Multiple substrate reactions

All enzymatic reactions are initiated by formation of a binary encounter complex between the enzyme and its substrate molecule (or one of its substrate molecules in the case of multiple substrate reactions see Section 2.6 below). Formation of this encounter complex is almost always driven by noncovalent interactions between the enzyme active site and the substrate. Hence the reaction represents a reversible equilibrium that can be described by a pseudo-first-order association rate constant (kon) and a first-order dissociation rate constant (kM) (see Appendix 1 for a refresher on biochemical reaction kinetics) ... 

Although the Michaelis-Menten equation is applicable to a wide variety of enzyme catalyzed reactions, it is not appropriate for reversible reactions and multiple-substrate reactions. However, the generalized steady-state analysis remains applicable. Consider the case of reversible decomposition of the enzyme-substrate complex into a product molecule and enzyme with mechanistic equations. 

The majority of such reactions entail the transfer of a functional group, such as a phosphoryl or an ammonium group, from one substrate to the other. In oxidation-reduction reactions, electrons are transferred between substrates. Multiple substrate reactions can be divided into two classes sequential displacement and double displacement. 

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). 

A similar analysis can be made for any other equilibrium reaction within a catalytic scheme, so that the effect of temperature on any inhibition constant or dissociation constants for multiple substrate reactions (see sections 3.3 and 3.4) can be determined accordingly. For any inhibition constant Kj ... 

Multiple substrate reactions are more frequently occurred than single substrate reactions. Thus, efforts have been made to develop two or more substrate enzymatic reaction mechanisms. One of the most important examples is lipid hydrolysis, where lipid and water molecules act as two substrates to produce two products namely fatty acids and glycerol. Cleland (1963) proposed three types of mechanism for two substrate reactions based on the order of adding the substrates and products release from the active site within the reaction sequence. These are ordered-sequential, random-sequential, and Ping-Pong, shown in Figure 4.1. 

Kinetic analysis of multiple substrate reactions could stop at this point. However, if more in-depth knowledge of the mechanism of a particular multisubstrate reaction is required, a more intricate kinetic analysis has to be carried out. There are a number of common reaction pathways through which two-substrate reactions can proceed, and the three major types are discussed in turn. 

In a complex enzyme reaction, multiple substrate-enzyme complexes are formed. Assume the following reaction mechanisms are taking place in three consecutive stages ... 

In what follows, enzyme reactions are treated as if they had only a single substrate and a single product. While most enzymes have more than one substrate, the principles discussed below apply with equal vaUdity to enzymes with multiple substrates. 

Note that multiple hydrozirconation reactions can be conducted on the same substrate in a one-pot reaction (Scheme 8-10) [90-93]. 

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 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. 

To simplify the complex reaction pathways of PET reactions and to make them more transferable to multiple substrates it is sometimes advisable to carry them out in a sensitized way. The sensitizer has three characteristics the substrate is excited for the primary PET process, its resulting radical ion or radical is so inert that it does not react with the substrate, and, in most cases, the sensitizer is regenerated by back-electron transfer. A simplified mechanism of a sensitized PET reaction is shown in Scheme 5. 

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. 

Enzymes that catalyze redox reactions often require a coenzyme such as NAD+ or FADH2 in addition to a substrate. These are all multiple substrate enzymes. Each substrate and coenzyme will have its own Km value. The substrates for glutamate dehydrogenase, an enzyme with three substrates in both forward and reverse directions, are shown in Scheme 4.10 with their K values.9... 

As most NRPS multienzymes are multidomain proteins with multiple activation domains, multiple sites may participate in the reactions assayed, and no clear result concerning a single specific site may result. In ACV synthetases, the nonadditivity of the initial rates has been observed in the S. clavuligerus enzyme [35] and the A. chrysogenum enzyme [1]. Two or more site activations of one substrate amino acid could be expected to depend on different binding constants, and thus be detectable by kinetic analysis. So far, however, no evidence for mixed types of concentration dependence has been found. It is thus not yet clear if nonadditivity results from misactivation or alteration of kinetic properties in the presence of multiple substrates. In the case of gramicidin S synthetase 2, evidence for misactivations has been reported [59],... 

If non-Michaelis-Menten kinetics for all P450 enzymes are a result of multiple substrates binding to the enzyme, then the reaction kinetics for the binding of two substrates to an active site can be complicated. A number of analyses of... 

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