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Analysis of receptor binding displacement curves by a nonhomologous ligand, on the basis of an equivalent competition principle
ANALYTICAL
BIOCHEMISTRY
200,3%-3%
(1992)
Analysis of Receptor Binding Displacement Curves by a Nonhomologous Ligand, on the Basis of an Equivalent Competition Principle Everardus Department
Received
J. J. van Zoelen of Cell Biology,
August
University
of Nijmegen,
Toernooiveld,
6525 ED Nimegen,
The Netherlands
2,199l
An exact method for the analysis of receptor-ligand binding data when labeled bound ligand is displaced by a nonhomologous ligand with a different dissociation constant is described. The present method, which is based on an equivalent competition principle for the homologous and the nonhomologous ligand, converts displacement curves into a linear form and is also applicable to situations in which free concentrations of ligand are significantly smaller than the added concentrations as a result of ligand binding. It is shown that the dissociation constant of the nonhomologous ligand is given directly by the concentration of this nonhomologous ligand added and the free concentration of unlabeled homologous ligand required to give the same level of displacement of labeled bound ligand. On the basis of these displacement characteristics, all binding parameters for receptor interaction of the nonhomologous ligand can be obtained and expressed, for example, in a Scatchard plot. The present method, which is referred to as the equivalent competition method, is also evaluated in this study with respect to the effects of nonspecific ligand binding and the presence of multiple receptor classes. 0 1992 Academic Press, Inc.
Studies on the structure-function relationship of hormone ligands require a detailed characterization of the receptor binding characteristics of structurally modified hormone analogues. The dissociation constant for the binding of hormone ligands to their receptors is generally determined experimentally from displacement curves, in which radiolabeled ligand bound to its receptor is competitively displaced by the addition of increasing concentrations of unlabeled ligand. Analysis of such data, for exampIe, in a Scatchard plot ( 1) , requires that the equilibrium binding of labeled ligand can be related to the concentration of free ligand in solution. In cases 0003-269’7192 $3.00 Copyright 0 1992 by Academic Press, All rights of reproduction in any form
in which radiolabeled ligand is displaced by a homologous ligand with a similar dissociation constant, this concentration of free ligand can be determined directly from the fraction of labeled ligand added that becomes bound after reaching equilibrium. If the radiolabeled ligand is displaced by a structurally modified ligand with a different dissociation constant, however, the free concentration of such a nonhomologous ligand cannot be determined experimentally in a direct way. For a simple interpretation of nonhomologous receptor displacement curves, Cheng and Prusoff (2) have used the assumption that the ligand molecules are present in a large excess over the receptor, implying that the free ligand concentration remains equal to that initially added throughout the displacement experiment. Chang et al. (3)) Hollemans and Bertina (4)) Horovitz and Levitzki ( 5 ) , and Martin et al. ( 6) have derived more general equations for the displacement of labeled bound ligand by a nonhomologous competitor without the above assumption, but due to the lack of a simple plotting strategy, these methods are not easily experimentally applicable. In this study the displacement curve for a nonhomologous ligand is compared with that of the homologous ligand. From the added concentrations of the two ligands required for an equivalent level of binding competition, a simple equation resulting in a linear relationship from which all the binding parameters of the nonhomologous ligand can be derived is obtained.
RESULTS
Theoretical Analysis
The equilibrium binding of a high-affinity ligand (referred to as the substrate ligand) in combination with a nonhomologous ligand (referred to as the inhibitor ligand) to a homogeneous set of receptor molecules in the 393
Inc. reserved.
394
EVERARDUS
absence of nonspecific equations
binding
J. J. VAN
is given by the set of B _ La - Fs Bi - Li - Fi *
a
(N-B,-Bi)F,=K;B,
ill
and
From (N - B, - Bi)Fi
= Ki.Bi,
2B
Bi
B, =
=
K-Fe
131
K,‘Fi of Bi it follows that
N-17, K, + F. + K,FilKi
N-F: K, + F: + F; + KsFilKi
’
of Eq. [ 31 and Eq. [ 71 it follows of B, / Bi that
Fi=Li Assuming that the added concentration of labeled substrate ligand (L,*) is a constant throughout the experiment, it follows from Eq. [ 51 that the same amount of labeled bound substrate is replaced in the presence of a free concentration of unlabeled substrate ligand F1,’ as that replaced in the presence of a free concentration of inhibitor (KS / Ki) Fi, since F: will also be similar under these two conditions. Therefore it follows under such a condition of equivalent competition that
[41
’
If the substrate is present as both labeled ligand (F:) and unlabeled ligand (Fy ) , the amount of labeled bound substrate (B:) is given by B: =
a comparison
121 upon elimination
in which B, and Bi are the amount of receptor-bound substrate and inhibitor, respectively, and N = B,,, the moles of receptor sites present. KS and Ki are the dissociation constants and F, and Fi the free concentrations of substrate and inhibitor, respectively. Upon division of these two equations it follows that
while upon elimination
ZOELEN
Fy = (K,IKi)Fi*
[91
By elimination of Fi from Eq. [ 81 and Eq. [ 91 the following equations for equivalent competition ( [lOA ] [lOD] ) are obtained:
WA1
[51
This equation describes the displacement curve of bound labeled substrate upon addition of either unlabeled substrate (homologous ligand) or inhibitor (nonhomologous ligand) . The free concentrations of substrate and inhibitor, required for evaluation of Eq. [ 51, differ from the added concentrations by the fraction bound to the receptor, according to
In this equation Li is the added concentration of inhibitor required for a certain level of displacement and F; the free concentration of unlabeled substrate required to obtain the same level of displacement. The second right-hand term of Eq. [lOA] gives the ratio of added and free substrate concentration and is similar for both labeled and unlabeled substrate molecules. When applied to unlabeled substrate molecules Eq. [ lOA] can be rewritten as
Fs = L, - pB,
Li - Ly Ki - K. F; = KS ’
[GA1
and Fi = Li - PBi,
[6Bl
in which La and Li are the added concentrations of substrate and inhibitor, respectively, and p is a dimensionconverting constant. If the ligand concentrations F and the concentrations of bound ligand B are both expressed in the same molar dimensions, p equals 1 without a dimension. If for a radiolabeled substrate F is expressed in nanograms per milliliter and B in cpm, 1 lp = SA X V, in which SA is the specific activity of the labeled ligand expressed in cpm per nanograms and V the incubation volume expressed in milliliters. From the combination these two equations it follows that
WBI
which is the most compact form of the equivalent competition equation, while upon introduction of Fi = Li(F:/L:) and F: = L: - pB,*, it follows that
L:(l
Li - Li -pB,*IL:)
=
Ki - K, KS ’
WC1
This latter equation relates the ratio of the dissociation constants Ki and KS only to added concentrations of labeled substrate (L,*), unlabeled substrate (Lt), and inhibitor ( Li) , in addition to the specific activity of the labeled ligand indicated by p and the experimentally ob-
EQUIVALENT
COMPETITION
ANALYSIS
OF
RECEPTOR
395
BINDING
E
’ ! \1 1
t N I 0 * *,”
i\ ., --\_
z
.
[added ligand]
1: ,L,
FIG. 1. Analysis of displacement curves upon ligand binding to a single receptor class. (A) Displacement curves for the competition of labeled ligand binding upon addition of homologous substrate ligand (0) or nonhomologous inhibitor ligand (0). Binding data were simulated by an iterative computer analysis based on Eq. (51 using initially added concentrations (L) instead of free concentrations (F). After the calculation of B values, corresponding F values for the substrate and inhibitor were calculated according to Eqs. [6A] and [ 6B] and were used again in Eq. [ 51 in an iterative way. The following values were used for simulation: N = 10,000; KS = 10; Ki = 30; p = 0.005; KA = 0; Lz = 1. The interpretation of the dashed horizontal and vertical lines, which form the basis of the equivalent competition method, is given in the text. (B) Equivalent competition plot, following Eq. [lOA], for the displacement curve of the nonhomologous inhibitor ligand. (C) Scatchard plot for receptor binding of the homologous substrate ligand (0) and the nonhomologous inhibitor ligand (0).
tained value for labeled bound substrate B:. Moreover, since (1 - pB:IL,*) equals the ratio of free and added labeled ligand, this equation can be I me written as L, - L; Ly (1 - cpmb”“d/cpmedd”d)
=
K - K K -
PODI
This equation is equivalent to Eqs. [ lOA]-[lOC] and permits the determination of Ki /K, also in cases in which the specific activity p of the labeled ligand is not exactly known. Equation [lOD] shows that the ratio of the dissociation constants of the inhibitor and substrate ligand is given by the ratio of the concentrations added to induce the same extent of ligand binding competition, provided that a correction term ( 1 - cpmb”““d / cpmaddd) is included to account for the fact that the free ligand concentrations can be significantly lower than the ligand concentrations added. Data Simulation
Figure 1 illustrates the use of the equivalent competition method with simulated receptor-ligand binding data, assuming a receptor affinity threefold lower for the inhibitor than that for the substrate molecules. The parameters used for data simulation are indicated in the
legend. Figure 1A shows the displacement curves for the homologous substrate ligand and the nonhomologous inhibitor as a function of the logarithm of the ligand concentration added. Under the conditions tested a significant fraction of added ligand is bound to the receptor, resulting in significant differences between added and free ligand concentrations. Each added concentration of inhibitor Li (dashed vertical line on the right) corresponds to a value for labeled bound ligandB,* (horizontal dashed line) and to an added concentration of unlabeled substrate L:, which gives rise to a similar value of B: (dashed vertical line on the left). By the use of this graphical method, values for B: and L; can be obtained for every applied inhibitor concentration Li, which after analysis according to Eq. [lOC ] or [lOD] results in a value for the ratio of the dissociation constants of the inhibitor and substrate ligand K,IK,. Alternatively, the homologous displacement curve of Fig. 1A can first be analyzed for the system parameters K, and N in a Scatchard plot (Fig. 1C). Given the values of K, and N, the value of Fi for homologous competition can now be calculated directly from Eq. [ 5 ] following F: = (N.F:lB:)
- KS - F;,
WI
in which F: = L,* - pB:. By applying Eq. [ 1OA 1, Li /Fr can now be plotted as a function of L,*/F:, as shown in
396
EVERARDUS
J. J. VAN ZOELEN
Fig. 1B. From the linearity of this plot it can be concluded that (Ki - KS) /KS is a constant throughout the displacement curve, and given KS, the value of Ki is obtained directly. The value of ( Ki - KS) /KS can also be obtained for individual points from the difference of the numeric values on the two axes. From the values of Ki and N, the binding characteristics of the inhibitor can now be represented in a Scatchard plot, as shown in Fig. 1C. The value of Bi for the inhibitor is given by N. Fil (Ki + Fi), in which Fi is given by Eq. [ 81, using L:IF: = L, / Fs. This demonstrates that by the use of the equivalent competition method the complete binding characteristics of a nonhomologous ligand can be obtained from its competing activity toward a labeled homologous ligand.
binding component, while in addition Eq. [ 91 for equivalent competition will also still hold. The added and free concentrations of substrate and inhibitor. however, are now related according to F. = L, - p(B, + KA.Fs)
[13A
1
Fi = Li - p(Bi + KA.Fi),
[13B
1
and
from which it follows that B >Ls-F,(l+p.K,) Bi - Li- Fi(1 + p-K*)
[I41
’
Effect of Nonspecific Binding
A major problem generally encountered during receptor-ligand binding studies is the presence of nonspecific binding, defined as a low-affinity ligand binding to nonreceptor domains that does not show any saturating behavior within the range of ligand concentrations used ( 7). As a consequence of such nonspecific interaction the binding of radiolabeled ligand cannot fully be displaced by either a homologous or a nonhomologous unlabeled ligand. In the presence of such nonspecific binding the displacement curve is now described by (7,8) B; =
N-F,* KS + F: + Ff + KsFilKi
+ KA.F:,
[12]
and following the approach described above, the following equation is derived:
(l+p.K,)+$. 8
A comparison with Eq. [lOA] shows that in the presence of nonspecific binding the equation for equivalent competition still describes a linear relationship between (Li lF:) and (L:/F:) as plotted in Fig. lB, but now with a different interpretation of the intercept. In addition Eq. [ll] for the calculation of the equivalent Fi is now derived from experimental data according to Fy = (N.F:/(B;
in which the experimentally obtained value for the total amount of labeled bound ligand (BG) is composed of a specific component B: (see Eq. [ 5 ] ) and a nonspecific component characterized by the constant KA. Both substrate and inhibitor will give rise to nonspecific interaction, but since substrate molecules are labeled, only the nonspecific component of this ligand will manifest itself in the displacement curves. However, due to nonspecific interaction of the inhibitor the free concentration of this ligand available for receptor competition will be reduced, resulting in a shift of the displacement curve to higher concentrations. Recently we have described a method for the linearization of homologous displacement curves that is not affected by the presence of nonspecific binding (8). From this so-called linear subtraction plot, not only the values for N and K, but also a value for the nonspecific binding constant KA, can be obtained. For simplicity it will be assumed further that KA is similar for both the substrate and the inhibitor molecules. In the presence of nonspecific binding, Eqs. [l] , [ 21, and [ 31 will still be valid, assuming that Bi and B, only refer to the specific
W-4
- KA. F,*)) - KS - F:.
Similar to above, Eq. [15A] is the expression of choice if the binding parameters N, KS, and KA of the homologous substrate ligand have already been determined experimentally. If these parameters have not been established, the ratio of the parameters Ki and K, can again be determined graphically from experimental displacement curves, similar to the approach described using Eqs. [lOC] and [lOD] . As a consequence of nonspecific interaction the binding of labeled bound substrate cannot be fully competed by addition of an excess concentration of unlabeled ligand, but will reach an asymptotic value according to Eq. [12] given by B*, = KA. F*,, in which F*, is the free concentration of labeled substrate in the presence of an excess unlabeled ligand and B*, the amount of labeled bound substrate under these conditions. From Eqs. [6A] and [13A] it follows that Fz = L,* -pB;= Lo*- pKA * F*,. From this latter consideration it is derived that ( 1 + p * KA) = L,* lF*,. Since the value of F*,IL,* is determined by the fraction of labeled ligand that is still bound after addition of an excess of unlabeled ligand as a result of nonspecific binding, Eq. [15A] can be rewritten in the format of Eq. [ lOD] , giving
EQUIVALENT
COMPETITION
ANALYSIS
Ki - KS = K,(
1 -
cpmFd
P5Bl
/cpmadded) ’
Classes
In cases in which substrate and inhibitor ligand can bind to both a high-affinity ( Nh) and a low-affinity (Ni) receptor class, the displacement curve of labeled substrate is described by NhF: KS,, + F,* + F: + Ksh.FilKih
+
[B:(K,,
F*
+ LWh+N)+& 2Bz
- Ksh) - (N, - N,JF:12
8
+ 4NhN,Fz2.
]I71
in which cpmpd equals the amount of labeled ligand binding in the presence of an excess of unlabeled ligand. This equation, which is fully equivalent to Eq. [15A], shows that the ratio of dissociation constants of the inhibitor and substrate ligand is given by the ratio of the concentrations required to give the same extent of binding competition, provided that correction terms are added for the fact that free ligand concentrations may be significantly lower than ligand concentrations added (correction term on the left-hand side of the equation) and for the presence of a n.onspecific binding component (correction term in the right-hand side of the equation). The value of both correction terms can be read directly from experimental displacement curves. Receptor
x
397
BINDING
F: = - 2 (KS,-, + K, + 24’:)
Lz ( 1 - Cpmbound/cpmadded)
B,* =
RECEPTOR
1
L’ - LI
Effect of Multiple
OF
Upon extrapolation of this linearity or by subtracting the numeric values on the two axes of Fig. 2B, the ratio of the two dissociation constants for inhibitor and substrate ligand is obtained. In addition a Scatchard plot for the binding of the inhibitor to the receptor can be obtained from these competition data using similar equations as those above for a single receptor class, resulting in the curvilinear Scatchard plot shown in Fig. 2C. In cases in which ( Kah / K,,) # ( Kih /K,) displacement curves will in general not give rise to linear equivalent competition plots, as shown by the example in Figs. 2A and 2B. Instead a sigmoid equivalent competition plot will be obtained with a linear part only at high (lefthand side of the curve) and low (right-hand side of the curve) ligand concentrations. It is not the aim of the present study to describe the complete characteristics of this curve, but the parameters determining these two linear parts can be understood as follows. As long as the ligand concentrations are very low compared to the dissociation constants of the low-affinity receptor class, binding to this receptor will not have a saturating character and will increase linearly with increasing ligand concentration ( FS i 4 KS, i,). As a consequence it follows according the considerations given in Eqs. [l] - [ 31 that
WC K,, + F,1; + F~ + K,, . FilKi,
’
‘16]
(Bs - NlFsIKd) _ &-J’s (Bi - N,FiIKi,)
in which Ksh and KS, are the high- and low-affinity dissociation constants for the substrate ligand and Kih and Ki, for the inhibitor ligand, respectively. Figure 2A shows a displacement curve for substrate competition from two receptor classes, which gives rise to a well-characterized curvilinear behavior in a Scatchard plot (Fig. 2C ) . If the above approach for a single receptor class is applied to multiple receptor classes, complex equations that in general do not permit direct data. linearization are obtained (9). Only in cases in which the ratios of the dissociation constants of the high- and low-affinity receptor class are similar for the substrate and the inhibitor ligand can Eq. [ 91 be applied under equivalent competition conditions for both receptor classes. Particularly in the case of structurally related ligand analogues, such a constant ratio of dissociation constants can be expected. Figure 2A shows a displacement curve under these conditions, which gives rise to a linear equivalent competition plot according to Eq. [lOA 1, as shown in Fig. 2B. On the basis of Eq. [ 161 the value of Fy for equivalent competition is calculated under these conditions by
Ksh.Fi
’
WI
in which B, and Bi refer to a combination of both lowand high-affinity receptor binding of the substrate and inhibitor molecules. In combination with Eqs. [6A] and [ 6B] and Eq. [ 91 for equivalent competition, which are both valid under these conditions, the following equation is derived:
$=(Kih;s~)+~+p-N,[&-&].
[19]
This equation shows that at low ligand concentrations the equivalent competition plot, derived by calculating values for Fr according to Eq. [17], is linear and gives rise to a difference in numeric values on the two axes equal to the ratio of the high-affinity dissociation constants, added to a correction term determined by all four dissociation constants and the number of low-affinity receptor sites. Only where ( KSh / Kd) = ( Kih / Kd)
398
EVERARDUS
J. J. VAN
ZOELEN
H i
L: IF:
c
i
c
i t
VI2 [added ligand]
103
c
101
L: ,L,
lb
6
2
t
10
EXlO-5
FIG. 2. Analysis of displacement curves upon ligand binding to both a high- and a low-affinity receptor class. (A) Displacement curves for the homologous substrate ligand (0) and for two nonhomologous ligands. Data were simulated using N,, = 10,000, ZQ, = 10, N, = 1,000,000, K,, = 1000, KA = 0, p = 0.005, L: = 1 and (0) I(ih = 100, Kg = 10,000 or (A ) Kih = 100, Ki, = 2500. (B) Equivalent competition plots, following Eq. [lOA], for the two inhibitor ligands, characterized by either Ksh / Kd = I(ih / Kil (0) or Kh / KS, # I(ih / Kil (A). (C ) Scatchard plots for ligand binding of the homologous substrate ligand (0) and the nonhomologous ligand (0) , giving rise to a linear equivalent competition plot in B.
will this correction term be equal to zero. Since the lowaffinity receptor binding is considered unsaturable under these conditions, Eq. [ 191 is converted to Eq. [ 15 ] for nonspecific binding if Ni I KS,= IV,I Kti = KA. In cases in which ligand concentrations used are two orders of magnitude higher than the dissociation constants for high-affinity receptor binding, labeled substrate molecules will be completely displaced from these high-affinity receptors. As a consequence the binding characteristics will be determined only by low-affinity receptor interaction under these conditions and thus Eq. [lOA] can now be used, in which the intercept of the equivalent competition plot will be given by ( Kil - Kd) / KB,,which explains the linear behavior of the plot in Fig. 2B at high ligand concentrations. By using these equations for very low and very high ligand concentrations direct estimates can therefore be made for both the high- and the low-affinity dissociation constant of the inhibitor ligand. In contrast to the conditions tested in Fig. 2, however, a conversion of such experimental data into a Scatchard plot can be achieved only in an indirect way, using the estimated Ki values obtained according to the above approach. DISCUSSION
In the present study theoretical equations have been derived for the binding competition of a labeled ligand by a nonhomologous unlabeled ligand, which are also
valid if the free concentration of ligand molecules differs considerably from the ligand concentration added. By comparing the displacement curve of the native unlabeled ligand with that of the nonhomologous inhibitor ligand, simple equations that on the basis of an equivalent competition principle, allow direct determination of the dissooiation constant of the inhibitor ligand are obtained. In its most compact form (Eq. [lOB ] ) it has been derived that the ratio of the dissociation constants of the inhibitor and substrate ligand is given by the difference in concentration of the two ligands added, to give the same extent of binding competition, divided by the free concentration of unlabeled substrate ligand under these conditions. This basic equation has been worked out to Eq. [lOA], with which the independence of the ratio of the dissociation constants from the ligand concentrations can be tested in a linear plot. Alternatively a method has been described following Eqs. [ 1OC ] and [ lOD] , with which the ratio of the dissociation constants can be derived graphically from displacement curves, irrespective of prior determination of the binding characteristics of the substrate ligand. These two parallel approaches have also been worked out for the presence of an additional nonspecific binding component, resulting in Eqs. [15A] and [ 15B]. In comparison with other methods (3-6) the present equations are relatively simple, while in addition experimental data can readily be converted into a linear form,
EQUIVALENT
COMPETITION
ANALYSIS
designated the equivalent competition plot (Figs. 1B and 2B). Not only can this plot be used to determine the dissociation constant of the inhibitor ligand directly, but is also very sensitive for detecting the presence of additional binding processes, such as multiple receptor classes of either the substrate or the inhibitor ligand (Fig. 2B). Under these conditions Eq. [19] is generally applicable at low ligand concentrations, which, combined with data at high ligand concentrations, gives a good estimate of the dissociation constants involved. Also if one ligand binds to a single receptor class and the other to multiple classes, Eq. [19] can still be used, assuming an infinite value for the low-affinity dissociation constant of the ligand interacting with a single receptor class. In this way complex competition patterns, for example, in the case of competition of different polypeptide growth factors for binding to multiple receptor classes ( lo), can be analyzed successfully.
OF
RECEPTOR
BINDING
399
REFERENCES 1. Scatchard,
2. Cheng,
G. (1949) Y., and Prusoff,
Ann. N.Y. Acad. Sci. 61,660-672. W. M. (1973) Biochem. Pharmacol. 22,
3099-3108.
3. Chang, K. J., Jacobs, S., and Cuatrecasas, P. (1975) Biochim. Biophys. Aeta 406,294-303. 4. Hollemans, H. J. G., and Bertina, R. M. ( 1975) Clin. Chem. 21, 1769-1773.
5. Horovitz, A., and Levitzki, A. ( 1987) Proc. Nat&. Acad. Sci. USA 84,6654-6658. 6. Martin, R. L., Renosto, F., and Segel, I. H. ( 1991) Arch. Biochem. Biophys. 284,26-29. 7. Mendel, C. M., and Mendel, D. B. (1985) Biochem. J. 228,269272. 8. Van Zoelen, E. J. J. (1989) Biochem. J. 262,549-556. 9. Almagor, H., and Levitzki, A. (1990) Proc. Nati. Acad. Sci. USA 87,6482-6486. 10. Van Zoelen, E. J. J. (1990) Prog. Growth Fact. Res. 2, 131-152.
BIOCHEMISTRY
200,3%-3%
(1992)
Analysis of Receptor Binding Displacement Curves by a Nonhomologous Ligand, on the Basis of an Equivalent Competition Principle Everardus Department
Received
J. J. van Zoelen of Cell Biology,
August
University
of Nijmegen,
Toernooiveld,
6525 ED Nimegen,
The Netherlands
2,199l
An exact method for the analysis of receptor-ligand binding data when labeled bound ligand is displaced by a nonhomologous ligand with a different dissociation constant is described. The present method, which is based on an equivalent competition principle for the homologous and the nonhomologous ligand, converts displacement curves into a linear form and is also applicable to situations in which free concentrations of ligand are significantly smaller than the added concentrations as a result of ligand binding. It is shown that the dissociation constant of the nonhomologous ligand is given directly by the concentration of this nonhomologous ligand added and the free concentration of unlabeled homologous ligand required to give the same level of displacement of labeled bound ligand. On the basis of these displacement characteristics, all binding parameters for receptor interaction of the nonhomologous ligand can be obtained and expressed, for example, in a Scatchard plot. The present method, which is referred to as the equivalent competition method, is also evaluated in this study with respect to the effects of nonspecific ligand binding and the presence of multiple receptor classes. 0 1992 Academic Press, Inc.
Studies on the structure-function relationship of hormone ligands require a detailed characterization of the receptor binding characteristics of structurally modified hormone analogues. The dissociation constant for the binding of hormone ligands to their receptors is generally determined experimentally from displacement curves, in which radiolabeled ligand bound to its receptor is competitively displaced by the addition of increasing concentrations of unlabeled ligand. Analysis of such data, for exampIe, in a Scatchard plot ( 1) , requires that the equilibrium binding of labeled ligand can be related to the concentration of free ligand in solution. In cases 0003-269’7192 $3.00 Copyright 0 1992 by Academic Press, All rights of reproduction in any form
in which radiolabeled ligand is displaced by a homologous ligand with a similar dissociation constant, this concentration of free ligand can be determined directly from the fraction of labeled ligand added that becomes bound after reaching equilibrium. If the radiolabeled ligand is displaced by a structurally modified ligand with a different dissociation constant, however, the free concentration of such a nonhomologous ligand cannot be determined experimentally in a direct way. For a simple interpretation of nonhomologous receptor displacement curves, Cheng and Prusoff (2) have used the assumption that the ligand molecules are present in a large excess over the receptor, implying that the free ligand concentration remains equal to that initially added throughout the displacement experiment. Chang et al. (3)) Hollemans and Bertina (4)) Horovitz and Levitzki ( 5 ) , and Martin et al. ( 6) have derived more general equations for the displacement of labeled bound ligand by a nonhomologous competitor without the above assumption, but due to the lack of a simple plotting strategy, these methods are not easily experimentally applicable. In this study the displacement curve for a nonhomologous ligand is compared with that of the homologous ligand. From the added concentrations of the two ligands required for an equivalent level of binding competition, a simple equation resulting in a linear relationship from which all the binding parameters of the nonhomologous ligand can be derived is obtained.
RESULTS
Theoretical Analysis
The equilibrium binding of a high-affinity ligand (referred to as the substrate ligand) in combination with a nonhomologous ligand (referred to as the inhibitor ligand) to a homogeneous set of receptor molecules in the 393
Inc. reserved.
394
EVERARDUS
absence of nonspecific equations
binding
J. J. VAN
is given by the set of B _ La - Fs Bi - Li - Fi *
a
(N-B,-Bi)F,=K;B,
ill
and
From (N - B, - Bi)Fi
= Ki.Bi,
2B
Bi
B, =
=
K-Fe
131
K,‘Fi of Bi it follows that
N-17, K, + F. + K,FilKi
N-F: K, + F: + F; + KsFilKi
’
of Eq. [ 31 and Eq. [ 71 it follows of B, / Bi that
Fi=Li Assuming that the added concentration of labeled substrate ligand (L,*) is a constant throughout the experiment, it follows from Eq. [ 51 that the same amount of labeled bound substrate is replaced in the presence of a free concentration of unlabeled substrate ligand F1,’ as that replaced in the presence of a free concentration of inhibitor (KS / Ki) Fi, since F: will also be similar under these two conditions. Therefore it follows under such a condition of equivalent competition that
[41
’
If the substrate is present as both labeled ligand (F:) and unlabeled ligand (Fy ) , the amount of labeled bound substrate (B:) is given by B: =
a comparison
121 upon elimination
in which B, and Bi are the amount of receptor-bound substrate and inhibitor, respectively, and N = B,,, the moles of receptor sites present. KS and Ki are the dissociation constants and F, and Fi the free concentrations of substrate and inhibitor, respectively. Upon division of these two equations it follows that
while upon elimination
ZOELEN
Fy = (K,IKi)Fi*
[91
By elimination of Fi from Eq. [ 81 and Eq. [ 91 the following equations for equivalent competition ( [lOA ] [lOD] ) are obtained:
WA1
[51
This equation describes the displacement curve of bound labeled substrate upon addition of either unlabeled substrate (homologous ligand) or inhibitor (nonhomologous ligand) . The free concentrations of substrate and inhibitor, required for evaluation of Eq. [ 51, differ from the added concentrations by the fraction bound to the receptor, according to
In this equation Li is the added concentration of inhibitor required for a certain level of displacement and F; the free concentration of unlabeled substrate required to obtain the same level of displacement. The second right-hand term of Eq. [lOA] gives the ratio of added and free substrate concentration and is similar for both labeled and unlabeled substrate molecules. When applied to unlabeled substrate molecules Eq. [ lOA] can be rewritten as
Fs = L, - pB,
Li - Ly Ki - K. F; = KS ’
[GA1
and Fi = Li - PBi,
[6Bl
in which La and Li are the added concentrations of substrate and inhibitor, respectively, and p is a dimensionconverting constant. If the ligand concentrations F and the concentrations of bound ligand B are both expressed in the same molar dimensions, p equals 1 without a dimension. If for a radiolabeled substrate F is expressed in nanograms per milliliter and B in cpm, 1 lp = SA X V, in which SA is the specific activity of the labeled ligand expressed in cpm per nanograms and V the incubation volume expressed in milliliters. From the combination these two equations it follows that
WBI
which is the most compact form of the equivalent competition equation, while upon introduction of Fi = Li(F:/L:) and F: = L: - pB,*, it follows that
L:(l
Li - Li -pB,*IL:)
=
Ki - K, KS ’
WC1
This latter equation relates the ratio of the dissociation constants Ki and KS only to added concentrations of labeled substrate (L,*), unlabeled substrate (Lt), and inhibitor ( Li) , in addition to the specific activity of the labeled ligand indicated by p and the experimentally ob-
EQUIVALENT
COMPETITION
ANALYSIS
OF
RECEPTOR
395
BINDING
E
’ ! \1 1
t N I 0 * *,”
i\ ., --\_
z
.
[added ligand]
1: ,L,
FIG. 1. Analysis of displacement curves upon ligand binding to a single receptor class. (A) Displacement curves for the competition of labeled ligand binding upon addition of homologous substrate ligand (0) or nonhomologous inhibitor ligand (0). Binding data were simulated by an iterative computer analysis based on Eq. (51 using initially added concentrations (L) instead of free concentrations (F). After the calculation of B values, corresponding F values for the substrate and inhibitor were calculated according to Eqs. [6A] and [ 6B] and were used again in Eq. [ 51 in an iterative way. The following values were used for simulation: N = 10,000; KS = 10; Ki = 30; p = 0.005; KA = 0; Lz = 1. The interpretation of the dashed horizontal and vertical lines, which form the basis of the equivalent competition method, is given in the text. (B) Equivalent competition plot, following Eq. [lOA], for the displacement curve of the nonhomologous inhibitor ligand. (C) Scatchard plot for receptor binding of the homologous substrate ligand (0) and the nonhomologous inhibitor ligand (0).
tained value for labeled bound substrate B:. Moreover, since (1 - pB:IL,*) equals the ratio of free and added labeled ligand, this equation can be I me written as L, - L; Ly (1 - cpmb”“d/cpmedd”d)
=
K - K K -
PODI
This equation is equivalent to Eqs. [ lOA]-[lOC] and permits the determination of Ki /K, also in cases in which the specific activity p of the labeled ligand is not exactly known. Equation [lOD] shows that the ratio of the dissociation constants of the inhibitor and substrate ligand is given by the ratio of the concentrations added to induce the same extent of ligand binding competition, provided that a correction term ( 1 - cpmb”““d / cpmaddd) is included to account for the fact that the free ligand concentrations can be significantly lower than the ligand concentrations added. Data Simulation
Figure 1 illustrates the use of the equivalent competition method with simulated receptor-ligand binding data, assuming a receptor affinity threefold lower for the inhibitor than that for the substrate molecules. The parameters used for data simulation are indicated in the
legend. Figure 1A shows the displacement curves for the homologous substrate ligand and the nonhomologous inhibitor as a function of the logarithm of the ligand concentration added. Under the conditions tested a significant fraction of added ligand is bound to the receptor, resulting in significant differences between added and free ligand concentrations. Each added concentration of inhibitor Li (dashed vertical line on the right) corresponds to a value for labeled bound ligandB,* (horizontal dashed line) and to an added concentration of unlabeled substrate L:, which gives rise to a similar value of B: (dashed vertical line on the left). By the use of this graphical method, values for B: and L; can be obtained for every applied inhibitor concentration Li, which after analysis according to Eq. [lOC ] or [lOD] results in a value for the ratio of the dissociation constants of the inhibitor and substrate ligand K,IK,. Alternatively, the homologous displacement curve of Fig. 1A can first be analyzed for the system parameters K, and N in a Scatchard plot (Fig. 1C). Given the values of K, and N, the value of Fi for homologous competition can now be calculated directly from Eq. [ 5 ] following F: = (N.F:lB:)
- KS - F;,
WI
in which F: = L,* - pB:. By applying Eq. [ 1OA 1, Li /Fr can now be plotted as a function of L,*/F:, as shown in
396
EVERARDUS
J. J. VAN ZOELEN
Fig. 1B. From the linearity of this plot it can be concluded that (Ki - KS) /KS is a constant throughout the displacement curve, and given KS, the value of Ki is obtained directly. The value of ( Ki - KS) /KS can also be obtained for individual points from the difference of the numeric values on the two axes. From the values of Ki and N, the binding characteristics of the inhibitor can now be represented in a Scatchard plot, as shown in Fig. 1C. The value of Bi for the inhibitor is given by N. Fil (Ki + Fi), in which Fi is given by Eq. [ 81, using L:IF: = L, / Fs. This demonstrates that by the use of the equivalent competition method the complete binding characteristics of a nonhomologous ligand can be obtained from its competing activity toward a labeled homologous ligand.
binding component, while in addition Eq. [ 91 for equivalent competition will also still hold. The added and free concentrations of substrate and inhibitor. however, are now related according to F. = L, - p(B, + KA.Fs)
[13A
1
Fi = Li - p(Bi + KA.Fi),
[13B
1
and
from which it follows that B >Ls-F,(l+p.K,) Bi - Li- Fi(1 + p-K*)
[I41
’
Effect of Nonspecific Binding
A major problem generally encountered during receptor-ligand binding studies is the presence of nonspecific binding, defined as a low-affinity ligand binding to nonreceptor domains that does not show any saturating behavior within the range of ligand concentrations used ( 7). As a consequence of such nonspecific interaction the binding of radiolabeled ligand cannot fully be displaced by either a homologous or a nonhomologous unlabeled ligand. In the presence of such nonspecific binding the displacement curve is now described by (7,8) B; =
N-F,* KS + F: + Ff + KsFilKi
+ KA.F:,
[12]
and following the approach described above, the following equation is derived:
(l+p.K,)+$. 8
A comparison with Eq. [lOA] shows that in the presence of nonspecific binding the equation for equivalent competition still describes a linear relationship between (Li lF:) and (L:/F:) as plotted in Fig. lB, but now with a different interpretation of the intercept. In addition Eq. [ll] for the calculation of the equivalent Fi is now derived from experimental data according to Fy = (N.F:/(B;
in which the experimentally obtained value for the total amount of labeled bound ligand (BG) is composed of a specific component B: (see Eq. [ 5 ] ) and a nonspecific component characterized by the constant KA. Both substrate and inhibitor will give rise to nonspecific interaction, but since substrate molecules are labeled, only the nonspecific component of this ligand will manifest itself in the displacement curves. However, due to nonspecific interaction of the inhibitor the free concentration of this ligand available for receptor competition will be reduced, resulting in a shift of the displacement curve to higher concentrations. Recently we have described a method for the linearization of homologous displacement curves that is not affected by the presence of nonspecific binding (8). From this so-called linear subtraction plot, not only the values for N and K, but also a value for the nonspecific binding constant KA, can be obtained. For simplicity it will be assumed further that KA is similar for both the substrate and the inhibitor molecules. In the presence of nonspecific binding, Eqs. [l] , [ 21, and [ 31 will still be valid, assuming that Bi and B, only refer to the specific
W-4
- KA. F,*)) - KS - F:.
Similar to above, Eq. [15A] is the expression of choice if the binding parameters N, KS, and KA of the homologous substrate ligand have already been determined experimentally. If these parameters have not been established, the ratio of the parameters Ki and K, can again be determined graphically from experimental displacement curves, similar to the approach described using Eqs. [lOC] and [lOD] . As a consequence of nonspecific interaction the binding of labeled bound substrate cannot be fully competed by addition of an excess concentration of unlabeled ligand, but will reach an asymptotic value according to Eq. [12] given by B*, = KA. F*,, in which F*, is the free concentration of labeled substrate in the presence of an excess unlabeled ligand and B*, the amount of labeled bound substrate under these conditions. From Eqs. [6A] and [13A] it follows that Fz = L,* -pB;= Lo*- pKA * F*,. From this latter consideration it is derived that ( 1 + p * KA) = L,* lF*,. Since the value of F*,IL,* is determined by the fraction of labeled ligand that is still bound after addition of an excess of unlabeled ligand as a result of nonspecific binding, Eq. [15A] can be rewritten in the format of Eq. [ lOD] , giving
EQUIVALENT
COMPETITION
ANALYSIS
Ki - KS = K,(
1 -
cpmFd
P5Bl
/cpmadded) ’
Classes
In cases in which substrate and inhibitor ligand can bind to both a high-affinity ( Nh) and a low-affinity (Ni) receptor class, the displacement curve of labeled substrate is described by NhF: KS,, + F,* + F: + Ksh.FilKih
+
[B:(K,,
F*
+ LWh+N)+& 2Bz
- Ksh) - (N, - N,JF:12
8
+ 4NhN,Fz2.
]I71
in which cpmpd equals the amount of labeled ligand binding in the presence of an excess of unlabeled ligand. This equation, which is fully equivalent to Eq. [15A], shows that the ratio of dissociation constants of the inhibitor and substrate ligand is given by the ratio of the concentrations required to give the same extent of binding competition, provided that correction terms are added for the fact that free ligand concentrations may be significantly lower than ligand concentrations added (correction term on the left-hand side of the equation) and for the presence of a n.onspecific binding component (correction term in the right-hand side of the equation). The value of both correction terms can be read directly from experimental displacement curves. Receptor
x
397
BINDING
F: = - 2 (KS,-, + K, + 24’:)
Lz ( 1 - Cpmbound/cpmadded)
B,* =
RECEPTOR
1
L’ - LI
Effect of Multiple
OF
Upon extrapolation of this linearity or by subtracting the numeric values on the two axes of Fig. 2B, the ratio of the two dissociation constants for inhibitor and substrate ligand is obtained. In addition a Scatchard plot for the binding of the inhibitor to the receptor can be obtained from these competition data using similar equations as those above for a single receptor class, resulting in the curvilinear Scatchard plot shown in Fig. 2C. In cases in which ( Kah / K,,) # ( Kih /K,) displacement curves will in general not give rise to linear equivalent competition plots, as shown by the example in Figs. 2A and 2B. Instead a sigmoid equivalent competition plot will be obtained with a linear part only at high (lefthand side of the curve) and low (right-hand side of the curve) ligand concentrations. It is not the aim of the present study to describe the complete characteristics of this curve, but the parameters determining these two linear parts can be understood as follows. As long as the ligand concentrations are very low compared to the dissociation constants of the low-affinity receptor class, binding to this receptor will not have a saturating character and will increase linearly with increasing ligand concentration ( FS i 4 KS, i,). As a consequence it follows according the considerations given in Eqs. [l] - [ 31 that
WC K,, + F,1; + F~ + K,, . FilKi,
’
‘16]
(Bs - NlFsIKd) _ &-J’s (Bi - N,FiIKi,)
in which Ksh and KS, are the high- and low-affinity dissociation constants for the substrate ligand and Kih and Ki, for the inhibitor ligand, respectively. Figure 2A shows a displacement curve for substrate competition from two receptor classes, which gives rise to a well-characterized curvilinear behavior in a Scatchard plot (Fig. 2C ) . If the above approach for a single receptor class is applied to multiple receptor classes, complex equations that in general do not permit direct data. linearization are obtained (9). Only in cases in which the ratios of the dissociation constants of the high- and low-affinity receptor class are similar for the substrate and the inhibitor ligand can Eq. [ 91 be applied under equivalent competition conditions for both receptor classes. Particularly in the case of structurally related ligand analogues, such a constant ratio of dissociation constants can be expected. Figure 2A shows a displacement curve under these conditions, which gives rise to a linear equivalent competition plot according to Eq. [lOA 1, as shown in Fig. 2B. On the basis of Eq. [ 161 the value of Fy for equivalent competition is calculated under these conditions by
Ksh.Fi
’
WI
in which B, and Bi refer to a combination of both lowand high-affinity receptor binding of the substrate and inhibitor molecules. In combination with Eqs. [6A] and [ 6B] and Eq. [ 91 for equivalent competition, which are both valid under these conditions, the following equation is derived:
$=(Kih;s~)+~+p-N,[&-&].
[19]
This equation shows that at low ligand concentrations the equivalent competition plot, derived by calculating values for Fr according to Eq. [17], is linear and gives rise to a difference in numeric values on the two axes equal to the ratio of the high-affinity dissociation constants, added to a correction term determined by all four dissociation constants and the number of low-affinity receptor sites. Only where ( KSh / Kd) = ( Kih / Kd)
398
EVERARDUS
J. J. VAN
ZOELEN
H i
L: IF:
c
i
c
i t
VI2 [added ligand]
103
c
101
L: ,L,
lb
6
2
t
10
EXlO-5
FIG. 2. Analysis of displacement curves upon ligand binding to both a high- and a low-affinity receptor class. (A) Displacement curves for the homologous substrate ligand (0) and for two nonhomologous ligands. Data were simulated using N,, = 10,000, ZQ, = 10, N, = 1,000,000, K,, = 1000, KA = 0, p = 0.005, L: = 1 and (0) I(ih = 100, Kg = 10,000 or (A ) Kih = 100, Ki, = 2500. (B) Equivalent competition plots, following Eq. [lOA], for the two inhibitor ligands, characterized by either Ksh / Kd = I(ih / Kil (0) or Kh / KS, # I(ih / Kil (A). (C ) Scatchard plots for ligand binding of the homologous substrate ligand (0) and the nonhomologous ligand (0) , giving rise to a linear equivalent competition plot in B.
will this correction term be equal to zero. Since the lowaffinity receptor binding is considered unsaturable under these conditions, Eq. [ 191 is converted to Eq. [ 15 ] for nonspecific binding if Ni I KS,= IV,I Kti = KA. In cases in which ligand concentrations used are two orders of magnitude higher than the dissociation constants for high-affinity receptor binding, labeled substrate molecules will be completely displaced from these high-affinity receptors. As a consequence the binding characteristics will be determined only by low-affinity receptor interaction under these conditions and thus Eq. [lOA] can now be used, in which the intercept of the equivalent competition plot will be given by ( Kil - Kd) / KB,,which explains the linear behavior of the plot in Fig. 2B at high ligand concentrations. By using these equations for very low and very high ligand concentrations direct estimates can therefore be made for both the high- and the low-affinity dissociation constant of the inhibitor ligand. In contrast to the conditions tested in Fig. 2, however, a conversion of such experimental data into a Scatchard plot can be achieved only in an indirect way, using the estimated Ki values obtained according to the above approach. DISCUSSION
In the present study theoretical equations have been derived for the binding competition of a labeled ligand by a nonhomologous unlabeled ligand, which are also
valid if the free concentration of ligand molecules differs considerably from the ligand concentration added. By comparing the displacement curve of the native unlabeled ligand with that of the nonhomologous inhibitor ligand, simple equations that on the basis of an equivalent competition principle, allow direct determination of the dissooiation constant of the inhibitor ligand are obtained. In its most compact form (Eq. [lOB ] ) it has been derived that the ratio of the dissociation constants of the inhibitor and substrate ligand is given by the difference in concentration of the two ligands added, to give the same extent of binding competition, divided by the free concentration of unlabeled substrate ligand under these conditions. This basic equation has been worked out to Eq. [lOA], with which the independence of the ratio of the dissociation constants from the ligand concentrations can be tested in a linear plot. Alternatively a method has been described following Eqs. [ 1OC ] and [ lOD] , with which the ratio of the dissociation constants can be derived graphically from displacement curves, irrespective of prior determination of the binding characteristics of the substrate ligand. These two parallel approaches have also been worked out for the presence of an additional nonspecific binding component, resulting in Eqs. [15A] and [ 15B]. In comparison with other methods (3-6) the present equations are relatively simple, while in addition experimental data can readily be converted into a linear form,
EQUIVALENT
COMPETITION
ANALYSIS
designated the equivalent competition plot (Figs. 1B and 2B). Not only can this plot be used to determine the dissociation constant of the inhibitor ligand directly, but is also very sensitive for detecting the presence of additional binding processes, such as multiple receptor classes of either the substrate or the inhibitor ligand (Fig. 2B). Under these conditions Eq. [19] is generally applicable at low ligand concentrations, which, combined with data at high ligand concentrations, gives a good estimate of the dissociation constants involved. Also if one ligand binds to a single receptor class and the other to multiple classes, Eq. [19] can still be used, assuming an infinite value for the low-affinity dissociation constant of the ligand interacting with a single receptor class. In this way complex competition patterns, for example, in the case of competition of different polypeptide growth factors for binding to multiple receptor classes ( lo), can be analyzed successfully.
OF
RECEPTOR
BINDING
399
REFERENCES 1. Scatchard,
2. Cheng,
G. (1949) Y., and Prusoff,
Ann. N.Y. Acad. Sci. 61,660-672. W. M. (1973) Biochem. Pharmacol. 22,
3099-3108.
3. Chang, K. J., Jacobs, S., and Cuatrecasas, P. (1975) Biochim. Biophys. Aeta 406,294-303. 4. Hollemans, H. J. G., and Bertina, R. M. ( 1975) Clin. Chem. 21, 1769-1773.
5. Horovitz, A., and Levitzki, A. ( 1987) Proc. Nat&. Acad. Sci. USA 84,6654-6658. 6. Martin, R. L., Renosto, F., and Segel, I. H. ( 1991) Arch. Biochem. Biophys. 284,26-29. 7. Mendel, C. M., and Mendel, D. B. (1985) Biochem. J. 228,269272. 8. Van Zoelen, E. J. J. (1989) Biochem. J. 262,549-556. 9. Almagor, H., and Levitzki, A. (1990) Proc. Nati. Acad. Sci. USA 87,6482-6486. 10. Van Zoelen, E. J. J. (1990) Prog. Growth Fact. Res. 2, 131-152.