Kinetics of Enzyme-Catalyzed Reactions

5 KINETICS OF ENZYME-CATALYZED REACTIONS 5.5.1 Units of Enzyme Activity 

The most commonly used unit of enzyme activity has been defined as the amount of activity that catalyzes the transformation of 1 (xmol of substrate per minute under specified assay conditions. Specific activity is the number of enzyme units per milligram of protein ((imol/min per milligram of protein). The turnover number or catalytic constant is equal to the units of enzyme activity per nmole of enzyme ((xmol/min per nmol of enzyme). 

A new international unit has been recommended. The katal (kat) is defined as the amount of enzyme activity that transforms 1 mole of substrate per second. Activities can also be given in millikatals (mkat), microkatals (fxkat), nanokatals (nkat), etc. Both specific activity and turnover number can also be expressed in these units. 

In this formulation, the k s are the specific rate constants for the two reversible reactions. The velocity of a reaction is equal to the rate of formation of P minus the rate of disappearance of P and is given by 

However, when the initial velocity (V0) is measured, the concentration of P will be very low so that k4 [E][P] [ES], and V0 may be approximated by the first-order relationship 

Oscillations such as in Fig. 2.15 are quite regular and can be sustained for hours if the conditions are kept the same. Depending on the feed rate of the reactants, which determines how far the system deviates from equilibrium, the oscillations may become more complex, and develop into chaotic oscillations (see, for example, P.D. Cobden, J. Siera, and B.E. Nieuwenhuys, J. Vac. Sci. Technol. A10 (1992) 2487). 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

Enzymes are highly specific catalysts in biological systems. They are proteins that consist of many amino acids coupled to each other by peptide bonds. The rather small enzyme insulin, for example, consists of 51 amino acids. The chain of amino 

Amino adds contain two adive groups, namely a carboxylic (COOH) and an amino (NH2) group. 

Twenty different amino adds are known. They combine to give proteins by forming an amide or peptide bond between the carbon from COOH and nitrogen from NH2. 

All enzymes are proteins that catalyze many biochemical reactions. They are unbranched polymers of a-amino acids of the general formula 

The catalytic action is specific and may be affected by the presence of other substances both as inhibitors and as coenzymes. Most enzymes are named in terms of the reactions they catalyze (see Chapter 1). There are three major types of enzyme reactions, namely  

The predominant activity in the study of enzymes has been in relation to biological reactions. This is because specific enzymes have bodi controlled and catalyzed syndietic and degradation reactions in all living cells. Many of diese reactions are homogeneous in the liquid phase (i.e., type 3 reactions). 

This is used in the treatment of some cancer cells, where L-asparaginase is used to remove an essential nutrient, tlius inliibiting their growth. 

The hydrogen peroxide produced in the glucose oxidase catalyzed reaction has an antibacterial action. If the presence of hydrogen peroxide is undesirable in the product, catalase is added to remove the peroxide. 

The time course of an enzymatic reaction permits one to deduce the substrate affinity, the catalytic mechanism in the active center, and the efficiency of the enzyme (maximum rate, turnover number). 

The rate of an enzyme-catalyzed single reactant reaction depends on the concentration of substrate and product, respectively  

The initial rate (vo) is determined by extrapolating the slope of the time course of the substrate or product concentration to time zero (Fig. 22). The dependence of vo on the substrate concentration, S (at constant enzyme concentration), is shown in Fig. 23. It reflects the typical substrate saturation. At first, vo increases proportionally to the amount of substrate. Upon further enhancement of substrate concentration vo levels off. The curve asymptotically approaches a maximum value, i max. When this plateau is reached, a change of S does not lead to a measurable change of vo the enzyme is saturated by substrate and has thus reached the limit of its efficiency. 

These kinetics result from the fast and reversible formation of an enzyme-substrate complex, ES, which dissociates in a second, slower reaction under liberation of the product, P  

Because the second reaction is rate-limiting, at very high substrate concentration almost all enzyme is present as enzyme-substrate complex. Under these conditions a steady state is reached in which the enzyme is steadily saturated by substrate and the initial rate is at a 

What we have seen is, that enzymes are highly specific catalysts in biological systems. Enzymes are catalytic proteins, they represent the most efficient class of catalysts. Their active site can, for example, be a carboxylic or an amino group, embedded in a specific geometry. Several weak interactions (electrostatic, H-bonds, van der Waals) help in establishing the highly specific manner in which a substrate mulecule binds to the active site. 

The kinetics of enzyme-catalyzed reactions resemble those of the heterogeneous reactions. However, because in practice there are a few characteristic differences in how the equations are handled, we will treat the encymatic case as follows. 

The enzyme, E, acts by forming a complex with the reactant, S, (commonly referred to as substrate), to give a product, P, according to the following scheme (Eqs. 4-1 and 4-2)  

Although we can easily measure the total concentration of enzyme [E],ot, it is difficult to measure the concentration of free enzyme [E]. Because enzyme, substrate and product are all in the same medimn we can conveniently work with concentrations. With the total enzyme concentration [E]tot the conservation of active species requires that 

The rate of product formation (Eq. 4-4) follows from the reaction equation (Eq. 4-2) d[P] 

Chemical reactions between biochemical compounds are enhanced by biological catalysts called enzymes, which consist mostly or entirely of globular proteins. In many cases a cofactor is needed to combine with an otherwise inactive protein to produce the catalytically active enzyme complex. The two distinct varieties of cofactors are coenzymes, which are complex organic molecules, and metal ions. Enzymes catalyze six major classes of reactions 1) Oxidoreductases (oxidation-reduction reactions), 2) Transferases (transfer of functional groups), 3) Hydrolases (hydrolysis reactions), 4) Lyases (addition to double bonds, 5) Isomerases (isomerization reactions) and 6) Ligases (formation of bonds with ATP (adenosine triphosphate) cleavage) [1]. 

This is the mechanism of catalysis by aldolases which occur in plant and animal tissues (lysine aldolases or class I aldolases). A second group of these enzymes often produced by microorganisms contains a metal ion (metallo-aldolases). This group is involved in accelerating retroaldol condensations through electrophihc reactions with carbonyl groups  

Other examples of electrophilic metal catalysis are given under section 2.3.3.1. Electrophilic reactions are also carried out by enzymes which have an a-keto acid (pyruvic acid or a-keto butyric acid) at the transforming locus of the active site. One example of such an enzyme is histidine decarboxylase in which the N-terminal amino acid residue is bound to pyruvate. Histidine decarboxylation is initiated by the formation of a Schiff base by the reaction mechanism in Fig. 2.20. 

The hypotheses discussed here allow some understanding of the fundamentals involved in the action of enzymes. However, the knowledge is far from the point where the individual or combined effects which regulate the rates of enzyme-catalyzed reactions can be calculated. 

Kinetic analysis was used to characterize enzyme-catalyzed reactions even before enzymes had been isolated in pure form. As a rule, kinetic measurements are made on purified enzymes in vitro. But the properties so determined must be referred back to the situation in vivo to ensure they are physiologically relevant. This is important because the rate of an enzymatic reaction can depend strongly on the concentrations of the substrates and products, and also on temperature, pH, and the concentrations of other molecules that activate or inhibit the enzyme. Kinetic analysis of such effects is indispensable to a comprehensive picture of an enzyme. 

Kinetic Parameters Are Determined by Measuring the Initial Reaction Velocity as a Function of the Substrate Concentration 

The usual procedure for measuring the rate of an enzymatic reaction is to mix enzyme with substrate and observe the formation of product or disappearance of substrate as soon as possible after mixing, when the substrate concentration is still close to its initial value and the product concentration is small. The measurements usually are repeated over a range of substrate concentrations to map out how the initial rate depends on concentration. Spectro-photometric techniques are used commonly in such experiments because in many cases they allow the concentration of a substrate or product in the mixture to be measured continuously as a function of time. 

Measurements of reactions that occur in less than a few seconds require special techniques to speed up the mixing of the enzyme and substrate. One way to achieve this is to place solutions containing the enzyme and the substrate in two separate syringes. A pneumatic device then is used to inject the contents of both syringes rapidly into a common chamber that resides in a spectrophotometer for measuring the course of the reaction (fig. 7.5). Such an apparatus is referred to as a stopped-flow device because the flow stops abruptly when the movement of the pneumatic driver is arrested. In this type of apparatus it is possible to make kinetic measurements within about 1 ms after mixing of enzyme and substrate. 

The stopped-flow apparatus for measuring enzyme-catalyzed reactions very soon after mixing enzyme and substrate. 

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Perez-Bendito, D. Silva, M. Kinetic Methods in Analytical Chemistry. Ellis Horwood Chichester, England, 1988. Additional information on the kinetics of enzyme catalyzed reactions maybe found in the following texts. 

Kinetics of Enzyme-Catalyzed Reactions Involving Two or More Substrates... 

For these reasons, in the experimental study of the kinetics of enzyme-catalyzed reactions, T, shear and PH are carefully controlled, the last by use of buffered solutions. In the development, examples, and problems to follow, we assume that both T and pH... 

W. W. Cleland. The kinetics of enzyme catalyzed reactions with two or more substrates or products. I. Nomenclature and rate equations. Biochim. Biopkys. Acta. 67, 104 137 (1963). 

The kinetics of enzyme-catalyzed reactions (i. e the dependence of the reaction rate on the reaction conditions) is mainly determined by the properties of the catalyst, it is therefore more complex than the kinetics of an uncatalyzed reaction (see p.22). Here we discuss these issues using the example of a simple first-order reaction (see p.22)... 

A linear graphical method for analyzing the initial rate kinetics of enzyme-catalyzed reactions. In the Hanes plot, [A]/v is plotted as a function of [A], where v is the initial rate and [A] is the substrate concentration ". ... 

There are methods used Lo study enzymes other than those of chemical instrumental analysis, such as chromatography, that have already been mentioned. Many enzymes can be crystallized, and their structure investigated by x-ray or electron diffraction methods. Studies of the kinetics of enzyme-catalyzed reactions often yield useful data, much of this work being based on the Michaelis-Menten treatment. Basic to this approach is the concept (hat the action of enzymes depends upon the formation by the enzyme and substrate molecules of a complex, which has a definite, though transient, existence, and then decomposes into the products, of the reaction. Note that this point of view was the basis of the discussion of the specilicity of the active sites discussed abuve. 

A Critical Amount of Energy Is Needed for the Reactants to Reach the Transition State Catalysts Speed up Reactions by Lowering the Free Energy of Activation Kinetics of Enzyme-Catalyzed Reactions... 

Appropriate expressions for the fluxes of each of the reactions in the system must be determined. Typically, biochemical reactions proceed through multiple-step catalytic mechanisms, as described in Chapter 4, and simulations are based on the quasi-steady state approximations for the fluxes through enzyme-catalyzed reactions. (See Section 3.1.3.2 and Chapter 4 for treatments on the kinetics of enzyme catalyzed reactions.)... 

The Journal of Chemical Education The Kinetics of Enzyme Catalyzed Reactions The Enzymes (p. 22) Volume 34, Number 1, January 1951... 

Cleland, W.W. (1963b). The kinetics of enzyme-catalyzed reactions with two or more substrates orproducts. 

If the product dissociates rapidly from the enzyme (i.e., kj, is large compared to 2 and k-2), then a simplified sequence is obtained and is the one most commonly employed to describe the kinetics of enzyme catalyzed reactions. For this case. 

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