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Rates of Enzymatic Reactions CHAPTER 14 JWCL281_c14_482-505.qxd 2/19/10 2:21 PM Page 482 Rates of Enzymatic Reactions CHAPTER 14 1 Chemical Kinetics 1. It is through kinetic studies that the binding affinities A. Elementary Reactions of substrates and inhibitors to an enzyme can be deter- B. Rates of Reactions mined and that the maximum catalytic rate of an enzyme C. Transition State Theory can be established. 2 Enzyme Kinetics 2. By observing how the rate of an enzymatic reaction A. The Michaelis–Menten Equation varies with the reaction conditions and combining this in- B. Analysis of Kinetic Data formation with that obtained from chemical and structural C. Reversible Reactions studies of the enzyme, the enzyme’s catalytic mechanism 3 Inhibition may be elucidated. A. Competitive Inhibition B. Uncompetitive Inhibition 3. Most enzymes, as we shall see in later chapters, func- C. Mixed Inhibition tion as members of metabolic pathways. The study of the 4 Effects of pH kinetics of an enzymatic reaction leads to an understanding of that enzyme’s role in an overall metabolic process. 5 Bisubstrate Reactions A. Terminology 4. Under the proper conditions, the rate of an enzymat- B. Rate Equations ically catalyzed reaction is proportional to the amount of C. Differentiating Bisubstrate Mechanisms the enzyme present, and therefore most enzyme assays D. Isotope Exchange (measurements of the amount of enzyme present) are Appendix: Derivations of Michaelis–Menten based on kinetic studies of the enzyme. Measurements of Equation Variants enzymatically catalyzed reaction rates are therefore among A. The Michaelis–Menten Equation for Reversible Reactions— the most commonly employed procedures in biochemical Equation [14.30] and clinical analyses. B. The Michaelis–Menten Equation for Uncompetitive Inhibition—Equation [14.41] We begin our consideration of enzyme kinetics by re- C. The Michaelis–Menten Equation for Mixed Inhibition— viewing chemical kinetics because enzyme kinetics is based Equation [14.45] on this formalism. Following that, we derive the basic equa- D. The Michaelis–Menten Equation for Ionizable Enzymes— tions of enzyme kinetics, describe the effects of inhibitors Equation [14.47] on enzymes, and consider how the rates of enzymatic reac- tions vary with pH. We end by outlining the kinetics of complex enzymatic reactions. Kinetics is, by and large, a mathematical subject. Al- though the derivations of kinetic equations are occasion- Kinetics is the study of the rates at which chemical reactions ally rather detailed, the level of mathematical skills it re- occur. A major purpose of such a study is to gain an un- quires should not challenge anyone who has studied derstanding of a reaction mechanism, that is, a detailed elementary calculus. Nevertheless, to prevent mathemati- description of the various steps in a reaction process and cal detail from obscuring the underlying enzymological the sequence with which they occur. Thermodynamics, as principles, the derivations of all but the most important ki- we saw in Chapter 3, tells us whether a given process can netic equations have been collected in the appendix to this occur spontaneously but provides little indication as to the chapter. Those who wish to cultivate a deeper understand- nature or even the existence of its component steps. In con- ing of enzyme kinetics are urged to consult this appendix. trast, the rate of a reaction and how this rate changes in response to different conditions is intimately related to the path followed by the reaction and is therefore indicative of its reaction mechanism. 1 CHEMICAL KINETICS In this chapter, we take up the study of enzyme kinetics, a subject that is of enormous practical importance in bio- Enzyme kinetics is a branch of chemical kinetics and, as chemistry because: such, shares much of the same formalism. In this section we 482 JWCL281_c14_482-505.qxd 2/22/10 8:44 AM Page 483 Section 14-1. Chemical Kinetics 483 shall therefore review the principles of chemical kinetics so where v is the instantaneous rate or velocity of the reac- that, in later sections, we can apply them to enzymatically tion. For the first-order reaction A S P: catalyzed reactions. d[A] v ϭϪ ϭ k[A] [14.3a] dt A. Elementary Reactions A reaction of overall stoichiometry For second-order reactions such as 2A S P: A ¡ P d[A] 2 v ϭϪ ϭ k[A] [14.3b] dt may actually occur through a sequence of elementary reac- tions (simple molecular processes) such as whereas for A ϩ B S P, a second-order reaction that is ¡ ¡ ¡ first order in [A] and first order in [B], A I1 I2 P d[A] d[B] Here A represents reactants, P products, and I1 and I2 sym- v ϭϪ ϭϪ ϭ k[A][B] [14.3c] bolize intermediates in the reaction. The characterization dt dt of the elementary reactions comprising an overall reaction process constitutes its mechanistic description. The rate constants of first- and second-order reactions must have different units. In terms of units, v in Eq. [14.3a] a. Rate Equations is expressed as M ؒ sϪ1 ϭ kM. Therefore, k must have At constant temperature, elementary reaction rates units of reciprocal seconds (sϪ1) in order for Eq. [14.3a] to ,vary with reactant concentration in a simple manner. Con- balance. Similarly, for second-order reactions, M ؒ sϪ1 ϭ kM2 sider the general elementary reaction: so that k has the units MϪ1 sϪ1. The order of a specific reaction can be determined by p aA ϩ bB ϩ ϩ zZ ¡ P measuring the reactant or product concentrations as a function of time and comparing the fit of these data The rate of this process is proportional to the frequency with to equations describing this behavior for reactions of which the reacting molecules simultaneously come together, various orders. To do this we must first derive these that is, to the products of the concentrations of the reactants. equations. This is expressed by the following rate equation a. First-Order Rate Equation a b p z Rate ϭ k[A] [B] [Z] [14.1] The equation for [A] as a function of time for a first- order reaction, A S P, is obtained by rearranging Eq. where k is a proportionality constant known as a rate con- [14.3a] stant. The order of a reaction is defined as (a ϩ b ϩ p ϩ z), the sum of the exponents in the rate equation. For an d[A] elementary reaction, the order corresponds to the molecu- ϵ d ln [A] ϭϪk dt larity of the reaction, the number of molecules that must [A] simultaneously collide in the elementary reaction. Thus the elementary reaction A S P is an example of a first-order or and integrating it from [A]o, the initial concentration of A, unimolecular reaction, whereas the elementary reactions to [A], the concentration of A at time t: 2A S P and A ϩ B S P are examples of second-order or bimolecular reactions. Unimolecular and bimolecular reac- [A] t tions are common. Termolecular reactions are unusual and Ύ d ln[A] ϭϪk Ύ dt fourth- and higher order elementary reactions are unknown. [A]o 0 This is because the simultaneous collision of three mole- cules is a rare event; that of four or more molecules essen- This results in tially never occurs. ϭ Ϫ ln[A] ln[A] o kt [14.4a] B. Rates of Reactions or, by taking the antilogs of both sides, We can experimentally determine the order of a reaction by measuring [A] or [P] as a function of time; that is, ϭ Ϫkt [A] [A]o e [14.4b] d[A] d[P] v ϭϪ ϭ [14.2] dt dt Equation [14.4a] is a linear equation in terms of the JWCL281_c14_482-505.qxd 6/3/10 12:07 PM Page 484 484 Chapter 14. Rates of Enzymatic Reactions [A] 2 o> ϭϪ ln[A] ln kt1 2 o > a [A]o b Thus ϭ ln 2 ϭ 0.693 t1 2 [14.5] > k k Slope = –k In order to appreciate the course of a first-order reac- tion, let us consider the decomposition of 32P,a radioactive ln[A] isotope that is widely used in biochemical research. It has a half-life of 14 days. Thus, after 2 weeks, one-half of the 32P initially present in a given sample will have decomposed; after another 2 weeks, one-half of the remainder, or three- quarters of the original sample, will have decomposed; etc. The long-term storage of waste 32P therefore presents little problem, since after 1 year (26 half-lives), only 1 part in 226 ϭ 67 million of the original sample will remain. How much will remain after 2 years? In contrast, 14C, another 0 Time commonly employed radioactive tracer, has a half-life of 14 Figure 14-1 Plot of ln[A] versus time for a first-order reac- 5715 years. Only a small fraction of a given quantity of C tion. This illustrates the graphical determination of the rate will decompose over the course of a human lifetime. constant k using Eq. [14.4a]. b. Second-Order Rate Equation for One Reactant In a second-order reaction with one type of reactant, variables ln[A] and t as is diagrammed in Fig. 14-1. There- 2A S P, the variation of [A] with time is quite different fore, if a reaction is first order, a plot of ln[A] versus t will from that in a first-order reaction. Rearranging Eq. [14.3b] yield a straight line whose slope is Ϫk, the negative of the and integrating it over the same limits used for the first- first-order rate constant, and whose intercept on the ln[A] order reaction yields [A] t axis is ln[A]o.
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