Some Simple Enzyme Kinetics

Although the mechanisms may be complicated and varied, some simple equations can often describe the reaction kinetics of common enzymatic reac tions qiiite well. Each enzyme molecule is considered to have an active site that must first encounter the substrate (reactant) to form a complex so that the enzyme can function. Accordingly, the following reaction scheme is written ... 

Obtaining quasi-steady approximations for fluxes through reaction mechanisms, including mechanisms more complex than the simple Michaelis-Menten system studied in this section, is a major component of the study of enzyme kinetics. This topic will be treated in some detail in Chapter 4. 

In the following sections the extension of Eq. (18) to more complex reaction schemes is described. Again the rapid equilibrium assumption is used to show how more complex rate equations are derived from simple Michaelis-Menten kinetics. Attention is focused on some typical rate equations that are useful to describe enzyme kinetics with respect to a desired process optimization. The whole complexity of enzyme kinetics is of importance for a basic understanding of the enzyme mechanism, but it is not necessary for the fitting of kinetic data and the calculation of reactor performance. 

Figure 7-3 shows that glucose uptake by erythrocytes and liver cells exhibits kinetics characteristic of a simple enzyme-catalyzed reaction involving a single substrate. The kinetics of transport reactions mediated by other types of proteins are more complicated than for uniporters. Nonetheless, all protein-assisted transport reactions occur faster than allowed by passive diffusion, are substrate-specific as reflected in lower Kjn values for some substrates than others, and exhibit a maximal rate (Vjjjax)-... 

Although computer software is now readily available to fit enzyme kinetic data to the Michaelis-Menten and related equations, it can be instructive to use simple graphical methods in some cases. The most convenient of these (though not necessarily the most accurate) are based on doublereciprocal methods that convert the hyperbolic rate equations into much simpler linear forms for plotting. 

Kinetic analysis of reactions catalyzed by enzymes is a difficult subject. Flowever, many systems can be represented by rather simple kinetic models that have been successfully appUed by many workers. While we will not treat some of the more esoteric and advanced topics associated with enzyme kinetics, a knowledge of the basic concepts is necessary for students in chemical kinetics and biochemistry. We will now describe these concepts in sufficient detail to provide a basis for further study of this important field. 

This result is to be expected on the analogy between transport kinetics and enzyme kinetics. It is a well-known result in this latter discipline [4] that the introduction of intermediate forms in a kinetic scheme will not affect the steady-state predictions of that scheme, if these intermediate forms are not able to combine with a substrate or product species (or some modifier). Now, the transition between ES, and ES2 in Fig. 5 is just such a transition between forms which do not combine with substrate or product, and hence this step cannot be seen by steady-state methods. The simple pore of Fig. 4 is thus kineticaly equivalent at the steady-state level to the more complex pore of Fig. 5 and indeed to any more complex pore involving an indefinite number of such intermediate transitional forms between ES, and ESj. 

The studies of the kinetics of bioelectrocatalytic transformations show that in some systems (for instance, adsorbed laccase ) the kinetic parameters correspond to the phenomenology of electrochemical kinetics, while in other systems (for instance, lactate oxidation they fit the phenomenology of enzymatic catalysis. In the latter case, we observe a hyperbolic dependence of anode current on the substrate concentration, as expected from the Michaelis-Menten equation. The absence of a general theory of bioelectrocatalysis does not permit us to examine the kinetics of electrochemical reactions in the presence of enzymes under different conditions. At present we can only try to estimate the scope of possible accelerations of electrochemical reactions by making some simple assumptions. 

The activity of majority of enzymes is regulated by the concentration of their own substrates in a MichaeUs-Menten fashion. However, such a control may be insufficient for some metabolic purposes. For example, in order to increase the velocity of a simple Michaelian enzyme from o.iV to 0.9V, it is necessary to increase the substrate concentration from Xm/9 to that is, an 81-fold increase. Similarly, an 81-fold increase in inhibitor concentration is required to reduce the velocity from 90% to 10% of the uninhibited value. On one hand, the concentration of metabolites in vivo vary within relatively narrow limits wltile, on the other hand, the activity of specific enzymes must be increased or decreased within very large limits. Consequently, in addition to a simple Michaelian kinetics, nature has a need for additional control mechanisms for the regulation of enzyme activity in vivo. 

The enzyme catalyzing the condensation of two molecules of 5-AL to one of PBG (Fig. 11) is widely distributed in animal tissues (72-75), and the use of 5-AL in porphyrin biosynthesis has been shown in spinach (74), yeast (74), and bacteria (59, 74, 76). The preparation from ox liver (74) and that from rabbit reticulocytes (75) have been considerably purified and show very similar characteristics. The enzyme from ox liver is rather specific for its substrate. 2,5-Diamino-4-ketopentanoic acid, 6-amino-5-ketohexanoic acid, and aminoacetone form no pyrrole with this enzyme, and are only weakly inhibitory if at all. The rabbit enzyme does not act on amino acetone, nor does it seem to form a mixed pyrrole in the presence of both 5-AL and aminoacetone. The kinetics of this enzyme are of interest since it is a rare case of an enzyme catalyzing a reaction between two identical. substrate molecules. Both the rabbit and ox liver enzymes follow simple Michaelis kinetics over the range 10 to 2 X M substrate concentration. The simplest interpretation of these data, together with some related chemical evidence (75), is that the enzyme binds both molecules of substrate specifically, the first more tightly than the second. In this respect it resembles most the enzyme-coenzyme substrate reactions. The reaction... 

For the sake of completeness, let us start by recalling some basic equations governing enzyme kinetics. A simple mechanism consistent with experimentally observed kinetic data is the Michaelis-Menten relation ° ... 

In nature, aminotransferases participate in a number of metabolic pathways [4[. They catalyze the transfer of an amino group originating from an amino acid donor to a 2-ketoacid acceptor by a simple mechanism. First, an amino group from the donor is transferred to the cofactor pyridoxal phosphate with formation of a 2-keto add and an enzyme-bound pyridoxamine phosphate intermediate. Second, this intermediate transfers the amino group to the 2-keto add acceptor. The readion is reversible, shows ping-pong kinetics, and has been used industrially in the production ofamino acids [69]. It can be driven in one direction by the appropriate choice of conditions (e.g. substrate concentration). Some of the aminotransferases accept simple amines instead of amino acids as amine donors, and highly enantioselective cases have been reported [70]. 

The order of reaction is not always integral and sometimes the rate cannot be expressed in a simple form. For example, some enzyme-catalysed reactions obey Michaelis—Menten kinetics and the rate is given by... 

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