Proc. Nati. Acad. Sci. USA
Vol. 74, No. 11, pp. 4959-4963, November, 1977 Biochemistry
Analytical and graphical examination of strong binding by half-of-sites in proteins: Illustration with aspartate transcarbamylase (binding constants/affinity profiles/ligand affinities/interactions between sites)
IRVING M. KLOTZ* AND DONALD L. HUNSTONt t * Department of Chemistry and Department of Biochemistry and Molecular Biology, Northwestern University, Evanston, Illinois 60201; and t Chemistry Division, Naval Research Laboratory, Washington, D.C. 20375
Contributed by Irving M. Klotz, September 6, 1977
In multiple binding of ligands to a protein, the ABSTRACT binding sites may seem to behave as if they are partitioned equally between two modalities. This paper analyzes three different molecular situations in which two actual assemblages appear: (i) two classes of sites exist at the outset in the ligand-free macromolecule; (ii) all sites are initially identical but after half are occupied, the affinity of the residual ones is altered; (iii) all sites are initially identical but they interact in a pairwise manner. The contours of affinity profiles-graphs of normalized stoichiometric binding constants (iK;) versus stoichiometric step number i-are examined for each situation to provide a basis for discriminating among them. Proper procedures for evaluating the site binding constants are then described. To illustrate these procedures, published experimental data for two real systems, binding of substrate or modifier by the enzyme aspartate transcarbamylase (carbamoylphosphate: L-aspartate carbamoyltransferase, EC 2.1.3.2), are scrutinized and the meaning of the calculated binding parameters is examined. The results demonstrate concretely that site binding constants cannot be specified without assuming a particular molecular model, but the stoichiometric constants can be assigned unambiguously without regard to the type of behavior at the individual sites.
The term "half-of-sites binding" § connotes, to different individuals or under various circumstances, different types of relationships between binding sites (1, 2). In some situations the macromolecule is presumed to have at the outset two different classes of independent sites with the affinity of one class being much weaker than that of the other. Alternatively, the macromolecule may be constituted of identical sites of equal affinity but after half are occupied the residual sites change markedly in affinity for ligand. A third possibility is that in which the initially identical sites interact in a pairwise fashion so that when either partner of a pair is occupied by ligand the affinity of the unoccupied one changes strikingly. This ambiguity in meaning leads to misunderstandings in the analysis of equilibrium binding experiments and often to incorrect interpretation of the parameters derived. For example, there is a widespread impression that if the equilibrium binding, r, of substrate (ligand), A, by an enzyme (protein) with n sites can be fitted to an equation with two hyperbolic terms$ [k1] r = (n/2)ka(A) + (n /2) 1 + kc,(A) 1 + ko(A) then ka and k,6 are always the intrinsic binding constants for two sets of sites. For systems in which the enzyme is constituted of identical protomers and for which ka is found to be much larger than k,6, it is also generally assumed that ka is the intrinsic The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
binding constant of the initially identical active sites. However, such conclusions are unwarranted. As has been pointed out recently (3, 4), the use of a linear combination of hyperbolic terms to represent binding of ligands by macromolecules yields empirical parameters ka, ko, ..., kx that cannot correspond to site binding constants when there are interactions between sites. When such interactions exist, the affinity of each particular site changes with occupancy of other sites, and more than one site binding constant must be assigned to each particular site. The number of site binding constants that must be specified increases dramatically with increasing n, and generalized equations in the format of Eq. 1 collapse completely for evaluation of such constants. In contrast, even when there are interactions between binding sites, the number of stoichiometric constants (3, 4) is invariant, so long as the total number of sites is fixed. Moreover, the experimentally observed relationships between the stoichiometric constants reflect the nature of site binding interactions in important cases of special interest. The contours of affinity profiles, graphs of normalized stoichiometric constants as functions of the successive stoichiometric steps, are particularly concise and revealing in these situations. To elucidate these features, with special reference to "halfof-sites binding," we shall apply a completely general analysis for all types of multiple ligand equilibria (4) to each of three special cases for a six-site macromolecule. We shall then treat two concrete examples, binding of substrate or modifier by the enzyme aspartate transcarbamylase (carbamoylphosphate: L-aspartate carbamoyltransferase, EC 2.1.3.2).
Definitions We start by restating some definitions. Focussing on individual binding sites in the multiple equilibria between ligands bound to a macromolecule P and free ligand A in bulk solvent, we may define a site equilibrium constant, kj, for the equilibrium at each specific site jP: (kP)(A jP+A=jPA; + (IPA)
[21
If interactions are present, however, it will be necessary to designate more than one site binding constant for at least some * To whom requests for reprints should be addressed at: Naval Research
Laboratory, Chemistry Division (6170), Washington DC 20375.
§ If the kinetic consequences are being observed, this is generally called
"half-of-sites reactivity" (1, 2).
¶ In Eq. 1, r represents the moles of bound ligand per mol of total
protein, (A) is the concentration of unbound ligand at equilibrium with bound, and n is the moles of bound ligand at saturation of the protein.
4959
4960
Biochemistry: Klotz and Hunston
Proc. Natl. Acad. Sci. USA 74 (1977)
indeed correspond to Kl and K2, respectively. Stoichiometric constants for these circumstances can be derived for the general case of n total sites, by an extension of the polynominal analysis of Fletcher et al. (5), in terms of K, and combinational constants that are functions of the affinity ratio p
=
[5]
K2/K1.
For n = 6, the successive stoichiometric constants are given by the following general equations: K1
=
K(3S+3p)
K2=Ki(3+9p+3p2)/(3+3p)
KI(1 + 9p + 9p2 + p3)/(3 + 9p + 3p2) K4 = Klp(3 + 9p + 3p2)/(1 + 9p + 9p2 + p3) K3 =
FIG. 1. Affinity profiles for hexameric system with two independent, noninteracting classes of binding sites, with different affinity ratios p. The limiting line is reached as p - 0.
of the sites, because the value of kj will depend on which other sites are occupied. Alternatively, we may detail the multiple equilibria in terms of the stoichiometric equilibrium constants, Ki, for the formation of the sequential stoichiometric macromolecule-ligand species PA,, PA2, etc.;
PAjj + A = PA1; Kj= (PA( ) [31 The definitions of Kis are valid for any manner of interaction
between binding sites, because no assumptions regarding the mode of such interactions underlie Eq. 3. In terms of stoichiometric binding constants, the moles of bound ligand per mole of total protein, r, may be expressed by r = KI(A) + 2K1K2(A)2 +* [4 [ 1 + K1(A) + K1K2(A)2 +From experimental data for r as a function of (A), one can evaluate the successive stoichiometric constants KI, K2, etc. Their relationships may then be visualized in an affinity profile-that is, a graph of iKj versus i (3, 4). Affinity profiles for a hexameric system Let us now examine the appearances of some affinity profiles for a macromolecule capable of binding six molecules of ligand at saturation (i.e., n = 6). We shall consider in detail three situations of particular interest: (i) the macromolecule has at the outset two classes of independent sites (three sites being in each class) and the site affinities are unchanged with occupancy; (ii) the sites are initially identical but when three sites are filled (with unchanged affinities) an interaction arises and alters the affinities of the unoccupied sites to a new value, which remains constant for the binding of the last three ligands; (iii) the sites are initially identical in affinities, but they interact with each other in pairwise fashion-i.e., as one is occupied, its unoccupied pair-partner changes in affinity but the unoccupied pair(s) are unaffected. For situation (i), three binding sites have identical site affinity constants designated K1 and three have identical affinity constants designated K2. For this situation, ka and k3 (Eq. 1) do
[6]
K5 = Kip(3 + 3p)/(3 + 9P + 3P2) K6 = Kjp/(3 + 3P). A specific example is provided by the limiting case of small p (i.e., p < 0.001) for which K1 = 3K1 4K4= 4 3K2 5K5 = 5* K2 2K2 = 2 K1 [7] = 3K3= 3 * K/3 6K6 6 K2/3Successive values of iKj versus i, for a graph of the affinity profile, are readily determined with Eq. 6. Such profiles for various values of p are illustrated in Fig. 1. Note that for small p the first three points (Eq. 7) approach a straight line. Moreover, this line is that which would be followed by a system with only three sites because it intersects the abscissa at i = (n/2) + 1. On the other hand, the points 4 < i < 6 do not fit on a straight line because the stoichiometric multipliers convert the coefficients of successive steps 4-6 into 12, 5, and 2, respectively (see Eq. 7). An alternative profile plotting (i - 3)Kj versus (i - 3), however, would be linear. For situation (ii) the site constants for binding of the first three ligand molecules are the same and are designated K1'; the site constants for binding of the last threell are designated K2'. From previous analyses (4), it follows that -
-
-
K1 = 6K1'
2K2 = 2 -22 K1' 3K3 = 3
4K4 =4*K2 4 4
5K5 = 5 -52 K2'
[8]
-4
6K6 = 6. - K2'. 6 3 j we can now define a pato of situation (i) Corresponding p
rameter
a7 = K2 /K/. [9] An affinity profile for a = 0.1 is illustrated in Fig. 2. Its contour is markedly different from that for p = 0.1 in situation (i) (Fig. 1). In situation (ii) both sets of successive constants adhere to two straight lines with the same intercept on the abscissa, i = n + 1 (Fig. 2). The discrimination between the two sets of binding affinities is much more distinct in Fig. 2 than in Fig. 1. Fig. 2 shows a sharp break between stoichiometric steps 1 Note that each specific site has two possible site binding constants.
Consequently it is misleading to speak of high-affinity and low-affinity sites. One should say, rather, that each site has a high-affinity state and a low-affinity state.
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Klotz and Hunston
4961
3 and 4, in contrast to the continued change in curvature observed for the curve with a corresponding value of p in Fig. 1. Fig. 2 also demonstrates that plots for a > 1 are not monotonically decreasing and thus are clearly distinguishable. Consequently, an affinity profile could definitely distinguish between situations (i) and (ii). Turning to situation (iMi) we index the pairs as 1 and 4, 2 and 5, and 3 and 6. Initially, all sites have identical site binding constants, Kid, and these change to K2 when the partner is occupied. 1 To evaluate the stoichiometric constants, we first define kjlj,2.,,..* as the site binding constant for site le when sites 1112. * * *, Je-i are occupied. We can then write [10] kjli2,. -,,i= (K2" if lie -jm = 3 for any value of m between 1 and (e-l)-that is, when the partner site is occupied; otherwise the site constants are [11] kjl,...,J.e KI'v. If we then define =
K2-= #
[12]
and use the analysis developed in a previous paper (4), the stoichiometric binding constants can be written as follows: K1 = K1"6 K2 = Kj"(12 + 3r)/6 K3 = K1j(8 + 12r)/(12 + 3r) K4 = Kl"T(12 + 3r)/(8 + 12r) K5 = KI"r6/(12 + 3T) K6 = KlT/6.
[13]
For small values of r, the normalized stoichiometric constants 1K1 approach the following: K1 =6K1 2K2 = 2 Kj#
FIG. 2. Affinity profiles for hexameric system in which all six sites are initially identical and in which a change in affinity occurs after half of sites are occupied.
Some affinity profiles for different values of the interaction parameter r are illustrated in Fig. 3. It is readily evident by comparison with Fig. 2 that the contours of the profiles for situations (ii) and (TV) are readily distinguishable. This is true for situations in which r > 1 as well as those for r < 1. A comparison of contours for p ko, moreover, situations (i) and (iii) are indistinguishable,** unless information is available to supplement the binding data, even though the site binding constants and site models for the two situations are very different. If one correlates the binding data [i.e., r versus (A)] in terms of a best fit in a linear combination of six hyperbolic terms rather than just the two terms of Eq. 1, one obtains the following parameters: ka = 1.51 X 106 k= 1.51 X 106 ky = 0.35 X 106 k5=0.36X 105 ke = 0.36 X 105 kr = 0.65 X 104 Clearly, the best fit requires four (not two) categories of these parameters, but here again they do not correspond (4) to site binding constants if interactions are present. On the other hand, it should be emphasized that the stoichiometric binding constants are valid equilibrium constants of successive stoichiometric binding steps in any circumstance-i.e., whether molecular situation (i) or (iii) or one involving some other disposition and interaction of site binding constants is applicable to these enzyme-ligand complexes. For the binding of substrate carbamyl phosphate by aspartate transcarbamylase (7), the original data, supplied by the authors, lead to the following stoichiometric binding constants: K1 = 1.45 x 106 K2 = 0.403 x 106 If Eq. 1 is converted into a polynomial algebraic form similar to Eq. 4, the two equations produce different relationships between r and A for the circumstances in situation (iii). The values of ka and k, therefore, depend on what criterion is used to select the best-fit curve. For the special case r > ko, however, both Eq. 1 and the constants for situation (iii) reduce to the same expression with Kl" ;- 1/2ka and K2"' 2ko.
**
Biochemistry:
Klotz and Hunston
Proc. Natl. Acad. Sci. USA 74 (1977)
1.4 1.2 1.0 _
'&
%%
S
0 3-
0.8 -
IF
0.6-
\
0.4 -
%
-
0.2
1
%
2
3
4
5
6
7
FIG. 5. Affinity profile for binding of carbamyl phosphate by aspartate transcarbamylase.
K3=0.104X 106 K4 = 0.376 X 105 K,5 = 0.151 X 105 K6 = 0.821 X 104. The corresponding affinity profile is illustrated in Fig. 5. Once again it is evident, from the fact that the points for i = 2 and 3 fall below the limiting (dashed) line, that there is a decrease in affinity for ligand as one proceeds from the first to the second stoichiometric step (4), and from the second to the third. In fact, for carbamyl phosphate binding, the points for i = 2 and i = 3 fall distinctly farther below the limiting line for situations (i) and (ii) than do the corresponding points for CTP. It is also apparent that the drop in normalized stoichiometric coefficient iK( between i = 3 and i = 4 is much less for carbamyl phosphate binding (Fig. 5) than for CTP binding. For binding of substrate
by aspartate transcarbamylase, neither initially independent classes of sites nor pairwise behavior provides a good model of the disposition of site binding constants. If one correlates the carbamyl phosphate binding data in terms of a best fit for r versus A in a linear combination of six hyperbolic terms rather than just the two terms of Eq. 1, one obtains the following parameters: ka = 0.884 X 106 k# = 0.488 X 106 ky = 0.522 X 105 kb = 0.259 X 105 ke = 0.218 X 105 kit= 0.201 X 105.
4963
Clearly, the last three parameters, and possibly also kz, fall into one (three- or four-member) category, and it would not be unreasonable to assign ka and k,6 to a common two-member second category. Under these circumstances, a two-hyperbolic-term equation is applicable, but it differs from Eq. 1 in that the coefficient of n is not the same 1/2 for each term but instead is 1/3 and %, respectively. The experimental binding data have also been correlated (7) in terms of Eq. 1 with each hyperbolic term having the same n/2 factor. With these constraints the values of ka and k reported (7) are 0.44 X 106 and 0.16 X 105, respectively, but these are unlikely to be the site binding constants because the interactions do not seem to be of a simple type, such as one of those described in situations (i)-(ifi) above. Thus, a careful examination of ligand-binding data for aspartate transcarbamylase indicates that there is an interaction between sites at each successive stoichiometric step. Such interactions are revealed in the contours of an affinity profile, a graph of normalized stoichiometric binding constants 0K( versus the sequential stoichiometric step i. With varying degrees of approximation, interacting binding sites may appear to behave as if they are partitioned between two equal classes, one with much stronger affinity than the other for ligand. Under these circumstances, there is a general tendency to correlate binding data by means of an algebraic linear combination of two hyperbolic terms, Eq. 1. What is not generally recognized, however, is that the binding parameters ka and ki of the individual hyperbolic terms are not site binding constants, except under very limited circumstances. Only in these special circumstances can ka and k,6 be related to site binding constants. On the other hand, stoichiometric binding constants can be assigned unambiguously without assuming any particular molecular model of relationships among- binding sites.t t We are very grateful to Dr. John E. Fletcher for his assistance in setting up the computerized procedures for the computation of ligand-binding constants and to Drs. C. W. Gray, M. J. Chamberlin, and J. P. Rosenbusch for their kindness in providing us with original binding data. This investigation was supported in part by a grant from the National Science Foundation. 1. MacQuarrie, R. A. & Bernhard, S. A. (1971) J. Mol. Biol. 55, 181-192. 2. Levitzki, A., Stallcup, W. B. & Koshland, D. E., Jr. (1971) Biochemistry 10, 3371-3378. 3. Klotz, I. M. (1974) Acc. Chem. Res. 7, 162-168. 4. Klotz, I. M. & Hunston, D. L. (1975) J. Biol. Chem. 250,30013009. 5. Fletcher, J. E., Spector, A. A. & Ashbrook, J. D. (1970) Biochemistry 9, 4580-4587. 6. Winlund, C. C. & Chamberlin, M. J. (1970) Biochem. Biophys. Research Commun. 60,43-49. 7. Suter, P. & Rosenbusch, J. (1976) J. Biol. Chem. 251, 59865991. t t To a limited degree we are prepared to provide computer-calculated
stoichiometric binding constants to recognized investigators who supply us with reliable experimental binding data of their own.
Vol. 74, No. 11, pp. 4959-4963, November, 1977 Biochemistry
Analytical and graphical examination of strong binding by half-of-sites in proteins: Illustration with aspartate transcarbamylase (binding constants/affinity profiles/ligand affinities/interactions between sites)
IRVING M. KLOTZ* AND DONALD L. HUNSTONt t * Department of Chemistry and Department of Biochemistry and Molecular Biology, Northwestern University, Evanston, Illinois 60201; and t Chemistry Division, Naval Research Laboratory, Washington, D.C. 20375
Contributed by Irving M. Klotz, September 6, 1977
In multiple binding of ligands to a protein, the ABSTRACT binding sites may seem to behave as if they are partitioned equally between two modalities. This paper analyzes three different molecular situations in which two actual assemblages appear: (i) two classes of sites exist at the outset in the ligand-free macromolecule; (ii) all sites are initially identical but after half are occupied, the affinity of the residual ones is altered; (iii) all sites are initially identical but they interact in a pairwise manner. The contours of affinity profiles-graphs of normalized stoichiometric binding constants (iK;) versus stoichiometric step number i-are examined for each situation to provide a basis for discriminating among them. Proper procedures for evaluating the site binding constants are then described. To illustrate these procedures, published experimental data for two real systems, binding of substrate or modifier by the enzyme aspartate transcarbamylase (carbamoylphosphate: L-aspartate carbamoyltransferase, EC 2.1.3.2), are scrutinized and the meaning of the calculated binding parameters is examined. The results demonstrate concretely that site binding constants cannot be specified without assuming a particular molecular model, but the stoichiometric constants can be assigned unambiguously without regard to the type of behavior at the individual sites.
The term "half-of-sites binding" § connotes, to different individuals or under various circumstances, different types of relationships between binding sites (1, 2). In some situations the macromolecule is presumed to have at the outset two different classes of independent sites with the affinity of one class being much weaker than that of the other. Alternatively, the macromolecule may be constituted of identical sites of equal affinity but after half are occupied the residual sites change markedly in affinity for ligand. A third possibility is that in which the initially identical sites interact in a pairwise fashion so that when either partner of a pair is occupied by ligand the affinity of the unoccupied one changes strikingly. This ambiguity in meaning leads to misunderstandings in the analysis of equilibrium binding experiments and often to incorrect interpretation of the parameters derived. For example, there is a widespread impression that if the equilibrium binding, r, of substrate (ligand), A, by an enzyme (protein) with n sites can be fitted to an equation with two hyperbolic terms$ [k1] r = (n/2)ka(A) + (n /2) 1 + kc,(A) 1 + ko(A) then ka and k,6 are always the intrinsic binding constants for two sets of sites. For systems in which the enzyme is constituted of identical protomers and for which ka is found to be much larger than k,6, it is also generally assumed that ka is the intrinsic The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
binding constant of the initially identical active sites. However, such conclusions are unwarranted. As has been pointed out recently (3, 4), the use of a linear combination of hyperbolic terms to represent binding of ligands by macromolecules yields empirical parameters ka, ko, ..., kx that cannot correspond to site binding constants when there are interactions between sites. When such interactions exist, the affinity of each particular site changes with occupancy of other sites, and more than one site binding constant must be assigned to each particular site. The number of site binding constants that must be specified increases dramatically with increasing n, and generalized equations in the format of Eq. 1 collapse completely for evaluation of such constants. In contrast, even when there are interactions between binding sites, the number of stoichiometric constants (3, 4) is invariant, so long as the total number of sites is fixed. Moreover, the experimentally observed relationships between the stoichiometric constants reflect the nature of site binding interactions in important cases of special interest. The contours of affinity profiles, graphs of normalized stoichiometric constants as functions of the successive stoichiometric steps, are particularly concise and revealing in these situations. To elucidate these features, with special reference to "halfof-sites binding," we shall apply a completely general analysis for all types of multiple ligand equilibria (4) to each of three special cases for a six-site macromolecule. We shall then treat two concrete examples, binding of substrate or modifier by the enzyme aspartate transcarbamylase (carbamoylphosphate: L-aspartate carbamoyltransferase, EC 2.1.3.2).
Definitions We start by restating some definitions. Focussing on individual binding sites in the multiple equilibria between ligands bound to a macromolecule P and free ligand A in bulk solvent, we may define a site equilibrium constant, kj, for the equilibrium at each specific site jP: (kP)(A jP+A=jPA; + (IPA)
[21
If interactions are present, however, it will be necessary to designate more than one site binding constant for at least some * To whom requests for reprints should be addressed at: Naval Research
Laboratory, Chemistry Division (6170), Washington DC 20375.
§ If the kinetic consequences are being observed, this is generally called
"half-of-sites reactivity" (1, 2).
¶ In Eq. 1, r represents the moles of bound ligand per mol of total
protein, (A) is the concentration of unbound ligand at equilibrium with bound, and n is the moles of bound ligand at saturation of the protein.
4959
4960
Biochemistry: Klotz and Hunston
Proc. Natl. Acad. Sci. USA 74 (1977)
indeed correspond to Kl and K2, respectively. Stoichiometric constants for these circumstances can be derived for the general case of n total sites, by an extension of the polynominal analysis of Fletcher et al. (5), in terms of K, and combinational constants that are functions of the affinity ratio p
=
[5]
K2/K1.
For n = 6, the successive stoichiometric constants are given by the following general equations: K1
=
K(3S+3p)
K2=Ki(3+9p+3p2)/(3+3p)
KI(1 + 9p + 9p2 + p3)/(3 + 9p + 3p2) K4 = Klp(3 + 9p + 3p2)/(1 + 9p + 9p2 + p3) K3 =
FIG. 1. Affinity profiles for hexameric system with two independent, noninteracting classes of binding sites, with different affinity ratios p. The limiting line is reached as p - 0.
of the sites, because the value of kj will depend on which other sites are occupied. Alternatively, we may detail the multiple equilibria in terms of the stoichiometric equilibrium constants, Ki, for the formation of the sequential stoichiometric macromolecule-ligand species PA,, PA2, etc.;
PAjj + A = PA1; Kj= (PA( ) [31 The definitions of Kis are valid for any manner of interaction
between binding sites, because no assumptions regarding the mode of such interactions underlie Eq. 3. In terms of stoichiometric binding constants, the moles of bound ligand per mole of total protein, r, may be expressed by r = KI(A) + 2K1K2(A)2 +* [4 [ 1 + K1(A) + K1K2(A)2 +From experimental data for r as a function of (A), one can evaluate the successive stoichiometric constants KI, K2, etc. Their relationships may then be visualized in an affinity profile-that is, a graph of iKj versus i (3, 4). Affinity profiles for a hexameric system Let us now examine the appearances of some affinity profiles for a macromolecule capable of binding six molecules of ligand at saturation (i.e., n = 6). We shall consider in detail three situations of particular interest: (i) the macromolecule has at the outset two classes of independent sites (three sites being in each class) and the site affinities are unchanged with occupancy; (ii) the sites are initially identical but when three sites are filled (with unchanged affinities) an interaction arises and alters the affinities of the unoccupied sites to a new value, which remains constant for the binding of the last three ligands; (iii) the sites are initially identical in affinities, but they interact with each other in pairwise fashion-i.e., as one is occupied, its unoccupied pair-partner changes in affinity but the unoccupied pair(s) are unaffected. For situation (i), three binding sites have identical site affinity constants designated K1 and three have identical affinity constants designated K2. For this situation, ka and k3 (Eq. 1) do
[6]
K5 = Kip(3 + 3p)/(3 + 9P + 3P2) K6 = Kjp/(3 + 3P). A specific example is provided by the limiting case of small p (i.e., p < 0.001) for which K1 = 3K1 4K4= 4 3K2 5K5 = 5* K2 2K2 = 2 K1 [7] = 3K3= 3 * K/3 6K6 6 K2/3Successive values of iKj versus i, for a graph of the affinity profile, are readily determined with Eq. 6. Such profiles for various values of p are illustrated in Fig. 1. Note that for small p the first three points (Eq. 7) approach a straight line. Moreover, this line is that which would be followed by a system with only three sites because it intersects the abscissa at i = (n/2) + 1. On the other hand, the points 4 < i < 6 do not fit on a straight line because the stoichiometric multipliers convert the coefficients of successive steps 4-6 into 12, 5, and 2, respectively (see Eq. 7). An alternative profile plotting (i - 3)Kj versus (i - 3), however, would be linear. For situation (ii) the site constants for binding of the first three ligand molecules are the same and are designated K1'; the site constants for binding of the last threell are designated K2'. From previous analyses (4), it follows that -
-
-
K1 = 6K1'
2K2 = 2 -22 K1' 3K3 = 3
4K4 =4*K2 4 4
5K5 = 5 -52 K2'
[8]
-4
6K6 = 6. - K2'. 6 3 j we can now define a pato of situation (i) Corresponding p
rameter
a7 = K2 /K/. [9] An affinity profile for a = 0.1 is illustrated in Fig. 2. Its contour is markedly different from that for p = 0.1 in situation (i) (Fig. 1). In situation (ii) both sets of successive constants adhere to two straight lines with the same intercept on the abscissa, i = n + 1 (Fig. 2). The discrimination between the two sets of binding affinities is much more distinct in Fig. 2 than in Fig. 1. Fig. 2 shows a sharp break between stoichiometric steps 1 Note that each specific site has two possible site binding constants.
Consequently it is misleading to speak of high-affinity and low-affinity sites. One should say, rather, that each site has a high-affinity state and a low-affinity state.
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Klotz and Hunston
4961
3 and 4, in contrast to the continued change in curvature observed for the curve with a corresponding value of p in Fig. 1. Fig. 2 also demonstrates that plots for a > 1 are not monotonically decreasing and thus are clearly distinguishable. Consequently, an affinity profile could definitely distinguish between situations (i) and (ii). Turning to situation (iMi) we index the pairs as 1 and 4, 2 and 5, and 3 and 6. Initially, all sites have identical site binding constants, Kid, and these change to K2 when the partner is occupied. 1 To evaluate the stoichiometric constants, we first define kjlj,2.,,..* as the site binding constant for site le when sites 1112. * * *, Je-i are occupied. We can then write [10] kjli2,. -,,i= (K2" if lie -jm = 3 for any value of m between 1 and (e-l)-that is, when the partner site is occupied; otherwise the site constants are [11] kjl,...,J.e KI'v. If we then define =
K2-= #
[12]
and use the analysis developed in a previous paper (4), the stoichiometric binding constants can be written as follows: K1 = K1"6 K2 = Kj"(12 + 3r)/6 K3 = K1j(8 + 12r)/(12 + 3r) K4 = Kl"T(12 + 3r)/(8 + 12r) K5 = KI"r6/(12 + 3T) K6 = KlT/6.
[13]
For small values of r, the normalized stoichiometric constants 1K1 approach the following: K1 =6K1 2K2 = 2 Kj#
FIG. 2. Affinity profiles for hexameric system in which all six sites are initially identical and in which a change in affinity occurs after half of sites are occupied.
Some affinity profiles for different values of the interaction parameter r are illustrated in Fig. 3. It is readily evident by comparison with Fig. 2 that the contours of the profiles for situations (ii) and (TV) are readily distinguishable. This is true for situations in which r > 1 as well as those for r < 1. A comparison of contours for p ko, moreover, situations (i) and (iii) are indistinguishable,** unless information is available to supplement the binding data, even though the site binding constants and site models for the two situations are very different. If one correlates the binding data [i.e., r versus (A)] in terms of a best fit in a linear combination of six hyperbolic terms rather than just the two terms of Eq. 1, one obtains the following parameters: ka = 1.51 X 106 k= 1.51 X 106 ky = 0.35 X 106 k5=0.36X 105 ke = 0.36 X 105 kr = 0.65 X 104 Clearly, the best fit requires four (not two) categories of these parameters, but here again they do not correspond (4) to site binding constants if interactions are present. On the other hand, it should be emphasized that the stoichiometric binding constants are valid equilibrium constants of successive stoichiometric binding steps in any circumstance-i.e., whether molecular situation (i) or (iii) or one involving some other disposition and interaction of site binding constants is applicable to these enzyme-ligand complexes. For the binding of substrate carbamyl phosphate by aspartate transcarbamylase (7), the original data, supplied by the authors, lead to the following stoichiometric binding constants: K1 = 1.45 x 106 K2 = 0.403 x 106 If Eq. 1 is converted into a polynomial algebraic form similar to Eq. 4, the two equations produce different relationships between r and A for the circumstances in situation (iii). The values of ka and k, therefore, depend on what criterion is used to select the best-fit curve. For the special case r > ko, however, both Eq. 1 and the constants for situation (iii) reduce to the same expression with Kl" ;- 1/2ka and K2"' 2ko.
**
Biochemistry:
Klotz and Hunston
Proc. Natl. Acad. Sci. USA 74 (1977)
1.4 1.2 1.0 _
'&
%%
S
0 3-
0.8 -
IF
0.6-
\
0.4 -
%
-
0.2
1
%
2
3
4
5
6
7
FIG. 5. Affinity profile for binding of carbamyl phosphate by aspartate transcarbamylase.
K3=0.104X 106 K4 = 0.376 X 105 K,5 = 0.151 X 105 K6 = 0.821 X 104. The corresponding affinity profile is illustrated in Fig. 5. Once again it is evident, from the fact that the points for i = 2 and 3 fall below the limiting (dashed) line, that there is a decrease in affinity for ligand as one proceeds from the first to the second stoichiometric step (4), and from the second to the third. In fact, for carbamyl phosphate binding, the points for i = 2 and i = 3 fall distinctly farther below the limiting line for situations (i) and (ii) than do the corresponding points for CTP. It is also apparent that the drop in normalized stoichiometric coefficient iK( between i = 3 and i = 4 is much less for carbamyl phosphate binding (Fig. 5) than for CTP binding. For binding of substrate
by aspartate transcarbamylase, neither initially independent classes of sites nor pairwise behavior provides a good model of the disposition of site binding constants. If one correlates the carbamyl phosphate binding data in terms of a best fit for r versus A in a linear combination of six hyperbolic terms rather than just the two terms of Eq. 1, one obtains the following parameters: ka = 0.884 X 106 k# = 0.488 X 106 ky = 0.522 X 105 kb = 0.259 X 105 ke = 0.218 X 105 kit= 0.201 X 105.
4963
Clearly, the last three parameters, and possibly also kz, fall into one (three- or four-member) category, and it would not be unreasonable to assign ka and k,6 to a common two-member second category. Under these circumstances, a two-hyperbolic-term equation is applicable, but it differs from Eq. 1 in that the coefficient of n is not the same 1/2 for each term but instead is 1/3 and %, respectively. The experimental binding data have also been correlated (7) in terms of Eq. 1 with each hyperbolic term having the same n/2 factor. With these constraints the values of ka and k reported (7) are 0.44 X 106 and 0.16 X 105, respectively, but these are unlikely to be the site binding constants because the interactions do not seem to be of a simple type, such as one of those described in situations (i)-(ifi) above. Thus, a careful examination of ligand-binding data for aspartate transcarbamylase indicates that there is an interaction between sites at each successive stoichiometric step. Such interactions are revealed in the contours of an affinity profile, a graph of normalized stoichiometric binding constants 0K( versus the sequential stoichiometric step i. With varying degrees of approximation, interacting binding sites may appear to behave as if they are partitioned between two equal classes, one with much stronger affinity than the other for ligand. Under these circumstances, there is a general tendency to correlate binding data by means of an algebraic linear combination of two hyperbolic terms, Eq. 1. What is not generally recognized, however, is that the binding parameters ka and ki of the individual hyperbolic terms are not site binding constants, except under very limited circumstances. Only in these special circumstances can ka and k,6 be related to site binding constants. On the other hand, stoichiometric binding constants can be assigned unambiguously without assuming any particular molecular model of relationships among- binding sites.t t We are very grateful to Dr. John E. Fletcher for his assistance in setting up the computerized procedures for the computation of ligand-binding constants and to Drs. C. W. Gray, M. J. Chamberlin, and J. P. Rosenbusch for their kindness in providing us with original binding data. This investigation was supported in part by a grant from the National Science Foundation. 1. MacQuarrie, R. A. & Bernhard, S. A. (1971) J. Mol. Biol. 55, 181-192. 2. Levitzki, A., Stallcup, W. B. & Koshland, D. E., Jr. (1971) Biochemistry 10, 3371-3378. 3. Klotz, I. M. (1974) Acc. Chem. Res. 7, 162-168. 4. Klotz, I. M. & Hunston, D. L. (1975) J. Biol. Chem. 250,30013009. 5. Fletcher, J. E., Spector, A. A. & Ashbrook, J. D. (1970) Biochemistry 9, 4580-4587. 6. Winlund, C. C. & Chamberlin, M. J. (1970) Biochem. Biophys. Research Commun. 60,43-49. 7. Suter, P. & Rosenbusch, J. (1976) J. Biol. Chem. 251, 59865991. t t To a limited degree we are prepared to provide computer-calculated
stoichiometric binding constants to recognized investigators who supply us with reliable experimental binding data of their own.
