ANALYTICAL
BIOCHEMISTRY
206,394-399
(1992)
A Steady-State Kinetic Method for the Verification of the Rapid-Equilibrium Assumption in Allosteric Enzymes’ Marina M. Symcox2 and Gregory D. Reinhart Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019
Received
June
11,1992
A method for testing the validity of the rapid-equilibrium assumption as it might apply to allosteric enzymes using exclusively steady-state kinetic data is presented. The method is based upon a recognition that the ratio of apparent dissociation constants for the allosteric ligand, obtained under conditions of limiting and saturating substrate concentration, must yield the thermodynamic value for the coupling parameter between the substrate and allosteric ligand even in the general steady-state case. If this value is found to be equal to the apparent coupling parameter determined from the ratio of limiting values of the Michaelis constant for substrate obtained in the absence and saturating presence of the allosteric ligand, then the substrate can be correctly viewed as effectively achieving a binding equilibrium with the enzyme in the steady-state. The utility and limitations of this method are demonstrated by examining the ADP activation of beef heart mitochondrial NAD-dependent isocitrate dehydrogenase. o 1002 Academic
Press, Inc.
When studying the steady-state kinetic properties of allosteric enzymes, investigators almost always, explicitly or implicitly, assume that the enzyme and substrate achieve a near binding equilibrium in the steady-state. This rapid-equilibrium assumption greatly simplifies resulting kinetic expressions and transforms the interpretation of enzymatic responsiveness toward substrate and effector concentration to the thermodynamic
i This work was supported by Grant GM 33216 from the National Institutes of Health. ’ Current address: Department of Biochemistry andMolecular Biology, University of Oklahoma Health Sciences Center, Oklahoma City, OK. 3 Recipient of an Established Investigator Award from the American Heart Association, and to whom correspondence should be addressed. 394
realm. Consequently interpretations of the mechanistic causes of allosteric behavior are usually cast in terms of competing equilibria, either between various functionally defined enzyme forms (1,2) or between enzyme forms defined by their state of ligation (3,4). Unfortunately, the simplifications brought on by the rapid-equilibrium assumption, and its attendant conceptually straightforward models, provide such an enticing framework that whether the assumption isjustifiable can be overlooked. If the parameters obtained from steady-state kinetics measurements are kinetically constrained, and hence not thermodynamically based, then the relevance of the aforementioned models becomes dubious. Consequently, one must evaluate the validity of making the rapid-equilibrium assumption before giving credence to thermodynamically derived models. Several methods have been utilized to test for the validity of the rapid-equilibrium assumption, including isotope partitioning and use of a solvent viscogen (5). These techniques can present difficulties, particularly when applied to allosteric enzymes. Allosteric enzymes, for example, are characterized by conformations that are easily perturbed, and therefore adding viscogens to the solution can lead to complex results that are not easily interpretable. The analysis of isotope partitioning is complex when the enzyme exhibits nonhyperbolic saturation profiles, as well as often requiring the synthesis of isotopically labeled substrates. Of course one can also perform a thorough analysis of pre-steady-state kinetics in an effort to directly evaluate the magnitude of the rate constants on which the rapid-equilibrium assumption explicitly depends. However, this approach usually requires specialized instrumentation, copious amounts of enzyme, and an assay sensitive to the specific formation of the enzyme-substrate complex. We present below a method for assessingthe validity of the rapid-equilibrium assumption as it pertains to an allosteric enzyme using only initial velocity steady-state kinetics data. Although we derive the method from an 0003-2697/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
VERIFICATION
OF
THE
RAPID-EQUILIBRIUM
explicit analysis of the general rate equation resulting from a single-substrate, single-modifier mechanism, the principle behind the method is extrapolatable to oligomerit enzymes demonstrating cooperative substrate binding profiles. It only requires that the effect of the allosteric ligand be modest enough that the coupling free energy between the substrate and allosteric ligand can be measured explicitly. We then apply this method to the enzyme ICDH* and thereby demonstrate both the utility and certain limitations of this approach. THEORY
395
ASSUMPTION
We have defined these identical ratios to equal the thermodynamic coupling parameter Q. If neither A nor X achieves binding equilibrium in the steady-state, a rate equation much more complicated than Eq. [11, containing squared terms in both [A] and [Xl, is the correct solution to Scheme I. The general steady-state solution has been presented by Frieden (7). Using the notation defined above, the enzyme-form distribution equations that follow from the general solution are given by
Consider the following mechanism that describes in general the action of an allosteric ligand, X, on an enzyme catalyzing the reaction A + P:
sE+P ke 1 t k,
XE
[EAI = kW,,[Al k,, 11 k,,
+A% ;;- XEA 8
sXE+P
[A] + + [X] + & 1
(ks + kg + k,,) I
12
Scheme I It is well recognized (3,6,7) that if the rapid-equilibrium assumption is valid for the above mechanism, then the equation describing the rate of the reaction as a function of [A] and [X’j is given by
-=V Et
KO,K:,
V”%&U
+ QWAl[Xl
+ K&[A]
+ KL[X]
+ Q[A][X]
’
PI
where V” = maximal velocity in the absence of X, Kf., = the dissociation constant for A in the absence of X, Kf, = the dissociation constant for X in the absence of A, and W = V*lV’, where V” = the maximal velocity when X is fully saturating and Q = K&/KS, where Kz is the dissociation constant for A when X is fully saturating (3). Q is of considerable interest for so-called “K-type” modifiers because of its relation to the coupling free energy, AG,, between A and X: AGAx = -RT
In(Q).
PI
Regardless of whether the rapid-equilibrium assumption is valid, the following must be true, for it is merely a restatement of the principle of thermodynamic linkage (g-10):
used: ICDH, isocitrate 3-(morpholino)propane
dehydrogenase; sulfonic acid,
MgIC, DTT,
Mgdith-
151
1
[61
[XEAI = WddAl[Xl [A] + :
[X] + 2 1
7
12
k, + k,, + k&s 7 2
I
. 171
In the steady-state the apparent dissociation constants for A and X, K,, and K,, respectively, are given by
K = ([El + [XEl)[Al A WA1 + [XEAl) K = WI + [EAl)[Xl x
([XE]
+ [XEA])
PI
.
The limiting values of KA, obtained as [X] approaches either 0 or cc, are given by the following according to Eqs. [4-71:
x--o
WI
A b-1
lim (K A ) = [XE1[A1 X--CC
’ Abbreviations isocitrate; Mops, iothreitol.
1
[XEA]
k,
=-
ks + Ks~ K” k,
A’
WI
Consequently Q, as defined in Eq. [3], is obtained from the ratio of steady-state dissociation parameters per-
396
SYMCOX
taining to substrate (i.e., Michaelis constants) only the rapid-equilibrium assumption is satisfied, i.e. if k, k3 and k8 B b, in which case pA = K$ and Kz = Kg Next we consider the apparent steady-state values the dissociation constant of X, K,, as [A] approaches and co. Using Eqs. [4-71, as well as Eq. [3], we obtain lim
(K
)
=
X A-c0
lim
A--CO
(K
X
)
=
‘EI’XI ---c-z=-=
k,k,k,,
k,
LW
k&t&,,
ks
[EAI[JQ
[XEA]
=
k,,
k,,
=
KF
‘=
KO rx
AND
REINHART
if b : of 0
MATERIALS
[12]
[I31
Therefore, determining Q in this way must yield the true thermodynamic value even in the general case, and the rapid-equilibrium assumption can be evaluated by ascertaining if the apparent Q determined from KA is equal to the true value determined from K,. Mechanisms
Not Conforming
to Scheme I
Obviously we do not expect the preceding equations to explicitly apply to any mechanism other than that depicted in Scheme I. However, Scheme I serves as a paradigm for more complex allosteric mechanisms that often result from the oligomeric nature of most allosteric enzymes. As the concentration of substrate approaches zero, the enzyme is trapped in an equilibrium binding reaction with the allosteric ligand as depicted for the monomeric case by the left vertical equilibrium in Scheme I. As the concentration of substrate approaches saturation, the enzyme is also trapped in an equilibrium binding reaction with X as depicted by the right vertical equilibrium in Scheme I. For either zero or saturating levels of substrate, therefore, the rate constants pertaining to reactions drawn in the scheme in the horizontal direction, i.e., those involved in the binding of substrate and its turnover, will not influence the apparent binding interaction of X with the enzyme. These general features should be shared by mechanisms involving the binding of multiple equivalents of A and/ or X as is typical of oligomeric allosteric enzymes. Consequently evaluating whether Q derived from KA is equal to Q obtained from K,, while recognizing that these apparent dissociation parameters represent geometric mean values of individual ligand dissociation parameters in a multisite enzyme, should generally test the average applicability of the rapid-equilibrium assumption even in complex oligomeric cases.
‘This method can fail if &I& = $/ks. However, we know of no example of apparent rapid-equilibrium behavior that can be attributed to this circumstance, and even if this special condition were to be met, a thermodynamic value for Q is still realized from either the I$ or Kx data as described.
AND
METHODS
NAD+ (Grade III), Mops, DTT, sodium salts of threoD,,L,-isocitrate (Grade III), and ADP (Grade III) were purchased from Sigma Chemical Co. Other chemicals were analytical grade, and deionized, distilled water was used throughout. NAD-dependent isocitrate dehydrogenase was purified from beef heart mitochondria as described elsewhere (11). The enzyme was diluted, from an ammonium sulfate suspension stored at 4”C, into a buffer containing 0.8 M L&SO,, 90 mM Mops-NaOH, pH 7.2,l mM DTT and was stored on ice during the course of a day-long experiment without loss of activity (12). All kinetic assays were performed at 25°C in a final volume of 1.0 ml. Assay mixtures contained 90 mM Mops-NaOH, pH 7.4, 0.1 mM DTT, and variable concentrations of MgSO,, isocitrate, ADP, and NAD+. Reported concentrations of free isocitrate, free ADP, free Mg2+, and MgIC were calculated from stability constants for the metal complexation equilibria determined as described elsewhere (11). Unless otherwise qualified, the term ADP refers to free ADP, the species not complexed with Mg2+. Reported concentrations of NAD+ are total amounts, since NAD+ complexes free Mg2+ relatively weakly. Activity is expressed as a ratio of observed activity to maximal activity at 25°C. NADH production was monitored by following the increase in absorbance at 340 nm with a strip chart recorder. Initial velocities were calculated from estimated tangents of the early part of the recorder tracings. Typically, the final enzyme concentration varied from 0.1 to 0.5 pg/ml in the assay mixture. The half-saturation constants for MgIC were determined from Hill plot analysis (13). Approximately 12 initial velocity data points were used to define each MgIC saturation curve. V,, was estimated from plots of initial velocity versus [MgIC]. Rates falling between 10 and 90% of V,, were fit by least-squares linear regression to the Hill equation Wu~(V,,
- U)) = n log[MgIC]
-
n lOg(KA),
[14]
where the intercept on the abscissa equals log(K,) and the slope equals the Hill coefficient, n. Quoted values of KA pertain to the racemic mixture of threo-D,,L,-isocitrate, although threo-D,-isocitrate is the only active stereoisomer. The apparent dissociation constant for ADP at any given concentration of MgIC, K,, was taken to be equal to the concentration of ADP that produced half of the total enhancement in velocity at that Mg-isocitrate concentration. The enzyme was titrated with ADP at fixed levels of MgIC and NAD+. The dependence of the activation on the concentration of ADP was found to be hyperbolic, so Kx was determined from linear least-
VERIFICATION
OF
THE
I
I
I
-4.0
-3.0
-2.0
RAPID-EQUILIBRIUM
Log [Mg-isocitratel FIG. 1. Activation isocitrate dehydrogenase rithm of [Mg-isocitrate]. and the concentration tively.
of beef heart mitochondrial NAD-dependent by ADP expressed as a function of the logaThe concentration of NAD+ equals 0.15 mM, of ADP equals 0 (0) and 2.0 mM (O), respec-
squares fit of 15 to 20 velocity points to the double-reciprocal equation 1
KY
-= u - uo
(V,,
-
- uo) WPI
1
(Vm,, - ~0) ’
[151
397
ASSUMPTION
[ 141corresponds to the concentration of MgIC that produces an activity equal to one-half the maximal velocity. Figure 2 illustrates how the binding of ADP affects the initial velocity of ICDH when the concentration of ADP is varied at fixed concentrations of the substrate MgIC. Since ADP acts exclusively by decreasing the apparent KA for MgIC (Fig. l), its effect is greatest at intermediate concentrations of MgIC, as represented by 0.15 mM in Fig. 2, when ADP is able to shift the KA from above to below that concentration of substrate. At relatively high concentrations of substrate, such as 0.8 mM, which approach saturation even in the absence of ADP, the activity is not enhanced greatly. At the other extreme, when MgIC is substantially below KA even in the presence of ADP, little activation is also seen. Nevertheless, for any substrate concentration where activation can be measured, a value of ADP at which one-half of the activation is achieved can be determined. This concentration of ADP, determined in the present case by fitting the data in double reciprocal fashion to Eq. [151, is taken to be K,. Figure 3A shows the variation of KA with ADP while Fig. 3B gives the variation of K, with Mg-isocitrate at subsaturating concentrations of the cosubstrate NAD+ ([NAD+] = 0.15 mM, one-half its Michaelis constant). As we have found to be the case for other complex allosteric enzymes such as rat liver phosphofructokinase (14,15), the variation of KA with ADP can be fit well to
where u. is the initial velocity in the absence of ADP. Data describing the dependence of KA on the concentration of ADP were fit via nonlinear regression analysis to the following equation assuming a constant relative error as previously described (14,15): K
WI
A
RESULTS
ICDH catalyzes the formation of NADH, a-ketoglutarate, and CO, from NAD+ and isocitrate according to the overall reaction isocitrate + NAD+
= a-ketoglutarate
h .= :: 2
.-zf a
+ CO, + NADH
The absolute requirement of this reaction for Mg2+ results at least in part from the fact that MgIC is the true substrate of the enzyme (11). Free ADP activates ICDH by increasing the apparent affinity of the enzyme for MgIC as shown in Fig. 1. ADP has no effect on maximal velocity. The saturation curves both with and without ADP exhibit positive cooperativity, but the data can be described well by the Hill equation (Eq. [14]). The value of KA that results from fitting data such as these to Eq.
Log
[ADPI
FIG. 2. Influence of [ADP] on relative velocity drogenase with fixed [Mg-isocitrate] and [NAD+] citrate] = 0.80 mM (Cl), 0.15 mM (m), 0.05 mM (0). [15] by the method of linear least squares.
of isocitrate dehy= 5.0 mhi. [Mg-isoData were fit to Eq.
398
SYMCOX
AND
Eq. [16] as indicated by the solid curve in Fig. 3A. Fitting the data in Fig. 3A to Eq. [ 161 provides an accurate determination of the apparent Q, which we find to be equal to 9.2 f 0.3. Unfortunately an equation analogous to Eq. [ 161 does not describe the dependence of Kx on MgIC seen in Fig. 3B because of the cooperative nature of the MgIC binding. Moreover, the data in Fig. 3B are significantly scattered at low and high concentrations of MgIC, although a clear tendency of the data to form a plateau at each extreme is evident. The length of the double-headed arrow in Fig. 3B corresponds to the logarithm of the apparent Q determined from KA, i.e. the distance between the plateaus in Fig. 3A. It is apparent that this value agrees reasonably well with the distance between the two plateaus in Fig. 3B, particularly given the scatter in the data. Figures 4A and 4B show a similar comparison with a saturating concentration of NAD+ (5 mM). Once again there is agreement between the apparent Q determined
REINHART
-3.6 z 2 ~2
-3.8
1
-4.2 )I/ (-IW
-4.5
-3.5
Log
-2.5
[ADP]
-2.8 1
A 0
-3.2
a Y
-3.4
2 4
-3.6
-4.0 -4.15
-4.25
-3.15
Log
-4.5
-3.5
Log
[Mg-isocitrate]
FIG. 4. Mutual influence of the binding of MgIC and ADP with [NAD+] = 5.0 mM. (A) Apparent KA for Mg-isocitrate as a function of [ADP]. (B) Apparent Kx for ADP as a function of [MgIC]. The double-headed arrow in B corresponds to the logarithm of the apparent Q determined from a fit of data in A to Eq. [IS].
-3.8
-3.0:[ -4.0 t-INF)
-3.25
-2.5
[ADP]
-2.5 D
-4.5 -4.5
B
-4.0
-3.5
.
-3.0
Log [Mg-isocitrate] FIG. 3. Mutual influence of the binding of MgIC and ADP with [NAD+] = 0.15 mM. (A) Apparent KA for Mg-isocitrate as a function of [ADPJ. (B) Apparent Kx for ADP as a function of [Mg-isocitrate]. The double-headed arrow in B corresponds to the logarithm of the apparent Q determined from a fit of data in A to Eq. [16].
in Fig. 4A and the true value of Q indicated by the data in Fig. 4B. The apparent value of Q determined from the KA data presented in Fig. 4A is equal to 6.1 + 0.2. Despite significant scatter in Fig. 4B, limiting plateaus are evident. The distance between these two plateaus agrees with the magnitude of the log of the apparent Q determined in Fig. 4A as indicated by the length of the double-headed arrow. The results indicate that the rapid-equilibrium assumption is justified for both low and high concentrations of NAD+. DISCUSSION
We have previously estimated, on the basis of the magnitudes of r and KF for ICDH, that the rapidequilibrium assumption is likely to be valid unless the on-rate constants for MgIC (K, and & in Scheme I) are unusually small (16). Also in a two-substrate mecha-
VERIFICATION
OF
THE
RAPID-EQUILIBRIUM
nism when the second substrate (NAD+ in this case) is limiting, the rapid-equilibrium assumption with respect to MgIC is forced (unless NAD+ binds first in an ordered mechanism, which is not the case for ICDH (17,18)). However the data presented in Figs. 3 and 4 convey more explicit proof that the rapid-equilibrium assumption is appropriate at both low and high concentrations of NAD+. One important advantage of the method that we propose is that it uses essentially the same general assay conditions as routine measurements of steady-state kinetic activity. This not only makes this procedure easier to implement but also ensures that the validity of the rapid-equilibrium assumption will be assessed under the same experimental conditions in which the kinetic properties are otherwise being delineated. This latter consideration is particularly important for allosteric enzymes that are generally very sensitive to their environment. For example, when we attempted to use the viscosity-variation technique (19,20) to evaluate the rapid-equilibrium assumption, we found that the viscogens sucrose and glycerol caused decreases in the KA for MgIC that could not be explained by the effect that viscosity has on second-order rate constants (data not shown). Isotope partitioning experiments provide an alternative approach (5) but would have required in this case the time-consuming synthesis of radiolabeled isocitrate. The obvious difficulty associated with the method described herein is that the K, parameter can be difficult to determine as the limiting values of K& or Ki"f are approached since as these limits are approached the activation itself vanishes. In this case the velocity enhancement effected by ADP eventually becomes so small that random experimental error prohibits accurate identification of the point of half-maximal enhancement corresponding to Kx. This gives rise to the scatter apparent in Figs. 3B and 4B. The scatter that we observed should be a general feature of this method and necessitates an overdetermination of Kx at low and high substrate concentrations to define the plateau values.
399
ASSUMPTION
The observed reaction velocities can also need careful correction for any substantial blank rates, etc., particularly for those measurements made at low substrate concentration where the rates being measured are small. Nonetheless, with reasonable effort K& and Kg can be defined with a precision sufficient to provide for a satisfactory estimate of the true value of the coupling parameter Q. This value can then be used to check the validity of the rapid-equilibrium assumption. REFERENCES 1. Monod,
J., Wyman,
J., and Changeux,
J.-P.
(1965)
J. Mol.
Biol.
12,88-118. 2. Koshland, D. E., Jr., Nemethy, G., and Filmer, D. (1966) Biochemz+y6,365-385. 3. Reinhart, G. D. (1983) Arch. Biochem. Biophys. 224,389-401. 4. Reinhart, G. D. (1989) Biophys. Chem. 30,X9-172. 5. Cleland, W. W. (1986) in Techniques of Chemistry, (Bernasconi, C. F., Ed.),
Part
I, Vol VI, pp. 791-870,
Wiley,
New
6. Botts, J., and Morales, M. (1953) Trans. Faraday 707. 7. Frieden, C. (1964) J. Biol. Chem. 239, 3522-3531. 8. Wyman, J. (1948) Adv. Prot. Chem. 4,407-531. 9. Weber, G. (1972) Biochemistry 11, 864-878. 10. Weber, 11. Symcox,
G. (1975) Adv. M. M. (1991)
Prot. Chem. 29, l-83. Ph.D. thesis, University
12. Plaut, G. W. E., Schramm, V. L., and Aogaichi, Chem. 249, 1848-1856. 13. Hill, A. V. (1910) J. Physiol. London 40, iv-vii. 14. Reinhart,
G. D. (1985)
Biochemistry
24,
York. Sot. 49,
696-
of Oklahoma. T. (1974)
J. Biol.
7166-7172.
15. Reinhart, G. D., and Hartleip, S. B. (1986) Biochemistry 7308-7313. 16. Reinhart, G. D., Hartleip, S. B., and Symcox, M. M. (1989) Natl. Acad. Sci. USA 86,4032-4036.
25, Proc.
17. Rushbrook,
J. I. (1978) Ph.D. thesis, Rutgers University. 18. Gabriel, J. L., and Plaut, G. W. E. (1982) J. Bial. Chem. 8021-8029. 19. Brouwer,
A. C., and Kirsch,
J. F. (1982)
Biochemistry
257,
21, 1302-
1307. 20. Blacklow, Knowles,
S. C., Raines, R. T., Lin, W. A., Zamore, J. R. (1988) Biochemistry 27, 1158-1167.
P. D., and
BIOCHEMISTRY
206,394-399
(1992)
A Steady-State Kinetic Method for the Verification of the Rapid-Equilibrium Assumption in Allosteric Enzymes’ Marina M. Symcox2 and Gregory D. Reinhart Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019
Received
June
11,1992
A method for testing the validity of the rapid-equilibrium assumption as it might apply to allosteric enzymes using exclusively steady-state kinetic data is presented. The method is based upon a recognition that the ratio of apparent dissociation constants for the allosteric ligand, obtained under conditions of limiting and saturating substrate concentration, must yield the thermodynamic value for the coupling parameter between the substrate and allosteric ligand even in the general steady-state case. If this value is found to be equal to the apparent coupling parameter determined from the ratio of limiting values of the Michaelis constant for substrate obtained in the absence and saturating presence of the allosteric ligand, then the substrate can be correctly viewed as effectively achieving a binding equilibrium with the enzyme in the steady-state. The utility and limitations of this method are demonstrated by examining the ADP activation of beef heart mitochondrial NAD-dependent isocitrate dehydrogenase. o 1002 Academic
Press, Inc.
When studying the steady-state kinetic properties of allosteric enzymes, investigators almost always, explicitly or implicitly, assume that the enzyme and substrate achieve a near binding equilibrium in the steady-state. This rapid-equilibrium assumption greatly simplifies resulting kinetic expressions and transforms the interpretation of enzymatic responsiveness toward substrate and effector concentration to the thermodynamic
i This work was supported by Grant GM 33216 from the National Institutes of Health. ’ Current address: Department of Biochemistry andMolecular Biology, University of Oklahoma Health Sciences Center, Oklahoma City, OK. 3 Recipient of an Established Investigator Award from the American Heart Association, and to whom correspondence should be addressed. 394
realm. Consequently interpretations of the mechanistic causes of allosteric behavior are usually cast in terms of competing equilibria, either between various functionally defined enzyme forms (1,2) or between enzyme forms defined by their state of ligation (3,4). Unfortunately, the simplifications brought on by the rapid-equilibrium assumption, and its attendant conceptually straightforward models, provide such an enticing framework that whether the assumption isjustifiable can be overlooked. If the parameters obtained from steady-state kinetics measurements are kinetically constrained, and hence not thermodynamically based, then the relevance of the aforementioned models becomes dubious. Consequently, one must evaluate the validity of making the rapid-equilibrium assumption before giving credence to thermodynamically derived models. Several methods have been utilized to test for the validity of the rapid-equilibrium assumption, including isotope partitioning and use of a solvent viscogen (5). These techniques can present difficulties, particularly when applied to allosteric enzymes. Allosteric enzymes, for example, are characterized by conformations that are easily perturbed, and therefore adding viscogens to the solution can lead to complex results that are not easily interpretable. The analysis of isotope partitioning is complex when the enzyme exhibits nonhyperbolic saturation profiles, as well as often requiring the synthesis of isotopically labeled substrates. Of course one can also perform a thorough analysis of pre-steady-state kinetics in an effort to directly evaluate the magnitude of the rate constants on which the rapid-equilibrium assumption explicitly depends. However, this approach usually requires specialized instrumentation, copious amounts of enzyme, and an assay sensitive to the specific formation of the enzyme-substrate complex. We present below a method for assessingthe validity of the rapid-equilibrium assumption as it pertains to an allosteric enzyme using only initial velocity steady-state kinetics data. Although we derive the method from an 0003-2697/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
VERIFICATION
OF
THE
RAPID-EQUILIBRIUM
explicit analysis of the general rate equation resulting from a single-substrate, single-modifier mechanism, the principle behind the method is extrapolatable to oligomerit enzymes demonstrating cooperative substrate binding profiles. It only requires that the effect of the allosteric ligand be modest enough that the coupling free energy between the substrate and allosteric ligand can be measured explicitly. We then apply this method to the enzyme ICDH* and thereby demonstrate both the utility and certain limitations of this approach. THEORY
395
ASSUMPTION
We have defined these identical ratios to equal the thermodynamic coupling parameter Q. If neither A nor X achieves binding equilibrium in the steady-state, a rate equation much more complicated than Eq. [11, containing squared terms in both [A] and [Xl, is the correct solution to Scheme I. The general steady-state solution has been presented by Frieden (7). Using the notation defined above, the enzyme-form distribution equations that follow from the general solution are given by
Consider the following mechanism that describes in general the action of an allosteric ligand, X, on an enzyme catalyzing the reaction A + P:
sE+P ke 1 t k,
XE
[EAI = kW,,[Al k,, 11 k,,
+A% ;;- XEA 8
sXE+P
[A] + + [X] + & 1
(ks + kg + k,,) I
12
Scheme I It is well recognized (3,6,7) that if the rapid-equilibrium assumption is valid for the above mechanism, then the equation describing the rate of the reaction as a function of [A] and [X’j is given by
-=V Et
KO,K:,
V”%&U
+ QWAl[Xl
+ K&[A]
+ KL[X]
+ Q[A][X]
’
PI
where V” = maximal velocity in the absence of X, Kf., = the dissociation constant for A in the absence of X, Kf, = the dissociation constant for X in the absence of A, and W = V*lV’, where V” = the maximal velocity when X is fully saturating and Q = K&/KS, where Kz is the dissociation constant for A when X is fully saturating (3). Q is of considerable interest for so-called “K-type” modifiers because of its relation to the coupling free energy, AG,, between A and X: AGAx = -RT
In(Q).
PI
Regardless of whether the rapid-equilibrium assumption is valid, the following must be true, for it is merely a restatement of the principle of thermodynamic linkage (g-10):
used: ICDH, isocitrate 3-(morpholino)propane
dehydrogenase; sulfonic acid,
MgIC, DTT,
Mgdith-
151
1
[61
[XEAI = WddAl[Xl [A] + :
[X] + 2 1
7
12
k, + k,, + k&s 7 2
I
. 171
In the steady-state the apparent dissociation constants for A and X, K,, and K,, respectively, are given by
K = ([El + [XEl)[Al A WA1 + [XEAl) K = WI + [EAl)[Xl x
([XE]
+ [XEA])
PI
.
The limiting values of KA, obtained as [X] approaches either 0 or cc, are given by the following according to Eqs. [4-71:
x--o
WI
A b-1
lim (K A ) = [XE1[A1 X--CC
’ Abbreviations isocitrate; Mops, iothreitol.
1
[XEA]
k,
=-
ks + Ks~ K” k,
A’
WI
Consequently Q, as defined in Eq. [3], is obtained from the ratio of steady-state dissociation parameters per-
396
SYMCOX
taining to substrate (i.e., Michaelis constants) only the rapid-equilibrium assumption is satisfied, i.e. if k, k3 and k8 B b, in which case pA = K$ and Kz = Kg Next we consider the apparent steady-state values the dissociation constant of X, K,, as [A] approaches and co. Using Eqs. [4-71, as well as Eq. [3], we obtain lim
(K
)
=
X A-c0
lim
A--CO
(K
X
)
=
‘EI’XI ---c-z=-=
k,k,k,,
k,
LW
k&t&,,
ks
[EAI[JQ
[XEA]
=
k,,
k,,
=
KF
‘=
KO rx
AND
REINHART
if b : of 0
MATERIALS
[12]
[I31
Therefore, determining Q in this way must yield the true thermodynamic value even in the general case, and the rapid-equilibrium assumption can be evaluated by ascertaining if the apparent Q determined from KA is equal to the true value determined from K,. Mechanisms
Not Conforming
to Scheme I
Obviously we do not expect the preceding equations to explicitly apply to any mechanism other than that depicted in Scheme I. However, Scheme I serves as a paradigm for more complex allosteric mechanisms that often result from the oligomeric nature of most allosteric enzymes. As the concentration of substrate approaches zero, the enzyme is trapped in an equilibrium binding reaction with the allosteric ligand as depicted for the monomeric case by the left vertical equilibrium in Scheme I. As the concentration of substrate approaches saturation, the enzyme is also trapped in an equilibrium binding reaction with X as depicted by the right vertical equilibrium in Scheme I. For either zero or saturating levels of substrate, therefore, the rate constants pertaining to reactions drawn in the scheme in the horizontal direction, i.e., those involved in the binding of substrate and its turnover, will not influence the apparent binding interaction of X with the enzyme. These general features should be shared by mechanisms involving the binding of multiple equivalents of A and/ or X as is typical of oligomeric allosteric enzymes. Consequently evaluating whether Q derived from KA is equal to Q obtained from K,, while recognizing that these apparent dissociation parameters represent geometric mean values of individual ligand dissociation parameters in a multisite enzyme, should generally test the average applicability of the rapid-equilibrium assumption even in complex oligomeric cases.
‘This method can fail if &I& = $/ks. However, we know of no example of apparent rapid-equilibrium behavior that can be attributed to this circumstance, and even if this special condition were to be met, a thermodynamic value for Q is still realized from either the I$ or Kx data as described.
AND
METHODS
NAD+ (Grade III), Mops, DTT, sodium salts of threoD,,L,-isocitrate (Grade III), and ADP (Grade III) were purchased from Sigma Chemical Co. Other chemicals were analytical grade, and deionized, distilled water was used throughout. NAD-dependent isocitrate dehydrogenase was purified from beef heart mitochondria as described elsewhere (11). The enzyme was diluted, from an ammonium sulfate suspension stored at 4”C, into a buffer containing 0.8 M L&SO,, 90 mM Mops-NaOH, pH 7.2,l mM DTT and was stored on ice during the course of a day-long experiment without loss of activity (12). All kinetic assays were performed at 25°C in a final volume of 1.0 ml. Assay mixtures contained 90 mM Mops-NaOH, pH 7.4, 0.1 mM DTT, and variable concentrations of MgSO,, isocitrate, ADP, and NAD+. Reported concentrations of free isocitrate, free ADP, free Mg2+, and MgIC were calculated from stability constants for the metal complexation equilibria determined as described elsewhere (11). Unless otherwise qualified, the term ADP refers to free ADP, the species not complexed with Mg2+. Reported concentrations of NAD+ are total amounts, since NAD+ complexes free Mg2+ relatively weakly. Activity is expressed as a ratio of observed activity to maximal activity at 25°C. NADH production was monitored by following the increase in absorbance at 340 nm with a strip chart recorder. Initial velocities were calculated from estimated tangents of the early part of the recorder tracings. Typically, the final enzyme concentration varied from 0.1 to 0.5 pg/ml in the assay mixture. The half-saturation constants for MgIC were determined from Hill plot analysis (13). Approximately 12 initial velocity data points were used to define each MgIC saturation curve. V,, was estimated from plots of initial velocity versus [MgIC]. Rates falling between 10 and 90% of V,, were fit by least-squares linear regression to the Hill equation Wu~(V,,
- U)) = n log[MgIC]
-
n lOg(KA),
[14]
where the intercept on the abscissa equals log(K,) and the slope equals the Hill coefficient, n. Quoted values of KA pertain to the racemic mixture of threo-D,,L,-isocitrate, although threo-D,-isocitrate is the only active stereoisomer. The apparent dissociation constant for ADP at any given concentration of MgIC, K,, was taken to be equal to the concentration of ADP that produced half of the total enhancement in velocity at that Mg-isocitrate concentration. The enzyme was titrated with ADP at fixed levels of MgIC and NAD+. The dependence of the activation on the concentration of ADP was found to be hyperbolic, so Kx was determined from linear least-
VERIFICATION
OF
THE
I
I
I
-4.0
-3.0
-2.0
RAPID-EQUILIBRIUM
Log [Mg-isocitratel FIG. 1. Activation isocitrate dehydrogenase rithm of [Mg-isocitrate]. and the concentration tively.
of beef heart mitochondrial NAD-dependent by ADP expressed as a function of the logaThe concentration of NAD+ equals 0.15 mM, of ADP equals 0 (0) and 2.0 mM (O), respec-
squares fit of 15 to 20 velocity points to the double-reciprocal equation 1
KY
-= u - uo
(V,,
-
- uo) WPI
1
(Vm,, - ~0) ’
[151
397
ASSUMPTION
[ 141corresponds to the concentration of MgIC that produces an activity equal to one-half the maximal velocity. Figure 2 illustrates how the binding of ADP affects the initial velocity of ICDH when the concentration of ADP is varied at fixed concentrations of the substrate MgIC. Since ADP acts exclusively by decreasing the apparent KA for MgIC (Fig. l), its effect is greatest at intermediate concentrations of MgIC, as represented by 0.15 mM in Fig. 2, when ADP is able to shift the KA from above to below that concentration of substrate. At relatively high concentrations of substrate, such as 0.8 mM, which approach saturation even in the absence of ADP, the activity is not enhanced greatly. At the other extreme, when MgIC is substantially below KA even in the presence of ADP, little activation is also seen. Nevertheless, for any substrate concentration where activation can be measured, a value of ADP at which one-half of the activation is achieved can be determined. This concentration of ADP, determined in the present case by fitting the data in double reciprocal fashion to Eq. [151, is taken to be K,. Figure 3A shows the variation of KA with ADP while Fig. 3B gives the variation of K, with Mg-isocitrate at subsaturating concentrations of the cosubstrate NAD+ ([NAD+] = 0.15 mM, one-half its Michaelis constant). As we have found to be the case for other complex allosteric enzymes such as rat liver phosphofructokinase (14,15), the variation of KA with ADP can be fit well to
where u. is the initial velocity in the absence of ADP. Data describing the dependence of KA on the concentration of ADP were fit via nonlinear regression analysis to the following equation assuming a constant relative error as previously described (14,15): K
WI
A
RESULTS
ICDH catalyzes the formation of NADH, a-ketoglutarate, and CO, from NAD+ and isocitrate according to the overall reaction isocitrate + NAD+
= a-ketoglutarate
h .= :: 2
.-zf a
+ CO, + NADH
The absolute requirement of this reaction for Mg2+ results at least in part from the fact that MgIC is the true substrate of the enzyme (11). Free ADP activates ICDH by increasing the apparent affinity of the enzyme for MgIC as shown in Fig. 1. ADP has no effect on maximal velocity. The saturation curves both with and without ADP exhibit positive cooperativity, but the data can be described well by the Hill equation (Eq. [14]). The value of KA that results from fitting data such as these to Eq.
Log
[ADPI
FIG. 2. Influence of [ADP] on relative velocity drogenase with fixed [Mg-isocitrate] and [NAD+] citrate] = 0.80 mM (Cl), 0.15 mM (m), 0.05 mM (0). [15] by the method of linear least squares.
of isocitrate dehy= 5.0 mhi. [Mg-isoData were fit to Eq.
398
SYMCOX
AND
Eq. [16] as indicated by the solid curve in Fig. 3A. Fitting the data in Fig. 3A to Eq. [ 161 provides an accurate determination of the apparent Q, which we find to be equal to 9.2 f 0.3. Unfortunately an equation analogous to Eq. [ 161 does not describe the dependence of Kx on MgIC seen in Fig. 3B because of the cooperative nature of the MgIC binding. Moreover, the data in Fig. 3B are significantly scattered at low and high concentrations of MgIC, although a clear tendency of the data to form a plateau at each extreme is evident. The length of the double-headed arrow in Fig. 3B corresponds to the logarithm of the apparent Q determined from KA, i.e. the distance between the plateaus in Fig. 3A. It is apparent that this value agrees reasonably well with the distance between the two plateaus in Fig. 3B, particularly given the scatter in the data. Figures 4A and 4B show a similar comparison with a saturating concentration of NAD+ (5 mM). Once again there is agreement between the apparent Q determined
REINHART
-3.6 z 2 ~2
-3.8
1
-4.2 )I/ (-IW
-4.5
-3.5
Log
-2.5
[ADP]
-2.8 1
A 0
-3.2
a Y
-3.4
2 4
-3.6
-4.0 -4.15
-4.25
-3.15
Log
-4.5
-3.5
Log
[Mg-isocitrate]
FIG. 4. Mutual influence of the binding of MgIC and ADP with [NAD+] = 5.0 mM. (A) Apparent KA for Mg-isocitrate as a function of [ADP]. (B) Apparent Kx for ADP as a function of [MgIC]. The double-headed arrow in B corresponds to the logarithm of the apparent Q determined from a fit of data in A to Eq. [IS].
-3.8
-3.0:[ -4.0 t-INF)
-3.25
-2.5
[ADP]
-2.5 D
-4.5 -4.5
B
-4.0
-3.5
.
-3.0
Log [Mg-isocitrate] FIG. 3. Mutual influence of the binding of MgIC and ADP with [NAD+] = 0.15 mM. (A) Apparent KA for Mg-isocitrate as a function of [ADPJ. (B) Apparent Kx for ADP as a function of [Mg-isocitrate]. The double-headed arrow in B corresponds to the logarithm of the apparent Q determined from a fit of data in A to Eq. [16].
in Fig. 4A and the true value of Q indicated by the data in Fig. 4B. The apparent value of Q determined from the KA data presented in Fig. 4A is equal to 6.1 + 0.2. Despite significant scatter in Fig. 4B, limiting plateaus are evident. The distance between these two plateaus agrees with the magnitude of the log of the apparent Q determined in Fig. 4A as indicated by the length of the double-headed arrow. The results indicate that the rapid-equilibrium assumption is justified for both low and high concentrations of NAD+. DISCUSSION
We have previously estimated, on the basis of the magnitudes of r and KF for ICDH, that the rapidequilibrium assumption is likely to be valid unless the on-rate constants for MgIC (K, and & in Scheme I) are unusually small (16). Also in a two-substrate mecha-
VERIFICATION
OF
THE
RAPID-EQUILIBRIUM
nism when the second substrate (NAD+ in this case) is limiting, the rapid-equilibrium assumption with respect to MgIC is forced (unless NAD+ binds first in an ordered mechanism, which is not the case for ICDH (17,18)). However the data presented in Figs. 3 and 4 convey more explicit proof that the rapid-equilibrium assumption is appropriate at both low and high concentrations of NAD+. One important advantage of the method that we propose is that it uses essentially the same general assay conditions as routine measurements of steady-state kinetic activity. This not only makes this procedure easier to implement but also ensures that the validity of the rapid-equilibrium assumption will be assessed under the same experimental conditions in which the kinetic properties are otherwise being delineated. This latter consideration is particularly important for allosteric enzymes that are generally very sensitive to their environment. For example, when we attempted to use the viscosity-variation technique (19,20) to evaluate the rapid-equilibrium assumption, we found that the viscogens sucrose and glycerol caused decreases in the KA for MgIC that could not be explained by the effect that viscosity has on second-order rate constants (data not shown). Isotope partitioning experiments provide an alternative approach (5) but would have required in this case the time-consuming synthesis of radiolabeled isocitrate. The obvious difficulty associated with the method described herein is that the K, parameter can be difficult to determine as the limiting values of K& or Ki"f are approached since as these limits are approached the activation itself vanishes. In this case the velocity enhancement effected by ADP eventually becomes so small that random experimental error prohibits accurate identification of the point of half-maximal enhancement corresponding to Kx. This gives rise to the scatter apparent in Figs. 3B and 4B. The scatter that we observed should be a general feature of this method and necessitates an overdetermination of Kx at low and high substrate concentrations to define the plateau values.
399
ASSUMPTION
The observed reaction velocities can also need careful correction for any substantial blank rates, etc., particularly for those measurements made at low substrate concentration where the rates being measured are small. Nonetheless, with reasonable effort K& and Kg can be defined with a precision sufficient to provide for a satisfactory estimate of the true value of the coupling parameter Q. This value can then be used to check the validity of the rapid-equilibrium assumption. REFERENCES 1. Monod,
J., Wyman,
J., and Changeux,
J.-P.
(1965)
J. Mol.
Biol.
12,88-118. 2. Koshland, D. E., Jr., Nemethy, G., and Filmer, D. (1966) Biochemz+y6,365-385. 3. Reinhart, G. D. (1983) Arch. Biochem. Biophys. 224,389-401. 4. Reinhart, G. D. (1989) Biophys. Chem. 30,X9-172. 5. Cleland, W. W. (1986) in Techniques of Chemistry, (Bernasconi, C. F., Ed.),
Part
I, Vol VI, pp. 791-870,
Wiley,
New
6. Botts, J., and Morales, M. (1953) Trans. Faraday 707. 7. Frieden, C. (1964) J. Biol. Chem. 239, 3522-3531. 8. Wyman, J. (1948) Adv. Prot. Chem. 4,407-531. 9. Weber, G. (1972) Biochemistry 11, 864-878. 10. Weber, 11. Symcox,
G. (1975) Adv. M. M. (1991)
Prot. Chem. 29, l-83. Ph.D. thesis, University
12. Plaut, G. W. E., Schramm, V. L., and Aogaichi, Chem. 249, 1848-1856. 13. Hill, A. V. (1910) J. Physiol. London 40, iv-vii. 14. Reinhart,
G. D. (1985)
Biochemistry
24,
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15. Reinhart, G. D., and Hartleip, S. B. (1986) Biochemistry 7308-7313. 16. Reinhart, G. D., Hartleip, S. B., and Symcox, M. M. (1989) Natl. Acad. Sci. USA 86,4032-4036.
25, Proc.
17. Rushbrook,
J. I. (1978) Ph.D. thesis, Rutgers University. 18. Gabriel, J. L., and Plaut, G. W. E. (1982) J. Bial. Chem. 8021-8029. 19. Brouwer,
A. C., and Kirsch,
J. F. (1982)
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P. D., and
