Enzymatic reactions expressing

Chapter 10 begins a more detailed treatment of heterogeneous reactors. This chapter continues the use of pseudohomogeneous models for steady-state, packed-bed reactors, but derives expressions for the reaction rate that reflect the underlying kinetics of surface-catalyzed reactions. The kinetic models are site-competition models that apply to a variety of catalytic systems, including the enzymatic reactions treated in Chapter 12. Here in Chapter 10, the example system is a solid-catalyzed gas reaction that is typical of the traditional chemical industry. A few important examples are listed here ... 

Second-order enzymatic reactions require two adsorption events at the same site. For the reaction A + B — P, there may be a compulsory order of adsorption (e.g., first A, then B) or the two reactants may adsorb in a random order. Different assumptions lead to slightly different kinetic expressions, but a general form with theoretical underpinnings is... 

This is the Michaelis-Menten expression (which dates back to 1913) for the rate of an enzymatic reaction. 

We begin our discussion with path integral quantum transition state theory (QTST) [14], which is the theoretical model that we use to model enzymatic reactions. In QTST, the exact rate constant is expressed by the QTST rate constant, qtst, multiplied by a transmission coefficient yq ... 

Rate Expressions for Enzyme Catalyzed Single-Substrate Reactions. The vast majority of the reactions catalyzed by enzymes are believed to involve a series of bimolecular or unimolecular steps. The simplest type of enzymatic reaction involves only a single reactant or substrate. The substrate forms an unstable complex with the enzyme, which subsequently undergoes decomposition to release the product species or to regenerate the substrate. 

Reaction rate expressions for enzymatic reactions are usually derived by making the Bo-denstein steady-state approximation for the intermediate enzyme-substrate complexes. This is an appropriate assumption when the substrate concentration greatly exceeds that of the enzyme (the usual laboratory situation) or when there is both a continuous supply of reactant and a continuous removal of products (the usual cellular situation). 

As in the homogeneous case, expression of the plateau current in equation (5.20) gives a simple representation of the competition between substrate and cosubstrate in the kinetic control of the enzymatic reaction. Equation (5.19) suggests the construction of primary and secondary plots allowing the derivation of the kinetic constants, as will be shown in the next section. 

Figure 1.1 illustrates a condensed version of the classical pathway of bile-acid synthesis, a series of 12 enzymatic reactions that convert cholesterol, which is insoluble, into BAs, which are water soluble. The cholesterol is first converted to 7 alpha-hydroxy cholesterol, followed by the series of enzymatic transformations, eventually producing cholic and chenodeoxycholic acids (not all steps shown). The rate-limiting enzyme in this pathway is cholesterol 7 alpha-hydroxylase (CYP 7A1), which originates from microsomal cytochrome P-450 enzymes, expressed only in the liver hepatocytes. 

Transporters recognize and bind the molecules to be transported and help them to pass through the membrane as a result of a conformational change. These proteins (permeases) are thus comparable with enzymes—although with the difference that they catalyze vectorial transport rather than an enzymatic reaction. Like enzymes, they show a certain affinity for each molecule transported (expressed as the dissociation constant, in mol L ) and a maximum transport capacity (V). 

Expression of the catalytic capacity of the immobilized laccase was also observed in more than a dozen different solvents, provided that they were either saturated with water or, in the case of solvents miscible with water, small amounts of water had been added (Table III). No enzymatic reaction was observed when the solvents tested were free of water. No correlation was found between the activity of the immobilized laccase and the hydrophobicity of the solvent in which the reaction took place. The rate of laccase reaction in ethylacetate was only twice that in toluene, despite the fact that water-saturated ethylacetate contains 50 times more water than... 

When choosing reaction conditions, such as water activity and solvent, for an enzymatic reaction, possible effects on the equilibrium position of the reaction should be considered. When the aim is to produce an equilibrium mixture as the final product, the position of this equilibrium is of course of vital importance. It is, however, also important that the biocatalyst expresses sufficient catalytic activity under the conditions used, so that equilibrium is reached within a reasonable time. In practice, it often happens that a compromise must be made between high reaction rate and high equilibrium conversion. 

Selectivity is an intrinsic properly of enzymatic catalysis. [3] Following the nomenclature proposed by Cleland [24, 25], the pseudo second-order rate constant for the reaction of a substrate with an enzyme, kml/KM, is known as the specificity constant, ksp. [26] To express the relative rates of competing enzymatic reactions, involving any type of substrates, the ratio of the specificity constants appears to be the parameter of choice [3]. Since the authoritative proposition by Sih and coworkers [27], the ratio of specificity constants for the catalytic conversion of enantiomeric substrates, R and S, is commonly known as the enantiomeric ratio or E -value (Equation 1) ... 

Let s assume that the rate constant kcat for the formation of products on either subunit is the same, whether only that site or both catalytic sites are occupied. Suppose also that ES, SE, and SES are in equilibrium with the free enzyme and substrate. By following the same procedure that led to the Henri-Michaelis-Menten equation in chapter 7, we can derive an expression for the rate of the enzymatic reaction in terms of [S], AT], and K2. Here we just give the result. 

The inhibition effect of poly (vinyl alcohol) on the amylose hydrolysis was investigated. Figure 7 shows Lineweaver-Burk plots of the amylose hydrolysis rates catalyzed by the random copolymer in the presence of poly (vinyl alcohol). The reaction rate is found to decrease with increasing the concentration of poly (vinyl alcohol), and all of the straight lines obtained in the plots cross with each other at a point on the ordinate. This is a feature of the competitive inhibition in the enzymatic reactions. In the present reaction system, however, it is inferred to suggest that the copolymer and poly (vinyl alcohol) molecules competitively absorb the substrate molecules. The elementary reaction can be described in the most simplified form as in Equation 3 where Z, SI, and Kj[ are inhibitor, nonproductive complex, and inhibitor constant, respectively. Then the reaction rate is expressed with Equation 4. 

For expressing enzymatic reactions, the sequence is written from left to right with a horizontal line or group of lines representing the enzyme in its various forms. Substrate additions and product releases are indicated by the downward ( ) and upward ( ) vertical arrows, respectively. 

For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively (Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1 1 function in both A and B (i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2 2 function (i.e., second order in both A and B). The 2 2 function rate equation results in nonlinear kinetics that should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966 Tsai, 1978). 

Payens (1976, 1977) realized that in order to derive an expression for the clotting time of milk, a kinetic description of the enzyme reaction must be combined with the kinetics of aggregation of destabilized colloidal particles. In early versions of his theory, the enzymatic reaction was described by Michaelis—Menten kinetics with the... 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate complexes derives reaction rate expressions for enzymatic reactions. A possible mechanism of a closed sequence reaction is ... 

In carrying out an enzyme assay it may be convenient to introduce an auxiliary enzyme to the system to affect the removal of a product produced by the first enzymatic reaction. The kinetics of these coupled enzyme assays can be expressed by ... 

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