8885dc062052/2/042:51 PM Page205mac76mac76:385reb 6.3 Enzyme Kinetics as an Approach to Understanding Mechanism 205 Km=[s when Vo =vMax (eqn 6-23)holds for all en- zymes that follow Michaelis-Menten kinetics. (The most important exceptions to Michaelis-Menten kinetics are the regulatory enzymes, discussed in Section 6.5.) How- ever, the Michaelis-Menten equation does not depend 3 on the relatively simple two-step reaction mechanism proposed by Michaelis and Menten (Eqn 6-10). Many enzymes that follow Michaelis-Menten kinetics have quite different reaction mechanisms, and enzymes that catalyze reactions with six or eight identifiable steps of- ten exhibit the same steady-state kinetic behavior. Even though Equation 6-23 holds true for many enzymes both the magnitude and the real meaning of Vmax and Km can differ from one enzyme to the next. This is important limitation of the steady-state approach to en- FIGURE 6-12 Dependence of initial velocity on substrate concen- zyme kinetics. The parameters Vmax and Km can be ob- tration. This graph shows the kinetic parameters that define the limits tained experimentally for any given enzyme, but by of the curve at high and low [S]. At low IS), Km >>[S] and the iS] themselves they provide little information about the term in the denominator of the Michaelis-Menten equation(Eqn 6-9) number, rates, or chemical nature of discrete steps in becomes insignificant. The equation simplifies to Vo=VmaxIS]km and the reaction. Steady-state kinetics nevertheless is the o exhibits a linear dependence on [Sl, as observed here. At high standard language by which biochemists compare and where ISI>>Km, the km term in the denominator of the Michaelis. characterize the catalytic efficiencies of enzymes Menten equation becomes insignificant and the equation simplifies to Vo=Vmax this is consistent with the plateau observed at high (S). The Interpreting Vmax and Km Figure 6-12 shows a simple lichaelis-Menten equation is therefore consistent with the observed graphical method for obtaining an approximate value dependence of Vo on (S), and the shape of the curve is defined by the for Km. A more convenient procedure, using a double terms Vma/Km at low [S] and Vmax at high [S] reciprocal plot, is presented in Box 6-1. The Km can vary greatly from enzyme to enzyme, and even for dif- On dividing by Vmax, we obtain ferent substrates of the same enzyme (table 6-6). The term is sometimes used (often inappropriately) as al (6-22) indicator of the affinity of an enzyme for its substrate. The actual meaning of Km depends on specific aspects Solving for Km, we get Km+[s=2S,or of the reaction mechanism such as the number and rel- ative rates of the individual steps. For reactions with Km=S], when Vo =vmax wo steps, This is a very useful, practical definition of Km: Km is (6-24) equivalent to the substrate concentration at which Vo one-half v. When k2 is rate-limiting, k2 <<k-l and Km reduces to The Michaelis-Menten equation(Eqn 6-9)can be k-1/kl, which is defined as the dissociation constant, algebraically transformed into versions that are useful Kd, of the ES complex. Where these conditions hold, K in the practical determination of Km and Vmax(Box 6-1) does represent a measure of the affinity of the enzyme and, as we describe later, in the analysis of inhibitor action(see Box 6-2 on page 210). TABLE 6-6 Km for Some Enzymes and Substrates Kinetic Parameters Are Used to Compare Substrate Enzyme Activities Hexokinase(brain) 0.4 important to distinguish between the D-Glucose 0.05 D-Fructose Michaelis-Menten equation and the specific kinetic mechanism on which it was originally Carbonic anhydrase ased. The equation describes the kinetic be- Chymotrypsin Glycyltyrosinylglycine avior of a great many enzymes, and all en- N-Benzoyltyrosinamide 2.5 zymes that exhibit a hyperbolic dependence B-Galactosidase 4.0 L-Threonine of Vo on [s are said to follow Michaelis- Threonine dehydratase Menten kinetics. The practical rule that
On dividing by Vmax, we obtain 1 2 Km [ S] [S] (6–22) Solving for Km, we get Km [S] 2[S], or Km [S], when V0 1 2 Vmax (6–23) This is a very useful, practical definition of Km: Km is equivalent to the substrate concentration at which V0 is one-half Vmax. The Michaelis-Menten equation (Eqn 6–9) can be algebraically transformed into versions that are useful in the practical determination of Km and Vmax (Box 6–1) and, as we describe later, in the analysis of inhibitor action (see Box 6–2 on page 210). Kinetic Parameters Are Used to Compare Enzyme Activities It is important to distinguish between the Michaelis-Menten equation and the specific kinetic mechanism on which it was originally based. The equation describes the kinetic behavior of a great many enzymes, and all enzymes that exhibit a hyperbolic dependence of V0 on [S] are said to follow MichaelisMenten kinetics. The practical rule that Km [S] when V0 1 ⁄ 2Vmax (Eqn 6–23) holds for all enzymes that follow Michaelis-Menten kinetics. (The most important exceptions to Michaelis-Menten kinetics are the regulatory enzymes, discussed in Section 6.5.) However, the Michaelis-Menten equation does not depend on the relatively simple two-step reaction mechanism proposed by Michaelis and Menten (Eqn 6–10). Many enzymes that follow Michaelis-Menten kinetics have quite different reaction mechanisms, and enzymes that catalyze reactions with six or eight identifiable steps often exhibit the same steady-state kinetic behavior. Even though Equation 6–23 holds true for many enzymes, both the magnitude and the real meaning of Vmax and Km can differ from one enzyme to the next. This is an important limitation of the steady-state approach to enzyme kinetics. The parameters Vmax and Km can be obtained experimentally for any given enzyme, but by themselves they provide little information about the number, rates, or chemical nature of discrete steps in the reaction. Steady-state kinetics nevertheless is the standard language by which biochemists compare and characterize the catalytic efficiencies of enzymes. Interpreting Vmax and Km Figure 6–12 shows a simple graphical method for obtaining an approximate value for Km. A more convenient procedure, using a doublereciprocal plot, is presented in Box 6–1. The Km can vary greatly from enzyme to enzyme, and even for different substrates of the same enzyme (Table 6–6). The term is sometimes used (often inappropriately) as an indicator of the affinity of an enzyme for its substrate. The actual meaning of Km depends on specific aspects of the reaction mechanism such as the number and relative rates of the individual steps. For reactions with two steps, Km k2 k1 k1 (6–24) When k2 is rate-limiting, k2 k1 and Km reduces to k1/k1, which is defined as the dissociation constant, Kd, of the ES complex. Where these conditions hold, Km does represent a measure of the affinity of the enzyme 6.3 Enzyme Kinetics as an Approach to Understanding Mechanism 205 V0 ( M/min) V0 Vmax V0 [S] (mM) Km Vmax 1 2 Vmax [S] Km FIGURE 6–12 Dependence of initial velocity on substrate concentration. This graph shows the kinetic parameters that define the limits of the curve at high and low [S]. At low [S], Km [S] and the [S] term in the denominator of the Michaelis-Menten equation (Eqn 6–9) becomes insignificant. The equation simplifies to V0 Vmax[S]/Km and V0 exhibits a linear dependence on [S], as observed here. At high [S], where [S] Km, the Km term in the denominator of the MichaelisMenten equation becomes insignificant and the equation simplifies to V0 Vmax; this is consistent with the plateau observed at high [S]. The Michaelis-Menten equation is therefore consistent with the observed dependence of V0 on [S], and the shape of the curve is defined by the terms Vmax/Km at low [S] and Vmax at high [S]. Enzyme Substrate Km (mM) Hexokinase (brain) ATP 0.4 D-Glucose 0.05 D-Fructose 1.5 Carbonic anhydrase HCO3 26 Chymotrypsin Glycyltyrosinylglycine 108 N-Benzoyltyrosinamide 2.5 -Galactosidase D-Lactose 4.0 Threonine dehydratase L-Threonine 5.0 TABLE 6–6 Km for Some Enzymes and Substrates 8885d_c06_205 2/2/04 2:51 PM Page 205 mac76 mac76:385_reb:
8885dc06_190-2371/27/047:13 AM Page206mac76mac76:385 206 Chapter 6 Enzymes BOX 6-1 WORKING IN BIOCHEMISTRY Transformations of the Michaelis-Menten of -1/Km on the 1/S axis. The double-reciprocal pres- Equation: The Double-Reciprocal Plot entation, also called a Lineweaver-Burk plot, has the The Michaelis-Menten equation great advantage of allowing a more accurate determi nation of Vmax, which can only be approximated from a simple plot of Vo versus s(see Fig 6-12) Other transformations of the Michaelis-Menten can be algebraically transformed into equations that equation have been derived, each with some particu- are more useful in plotting experimental data. One ar advantage in analyzing enzyme kinetic data. ( See common transformation is derived simply by taking Problem 1l at the end of this chapter) the reciprocal of both sides of the Michaelis-Menten The double-reciprocal plot of enzyme reaction rates is very useful in distinguishing between certain types of enzymatic reaction mechanisms(see Fig 6-14) and in analyzing enzyme inhibition(see Box 6-2) Separating the components of the numerator on the right side of the equation gives S which simplifies to This form of the Michaelis-Menten equation is called the lineweaver- Burk equation. For enzymes obey ing the Michaelis-Menten relationship, a plot of 1/vo versus 1/S(the"double reciprocal"of the Vo versus [s plot we have been using to this point) yields a straight line(Fig. 1). This line has a slope of Km/Vmax, an in- tercept of 1/Vmax on the 1/Vo axis, and an intercept FIGURE 1 A double-reciprocal or Lineweaver-Burk plot for its substrate in the Es complex. However, this sce- nario does not apply for most enzymes. Sometimes E+S: ES K ep k et p(6-25) k2>>k-l, and then Km=ky/k In other cases, k2 and k-l are comparable and km remains a more complex In this case, most of the enzyme is in the EP form at function of all three rate constants (Eqn 6-24). The saturation, and Vmax= k3lEt ] It is useful to define a Michaelis-Menten equation and the characteristic satu- more general rate constant, keat, to describe the limit- ration behavior of the enzyme still apply, but Km cannot ing rate of any enzyme-catalyzed reaction at saturation. ple measure of substrate affinity. If the reaction has several steps and one is clearly rate Even more common are cases in which the reaction goes limiting, kcat is equivalent to the rate constant for that through several steps after formation of Es; Km can then limiting step. For the simple reaction of Equation 6-10 become a very complex function of many rate constants. kcat=k2. For the reaction of Equation 6-25, kcat=k3 The quantity Vmax also varies greatly from one en- When several steps are partially rate-limiting, kcat ca zyme to the next. If an enzyme reacts by the two-step become a complex function of several of the rate con- Michaelis-Menten mechanism, Vmax k2Eth, where k2 stants that define each individual reaction step. In the is rate-limiting. However, the number of reaction steps Michaelis-Menten equation, kcat=Vma/Et1, and Equa- and the identity of the rate-limiting step(s) can vary tion 6-9 becomes from enzyme to enzyme. For example, consider the quite common situation where product release keat eusi (6-26) EPE+ P, is rate-limiting. Early in the reaction(when IPI is low), the overall reaction can be described by the The constant kcat is a first-order rate constant scheme hence has units of reciprocal time. It is also called
for its substrate in the ES complex. However, this scenario does not apply for most enzymes. Sometimes k2 k1, and then Km k2/k1. In other cases, k2 and k1 are comparable and Km remains a more complex function of all three rate constants (Eqn 6–24). The Michaelis-Menten equation and the characteristic saturation behavior of the enzyme still apply, but Km cannot be considered a simple measure of substrate affinity. Even more common are cases in which the reaction goes through several steps after formation of ES; Km can then become a very complex function of many rate constants. The quantity Vmax also varies greatly from one enzyme to the next. If an enzyme reacts by the two-step Michaelis-Menten mechanism, Vmax k2[Et], where k2 is rate-limiting. However, the number of reaction steps and the identity of the rate-limiting step(s) can vary from enzyme to enzyme. For example, consider the quite common situation where product release, EP n E P, is rate-limiting. Early in the reaction (when [P] is low), the overall reaction can be described by the scheme k1 k2 k3 E S ES EP E P (6–25) k1 k2 In this case, most of the enzyme is in the EP form at saturation, and Vmax k3[Et]. It is useful to define a more general rate constant, kcat, to describe the limiting rate of any enzyme-catalyzed reaction at saturation. If the reaction has several steps and one is clearly ratelimiting, kcat is equivalent to the rate constant for that limiting step. For the simple reaction of Equation 6–10, kcat k2. For the reaction of Equation 6–25, kcat k3. When several steps are partially rate-limiting, kcat can become a complex function of several of the rate constants that define each individual reaction step. In the Michaelis-Menten equation, kcat Vmax/[Et], and Equation 6–9 becomes V0 k K ca m t [ Et [ ] S [S ] ] (6–26) The constant kcat is a first-order rate constant and hence has units of reciprocal time. It is also called the yz yz yz 206 Chapter 6 Enzymes BOX 6–1 WORKING IN BIOCHEMISTRY Transformations of the Michaelis-Menten Equation: The Double-Reciprocal Plot The Michaelis-Menten equation V0 K V m ma x [ [ S S ] ] can be algebraically transformed into equations that are more useful in plotting experimental data. One common transformation is derived simply by taking the reciprocal of both sides of the Michaelis-Menten equation: V 1 0 K V m ma x [ [ S S ] ] Separating the components of the numerator on the right side of the equation gives V 1 0 Vm K ax m [S] Vm [ a S x ] [S] which simplifies to V 1 0 Vm K ax m [S] Vm 1 ax This form of the Michaelis-Menten equation is called the Lineweaver-Burk equation. For enzymes obeying the Michaelis-Menten relationship, a plot of 1/V0 versus 1/[S] (the “double reciprocal” of the V0 versus [S] plot we have been using to this point) yields a straight line (Fig. 1). This line has a slope of Km/Vmax, an intercept of 1/Vmax on the 1/V0 axis, and an intercept of 1/Km on the 1/[S] axis. The double-reciprocal presentation, also called a Lineweaver-Burk plot, has the great advantage of allowing a more accurate determination of Vmax, which can only be approximated from a simple plot of V0 versus [S] (see Fig. 6–12). Other transformations of the Michaelis-Menten equation have been derived, each with some particular advantage in analyzing enzyme kinetic data. (See Problem 11 at the end of this chapter.) The double-reciprocal plot of enzyme reaction rates is very useful in distinguishing between certain types of enzymatic reaction mechanisms (see Fig. 6–14) and in analyzing enzyme inhibition (see Box 6–2). 1 V0 1 M/min ( ) 1 [S] 1 ( ) mM 1 Km Slope Vmax Km Vmax 1 FIGURE 1 A double-reciprocal or Lineweaver-Burk plot. 8885d_c06_190-237 1/27/04 7:13 AM Page 206 mac76 mac76:385_reb: