Enzyme-catalyzed bimolecular reactions

Previous sections of this chapter have focused on developing general principles for enzyme-catalyzed reactions based on analysis of single-substrate enzyme systems. Yet the majority of biochemical reactions involve multiple substrates and products. With multiple binding steps, competitive and uncompetitive binding interactions, and allosteric and covalent activations and inhibitions possible, the complete set of possible kinetic mechanisms is vast. For extensive treatments on a great number of mechanisms, we point readers to Segel s book [183], Here we review a handful of two-substrate reaction mechanisms, with detailed analysis of the compulsory-order ternary mechanism and a cursory overview of several other mechanisms. 

1 Compulsory-order ternary mechanism Consider first the case where two substrates (A and B) bind to an enzyme in an ordered manner and two products (P and Q) dissociate in an ordered manner as well  

This mechanism is known as the ordered bi-bi mechanism ( bi-bi denotes a bi-substrate bi-product reaction), or the compulsory-order ternary mechanism , where the term ternary refers to the three-species complex formed by the binding of two substrates to the enzyme. 

The constant Keq is the equilibrium constant for the reaction E0 = [E] + [E A] + [E AB] + [E Q] is the total enzyme concentration and the /, factors in Equation (4.52) account for non-productive binding (inhibition) of inhibitors to each of the enzyme states. These inhibition factors are computed 

The flux equation is J = n/d, where the numerator and denominator written in terms of kinetic constants are 

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Rate Expressions for Enzyme Catalyzed Single-Substrate Reactions. The vast majority of the reactions catalyzed by enzymes are believed to involve a series of bimolecular or unimolecular steps. The simplest type of enzymatic reaction involves only a single reactant or substrate. The substrate forms an unstable complex with the enzyme, which subsequently undergoes decomposition to release the product species or to regenerate the substrate. 

Bimolecular reactions of two molecules, A and B, to give two products, P and Q, are catalyzed by many enzymes. For some enzymes the substrates A and B bind into the active site in an ordered sequence while for others, bindingmay be iii a random order. The scheme shown here is described as random Bi Bi in a classification introduced by Cleland. Eighteen rate constants, some second order and some first order, describe the reversible system. Determination of these kinetic parameters is often accomplished using a series of double reciprocal plots (Lineweaver-Burk plots), such as those at the right. 

Reymond and Chen88 have investigated the same set of antibodies for their ability to catalyze bimolecular aldol condensation reactions. The antibodies were assayed individually at pH 8.0 for the formation of aldol 111 from aldehyde 109 and acetone. None catalyzed the direct reaction, but in the presence of amine 110 three anti-52a and three anti-52b antibodies showed modest activity. In analogy with natural type I aldolase enzymes, the reaction is believed to occur by formation of an enamine from acetone and the amine, followed by rate-determining condensation of the enamine with the aldehyde. As in the previous example, the catalyst, which was characterized in detail, is not very efficient in absolute terms ( cat = 3 x 10-6 s 1 for the anti-52b antibody 72D4), but it is approximately 600 times more effective than amine alone. Moreover, the reactions with the antibody are stereoselective The enamine adds only to the si face of the aldehyde to give... 

Because the general principles of chemical kinetics apply to enzyme-catalyzed reactions, a brief discussion of basic chemical kinetics is useful at this point. Chemical reactions may be classified on the basis of the number of molecules that react to form the products. Monomolecular, bimolecular, and termolecular reactions are reactions involving one, two, or three molecules, respectively. 

The type of reaction which is probably of most importance in the enzymatic degradation of polymers is the bimolecular reaction illustrated above, in which the enzyme catalyzes the interaction of the polymer and a low molecular reagent (such as water in a hydrolysis reaction). These reactions can occur by either a single displacement or a double displacement mechanism. In the former, both substrates, A and B below, are bound to the enzyme by consecutive, reversible reactions, after which the final complex, EAB, dissociates into the products, C and D, and the free enzyme, as follows ... 

An obvious piece of evidence is when the kinetic law is more complex than would be consistent with the occurrence of the reaction in a single step two examples of this have already been noted. Enzyme-catalyzed reactions provide additional examples. Thus, if an enzyme reaction involving a single substrate occurred by a simple bimolecular reaction between enzyme and substrate, the kinetics would be first-order with respect to substrate. However, the behavior as far as the substrate is concerned is rarely so simple, and it can therefore be concluded that the reaction occurs in more than one stage. 

The intermolecular electrostatic interactions are found in bimolecular reactions of a charged reactant approaching a molecule with strong dipolar bonds or even charges (e.g., in enzyme-catalyzed reactions, where they are used not only to properly position a substrate in the active site of an enzyme but also to lower the activation energy barrier for the subsequent chemical transformation of a substrate). 

For a simple reaction that consists of a single step, or for each step in a complex reaction, the order is usually the same as the molecularity. However, many chemical reactions consist of sequences of unimolecular and bimolecular steps and the molecularity of the complete reaction need not be the same as its order. Complex chemical reactions often have no meaningful order, as the rate often cannot be expressed as a product of concentration terms. This is almost universal in enzyme kinetics, and even the simplest enzyme-catalyzed reactions... 

Many substitution reactions in organic chemistry (for instance, Sn2 nucleophilic substitutions) are bimolecular and involve an activated complex that is formed from two reactant species. Many enzyme-catalyzed reactions can be regarded, to a good approximation, as bimolecular in the sense that they depend on the encounter of a substrate molecule and an enzyme molecule. 

In this reaction (demonstrated in vitro), one of the two radicals is oxidizing while the other is reducing. In vivo, this reaction is catalyzed by one of several isoforms of an enzyme known as superoxide dismutase (SOD). As shown above, hydrogen peroxide may form as a result of the superoxide anion s dismutation reaction however, it may also be produced from a bivalent reduction of 02. The addition of the second electron leads to the formation of hydrogen peroxide, which is a powerful oxidizing agent. Due to the unpaired electrons in their outer shells, free radicals are favored to pair with other molecules during bimolecular collisions. 

We begin by discussing the general strategy for computation design of an enzyme. We describe the process apphed specifically to the design of an enzyme to catalyze a bimolecular Diels-Alder reaction and follow that with a discussion of the other designed enzymes. 

The rate of a-chymotrypsin-catalyzed hydrolysis as a function of overall GPANA concentration in CTAB reversed micelles and in aqueous solution are shown in Figure 5. It is apparent that the reaction rate in the reversed micellar solution is on the order of 50 times more rapid than in the aqueous system. Furthermore, in the reversed micellar system there is no indication of enzyme saturation as the reaction is first order in substrate concentration. As enzyme saturation kinetics are not observed, it is impossible to differentiate between the parameters kcat and Kg. Instead a second order bimolecular rate constant for both the micelle interior ( micelle) and for what is experimentally observed ( observed) is defined. 

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