Enzymes (Second Edition)

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Figure 5.1 Reaction progress curves for the loss of substrate [S] and production of product [P] during an enzyme-catalyzed reaction.

amount of product produced over time, one will observe progress curves similar to those shown in Figure 5.1. Note that the substrate depletion curve is the mirror image of the product appearance curve. At early times substrate loss and product appearance change rapidly with time but as time increases these rates diminish, reaching zero when all the substrate has been converted to product by the enzyme. Such time courses are well modeled by first-order kinetics, as discussed in Chapter 2:

where [S] is the substrate concentration remaining at time t, [S ] is the starting substrate concentration, and k is the pseudo-first-order rate constant for the reaction. The velocity v of such a reaction is thus given by:

Let us look more carefully at the product appearance profile for an enzymecatalyzed reaction (Figure 5.2). If we restrict our attention to the very early portion of this plot (shaded area), we see that the increase in product formation (and substrate depletion as well) tracks approximately linear with time. For this limited time period, the initial velocity v can be approximated as the slope (change in y over change in x) of the linear plot of [S] or [P] as a function of time:

Figure 5.2 Reaction progress curve for the production of product during an enzyme-catalyzed reaction. Inset highlights the early time points at which the initial velocity can be determined from the slope of the linear plot of [P] versus time.

Experimentally one finds that the time course of product appearance and substrate depletion is well modeled by a linear function up to the time when about 10% of the initial substrate concentration has been converted to product (Chapter 2). We shall see in Chapter 7 that by varying solution conditions, we can alter the length of time over which an enzyme-catalyzed reaction will display linear kinetics. For the rest of this chapter we shall assume that the reaction velocity is measured during this early phase of the reaction, which means that from here v v , the initial velocity.

5.2 EFFECTS OF SUBSTRATE CONCENTRATION ON VELOCITY

From Equation 5.2, one would expect the velocity of a pseudo-first-order reaction to depend linearly on the initial substrate concentration. When early studies were performed on enzyme-catalyzed reactions, however, scientists found instead that the reactions followed the substrate dependence illustrated in Figure 5.3. Figure 5.3A illustrates the time course of the enzyme-catalyzed reaction observed at different starting concentrations of substrate; the velocities for each experiment are measured as the slopes of the plots of [P] versus time. Figure 5.3B replots these data as the initial velocity v as a function of [S], the starting concentration of substrate. Rather than observing the linear relationship expected for first-order kinetics, we find the velocity apparently saturable at high substrate concentrations. This behavior puzzled early enzymologists.

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Figure 5.3 (A) Progress curves for a set of enzyme-catalyzed reactions with different starting concentrations of substrate [S]. (B) Plot of the reaction velocities, measured as the slopes of the lines from (A), as a function of [S].

Three distinct regions of this curve can be identified: at low substrate concentrations the velocity appears to display first-order behavior, tracking linearly with substrate concentration; at very high concentrations of substrate, the velocity switches to zero-order behavior, displaying no dependence on substrate concentration; and in the intermediate region, the velocity displays a curvilinear dependence on substrate concentration. How can one rationalize these experimental observations?

A qualitative explanation for the substrate dependence of enzyme-catalyzed reaction velocities was provided by Brown (1902). At the same time that the

kinetic characteristics of enzyme reactions were being explored, evidence for complex formation between enzymes and their substrates was also accumulating. Brown thus argued that enzyme-catalyzed reactions could best be described by the following reaction scheme:

E S ES E P

This scheme predicts that the reaction velocity will be proportional to the concentration of the ES complex as: v k [ES]. Suppose that we held the total enzyme concentration constant at some low level and varied the concentration of S. At low concentrations of S the concentration of ES would be directly proportional to [S]; hence the velocity would depend on [S] in an apparent first-order fashion. At very high concentrations of S, however, practically all the enzyme would be present in the form of the ES complex. Under such conditions the velocity depends of the rate of the chemical transformations that convert ES to EP and the subsequent release of product to re-form free enzyme. Adding more substrate under these conditions would not effect a change in reaction velocity; hence the slope of the plot of velocity versus [S] would approach zero (as seen in Figure 5.3B). The complete [S] dependence of the reaction velocity (Figure 5.3B) predicted by the model of Brown resembles the results seen from the Langmuir isotherm Equation (Chapter 4) for equilibrium binding of ligands to receptors. This is not surprising, since in the model of Brown, catalysis is critically dependent on initial formation of a binary ES complex through equilibrium binding.

5.3 THE RAPID EQUILIBRIUM MODEL OF ENZYME KINETICS

Although the model of Brown provided a useful qualitative picture of enzyme reactions, to be fully utilized by experimental scientists, it needed to be put into a rigorous mathematical framework. This was accomplished first by Henri (1903) and subsequently by Michaelis and Menten (1913). Ironically, Michaelis and Menten are more widely recognized for this contribution, although they themselves acknowledged the prior work of Henri. The basic rate equation derived in this section is commonly referred to as the Michaelis—Menten equation. Several writers have recently taken to referring to the equation as the Henri—Michaelis—Menten equation, in an attempt to correct this neglect of Henri’s contributions. The reader should be aware, however, that the majority of the scientific literature continues to use the traditional terminology.

The Henri—Michaelis—Menten approach assumes that a rapid equilibrium is established between the reactants (E S) and the ES complex, followed by slower conversion of the ES complex back to free enzyme and product(s); that is, this model assumes that k k in the scheme presented in Section 5.2. In

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this model, the free enzyme E first combines with the substrate S to form the binary ES complex. Since substrate is present in large excess over enzyme, we can use the assumption that the free substrate concentration [S] is well approximated by the total substrate concentration added to the reaction [S]. Hence, the equilibrium dissociation constant for this complex is given by:

Similar to the treatment of receptor—ligand binding in Chapter 4, here the free enzyme concentration is given by the difference between the total enzyme concentration [E] and the concentration of the binary complex [ES]:

This can be rearranged to give an expression for [ES]:

[E][S]

[ES] (5.7) K [S]

Next, the ES complex is transformed by various chemical steps to yield the product of the reaction and to recover the free enzyme. In the simplest case, a single chemical step, defined by the first-order rate constant k , results in product formation. More likely, however, there will be a series of rapid chemical events following ES complex formation. For simplicity, the overall rate for these collective chemical steps can be described by a single first-order rate constant k . Hence:

E S ES E P

and the rate of product formation is thus given by the first-order equation:

Equation 5.9 is similar to the equation for a Langmuir isotherm, as derived in Chapter 4 (Equation 4.21). This, then, describes the reaction velocity as a hyperbolic function of [S], with a maximum value of k [E] at infinite [S]. We refer to this value as the maximum reaction velocity, or V .

Equation 5.11 is the final equation derived independently by Henri and Michaelis and Menten to describe enzyme kinetic data. Note the striking similarity between this equation and the forms of the Langmuir isotherm equation presented in Chapter 4 (Equations 4.21 and 4.22). Thus, much of enzyme kinetics can be explained in terms of a simple equilibrium model involving rapid equilibrium between free enzyme and substrate to form the binary ES complex, followed by chemical transformation steps to produce and release product.

5.4 THE STEADY STATE MODEL OF ENZYME KINETICS

The original derivations by Henri and by Michaelis and Menten depended on a rapid equilibrium approach to enzyme reactions. This approach is quite useful in rapid kinetic measurements, such as single-turnover reactions, as described later in this chapter. The majority of experimental measurements of enzyme reactions, however, occur when the ES complex is present at a constant, steady state concentration (as defined below). Briggs and Haldane (1925) recognized that the equilibrium-binding approach of Henri and Michaelis and Menten could be described more generally by a steady state approach that did not require k k . The following discussion is based on this description by Briggs and Haldane. As we shall see, the final equation that results from this treatment is very similar to Equation 5.11, and despite the differences between the rapid equilibrium and steady state approaches, the final steady state equation is commonly referred to as the Henri—Michaelis—Menten equation.

Steady state refers to a time period of the enzymatic reaction during which the rate of formation of the ES complex is exactly matched by its rate of decay to free enzyme and products. This kinetic phase can be attained when the concentration of substrate molecules is in great excess of the free enzyme concentration. To achieve a steady state, certain condition must be met, and

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these conditions allow us to make some reasonable assumption, which greatly simplify the mathematical treatment of the kinetics. These assumptions are as follows:

1.During the initial phase of the reaction progress curve (i.e., conditions under which we are measuring the linear initial velocity), there is no appreciable buildup of any intermediates other than the ES complex. Hence, all the enzyme molecules can be accounted for by either the free enzyme or by the enzyme—substrate complex. The total enzyme concentration [E] is therefore given by:

2. As in the rapid equilibrium treatment, we assume that the enzyme is acting catalytically, so that it is present in very low concentration relative to substrate, that is, [S] [E]. Hence, formation of the ES complex does not significantly diminish the concentration of free substrate. We can therefore make the approximation: [S] [S], where [S] is the free substrate concentration and [S] is the total substrate concentration).

3.During the initial phase of the progress curve, very little product is

formed relative to the total concentration of substrate. Hence, during this early phase [P] 0 and therefore depletion of [S] is minimal. At the initiation of the reaction there will be a rapid burst of formation of the ES complex followed by a kinetic phase in which the rate of formation of new ES complex is balanced by the rate of its decomposition back to free enzyme and product. In other words, during this phase the concentration of ES is constant. We refer to this kinetic phase as the steady state, which is defined by:

Figure 5.4 illustrates the development and duration of the steady state for the enzyme cytochrome c oxidase interacting with its substrates cytochrome c and molecular oxygen. As soon as the substrates and enzyme are mixed, we see a rapid pre—steady state buildup of ES complex, followed by a long time window in which the concentration of ES does not change (the steady state phase), and finally a post—steady state phase characterized by significant depletion of the starting substrate concentration.

With these assumptions made, we can now work out an expression for the enzyme velocity under steady state conditions. As stated previously, for the simplest of reaction schemes, the pseudo-first-order progress curve for an enzymatic reaction can be described by:

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At this point let us define the term K as an abbreviation for the kinetic constants in the denominatior of the right-hand side of Equation 5.17:

For now we will consider K to be merely an abbreviation to make our subsequent mathematical expressions less cumbersome. Later, however, we shall see that K has a more significant meaning. Substituting Equation 5.18 into Equation 5.17 we obtain:

Now, since substrate depletion is insignificant during the steady state phase, we can replace the term [S] by the total substrate concentration [S] (which is much more easily measured in real experimental situations). We can also use the equality of Equation 5.12 to replace [E] by ([E] [ES]). With these substitutions, Equation 5.19 can be recast as follows:

[S]

[ES] [E] (5.20) [S] K

If we now combine this expression for [ES] with the velocity expression of Equation 5.14, we obtain:

Or, we can generalize Equation 5.21 for more complex reaction schemes by

As described earlier, as the concentration of substrate goes towards infinity, the

and thus we again arrive at Equation 5.10: V k [E]. Combining this with

Equation 5.23 we finally arrive at an expression very similar to that first described by Henri and Michaelis and Menten (i.e., similar to Equation 5.11):

This is the central expression for steady state enzyme kinetics. While it differs from the equilibrium expression derived by Henri and by Michaelis and Menten, it is nevertheless universally referred to as the Michaelis—Menten or Henri—Michaelis—Menten equation.

In our definition of K (Equation 5.18), we combined first-order rate constants (k and k , which have units of reciprocal time) with a second-order rate constant (k , which has units of reciprocal molarity, reciprocal time) in such a way that the resulting K has units of molarity, as does [S]. If we set up our experimental system so that the concentration of substrate exactly matches K , Equation 5.24 will reduce to:

This provides us with a working definition of K : T he K is the substrate concentration that provides a reaction velocity that is half of the maximal velocity obtained under saturating substrate conditions. The K value is often referred to in the literature as the Michaelis constant. In comparing Equation 5.24 for steady state kinetics with Equation 5.11 for the rapid equilibrium treatment, we see that the equations are identical except for the substitution of K for K in the steady state treatment. It is therefore easy to confuse these terms and to treat K as if it were the thermodynamic dissociation constant for the ES complex. However, the two constants are not always equal, even in considerations of the simplest of reactions schemes, as here. Recall that K can be defined by the rato of the reverse and forward reaction rate constants:

This value is not identical to the expression for K given in Equation 5.18.

Only under the specific conditions that k k are K and K equivalent. For more complex reaction schemes one would replace the k term in Equation 5.18 by k . Recall that k reflects a summation of multiple chemical steps in catalysis. Hence, depending on the details of the reaction mechanism, and the

values of the individual rate constants, situations can arise in which the value of K is less than, greater than, or equal to K . Therefore, K should generally be considered as a kinetic, not thermodynamic, constant.