Reactions Involving Multiple Substrates

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. 

Distinguish sequential displacement and double displacement in reactions involving multiple substrates. Provide examples of enzymes using each mechanism. 

The path computation has been most successful when applied to a specific class of binary relations, namely the substrate-product relations of enzymatic reactions. They constitute a well-characterized set of binary relations, and the amount of available data is relatively large. There are about 3,500 main reactions between the main compounds that are represented in the KEGG pathway diagrams. An enzymatic reaction generally involves multiple substrates and multiple products, so that it must first be decomposed into all possible substrate-product binary relations. However, because the relations involving ubiquitous compounds such as water and ATP will make many undesired connections, it is better to limit to main compounds for practical purposes. 

Still another possibility is that the inhibitor binds only to the enzyme-substrate complex and not to the free enzyme (fig. 7.14c). This reaction is called uncompetitive inhibition. Uncompetitive inhibition is rare in reactions that involve a single substrate but more common in reactions with multiple substrates. Plots of 1/v versus 1/[S] at different concentrations of an uncompetitive inhibitor give a series of parallel lines. 

When an enzyme contains multiple subunits and the reaction which it catalyzes involves multiple substrates, detailed steps in the cataljrtic processes may become very complicated. The malate dehydrogenases from mammals contain two subunits, and the substrates include both a form of the dicarboxylic acid and the dinucleotide coenzyme. The third... 

Many enzymes can generate several intermediates as they process a substrate into one or more products. An example is the enzyme chymotrypsin, which we treat in detail in Case study 8.1. Other enzymes act on multiple substrates. An excunple is hexokinase, which catalyzes the reaction between ATP cuid glucose (the two substrates of the enzyme), the first step of glycolysis (Section 4.8). The same strategies developed in Section 8.1 Ccui be used to deal with such complex reaction schemes, and we shall focus on reactions involving two substrates. 

Recently, Wang et al. reported anovel palladium-catalyzed sequential Sonogashira/ carbopalladative cyclization/Suzuki reactions involving multiple carbon-carbon bond formation using protected homopropargyl alcohol 155 under mild conditions [59] (Scheme 6.40). Various indene derivatives 156 could be constructed efficiently with good yields in this transformation. Moreover, this reaction has a wide tolerance of various substituents in the substrates. 

A titrametric assay of PLCSc, alternatively called the pH-stat method, was the workhorse in early studies [28]. This method simply involves titrating the acidic product of the PLC reaction as it is formed with a solution of standard base. An advantage of this continuous assay is that it can be used to detect the turnover of both synthetic and natural substrates, and its sensitivity has been estimated to be in the 20-100 nmol range. However, the pH-stat assay has low throughput capability, and it cannot be easily performed in a parallel fashion with multiple substrate concentrations. It is also necessary to exclude atmospheric carbon dioxide from the aqueous media containing the enzyme and substrate. 

Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. 

The simplest enzymatic system is the conversion of a single substrate to a single product. Even this straightforward case involves a minimum of three steps binding of the substrate by the enzyme, conversion of the substrate to the product, and release of the product by the enzyme (Scheme 4.6). Each step has its own forward and reverse rate constant. Based on the induced fit hypothesis, the binding step alone can involve multiple distinct steps. The substrate-to-product reaction is also typically a multistep reaction. Kinetically, the most important step is the rate-determining step, which limits the rate of conversion. 

Photocurrent multiplication has been observed for a variety of semiconductors including Ge [269], Si [268-271], ZnO [272-278], Ti02 [279-282], CdS [283, 284], GaP [285], InP [286] and GaAs [287-289]. These studies have included both n- and p-type semiconductors, and have spanned a range of substrates, both organic and inorganic. As in the Si case, this phenomenon can also be caused by photodissolution reactions involving the semiconductor itself. The earlier studies have mainly employed voltammetry, particularly hydrodynamic voltammetry (see, e.g.. Ref. [282]). 

The stereochemical outcome of an aldol reaction involving more than one chiral component is consistent with the rule of approximate multiplicativity of diastereofacial selectivities intrinsic to the chiral reactants. For a matched case, the diastereoselectivity approximates (substrate DS) X (reagent DS). For a mismatched case, the diastereoselectivity is (substrate DS) (reagent DS). Double asymmetric induction also can be used to enforce the inherent facial selectivity of a chiral aldehyde, as shown below. 

The reactions of alkynes and alkenes with metal-metal multiple bonds in dinuclear complexs have important catalytic implications, but less attention is paid to their insertions into metal-metal single bonds. A 1965 publication comprehensively reviewing insertions of compounds of metals and metalloids involving unsaturated substrates lists insertions into metal-metal bonds prior to 1965. 

There are many more complex mechanisms for enzyme reactions involving reversible reactions, multiple substrates, activators, and inhibitors. Techniques for modeling and analyzing these systems are available. ... 

Some of the natural extensions of this classical approach include the treatment of mechanisms with multiple intermediate complexes and near-equilibrium conditions (e.g., Peller and Alberty, 1959). Enzyme-catalyzed reactions that involve two substrates and two products are among the most common mechanisms found in biochemistry (about 90% of all enzymatic reactions according to Webb, 1963). It is not surprising, then, that this class of mechanisms also has received a great deal of attention (e.g., Dalziel, 1957,1969 Peller and Alberty, 1959 Bloomfield et al., 1962a,b Cleland, 1963a,b,c). This class includes mechanisms in which reactant molecules enter and exit a single pathway in fixed order and mechanisms with parallel pathways in which reactant molecules enter and exit in a random order (Cleland, 1970). 

Although the MM equation is a powerful kinetic form to which the vast majority of enzyme kinetics has been fitted, one should not forget the assumptions and limitations of the model. As a basic example, feedback inhibition, whereby the product of the reaction inhibits the enzyme-substrate cooperativity, multiple-substrate reactions, allosteric modifications, and other deviations from the reaction scheme in equation (1) are treated only adequately by the MM formalism under certain experimental conditions. In other words, enzyme kinetics are often bent to conform to the MM formalism for the sake of obtaining a set of parameters easily recognizable by most biochemists. The expUcit mathematical and experimental treatment of reaction mechanisms more complex than that shown in equation (1) is highly involved, although a mathematical automated kinetic equation derivation framework for an arbitrary mechanism has been described in the past (e.g., ref. 6). 

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