## B3 Enzyme Kinetics MichaelisMenten Model

The Michaelis-Menten equation is the standard model used to analyze kinetics of simple enzyme reactions . The theory (shown in Box 6.1) assumes that the substrate (5) first binds to the enzyme (E). The enzymesubstrate complex then reacts to yield product P.

 k+l E + S ^ ES (6.35) k-i k+2 ES ^ E + P (6.36) k_2

Box 6.1 Chemical basis of the Michaelis-Menten model.

Box 6.1 Chemical basis of the Michaelis-Menten model.

The Michaelis-Menten constant KM is the dissociation constant for eq. (6.35) and is given by

The initial rate of an enzyme reaction is 

At high substrate concentrations, the enzyme becomes saturated, and [■&V] = CE, where CE is the total enzyme concentration. The rate of the enzyme reaction is then at its maximum value Vmax:

Using these definitions of Vmilx and KM, the Michaelis-Menten equation takes the form

Equation (6.40) is usually linearized to eq. (6.41) for the estimation of Vmax and Km:

A graph of 1/V0 vs. 1/[S] provides KM/Vm&x as the slope and 1/Vmax as the intercept on the 1 /V0 axis. This is called a Lineweaver-Burke plot .

Data analysis problems created by linearization of equations were discussed in Section 2.A.3. In this case, we realize that the Michaelis-Menten model for enzyme kinetics can be formulated in terms of the expressions in Table 6.1. To make this clear, we express the scheme in Box 6.1 in terms of the symbols we have been using throughout this chapter. This is shown in Box 6.2. Because only bound enzyme leads to products, we can express the initial rate V0 in the form:

Box 6.2 Michaelis-Menten model using generalized symbols.

Expanding this expression using the definition of fM,\ leads to the model in Table 6.8. The model is simply a different formulation of the MichaelisMenten equation and in fact can be derived from eq. (6.40). The constant

Farrell applied the model in Table 6.8 to the estimation of KM and Vmax for bovine NADP+:isocitrate dehydrogenase in the presence of several monovalent cations . Analysis of a particular set of data obtained from the author shows that the model provides a good description of the data (Figure 6.6). Nonlinear regression analysis gave KM = 2.26 ± 0.26 ¡jlM and Vmax = 163 ± 3 /xmol mkr'mg1 (relative units).

The traditional double reciprocal plot of the same data (Figure 6.7) shows considerably more scatter of data points around the regression line than the nonlinear regression results. Values obtained from the linear analysis were KM = 2.1 ± 0.1 y,M and Vmax = 161 ± 3 /umol min_1mg_1. These results are similar to those from nonlinear regression. We shall see in later sections of this chapter that the linear regression method has limited ability in the analysis of more complex enzyme kinetics, such as those involving inhibition.

B.4. The pH Dependence of Enzyme Activity

The influence of pH on the activity of an enzyme depends on the acid-base properties of the enzyme and the substrate. The shape of the pH vs.

Table 6.8 Model for Simple Michaelis-Menten Enzyme Kinetics

Assumptions: Michaelis-Menten enzyme kinetics (see ) Cx = [X] Regression equation:

Regression parameters:

K(=1IKM) Vaax Special considerations: K = 11KM Figure 6.6 Influence of the concentration of DL-isocitrate in ¡j.M on the relative initial velocity of the isocitrate dehydrogenase catalyzed reaction in presence of 100 ¡xM Mn2+. Points are experimental data; line computed as best fit onto model in Table 6.8 (The authors thank Dr. H. M. Farrell for the original data.)

Figure 6.6 Influence of the concentration of DL-isocitrate in ¡j.M on the relative initial velocity of the isocitrate dehydrogenase catalyzed reaction in presence of 100 ¡xM Mn2+. Points are experimental data; line computed as best fit onto model in Table 6.8 (The authors thank Dr. H. M. Farrell for the original data.) Figure 6.7 Double reciprocal Lineweaver-Burke plot of data in Figure 6.6 (Dashed lines show confidence intervals).

activity profile is often roughly bell shaped but may vary considerably depending on the system , By working under conditions where the enzyme is saturated with substrate, we can employ the formalism that addition or removal or protons will change the activity of the enzyme.

Data on bovine NADP+:isocitrate dehydrogenase will be used  to illustrate the linkage approach to analysis of an enzyme pH-activity profile. The roughly peak-shaped curve (Figure 6.8) suggests that an initial protonation activates the system, whereas a second protonation of this active form deactivates the system. The model can be expressed as

This is a stepwise or cooperative model, and the K represents association constants. That is, one proton binds before the second one can bind. According to the linkage approach, the activity v4obs is given by 

where Ax ,A2, and A3 are the activities of M, MH+, and MH2+, respectively.

The full model is summarized in Table 6.9. This model was used to analyzed pH dependence data on the relative activity of bovine NADP+:isocitrate dehydrogenase. The results are presented graphically in Figure 6.8. Parameter values from the analysis of these data follow:

log Ki = 8.6 ± 0.2 log K2 = 7.0 ± 0.4 /t, = 0 A2 = 68 ± 14 A3-A2= -35 ± 12