Rate Equations in the Absence of Products

In this section, we shall reviewthe rate equations forthe majortypes of trisubstrate mechanisms, written in the absence of products (Cleland, 1963 Plowman, 1972 Fromm, 1975,1979). All trisubstrate mechanisms in the rapid equilibrium category are relatively rare and the steady-state mechanisms are more common. However, the derivation of rate equations for rapid equilibrium mechanisms, in the absence of products, is less demanding, as it requires only the rapid equilibrium assumptions and, therefore, the resulting rate equations are relatively simple. 

In Sections 12.2.1 and 12.2.2, we shall divide the rapid equilibrium trisubstrate mechanisms into the following major t)q)es  

Completely random Stricfly ordered Random A-5, Ordered C Ordered A, Random B-C, and Random A-C, Ordered B. 

Much more realistic are the steady-stale trisubstrale mechanisms, that occur very frequently. In the Ter Bi Mechanisms Section, we shall develop the rate equations, in the absence of all products of reaction, for major Ter Bi mechanisms  

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Rate equations in the absence of products. When both P and Q are zero, we obtain a usual Michaelis-Menten equation Uo — + -A). 

Rate equation in the absence of product. If P reacts with Q, and A = 0, the rate equation in the reverse direction is... 

Initial rate equation, in the absence of products P and Q, is given by... 

Isomerization of a stable enzyme form does not affect the algebraic form of the rate equation in the absence of products, but product inhibition patterns are modified so that the order of addition of substrates and release of products can not be determined by steady-state kinetic experiments. Rate constants for steps involving the isomerizing stable form or any central complex are not determinable, and steady-state distributions can be calculated only for non-isomerizing... 

The rate equation is obtained in the following manner. In the rate equation in the absence of products (Eq. (9.15)), those terms in the denominator representing free enzyme are multiplied by ii+B/KO, where Jf is the dissociation constant of the EB complex. The terms which represent the free enzyme are found in distribution equations for this system in the absence of products, these terms are JCiAlifB and KaB (found in Eq. (9.13)). Thus, the velocity equation becomes... 

The general rate equations for the steady-state sequential mechanisms, in the presence of products, are developed in the following sections. The rate equations in the absence of products can be written down directly from the general rate equations, simply by omitting the terms in the denominator and the numerator that contain the concentration terms for products P, Q, and R, and eliminating the JKeq with the aid of Haldane relationships. In all trisubstrate systems, the inhibition constants will always represent tme dissociation constants. 

Although each trisubstrate mechanism has a unique general rate equation, the rate equations in the absence of the products of reaction sometimes have identical forms for several Ter Bi and Ter Ter mechanisms. Therefore, in order to identify unequivocally the mechanism, we must revert to the product inhibition analysis and the use of dead-end inhibitors. 

An enzyme reaction scheme in which there are two substrates (A and B) and three products (P, Q, and R) and in which the substrates bind and the products are released in an ordered fashion. This reaction scheme is exemphfied by the malic enzyme The initial rate expression, in the absence of abortive complexes and products, is identical to the corresponding equation for the ordered Bi Bi mechanism. See Multisubstrate Mechanisms Ordered Bi Bi Mechanism... 

In that work, the kinetic equation obtained for the whole range of compositions is based on a LHHW mechanism in which IB and MeOH, both adsorbed on the resin, react to form MTBE. The rate-controlling step is surface reaction. As a simplification in the activities calculation, the authors consider that the reaction mixture is composed by three compounds methanol, MTBE and a C4 pseudo-compound that includes all hydrocarbons. In the absence of product, the proposed kinetic equation is... 

The Michaelis-Menten equation (8.8) and the irreversible Uni Uni kinetic scheme (Scheme 8.1) are only really applicable to an irreversible biocatalytic process involving a single substrate interacting with a biocatalyst that comprises a single catalytic site. Hence with reference to the biocatalyst examples given in Section 8.1, Equation (8.8), the Uni Uni kinetic scheme is only really directly applicable to the steady state kinetic analysis of TIM biocatalysis (Figure 8.1, Table 8.1). Furthermore, even this statement is only valid with the proviso that all biocatalytic initial rate values are determined in the absence of product. Similarly, the Uni Uni kinetic schemes for competitive, uncompetitive and non-competitive inhibition are only really applicable directly for the steady state kinetic analysis for the inhibition of TIM (Table 8.1). Therefore, why are Equation (8.8) and the irreversible Uni Uni kinetic scheme apparently used so widely for the steady state analysis of many different biocatalytic processes A main reason for this is that Equation (8.8) is simple to use and measured k t and Km parameters can be easily interpreted. There is only a necessity to adapt catalysis conditions such that... 

Thus, the term in the denominator represents the EA complex, and we must multiply the JTbA term in the rate Eq. (9.15) (Chapter 9) with Accordingly, in the absence of products, the rate equation in the presence of I becomes... 

Even in the absence of product P, the rate equation is complex, a rational polynomial of the order 2 2 with respect to both substrates. The reciprocal plots are nonlinear, but the departure from linearity may be very difficult to detect if both routes to EAB are about equally favorable. 

The general form of the rate equation for aU tiisubstrate mechanisms, in the absence of products and assuming linear reciprocal plots, is... 

The rate Eq. (12.83) can be written down directly from the general rate equation for the Ordered Ter Ter mechanism (Table 4), simply by omitting all the concentration terms that contain P and R and eliminating the Keq with the aid of Haldane relationships. Equation (12.83) is identical with the rate Eq. (12.26) for the Ordered Ter Ter mechanism, in the absence of products, except that a new term, 4>ABCQ, is added in the denominator ... 

Initially, the reaction proceeds forward in the absence of product. The rate of the reaction at this stage (v=v ) depends only on the substrate concentration and follows the Michaelis-Menten equation. 

Lipase-catalyzed reactions do not necessarily require water as the second reactant. An immobilized enzyme is often stable in anhydrous media (Sharma et al 2001) and can use different alcohols as the secondary acceptors of fatty acids. The concentration of alcohol is relatively low and cannot be treated as a constant. Consequently, the reaction obeys the bisubstrate ping-pong mechanism (Cheirsilp et al 2008 Mitchell et al., 2008 Xiong et al., 2008) as shown in Figure 14.3. The corresponding equation of the reaction rate in the absence of products is ... 

Consider first the case with v = k[A][P], where [P] represents the concentration of a product. Taken literally, this equation implies that the reaction will not start in the absence of P. With an initially preset concentration of P designated as [P]o, the rate law becomes... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

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