Article pubs.acs.org/jchemeduc
Illustrating Enzyme Inhibition Using Gibbs Energy Profiles Stephen L. Bearne* Department of Biochemistry & Molecular Biology and Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada S Supporting Information *
ABSTRACT: Gibbs energy profiles have great utility as teaching and learning tools because they present students with a visual representation of the energy changes that occur during enzyme catalysis. Unfortunately, most textbooks divorce discussions of traditional kinetic topics, such as enzyme inhibition, from discussions of these same topics in terms of Gibbs energy profiles. Examination of the changes in the values of the apparent kinetic parameters KSapp, kcatapp, and (kcat/KM)app in response to various modes of inhibition may be informative to students when presented in combination with Gibbs energy profiles. Herein, the symbolism of standard Gibbs energy profiles is utilized to derive expressions for the changes in Gibbs energy associated with the apparent kinetic parameters and to describe their behavior in the presence of either a competitive, uncompetitive, noncompetitive, or linear mixed-type inhibitor under rapid equilibrium conditions. The approach is intuitive and complementary to the traditional derivations of enzyme kinetic equations. KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Biochemistry, Inquiry-Based/Discovery Learning, Enzymes, Kinetics, Thermodynamics
G
to product (P) under initial velocity conditions (negligible back reaction from products). Although the ordinate is the standard Gibbs energy (G°′)3,7,10,11 and the abscissa represents the progress of the reaction, a number of authors have pointed out various difficulties in using Gibbs energy prof iles to represent enzyme-catalyzed reactions. First, the abscissa is best considered as some sort of progress variable for the stages in a mechanism as a variety of reaction coordinates probably apply in multibarrier mechanisms.11,20 Consequently, referring to the abscissa as the “reaction coordinate” or “extent of reaction” is not formally correct.6 Second, the ordinate may be defined as the standard Gibbs energy at pH 7 (G°′)3,6,7,10,11 so that changes in ground-state Gibbs energies or activation barriers are denoted as ΔG°′ or ΔG°′⧧, respectively.6 Some authors, however, point out that the use of chemical potentials might be more appropriate.4,6 Because of the problems of representing first-order rate processes and second-order rate processes on the same diagram and allowing for varying concentrations of the substrate and product, some authors favor the use of kinetic barrier diagrams.5,13,17,20,21 In these diagrams, the ordinate is usually equal to log(1/k) where k is a first-order rate constant and the resulting barriers represent the relative slowness of the reaction. Second-order rate constants are then written as pseudo-first-order rate constants by including the actual substrate or product concentrations,21 and equilibrium constants can be written as apparent equilibrium constants (e.g., KSapp = k−1/k1[S]).22 Despite these caveats, many authors
ibbs energy profiles are often used to describe the effects on enzyme catalysis that arise from various perturbations, such as mutations, isotopic substitutions, substrate alterations, and changes in solvent viscosity. Indeed, the use of Gibbs energy profiles offers an alternative to enzyme kinetic equations as a pedagogic paradigm for thinking about enzyme catalysis.1−19 Figure 1 shows a Gibbs energy profile for an enzyme (E) catalyzing the conversion of a single substrate (S)
Figure 1. Standard Gibbs energy profile for a single substrate enzyme. The abscissa is not to scale and is intended merely to show the temporal conversion of free enzyme (E) and substrate (S) to enzyme− substrate complex (ES), and subsequently to altered enzyme-bound substrate in the transition state (ES ⧧), and then to the free enzyme and product (P) under initial velocity conditions. Transition states for substrate and product dissociations are not shown and are assumed not to be rate-limiting. Hence, only the transition state for the chemical transformation of ES is shown. © 2012 American Chemical Society and Division of Chemical Education, Inc.
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Figure 2. Summary of kinetic mechanisms, Lineweaver−Burk plots, changes in the apparent kinetic parameters, and changes in Gibbs energy associated with various modes of inhibition. In the Gibbs energy profiles, each energy level (indicated by a horizontal bar) corresponds to a specific enzyme species, i.e., E, ES, ES⧧, EI, and ESI. Enzyme species with bound inhibitor (i.e., EI and ESI) lie off the enzyme-catalyzed reaction pathway and therefore have been shown to the left side of the Gibbs energy profile. The dark blue profiles show the enzyme-catalyzed reaction under standard state conditions (i.e., [E] = [S] = [ES] = [I] = 1 M) in the presence of inhibitor, but the inhibitor has not bound to the enzyme species. The light blue profiles show the enzyme-catalyzed reaction when the inhibitor has bound to specific enzyme species and altered their concentration from the standard state concentration. Because E, S, and I are present in both the dark and light blue profiles, with inhibitor binding to enzyme species only in the latter profile, the total standard Gibbs energy of both profiles is equal and hence readily comparable by setting the Gibbs energy of the initial state equal to zero (i.e., GE°′ + GS°′ + GI°′ = 0). In the absence of inhibitor binding (dark blue profiles), the E and ES species correspond to EO and ESO, respectively. When the inhibitor is bound (light blue profiles), the E and ES species correspond to EI and ESI, respectively. The ordinates on the Gibbs energy profiles are labeled G′ to accommodate ΔG°′⧧ and ΔG°′ values (dark blue arrows) and ΔGapp′⧧ and ΔGapp′ values (light blue arrows). The prime symbol indicates that the Gibbs energy changes are at pH 7. Changes in Gibbs energy are drawn to scale with kcat = 102 s−1, KS = 10−5 M, Ki = 10−3 M, and αKi = 10−2 M (i.e., α = 10). See text for further discussion.
seem to agree that Gibbs energy profiles do offer a convenient and intuitive short-hand symbolism for complex reactions.1,2,15,20 Because the appearance of Gibbs energy profiles depends on the choice of standard state,1 the standard state will be defined as [E]total = [S] = [P] = [I] = 1 M with the temperature equal to 25 °C and pH = 7. The change in Gibbs energy accompanying the association of E and S (ΔGS°′) to form ES is given by eq 1, where KS is the dissociation constant for the ES complex. The Gibbs energies of activation for conversion of enzyme-bound substrate in the ground state (ES) to altered enzyme-bound substrate in the transition state (ES⧧) (i.e., ΔGES°′⧧) and for conversion of free E and free S to ES⧧ (i.e., ΔGE°′⧧) are given by transition-state theory as shown in eqs 2 and 3, respectively, where R is the gas constant, kB is the Boltzmann constant, h is Planck’s constant, and T is the absolute temperature.
ΔGS° ′ = −RT ln(1/KS)
(1)
⎛ kT ⎞ ° ⧧ ′ = RT ⎜ln B − ln kcat⎟ ΔG ES ⎝ ⎠ h
(2)
⎛ kT k ⎞ ΔG E° ′ ⧧ = RT ⎜ln B − ln cat ⎟ h KS ⎠ ⎝
(3)
These equations show that ΔGS°′, ΔGES°′⧧, and ΔGE°′⧧ depend on the reciprocals of 1/KS, kcat, and kcat/KS, respectively. Under standard-state conditions, the kinetic parameters KS, kcat, and kcat/KS (or the thermodynamic parameters ΔGS°′, ΔGES°′⧧, and ΔGE°′⧧) do not change; however, their apparent values may be altered as a function of inhibitor concentration.
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INHIBITION KINETICS To see how the apparent kinetic parameters are affected by the binding of a reversible inhibitor, the four types of inhibition 733
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noncompetitive inhibition pattern of lines with a common intersection on the x axis of a Lineweaver−Burk plot (i.e., at −1/KSapp) (Figure 2).
usually discussed in undergraduate biochemistry courses and textbooks will be examined: competitive, uncompetitive, noncompetitive, and linear mixed-type inhibition.23 The corresponding kinetic mechanisms are shown in Figure 2. For each kinetic mechanism, an initial velocity (vi) equation of the general form shown in eq 4 may be derived, where KSapp is the apparent dissociation constant for the enzyme−substrate complex (ES), kcatapp is the apparent first-order rate constant for the conversion of bound substrate into products, [E]T is the total enzyme concentration, and the coefficient of the substrate concentration ([S]) in the denominator is unity. The initial velocity equations discussed below are those derived using the rapid equilibrium assumption, that is, the rate constants for association of E and S (k1) and dissociation of ES (k−1) are much greater than the turnover number kcat (see Figure 1). As illustrated in the following discussion, both KSapp and kcatapp will differ from KS and kcat depending on the mode of inhibition. Detailed derivations of the initial velocity equations for the various modes of inhibition may be found elsewhere.23−32 vi =
app app V max [S] kcat [E]T [S] = app app KS + [S] KS + [S]
kcat
vi =
(
KS 1 + app kcat
and
[I] Ki
) + [S]
kcat
vi =
(1 + ) [I] Ki
(1 + ) app kcat =
+ [S]
where
KSapp =
KS
(
1+
[I] Ki
)
kcat
vi =
(1 + )
[I] Ki
(
[I] αK i
)
KSapp = KS
where
and
)
(9)
(1 + ) KS (1 + )
and
■
) + [S](1 + ) [I] αK i
[E]T [S]
[I] Ki
[I] αK i
app = kcat
where + [S]
KSapp
(10)
(1 + ) =K (1 + ) [I] Ki
S
[I] αK i
kcat
(1 + ) [I] αK i
(11)
GIBBS ENERGY PROFILES IN THE PRESENCE OF INHIBITORS When a given enzyme species binds an inhibitor, the concentration of that enzyme species is effectively altered resulting in a change in its apparent Gibbs energy. The effect of the various modes of inhibition shown in Figure 2 on the apparent Gibbs energy profiles that arise in the presence of inhibitor binding will now be examined. These profiles will be employed as an alternative approach to using initial velocity equations for deriving and describing the effect of inhibitor binding on KSapp, kcatapp, and (kcat/KS)app.
and
Competitive Inhibition Profile
As illustrated in Figure 2, a competitive inhibitor binds to free E, giving rise to an EI complex. This serves to lower the concentration of E from its initial value in the absence of inhibitor binding ([E]O) to a new value in the presence of inhibitor binding ([E]I) as shown by eq 12. It is important to note that contradictions may arise from Gibbs energy profiles when the standard states are not defined or are not uniform.13 When the inhibitor I is added to the system, the standard Gibbs energy (G°′) of all states, including transition states, will
kcat [I] Ki
i
(
1+
(6)
[E]T [S]
KS [I] Ki
)
[E]T [S]
KS 1 +
(5)
[I] Ki
(8)
kcat[E]T [S]
vi =
kcat[E]T [S]
(
[I] Ki
The kinetic mechanism for linear mixed-type inhibition is similar to that for noncompetitive inhibition except that the enzyme’s affinity for the inhibitor is altered by the factor α when substrate is bound. Equation 10 shows the initial velocity equation describing linear mixed-type inhibition.23 From eq 11, it can be seen that kcatapp will decrease with increasing concentration of inhibitor, while KSapp will increase relative to KS for 1 < α < ∞.23 This gives rise to a series of lines with a common intersection in the left quadrant of a Lineweaver− Burk plot (Figure 2).
Equation 6 shows the initial velocity equation describing uncompetitive inhibition.23 From eq 7, it can be seen that both kcatapp and KSapp will decrease by the same factor with increasing concentration of inhibitor. However, the ratio (kcat/KS)app does not change, giving rise to the classical uncompetitive inhibition pattern of parallel lines (i.e., slope = (KS/kcat)app) on a Lineweaver−Burk plot (Figure 2).
KS + [S] 1 +
)
) + [S](1 + )
Linear Mixed-Type Inhibition
Uncompetitive Inhibition
vi =
[I] Ki
(
⎛ [I] ⎞ KSapp = KS⎜1 + ⎟ Ki ⎠ ⎝
= kcat
[I] Ki
KS + [S] kcat app kcat = [I] 1+ K
(4)
where
(
1+
From the initial velocity eq 5, where Ki is the dissociation constant for the enzyme−inhibitor complex (EI), it can be seen that kcatapp is not affected by the concentration of inhibitor ([I]).23 However, KSapp increases with increasing concentration of inhibitor. This gives rise to the classical competitive inhibition pattern of intersecting lines on the ordinate of a Lineweaver−Burk (double reciprocal) plot33 (Figure 2). kcat[E]T [S]
(
KS 1 +
Competitive Inhibition
vi =
kcat[E]T [S]
vi =
(7)
Noncompetitive Inhibition
Equation 8 shows the initial velocity equation describing noncompetitive inhibition.23 From eq 9, it can be seen that kcatapp will decrease with increasing concentration of inhibitor, while KSapp remains unchanged. This gives rise to the classical 734
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⎛ [I] ⎞ KSapp = KS⎜1 + ⎟ Ki ⎠ ⎝
increase uniformly by the amount GI°′ + RT ln[I] (or by the amount GI°′ assuming standard state conditions with [I] = 1 M) except for that of the bound enzyme−inhibitor species (i.e., EI or ESI).19 For the present discussion, the standard Gibbs energy of the state E + S + I is arbitrarily set at 0 on the ordinate (i.e., GE°′ + GS°′ + GI°′ = 0). Thus, in Figure 2, the dark blue profiles portray the standard Gibbs energy profiles in which all enzyme species (E and ES) are present in the presence of I but no binding of I has occurred. The light blue profiles, on the other hand, are the apparent Gibbs energy profiles when binding of I occurs.
[E]I = [E]O − [EI]
Thus, upon binding of a competitive inhibitor to the enzyme, the value of ΔGSapp′ becomes more positive relative to ΔGS°′, consistent with decreased apparent binding affinity (i.e., KSapp > KS), and the apparent Gibbs energy level for E is lowered relative to the value for the standard state as shown in Figure 2. In other words, the change in KSapp accompanying competitive inhibition is due to E being lower in Gibbs energy whereas the Gibbs energy of ES remains the same. Consequently, it is the formation of ES that appears less favorable due to E being lower in energy. The value of (kcat/KS)app is therefore decreased because its reciprocal is exponentially proportional to the Gibbs energy difference between E and ES⧧ (i.e., ΔGEapp′⧧ > ΔGE°′⧧), while the value of kcatapp is unchanged (i.e., ΔGESapp′⧧ = ΔGES°′⧧) as shown in Figure 2.
(12)
Lowering the concentration of E from [E]O to [E]I results in a decrease in the apparent Gibbs energy of the free E species. To determine the magnitude of this effect, the value of KSapp in the presence of inhibitor binding can be calculated. Substitution of the expression for the competitive inhibition constant Ki (eq 13) into eq 12 reveals that [E]O is reduced by the factor (1 + [I]/Ki) (eqs 14 and 15). Ki =
[E]I [I] [EI]
[E]I =
Uncompetitive Inhibition Profile
As illustrated in Figure 2, an uncompetitive inhibitor binds to the ES complex, giving rise to the ternary ESI complex. This serves to lower the concentration of ES from its initial value in the absence of inhibitor binding ([ES]O) to a new value in the presence of inhibitor binding ([ES]I) as shown in eq 22.
(13)
[E]I = [E]O −
[E]I [I] Ki
[ES]I = [ES]O − [ESI] (14)
(1 + ) [I] Ki
(15)
The Gibbs energy change associated with E binding S in the absence of inhibitor binding (ΔGSapp′) is given by eq 16. Under standard state conditions (i.e., [E]O = [S] = [ES] = 1 M), the second term on the right-hand side is zero and ΔGSapp′ = ΔGS°′. [E]O [S] [ES]
Ki =
⎧ [E]O [S] ⎪ [ES] ΔGSapp ′ = ΔGS°′ − RT ln⎨ ⎪ 1 + [I] Ki ⎩
ΔGSapp ′
) )
⎫ ⎪ ⎬ ⎪ ⎭
⎛ [I] ⎞ = ΔGS°′ + RT ln⎜1 + ⎟ Ki ⎠ ⎝
[ES]I =
[ES]I [I] Ki
(24)
[ES]O
(1 + ) [I] Ki
(25)
⎧⎛ [E][S] ⎞⎫ ΔGSapp ′ = ΔGS°′ − RT ln⎨⎜ ⎟⎬ ⎩⎝ [ES]I ⎠⎭
(17)
(18)
⎪
⎪
⎪
⎪
(26)
⎧⎛ [E][S] ⎞⎛ [I] ⎞⎫ ΔGSapp ′ = ΔGS°′ − RT ln⎨⎜ ⎟⎜1 + ⎟⎬ K i ⎠⎭ ⎩⎝ [ES]O ⎠⎝
(27)
⎛ [I] ⎞ ΔGSapp ′ = ΔGS°′ − RT ln⎜1 + ⎟ Ki ⎠ ⎝
(28)
⎪
⎪
⎪
⎪
Converting the changes in Gibbs energy into equilibrium constants yields eq 29, which is the same expression for KSapp as obtained from the initial velocity expression for uncompetitive inhibition using the rapid equilibrium approach (eq 7).
(19)
Converting the Gibbs energy changes into equilibrium constants via eq 20 yields eq 21, which is the same expression for KSapp as obtained from the initial velocity expression for competitive inhibition using the rapid equilibrium approach (eq 5). ⎛ [I] ⎞ RT ln KSapp = RT ln KS + RT ln⎜1 + ⎟ Ki ⎠ ⎝
(23)
[ES]I = [ES]O −
However, in the presence of a competitive inhibitor, eq 16 becomes eq 17 where [E]I < [E]O; and substitution of eq 15 into eq 17 yields eqs 18 and 19 (as [E]O = [S] = [ES] = 1 M).
( (
[ES]I [I] [ESI]
(16)
⎧⎛ [E] [S] ⎞⎫ ΔGSapp ′ = ΔGS°′ − RT ln⎨⎜ I ⎟⎬ ⎩⎝ [ES] ⎠⎭
(22)
Lowering the concentration of ES from [ES]O to [ES]I results in a decrease in the apparent Gibbs energy of the ES species. Using the same approach as outlined above for competitive inhibition, the value of K Sapp in the presence of an uncompetitive inhibitor can be calculated (eqs 23−28).
[E]O
ΔGSapp ′ = ΔGS°′ − RT ln
(21)
KSapp =
KS
(1 + ) [I] Ki
(29)
As shown in Figure 2, the apparent Gibbs energy level for ES is lowered relative to the value for the standard state, making formation of ES appear more favorable. Thus, upon binding of
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an uncompetitive inhibitor, the value of ΔGSapp′ becomes more negative, consistent with an increase in the apparent binding affinity (i.e., KSapp < KS). The value of (kcat/KS)app does not change because the standard Gibbs energy of the free E species is not altered (i.e., ΔGEapp′⧧ = ΔGE°′⧧); however, the value of kcatapp is decreased because its reciprocal is exponentially proportional to the difference in Gibbs energy between ES and ES⧧ (i.e., ΔGESapp′⧧ > ΔGES°′⧧ as shown is Figure 2).
Substitution of eqs 34 and 35 into eq 36 yields eqs 37 and 38. ⎧⎛ [E] [S] ⎞⎫ ΔGSapp ′ = ΔGS°′ − RT ln⎨⎜ I ⎟⎬ ⎩⎝ [ES]I ⎠⎭
ΔGSapp ′
Noncompetitive Inhibition Profile
A noncompetitive inhibitor binds to both E and ES, giving rise to the EI and ESI complexes, respectively. The binding affinity is not affected by the presence of the substrate. This serves to lower the concentration of EO to EI and ESO to ESI to the same extent as indicated by eqs 15 and 25, respectively. This results in a decrease in the apparent Gibbs energy of both the E and ES species. Using the same approach as outlined above for competitive and uncompetitive inhibition, the value of KSapp observed upon binding of a noncompetitive inhibitor can be calculated (eqs 30−32). ⎧⎛ [E] [S] ⎞⎫ ΔGSapp ′ = ΔGS°′ − RT ln⎨⎜ I ⎟⎬ ⎩⎝ [ES]I ⎠⎭ ⎪
⎪
⎪
⎪
⎧⎧ [E]O [S] ⎪⎪ [ES]O app ΔGS ′ = ΔGS°′ − RT ln⎨⎨ ⎪⎪ 1 + [I] Ki ⎩⎩
( (
) )
ΔGSapp ′
= ΔGS°′
KSapp
(30)
⎫ ⎫ ⎪⎛ [I] ⎞⎪ ⎬⎜1 + ⎟⎬ K i ⎠⎪ ⎪⎝ ⎭ ⎭ (32)
[E]I =
■
[ES]I =
[I] Ki
[I] αK i
(38)
( (
[I] Ki [I] αK i
) ⎫⎪⎬ ) ⎪⎭
(
KS 1 + =
(1 +
) )
[I] Ki
[I] αK i
(39)
ASSOCIATED CONTENT
A worked example to demonstrate the construction of a Gibbs energy profile in the presence of a competitive inhibitor. This material is available via the Internet at http://pubs.acs.org.
■
(34)
[ES]O
(1 + )
⎧ ⎪ 1+ = ΔGS°′ + RT ln⎨ ⎪ 1+ ⎩
) )
S Supporting Information *
(33)
[E]O
(1 + )
(37)
( (
SUMMARY Although the traditional approach to illustrate the changes in the apparent kinetic constants (i.e., KSapp, kcatapp, and (kcat/ KS)app) has focused on examination of the initial velocity equations, the present work describes a related approach that focuses on describing how the Gibbs energy profile for an enzyme-catalyzed reaction is altered upon inhibitor binding. This approach assumes a rapid equilibrium for both inhibitor and substrate binding relative to the rate of turnover. By adjusting the Gibbs energies of E and ES in accord with changes in concentration of these species away from their standard state concentrations that result from the formation of inhibitor bound species (i.e., EI and ESI), it is possible to determine the resulting changes in the apparent kinetic parameters without necessarily using initial velocity equations. A pedagogic advantage of this approach is that it provides students with a striking visual indication of the Gibbs energy changes that accompany enzyme inhibition.
A linear mixed-type inhibitor binds to both E and ES, giving rise to the EI and ESI complexes, respectively; however, unlike a noncompetitive inhibitor, a linear mixed-type inhibitor binds to E and ES with different affinities. Using the equations defining the inhibition constants describing linear mixed-type inhibition (Ki and αKi in eq 33), it can be shown that the concentration of both E and ES are both lowered in the presence of a linear mixed-type inhibitor, but each to a dif ferent extent as shown in eqs 34 and 35. [ES]I [I] [ESI]
⎫ ⎧ [E]O [S] ⎪ [ES]O ⎛ [I] ⎞⎪ = ΔGS°′ − RT ln⎨ ⎟⎬ ⎜1 + α K i ⎠⎪ ⎪ 1 + [I] ⎝ Ki ⎩ ⎭
■
Linear Mixed-Type Inhibition Profile
αK i =
(36)
Thus, for a linear mixed-type inhibitor, > KS for 1 < α < ∞.23 The apparent Gibbs energy levels for E and ES are lowered relative to the values for the standard state, but to different degrees (i.e., ΔGEapp′⧧ > ΔGE°′⧧ and ΔGESapp′⧧ > ΔGES°′⧧, but each is increased by a different amount) as shown in Figure 2. As a result, the values of kcatapp and (kcat/KS)app are decreased to different degrees.
From eq 32, it is evident that = KS upon binding of a noncompetitive inhibitor. Because the apparent Gibbs energy levels for E and ES are both lowered relative to the values for the standard state, the formation of ES appears neither less favorable nor more favorable. However, the values of both (kcat/ KS)app and kcatapp are decreased because their reciprocals are exponentially proportional to the difference in Gibbs energy between E and ES⧧, and between ES and ES⧧, respectively (i.e., ΔGEapp′⧧ > ΔGE°′⧧ and ΔGESapp′⧧ > ΔGES°′⧧ as shown in Figure 2).
[E]I [I] [EI]
⎪
KSapp
KSapp
Ki =
⎪
⎪
Converting the changes in Gibbs energy into equilibrium constants yields eq 39, which is the same expression for KSapp as obtained from the initial velocity expression for linear mixedtype inhibition using the rapid equilibrium approach (eq 11).
(31)
ΔGSapp ′
⎪
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
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ACKNOWLEDGMENTS This work was supported by Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
■
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