One-Substrate Reaction

The effect of substrate concentration on enzymatic reaction was first put forward in 1903 (Henri, 1903), where the conversion into the product involved a reaction between the enzyme and the substrate to form a substrate-enzyme complex that is then converted to the product. However, the reversibility of the substrate-enzyme complex and its final breakdown into the substrate and free enzyme regeneration was generally ignored. In 1913, Michaelis and Menten took this into consideration and proposed the scheme shown in Equation 4.1 for a one-substrate enzymatic reaction. Experimental data, that is, the initial reaction rates, were collected to support their analysis. The reaction mechanism, which is one of the most common mechanisms in enzymatic reactions, was based on the assumption that only a single substrate and product are involved in the reaction. 

Although the reaction mechanism shown in Equation 4.1 is commonly used, a more representative mechanism should have some degree of reversibility in considering the enzyme-substrate complex s reversible conversion to the product, as shown in Equation 4.2  

---

There are obvious special cases of this equation, e.g., at either high [A] or [B], whereby it reduces to pseudo-one-substrate reactions as treated above. Let us now consider some typical mechanisms as examples. 

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. 

The kinetic scheme according to Michaelis-Menten for a one-substrate reaction (Michaelis, 1913) assumes three possible elementary reaction steps (i) formation of an enzyme-substrate complex (ES complex), (ii) dissociation of the ES complex into E and S, and (iii) irreversible reaction to product P. In this scheme, the product formation step from ES to E + P is assumed to be rate-limiting, so the ES complex is modeled to react directly to the free enzyme and the product molecule, which is assumed to dissociate from the enzyme without the formation of an enzyme-product (EP) complex [Eq. (2.2)]. 

The steady-state treatment of enzyme kinetics assumes that concentrations of the enzyme-containing intermediates remain constant during the period over which an initial velocity of the reaction is measured. Thus, the rates of changes in the concentrations of the enzyme-containing species equal zero. Under the same experimental conditions (i.e., [S]0 [E]0 and the velocity is measured during the very early stage of the reaction), the rate equation for one substrate reaction (uni uni reaction), if expressed in kinetic parameters (V and Ks), has the form identical to the Michaelis-Menten equation. However, it is important to note the differences in the Michaelis constant that is, Ks = k2/k1 for the quasi-equilibrium treatment whereas Ks = (k2 + k3)/k i for the steady-state treatment. 

Many enzymes catalyze reactions with two interacting substrates, and although the kinetics of these reactions are more complex than those of one-substrate reactions, they still obey Michaelis-Menten kinetics. Reactions of the type A + B <= P + Q usually fall within either of two classes with respect to kinetic behavior and mechanism of action. 

In this model, the enzyme-inhibitor complex is completely inactive, but is in equilibrium with the active form of the enzyme. For a simple one-substrate reaction, the effect of a competitive inhibitor on the initial rate of the reaction is described by Eq. 2.40. 

When one of the substrates is water (i.e., when the process is one of hydrolysis), with the reaction taking place in aqueous solution, only a fraction of the total number of water molecules present participates in the reaction. The small change in the concentration of water has no effect on the rate of reaction and these pseudo-one substrate reactions are described by one-substi ate kinetics. More generally the concentrations of both substrates may be variable, and both may affect the rate of reaction. Among the bisubstrate reactions important in clinical enzymology are the reactions catalyzed by dehydrogenases, in which the second substrate is a specific coenzyme, such as the oxidized or reduced forms of nicotinamide adenine dinucleotide, (NADH), or nicotinamide adenine dinucleotide phosphate, (NADPH), and the amino-group transfers catalyzed by the aminotransferases. 

Most enzymatic reactions involving two-substrate reactions show more complex kinetics than do one-substrate reactions. Examples are catalyzed by dehydrogenases and aminotransferases. Hydrolytic reactions are bisubstrate reactions in which water is one of the substrates. The change in water concentration is negligible and has no effect on the rate of reaction. A two-suhstrate reaction can be written as... 

An enzymatic reaction may be described by the following steps first, binding of enzyme E and substrate S occurs second, while bound to the enzyme the substrate will be converted to the product P finally, the product is released from the enzyme and free enzyme becomes available for the next cycle. In the simple case of a one substrate reaction this can be described by the following equations... 

First it should be noted that the equilibrium constants for binding A and Q can be measured by standard equilibrium methods, thus the problem reduces to the central reactions that parallel the simple one-substrate reaction described above. Moreover, one can measure the overall and internal equilibria by placing radiolabel in the substrate A and quantitating its conversion to product Q. For the external equilibrium measurement, the ratio of unlabeled [P]/[B] can be altered to bring the ratio of labeled [Q]/[A] close to unity to allow more accurate measurement (8), allowing the overall equilibrium constant to be calculated from Tfne. = [P][Q]/[A][B]. 

Many enzyme reactions in the body involve two substrates. In these cases, the simple reaction scheme used to describe one-substrate reactions is inapplicable. However, the concepts of ACM and kmax can still be applied to two-substrate reactions as follows. If there are two substrates, A and B, and two products, P and Q, the overall reaction is ... 

EC 4 Lyases Additions/Eliminations Cleavage of C-C-, C-O-, C-N-bonds one-substrate reactions two-substrate reactions - no co-factors required... 

Eqs. 3.11 and 3.14 represent the kinetics of irreversible and reversible one substrate reactions respectively. This simple kinetic models are however quite relevant since they represent a good portion of the reactions of industrial relevance. As said before (see section 1.6), most of the traditional enzyme technology refers to hydrolytic reactions performed in aqueous medium with hydrolases. Even though hydrolytic reactions are strictly speaking two substrate reactions, water plays the role of the solvent being in large stoichiometric excess, so its effect on enzyme kinetics can... 

Determination of Kinetic Parameters for Irreversible and Reversible One-Substrate Reactions... 

For simple kinetic mechanisms, like irreversible one-substrate reactions, both rapid equilibrium and steady-state hypothesis lead to rate equations that are formally equal in parametric terms, so when those parameters are experimentally determined, results are the same no matter what hypothesis is considered. Kinetic parameters are to be experimentally determined to obtain validated rate expressions to be used for the design or performance evaluation of enzyme reactors. 

Development of a Generalized Kinetic Model for One-Substrate Reactions Under Inhibition... 

Inhibition by products and/or substrates is quite relevant so that a generalized equation will be derived considering most of the situations of inhibition kinetics for one substrate reactions (Siimer 1978). The following reaction scheme will be considered ... 

A generalized kinetic expression for one-substrate reactions was presented in section 3.3.2 that considers all possible mechanisms for one-substrate reactions (or reactions in which one of the substrates is clearly limiting as it occurs in most hydrolytic reactions in aqueous medium) ... 

Assuming plug-flow regime through the catalyst bed, a steady-state material balance for a one-substrate reaction renders ... 

The corresponding kinetic expressions and the values of a, b and c for the mechanism of one-substrate reactions are in section 3.3.2. The expressions for CPBR performance for the more common kinetic mechanism are presented in Table 5.1. 

From Eq. 5.6, the model for a batch reactor performance considering one-substrate reactions (Eq. 5.3) and one-stage first-order inactivation mechanism (Eq. 5.46) is ... 

All enzymatic reactions involving co-enzymes are two-step reactions. However, if one of the substrates is present in a very high concentration in relation to its Michaelis constant, two-substrate reactions can be treated kinetically as one-substrate reactions. These conditions are also desirable for endpoint determinations with co-enzymes, simply to achieve a high reaction rate. However, if NAD(P)H is the second substrate, the degree to which its concentration can be increased is limited by its high absorbance. On the basis of experience, relatively large quantities of enzyme are used with relatively little substrate, so that the reaction proceeds rapidly to completion. The absorbance should thus be easily readable (neither too low nor too high). 

Uncompetitive inhibitors bind only to a formed ES complex. This type of inhibition, characterized by equal effects on both V and K , is rare in one-substrate reactions but may occur as a type of product inhibition in reactions with multiple substrates and products. Figure 6.10 illustrates mechanistic and plot differences between the discussed inhibitions. Table 6.1 summarizes the effect of inhibitors on Lineweaver-Burk plot parameters. Graphical methods are available for the estimation of Aj, the inhibition constant. In competitive inhibition, for example jgjjjj... 

In the same way, a new target can be reached by enhancing the scope of donor substrates. Some studies have already been done on one-substrate reactions (CJO 71 or Xyl5P 68 are then used as donor and acceptor) or on new ketoacid donors (as halogenated keto-acid ). 

---