In the Classroom edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Reformulation of the Michaelis–Menten Equation: How Enzyme-Catalyzed Reactions Depend on Gibbs Energy
W
Brian J. Bozlee Department of Chemistry, Hawaii Pacific University, Kaneohe, HI 96744- 5297;
[email protected] Textbooks in biochemistry universally present a simplified two-step reaction mechanism for enzyme-catalyzed reactions (1), sometimes accompanied by the corresponding reaction profile shown in Figure 1 (2). The enzyme, E, binds the substrate, S, reversibly to form an intermediate species called the enzyme–substrate complex, ES. The intermediate then breaks down irreversibly to form product, P, and regenerate the enzyme catalyst. The net reaction is simply transformation of substrate to product, S → P. Textbooks in biochemistry (3) point out that the second step of the mechanism is rate determining. Thus they argue that the second step is all that needs to be considered in determining the overall rate of product formation. The rate of the second step of the reaction mechanism will be increased if the enzyme can lower the activation energy, G4 − G3 . This may be done in two ways: (i) by raising the Gibbs energy of the ES intermediate, G3 , or (ii) by lowering the Gibbs energy of the second transition state, G4 . The chemical means by which this is accomplished is outside the scope of the present discussion. The argument that only the activation energy of the second step is important in determining the rate of product formation may be an oversimplification. The velocity of the rate-determining step does not depend on subsequent events but does depend on previous steps. In this context one might wonder if the steady-state concentration of the ES intermediate is decreased when G2 and G3 are raised, thus slowing the second step and overall production of product. Is it possible that raising G3 could both speed and slow the rate of product formation? This is the question that is addressed in this article.
Michaelis–Menton Kinetics and Gibbs Energies The velocity of reaction, v, is the rate of product formation. A kinetic analysis of the mechanism in Figure 1 results in the well-known Michaelis–Menten equation (4): v =
vm [ S ] K M + [S ]
(1)
where [S] is concentration of substrate, vm is the maximum velocity of reaction when the enzyme is flooded with excess substrate, and KM is a characteristic constant for the enzyme. Both vm and KM depend on the rate constants shown in Figure 1
v m = k 2 [E ] T
(2)
where [E]T is the total concentration of bound and unbound enzyme and k −1 + k 2 k1
KM =
(3)
A simple way to illustrate the dependence of reaction rate on the Gibbs energies G2, G3, and G4 is to replace the rate constants in the Michaelis–Menten equation (see the Supplemental MaterialW) with the following Arrhenius expressions: k1 = A1
− G2 e RT
k −1 = A −1 k 2 = A2
− (G 2 − G 3 ) e RT
− (G 4 − G3 ) e RT
We have chosen G1 to be the reference Gibbs energy, set arbitrarily to zero. The result of such a substitution, after rearrangement, is
v m = A2 [ E ]T e
−(G 4 − G 3 ) RT
(4)
and G3
K M = e RT Figure 1. Reaction profile for a two-step enzyme-catalyzed reaction. The k values are rate constants for each step in the mechanism.
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A −1 A1
+
A2 e A1
− (G 4 − G 2 ) RT
(5)
Equation 4 verifies that the maximal velocity of reaction does depend solely on the activation energy of the second
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step (G4 − G3). But it is interesting to note that KM (eq 5) depends on G2 and G3 in such a way that as they increase, KM also increases. This leads to a slowing of the reaction velocity (eq 1). This effect is particularily apparent when the substrate concentration is low. In short, if an enzyme raises G3 (and thereby G2 also by the Hammond postulate) it will speed the reaction by increasing vm, but it will also slow the reaction by increasing KM. Thus there is a complicated interplay between two competing effects. Assessing the Magnitude of the Effect To assess the theoretical effect of raising the Gibbs energy of the ES complex alone, it might be desirable to use an initial value of G3 consistent with laboratory measurements. To this end, eqs 4 and 5 may be used to fit experimental data in which vm and KM are measured as a function of temperature. For example Bhuiya et al. (5) have measured the kinetic parameters of a hyperthermophylic glutamate dehydrogenase found in Aeropyrum pernix K1, which binds its substrate, L-glutamate, and causes oxidative deamination. Their data are shown in Table 1 along with calculated values based on eqs 4 and 5. The KM values in Table 1 were reported directly by Bhuiya et al. (5). The vm values in the table were calculated from their reported catalytic efficiencies, defined as vm兾KM. The vm data were fit by eq 4 assuming A2 = 3.74 × 1010 min᎑1 and (G4 − G3 ) = 59.3 kJ. The KM data were fit by assuming that the second term of eq 5 does not contribute significantly in this case, since KM is decreasing with temperature rather than increasing. Dropping the second term and using (A᎑1兾A1) = 0.00355 with G3 = 15.2 kJ resulted in a reasonable fit to the data, except at 323 K. The source of the discrepancy at 323 K is unknown. Assuming the fitting parameters are reasonable we have the following approximate Gibbs energies for glutamate dehydrogenase: G3 = 15.2 kJ and G4 = 74.5 kJ. G2 was left undetermined in this analysis but should be close to G3. It is now of interest to see how changing G3 alone might affect the kinetics of this particular enzyme. Using the fitting constants determined here we find the following at 333 K (Table 2): A hypothetical 4.8 kJ increase in the ES Gibbs energy (G3 ) would result in about a 6-fold increase in both vm and KM. Figure 2 shows how the Michaelis–Menten plot of eq 1 would be altered by increasing G3. Conclusions If an enzyme raises the Gibbs energy of the enzyme–substrate complex (G3 ) it will increase both vm and KM. The former effect causes the rate of product formation to increase, whereas the latter effect causes the rate of product formation to decrease. The net result of these two competing effects appears to be that the rate of reaction does increase when G3 is increased as stated in the biochemistry textbooks, although a higher concentration of substrate is required to saturate the enzyme. WSupplemental
Material Discussion of the temperature dependence of the Gibbs terms in the Arrhenius equation is available in this issue of JCE Online. www.JCE.DivCHED.org
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Figure 2. Hypothetical effect on glutamate dehydrogenase kinetics if G3 is increased from 15.2 kJ (A) to 20.0 kJ (B). The assumed temperature is 333 K (60 ⬚C). The vertical dotted lines indicate the KM values and the horizontal dotted line indicates the vm of curve B.
Table 1. Experimental and Calculated Values of vm and KM T/K
vm/(µmol min᎑1)
KM/mM Exp
Calc
Exp
Calc
323
2.02
1.00
08
08
333
0.89
0.86
16
16
343
0.71
0.74
32
30
353
0.63
0.63
64
55
363
0.56
0.55
84
96
Note: Experimental data is from Bhuiya et al. (5).
Table 2. Calculated Effects of Changing G3 on vm and KM T/K
G3/kJ
G4/kJ
vm/(µmol min᎑1)
KM/mM
333
15.2
74.5
016
0.87
333
20.0
74.5
106
4.86
Literature Cited 1. Horton, H.; Moran, L.; Ochs, R.; Rawn, J.; Scrimgeour, K. Principles of Biochemistry, 2nd ed.; Prentiss–Hall: New Jersey, 1996; p 123. 2. Hansen, D.; Raines, R. J. Chem. Educ. 1990, 67, 483– 489. 3. Horton, H.; Moran, L.; Ochs, R.; Rawn, J.; Scrimgeour, K. Principles of Biochemistry, 2nd ed.; Prentiss–Hall: New Jersey, 1996; p 151, Figure 6-3. 4. Horton, H.; Moran, L.; Ochs, R.; Rawn, J.; Scrimgeour, K. Principles of Biochemistry, 2nd ed.; Prentiss–Hall: New Jersey, 1996; p 125. 5. Bhuiya, M.; Sakuraba, H.; Ohshima, T. Biosci. Biotechnol. Biochem. 2002, 66 (4), 873–876.
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