Initial rate equations

For the reactions, A+B Products, of Tables 6.1 and 6.2, develop initial rate equations and plots and use them to identify possible valid mechanisms. 

This concentration remains practically the same over the full range of HC1 concentrations. Rearrange the initial rate equation into a linear form. 

The derivation of initial velocity equations invariably entails certain assumptions. In fact, these assumptions are often conditions that must be fulfilled for the equations to be valid. Initial velocity is defined as the reaction rate at the early phase of enzymic catalysis during which the formation of product is linear with respect to time. This linear phase is achieved when the enzyme and substrate intermediates reach a steady state or quasi-equilibrium. Other assumptions basic to the derivation of initial rate equations are as follows ... 

Symbol for the dissociation constant of an inhibitor with respect to a particular form of the enzyme. This dissociation constant is associated with the intercept term in the double-reciprocal form of the initial-rate equation. For example, consider an inhibitor that can bind to either the free enzyme, E, or the binary central complex, EX, of a Uni Uni mechanism. Ka would be the dissociation constant for the EX -t 1 EXl step and is equal to [EX][1]/[EX1]. The binding of 1 to the free enzyme (i.e., E -t 1 El) is governed by Kis (equal to [E][1]/[E1]). 

Symbol for the dissociation constant of an inhibitor with respect to the free enzyme, E. In the double-reciprocal form of the initial-rate equation, this term is associated with the slope portion of the expression. See Ka... 

Rapid Equilibrium Mechanism. If the rate-determining step is the catalytic step and all binding steps can be described by dissociation constants (e.g., K = [E][A]/ [EA]), then, in the absense of products i.e., [P] and [Q] 0), the initial rate equation for the rapid equihbrium Uni Bi mechanism is identical to that of the Uni Uni... 

Both schemes yield the same generalized steady-state initial rate equation ... 

ENZYME RATE EQUATIONS (2. Derivation of Initial Rate Equations)... 

Using the method of graphs, write the initial rate equation for the following system with A, B, P, and Q present. 

For example, in the acylation of veratrole with benzoic anydride,[14] following this mechanism, we assume that the veratrole chemisorption reduces the number of acid sites available for benzoic anydride, but that the reaction does not proceed between the two adsorbed species. Such an assumption leads to the corresponding initial rate equation as follows ... 

The kinetic model of styrene auto-initiation proposed by Hui and Hameilec [27] was used as a starting point for this work. The Mayo initiation mechanism was assumed (Figure 7.2) but the acid reaction was of course omitted. After invoking the quasi-steady-state assumption (QSSA) to approximate the reactive dimer concentration, Hui and Hameilec used different simplifying assumptions to derive initiation rate equations that are second and third order in monomer concentration. 

Integration of the initial-rate equation gives Gj as a function of time... 

Equation (A) for surface reaction controlling shows that the initial rate will be proportional to the square of the pressure at low pressures and will approach a constant value at high pressures. This type of relation is shown in Fig. 9-3n. The case for adsorption controlling is indicated in Fig. 9-3 >, and that for desorption controlling is shown in Fig. 9-3c. If the equilibrium constant were not very large for case (c), the initial rate equation would be as shown in Fig. 9-3a. 

B. Initial Rate Equations for Ordered and Random Mechanismi... 

A. Phenomenological Initial Rate Equations fob Two-Substrate and Thbee-Substbate Reactions (16)... 

Four mechanisms for the reaction A + B =tP + Q [A = NAD(P), P = NAD(P)H] consistent with initial rate measurements that conform to Eq. (1) will be described. Examination of the initial rate equations will indicate various ways in which the mechanisms may be distinguished and tested experimentally, and the results of such studies will be discussed in subsequent sections of this chapter. 

The simplest mechanism that gives an initial rate equation of the form of Eq. (1) is... 

The initial rate equation derived by steady-state analysis is Eq. (1) with the kinetic coefficients for the reaction from left to right as defined in Table I. (The kinetic coefficients for the reverse reaction, o, 0p, etc., are obtained by addition or deletion of primes on the rate constants.) The characteristic features of this mechanism 6) are that the individual velocity constants for the formation and dissociation of the enzyme-coenzyme compounds can be calculated from experimental values for the kinetic coefficients ... 

The initial rate equation is again of the form of Eq. (1) with the kinetic coefficients as in Table I, which shows that the mechanism differs from the simple ordered mechanism in three important respects. First, the isomerization steps are potentially rate-limiting evidence for such a rate-limiting step not attributable to product dissociation or the hydride-transfer step (fc) has been put forward for pig heart lactate dehydrogenase 25). Second, Eqs. (5) and (6) no longer apply in each case the function of kinetic coefficients will be smaller than the individual velocity constant (Table I). Third, because < ab/ a< b is smaller than it may also be smaller than the maximum specific rate of the reverse reaction that is, one of the maximum rate relations in Eq. (7) need not hold 26). This mechanism was in fact first suggested to account for anomalous maximum rate relations obtained with dehydrogenases for which there was other evidence for an ordered mechanism 27-29). 

The initial rate equation derived by steady-state analysis is of the second degree in A and B (SO). It simplifies to the form of Eq. (1) if the rates of dissociation of substrates and products from the complexes are assumed to be fast compared with the rates of interconversion of the ternary complexes k, k )] thus, the steady-state concentrations of the complexes approximate to their equilibrium concentrations, as was first shown by Haldane (14)- The kinetic coefficients for this rapid equilibrium random mechanism (Table I), together with the thermodynamic relations KeaKeab — KebKeba and KepKepq — KeqKeqp, suffice for the calculation of k, k and all the dissociation constants Kea = k-i/ki, Keab = k-i/ki, etc. 

The initial rate equation is discussed in detail elsewhere (39,60), and a qualitative examination of the mechanism will suffice here to indicate its main features. If < A-i, fc-2, and k, substrate inhibition will occur at sufficiently large concentrations of B, and will be most pronounced when A is saturating. A large steady-state concentration of EP (k-i < k, k-2) will favor substrate inhibition. If k-t > fc i, substrate activation can occur but only if EP dissociation is the rate-limiting step. (Theorell-Chance mechanism). If k-e = 0, EPB is a dead-end complex, and only 00 in Eq. (1) will be affected by the inhibition which will therefore be uncompetitive with respect to A and complete at high concentration of... 

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