Enzyme-catalyzed reactions, equilibrium constants

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 ... 

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... 

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... 

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 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 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. 

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. 

A number of recent studies have shown that under certain conditions, FABMS indeed can very accurately measure the balance of ionic species in ongoing chemical reactions in solutions. These studies include the determination of acid dissociation constants (2), equilibrium constants for enzyme catalyzed reactions (1), metal-ligand association constants 03), and measurements of... 

Acid dissociation constants and dissociation constants of complex ions determine the concentrations of species that are present in a solution at equilibrium under specified conditions. Ionic dissociation reactions occur rapidly and tend to remain at equilibrium during an enzyme-catalyzed reaction. Since ATP (see Fig. 1.1) is the primary carrier of energy in biochemical systems and since a good deal is known about its binding properties, these properties are considered here in some detail. 

The procedure for calculating standard formation properties of species at zero ionic strength from measurements of apparent equilibrium constants is discussed in the next chapter. The future of the thermodynamics of species in aqueous solutions depends largely on the use of enzyme-catalyzed reactions. The reason that more complicated ions in aqueous solutions were not included in the NBS Tables (1992) is that it is difficult to determine equilibrium constants in systems where a number of reactions occur simultaneously. Since many enzymes catalyze clean-cut reactions, they make it possible to determine apparent equilibrium constants and heats of reaction between very complicated organic reactants that could not have been studied classically. 

If the apparent equilibrium constant K for an enzyme-catalyzed reaction has been determined at 298.15K and AfG ° values can be calculated at the experimental pH and ionic strength using known functions of pH and ionic strength for all the reactants but one, the AfG ° of that reactant under the experimental conditions can be calculated using equation 4.4-2. So far functions of pH and ionic strength that yield AfG ° are have been published for 131 reactants at 298.15 K (Alberty, 2001f). 

This field owes a tremendous debt to the experimentalists who have measured apparent equilibrium constants and heats of enzyme-catalyzed reactions and to those who have made previous thermodynamic tables that contain information needed in biochemical thermodynamics. 

Establishing the inhibition patterns in an enzyme-catalyzed reaction is usually an important step in elucidating the reaction mechanism. One complication in the interpretation of such data is the possible formation of dead-end complexes (i.e., a complex of the form EAP in the above scheme). This is especially important in rapid-equilibrium reactions [ones in which all steps except the rate constants for the central isomerization step (EAB EPQ in the above example) are very large]. 

Enzyme-catalyzed reactions do not always proceed to completion because of an unfavorable equilibrium constant or insufficient reaction time. Nevertheless, a compound can be determined by a total change method. The reaction may be pulled to completion by trapping the products or by coupling one of the products to a second enzyme reaction. In determining lactate with lactate dehydrogenase the equilibrium lies far to the left in the direction of lactate at pH 9.5 (K = 2.9 X 10"at 25 C). 

When enzyme-catalyzed reactions produce or eonsume hydrogen ions, their apparent equilibrium constants depend on the pH. The term apparent equilibrium constant and the symbol K are used to indicate that a biochemical reaction and the expression for its apparent equilibrium constant are written in terms of sums of species. Hydrogen ions are omitted in writing a biochemical reaction, and [H ] is omitted in writing the expression for the apparent equilibrium constant of a biochemical reaction because [H ] is specified. 

---