In the Classroom
Understanding Enzyme Inhibition Raymond S. Ochs Department of Pharmaceutical Sciences, St. John’s University, Jamaica, NY 11439;
[email protected] Reversible enzyme inhibition is a well-established phenomenon; there is little debate about its significance or the mathematical equations describing it. However, it is a surprisingly difficult subject for students to absorb; even established scientists have difficulties with enzyme inhibition. Most come away from this subject with a good idea of competitive inhibition and a fuzzy notion about everything else! I contend that the real difficulty is twofold. First, there exists no clear picture to conceptually illustrate inhibition types other than competitive. Second, most derivations, particularly at the elementary level, focus on the wrong kinetic concepts. A Mental Picture While it is important to formulate a mathematically unambiguous representation of the ideas of enzyme inhibition so that they can be put into practical use, the concept that should come first is often missing. The only currently available mental picture is that of a “competitive” inhibitor. Here, the substrate and inhibitor are in competition for the free enzyme. To arrive at a picture of the other types, we need to consider mechanistically what the other inhibition forms are. The basic Michaelis–Menten model describing the simplest enzyme mechanism is E+S
ES → E + P
(1)
The three types of inhibition are competitive, uncompetitive, and mixed.1 For each type, the equation is modified by the addition of equilibria between the inhibitor, I, and a form of the enzyme. For a competitive inhibitor, I binds exclusively to E, the free enzyme form, and the extra equation is E+I
EI
(2)
Now, if the inhibitor instead binds exclusively to ES, the bound enzyme form, then the equilibrium that modifies eq 1 is ES + I
ESI
(3)
Finally, an inhibitor that has a significant binding to both E and ES is called “mixed”. In summary, Inhibition type
Equations that describe it
Competitive
1 and 2
Uncompetitive
1 and 3
Mixed
1, 2, and 3
With these basic definitions, we can move to the mental image. Virtually all students immediately picture competitive inhibition. Substrates with structural similarity to inhibitors, but not used in the reaction, are often competitive inhibitors. It is easy to formulate the idea because we have so many analogies to draw on. In an ecosystem, different organisms compete for food; in academic life, students compete for grades; in financial life, businessmen compete for money. But what of uncompetitive inhibition? We need an everyday explanation for the idea that an inhibitor can bind
not to the free enzyme, but rather to the ES form. Consider the analogy of the honey trap. A bear attempts to get honey from a hive, but owing to a constriction in the top, he can’t get his paw out after it is filled with honey. He is stuck (inhibited). Note that the inhibition would not arise unless he tried for the honey and became “bound” to it. For this analogy, the bear is the enzyme E; the honey is substrate S; the hive is the inhibitor I. Alternatively, consider a law-enforcement “sting” operation. The suspect is offered an especially attractive deal involving illegal contraband. When she accepts it, an arrest is made (she becomes inhibited). Note that inhibition would not arise unless she tried for the contraband and became “bound” to it. For this analogy, the suspect is the enzyme E; the contraband is the substrate S; the law enforcement agent is the inhibitor I. Once the picture is grasped, the notion of uncompetitive inhibitors can immediately become an intuitive one; and since mixed inhibition merely combines competitive and uncompetitive effects, the mental picture for all forms of reversible inhibition is complete. To extend this understanding as it directly applies to enzymes, we need to consider the behavior of the velocity of the enzyme as a function of both changing substrate and changing inhibitor concentrations. Here too, I suggest we take an approach that is a departure from tradition but will lead to a much firmer understanding. Analysis of Enzyme Velocity Equations with Reversible Inhibitors Present Following the lead of virtually all biochemistry texts and eqs 1, 2, and 3, along with the appropriate conservation equations for the enzyme forms E, ES, EI, and ESI, we have three equations for the initial velocity, vi. For competitive inhibition, Vmax S vi = (5) I Km 1 + + S Ki for uncompetitive inhibition,
Vmax S
vi =
Km + S 1 +
I Ki
(6)
and for mixed inhibition
Vmax S
vi = Km 1 +
I I + S 1+ Ki Ki
(7)
Note that the last is actually for the case of equal affinities of I for E and ES; otherwise there would be two different constants (Kic and Kiu) representing the different affinities.
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In the Classroom
Figure 1. Competitive inhibition. In the direct plot of initial velocity (vi) vs [S], inhibition is maximal at low [S] (the Vmax/Km tangent line of the control) and diminishes to zero as [S] approaches infinity (the Vmax asymptote of the control).
Figure 2. Uncompetitive inhibition. In the direct plot (v i vs [S]) inhibition is maximal at high [S] (the Vmax asymptote of the control) and diminishes to zero as [S] approaches zero (the Vmax/ Km tangent line of the control).
Graphs of the results of the three types of inhibition are displayed in Figures 1–3. I have also plotted two straight lines, designated as Vmax and Vmax/Km. Vmax is the asymptote of the control (uninhibited) curve as [S] → ∞. Vmax/Km is the tangent to the curve at [S] = 0. Clearly, the parameters Vmax and Vmax /Km characterize the curve; it should also be noted that they have a simple yet important interpretation in terms of the mechanism of eq 1. Vmax is the velocity when all the enzyme is in the form of ES, at high [S]. Vmax/Km is the apparent first-order rate constant when essentially all the enzyme is in the form E, at low [S]. These kinetic constants are therefore our focal points and the key to understanding enzyme inhibition. Put another way, we are examining the extremes of substrate effects on velocity in direct space, rather in reciprocal space (the traditional approach; i.e., the Lineweaver–Burke plot). Consider first the competitive inhibition curve (Fig. 1). At low [S]—near the Vmax/Km line—the competitive inhibitor causes a substantial drop in enzyme activity; but the inhibitor has less influence as [S] increases, and in the limit of infinite [S], there is no inhibition. In the uncompetitive inhibition curve (Fig. 2), exactly the opposite behavior is apparent. Here, there is no effect of inhibitor at low [S]—near the Vmax/K m line—and inhibition increases as [S] increases, becoming maximally effective as [S] approaches infinity. While it is true that any ES that is formed even at low [S] can bind I, the formation of ESI displaces the equilibrium E + S = ES to the right, just offsetting the inhibition. As [S] becomes saturating, the displacement of the equilibrium is irrelevant; all the enzyme is in the ES form. As this is the form to which the uncompetitive inhibitor binds exclusively, inhibition is maximal. Finally, the mixed inhibition curve (Fig. 3) is just a combination of the two we have already considered. This follows from the fact that the inhibitor binds both the free (E) and bound (ES) forms of the enzyme. We can appreciate the result because we already understand these two events in their extremes—at zero and infinite substrate concentrations. At low substrate concentration, where the uninhibited velocity
approaches Vmax/K m, there is no contribution from the uncompetitive inhibitor, so here the mixed inhibitor acts like a competitive inhibitor; note how the mixed and competitive inhibition become the same as [S] → 0. At high [S], there is no contribution from the competitive type of inhibition; here mixed inhibition approaches uncompetitive. Thus, mixed and uncompetitive inhibition become the same as [S] → ∞. Another way of looking at reversible inhibition is that all three types of inhibition—competitive, uncompetitive, and mixed with equal affinities for (E) and (ES)—are just limiting cases of a single more general type in which the inhibitor has some affinity for both enzyme forms.
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Textbook Comparisons Examination of several current biochemistry textbooks (1–11) reveals that none have considered enzyme inhibition from the standpoint of the kinetic terms Vmax and Vmax/Km; all have used Vmax and Km instead. Virtually all employ double reciprocal plots as an implicitly essential part of the analysis. The use of double reciprocal plots for data analysis has serious disadvantages (12); for pedagogical purposes they are even more unsuitable, as they confuse more than they enlighten. Some treatments (e.g., 13) omit uncompetitive inhibition because it is rarely encountered. This ad populum argument is unsound when it is realized that competitive and uncompetitive types are extreme forms of inhibition, reflecting the affinity of an inhibitor for the E or ES form of the enzyme, respectively. Without understanding each individually, mixed inhibition has no conceptual basis. It may also be argued that uncompetitive inhibition is more generally that form arising when I binds any enzyme form other than the free form, E. Thus, this inhibition type is actually common in multisubstrate cases, which are, after all, also more realistic. Yet introducing multisubstrate kinetics is an enormous burden on those exposed to kinetics for the first time. It becomes easier to understand only after the basic idea of inhibition types is in place.
Journal of Chemical Education • Vol. 77 No. 11 November 2000 • JChemEd.chem.wisc.edu
In the Classroom
may allow more comprehensive analysis (17), although the aim here is really to extract constants rather than to explain fundamentals. Finally, there are extensions of enzyme kinetics to enzymes acting at membrane surfaces, where the kinetics are dominated by the surface interactions and become quite different (18). None of these approaches specifically addresses a new way to fundamentally view enzyme inhibition. The Advantages of Vmax/Km
Figure 3. Mixed inhibition. In the direct plot (vi vs [S]), inhibition is evident at all concentrations of substrate. Thus the inhibitor behaves as if it were competitive at low [S] (concentrations where the curve approaches the Vmax/Km tangent line of the control) and noncompetitive at high [S] (concentrations where the control graph approaches the Vmax asymptote).
Educational Literature There exists an extensive literature beyond the textbook treatment, which aims to extend and clarify enzyme kinetics and enzyme inhibition. The meanings of the kinetic constants Vmax and Vmax/Km were themselves recently explored (14 ), although not with an emphasis on enzyme inhibition. In this article, Northrop re-explains these constants as “release” and “capture” constants, which is more meaningful than the standard interpretation, particularly with more complex enzyme mechanisms than the one examined here. This treatment also shows the value of reciprocal plots for a deeper understanding of these constants. However, the approach is far more advanced and general than the one presented here and befits the advanced student rather than the beginner who needs a fundamental understanding of enzyme inhibition. Another approach is the presentation of computer programs to allow direct visualization of kinetic results (15, 16 ). Still others emphasize various types of graphical transformations that Table 1. Influence of an Enzyme Inhibitor on Kinetic Constant Pairs Vmax, Km and Vmax, Vmax /Km Inhibition Type Competitive
Note 1. There are also problems in nomenclature. Mixed inhibition is also called noncompetitive, which leads to some conflict with uncompetitive. Some differentiate mixed from noncompetitive. Since our objective is the most effective teaching of the concepts, the word mixed is used throughout this treatment. After the idea is clear, it is a simple matter for students to learn alternative definitions and more formal treatments.
Effect of Inhibitor on Vmax
Km
—
Vmax
Vmax /Km
— —
Uncompetitive Mixed
Table 1, modified from Cornish-Bowden (19), compares the use of the pair Vmax, Km with the pair Vmax, Vmax/Km in the analysis of reversible inhibitors. Note how the symmetry in the inhibition types emerges readily when Vmax/Km and Vmax are taken as kinetic constants. Interpretation of events using the pair Vmax and Km is far more complicated. A good deal of confusion arises from the more traditional focus on Km instead of Vmax/Km. Thus, Nahorski et al. (20) described uncompetitive inhibition as “unusual and counterintuitive”. Fell (21) similarly described uncompetitive inhibition as having “strange properties”, with “contradictory actions … reducing the apparent limiting rate (like a noncompetitive inhibitor), but at the same time it reduces the Km of the enzyme for its substrate, which would normally cause an activation”. This confusion of an inhibitor causing an “activation” stems entirely from trying to use Km rather than Vmax/Km to describe it. I realize it will be difficult for the more traditionally oriented to use direct velocity–substrate plots and embrace Vmax/Km for the understanding of enzyme inhibition. Moreover, many are more interested in discussing more elaborate mechanisms, of disdaining the simple Michaelis–Menten kinetic ideas in favor of more complex notions. However, all the more complex notions are, in one way or another, based on this bedrock. If it is not fully and clearly understood, our grasp of the more esoteric notions may seem lofty, but have no foundation.
—
N OTE: When enzyme inhibition is described in terms of the pair Vmax, Km, the changes are confusing. The common interpretation of Km as an affinity constant, in which changes are inverse to the expected rate, incurs the double problem of dealing with inverses and having an inhibitor exert an apparent stimulation (uncompetitive case). Moreover, the fact that Km is unchanged in the mixed case leads to no mechanistic insight. The opposing symmetry between competitive and uncompetitive is not apparent at all using the Vmax, Km pair. With the pair Vmax, Vmax/Km, all interpretations are direct, no assumption of affinity is needed, and symmetry is obvious.
Literature Cited 1. Horton, H. R.; Moran, L. A.; Ochs, R. S.; Rawn, J. D.; Scrimgeour, K. G. Principles of Biochemistry; Prentice Hall: Englewood Cliffs, NJ, 1996. 2. Murray, R. K.; Granner, D. K.; Mayes, P. A.; Rodwell, V. W. Harper’s Biochemistry; Appleton & Lange: Stamford, CT, 1996. 3. Marks, D. B.; Marks, A. D.; Smith, C. M. Basic Medical Biochemistry; Williams & Wilkins: Baltimore, 1996. 4. Matthews, H. R.; Freedland, R. A.; Miesfeld, R. L. Biochemistry. A Short Course; Wiley-Liss: New York, 1997. 5. Garrett, R. H.; Grisham, C. M. Biochemistry; Saunders: Fort Worth, TX, 1995. 6. Zubay, G. Biochemistry; Wm. C. Brown: Dubuque, IA, 1998.
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In the Classroom 7. Devlin, T. M. Textbook of Biochemistry with Clinical Correlations; Wiley-Liss: New York, 1997. 8. Tropp, B. E. Biochemistry. Concepts and Applications; ITP: Pacific Grove, CA, 1997. 9. Ritter, P. Biochemistry. A Foundation; ITP: Pacific Grove, CA, 1996. 10. Campbell, M. K. Biochemistry; Saunders: Philadelphia, 1999. 11. Meisenberg, G.; Simmons, W. H. Principles of Medical Biochemistry; Mosby: St. Louis, 1998. 12. Martin, R. B. J. Chem. Educ. 1997, 74, 1238–1240. 13. Garrett, R. H.; Grisham, C. M. Biochemistry; Saunders: Fort Worth, TX, 1995. 14. Northrop, D. B. J. Chem. Educ. 1998, 75, 1153–1157.
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15. Daron, H. H.; Aull, J. L. Comput. Appl. Biosci. 1986, 2, 207–209. 16. Czerlinski, G.; Sikorski, J. J. Chem. Inf. Comput. Sci. 1976, 16, 30–33. 17. Ehrlich, B. E.; Watras, J. Nature 1988, 336, 583–586. 18. Nelsestuen, G. L.; Martinez, M. B. Biochemistry 1997, 36, 9081–9086. 19. Cornish-Bowden, A. Fundamentals of Enzyme Kinetics; Portland: London, 1995. 20. Nahorski, S. R.; Ragan, I.; Challiss, R. A. J. TIPS 1991, 12, 297–303. 21. Fell, D. Understanding the Control of Metabolism; Portland: London, 1997.
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