Multisubstrate reactions

FIGURE 4.1 Graphical representation of (a) ordered-sequential, (b) random-sequential, and 

In the ordered-sequential mechanism, there is no formation of a second-substrate complex, as the enzyme initially has no site for binding with the second substrate. This site is formed only when the first substrate has been bound to the enzyme. On the other hand, in the random mechanism, the enzyme can bind with either the first or second substrate to produce the corresponding complex, which can then bind to the other substrate to generate the new complex, which later generates and releases the product. 

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Many other multisubstrate examples abound in metabolism. In effect, these situations are managed by realizing that the interaction of the enzyme with its many substrates can be treated as a series of uni- or bisubstrate steps in a multi-step reaction pathway. Thus, the complex mechanism of a multisubstrate reaction is resolved into a sequence of steps, each of which obeys the single- and double-displacement patterns just discussed. 

While many enzymes have a single substrate, many others have two—and sometimes more than two—substrates and products. The fundamental principles discussed above, while illustrated for single-substrate enzymes, apply also to multisubstrate enzymes. The mathematical expressions used to evaluate multisubstrate reactions are, however, complex. While detailed kinetic analysis of multisubstrate reactions exceeds the scope of this chapter, two-substrate, two-product reactions (termed Bi-Bi reactions) are considered below. 

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. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

A. J. Hanekom, Generic kinetic equations for modeling multisubstrate reactions in computa tional systems biology. Master s thesis, Stellenbosch University (2006). 

Occasionally, one may also wish to use an auxiliary enzyme not as an assay system but strictly as a means for maintaining the steady-state concentration of a primary reactant in a multisubstrate reaction system. For instance, acetate kinase (and its substrate acetyl phosphate) or creatine kinase (and its substrate creatine phos-... 

The use of Haldane relationships to verify the magnitude of the equilibrium constant or, conversely, to determine (or verify) one of the kinetic parameters requires that aU constants be measured under the same experimental conditions (eg., temperature, pH, buffer species, ionic strength, free metal ion concentrations, etc) If not, the Haldane relationship has no meaning. In addition, kinetic data are often limited in precision, unlike equilibrium measurements. For multisubstrate reactions, there are at least two different Haldane relationships. Thus,... 

A parameter that depends on the concentration of another substrate in a multisubstrate reaction or on one or more cofactors or substances that influence reaction rate. It is an approximation of the true Michaelis constant. 2. A parameter obtained under conditions that do not rigorously satisfy the requirements of initial rate measurements. 

The number of reactants partaking in an enzyme-catalyzed reaction. Because most enzyme reactions have an unequal number of substrates and products, one must specify the reactancy for a specified direction of the reaction. As an example, a multisubstrate reaction having two substrates and three products has a reactancy of two in the forward direction and three in the reverse direction. Cleland introduced the prefixes Uni , Bi , Ter , and Quad to indicate reactancies of one, two, three, and four, respectively. Thus, the example given above can be called a Bi Ter reaction. Water molecules and protons are not usually considered when specifying reactancy. 

Except for very simple systems, initial rate experiments of enzyme-catalyzed reactions are typically run in which the initial velocity is measured at a number of substrate concentrations while keeping all of the other components of the reaction mixture constant. The set of experiments is run again a number of times (typically, at least five) in which the concentration of one of those other components of the reaction mixture has been changed. When the initial rate data is plotted in a linear format (for example, in a double-reciprocal plot, 1/v vx. 1/[S]), a series of lines are obtained, each associated with a different concentration of the other component (for example, another substrate in a multisubstrate reaction, one of the products, an inhibitor or other effector, etc.). The slopes of each of these lines are replotted as a function of the concentration of the other component (e.g., slope vx. [other substrate] in a multisubstrate reaction slope vx. 1/[inhibitor] in an inhibition study etc.). Similar replots may be made with the vertical intercepts of the primary plots. The new slopes, vertical intercepts, and horizontal intercepts of these replots can provide estimates of the kinetic parameters for the system under study. In addition, linearity (or lack of) is a good check on whether the experimental protocols have valid steady-state conditions. Nonlinearity in replot data can often indicate cooperative events, slow binding steps, multiple binding, etc. 

The substance or reactant being acted upon by a catalyst. The substrate is often symbolized by S in one-substrate reactions. In multisubstrate reactions, the substrates are commonly symbolized by A, B, C, etc. 2. The base or foundation upon which an organism lives or grows. 3. The substance or compound of particular interest, with which a reaction with some other chemical reagent is under study. 

Uncompetitive inhibition can also be a possibility in multisubstrate reactions. For example, in an ordered Bi Bi reaction, a competitive inhibitor with respect to the second substrate B, will act as an uncompetitive inhibitor with respect to the first substrate, A. 

Where there is no subsequent turnover of a substrate, such as occurs on the omission of a cosubstrate in a multisubstrate reaction, or on inhibitor binding, the temperature-jump technique is generally the most useful tool for the determination of these constants. 

The most common enzymatic reactions are those with two or more substrates and as many products. But many of the simpler single-substrate schemes are valuable for the development of kinetic ideas concerning effects of pH, temperature, etc., on enzyme reaction rates. Although the mechanisms of multisubstrate reactions are complicated, their kinetics can often be described by an equation of the form ... 

The catalytic reaction of lipases follow the so called ping-pong bi-bi mechanism, a double displacement mechanism. This is a special multisubstrate reaction in which, for a two-substrate, two-product (i.e., bi-bi) system, an enzyme reacts with one substrate to form a product and a modified enzyme, the latter then reacting with a second substrate to form a second, final product, and regenerating the original enzyme (ping-pong). 

The Michaelis-Menten mechanism of enzyme activity models the enzyme with one active site that, weakly and reversibly, binds a substrate in homogeneous solution. It is a three-step mechanism. The first and second steps are the reversible formation of the enzyme-substrate complex (ES). The third step is the decay of the complex into the product. The steady-state approximation is applied to the concentration of the intermediate (ES) and its use simplifies the derivation of the final rate expression. However, the justification for the use of the approximation with this mechanism is suspect, in that both rate constants for the reversible steps may not be as large, in comparison to the rate constant for the decay to products, as they need to be for the approximation to be valid. The simplest form of the mechanism applies only when A h 2> k. Neverthele.ss, the form of the rate equation obtained does seem to match the principal experimental features of enzyme-catalyzed reactions it explains why there is a maximum in the reaction rate and provides a mechanistic understanding of the turnover number. The model may be expanded to include multisubstrate reaction rate and provides a mechanistic understanding of the turnover number. The model may be expanded to include multisubstrate reactions and inhibition. 

The earher recommendation of the Enz5nne Commission of the International Union of Biochemistry was that the Ks should be apphed for the Michaelis-Menten mechanism and Ku for the Briggs-Haldane mechanism (Enzyme Nomenclature, 1973) in this case, /Cm = Jfs + k /kf This practice must be discouraged because it leads to cumbersome and ambiguous expressions in multisubstrate reactions. 

The use of coefficients may be of only marginal help in making Eqs. (3.24) and (3.36) less formidable however, this nomenclature is of immense help in more complex cases, particularly with multisubstrate reactions. 

The steady-state kinetics of monosubstrate enzyme reactions has been described in Chapter 3. However, tme monosubstrate reactions are quite rare in nature and are restricted only to some isomerases and epimerases. The majority of enzyme reactions are multisubstrate reactions, with two or three substrates and one, two, or three products of reaction (lUBMB, 1992). 

This simple rule is also valid with multisubstrate reactions. If the enzyme is saturated with all substrates, except one (which is variable), the pH profile of V/K will always reflect dissociation in the free enzyme and the pH profile of V will always reflect dissociation in the enzyme-substrate complex. 

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