Simple Enzyme Kinetics

The reaction rate is proportional to the substrate concentration (that is, first-order reaction) when the substrate concentration is in the low range. 

The reaction rate does not depend on the substrate concentration when the substrate concentration is high, since the reaction rate changes gradually from first order to zero order as the substrate concentration is increased. 

The maximum reaction rate rmax is proportional to the enzyme concentration within the range of the enzyme tested. 

Henri observed this behavior in 1902 (Bailey and Ollis, p. 100,1986) and proposed the rate equation 

Brown (1902) proposed that an enzyme forms a complex with its substrate. The complex then breaks down to the products and regenerates the free enzyme. The mechanism of one substrate-enzyme reaction can be expressed as 

---

In 1975, Cleland introduced the net rate constant method to simplify the treatment of simple enzyme kinetic mechanisms that do not involve branched pathways (Cleland, 1975). This method can be successfully applied to obtain rate laws for isotope exchange. Since the net rate constant method allows one to obtain Vmax/ M and Ku in terms of the individual rate constants, this method has the greatest value for fee characterization of isotope effects on VmxJKu and ATm (Huang, 1979 Punch Allison, 2000). 

The individual reaction rates (r+ at p = 0 and r at s = 0) govern simple enzyme kinetics. The kinetic parameters, among others, are taken from Equ. 5.22 using special methods for experimental verification. 

Figure 6.24. Parameter estimation from integral reactors (DCSTR and CPFR) in the case of simple enzyme kinetics (a), substrate inhibition (b), and competitive (c) and noncompetitive product inhibition (d), according to Levenspiel (1979).

The simple enzyme kinetic system does not exhibit oscillatory behaviour. The existence, uniqueness and global asymptotic stability of a periodic solution of a Michaelis-Menten mechanism was proved (Dai, 1979) for the case when the reaction occurred in a volume bounded by a membrane. The permeability of the membrane to a given species was specified as the function of another species. The form of the model is... 

In concluding this short chapter on simple enzyme kinetics, several other aspects should be mentioned. First, the influence of pH as well as other activity modulators, particularly inhibitors and poisons, can be quantitatively accounted for using the approaches introduced here that can produce Michaelis-Menten-type rate expressions. Second, the apparent rate constant from many of these rate expressions obeys an Arrhenius form over a limited temperature range, but if the temperature becomes too high (ca. 325 K), the enzymes denature (fall apart). Finally, there has been, and continues to be,... 

Lineweaver-Burk plot A double-reciprocal plot used to determine the two constants featured in simple enzyme kinetic equations such as Michaelis-Menten kinetics, Monod kinetics, and in similar adsorption isotherm models such as the Langmuir adsorption isotherm. The constants are determined from the intercept with the y-axis and the gradient (see Fig. 26). It was devised and published in 1934 by American chemist Hans Lineweaver (1907-2009) and American biochemist Dean Burk (1904-1988). 

Enzyme Kinetics. A simple en2yme cataly2ed reaction can be described ... 

Equation 11-15 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions, which involve a single substrate. The interpretation of as an equilibrium constant is not universally valid, since the assumption that the reversible reaction as a fast equilibrium process often does not apply. 

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

Considerable progress has been made within the last decade in elucidating the effects of the microenvironment (such as electric charge, dielectric constant and lipophilic or hydrophilic nature) and of external and internal diffusion on the kinetics of immobilized enzymes (7). Taking these factors into consideration, quantitative expressions have been derived for the kinetic behavior of relatively simple enzyme systems. In all of these derivations the immobilized enzymes were treated as simple heterogeneous catalysts. 

The relationship between substrate concentration ([S]) and reaction velocity (v, equivalent to the degree of binding of substrate to the active site) is, in the absence of cooperativity, usually hyperbolic in nature, with binding behavior complying with the law of mass action. However, the equation describing the hyperbolic relationship between v and [S] can be simple or complex, depending on the enzyme, the identity of the substrate, and the reaction conditions. Quantitative analyses of these v versus [S] relationships are referred to as enzyme kinetics. 

A. The rate of the simple enzyme-catalyzed reaction shown in the equation below can be described by Michaelis-Menten kinetics. 

In Box 12.2, a simple model for a special kind of catalyzed reaction, the Michaelis-Menten enzyme kinetics, is presented, which leads to the following kinetic expression ... 

Handling rate equations for complex mechanisms. While steady-state rate equations can be derived easily for the simple cases discussed in the preceding sections, enzymes are often considerably more complex and the derivation of the correct rate equations can be extremely tedious. The topological theory of graphs, widely used in analysis of electrical networks, has been applied to both steady-state and nonsteady-state enzyme kinetics 45-50 The method employs diagrams of the type shown in Eq. 9-50. Here... 

The constant Km defined in equation (23) is called the Michaelis constant and is one of the key parameters in enzyme kinetics. It is a simple matter to proceed from this point to an expression comparable to the Henri-Michaelis-Menten equation (18), but with Km in place of Ks. First, rearranging equation (23) gives... 

This mechanism is important for compounds that lack sufficient lipid solubility to move rapidly across the membrane by simple diffusion. A membrane-associated protein is usually involved, specificity, competitive inhibition, and the saturation phenomenon and their kinetics are best described by Michaelis-Menton enzyme kinetic models. Membrane penetration by this mechanism is more rapid than simple diffusion and, in the case of active transport, may proceed beyond the point where concentrations are equal on both... 

For a detailed review of simple to complex enzyme kinetics, a book by Segel (21) is recommended. Most P450 oxidations show hyperbolic saturation kinetics and competitive inhibition between substrates. Therefore, both Km values and drug interactions can be predicted from inhibition studies. Competitive inhibition suggests that the enzymes have a single binding site and only one substrate can bind at any one time. For the inhibition of substrate A by substrate B to be competitive, the following must be observed ... 

---