Proc. Natl. Acad. Sci. USA Vol. 74, No. 1, pp. 139-143, January 1977
Biochemistry
Distinctions between the two-state and sequential models for cooperative ligand binding (acetylcholine/receptor)
RAYMOND E. GIBSON* t AND SIMON A. LEVINt * Section of Neurobiology and Behavior and t Section of Ecology and Systematics, Cornell University, Ithaca, New York 14853
Communicated by Roy Hertz, October 22, 1976
ABSTRACT The two-state and sequential models for positive cooperativity in ligand binding can produce significantly different theoretical binding curves when presented in a Scatchard plot. The conditions that produce the greatest differences have been examined. The theoretical differences have been used to select the two-state model as the best model for describing the binding of acetylcholine to acetylcholine receptors that have been solubilized by Triton X-100 and sodium cholate.
The cooperative binding of ligands to an oligomeric protein is usually described by a variant of one of two different models: the symmetry or two-state model of Monod et al. (1) and the sequential or induced-fit model of Koshland et al. (2). For data exhibiting positive cooperativity alone, the accuracy of most binding studies is such that it is very difficult to distinguish between the two models (3). The interchangeability of the two models has been suggested (4) since the three parameters in both models can be related to the parameters of the Adair model (5), an earlier model that gave way to the Pauling model (6) and subsequently to the induced-fit model. Although data that exhibit negative cooperativity or "half-of-sites" binding are generally described by the induced-fit model, such data may also be described by extensions of the two-state model involving nonidentical subunits with differing inherent affinities for the ligand (7, 8). However, even when subunit homogeneity is known, the induced-fit model has remained the model of choice. When the data exhibit only positive cooperativity, curvefitting procedures using either model will generally fit the data equally well (9, 10). However, in studies on the interaction of acetylcholine with the acetylcholine receptor from Torpedo californica electroplax, we noted and were able to capitalize on important theoretical differences between the two-state model and two geometries of the induced-fit model, thus enabling us to draw conclusions concerning preferred models. Basically, the concerted (two-state) model is more flexible, especially at low ligand concentrations; both the tetrahedral and square forms of the induced-fit model are controlled by a single "shape" parameter. In particular, when results exhibiting positive cooperativity are presented in a Scatchard plot (Y/x against Y, where x is ligand concentration and Y is amount bound), either model can provide for a maximum in the plot. However, when the two-state model is compared with either form of the induced-fit model, the two-state model is the only one that allows for the possibility of a minimum in the Scatchard plot; this occurs at a binding value Y less than that at which the maximum occurs. For one set of data we present, this flexibility is critical and provides the principal basis for our argument in favor of the two-state model. Furthermore, we establish that the induced-fit model has only two independent parameters, t Present address: Department of Radiology, The George Washington
University Medical Center, Washington, D.C. 20037. 139
one of which is a scaling constant; thus, when protein interaction constants (of which there are two in the induced-fit model) are not known, the ligand affinity constant cannot be determined.
MATERIALS AND METHODS [3H]Acetylcholine (170 mCi/mmol) was purchased from Amersham/Searle. Live Torpedo californica were purchased from Pacific Biomarine Supply Co. (Venice, Calif.). Preparations of acetylcholine receptor were made from the heavy membrane fraction of T. californica electroplax as described (8). The heavy membrane preparation was lyophilized and stored at -250 as long as nine months without deterioration of acetylcholine binding. The acetylcholine receptor was solubilized in either 1% sodium cholate or 1% Triton X-100 in Ringer's solution (115 mM NaCl, 4.6 mM KCl, 0.65 mM CaC12, 1.15 mM MgSO4, and 15.7 mM Na2HPO4 adjusted to pH 7.4 with HCI) as described (11). Acetylcholinesterase was inhibited by incubation with the anticholinesterase Tetram [oxalate salt of O,O-diethyl-S-(2-diethylaminoethyl)phosphorothiolate] at 10-4 M. Acetylcholine binding was determined by equilibrium dialysis assay; details of the assay have been described elsewhere (11).
The binding parameters for Fig. la and b were determined by an iterative procedure, using the equations for the models described below, which minimized the deviations in the observed B/x values (Fig. 1) and Y/x values (Fig. 2) in the Scatchard plot as judged by x2 analysis. Y is defined as the fractional saturation B/Bmax, when Bmax is the maximum binding. Minimizing the deviation with respect to Y values did not produce significantly different results. RESULTS The binding of acetylcholine to acetylcholine receptors that had been solubilized with cholate and Triton is presented in Fig. la and b. The data exhibit positive cooperativity, as indicated by the convexity in the Scatchard plots. The points indicated by open circles were considered to exceed normal scatter before curve-fitting procedures were undertaken. Subsequent analysis suggests that the deletions were valid, and those data points are not included in the final x2 analysis. Positive cooperativity in acetylcholine binding has been reported earlier (8), and the data were analyzed by the two-state model. Since the two-state and induced-fit models have been described as the two extremes of a more general model (12), we attempted curve-fitting procedures using both models. However, in the simplest forms of the models, we found that the two models produced very different theoretical binding curves. The differences appeared to result from the differences in the properties of the first and second derivatives of the saturation functions for each model. We therefore undertook a detailed analysis of the models.
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Proc. Nati. Acad. Sci. USA 74 (1977)
0
0.2
0.4.
0.6 0.8
1.0
1. 2
1.4.
1.6 1.8
2.0
2.2 2.4 2.6
B B FIG. 1. Scatchard plot of [3H]acetylcholine binding to Torpedo californica receptor. B is in nmol/g of original tissue; values forB/x should be multiplied by 1000. (a) Solubilized in sodium cholate. Curve fit by two-state model (-): KR = 2 nM, c = 0.018, and L = 325,000; X2(B/x) = 0.0932 and x2(B) = 0.0081. Curve fit by induced-fit model (- -): Kx = 500 X 106 M, KAB = 0.1323, and KBB = 0.040; X2(B/x) = 1.719 and x2(B) = 0.0731. (b) Solubilized by Triton X-100. Curve fit by two-state model: KR = 2 nM, c = 0.033, and L = 325,000; x2(B/x) = 0.5034 and X2(B) = 0.0075. Curve fit by induced-fit model: Kx = 500 X 106 M, KAB = 0.1822, and KBB = 0,0470; X2(B/x) = 2.022 and X2(B) = 0.1104. Theoretical curves were minimized on unweighted (Bix) values.
The saturation function, Y, for all binding models can be described by the general equation: [i] Y - (x/n)IJn F(x)T where x is the ligand concentration. F(x) is a function describing the total conformational states available in the presence of ligand, which is normalized to the unoccupied state (Ro). For example, F(x) = 1 + x in Eq. 1 gives the Langmuir isotherm (prime denotes differentiation). Cooperativity (positive or negative) may be defined as a positive or negative change in effective (average) binding constant as the amount bound increases. This will appear as (respectively) negative or positive curvature (essentially change in slope) in either the Scatchard or double-reciprocal plots of the (x,Y) data; in the presence of no cooperativity, both would be straight lines. Analytically, cooperativity is defined by the sign of the quantity
a
H2, (G)1-2 ( 1Flt where G = (In F)' and H = (ln G)'. It is easily confirmed that a> 0 if and only if the slope in either the Scatchard or double-reciprocal plot is decreasing, and that this corresponds to an increase in average effective binding constant. Two-state model In the two-state model, the oligomer of n subunits is assumed to exist in two conformations, the T or constrained state and the R or relaxed state. These two conformations are in equilibrium in the absence of ligand, with the equilibrium constant defined as L = (T)/(R). The T and R states exhibit different affinities for ligand, the dissociation constants being defined as KT and KR, respectively (c = KR/KT). When L > 1 and c < 1, positive cooperativity may be observed. The fraction of oligomers binding the ligand is given by Eq. 1 when F(x) is given by Eq.
(In
2: Fx)
_
(1+
ar
+
ca)n
(1 +
[2]
where a = X/KR. The parameters that produced the curve fits in Fig. la and b are presented in the legend to the figure. A family of theoretical curves generated from the Y-function of the two-state model (n = 4, c = 0.042, and L from 102 to 105) is presented in Fig. 2a. For L values from 1 to 1515, the Scatchard plot has the maximum characteristic of positive cooperativity. When 1515 < L < 23,100, both a minimum and a maximum are evident. At L values greater than 23,100, there is neither a maximum nor a minimim. Values for the intercepts, initial and final slopes, and initial and final curvatures can be quantitatively determined from the expressions in Table 1.
Induced-fit model The assumptions in the induced-fit model (2) are that each subunit can exist in either of two conformational states, the A and B states, with the equilibrium constant Kt = (B)/(A) characterizing the relative stability of the conformations; the binding of ligand causes a conformational change from the A to the B state. The intrinsic association constant is defined as Kx = 1/KR = (Bx)/(B)(x), where x is the ligand concentration. Additionally, the ligand-induced conformational change in one subunit alters its interaction with neighboring subunits. The subunit-subunit interactions 'are characterized by two equilibrium constants: KAB reflects the strength of the A-B interactions and KBB reflects the strength of the B-B interactions. The saturation function, Y, for the induced-fit model is dependent upon the geometry of the subunits as it reflects the differing number of neighboring interactions. For example, with n = 4 in a tetrahedral arrangement there are three neighboring interactions for each subunit, while in the square geometry there are only two neighboring interactions. The saturation functions for these two geometries are given by Eqs. 3 and 1 for the tetrahedral geometry; F(x) -1 + 4K3AeX + 6K4ABKBBC? + 4K3,AK3a3 + K6uma4
[3a]
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Gibson and Levin
0.1 0.2 0.3 0.4 0 5 0.6 0.7 0.8 0.9 By
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0
141
1.0
14
Y/x
0.5 By
FIG. 2. Scatchard plot of theoretical binding curves. (a) Two-state model (Eqs. 1 and 2): n = 4, KR = 5 nM, c = 0.042, and L is (A) 100, (B) 300, (C) 1000, (D) 3000, (E) 10,000, (F) 28,060, and (G) 100,000. (b) Induced-fit model (Eqs. 1 and 3): n = 4 in the square geometry, K. = 200, and (A) KAB = 0.2269, KBB = 0.3374; (B) KAB = 0.2121, KBB = 0.2375; (C) KAB = 0.2074, KBB = 0.1650; (D) KAB = 0.2062, KBB = 0.1200; (E) KAB = 0.2049, KBB = 0.0900; (F) KAB = 0.2049, KBB = 0.0725; and (G) KAB = 0.2049, KBB = 0.0550. (c) Scatchard plot of curves D and F from Figs. la (-) and 2b (- -).
and by Eqs. 4 and 1 for the square geometry: F(x) -1 + 4K2ABO + (2K4AB.+ 4K2ABKBB)a2
+4K2ABK2Ba3 + K4BjCa4. [4a] In both models, a = KKtx.
These forms are misleading in their apparent range, but the limitations become apparent if the single shape parameter S = K2AB/KBB is introduced. In this case, [3a] and [4a] may be rewritten, respectively, as [3b] F(X) 1+4S3/2y+6S2V+4S3/23 - 4
in the tetrahedral (with x = KXKtK3/2BBX), and [4b] F(x) = 1 + 4Si + 2(S2 + 2S)T2 + 4SX3 + X-4 in the square (with x = KxKtKBpx). Except for the scale factor XITBB, there is only one free parameter S. The parameters that produced the curve fits of Fig. la and b were obtained for the square geometry and the values for the parameters are presented in the legend to Fig. 1. The theoretical curves in Fig. 2b were generated from Eqs. 1 and 4 for the square geometry. The value of KxKt was assigned as 200 X 106 M to compare with the value of KR = 5 nM in Fig. 2a, and
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Proc. Natl. Acad. Sci. USA 74 (1977)
Table 1. Equations relating the initial intercept, initial slope, and final slope of the Scatchard plot of the two-state and induced-fit models Induced-fit model
1 I 1 + +LL KR
Initial intercept
L 1 V +LK ABKxKtt ABxxttKBK 1 +L TK3BK
(1 + cL)(1
-
Final slope
+
+
L) KR
C4 L KR
K3ABBK
values of KAB and KBB were chosen such that the generated curves in Fig. 2b would closely approximate the curves in Fig. 2a. For example, curve A in Fig. 2a has a Y-intercept at 10.4. Since the Y-intercept in the induced-fit model for the square geometry is given by KXKtK2AB (see Table 1), the value of KAB was determined to be 0.2269 for the first curve. From the fixed value of K1Kt the value of KBB was then varied to give the best fit of curve A in Fig. 2a, as judged by x2 analysis. Curves A, B, and C are qualitatively very similar to curves A through C in Fig. 2a. However, curves D, E, and F differ appreciably. Curves D and E from Fig. 2a and b are presented overlapping in Fig. 2c to highlight the differences. At very low values of Y the deviations appear as a result of the minimum on the two-state model. The deviations at high values of Y are quite significant in that experimental data obtained at high ligand concentrations (Y > 0.75) generally show less scatter than data obtained at low ligand concentration (Y < 0.25). Thus, for experimental data conforming to the shapes of the theoretical curves in curves D, E, and F, a distinction exists which may be large enough to be useful in distinguishing between the two-state and induced-fit models. For both geometries the values for the intercepts, initial slopes, and final slopes for the Scatchard plot are quantitatively given by the expressions in Table 1. Scatchard extrema The models presented above have been shown to generate different theoretical binding curves when curve-fitting procedures were used to produce the best fit of one theoretical data set to another. The primary differences appear to be in the presence of the minimum in the Scatchard plot and in the range of curvatures available for the two models. The conditions that produce extrema in the Scatchard plot for any model may be quantitatively determined, and correspond to zeros of Eq. 5: W=
[dF(x)/dxJ2 -F(x)d;F(x)/dxl.
3KBB(2/3 - S)KXKt
4KABKBB(3/4 - S)KxKt
1 + cL-(3 + 3- 8)
Initial slope
Square geometry
Tetrahedral geometry
Two-state model
K2ABBK
t
t
2a). If L < 7 + 4(3)1/2 = 13.93, the only alternative to this is the absence of any extrema. However, at this value of L a critical difference takes place, and for L > 13.93 it is possible for the Scatchard plot to exhibit both a maximum and a minimum (curves D-F). This takes place when (3)1/2 -1 (3L)1/2 -1 [7] (3L)'i + L
Biochemistry
Distinctions between the two-state and sequential models for cooperative ligand binding (acetylcholine/receptor)
RAYMOND E. GIBSON* t AND SIMON A. LEVINt * Section of Neurobiology and Behavior and t Section of Ecology and Systematics, Cornell University, Ithaca, New York 14853
Communicated by Roy Hertz, October 22, 1976
ABSTRACT The two-state and sequential models for positive cooperativity in ligand binding can produce significantly different theoretical binding curves when presented in a Scatchard plot. The conditions that produce the greatest differences have been examined. The theoretical differences have been used to select the two-state model as the best model for describing the binding of acetylcholine to acetylcholine receptors that have been solubilized by Triton X-100 and sodium cholate.
The cooperative binding of ligands to an oligomeric protein is usually described by a variant of one of two different models: the symmetry or two-state model of Monod et al. (1) and the sequential or induced-fit model of Koshland et al. (2). For data exhibiting positive cooperativity alone, the accuracy of most binding studies is such that it is very difficult to distinguish between the two models (3). The interchangeability of the two models has been suggested (4) since the three parameters in both models can be related to the parameters of the Adair model (5), an earlier model that gave way to the Pauling model (6) and subsequently to the induced-fit model. Although data that exhibit negative cooperativity or "half-of-sites" binding are generally described by the induced-fit model, such data may also be described by extensions of the two-state model involving nonidentical subunits with differing inherent affinities for the ligand (7, 8). However, even when subunit homogeneity is known, the induced-fit model has remained the model of choice. When the data exhibit only positive cooperativity, curvefitting procedures using either model will generally fit the data equally well (9, 10). However, in studies on the interaction of acetylcholine with the acetylcholine receptor from Torpedo californica electroplax, we noted and were able to capitalize on important theoretical differences between the two-state model and two geometries of the induced-fit model, thus enabling us to draw conclusions concerning preferred models. Basically, the concerted (two-state) model is more flexible, especially at low ligand concentrations; both the tetrahedral and square forms of the induced-fit model are controlled by a single "shape" parameter. In particular, when results exhibiting positive cooperativity are presented in a Scatchard plot (Y/x against Y, where x is ligand concentration and Y is amount bound), either model can provide for a maximum in the plot. However, when the two-state model is compared with either form of the induced-fit model, the two-state model is the only one that allows for the possibility of a minimum in the Scatchard plot; this occurs at a binding value Y less than that at which the maximum occurs. For one set of data we present, this flexibility is critical and provides the principal basis for our argument in favor of the two-state model. Furthermore, we establish that the induced-fit model has only two independent parameters, t Present address: Department of Radiology, The George Washington
University Medical Center, Washington, D.C. 20037. 139
one of which is a scaling constant; thus, when protein interaction constants (of which there are two in the induced-fit model) are not known, the ligand affinity constant cannot be determined.
MATERIALS AND METHODS [3H]Acetylcholine (170 mCi/mmol) was purchased from Amersham/Searle. Live Torpedo californica were purchased from Pacific Biomarine Supply Co. (Venice, Calif.). Preparations of acetylcholine receptor were made from the heavy membrane fraction of T. californica electroplax as described (8). The heavy membrane preparation was lyophilized and stored at -250 as long as nine months without deterioration of acetylcholine binding. The acetylcholine receptor was solubilized in either 1% sodium cholate or 1% Triton X-100 in Ringer's solution (115 mM NaCl, 4.6 mM KCl, 0.65 mM CaC12, 1.15 mM MgSO4, and 15.7 mM Na2HPO4 adjusted to pH 7.4 with HCI) as described (11). Acetylcholinesterase was inhibited by incubation with the anticholinesterase Tetram [oxalate salt of O,O-diethyl-S-(2-diethylaminoethyl)phosphorothiolate] at 10-4 M. Acetylcholine binding was determined by equilibrium dialysis assay; details of the assay have been described elsewhere (11).
The binding parameters for Fig. la and b were determined by an iterative procedure, using the equations for the models described below, which minimized the deviations in the observed B/x values (Fig. 1) and Y/x values (Fig. 2) in the Scatchard plot as judged by x2 analysis. Y is defined as the fractional saturation B/Bmax, when Bmax is the maximum binding. Minimizing the deviation with respect to Y values did not produce significantly different results. RESULTS The binding of acetylcholine to acetylcholine receptors that had been solubilized with cholate and Triton is presented in Fig. la and b. The data exhibit positive cooperativity, as indicated by the convexity in the Scatchard plots. The points indicated by open circles were considered to exceed normal scatter before curve-fitting procedures were undertaken. Subsequent analysis suggests that the deletions were valid, and those data points are not included in the final x2 analysis. Positive cooperativity in acetylcholine binding has been reported earlier (8), and the data were analyzed by the two-state model. Since the two-state and induced-fit models have been described as the two extremes of a more general model (12), we attempted curve-fitting procedures using both models. However, in the simplest forms of the models, we found that the two models produced very different theoretical binding curves. The differences appeared to result from the differences in the properties of the first and second derivatives of the saturation functions for each model. We therefore undertook a detailed analysis of the models.
140
1
Biochemistry: Gibson and Levin
Proc. Nati. Acad. Sci. USA 74 (1977)
0
0.2
0.4.
0.6 0.8
1.0
1. 2
1.4.
1.6 1.8
2.0
2.2 2.4 2.6
B B FIG. 1. Scatchard plot of [3H]acetylcholine binding to Torpedo californica receptor. B is in nmol/g of original tissue; values forB/x should be multiplied by 1000. (a) Solubilized in sodium cholate. Curve fit by two-state model (-): KR = 2 nM, c = 0.018, and L = 325,000; X2(B/x) = 0.0932 and x2(B) = 0.0081. Curve fit by induced-fit model (- -): Kx = 500 X 106 M, KAB = 0.1323, and KBB = 0.040; X2(B/x) = 1.719 and x2(B) = 0.0731. (b) Solubilized by Triton X-100. Curve fit by two-state model: KR = 2 nM, c = 0.033, and L = 325,000; x2(B/x) = 0.5034 and X2(B) = 0.0075. Curve fit by induced-fit model: Kx = 500 X 106 M, KAB = 0.1822, and KBB = 0,0470; X2(B/x) = 2.022 and X2(B) = 0.1104. Theoretical curves were minimized on unweighted (Bix) values.
The saturation function, Y, for all binding models can be described by the general equation: [i] Y - (x/n)IJn F(x)T where x is the ligand concentration. F(x) is a function describing the total conformational states available in the presence of ligand, which is normalized to the unoccupied state (Ro). For example, F(x) = 1 + x in Eq. 1 gives the Langmuir isotherm (prime denotes differentiation). Cooperativity (positive or negative) may be defined as a positive or negative change in effective (average) binding constant as the amount bound increases. This will appear as (respectively) negative or positive curvature (essentially change in slope) in either the Scatchard or double-reciprocal plots of the (x,Y) data; in the presence of no cooperativity, both would be straight lines. Analytically, cooperativity is defined by the sign of the quantity
a
H2, (G)1-2 ( 1Flt where G = (In F)' and H = (ln G)'. It is easily confirmed that a> 0 if and only if the slope in either the Scatchard or double-reciprocal plot is decreasing, and that this corresponds to an increase in average effective binding constant. Two-state model In the two-state model, the oligomer of n subunits is assumed to exist in two conformations, the T or constrained state and the R or relaxed state. These two conformations are in equilibrium in the absence of ligand, with the equilibrium constant defined as L = (T)/(R). The T and R states exhibit different affinities for ligand, the dissociation constants being defined as KT and KR, respectively (c = KR/KT). When L > 1 and c < 1, positive cooperativity may be observed. The fraction of oligomers binding the ligand is given by Eq. 1 when F(x) is given by Eq.
(In
2: Fx)
_
(1+
ar
+
ca)n
(1 +
[2]
where a = X/KR. The parameters that produced the curve fits in Fig. la and b are presented in the legend to the figure. A family of theoretical curves generated from the Y-function of the two-state model (n = 4, c = 0.042, and L from 102 to 105) is presented in Fig. 2a. For L values from 1 to 1515, the Scatchard plot has the maximum characteristic of positive cooperativity. When 1515 < L < 23,100, both a minimum and a maximum are evident. At L values greater than 23,100, there is neither a maximum nor a minimim. Values for the intercepts, initial and final slopes, and initial and final curvatures can be quantitatively determined from the expressions in Table 1.
Induced-fit model The assumptions in the induced-fit model (2) are that each subunit can exist in either of two conformational states, the A and B states, with the equilibrium constant Kt = (B)/(A) characterizing the relative stability of the conformations; the binding of ligand causes a conformational change from the A to the B state. The intrinsic association constant is defined as Kx = 1/KR = (Bx)/(B)(x), where x is the ligand concentration. Additionally, the ligand-induced conformational change in one subunit alters its interaction with neighboring subunits. The subunit-subunit interactions 'are characterized by two equilibrium constants: KAB reflects the strength of the A-B interactions and KBB reflects the strength of the B-B interactions. The saturation function, Y, for the induced-fit model is dependent upon the geometry of the subunits as it reflects the differing number of neighboring interactions. For example, with n = 4 in a tetrahedral arrangement there are three neighboring interactions for each subunit, while in the square geometry there are only two neighboring interactions. The saturation functions for these two geometries are given by Eqs. 3 and 1 for the tetrahedral geometry; F(x) -1 + 4K3AeX + 6K4ABKBBC? + 4K3,AK3a3 + K6uma4
[3a]
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Gibson and Levin
0.1 0.2 0.3 0.4 0 5 0.6 0.7 0.8 0.9 By
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0
141
1.0
14
Y/x
0.5 By
FIG. 2. Scatchard plot of theoretical binding curves. (a) Two-state model (Eqs. 1 and 2): n = 4, KR = 5 nM, c = 0.042, and L is (A) 100, (B) 300, (C) 1000, (D) 3000, (E) 10,000, (F) 28,060, and (G) 100,000. (b) Induced-fit model (Eqs. 1 and 3): n = 4 in the square geometry, K. = 200, and (A) KAB = 0.2269, KBB = 0.3374; (B) KAB = 0.2121, KBB = 0.2375; (C) KAB = 0.2074, KBB = 0.1650; (D) KAB = 0.2062, KBB = 0.1200; (E) KAB = 0.2049, KBB = 0.0900; (F) KAB = 0.2049, KBB = 0.0725; and (G) KAB = 0.2049, KBB = 0.0550. (c) Scatchard plot of curves D and F from Figs. la (-) and 2b (- -).
and by Eqs. 4 and 1 for the square geometry: F(x) -1 + 4K2ABO + (2K4AB.+ 4K2ABKBB)a2
+4K2ABK2Ba3 + K4BjCa4. [4a] In both models, a = KKtx.
These forms are misleading in their apparent range, but the limitations become apparent if the single shape parameter S = K2AB/KBB is introduced. In this case, [3a] and [4a] may be rewritten, respectively, as [3b] F(X) 1+4S3/2y+6S2V+4S3/23 - 4
in the tetrahedral (with x = KXKtK3/2BBX), and [4b] F(x) = 1 + 4Si + 2(S2 + 2S)T2 + 4SX3 + X-4 in the square (with x = KxKtKBpx). Except for the scale factor XITBB, there is only one free parameter S. The parameters that produced the curve fits of Fig. la and b were obtained for the square geometry and the values for the parameters are presented in the legend to Fig. 1. The theoretical curves in Fig. 2b were generated from Eqs. 1 and 4 for the square geometry. The value of KxKt was assigned as 200 X 106 M to compare with the value of KR = 5 nM in Fig. 2a, and
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Proc. Natl. Acad. Sci. USA 74 (1977)
Table 1. Equations relating the initial intercept, initial slope, and final slope of the Scatchard plot of the two-state and induced-fit models Induced-fit model
1 I 1 + +LL KR
Initial intercept
L 1 V +LK ABKxKtt ABxxttKBK 1 +L TK3BK
(1 + cL)(1
-
Final slope
+
+
L) KR
C4 L KR
K3ABBK
values of KAB and KBB were chosen such that the generated curves in Fig. 2b would closely approximate the curves in Fig. 2a. For example, curve A in Fig. 2a has a Y-intercept at 10.4. Since the Y-intercept in the induced-fit model for the square geometry is given by KXKtK2AB (see Table 1), the value of KAB was determined to be 0.2269 for the first curve. From the fixed value of K1Kt the value of KBB was then varied to give the best fit of curve A in Fig. 2a, as judged by x2 analysis. Curves A, B, and C are qualitatively very similar to curves A through C in Fig. 2a. However, curves D, E, and F differ appreciably. Curves D and E from Fig. 2a and b are presented overlapping in Fig. 2c to highlight the differences. At very low values of Y the deviations appear as a result of the minimum on the two-state model. The deviations at high values of Y are quite significant in that experimental data obtained at high ligand concentrations (Y > 0.75) generally show less scatter than data obtained at low ligand concentration (Y < 0.25). Thus, for experimental data conforming to the shapes of the theoretical curves in curves D, E, and F, a distinction exists which may be large enough to be useful in distinguishing between the two-state and induced-fit models. For both geometries the values for the intercepts, initial slopes, and final slopes for the Scatchard plot are quantitatively given by the expressions in Table 1. Scatchard extrema The models presented above have been shown to generate different theoretical binding curves when curve-fitting procedures were used to produce the best fit of one theoretical data set to another. The primary differences appear to be in the presence of the minimum in the Scatchard plot and in the range of curvatures available for the two models. The conditions that produce extrema in the Scatchard plot for any model may be quantitatively determined, and correspond to zeros of Eq. 5: W=
[dF(x)/dxJ2 -F(x)d;F(x)/dxl.
3KBB(2/3 - S)KXKt
4KABKBB(3/4 - S)KxKt
1 + cL-(3 + 3- 8)
Initial slope
Square geometry
Tetrahedral geometry
Two-state model
K2ABBK
t
t
2a). If L < 7 + 4(3)1/2 = 13.93, the only alternative to this is the absence of any extrema. However, at this value of L a critical difference takes place, and for L > 13.93 it is possible for the Scatchard plot to exhibit both a maximum and a minimum (curves D-F). This takes place when (3)1/2 -1 (3L)1/2 -1 [7] (3L)'i + L
