corresponding rate equation (47), which is modified by the additional equilibria ... 

Rapid equilibrium random. 1 1 fee k k.i k 1 Keba Keab KeaKeab 4M... 

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... 

One single concentration term missing, and the two noncorresponding Michaelis constants equal to zero (C,. BC, and ilTbAC terms missing). The mechanism proceeds by rapid equilibrium random addition of A and B, with C... 

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. 

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