Rapid Equilibrium Ordered mechanism

Unconsumed substrates are treated as substrates or essential activators in deriving rate equations and studying detailed mechanisms. Nonetheless, one must indicate whether an unconsumed substrate (U) remains bound to the enzyme or not (in this case, U also becomes an unaltered product) in the reaction scheme. In practice, unconsumed substrates are likely to be involved in all the typical multisubstrate kinetic mechanisms Only one case is illustrated here, namely that the unconsumed substrate Su activates catalysis when bound in a rapid-equilibrium ordered 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. 

This rate equation is identical to that for a rapid equilibrium ordered addition bisubstrate mechanism (/.c., a scheme where substrate A rapidly binds prior to the addition of the second substrate B). Huang has presented the theoretical basis for mechanisms giving rise to... 

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. 

Reduction of oxidized Cuh and Cum by ascorbate is rapid, and yields catalytically competent enzyme that can bind O2 and organic substrate." " For PHM, an equilibrium ordered mechanism... 

The velocity equation has now the same form as that for the total Rapid Equilibrium Ordered Bi Bi mechanism (Eq. (8.12)), except that the constants associated with B and P are the Michaelis constants and the constants associated with A and Q are the dissociation constants. 

The intersecting pattern for an Ordered bisubstrate mechanism in Fig. 1 (left) will also be obtained with a Rapid Equilibrium bisubstrate mechanism in each case, an intersecting point may be above, below, or on the axis. 

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. 

The Rapid Equilibrium Ordered Bi Bi system (Section 8.2) is a limiting case of the more realistic Steady-State Ordered Bi Bi system (Section 9.2). In bisubstrate mechanisms, the two approaches yield different velocity equations. As described... 

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. 

If the value of is very fast with respect to k B at any concentration of B used, the mechanism approximates rapid equilibrium order addition of A. Under these conditions, the isotope effect will be constant, that is, whatever substrate is varied at any concentration of the fixed substrate. 

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. 

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]. 

Figure 11.1 illustrates the behavior of Equation 11.6. By the assumption of rapid equilibrium the rate determining step is the unimolecular decomposition. At high substrate composition [S] KM and the rate becomes zero-order in substrate, v = Vmax = k3 [E0], the rate depends only on the initial enzyme concentration, and is at its maximum. We are dealing with saturation kinetics. The most convenient way to test mechanism is to invert Equation 11.6... 

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. 

Equation (6) is identical in form with Eq. (4). In fact, if 3 2, k-2, Eq. (6) reduces to Eq. (4). Although Eq. (5) is a more realistic mechanism compared with Eq. (1), especially when the rapid-equilibrium treatment is applied to the reversible reaction, the information obtainable from initial-rate studies of such unireactant system remains nevertheless the same Vi and K. This serves to justify the simplification used by the kineticist that is, the elimination of certain intermediates to maintain brevity of the rate equation (provided the mathematical form is unaltered). Thus, the forward reaction of an ordered Bi Bi mechanism is generally written as diagrammed below. 

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 term first introduced by Cleland to indicate that for ordered substrate binding mechanisms, addition of an inhibitor mimicking the first substrate may still permit binding of the second substrate. Hence, as long as the addition of the first substrate is not of the rapid equilibrium type, the presence of the inhibitor will induce substrate inhibition by the second substrate. An example of induced substrate inhibition is provided in the thymi-dylate synthase reaction where the second substrate methylene tetrahydrofolate becomes an inhibitor, but only in the presence of the inhibitor bromodeoxyuridine 5 -monophosphate. 

A three-substrate (A, B, and C), two-product (P and Q) enzyme reaction scheme in which all substrates and products bind and are released in an ordered fashion. Glyceraldehyde-3-phosphate dehydrogenase has been reported to have this reaction scheme. The steady-state and rapid equilibrium expressions, in the absence of products and abortive complexes, are identical to the ordered Ter Ter mechanism. See Ordered Ter Ter Mechanism... 

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-... 

A two-substrate, two-product enzyme-catalyzed reaction scheme in which both the substrates (A and B) and the products (P and Q) bind and are released in any order. Note that this definition does not imply that there is an equal preference for each order (that is, it is not a requirement that the flux of the reaction sequence in which A binds first has to equal the flux of the reaction sequence in which B binds first). In fact, except for rapid equilibrium schemes, this is rarely true. There usually is a distinct preference for a particular pathway in a random mechanism. A number of kinetic tools and protocols... 

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... 

Thus, for the conditions where second-order behavior is observed, the chemical circumstances indicate the cerium(IV) oxidation of each chromium complex will involve a rate-determining one-equivalent oxidation of the complex ion (or a species in rapid equilibrium with the complex ion) to an intermediate, followed by the rapid one-equivalent oxidation of the intermediate. Without reference to the role of water coordinated to the chromium, the most obvious mechanism in accord with these specifications is ... 

Here, N03 was assumed to be in rapid equilibrium with NO and 02, and N205 in equilibrium with N03 and N02. This was the only mechanism the investigators could find in which the order of the reaction, with respect to [NO], could exceed 2. Assuming steady-state conditions for the intermediates, one obtains... 

Reactions in which all the substrates bind to the enzyme before the first product is formed are called sequential. Reactions in which one or more products are released before all the substrates are added are called ping-pong. Sequential mechanisms are called ordered if the substrates combine with the enzyme and the products dissociate in an obligatory order. A random mechanism implies no obligatory order of combination or release. The term rapid equilibrium is applied when the chemical steps are slower than those for the binding of reagents. Some examples follow. 

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). 

Hine and his co-workers showed in the 1950s by kinetic and trapping experiments that dichloromethylene, CC12> is an intermediate in the reaction of haloforms with base in aqueous solution.144 Scheme 9 depicts for chloroform the mechanism they proposed. If the first step is a rapid equilibrium and k2 is ratedetermining, the observed second-order kinetics are consistent with the mechanism,145 as are a number of other results.146... 

Which step would be rate-determining for this mechanism It could not be step 1 since, if this were the case, then the rate law would be first-order with respect to the aldehyde rather than the observed second-order relationship. Also, if the reaction is carried out in water labelled with oxygen-18, the oxygen in the benzaldehyde exchanges with the 180 from the solvent much faster than the Cannizzaro reaction takes place. This can only be because of a rapid equilibrium in step 1 and so step 1 cannot be rate-determining. 

---