The Kinetics of Enzyme Action (With Diagram)

In most instances, the association of the enzyme with the substrate is so fleeting that the complex is ex­tremely difficult to detect.

Yet, as early as 1913, L. Michaelis and M. L. Menten postulated the existence of this transient complex.

On the basis of their obser­vations with the enzyme invertase, which catalyzes the hydrolysis of sucrose to glucose and fructose, they proposed that enzyme-catalyzed reactions were char­acterized by a sequence of phases that involves:

(1) The formation of a complex (ES) between the enzyme (E) and the substrate (S);

(2) The modification of the sub­strate to form the product (P) or products, which briefly remain associated with the enzyme (EP); and

(3) The release of the product or products from the en­zyme; that is,

In this equation, it is assumed that the combination of enzyme and substrate is reversible; is the rate con­stant for the formation of ES (the dimensions of the rate constant are seconds-¹), and k₂ is the rate con­stant for the dissociation of ES. After the ES complex is formed, S is converted to P with the rate constant k₃. As long as the concentration of P remains negligi­ble (as it would be at the outset of catalysis) or if P is in some manner quickly removed from the system, then it is not necessary to consider the negligible reverse flux from E + P to ES.

Figure 8-4 depicts the relationship that exists be­tween substrate concentration and the rate at which reaction products appear for enzyme-catalyzed reac­tions of this type. The curve describes the initial rate of product formation at a fixed enzyme concentration when the substrate concentration is varied on succes­sive trials.

At low concentrations of substrate, the ini­tial velocity of the reaction (i.e., v₀) is directly propor­tional to the substrate concentration (i.e., follows first-order kinetics). However, as the substrate con­centration is increased, the reaction velocity levels off, approaching a maximum value.

At this high substrate concentration, the reaction velocity is limited by the amount of time required to convert bound substrate (i.e., ES) to product (P) and free enzyme (E). The curve now follows zero-order kinetics. The curve of Figure 8-4 is called a Michaelis-Menten curve, and for an idealized enzyme-catalyzed reaction is a rectan­gular hyperbola.

Michaelis and Menten are also credited with the first mathematical study of the relationship between substrate concentration and reaction rates. They in­troduced two particularly useful mathematical expres­sions that for any enzyme relate [S] to V (velocity) and permit quick comparisons of various enzyme- catalyzed reactions; the two expressions are now called the Michaelis-Menten constant and the Michaelis-Menten equation. They are derived as fol­lows.

Let

[S] = the concentration of free substrate, S; (We may assume that the amount of available sub­strate is so great that the amount combined with the enzyme may be ignored in compari­son. Hence, total substrate concentration and the free substrate concentration are the same);

[E]_T =the total concentration of enzyme, E;

[E] = the concentration of free enzyme (that not complexed with substrate);

[ES] = the concentration of enzyme-substrate com­plex (Because the total amount of enzyme present is assumed to be very small in com­parison with the total amount of substrate, a significant proportion of the total enzyme may be involved in the ES complex. Hence, separate terms for the total and free enzyme concentrations are warranted);

[P] = the concentration of product, P.

The total enzyme concentration is given by:

The ratio of constants given in equation 8-22 may be set equal to a new constant, K_M, which is the Michaelis-Menten constant. Thus, the Michaelis- Menten constant is a constant that relates the steady- state concentrations of enzyme, enzyme-substrate complex, and substrate.

Each enzyme-catalyzed reaction reveals a charac­teristic K_M value, and this value is a measure of the tendency of the enzyme and the substrate to combine with each other. In this sense, the K_M value is an index of the affinity of the enzyme for its particular sub­strate.

Some representative K_M values are given in Table 8-2. It is to be stressed that the greater the af­finity of an enzyme for its substrate, the lower the K_M value. This is because the K_M value is numerically equal to the substrate concentration at which half of the enzyme molecules are associated with substrate (i.e., in the ES form).

The relationships derived above are based on reac­tions in which a single substrate molecule is bound to the enzyme. However, they also apply in situations where more than one substrate is bound to the en­zyme, as long as the concentrations of all but one sub­strate species are held constant or are not rate- limiting (i.e., present in large excess).

For example, even the pioneering studies of Michaelis and Menten involved a bimolecular enzyme-catalyzed reaction. They employed the enzyme invertase, which forms a complex with one molecule of sucrose and one mole­cule of water and releases glucose and fructose as products. However, in this reaction (as in nearly all hy­dropses), the concentration of water remains virtu­ally unaltered during the course of the reaction.

K_u values are seldom determined using the rela­tionship given in equation 8-22, because the enzyme- substrate concentration [ES] cannot easily be mea­sured. Some additional mathematical manipulations may be carried out to convert equation 8-22 into a more useful form.

As seen in Figure 8-4, the initial reaction velocity increases with substrate concentration until the en­zyme concentration becomes limiting, and at that time, the initial reaction velocity approaches a maxi­mum value (i.e., V_max). This occurs because nearly all the enzyme is maintained in the ES form. Thus, [ES] will equal [E]_T, and k₃[E]_T corresponds to V_max.

By substituting V_max for k₃[E]_T in equation 8-27, we obtain

Equation 8-28 is the Michaelis-Menten equation. From this equation, it may be seen that when the sub­strate concentration is numerically equal to the K_M value of the enzyme, then the reaction velocity is equal to one-half the maximum value. For example, if [S] and K_m are both equal to 3, then equation 8-27 simpli­fies to

Consequently, the Michaelis-Menten constant for an enzyme may be determined from the substrate con­centration at which the reaction velocity proceeds at one-half its maximum value (Fig. 8-5).

From the above, we may conclude that the maxi­mum velocity of an enzyme-catalyzed reaction de­pends on the affinity between the enzyme and its sub­strate (i.e., the K_M value). The higher the K_M value is, the lower the affinity of the enzyme for the substrate; and the smaller the K_M value, the greater the affinity.

In many instances, the same substrate may enter ei­ther of several different enzyme-catalyzed reactions occurring in cells. Which of the alternative reactions predominates in the cell depends in part on the K_M val­ues of the respective enzymes and the concentration of available substrate.

At very low substrate concen­trations, the specific reaction catalyzed by the enzyme with the lowest K_M will dominate, whereas at higher substrate concentrations, the reaction catalyzed by the enzyme having the greatest K_M value can domi­nate (if enough of the enzyme is present).

Therefore, which of several different metabolic pathways is actu­ally followed by the initial substrate in the course of a series of enzyme-catalyzed reactions may be regu­lated by controlling the amount of available substrate. The relationship between re­action velocity and substrate concentration for two en­zymes that act on the same substrate is depicted in Figure 8-6.

To obtain the K_M value of an enzyme experimen­tally, it is necessary to determine v₀ for a series of sub­strate concentrations. In practice the evaluation of K_M from a plot similar to that in Figure 8-6 is difficult, because the precise value of V_max cannot be deter­mined. This is because the curve is a rectangular hy­perbola that approaches V_max asymptotically. In 1934, H. Lineweaver and D. Burk introduced a different form of the Michaelis-Menten equation (8-28) that simplifies the determination of K_M from experimen­tally obtained values of [S] and v₀. In this form, the ac­quired data are used to construct a straight line rather than a hyperbola. By taking the inverse of equation 8- 28 we obtain

The terms are now arranged in the form of the gen­eral equation of a straight line, y = ax + b, in which a is the slope of the line and b is the intercept on the y axis. Thus, for equation 8-31, 1/v₀ serves as the y axis and 1/[S] as the x axis. Consequently, the y intercept will be 1/v_max, the slope will be K_M/V_max, and the x inter­cept will be -1/K_M. The graph showing this relation­ship (known as a double reciprocal plot) is presented in Figure 8-7.