- Email: [email protected]
Kinetics of the two-sited enzyme
J. theor. Biol. (1973) 39, 91-102
Kinetics of the Two-sited Enzyme II. A Method of Distinguishing between Anticooperative and Independent Active Sites Based on Competitive Inhibition? EDWIN
T. HARPERS
Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676, U.S.A
Department of Biochemistry, Indiana University School of Medicine, Indianapolis, Indiana 46202, U.S.A. (Received 2 September 1969, and in revisedform 20 April 1972) The presenceof two active sites on an enzyme leads to downwardly curving Lineweaver-Burk plots if (A) the sitesare independent,but have different Michaelis constants,or (B) if the sitesinteract anticooperatively to impair binding, but not catalysis,at the secondsite filled. CasesA and B are kinetically indistinguishablewhen only enzyme and substrate are present. However, equations derived by the rapid-equilibrium treatment show that the two caseshave different patterns of competitive inhibition and becomedistinguishablein the presenceof a suitable inhibitor. The inhibitor may decreaseor increasethe curvature of Lineweaver-Burk plots, but certain patterns have diagnostic value becausethey can occur only in case(B). In one type of diagnosticpattern, high concentrationsof inhibitor cause the Lineweaver-Burk plots to curve upward, and causethe corresponding saturation curves to becomesigmoid.The effect of the inhibitor is thus to make sites which are anticooperative appear to be cooperative. This suggeststhat the mere occurrence of sigmoid saturation curves is not necessarilyevidenceof cooperativebinding effects,and may have uncertain significancein considerationsof enzyme regulation.
1. Introduction The presenceof more than one active site on an enzyme is often manifested by curvature in Lineweaver-Burk, or equivalent, plots of kinetic data (cf. Klotz & Hunston (1971) and Harper (1971) and references cited there). t Presentedin part at the 158thNational Meetingof the AmericanChemicalSociety, NewYork, September,1969. $ Pleaseaddress inquiriesto Indianapolis. 91
92
E.
T.
HARPER
Downwardly curving Lineweaver-Burk plots are particularly striking, because they are relatively uncommon, and because they mistakenly suggest that the enzyme is somehow activated by the substrate. However, as the preceding paper (Harper, 1971) showed, the downward curvature is in fact characteristic of an inhibitory (anticooperative) effect of the substrate on the binding properties of the enzyme (K-inhibition). Moderate substrate activation, in the form of enhanced maximal velocity (V-activation), actually decreases the curvature, and strong activation (I’- or K-) reverses its direction (cf. Table I). It is also well known that downwardly curving Lineweaver-Burk plots can occur with sites that are independent, where there is neither activation nor inhibition by the substrate (Kistiakowsky & Rosenberg, 1952). The kinetics of anticooperative and independent sites are thus similar, and the two cases have been considered kinetically indistinguishable. This paper presents a method of distinguishing between them, based upon the different responses of the two kinds of sites to competitive inhibition.?
2. Model and Rate Equations A general model for a two-sited enzyme, subject to reversible inhibition by substrate analogs, is shown in equation (I), where E, S, and I represent enzyme, substrate, and inhibitor, respectively; C, and Cl are binary ES complexes involving “unprimed” and “primed” sites, respectivelyt; K
k: i
i',
311
s ,
I$
/iI* '
@
_ 51
k; i I
I,
kl
I'
Cl ( ,s-,
s,
.
I-K,
K2
E K*.
k2
c2 1.
s
(1)
C1, D,, X and x’ are ternary complexes with compositions ES2, EI,, and ESI, respectively. t “Anticooperative” is used here in a broad sense, and includes sites which influence each other because of effects on conformation, on aggregation of subunits, or in other ways. $ The author is grateful to a referee for suggesting that the sites be labeled in this way.
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
93
SITES
The initial rate equation for model (I), derived by the rapid equilibrium treatment (Cha, 1968), is
where u,, is the rate in the absence of products, S and I represent (invariant) concentrations of substrate and inhibitor, respectively, dissociation constants (K) are defined as shown in equation (l), and the maximal velocities (V) are products of the total enzyme concentration and the respective rate constants (k). We have also derived the steady-state rate equation for this model, but the completely random binding sequence makes it too complicated for the purpose of this paper. The treatment is therefore limited to enzymes with rapid binding equilibria. We previously (Harper, 1971) considered three well-known special cases of model (1): case A, with independent, non-equivalent sites; case B, with initially equivalent, interacting sites; and case C, with one catalytic site and one modifier-binding site. Since C is a further special case of B when the substrate is the modifier, it will not be developed further here but warrants separate treatment as the simplest model of an allosteric enzyme (Monod, Changeux & Jacob, 1963). Initial rate equations showing the effects of competitive inhibition on cases A and B may be derived from equation (2) as follows. In case A, the independence of the sites means that the same constants describe the reactivity of a given site in all states of the enzyme. Thus, for the “unprimed” site, K, = K, = K2 = K3, K, = K,, = f&,, and V= V,= V,= V,; and for the “primed” site, K: = K; = K; = K;, K; = K;, = K;,,, and V’ = Vi = Vi = Vi. The resulting equation, v[l+;(l+$)] O”= -r+$(l+$I)]
+V[l+%(l+;)] [l++(l+f,)]
’
(3)
is the same equation that is obtained by considering case A equivalent to a mixture of independently inhibited, simple enzymes. In case B, the initial equivalence of the sites means that all primed symbols are equivalent to unprimed ones. Model (1) is thus simplified to model (4) which corresponds to rate equation (5). The constants of equation (5) represent modifier effects of substrate and inhibitor. In the terminology of
94
E. ‘I-. HARPER
Monod, Wyman & Changeux (1965), X,/K, and K,,/K, are homotropic binding effects of substrate and inhibitor, respec!ively; k;/K, is the heterotropic effect of substrate on inhibitor binding, and K,,,/K, is the heterotropic effect of inhibitor on substrate binding; V2/V1 and VJY, are, respectively, homotropic and heterotropic effects on catalytic efficiency. 3. Shapes of Lineweaver-Burk The curvature
of Lineweaver-Burk
Plots
plots can be conveniently
described
:n
p,
(4)
by the ratio (r) of slopes of the pair of linear asymptotes to the curve (Table 1). The slopes, slope ratio, and point of intersection of the asymptotes were evaluated as before (Harper, 1971). They are functions of the inhibitor 1
TABLE
Characteristics
of curved Lineweaver-Burk effects
Slope ratiot ____-...---_-~~.r>l
l>rTO O>r
plots due to substrate
modifier
Essential modifier effect
Plot shape __-_
~~~~-.~
Downward curvature Upward curvature, (no negative slope) Negative slope at high substrate concentration
t Ratio of limiting slope at high substrate concentration substrate concentration.
~--__.
K-inhibition K-activation V-inhibition
to limiting
slope at low
ANTICOOPERATIVE
I’S’. INDEPENDENT
ACTIVE
SITES
95
concentration, as shown in Table 2. When Z = 0, the expressions simplify to those (e.g., re) of the uninhibited enzymes (Harper, 1971). When the concentration of inhibitor is much larger than any of the inhibition constants, the equations for r assume the limiting forms (r,) shown on the fourth line of Table 2. The expressions of Table 2 were verified using an IBM 7040 computer which was programmed to plot various forms of the overall rate equation on a Cal-Comp plotter?. Values of r measured from the LineweaverBurk plots agreed with those calculated from Table 2. TABLE 2
EfSect of competitive inhibition on asymptotes Limiting parameter
Case At
Slope (S-t~) Slope w+O)
VK,,,-I- V’K; tv+ IT
of Lineweaver-Burk plots
sA t Km = Ks(l+IIK,); K,i, = K;(l +Z/K;); V* = VI+ V,Z/K,,,. f r = ratio of limiting slope at infinite substrate concentration substrate concentration. 0 S, = S at the apparent intersection of the asymptotes.
to limiting slope at zero
In case A, the shapes of the plots are controlled by the relative selectivities of substrate and inhibitor for the two sites. The selectivity of a ligand is its power to discriminate between sites, and increases with increasing difference between the affinities of the ligand for the “primed” and “unprimed” sites. The relative selectivities of substrate and inhibitor can thus be defined by the ratio (K~/I(,)/(K~/KJ. [The discussion could aiso be framed in terms of the relative selectivities of the sites for the two ligands, defined by the ratio
K/G>/K/&I.I t Figs 1, 3, 4 and 5 are manual reproductions
of curves drawn by the machine.
96
E. T. HARPER
If the substrate and inhibitor are equally selective, then K:/K, = K;/K,, and Table 2 shows that ras = 1 (cf. Fig. IA and Fig. 2A, curve d). In practice, the slopes become indistinguishable before they become equal, and the difference in slope is less than 10% by the time the inhibitor concentration is three times larger than K;, the larger of the inhibition constants. Thus. apparently linear plots can be obtained with reasonable concentrations of any selective inhibitor that can be observed to nearly saturate the enzyme.
A
B-l
B-2
4r-----
FIG. 1. Effect of competitive inhibition on shapes of Lineweaver-Burk dent sites, KS = Kl = 10e4, K: = KI = 10m2. B-l: anticooperative lo-*, Ka = K,, = KIII = 10m2. B-2: Same except KIII = 4~10~~.
plots. A: indepensites, K1 = Kl =
If the inhibitor is more selective than the substrate (K;/K, > Kt/K,), plots become linear with a level of inhibitor given by
the
--I = K;/K, - 1 (6) K;/K, - K:/K; G With larger concentrations of inhibitor, downward curvature reappears, but at somewhat higher concentrations of substrate. Thus, a plot of the slope ratio against inhibitor concentration (Fig. 2A, curve c) shows a minimum, with rmin = 1 andr, > 1. If the inhibitor is bound less selectively than the substrate, linear plots are never obtained because the numerator in the expression for r reaches its limiting value before the inhibitor concentration can compensate for the inequality of K, and K: (Fig. 2A, curve b, r0 > rca > 1). Finally, if the substrate and inhibitor have opposite selectivities, i.e., Kr > K;, the curvature increases (rm > rc) with increasing inhibitor concentration (Fig. 2A, curve a). In case B, the shapes of Lineweaver-Burk plots are controlled primarily by the magnitude of the heterotropic binding effects of substrate and inhibitor relative to the homotropic effects: K3KnI/K,Kn. However, only
ANTICOOPERATlVE
VS.
INDEPENDENT
ACTIVE
SITES
97
three of the four effects are mathematically independent. Thus, KS/K1 = i&/K,, and the above relationship may also be expressed by the ratio uGId~IKn)/(~2/~1). All of the plot shapes of case A are observable in case B. However, Lineweaver-Burk plots are less likely to become linear with high concentrations of inhibitor in case B, than in case A, for two reasons. First, the
FIG. 2. slope ratios of Lineweaver-Burk plots as functions of inhibitor concentration. A: independent sites, K, = 10-3, K,’ = 10-2, r. = 3.025; curve a, Kr = lO-a, K; = 7X10m3,rm = 4; curve b, KI = 5X10s3, K{ = 10-2, rm = 1.8; curve c, KI = 10-3, Ki = lo-l, rm = 3.025; curve d, KI = lOma, K; = lOma, rm = 1. B: anticooperative sites, KI = 10e3, Kz = 10V2, r0 = 10; curve a, KI = 1.5x10-2, K,, = KIII = 10-2, rrq=15;curveb,KI=K,n=10-a,KII=l.5X10-a,r to = 15; curve c, KI = 5 x~O-~, K~~=lO~3,K~~~=10~2,rm=5;cu~ed,K~=KIII=3~~O-~,KII~g~lO-~rr~D33; curve e, KI = 1O-3, K,, = lo-l, KIII = 10-2, rm = lo; curve f, KI = K,, = 1O-2, KrII = 5 X 10e2, rm = 0.4. curvature is more pronounced in case B (Figs 1 and 2). Second, the conditions for rm = 1 are more restrictive in case B: slopes of the asymptotes become equal only if the substrate and inhibitor exert comparable modifying effects on both binding and catalytic properties of the sites, i.e., K,/K, = K&/K,K,, and V, = V, (Fig. lB-1). The more likely result of high inhibitor concentrations in caseB is that the Lineweaver-Burk plots will curve upward (r, < 1, Figs IB-2 and 2B, T.8. 7
98
E. I-. HARPER
curve f). This is in marked contrast to case A, where such curvature is not possible. The necessary condition for upward curvature is
(7) Since the right-hand side of inequality (7) has a maximum value of one, when Vz = V3, a sufficient condition for P,, < 1 is that the heterotropic binding effects are larger than the homotropic effects: K,K,,, > K2K,,. However, upwardly curving plots may also be obtained if the hcterotropic effects are somewhat smaller than the homotropic effects, provided that the substrate and inhibitor modify the maximal velocity of the second site to I 00 090 0 60
z 070 060 050 O 4ooI
I 50
1 I I 100 150 200 I /s(xlo3)
IO I
g [II= 10-4 -1
?-
/
676-
'0
25I
50 I
I 00 o-go
I 1 250
75 I
100 I
125 I
I/..(x10~)
[II= 10-i
0 60
3
060 050 ; O-70 ,I---0400 50
100
150 200
250
l/S FIG. 3. Doubly reversed curvature of Lineweaver-Burk plots with increasing inhibitor concentration. Anticooperative sites, KI = 10eB, Ka = 10e5, KI =- 5 x lo-‘, KI, = 2~10-~. KI,, = 10-4.
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
SITES
99
different extents. In this situation, the right-hand side of inequality (7) must be less than one, and the inequality can be satisfied with K&/&K,, -=cK,/K,. As in case A, the value of r can passthrough a minimum as the inhibitor concentration increases(Fig. 2B, curve e), but rmin is not necessarilyequal to one. Figure 3 shows an example in which r < 1 at the minimum, causing the Lineweaver-Burk plots to successively curve downward, upward, then downward again, with increasing inhibitor concentration. Other distinctions of case B are that the value of r may pass through a maximum (Fig. 2B, curve c), or exhibit an inflection point (Fig. 2B, curves b, d). The requirement for minima or maxima, in the absence of modifying effects on maximal velocities, is that the homotropic effect on inhibitor binding must be larger than the heterotropic one, i.e., KI > K,,, > K,, or K,, > K,,, > K,. The level of inhibitor at the extremum of r must then satisfy the equation (cf. Table 2), r G/K,
1+2z/K,,,+P/K:II =---.
1+21/K, + 12/KIK,,
63)
4. Application to Experimental Data The proposed method of distinguishing between independent and anticooperative active sites is based upon the differences in their patterns of competitive inhibition (Figs 1 and 2). The patterns were analyzed in the previous section in terms of Lineweaver-Burk plots, because they are commonly used and have easily interpeted shapes. In applying the method, however, Eadie plots (Fig. 4) are recommended, because of their better display of curvature &die, 1942; Hofstee, 1952). The value of r, the ratio of the limiting slope at high substrate concentration to that at low substrate
FIG. 4. Eadie plots corresponding
to Fig. 1.
100
E.
T.
HARPER
concentration, is also a useful measure of curvature with Eadie plots. As in the case of Lineweaver-Burk plots, r > 1 characterizes effects that appca; to be anti-cooperative, and the magnitude of r increases with the magnitude of the effect. Also, r < 1 characterizes effects that appear to be cooperative. In contrast to Lineweaver-Burk plots, however, r may become increasingly negative as the magnitude of the cooperative effect increases, causing 111e plots to appear concave left. There are several ways in which a plot of r us. inhibitor concentration can be used to distinguish between independent and anticooperative sites. The presence of independent sites may be excluded if the value of r depends on the concentration of inhibitor in any of the following ways: r < 1, under any conditions; r exhibits a minimum where r # 1; or r exhibits a maximum or an inflection point. None of these patterns is possible with independent sites, and may be taken as evidence of site interaction. Under certain conditions, it is possible to establish the presence of independent sites, by the failure to observe one of the above criteria of interacting sites with an appropriate inhibitor, i.e., one which is more selective than the substrate. One difficulty, however, is that the relative selectivities of substrates and inhibitors are not usually known. Nor can the selectivities be predicted from the relative affinities of the ligands. Because of the unique specificities of binding interactions at active sites, generalized selectivityreactivity relationships (cf. Bender, 1960; Leffler & Grunwald, 1963), are perhaps unlikely to be valid from one enzyme to another. We have approached this problem, with fructose diphosphate aldolase, in a way which may be of general utility (Harper, Hokse, Lu & Phelps, unpublished). The method involves the use of matched pairs of substrates and inhibitors: pairs of monophosphates and diphosphates in the case of aldolase. The substrate and inhibitor of each pair are structurally matched, so that their binding interactions with the enzyme are closely similar to each other, but different from those of the other pair. Thus, within each matched pair, the substrate and inhibitor may be expected to have comparable affinities and selectivities (or modifying effects, as the case may be), and should exhibit values of r which approach unity at high concentrations of inhibitor. Experimental values of the alhnities (measured by K,,,) and the behavior of I provide a test of the inference of comparable selectivities. Since the respective pairs have dissimilar interactions with the binding sites, members of one pair will appear to be more selective than members of the other pair. The test is then based on plots given by the mismatched pairs, of dissimilar substrate and inhibitor. In these plots, the limiting value of r, at high concentration of inhibitor, will be unequal to one. If the sites are independent, the limiting value of r will be greater than one with each mis-
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
SITES
101
matched pair, and r will exhibit a minimum value of one with that pair in which the inhibitor is more selective than the substrate. A general limitation of the method proposed in this paper is that it demands an extensive kinetic study. First, the substrate concentration should be varied over a range of no less than IOO-fold in order to clearly observe the curvature of the plot. Even further extensions of the range are required to measure the slope ratio of the asymptotes to the curve (Harper, 1971). The range of substrate concentrations must also be shifted to higher values as the inhibitor concentration is increased, because the center of curvature (l/S,, cf. Table 2) shifts to higher substrate concentrations with increasing concentration of inhibitor. Second, one or more inhibitors must be available (or less reactive substrates, cf. Trowbridge, Krehbiel & Laskowski, 1963) which are known to be purely competitive, and which can be used over a range of concentrations spanning at least the range of the inhibition constants. Of course, the curvature must not be due to artifacts of the system (Almond & Niemann, 1960). These requirements will obviously yield to experimental restrictions of observable rates and solubilities in many cases, but the method may be useful with certain enzymes. 5. Real and Apparent Cooperativity Much of the current interest in the kinetics of multi-sited enzymes is due to the suggestion of Monod et al. (1963), that cooperative binding effects produced by allosteric modifiers may provide metabolic control through the regulation of enzymic activity. Although cooperative binding has been known 2 00
1
I 60
I 20
0 80
0 00
0 10
OX
030
040
050
5
FIG. 5. Apparent cooperativity of anticooperative sites under the influence of cornpetitive inhibition. Kl = KI = IO- 4, Kz = KIrr = lo- 2, KI1 = 10-s.
102
E.
1‘.
HARPER
for some time to produce sigmoid saturation curves and upwardly curving Lineweaver-Burk plots (Botts, 1958), the converse is not known to be true, i.e., the occurrence of such curves may not necessarily indicate the presence of cooperative binding. Figures 1, 3 and 5 show that. in fact, it dots not: sigmoid saturation curves and upwardly curving Lineweaver-Burk plots can be obtained even when both substrate and inhibitor exert unticooperatice binding efSects. Thus, if an enzyme is subject to unsuspected inhibition, it is possible to mistakenly conclude that the substrate exerts a positive homotropic effect (K-activation) when in fact it is a negative modifier (K-inhibition). This unexpected result underscores a caveat which is also evident from the calculations of Koshland, Nemethy & Filmcr (1966), cf. Atkinson (1966); namely, that the shapes of kinetic plots alone may not pro\,idc unequivocal evidence for the idenrification of modifier effects as cooperative or anticooperative, and may not provide an adequate basis for inferences concerning the regulation of enzymic activity. The author is grateful to Louise and Henk Hokse for the experimental observations and stimulating discussion which led to this paper, and to Michael W. Trogdlen and Bruce H. Phelps for their assistance with the computer program. This research was supported by a Grant-in-Aid from the American Heart Association,
and
by
Grant
GB-5367
from
the
National
Science
Foundation.
Computations were carried out at the Indiana University Medica! Center Research Computation Center, which is supported in part by Public HeoI:h Service Medical Kesearch Grant FROOl67. REFERENCES ALMOND, H. R. & NIEIIANPL’, C. (1960). Biochem. biophys. dctu. 44, 143. ATKINSON, D. E. (1966). A. Rev. Biochem. 35,85. BENDER, M. L. (1960). Chem. Rev. 60, 53. Borrs, J. (1958). Trans. Faraday Sot. 54, 593. CHA, S. (1968). J. biol. Chem. 243, 820. EADIE, G. S. (1942). J. biol. Chem. 146, 85. HARPER, E. T. (1971). J. theor. Biol. 32,405. HOFSTEE. B. H. J. (1952). Science, N. Y. 116, 319. KISTIAKOWSKY, G. B. & ROSEFU‘BERG, A. J. (1952). J. .4m. c/rem. Sot. 74, 5010. KLOTZ, I. M. & HUNSTON, D. L. (1971). Biochemistry, N. Y. 10, 3065. KOSHLAND, JR., D. E., N~METHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. LEFFLER, J. E. & GRUNWALD, E. (1963). Rates and Equilibriu of Organic Reactions. pp. 162-168. New York: John Wiley and Sons, Inc. MONOD, J.. CHANGEUX, J.-P. CsrJACOB, F. (1963). J. molec. Biol. 6, 306. MONOD, J., WYMAN, J. & CHANGEU~, J.-P. (1965). J. nlolec. Biol. 12, 88. TROWBRIDGE, C. G., KREHBIEL, A. & LASKOWSKI, JR., M. (1963). Biochemistry, N. Y. 2. 843.
Kinetics of the Two-sited Enzyme II. A Method of Distinguishing between Anticooperative and Independent Active Sites Based on Competitive Inhibition? EDWIN
T. HARPERS
Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676, U.S.A
Department of Biochemistry, Indiana University School of Medicine, Indianapolis, Indiana 46202, U.S.A. (Received 2 September 1969, and in revisedform 20 April 1972) The presenceof two active sites on an enzyme leads to downwardly curving Lineweaver-Burk plots if (A) the sitesare independent,but have different Michaelis constants,or (B) if the sitesinteract anticooperatively to impair binding, but not catalysis,at the secondsite filled. CasesA and B are kinetically indistinguishablewhen only enzyme and substrate are present. However, equations derived by the rapid-equilibrium treatment show that the two caseshave different patterns of competitive inhibition and becomedistinguishablein the presenceof a suitable inhibitor. The inhibitor may decreaseor increasethe curvature of Lineweaver-Burk plots, but certain patterns have diagnostic value becausethey can occur only in case(B). In one type of diagnosticpattern, high concentrationsof inhibitor cause the Lineweaver-Burk plots to curve upward, and causethe corresponding saturation curves to becomesigmoid.The effect of the inhibitor is thus to make sites which are anticooperative appear to be cooperative. This suggeststhat the mere occurrence of sigmoid saturation curves is not necessarilyevidenceof cooperativebinding effects,and may have uncertain significancein considerationsof enzyme regulation.
1. Introduction The presenceof more than one active site on an enzyme is often manifested by curvature in Lineweaver-Burk, or equivalent, plots of kinetic data (cf. Klotz & Hunston (1971) and Harper (1971) and references cited there). t Presentedin part at the 158thNational Meetingof the AmericanChemicalSociety, NewYork, September,1969. $ Pleaseaddress inquiriesto Indianapolis. 91
92
E.
T.
HARPER
Downwardly curving Lineweaver-Burk plots are particularly striking, because they are relatively uncommon, and because they mistakenly suggest that the enzyme is somehow activated by the substrate. However, as the preceding paper (Harper, 1971) showed, the downward curvature is in fact characteristic of an inhibitory (anticooperative) effect of the substrate on the binding properties of the enzyme (K-inhibition). Moderate substrate activation, in the form of enhanced maximal velocity (V-activation), actually decreases the curvature, and strong activation (I’- or K-) reverses its direction (cf. Table I). It is also well known that downwardly curving Lineweaver-Burk plots can occur with sites that are independent, where there is neither activation nor inhibition by the substrate (Kistiakowsky & Rosenberg, 1952). The kinetics of anticooperative and independent sites are thus similar, and the two cases have been considered kinetically indistinguishable. This paper presents a method of distinguishing between them, based upon the different responses of the two kinds of sites to competitive inhibition.?
2. Model and Rate Equations A general model for a two-sited enzyme, subject to reversible inhibition by substrate analogs, is shown in equation (I), where E, S, and I represent enzyme, substrate, and inhibitor, respectively; C, and Cl are binary ES complexes involving “unprimed” and “primed” sites, respectivelyt; K
k: i
i',
311
s ,
I$
/iI* '
@
_ 51
k; i I
I,
kl
I'
Cl ( ,s-,
s,
.
I-K,
K2
E K*.
k2
c2 1.
s
(1)
C1, D,, X and x’ are ternary complexes with compositions ES2, EI,, and ESI, respectively. t “Anticooperative” is used here in a broad sense, and includes sites which influence each other because of effects on conformation, on aggregation of subunits, or in other ways. $ The author is grateful to a referee for suggesting that the sites be labeled in this way.
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
93
SITES
The initial rate equation for model (I), derived by the rapid equilibrium treatment (Cha, 1968), is
where u,, is the rate in the absence of products, S and I represent (invariant) concentrations of substrate and inhibitor, respectively, dissociation constants (K) are defined as shown in equation (l), and the maximal velocities (V) are products of the total enzyme concentration and the respective rate constants (k). We have also derived the steady-state rate equation for this model, but the completely random binding sequence makes it too complicated for the purpose of this paper. The treatment is therefore limited to enzymes with rapid binding equilibria. We previously (Harper, 1971) considered three well-known special cases of model (1): case A, with independent, non-equivalent sites; case B, with initially equivalent, interacting sites; and case C, with one catalytic site and one modifier-binding site. Since C is a further special case of B when the substrate is the modifier, it will not be developed further here but warrants separate treatment as the simplest model of an allosteric enzyme (Monod, Changeux & Jacob, 1963). Initial rate equations showing the effects of competitive inhibition on cases A and B may be derived from equation (2) as follows. In case A, the independence of the sites means that the same constants describe the reactivity of a given site in all states of the enzyme. Thus, for the “unprimed” site, K, = K, = K2 = K3, K, = K,, = f&,, and V= V,= V,= V,; and for the “primed” site, K: = K; = K; = K;, K; = K;, = K;,,, and V’ = Vi = Vi = Vi. The resulting equation, v[l+;(l+$)] O”= -r+$(l+$I)]
+V[l+%(l+;)] [l++(l+f,)]
’
(3)
is the same equation that is obtained by considering case A equivalent to a mixture of independently inhibited, simple enzymes. In case B, the initial equivalence of the sites means that all primed symbols are equivalent to unprimed ones. Model (1) is thus simplified to model (4) which corresponds to rate equation (5). The constants of equation (5) represent modifier effects of substrate and inhibitor. In the terminology of
94
E. ‘I-. HARPER
Monod, Wyman & Changeux (1965), X,/K, and K,,/K, are homotropic binding effects of substrate and inhibitor, respec!ively; k;/K, is the heterotropic effect of substrate on inhibitor binding, and K,,,/K, is the heterotropic effect of inhibitor on substrate binding; V2/V1 and VJY, are, respectively, homotropic and heterotropic effects on catalytic efficiency. 3. Shapes of Lineweaver-Burk The curvature
of Lineweaver-Burk
Plots
plots can be conveniently
described
:n
p,
(4)
by the ratio (r) of slopes of the pair of linear asymptotes to the curve (Table 1). The slopes, slope ratio, and point of intersection of the asymptotes were evaluated as before (Harper, 1971). They are functions of the inhibitor 1
TABLE
Characteristics
of curved Lineweaver-Burk effects
Slope ratiot ____-...---_-~~.r>l
l>rTO O>r
plots due to substrate
modifier
Essential modifier effect
Plot shape __-_
~~~~-.~
Downward curvature Upward curvature, (no negative slope) Negative slope at high substrate concentration
t Ratio of limiting slope at high substrate concentration substrate concentration.
~--__.
K-inhibition K-activation V-inhibition
to limiting
slope at low
ANTICOOPERATIVE
I’S’. INDEPENDENT
ACTIVE
SITES
95
concentration, as shown in Table 2. When Z = 0, the expressions simplify to those (e.g., re) of the uninhibited enzymes (Harper, 1971). When the concentration of inhibitor is much larger than any of the inhibition constants, the equations for r assume the limiting forms (r,) shown on the fourth line of Table 2. The expressions of Table 2 were verified using an IBM 7040 computer which was programmed to plot various forms of the overall rate equation on a Cal-Comp plotter?. Values of r measured from the LineweaverBurk plots agreed with those calculated from Table 2. TABLE 2
EfSect of competitive inhibition on asymptotes Limiting parameter
Case At
Slope (S-t~) Slope w+O)
VK,,,-I- V’K; tv+ IT
of Lineweaver-Burk plots
sA t Km = Ks(l+IIK,); K,i, = K;(l +Z/K;); V* = VI+ V,Z/K,,,. f r = ratio of limiting slope at infinite substrate concentration substrate concentration. 0 S, = S at the apparent intersection of the asymptotes.
to limiting slope at zero
In case A, the shapes of the plots are controlled by the relative selectivities of substrate and inhibitor for the two sites. The selectivity of a ligand is its power to discriminate between sites, and increases with increasing difference between the affinities of the ligand for the “primed” and “unprimed” sites. The relative selectivities of substrate and inhibitor can thus be defined by the ratio (K~/I(,)/(K~/KJ. [The discussion could aiso be framed in terms of the relative selectivities of the sites for the two ligands, defined by the ratio
K/G>/K/&I.I t Figs 1, 3, 4 and 5 are manual reproductions
of curves drawn by the machine.
96
E. T. HARPER
If the substrate and inhibitor are equally selective, then K:/K, = K;/K,, and Table 2 shows that ras = 1 (cf. Fig. IA and Fig. 2A, curve d). In practice, the slopes become indistinguishable before they become equal, and the difference in slope is less than 10% by the time the inhibitor concentration is three times larger than K;, the larger of the inhibition constants. Thus. apparently linear plots can be obtained with reasonable concentrations of any selective inhibitor that can be observed to nearly saturate the enzyme.
A
B-l
B-2
4r-----
FIG. 1. Effect of competitive inhibition on shapes of Lineweaver-Burk dent sites, KS = Kl = 10e4, K: = KI = 10m2. B-l: anticooperative lo-*, Ka = K,, = KIII = 10m2. B-2: Same except KIII = 4~10~~.
plots. A: indepensites, K1 = Kl =
If the inhibitor is more selective than the substrate (K;/K, > Kt/K,), plots become linear with a level of inhibitor given by
the
--I = K;/K, - 1 (6) K;/K, - K:/K; G With larger concentrations of inhibitor, downward curvature reappears, but at somewhat higher concentrations of substrate. Thus, a plot of the slope ratio against inhibitor concentration (Fig. 2A, curve c) shows a minimum, with rmin = 1 andr, > 1. If the inhibitor is bound less selectively than the substrate, linear plots are never obtained because the numerator in the expression for r reaches its limiting value before the inhibitor concentration can compensate for the inequality of K, and K: (Fig. 2A, curve b, r0 > rca > 1). Finally, if the substrate and inhibitor have opposite selectivities, i.e., Kr > K;, the curvature increases (rm > rc) with increasing inhibitor concentration (Fig. 2A, curve a). In case B, the shapes of Lineweaver-Burk plots are controlled primarily by the magnitude of the heterotropic binding effects of substrate and inhibitor relative to the homotropic effects: K3KnI/K,Kn. However, only
ANTICOOPERATlVE
VS.
INDEPENDENT
ACTIVE
SITES
97
three of the four effects are mathematically independent. Thus, KS/K1 = i&/K,, and the above relationship may also be expressed by the ratio uGId~IKn)/(~2/~1). All of the plot shapes of case A are observable in case B. However, Lineweaver-Burk plots are less likely to become linear with high concentrations of inhibitor in case B, than in case A, for two reasons. First, the
FIG. 2. slope ratios of Lineweaver-Burk plots as functions of inhibitor concentration. A: independent sites, K, = 10-3, K,’ = 10-2, r. = 3.025; curve a, Kr = lO-a, K; = 7X10m3,rm = 4; curve b, KI = 5X10s3, K{ = 10-2, rm = 1.8; curve c, KI = 10-3, Ki = lo-l, rm = 3.025; curve d, KI = lOma, K; = lOma, rm = 1. B: anticooperative sites, KI = 10e3, Kz = 10V2, r0 = 10; curve a, KI = 1.5x10-2, K,, = KIII = 10-2, rrq=15;curveb,KI=K,n=10-a,KII=l.5X10-a,r to = 15; curve c, KI = 5 x~O-~, K~~=lO~3,K~~~=10~2,rm=5;cu~ed,K~=KIII=3~~O-~,KII~g~lO-~rr~D33; curve e, KI = 1O-3, K,, = lo-l, KIII = 10-2, rm = lo; curve f, KI = K,, = 1O-2, KrII = 5 X 10e2, rm = 0.4. curvature is more pronounced in case B (Figs 1 and 2). Second, the conditions for rm = 1 are more restrictive in case B: slopes of the asymptotes become equal only if the substrate and inhibitor exert comparable modifying effects on both binding and catalytic properties of the sites, i.e., K,/K, = K&/K,K,, and V, = V, (Fig. lB-1). The more likely result of high inhibitor concentrations in caseB is that the Lineweaver-Burk plots will curve upward (r, < 1, Figs IB-2 and 2B, T.8. 7
98
E. I-. HARPER
curve f). This is in marked contrast to case A, where such curvature is not possible. The necessary condition for upward curvature is
(7) Since the right-hand side of inequality (7) has a maximum value of one, when Vz = V3, a sufficient condition for P,, < 1 is that the heterotropic binding effects are larger than the homotropic effects: K,K,,, > K2K,,. However, upwardly curving plots may also be obtained if the hcterotropic effects are somewhat smaller than the homotropic effects, provided that the substrate and inhibitor modify the maximal velocity of the second site to I 00 090 0 60
z 070 060 050 O 4ooI
I 50
1 I I 100 150 200 I /s(xlo3)
IO I
g [II= 10-4 -1
?-
/
676-
'0
25I
50 I
I 00 o-go
I 1 250
75 I
100 I
125 I
I/..(x10~)
[II= 10-i
0 60
3
060 050 ; O-70 ,I---0400 50
100
150 200
250
l/S FIG. 3. Doubly reversed curvature of Lineweaver-Burk plots with increasing inhibitor concentration. Anticooperative sites, KI = 10eB, Ka = 10e5, KI =- 5 x lo-‘, KI, = 2~10-~. KI,, = 10-4.
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
SITES
99
different extents. In this situation, the right-hand side of inequality (7) must be less than one, and the inequality can be satisfied with K&/&K,, -=cK,/K,. As in case A, the value of r can passthrough a minimum as the inhibitor concentration increases(Fig. 2B, curve e), but rmin is not necessarilyequal to one. Figure 3 shows an example in which r < 1 at the minimum, causing the Lineweaver-Burk plots to successively curve downward, upward, then downward again, with increasing inhibitor concentration. Other distinctions of case B are that the value of r may pass through a maximum (Fig. 2B, curve c), or exhibit an inflection point (Fig. 2B, curves b, d). The requirement for minima or maxima, in the absence of modifying effects on maximal velocities, is that the homotropic effect on inhibitor binding must be larger than the heterotropic one, i.e., KI > K,,, > K,, or K,, > K,,, > K,. The level of inhibitor at the extremum of r must then satisfy the equation (cf. Table 2), r G/K,
1+2z/K,,,+P/K:II =---.
1+21/K, + 12/KIK,,
63)
4. Application to Experimental Data The proposed method of distinguishing between independent and anticooperative active sites is based upon the differences in their patterns of competitive inhibition (Figs 1 and 2). The patterns were analyzed in the previous section in terms of Lineweaver-Burk plots, because they are commonly used and have easily interpeted shapes. In applying the method, however, Eadie plots (Fig. 4) are recommended, because of their better display of curvature &die, 1942; Hofstee, 1952). The value of r, the ratio of the limiting slope at high substrate concentration to that at low substrate
FIG. 4. Eadie plots corresponding
to Fig. 1.
100
E.
T.
HARPER
concentration, is also a useful measure of curvature with Eadie plots. As in the case of Lineweaver-Burk plots, r > 1 characterizes effects that appca; to be anti-cooperative, and the magnitude of r increases with the magnitude of the effect. Also, r < 1 characterizes effects that appear to be cooperative. In contrast to Lineweaver-Burk plots, however, r may become increasingly negative as the magnitude of the cooperative effect increases, causing 111e plots to appear concave left. There are several ways in which a plot of r us. inhibitor concentration can be used to distinguish between independent and anticooperative sites. The presence of independent sites may be excluded if the value of r depends on the concentration of inhibitor in any of the following ways: r < 1, under any conditions; r exhibits a minimum where r # 1; or r exhibits a maximum or an inflection point. None of these patterns is possible with independent sites, and may be taken as evidence of site interaction. Under certain conditions, it is possible to establish the presence of independent sites, by the failure to observe one of the above criteria of interacting sites with an appropriate inhibitor, i.e., one which is more selective than the substrate. One difficulty, however, is that the relative selectivities of substrates and inhibitors are not usually known. Nor can the selectivities be predicted from the relative affinities of the ligands. Because of the unique specificities of binding interactions at active sites, generalized selectivityreactivity relationships (cf. Bender, 1960; Leffler & Grunwald, 1963), are perhaps unlikely to be valid from one enzyme to another. We have approached this problem, with fructose diphosphate aldolase, in a way which may be of general utility (Harper, Hokse, Lu & Phelps, unpublished). The method involves the use of matched pairs of substrates and inhibitors: pairs of monophosphates and diphosphates in the case of aldolase. The substrate and inhibitor of each pair are structurally matched, so that their binding interactions with the enzyme are closely similar to each other, but different from those of the other pair. Thus, within each matched pair, the substrate and inhibitor may be expected to have comparable affinities and selectivities (or modifying effects, as the case may be), and should exhibit values of r which approach unity at high concentrations of inhibitor. Experimental values of the alhnities (measured by K,,,) and the behavior of I provide a test of the inference of comparable selectivities. Since the respective pairs have dissimilar interactions with the binding sites, members of one pair will appear to be more selective than members of the other pair. The test is then based on plots given by the mismatched pairs, of dissimilar substrate and inhibitor. In these plots, the limiting value of r, at high concentration of inhibitor, will be unequal to one. If the sites are independent, the limiting value of r will be greater than one with each mis-
ANTICOOPERATIVE
VS.
INDEPENDENT
ACTIVE
SITES
101
matched pair, and r will exhibit a minimum value of one with that pair in which the inhibitor is more selective than the substrate. A general limitation of the method proposed in this paper is that it demands an extensive kinetic study. First, the substrate concentration should be varied over a range of no less than IOO-fold in order to clearly observe the curvature of the plot. Even further extensions of the range are required to measure the slope ratio of the asymptotes to the curve (Harper, 1971). The range of substrate concentrations must also be shifted to higher values as the inhibitor concentration is increased, because the center of curvature (l/S,, cf. Table 2) shifts to higher substrate concentrations with increasing concentration of inhibitor. Second, one or more inhibitors must be available (or less reactive substrates, cf. Trowbridge, Krehbiel & Laskowski, 1963) which are known to be purely competitive, and which can be used over a range of concentrations spanning at least the range of the inhibition constants. Of course, the curvature must not be due to artifacts of the system (Almond & Niemann, 1960). These requirements will obviously yield to experimental restrictions of observable rates and solubilities in many cases, but the method may be useful with certain enzymes. 5. Real and Apparent Cooperativity Much of the current interest in the kinetics of multi-sited enzymes is due to the suggestion of Monod et al. (1963), that cooperative binding effects produced by allosteric modifiers may provide metabolic control through the regulation of enzymic activity. Although cooperative binding has been known 2 00
1
I 60
I 20
0 80
0 00
0 10
OX
030
040
050
5
FIG. 5. Apparent cooperativity of anticooperative sites under the influence of cornpetitive inhibition. Kl = KI = IO- 4, Kz = KIrr = lo- 2, KI1 = 10-s.
102
E.
1‘.
HARPER
for some time to produce sigmoid saturation curves and upwardly curving Lineweaver-Burk plots (Botts, 1958), the converse is not known to be true, i.e., the occurrence of such curves may not necessarily indicate the presence of cooperative binding. Figures 1, 3 and 5 show that. in fact, it dots not: sigmoid saturation curves and upwardly curving Lineweaver-Burk plots can be obtained even when both substrate and inhibitor exert unticooperatice binding efSects. Thus, if an enzyme is subject to unsuspected inhibition, it is possible to mistakenly conclude that the substrate exerts a positive homotropic effect (K-activation) when in fact it is a negative modifier (K-inhibition). This unexpected result underscores a caveat which is also evident from the calculations of Koshland, Nemethy & Filmcr (1966), cf. Atkinson (1966); namely, that the shapes of kinetic plots alone may not pro\,idc unequivocal evidence for the idenrification of modifier effects as cooperative or anticooperative, and may not provide an adequate basis for inferences concerning the regulation of enzymic activity. The author is grateful to Louise and Henk Hokse for the experimental observations and stimulating discussion which led to this paper, and to Michael W. Trogdlen and Bruce H. Phelps for their assistance with the computer program. This research was supported by a Grant-in-Aid from the American Heart Association,
and
by
Grant
GB-5367
from
the
National
Science
Foundation.
Computations were carried out at the Indiana University Medica! Center Research Computation Center, which is supported in part by Public HeoI:h Service Medical Kesearch Grant FROOl67. REFERENCES ALMOND, H. R. & NIEIIANPL’, C. (1960). Biochem. biophys. dctu. 44, 143. ATKINSON, D. E. (1966). A. Rev. Biochem. 35,85. BENDER, M. L. (1960). Chem. Rev. 60, 53. Borrs, J. (1958). Trans. Faraday Sot. 54, 593. CHA, S. (1968). J. biol. Chem. 243, 820. EADIE, G. S. (1942). J. biol. Chem. 146, 85. HARPER, E. T. (1971). J. theor. Biol. 32,405. HOFSTEE. B. H. J. (1952). Science, N. Y. 116, 319. KISTIAKOWSKY, G. B. & ROSEFU‘BERG, A. J. (1952). J. .4m. c/rem. Sot. 74, 5010. KLOTZ, I. M. & HUNSTON, D. L. (1971). Biochemistry, N. Y. 10, 3065. KOSHLAND, JR., D. E., N~METHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. LEFFLER, J. E. & GRUNWALD, E. (1963). Rates and Equilibriu of Organic Reactions. pp. 162-168. New York: John Wiley and Sons, Inc. MONOD, J.. CHANGEUX, J.-P. CsrJACOB, F. (1963). J. molec. Biol. 6, 306. MONOD, J., WYMAN, J. & CHANGEU~, J.-P. (1965). J. nlolec. Biol. 12, 88. TROWBRIDGE, C. G., KREHBIEL, A. & LASKOWSKI, JR., M. (1963). Biochemistry, N. Y. 2. 843.