Cleland-form rate equation

Another mechanism that may apply to a two substrate reaction is the ordered ternary-complex mechanism. For example, the complex EAB can be formed from EA by addition of B, but not from EB by addition of A the substrates must become attached in a particular order. This mechanism is represented in Figure 10.5, the second representation being that of Cleland. This mechanism also leads to a rate equation of the form of (10.26), but the significance of the constants is different. 

Unfortunately, the form of Equation (8.53) is a little way off the form of the Michaelis-Menten equation. For this reason, the King-Altman approach is usually supplemented by an approach developed by Cleland. The Cleland approach seeks to group kinetic rate constants together into numbers (num), coefficients (Coef) and constants (const) that themselves can be collectively defined as experimental steady-state kinetic parameters equivalent to fccat> Umax and fCm of the original Michaelis-Menten equation. After such substitutions, the result is that equations may be algebraically manipulated to reproduce the form of the Michaelis-Menten equation (8.8). Use of the Cleland approach is illustrated as follows. 

The reversible reaction involves an interconversion of the two enzyme complexes, namely the enzyme-substrate (EA) and the enzyme-product (EP) complexes. Therefore the steady-state assumption considers the steady-state concentrations of these two complexes, dEA/dt = 0 and dEP/dt = 0. The kinetic treatment gives a rate equation in the Cleland form ... 

The quasi-equilibrium treatment yields an identical rate equation if represented in the Cleland form. However, the quasi-equilibrium condition assumes ks and Ict being rate-limiting, which yields different expressions for the maximum velocities, Vi = ksEt, V2 = IctEt and Michaelis-Menten constants, Ka = ka/ki, Kp = kg/ks respectively. 

Notes 1. Vi, Ka, Kb and Kia are maximum velocity, Michaelis-Menten constants for A and B, and inhibition constant for A respectively. 2. The Order bi bi reactions may display zero (known as Theorell-Chance mechanism), one and two ternary complexes. All of them give the rate equation in the same Cleland form. 

Further division of reversible inhibitors is made according to their influence on the form of rate equations thus, we can make a difference between the linear and a nonlinear inhibition. In Chapter 5, we shall describe the linear inhibition and in Chapter 6 the main forms of the nonlinear inhibitioa Thus, we could distinguish tfie various types of inhibition stiU further by referring to competitive inhibition as linear, hyperbolic, or paraboUc inhibition. Fven more complex forms are possible (Cleland, 1970). 

The kinetics of substrates, products, and alternate products define the form of the rate equation, and are certainly necessary to deduce the kinetic mechanism. However, they are often not sufficient to do this unequivocally and other kinetic approaches are necessary, especially when reaction can be studied in only one direction. One of the most useful approaches in such cases involves the use of dead-end inhibitors (Cleland, 1970, 1979, 1990). 

The majority of steady-state rate equations described in this chapter, in its present form, have been derived by Cleland (Cleland, 1963,1967,1970,1977). 

The various Ter Ter mechanisms differ only by the composition of their denominators. Thus, Eq. (12.61) can be expanded into the general rate equation in rate constant form for each Ter Ter mechanism in turn, taking the appropriate denominator terms from Table 4 (Cleland, 1963). 

The Michaelis-Menten Formalism did not anticipate the type of enzyme-enzyme organization described above. One of its fundamental assumptions has been that complexes do not occur between different forms of an enzyme or between different enzymes (Segal, 1959 Webb, 1963 Cleland, 1970 Segel, 1975 Wong, 1975). From the derivation of the classical Michaelis-Menten rate law, it can be seen that such complexes must be excluded or they will destroy the linear structure of the kinetic equations. 

Let us compare the rate laws for the two reversible Michaelis-Menten mechanisms ((3.20) and (3.31)), that is, Eq. (3.24) with Eq. (3.36). We can rewrite both equations in a more general form, using the kinetic coefficients in a manner of Cleland (1963) ... 

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