Enzyme-catalyzed reactions, equilibrium

Isopentenyl pyrophosphate undergoes an enzyme catalyzed reaction that converts It m an equilibrium process to 3 methyl 2 butenyl pyrophosphate (dimethylallyl pyrophosphate)... 

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. 

A certain enzyme-catalyzed reaction in a biochemical cycle has an equilibrium constant that is 10 times the equilibrium constant of the next step in the cycle. If the standard Gibbs free energy of the first reaction is —200. k -mol 1, what is the standard Gihhs free energy of the second reaction ... 

In the last decade there were many papers published on the study of enzyme catalyzed reactions performed in so-called chromatographic reactors. The attractive feature of such systems is that during the course of the reaction the compounds are already separated, which can drive the reaction beyond the thermodynamic equilibrium as well as remove putative inhibitors. In this chapter, an overview of such chromatographic bioreactor systems is given. Besides, some immobilization techniques to improve enzyme activity are discussed together with modern chromatographic supports with improved hydrodynamic characteristics to be used in this context. 

Quantitative measurements of simple and enzyme-catalyzed reaction rates were under way by the 1850s. In that year Wilhelmy derived first order equations for acid-catalyzed hydrolysis of sucrose which he could follow by the inversion of rotation of plane polarized light. Berthellot (1862) derived second-order equations for the rates of ester formation and, shortly after, Harcourt observed that rates of reaction doubled for each 10 °C rise in temperature. Guldberg and Waage (1864-67) demonstrated that the equilibrium of the reaction was affected by the concentration ) of the reacting substance(s). By 1877 Arrhenius had derived the definition of the equilbrium constant for a reaction from the rate constants of the forward and backward reactions. Ostwald in 1884 showed that sucrose and ester hydrolyses were affected by H+ concentration (pH). 

Further experiments by Brown and particularly Henri were made with invertase. At that time the pH of the reactions was not controlled, mutarotation did not proceed to completion, and it is no longer possible to identify how much enzyme was used (Segal, 1959). Nevertheless, in a critical review of kinetic studies with invertase, Henri concluded (1903) that the rate of reaction was proportional to the amount of enzyme. He also stated that the equilibrium of the enzyme-catalyzed reaction was unaffected by the presence of the catalyst, whose concentration remained unchanged even after 10 hours of activity. When the concentration of the substrate [S] was sufficiently high the velocity became independent of [S]. Henri derived an equation relating the observed initial velocity of the reaction, Vq, to the initial concentration of the substrate, [S0], the equilibrium constant for the formation of an enzyme-substrate complex, Ks, and the rate of formation of the products, ky... 

The protocol described in Section 7.1.2 involves isotopic competition, but with the different isotopomers held in separate containers. Equations 7.10 to 7.13 apply equally well to a type of competition experiment known in biochemistry as the perturbation method for determining KIE s of reversible enzyme catalyzed reactions. The perturbation method differs from simultaneous non-competitive measurements in several important ways. One begins by mixing equilibrium concentrations of substrate and product but with one component (substrate or product) at a different isotopic composition than the other. Thus, the mixture is in chemical, but not isotopic equilibrium. At this stage no enzyme is present and the interconversion is... 

Cook, P.F., Blanchard, J.S. and Cleland, W.W. (1980). Primary and secondary deuterium isotope effects on equilibrium constants for enzyme-catalyzed reactions. Biochemistry 19, 4853-4858... 

Alberty analyzed the anion effect on pH-rate data. He first considered a one-substrate, one-product enzyme-catalyzed reaction in which all binding interactions were rapid equilibrium phenomena. He obtained rate expressions for effects on F ax and thereby demonstrating how an anion might alter a pH-rate profile. He also considered how anions may act as competitive inhibitors. The effect of anions on alcohol dehydrogenase has also been investigated. Chloride ions appear to affect the on- and off-rate constants for NAD and NADH binding. See also pH Studies Activation Optimum pH... 

Finally, yet another issue enters into the interpretation of nonlinear Arrhenius plots of enzyme-catalyzed reactions. As is seen in the examples above, one typically plots In y ax (or. In kcat) versus the reciprocal absolute temperature. This protocol is certainly valid for rapid equilibrium enzymes whose rate-determining step does not change throughout the temperature range studied (and, in addition, remains rapid equilibrium throughout this range). However, for steady-state enzymes, other factors can influence the interpretation of the nonlinear data. For example, for an ordered two-substrate, two-product reaction, kcat is equal to kskjl ks + k ) in which ks and k are the off-rate constants for the two products. If these two rate constants have a different temperature dependency (e.g., ks > ky at one temperature but not at another temperature), then a nonlinear Arrhenius plot may result. See Arrhenius Equation Owl Transition-State Theory van t Hoff Relationship... 

An enzyme-catalyzed reaction involving two substrates and one product. There are two basic Bi Uni mechanisms (not considering reactions containing abortive complexes or those catagorized as Iso mechanisms). These mechanisms are the ordered Bi Uni scheme, in which the two substrates bind in a specific order, and the random Bi Uni mechanism, in which either substrate can bind first. Each of these mechanisms can be either rapid equilibrium or steady-state systems. 

Many of the 60 known reactions catalyzed by monoclonal antibodies involve kinetically favored reactions e.g., ester hydrolysis), but abzymes can also speed up kinetically disfavored reactions. Stewart and Benkovic apphed transition-state theory to analyze the scope and limitations of antibody catalysis quantitatively. They found the observed rate accelerations can be predicted from the ratio of equilibrium binding constants of the reaction substrate and the transition-state analogue used to raise the antibody. This approach permitted them to rationalize product selectivity displayed in antibody catalysis of disfavored reactions, to predict the degree of rate acceleration that catalytic antibodies may ultimately afford, and to highlight some differences between the way that they and enzymes catalyze reactions. 

THE RAPID-EQUILIBRIUM TREATMENT. The first rate equation for an enzyme-catalyzed reaction was derived by Henri and by Michaelis and Menten, based on the rapid-equilibrium concept. With this treatment it is assumed that there is a slow catalytic conversion step and the combination and dissociation of enzyme and substrate are relatively fast, such that they reach a state of quasi-equilibrium or rapid equilibrium. 

The equilibrium constant of an enzyme-catalyzed reaction can depend greatly on reaction conditions. Because most substrates, products, and effectors are ionic species, the concentration and activity of each species is usually pH-dependent. This is particularly true for nucleotide-dependent enzymes which utilize substrates having pi a values near the pH value of the reaction. For example, both ATP" and HATP may be the nucleotide substrate for a phosphotransferase, albeit with different values. Thus, the equilibrium constant with ATP may be significantly different than that of HATP . In addition, most phosphotransferases do not utilize free nucleotides as the substrate but use the metal ion complexes. Both ATP" and HATP have different stability constants for Mg +. If the buffer (or any other constituent of the reaction mixture) also binds the metal ion, the buffer (or that other constituent) can also alter the observed equilibrium constant . ... 

A potential limitation encountered when one seeks to characterize the kinetic binding order of certain rapid equilibrium enzyme-catalyzed reactions containing specific abortive complexes. Frieden pointed out that initial rate kinetics alone were limited in the ability to distinguish a rapid equilibrium random Bi Bi mechanism from a rapid equilibrium ordered Bi Bi mechanism if the ordered mechanism could also form the EB and EP abortive complexes. Isotope exchange at equilibrium experiments would also be ineffective. However, such a dilemma would be a problem only for those rapid equilibrium enzymes having fccat values less than 30-50 sec h For those rapid equilibrium systems in which kcat is small, Frieden s dilemma necessitates the use of procedures other than standard initial rate kinetics. 

A mathematical equation indicating how the equilibrium constant of an enzyme-catalyzed reaction (or half-reaction in the case of so-called ping pong reaction mechanisms) is related to the various kinetic parameters for the reaction mechanism. In the Briggs-Haldane steady-state treatment of a Uni Uni reaction mechanism, the Haldane relation can be written as follows ... 

Equihbria involving the productively bound substrates and the products formed during an enzyme-catalyzed reaction. These equihbria can be treated in terms of internal equilibrium constants (i mt) between these enzyme-bound species. 

The quotient of rate constants obtained in steady-state treatments of enzyme behavior to define a substrate s interaction with an enzyme. While the Michaelis constant (with overall units of molarity) is a rate parameter, it is not itself a rate constant. Likewise, the Michaelis constant often is only a rough gauge of an enzyme s affinity for a substrate. 2. Historically, the term Michaelis constant referred to the true dissociation constant for the enzyme-substrate binary complex, and this parameter was obtained in the Michaelis-Menten rapid-equilibrium treatment of a one-substrate enzyme-catalyzed reaction. In this case, the Michaelis constant is usually symbolized by Ks. 3. The value equal to the concentration of substrate at which the initial rate, v, is one-half the maximum velocity (Lmax) of the enzyme-catalyzed reaction under steady state conditions. 

How then might one represent the facilitated exchange process The facilitation of nucleotide exchange is in all respects analogous to an enzyme-catalyzed reaction (Le., the rate of a reaction is accelerated without altering the equilibrium position of the reaction). In this way, one can represent the facilitated nucleotide exchange reaction as one would any enzymic process ... 

A two-substrate, two-product enzyme-catalyzed reaction scheme in which both the substrates (A and B) and the products (P and Q) bind and are released in any order. Note that this definition does not imply that there is an equal preference for each order (that is, it is not a requirement that the flux of the reaction sequence in which A binds first has to equal the flux of the reaction sequence in which B binds first). In fact, except for rapid equilibrium schemes, this is rarely true. There usually is a distinct preference for a particular pathway in a random mechanism. A number of kinetic tools and protocols... 

An enzyme-catalyzed reaction scheme in which the two substrates (A and B) can bind in any order, resulting in the formation of a single product of the enzyme-catalyzed reaction (hence, this reaction is the reverse of the random Uni Bi mechanism). Usually the mechanism is distinguished as to being rapid equilibrium (/.c., the ratedetermining step is the central complex interconversion, EAB EP) or steady-state (in which the substrate addition and/or product release steps are rate-contributing). See Multisubstrate Mechanisms... 

Multisubstrate or multiproduct enzyme-catalyzed reaction mechanisms in which one or more substrates and/ or products bind and/or are released in a random fashion. Note that this definition does not imply that there has to be an equal preference for any particular binding sequence. The flux through the different binding sequences could very easily be different. However, in rapid equilibrium random mechanisms, the flux rates are equivalent. See Multisubstrate Mechanisms... 

Mechanism for enzyme catalyzed reactions. To explain the kinetics of enzyme-substrate reactions, Michaelis and Menten (1913) came up with the following mechanism, which uses an equilibrium assumption... 

At any given instant in an enzyme-catalyzed reaction, the enzyme exists in two forms, the free or uncombined form E and the combined form ES. At low [S], most of the enzyme is in the uncombined form E. Here, the rate is proportional to [S] because the equilibrium of Equation 6-7 is pushed toward formation of more ES as [S] increases. The maximum initial rate of the catalyzed reaction (Prnax) is observed when virtually all the enzyme is present as the ES complex and [E] is vanishingly small. Under these conditions, the enzyme is saturated with its substrate, so that further increases in [S] have no effect on rate. This condition exists when [S] is sufficiently high that essentially all the free enzyme has been converted to the ES form. After the ES complex breaks down to yield the product P, the enzyme is free to catalyze reaction of another molecule of substrate. The saturation effect is a distinguishing characteristic of enzymatic catalysts and is responsible for the plateau observed in Figure 6-11. The pattern seen in Figure 6-11 is sometimes referred to as saturation kinetics. 

Calculate the standard free-eneigy changes of the following metabolically important enzyme-catalyzed reactions at 25 °C and pH 7.0, using the equilibrium constants given. 

For some enzyme-catalyzed reactions the equilibrium lies far to one side. However, many other reactions are freely reversible. Since a catalyst promotes reactions in both directions, we must consider the action of an enzyme on the reverse reaction. Let us designate the maximum velocity in the forward direction as Vf and that in the reverse direction as Vr There will be a Michaelis constant for reaction of enzyme with product Kmp, while Kms will refer to the reaction with substrate. 

As in any other chemical reaction, there is a relationship between the rate constants for forward and reverse enzyme-catalyzed reactions and the equilibrium constant. This relationship, first derived by the British kineticist J. B. S. Haldane and proposed in his book Enzymes41 in 1930, is known as the Haldane relationship. It is obtained by setting v( = vr for the condition that product and substrate concentrations are those at equilibrium. For a single substrate-single product system it is given by Eq. 9-42. 

Figure C1.1.2 Time course of product generation for typical enzyme-catalyzed reaction. Product concentration is shown to asymptotically approach its equilibrium value (horizontal dashed line). The diagonal dashed line illustrates the portion of the curve used to calculate initial velocity.

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