- Email: [email protected]
Linearization of two ligand-one binding site Scatchard plot and the “IC50” competitive inhibition plot: Application to the simplified graphical determination of equilibrium constants
Life Sciences, Vol. 42, pp. 231-237 Printed in the U.S.A.
Pergamon Journals
LINEARIZATION OF TWO LIGAND-ONE BINDING SITE SCATCHARD PLOT AND THE "1C50" COMPETITIVE INHIBITION PLOT: APPLICATION TO THE SIMPLIFIED GRAPHICAL DETERMINATION OF EQUILIBRIUM CONSTANTS Harry E. Gray and William G. Luttge
Depadmem of Neuroscience University of Flodda College of Medicine Gainesville, Florida 32610, USA (Received in final form November 13, 1987)
Summary A simple procedure for linearizing the curved, two ligand-one binding site Scatchard plot resulting from the presence of a constant concentration of competitive inhibitor is proposed; the same procedure may also be applied to the analysis of data derived from the "1C50"competitive inhibition experimental design. Furthermore, a useful generalization of the Cheng-Prusoff correction is presented. The affinity of an unlabeled ligand for a particular class of binding sites can be determined by measuring the bound fraction when increasing concentrations of a labeled ligand are incubated with a fixed total concentration of the binding sites, first in the absence and then in the presence of a fixed total concentration of the unlabeled competitive inhibitor (see 1 for review). It must be noted, however, that the concentration of the free (Le., unbound) form of the unlabeled competitive inhibitor must undergo a decrease as binding occurs, thus producing a curved Scatchard plot (2). In this communication we propose a simple procedure for linearizing the Scatchard plot resulting from this "Edsall-Wyman" experimental design (3). The very popular competitive inhibition experimental design (known as the "1C50"method) also generates data that are somewhat difficult to analyze in the laboratory without the aid of computerized nonlinear regression techniques. This design presents two major problems: The curvature of the inhibition plot makes the precise determination of IC50 difficult, and the IC50 itself is often quite different from, and difficult to relate to, the actual equilibrium dissociation constant of the competing ligand. In many situations the curvature of the competitive inhibition plot cannot be eliminated simply by performing a "logit-log" transformation (4) of the data. Furthermore, the use of approximations (e.g., the Cheng-Prusoff formula, 5) to relate the estimated IC50 to the equilibrium dissociation constant of the competitor is often inappropriate and can lead to substantial additional error. We suggest that if the initial receptor occupancy is not too high this experimental design can also be analyzed by a simple procedure (to be described) for linearization of the curved, two ligard-one binding site Scatchard plot obtained in the presence of a fixed concentration of competitive inhibitor. Furthermore, a more nearly exact approximation is suggested as an alternative to the Cheng-Prusoff formula. Th(~ory and ADplication The nomenclature for the two ligand-one binding site problem is as follows: BL and BC are, respectively, the concentrations of specifically bound labeled ligand L and competitive inhibitor C. The total concentration of binding sites is BMAX. The equilibrium dissociation constants for the binding of the labeled ligand and competitor are, respectively, KdL and KdC; and SL and SC are the total concentrations of these ligands. The free (unbound) concentrations of the ligands are FL and FC. In order to discuss the "1C5o" experimental design we let (BL)o and (FL)O represent the equilibrium values of BE and FL when Sc = O. 0024-3205/88 $3.00 + .00 Copyright (c) 1988 Pergamon Journals Ltd.
232
Linearization of Two Ligand Binding Data
Vol. 42, No. 3, 1988
IC50 is the value of SC when BL has been reduced by half (Le., when BL = (BL)O/2), and we let FC50 represent the corresponding value of FC (i.e., when Sc = IC50 and FC = FCso). As an illustrative example we have used the measured equilibrium dissociation constants that describe the binding of estradiol (E2) and estriol (E3) to the nonactivated calf uterine cytosol estrogen receptor (6) in order to generate the purely hypothetical Scatchard plots (Fig. 1) of theoretical data that would be observed for the binding of labeled E2, first in the absence and then in the presence of S¢ = 1 nM "cold" E3. The concentration of binding sites BMAX= 2.3 nM is also taken from the reference (6); the equilibrium constants are KdL = 1.7 x 1010 M (for E2) and KdC = 2.6 x 1010 M (for E3). The linear Scatchard plot (for the case Sc = 0) has slope (-1/KdL) and intercept BMAX on the x-axis (Fig. 1). The lower, curvilinear Scatchard plot (Fig. 1) for the purely competitive two ligand situation is a hyperbola whose geometric properties have been described (2) and whose equation is BL/FL = KdC[-1.(BL +SC.BMAX)/KdC+[(1.[BL +SC.BMAX]/KdC)2+4Sc/KdC]I/2]/2KdL .
[1]
The actual points on the curve are completely hypothetical and have been calculated and placed on the plot at equally spaced intervals of BE (0.275 nM) in order to indicate how specific binding values map into the modified coordinate systems to be discussed. 15-
,o
I0,
~,
2'5
FL
o
d.~
,;o BL
,i~
£o
-
215
(nM)
FIG. 1 Theoretical Scatchard plots lor a single ligand and for two ligands in competition for one class of noncooperative binding sites. Inset: linearization of the curved Scatchard plot by plotting the total concentration of occupied binding sites (BL+Bc) on the x-axis. The parameters used to generate the plots describe the binding of estradiol (E2) and estriol (Es) to the nonactivated calf uterine estrogen receptor and were taken from (6). Upper straight line: E2 alone, equilibrium dissociation constant KdL = 1.7 x 10-10M; lower curved plot E2 in the presence of 1 nM estriol (E3, equilibrium constant KdC = 2.6 x 10-tOM). The concentration of binding sites (BMAX) is 2.3 nM. The EdsalI-Wyman equation (3,7) can be rearranged to the convenient "Scatchard" form given by BL/FL =- (BL-BMAX)/KdL(I+Fc/KdC),
[2]
which predicts simply that the slope of the Scatchard plot will be reduced by a factor of 1/(I+Fc/KdC) in the presence of a constant concentration of free inhibitor FC. The error in the derived value of KdC resulting from the assumption that the concentration of free inhibitor FC is constant in this experiment is independent of KdL and vanishes under the ideal condition approached when KdC >> BMAX. This error, which may
Vol. 42, No. 3, 1988
Linearlzatlon of Two Binding Data
233
be large, can be avoided by analyzing the data from the same experimental design in a slightly different manner. At equilibrium the simple mass action equations for the binding of the two ligands may be written
[3]
(BMAX-BL-BC)FL = KdLBL and (BMAX-BL-Bc)Fc = KdCBC.
Thus, if BL/FL or Bc/Fc on the ordinate are plotted against (BL + BC) on the abscissa, then a linear "Scatchard-like" plot with slope (-l/KalE) or (-I/Kdc) and x-intercept BMAX results (insets to Figs. 1 and 2). Furthermore, KdC may be measured simply by plotting [BMAX-BL-BC, (free binding sites)] on the abscissa against Bc/Fc on the ordinate, as suggested by Equation 3. The resulting "free receptor" plot (Fig. 2) should pass near the origin and possess slope (1/KdC). This is the anlaysis we suggest for the determination of Kd¢ by this experimental method. 5.0
4.(~
__%_
3.0.
F¢ 6
2.0. FC
4
t.O
0.0
6.s ,:o ["MAX" o,..- ~:] (..)
,;s
FIG. 2 Linearization of the curved Scatchard plot shown in Rg. 1. The same hypothetical data points presented in Fig. 1 are shown plotted in the coordinate system recommended for the determination of KdC. The y-axis (Bc/Fc) refers explicitly to the concentrations of bound and free competitor (E3), whereas the x-axis (BMAX-BL-BC) refers to the concentration of free binding sites. The plot passes through the origin, and the slope (1/KdC) is the reciprocal of the dissociation constant of the competitor (E3). The inset is analogous to the inset of Fig. 1: the points are plotted with the total concentration of bound sites (BL+BC) on the x-axis. In this "Scatchard format" the slope is (-1/KdC) and the x-intercept is again BMAX.
The implementation of this analysis is very simple. We note that BMAX and KdL have been determined already from an initial Scatchard plot. Equation 3 is now rearranged to give BMAX-BL-BC= KdLBL/FL.
[4]
Thus, the abscissa of the recommended plot is determined directly from Equation 4, as is the value of Bc = [BMAX-BL-(KdLBL/FL)]. Since SC is known, FC = Sc-Bc. Note that this approximation ignores the nonspecific binding of the competitor. If this assumption is unwarranted, the nonspecific binding can easily be determined empirically with a radiolabeled form of the competitor. Given these values, the best-fitting straight line is then fit to the points as shown (Fig. 2), and the reciprocal of its slope determined. Although this line should pass through the origin, we do not recommend forcing this relationship when working with real data since it reduces one of the potential variables by inserting an additional, and not experimentally derived, obligatory value to the data set. The user can thus either force the graph to go through zero or extrapolate it to go as near to zero as a simple 2-parameter least squares linear regression permits. Since the plot of Bc/Fc vs. [free binding sites] is used to estimate only one binding parameter (Kdc), data sets
234
Linearizatlon of Two Ligand Binding Data
Vol. 42, No. 3, 1988
arising from experimental preparations differing in total binding site concentration may be merged prior to the final regression. We now turn to an analysis of the "1C50" experimental design. Figure 3 depicts theoretical competitive inhibition curves generated by the same binding parameters (taken from 5) that were employed in the construction of Figs. 1 and 2 and used in the above analysis of the design in which the inhibitor is present at a single concentration (the "Edsall-Wyman" design). Furthermore, it is presumed that KalEand BMAXhave previously been measured in the absence of competitor by constructing the one-ligand isotherm. If the intial receptor occupancy [(BL)O/BMAX] is not too high, it is obvious that linearization of the competition curves may be achieved by plotting the data in the same BC/Fc vs. [BMAX-BE -Bc, free binding site~] coordinate system discussed above in connection with the Edsall-Wyman experimental design, using the same data manipulations to determine the abscissa and ordinate of the plot (shown as inset to Fig. 3). If the initial receptor occupancy is too high, then the range of the plot will be compressed severely and this method will not be useful. 8-
-I.0
'-
"°"
-0.6
1@1
StF--"L"
"
•
El) ~t.
4-
[~.)o -0.4
-O.Z
;5
9:o
~ -UOG
8:o
~5
Sc
FIG, 3 Theoretical curves for the competitive Inhibition of ligand L by ligand C [rom a single class of noninteracting binding sites. Upper curve: fraction BL/(BL)o of initial bound ligand L remaining bound at equilibrium in the presence of different concentrations (Sc) of competitive inhibitor C;:lower curve: the bound/free ratio for ligand L at various concentrations of competing ligand C. Inset: linearization of the competition curves by plotting the same,data in the BC/Fc vs. [BMAX-BE -Bc, free binding sites] coordinate system. Again, the slope is (1/Kclc). Thebinding parameters used to generate the plots are listed in the legend,to Rg. 1 and are the same in all figures. The maximal (Sc = 0) receptor occupancy depicted in the figure is 50%, and the concentrations of competitor C are spaced equally on a logarithmic scale.
Other methods must be used to determine KdC from the competition plot when the intitial receptor occupancy is too high. The~dJrectestimation of an IC50 value from the curved displacement plot (Fig. 3) is statistically naive, but a "l~udo-Hill" or Iogit-log transformation of the data (Fig. 4) may be used to approximate a straight line for the estimation of IC50 by simple linear regression. The subsequent calculation of KdC from the IC50 estimate presents further difficulties. We shall consider sequentially the problems of estimating IC50 and then using thisrestimate to calculate KdC. The "logistic" equations (See 4 for review) used to transform the competition displacement data of Fig. 3 into the approximately linear 10git-log plots shown in Fig. 4 are the following:
Vol. 42, No. 3, 1988
Linearization of Two Ligand Binding Data
235
BL = (BL)O/(I+Sc/IC50) 1"0 and
[5]
BL = (BL)O/(I+Fc/FC50) 1'°.
[6]
ZO-
I.OLOG
BL
[~_~-~_] 05O0 ............................
-0.5-
tO.O
9.5 9.0 8.5 -LOG Fc(e) o~d-LOG 5c (IB)
80
7.5
FIG. 4 Logit.log transfomlations of the competitive inhibition data shown in Fig. 3. Both curves share the same y-axis (Iogit [BL/(BL)O]), but the x-axes refer respectively to the concentration of free competitor C (Fc, lower curve) and to the total concentraton of ligand C (Sc, upper curve). The intersections with the dotted line (x-intercepts) define the IC50 and FC50 for the given conditions, but neither is a good estimate of KdC. The slopes of the IC50 and FC50 plots (pseudo-Hill coefficients) determined by simple linear regression are, respectively, -1.22 and -0.91. The nonlinearity of the IC50 Iogit-log plot is quite apparent. The use of estimates of IC50 and FC50 to determine KdC is discussed in the text.
These logistic equations (appropriate for non-interacting binding sites) may be transformed immediately into the "pseudo-Hill" or Iogit-log expressions Iogit [BL/(BL)o] - log (BL/(BL)O-BL]) = log ICso- log Sc and
L7]
Iogit [BL/(BL)O] - log (BL/(BL)o-BL]) = log FC50- log FC,
[8]
which may be fit (approximately) to the binding data by simple linear regression. If Equations 5 and 7 were exact, then the IC50 plot (upper curve, Fig. 4) would be linear; if Equations 6 and 8 were exact, then the FC50 plot (lower curve, Fig. 4) would be linear. Equations 6 and 8 are always better approximations than Equations 5 and 7. In the example under consideration Equations 6 and 8 provide an excellent near-linear transformation of the binding data, as one can see upon examination of Fig. 4; the nonlinearity of the IC50 plot, however, is quite apparent in this example. Although the approximation that leads to linearization of the Iogit-log FC plot (Le., FL -= (FL)O) is derived from the initial (i.e., maximal) occupancy condition (BE)O<< SL, the approximate finearity of the plot is fairly robust over a broad spectrum of experimental conditions and depends only on the initial conditions relating to the labeled ligand L. Specifically, the approximate linearity of the Iogit-log FC plot does not depend on the relative affinity of the two ligands, (KdC/KdL). The Iogit-log plot containing log SC as abscissa (the "1C50"Iogit-log SC plot), however, departs significantly from linearity because the approximation FC = SC is a poor one at low values of Sc. If KdC >> KdL then the large values of SC required to achieve ligand displacement will also make this formula approximately valid and thus lead to linearization of the simpler IC50 plot. The calculation of FC for the construction of the Iogit-log FC plot from the measured data has been described above (Equation 4 combined with the relation FC = S¢ - BC), and the initial (Le., maximal) binding (BL)O may either be measured directly or calcula-
236
Linearization of Two Ligand Binding Data
Vol. 42, No. 3, 1988
ted from the values of KdL and BMAX(in combination with the known SL) measured previously. In the specific example under consideration simple linear regressions of the theoretical Iogit-log data of Fig. 4 yield the following: [IC50 (lin. regress.)/"exact" IC50] = 1.12 (13% error), and [FC50 (lin. regress.)/"exact" FC50] = 0.999 (0.1% error). The "exact" values of IC50 and FC50are, respectively, 4.80 and 3.20 nM.) Further analysis is, of course, required to calculate KdC from the estimates of either IC50 or FC50 obtained from the Iogit-log plots discussed above. The Cheng-Prusoff correction (5) is given by
[9]
KdC -= IC50/(I+SL/KdL).
This formula may also be derived from Equations 1 and 2 above by using the definition of IC50 (Le., that (BL)O = 2BE when SC = IC50) and applying the drastic approximation that both FC = SC and FL = SL. As the illustrative example will demonstrate, this does not provide a good estimate of KdC when the affinity of the competing ligand is too high. This Cheng-Prusoff correction can be improved substantially by including in the calculation of the value of (BE)O,which easily can be measured experimentally or calculated from the measured values of KdL and BMAX. Combining the above Equations 2 and 6 and obvious initial conditions yields, upon elimination of BMAX,an expression which, when evaluated at the 50% displacement point, eventually becomes [10]
KdC --- FC50x 2KdL[SL-(BL)O]/[(BL)O2 + 2SL2 + 2SLKdL-3SL(BL)O].
This is a much better approximation than Equation 9 (Le., the Cheng-Prusoff correction), to which it reduces when (BL)O is neglected and FC50is replaced by ICso. Equation 10 remains approximately valid when IC50 is substituted for FC50 [11]
KdC = IC50 x 2KdL[SL-(BL)O]/[(BL)O2 + 2SL2 + 2SLKdL -3SL(BL)O].
Equation 10, will, however, always be superior to Equation 11; similarly, the Cheng-Prusoff expression itself will always be more nearly exact if a good estimate of FC50 is substituted for the estimate of iC50. Table I lists, for the example that we have been considering, the KdC estimates derived from the two different approximations; each method has been used in combination with both the estimated and the exact values of FCsoand IC50 listed above. It will be seen that the retention of (BL)O in the approximation is required for the accurate derivationof KdC from the competitive inhibition data under consideration here. The ease of implementation of Equation 10 suggests that it (or at least Equation 11) should be used instead of the Cheng-Prusoff approximation. TABLE I Comparison of estimatesof KdCderived from the Cheng-Prusoffcorrection (Equation 9) and fromthe approximationsdescribedby Equations10 and 11. Input IC50
Input FC50
Eauation
Exact
Estimated
Exact
Estimated
9
5.462 x 10-10M
6.107 x 1010 M
3.648 x 10-1° M
3.644 x 10-1° M
10and11
3.941 xl0lOM
4.406x10-10M
2.632x10-10M
2.629x10-10M
The assumed exact value for KdC is 2.632 x 10-10 M. The inputs to the equations include the exact values of FC50 (3.20 nM) and IC50 (4.80 nM) for the example discussed in the text as well as the estimated values for FC50 (3.20 nM) and IC50 (5.38 nM) derived as described from the Iogit-log plots of Fig. 4.
Vol. 42, No. 3, 1988
Linearization of Two Ligand Binding Data
237
Acknowledoement~ Supported in part by N.I.N.C.D.S. Grant NS 24404 awarded to Dr. W.G. Luttge. The expert secretarial assistance provided by Mrs. Teresa L. McCord is also gratefully acknowledged.
References . .
3. 4.
5. 6. 7.
D. RODBARD, in Receotors for Re0roductive Hormones (Eds. B.W. O'Malley and A.R. Means), p. 289-326. Plenum Press, New York (1973). H.A. FELDMAN, Anal. Biochem. 48,317-338 (1972). J.T. EDSALL and J. WYMAN, Biophysical Chemistry. Vol. 1, p. 651, Academic Press, New York (1958). A. DELEAN, P. MUNSON, and D. RODBARD, Amer. J. Physiol., 235, E97-E102 (1978). Y. CH ENG and W.H. PRUSOFF, Biochem. Pharmac. 22, 3099-3108 (1973). B.M. WEICHMAN and A.C. NOTIDES, Endocrinology 106, 434-439 (1980). C.R. CANTOR and P.R. SCHIMMEL, Bi0Dhvsical Chemistry. Part IIh The Behavior of Biolooical Macromolecules. p. 943, Freeman and Col, San Francisco ('i980).
Pergamon Journals
LINEARIZATION OF TWO LIGAND-ONE BINDING SITE SCATCHARD PLOT AND THE "1C50" COMPETITIVE INHIBITION PLOT: APPLICATION TO THE SIMPLIFIED GRAPHICAL DETERMINATION OF EQUILIBRIUM CONSTANTS Harry E. Gray and William G. Luttge
Depadmem of Neuroscience University of Flodda College of Medicine Gainesville, Florida 32610, USA (Received in final form November 13, 1987)
Summary A simple procedure for linearizing the curved, two ligand-one binding site Scatchard plot resulting from the presence of a constant concentration of competitive inhibitor is proposed; the same procedure may also be applied to the analysis of data derived from the "1C50"competitive inhibition experimental design. Furthermore, a useful generalization of the Cheng-Prusoff correction is presented. The affinity of an unlabeled ligand for a particular class of binding sites can be determined by measuring the bound fraction when increasing concentrations of a labeled ligand are incubated with a fixed total concentration of the binding sites, first in the absence and then in the presence of a fixed total concentration of the unlabeled competitive inhibitor (see 1 for review). It must be noted, however, that the concentration of the free (Le., unbound) form of the unlabeled competitive inhibitor must undergo a decrease as binding occurs, thus producing a curved Scatchard plot (2). In this communication we propose a simple procedure for linearizing the Scatchard plot resulting from this "Edsall-Wyman" experimental design (3). The very popular competitive inhibition experimental design (known as the "1C50"method) also generates data that are somewhat difficult to analyze in the laboratory without the aid of computerized nonlinear regression techniques. This design presents two major problems: The curvature of the inhibition plot makes the precise determination of IC50 difficult, and the IC50 itself is often quite different from, and difficult to relate to, the actual equilibrium dissociation constant of the competing ligand. In many situations the curvature of the competitive inhibition plot cannot be eliminated simply by performing a "logit-log" transformation (4) of the data. Furthermore, the use of approximations (e.g., the Cheng-Prusoff formula, 5) to relate the estimated IC50 to the equilibrium dissociation constant of the competitor is often inappropriate and can lead to substantial additional error. We suggest that if the initial receptor occupancy is not too high this experimental design can also be analyzed by a simple procedure (to be described) for linearization of the curved, two ligard-one binding site Scatchard plot obtained in the presence of a fixed concentration of competitive inhibitor. Furthermore, a more nearly exact approximation is suggested as an alternative to the Cheng-Prusoff formula. Th(~ory and ADplication The nomenclature for the two ligand-one binding site problem is as follows: BL and BC are, respectively, the concentrations of specifically bound labeled ligand L and competitive inhibitor C. The total concentration of binding sites is BMAX. The equilibrium dissociation constants for the binding of the labeled ligand and competitor are, respectively, KdL and KdC; and SL and SC are the total concentrations of these ligands. The free (unbound) concentrations of the ligands are FL and FC. In order to discuss the "1C5o" experimental design we let (BL)o and (FL)O represent the equilibrium values of BE and FL when Sc = O. 0024-3205/88 $3.00 + .00 Copyright (c) 1988 Pergamon Journals Ltd.
232
Linearization of Two Ligand Binding Data
Vol. 42, No. 3, 1988
IC50 is the value of SC when BL has been reduced by half (Le., when BL = (BL)O/2), and we let FC50 represent the corresponding value of FC (i.e., when Sc = IC50 and FC = FCso). As an illustrative example we have used the measured equilibrium dissociation constants that describe the binding of estradiol (E2) and estriol (E3) to the nonactivated calf uterine cytosol estrogen receptor (6) in order to generate the purely hypothetical Scatchard plots (Fig. 1) of theoretical data that would be observed for the binding of labeled E2, first in the absence and then in the presence of S¢ = 1 nM "cold" E3. The concentration of binding sites BMAX= 2.3 nM is also taken from the reference (6); the equilibrium constants are KdL = 1.7 x 1010 M (for E2) and KdC = 2.6 x 1010 M (for E3). The linear Scatchard plot (for the case Sc = 0) has slope (-1/KdL) and intercept BMAX on the x-axis (Fig. 1). The lower, curvilinear Scatchard plot (Fig. 1) for the purely competitive two ligand situation is a hyperbola whose geometric properties have been described (2) and whose equation is BL/FL = KdC[-1.(BL +SC.BMAX)/KdC+[(1.[BL +SC.BMAX]/KdC)2+4Sc/KdC]I/2]/2KdL .
[1]
The actual points on the curve are completely hypothetical and have been calculated and placed on the plot at equally spaced intervals of BE (0.275 nM) in order to indicate how specific binding values map into the modified coordinate systems to be discussed. 15-
,o
I0,
~,
2'5
FL
o
d.~
,;o BL
,i~
£o
-
215
(nM)
FIG. 1 Theoretical Scatchard plots lor a single ligand and for two ligands in competition for one class of noncooperative binding sites. Inset: linearization of the curved Scatchard plot by plotting the total concentration of occupied binding sites (BL+Bc) on the x-axis. The parameters used to generate the plots describe the binding of estradiol (E2) and estriol (Es) to the nonactivated calf uterine estrogen receptor and were taken from (6). Upper straight line: E2 alone, equilibrium dissociation constant KdL = 1.7 x 10-10M; lower curved plot E2 in the presence of 1 nM estriol (E3, equilibrium constant KdC = 2.6 x 10-tOM). The concentration of binding sites (BMAX) is 2.3 nM. The EdsalI-Wyman equation (3,7) can be rearranged to the convenient "Scatchard" form given by BL/FL =- (BL-BMAX)/KdL(I+Fc/KdC),
[2]
which predicts simply that the slope of the Scatchard plot will be reduced by a factor of 1/(I+Fc/KdC) in the presence of a constant concentration of free inhibitor FC. The error in the derived value of KdC resulting from the assumption that the concentration of free inhibitor FC is constant in this experiment is independent of KdL and vanishes under the ideal condition approached when KdC >> BMAX. This error, which may
Vol. 42, No. 3, 1988
Linearlzatlon of Two Binding Data
233
be large, can be avoided by analyzing the data from the same experimental design in a slightly different manner. At equilibrium the simple mass action equations for the binding of the two ligands may be written
[3]
(BMAX-BL-BC)FL = KdLBL and (BMAX-BL-Bc)Fc = KdCBC.
Thus, if BL/FL or Bc/Fc on the ordinate are plotted against (BL + BC) on the abscissa, then a linear "Scatchard-like" plot with slope (-l/KalE) or (-I/Kdc) and x-intercept BMAX results (insets to Figs. 1 and 2). Furthermore, KdC may be measured simply by plotting [BMAX-BL-BC, (free binding sites)] on the abscissa against Bc/Fc on the ordinate, as suggested by Equation 3. The resulting "free receptor" plot (Fig. 2) should pass near the origin and possess slope (1/KdC). This is the anlaysis we suggest for the determination of Kd¢ by this experimental method. 5.0
4.(~
__%_
3.0.
F¢ 6
2.0. FC
4
t.O
0.0
6.s ,:o ["MAX" o,..- ~:] (..)
,;s
FIG. 2 Linearization of the curved Scatchard plot shown in Rg. 1. The same hypothetical data points presented in Fig. 1 are shown plotted in the coordinate system recommended for the determination of KdC. The y-axis (Bc/Fc) refers explicitly to the concentrations of bound and free competitor (E3), whereas the x-axis (BMAX-BL-BC) refers to the concentration of free binding sites. The plot passes through the origin, and the slope (1/KdC) is the reciprocal of the dissociation constant of the competitor (E3). The inset is analogous to the inset of Fig. 1: the points are plotted with the total concentration of bound sites (BL+BC) on the x-axis. In this "Scatchard format" the slope is (-1/KdC) and the x-intercept is again BMAX.
The implementation of this analysis is very simple. We note that BMAX and KdL have been determined already from an initial Scatchard plot. Equation 3 is now rearranged to give BMAX-BL-BC= KdLBL/FL.
[4]
Thus, the abscissa of the recommended plot is determined directly from Equation 4, as is the value of Bc = [BMAX-BL-(KdLBL/FL)]. Since SC is known, FC = Sc-Bc. Note that this approximation ignores the nonspecific binding of the competitor. If this assumption is unwarranted, the nonspecific binding can easily be determined empirically with a radiolabeled form of the competitor. Given these values, the best-fitting straight line is then fit to the points as shown (Fig. 2), and the reciprocal of its slope determined. Although this line should pass through the origin, we do not recommend forcing this relationship when working with real data since it reduces one of the potential variables by inserting an additional, and not experimentally derived, obligatory value to the data set. The user can thus either force the graph to go through zero or extrapolate it to go as near to zero as a simple 2-parameter least squares linear regression permits. Since the plot of Bc/Fc vs. [free binding sites] is used to estimate only one binding parameter (Kdc), data sets
234
Linearizatlon of Two Ligand Binding Data
Vol. 42, No. 3, 1988
arising from experimental preparations differing in total binding site concentration may be merged prior to the final regression. We now turn to an analysis of the "1C50" experimental design. Figure 3 depicts theoretical competitive inhibition curves generated by the same binding parameters (taken from 5) that were employed in the construction of Figs. 1 and 2 and used in the above analysis of the design in which the inhibitor is present at a single concentration (the "Edsall-Wyman" design). Furthermore, it is presumed that KalEand BMAXhave previously been measured in the absence of competitor by constructing the one-ligand isotherm. If the intial receptor occupancy [(BL)O/BMAX] is not too high, it is obvious that linearization of the competition curves may be achieved by plotting the data in the same BC/Fc vs. [BMAX-BE -Bc, free binding site~] coordinate system discussed above in connection with the Edsall-Wyman experimental design, using the same data manipulations to determine the abscissa and ordinate of the plot (shown as inset to Fig. 3). If the initial receptor occupancy is too high, then the range of the plot will be compressed severely and this method will not be useful. 8-
-I.0
'-
"°"
-0.6
1@1
StF--"L"
"
•
El) ~t.
4-
[~.)o -0.4
-O.Z
;5
9:o
~ -UOG
8:o
~5
Sc
FIG, 3 Theoretical curves for the competitive Inhibition of ligand L by ligand C [rom a single class of noninteracting binding sites. Upper curve: fraction BL/(BL)o of initial bound ligand L remaining bound at equilibrium in the presence of different concentrations (Sc) of competitive inhibitor C;:lower curve: the bound/free ratio for ligand L at various concentrations of competing ligand C. Inset: linearization of the competition curves by plotting the same,data in the BC/Fc vs. [BMAX-BE -Bc, free binding sites] coordinate system. Again, the slope is (1/Kclc). Thebinding parameters used to generate the plots are listed in the legend,to Rg. 1 and are the same in all figures. The maximal (Sc = 0) receptor occupancy depicted in the figure is 50%, and the concentrations of competitor C are spaced equally on a logarithmic scale.
Other methods must be used to determine KdC from the competition plot when the intitial receptor occupancy is too high. The~dJrectestimation of an IC50 value from the curved displacement plot (Fig. 3) is statistically naive, but a "l~udo-Hill" or Iogit-log transformation of the data (Fig. 4) may be used to approximate a straight line for the estimation of IC50 by simple linear regression. The subsequent calculation of KdC from the IC50 estimate presents further difficulties. We shall consider sequentially the problems of estimating IC50 and then using thisrestimate to calculate KdC. The "logistic" equations (See 4 for review) used to transform the competition displacement data of Fig. 3 into the approximately linear 10git-log plots shown in Fig. 4 are the following:
Vol. 42, No. 3, 1988
Linearization of Two Ligand Binding Data
235
BL = (BL)O/(I+Sc/IC50) 1"0 and
[5]
BL = (BL)O/(I+Fc/FC50) 1'°.
[6]
ZO-
I.OLOG
BL
[~_~-~_] 05O0 ............................
-0.5-
tO.O
9.5 9.0 8.5 -LOG Fc(e) o~d-LOG 5c (IB)
80
7.5
FIG. 4 Logit.log transfomlations of the competitive inhibition data shown in Fig. 3. Both curves share the same y-axis (Iogit [BL/(BL)O]), but the x-axes refer respectively to the concentration of free competitor C (Fc, lower curve) and to the total concentraton of ligand C (Sc, upper curve). The intersections with the dotted line (x-intercepts) define the IC50 and FC50 for the given conditions, but neither is a good estimate of KdC. The slopes of the IC50 and FC50 plots (pseudo-Hill coefficients) determined by simple linear regression are, respectively, -1.22 and -0.91. The nonlinearity of the IC50 Iogit-log plot is quite apparent. The use of estimates of IC50 and FC50 to determine KdC is discussed in the text.
These logistic equations (appropriate for non-interacting binding sites) may be transformed immediately into the "pseudo-Hill" or Iogit-log expressions Iogit [BL/(BL)o] - log (BL/(BL)O-BL]) = log ICso- log Sc and
L7]
Iogit [BL/(BL)O] - log (BL/(BL)o-BL]) = log FC50- log FC,
[8]
which may be fit (approximately) to the binding data by simple linear regression. If Equations 5 and 7 were exact, then the IC50 plot (upper curve, Fig. 4) would be linear; if Equations 6 and 8 were exact, then the FC50 plot (lower curve, Fig. 4) would be linear. Equations 6 and 8 are always better approximations than Equations 5 and 7. In the example under consideration Equations 6 and 8 provide an excellent near-linear transformation of the binding data, as one can see upon examination of Fig. 4; the nonlinearity of the IC50 plot, however, is quite apparent in this example. Although the approximation that leads to linearization of the Iogit-log FC plot (Le., FL -= (FL)O) is derived from the initial (i.e., maximal) occupancy condition (BE)O<< SL, the approximate finearity of the plot is fairly robust over a broad spectrum of experimental conditions and depends only on the initial conditions relating to the labeled ligand L. Specifically, the approximate linearity of the Iogit-log FC plot does not depend on the relative affinity of the two ligands, (KdC/KdL). The Iogit-log plot containing log SC as abscissa (the "1C50"Iogit-log SC plot), however, departs significantly from linearity because the approximation FC = SC is a poor one at low values of Sc. If KdC >> KdL then the large values of SC required to achieve ligand displacement will also make this formula approximately valid and thus lead to linearization of the simpler IC50 plot. The calculation of FC for the construction of the Iogit-log FC plot from the measured data has been described above (Equation 4 combined with the relation FC = S¢ - BC), and the initial (Le., maximal) binding (BL)O may either be measured directly or calcula-
236
Linearization of Two Ligand Binding Data
Vol. 42, No. 3, 1988
ted from the values of KdL and BMAX(in combination with the known SL) measured previously. In the specific example under consideration simple linear regressions of the theoretical Iogit-log data of Fig. 4 yield the following: [IC50 (lin. regress.)/"exact" IC50] = 1.12 (13% error), and [FC50 (lin. regress.)/"exact" FC50] = 0.999 (0.1% error). The "exact" values of IC50 and FC50are, respectively, 4.80 and 3.20 nM.) Further analysis is, of course, required to calculate KdC from the estimates of either IC50 or FC50 obtained from the Iogit-log plots discussed above. The Cheng-Prusoff correction (5) is given by
[9]
KdC -= IC50/(I+SL/KdL).
This formula may also be derived from Equations 1 and 2 above by using the definition of IC50 (Le., that (BL)O = 2BE when SC = IC50) and applying the drastic approximation that both FC = SC and FL = SL. As the illustrative example will demonstrate, this does not provide a good estimate of KdC when the affinity of the competing ligand is too high. This Cheng-Prusoff correction can be improved substantially by including in the calculation of the value of (BE)O,which easily can be measured experimentally or calculated from the measured values of KdL and BMAX. Combining the above Equations 2 and 6 and obvious initial conditions yields, upon elimination of BMAX,an expression which, when evaluated at the 50% displacement point, eventually becomes [10]
KdC --- FC50x 2KdL[SL-(BL)O]/[(BL)O2 + 2SL2 + 2SLKdL-3SL(BL)O].
This is a much better approximation than Equation 9 (Le., the Cheng-Prusoff correction), to which it reduces when (BL)O is neglected and FC50is replaced by ICso. Equation 10 remains approximately valid when IC50 is substituted for FC50 [11]
KdC = IC50 x 2KdL[SL-(BL)O]/[(BL)O2 + 2SL2 + 2SLKdL -3SL(BL)O].
Equation 10, will, however, always be superior to Equation 11; similarly, the Cheng-Prusoff expression itself will always be more nearly exact if a good estimate of FC50 is substituted for the estimate of iC50. Table I lists, for the example that we have been considering, the KdC estimates derived from the two different approximations; each method has been used in combination with both the estimated and the exact values of FCsoand IC50 listed above. It will be seen that the retention of (BL)O in the approximation is required for the accurate derivationof KdC from the competitive inhibition data under consideration here. The ease of implementation of Equation 10 suggests that it (or at least Equation 11) should be used instead of the Cheng-Prusoff approximation. TABLE I Comparison of estimatesof KdCderived from the Cheng-Prusoffcorrection (Equation 9) and fromthe approximationsdescribedby Equations10 and 11. Input IC50
Input FC50
Eauation
Exact
Estimated
Exact
Estimated
9
5.462 x 10-10M
6.107 x 1010 M
3.648 x 10-1° M
3.644 x 10-1° M
10and11
3.941 xl0lOM
4.406x10-10M
2.632x10-10M
2.629x10-10M
The assumed exact value for KdC is 2.632 x 10-10 M. The inputs to the equations include the exact values of FC50 (3.20 nM) and IC50 (4.80 nM) for the example discussed in the text as well as the estimated values for FC50 (3.20 nM) and IC50 (5.38 nM) derived as described from the Iogit-log plots of Fig. 4.
Vol. 42, No. 3, 1988
Linearization of Two Ligand Binding Data
237
Acknowledoement~ Supported in part by N.I.N.C.D.S. Grant NS 24404 awarded to Dr. W.G. Luttge. The expert secretarial assistance provided by Mrs. Teresa L. McCord is also gratefully acknowledged.
References . .
3. 4.
5. 6. 7.
D. RODBARD, in Receotors for Re0roductive Hormones (Eds. B.W. O'Malley and A.R. Means), p. 289-326. Plenum Press, New York (1973). H.A. FELDMAN, Anal. Biochem. 48,317-338 (1972). J.T. EDSALL and J. WYMAN, Biophysical Chemistry. Vol. 1, p. 651, Academic Press, New York (1958). A. DELEAN, P. MUNSON, and D. RODBARD, Amer. J. Physiol., 235, E97-E102 (1978). Y. CH ENG and W.H. PRUSOFF, Biochem. Pharmac. 22, 3099-3108 (1973). B.M. WEICHMAN and A.C. NOTIDES, Endocrinology 106, 434-439 (1980). C.R. CANTOR and P.R. SCHIMMEL, Bi0Dhvsical Chemistry. Part IIh The Behavior of Biolooical Macromolecules. p. 943, Freeman and Col, San Francisco ('i980).