1. Using the Michaelis-Menten Equation, When k2 is very small compared to K1 and

Question

1. Using the Michaelis-Menten Equation, When k2 is very small compared to K1 and K-, What process is the rate determining in product formation? 2. In a multi-step reaction, what step determines the overall reaction rate? 3. What step in M-M Kinetics determines the overall reaction rate? 4. The term used for the overall rate constant for multi-step reaction is Kcat. Discuss why it makes sense that kcat=k2 in simple Michaelis-Menten reactions. 5. In simple M-M kinetics, the units of Kcat are S^-1. Discuss how the units are consistent with the name "turnover number". If Kcat is large, what does that imply about the enzyme? 7. The term efficiency is often used in describing enzymes. What does it mean for an enzyme to be efficient?

Explanation / Answer

FOLLOW THIS The effect on V0of varying [S] when the enzyme concentration is held constant is shown in Figure 8-11. At relatively low concentrations of substrate, V0 increases almost linearly with an increase in [S]. At higher substrate concentrations, V0 increases by smaller and smaller amounts in response to increases in [S]. Finally, a point is reached beyond which there are only vanishingly small increases in V0 with increasing [S] (Fig. 8-11). This plateau is called the maximum velocity, Vmax. The ES complex is the key to understanding this kinetic behavior, just as it represented a starting point for the discussion of catalysis. The kinetic pattern in Figure 8-11 led Victor Henri to propose in 1903 that an enzyme combines with its substrate molecule to form the ES complex as a necessary step in enzyme catalysis. This idea was expanded into a general theory of enzyme action, particularly by Leonor Michaelis and Maud Menten in 1913. They postulated that the enzyme first combines reversibly with its substrate to form an enzymesubstrate complex in a relatively fast reversible step: The ES complex then breaks down in a slower second step to yield the free enzyme and the reaction product P: In this model the second reaction (Eqn 8-8) is slower and therefore limits the rate of the overall reaction. It follows that the overall rate of the enzyme-catalyzed reaction must be proportional to the concentration of the species that reacts in the second step, that is, ES. 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 will be in the uncombined form E. Here, the rate will be proportional to [SJ because the equilibrium of Equation 8-7 will be pushed toward formation of more ES as [S] is increased. The maximum initial rate of the catalyzed reaction (Vmax) is observed when virtually all of the enzyme is present as the ES complex and the concentration of 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 will exist when [S] is sufficiently high that essentially all the free enzyme will have been converted into the ES form. After the ES complex breaks down to yield the product P, the enzyme is free to catalyze another reaction. The saturation effect is a distinguishing characteristic of enzyme catalysts and is responsible for the plateau observed in Figure 8-11. Figure 8-11 Effect of substrate concentration on the initial velocity of an enzyme-catalyzed reaction. Vmax can only be approximated from such a plot, because V0 will approach but never quite reach Vmax. The substrate concentration at which V0 is half maximal is Km, the Michaelis-Menten constant. The concentration of enzyme E in an experiment such as this is generally so low that [S]>>[E] even when [S] is described as low or relatively low. The units given are typical for enzyme-catalyzed reactions and are presented only to help illustrate the meaning of V0 and [S]. (Note that the curve describes part of a rectangular hyperbola, with one asymptote at Vmax. If the curve were continued below [S] = 0, it would approach a vertical asymptote at [S] = -Km. ) When the enzyme is first mixed with a large excess of substrate, there is an initial period called the pre-steady state during which the concentration of the ES complex builds up. The pre-steady state is usually too short to be easily observed. The reaction quickly achieves a steady state in which [ES] (and the concentration of any other intermediates) remains approximately constant over time. The measured V0 generally reflects the steady state even though V0 is limited to early times in the course of the reaction. Michaelis and Menten concerned themselves with the steady-state rate, and this type of analysis is referred to as steady-state kinetics. The Relationship between Substrate Concentration and Enzymatic Reaction Rate Can Be Expressed Quantitatively Figure 8-11 shows the relationship between [S] and V0 for an enzymatic reaction. The curve expressing this relationship has the same general shape for most enzymes (it approaches a rectangular hyperbola ). The hyperbolic shape of this curve can be expressed algebraically by the Michaelis-Menten equation, derived by these workers starting from their basic hypothesis that the rate-limiting step in enzymatic reactions is the breakdown of the ES complex to form the product and the free enzyme. The important terms are [S], V0, Vmax, and a constant called the Michaelis-Menten constant or Km. All of these terms are readily measured experimentally. Here we shall develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation. The derivation starts with the two basic reactions involved in the formation and breakdown of ES (Eqns 8-7 and 8-8). At early times in the reaction, the concentration of the product [P] is negligible and the simplifying assumption is made that h-z can be ignored. The overall reaction then reduces to V0 is determined by the breakdown of ES to give product, which is determined by [ES]: As [ES] in Equation 8-10 is not easily measured experimentally, we must begin by finding an alternative expression for [ES]. First, we will introduce the term [Et], representing the total enzyme concentration (the sum of the free and substrate-bound enzyme). Free or unbound enzyme can then be represented by [Et] - [ES]. Also, because [S] is ordinarily far greater than [Et], the amount of substrate bound by the enzyme at any given time is negligible compared with the total [S]. With these in mind, the following steps will lead us to an expression for V0 in terms of parameters that are easily measured. Step l. The rates of formation and breakdown of ES are determined by the steps governed by the rate constants kl (formation) and k-1 + k2 (breakdown), according to the expressions Step 2. An important assumption is now made that the initial rate of reaction reflects a steady state in which [ES] is constant, i.e., the rate of formation of ES is equal to its rate of breakdown. This is called the steady-state assumption. The expressions in Equations 8-11 and 8-12 can be equated at the steady state, giving Step 3. A series of algebraic steps is now taken to solve Equation 8-13 for [ES]. The left side is multiplied out and the right side is simplified to give Adding the term k1[ES][S] to both sides of the equation and simplifying gives Solving this equation for [ES] gives This can now be simplified further, in such a way as to combine the rate constants into one expression: The term (k2 + k-1)/k1 is defined as the Michaelis-Menten constant, Km. Substituting this into Equation 8-17 simplifies the expression to Step 4. V0 can now be expressed in terms of [ES]. Equation 8-18 is used to substitute for [ES] in Equation 8-10, giving This equation can be further simplified. Because the maximum velocity will occur when the enzyme is saturated and [ES] = [Et], Vmax can be defined as k2[Et]. Substituting this in Equation 8-19 gives This is the Michaelis-Menten equation, the rate equation for a onesubstrate, enzyme-catalyzed reaction. It is a statement of the quantitative relationship between the initial velocity V0, the maximum initial Velocity Vmax, and the initial substrate concentration [S], all related through the Michaelis-Menten constant Km. Does the equation fit the facts? Yes; we can confirm this by considering the limiting situations where [S] is very high or very low, as shown in Figure 8-12. Figure 8-12 Dependence of initial velocity on substrate concentration, showing the kinetic parameters that define the limits of the curve at high and low ISI. At low [S], Km>>[S], and the [S] term in the denominator of the Michaelis-Menten equation (Eqn 8-20) becomes insignificant; the equation simplifies to V0 = Vmax[S]/Km and V0, exhibits a linear dependence on [S], as observed. At high [S], where [S]>>Km, the Km term in the denominator of the Michaelis-Menten 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], with the shape of the curve defined by the terms Vmax/Km, at low [S] and Vmax at high [S]. An important numerical relationship emerges from the MichaelisMenten equation in the special case when V0 is exactly one-half Vmax (Fig. 8-12). Then On dividing by Vmax, we obtain Solving for Km, we get Km + [S] = 2[S], or This represents a very useful, practical definition of Km: Km is equivalent to that substrate concentration at which V0 is one-half Vmax. Note that Km has units of molarity. The Michaelis-Menten equation (8-20) can be algebraically transformed into forms that are useful in the practical determination of Km and Vmax (Box 8-1) and, as we will describe later, in the analysis of inhibitor action ( see Box 8-2 ). BOX 8-1 Transformations of the Michaelis-Menten Equation: The Douhle-Reciprocal Plot The Michaelis-Menten equation: can be algebraically transformed into forms that are more useful in plotting experimental data. One common transformation is derived simply by taking the reciprocal of both sides of the MichaelisMenten equation to give Separating the components of the numerator on the right side of the equation gives Figure 1 A double-reciprocal, or Lineweaver-Burk, plot. which simplifies to This equation is a transform of the MichaelisMenten equation called the Lineweaver-Burk equation. For enzymes obeying the MichaelisMenten 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 will have 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. 8-12). Other transformations of the MichaelisMenten equation have been derived and used. Each has some particular advantage in analyzing enzyme kinetic data. The double-reciprocal plot of enzyme reaction rates is very useful in distinguishing between certain types of enzymatic reaction mechanisms (see Fig. 8-14) and in analyzing enzyme inhibition (see Box 8-2). The Meaning of Vmax and Km Is Unique for Each Enzyme It is important to distinguish between the Michaelis-Menten equation and the specific kinetic mechanism upon 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 Michaelis-Menten kinetics. The practical rule that Km = [S] when V0 =1/2Vmax (Eqn 8-23) holds for all enzymes that follow Michaelis-Menten kinetics (the major exceptions to MichaelisMenten kinetics are the regulatory enzymes, discussed at the end of this chapter). However, this equation does not depend on the relatively simple two-step reaction mechanism proposed by Michaelis and Menten (Eqn 8-9). Many enzymes that follow Michaelis-Menten kinetics have quite different reaction mechanisms, and enzymes that catalyze reactions with six or eight identifiable steps will often exhibit the same steady-state kinetic behavior. Even though Equation 8-23 holds true for many enzymes, both the magnitude and the real meaning of Vmax and Km can change from one enzyme to the next. This is an important limitation of the steady-state approach to enzyme kinetics. Vmax and Km are parameters that 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 represents the standard language by which the catalytic efficiencies of enzymes are characterized and compared. We now turn to the application and interpretation of the terms Vmax and Km. A simple graphical method for obtaining an approximate value for Km is shown in Figure 8-12. A more convenient procedure, using a double-reciprocal plot, is presented in Box 8-1. The Km can vary greatly from enzyme to enzyme, and even for different substrates of the same enzyme (Table 8-6). The term is sometimes used (inappropriately) as an indication of the affmity 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 of the reaction. Here we will consider reactions with two steps. On page 214 Kmis defined by the expression For the Michaelis-Menten reaction, k2 is rate-limiting; thus k2 k-l, and then Km= k2/kl. In other cases, k2 and k-l are comparable, and Km remains a more complex function of all three rate constants (Eqn 8-24). These situations were first analyzed by Haldane along with George E. Briggs in 1925. The Michaelis-Menten equation and the characteristic saturation behavior of the enzyme still apply, but Km cannot be considered a simple measure of substrate af fmity. Even more common are cases in which the reaction goes through multiple steps after formation of the ES complex; Km can then become a very complex function of many rate constants. Vmax also varies greatly from one enzyme to the next. If an enzyme reacts by the two-step Michaelis-Menten mechanism, Vmax is equivalent to k2[Et], where k2 is the rate-limiting step. 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?E + P, is rate-limiting: 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 there are several steps in the reaction, and one is clearly rate-limiting, kcat is equivalent to the rate constant for that limiting step. For the Michaelis-Menten reaction, kcat = k2. For the reaction of Equation 8-25, 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 8-19 becomes The constant km is a first-order rate constant with units of reciprocal time, and is also called the turnover number. It is equi

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