- Email: [email protected]
Multidimensional analysis of ligand binding data: Application to opioid receptors
Neuropeptides
3: 367 - 377, 1983
MULTIDIMENSIONAL ANALYSIS OF LIGAND BINDING DATA: APPLICATION TO OPIOID RECEPTORS Richard B. Rothman', Ronald W. Barrett2 and Jeffry L. Vaught' 'Clinical Neuroscience Branch , Section on Brain Bjochemistry, Bld 10, Rm 3N256, NIMH, Bethesda, Maryland 20205 and Dept. of Pharmacology and Toxicology, College of Pharmacy, Rutgers University, Piscataway, N.J. 08854. [Reprints requests to RBRI. The existence of distinct mu and delta opioid receptors is now well accepted. Most investigators favor the hypothesis that these receptors are physically distinct and that the enkephalins are only 2-10 fold selective for the delta receptor. Rothman and Westfall (Mol. Pharmacol. 21:548-557) recently challenged this hypothesis, proposing that at least some population of mu and delta receptors coexist in an opioid receptor complex and that the enkephalins are at least 100 fold selective for the delta receptor. In this paper we describe a generally applicable method we have used to design and analyze ligand binding experiments which distinguish between the two different models. The opiate binding field is remarkable for the diversity of opinion regarding the interaction of various opioids with putative subclasses of opioid receptors (l-10). Given that most investigators use similar methodology, it seems likely that this diversity mainly reflects differences in the interpretation of data. In our laboratories, we have attempted to closely analyze the binding of various opioids utilizing computerized curvefitting of experimental binding data as well as simulations of various binding models. In these studies, we have often encountered the problem of model redundancy: i.e., two binding models describe equally well the same set of data. In this paper, we describe the approach used in our laboratories to design and analyze ligand binding experiments. We believe this approach may be a generally applicable methodology for distinguishing between apparently redundant binding models. In its simplest form, the binding of a ligand to its receptor at equilibrium can be described by the function: 367
B = F(L) In this case, the amount bound (B) is defined as a function of the ligand concentration (L). A major goal of binding studies is to determine the functional form of F. For example, is L binding to one or two classes of binding sites? Clearly, the more information available about the binding of L, the easier it will be to determine the form of F. Thus, the measurement of binding at a single concentration of L provides considerably less information about F than examining the binding of L over a broad range of ligand concentrations. Likewise, F can be more precisely defined by the inclusion of additional independent variables. To fully define F requires that it be "mapped" as a function of its independent variables. When one measures the binding of L at various concentrations [Eq. 11, F maps into 2-dimensional space , with one axis being ligand concentration and the other being the observed amount of binding. However, when F is defined as a function of two independent variables, B=
F(L,I)
r21
where I is an inhibitor, the function F maps into 3-dimensional space. The three axes are ligand concentration, inhibitor concentration and observed binding. As defined by Eq.[21, the function F maps a surface of which Eq.[ll (equivilent to I=01 is but a 2-dimensional slice. Similarly, experiments conducted with a fixed concentration of I or L while varying the other generate slices of the 3-dimensional surface defined by Eq.[21, and thus provide much more information. Further, with F defined as a function of 3 independent variables: B=
F(L,I,J)
[31
where J is a second inhibitor, F maps into I-dimensional space. Eq.131 defines a series of surfaces of which Eq.[21 is but one (J=O), and thus contains considerably more information about F. The above formulation is particularly useful regarding model redundency. Two models which generate the same data when examined as a function of one independent variable may generate different surfaces when examined as a function of two independent variables and be distinguished on that basis. Alternatively, two models which may look identical as a function of two independent variables may be distinguished from each other by the inclusion of an additional variable as defined by Eq.131. As alluded to previously, several models have been proposed to explain the interaction of opioid ligands with mu and delta receptors (3-10). Most investigators favor the hypothesis that: 1. mu and delta receptors are physically distinct; 2. a two-site competitive model describes the interactions of various opioids
368
with these receptors; 3. the enkephalins are only 2-10 fold selective for the delta receptor (6,8). On the other hand, Rothman and Westfall (l-4) recently proposed that: 1. some proportion of these receptors coexist in an opioid receptor complex; 2. mu and delta receptor ligands can interact in a noncompetitive manner: 3. the enkephalins are at least 100 fold selective for the delta receptor. Using computer simulations, we examine in this paper these two models according to the principles outlined above and demonstrate that they can be distinguished with properly designed and analyzed experiments. In doing so, we hope to provide a theoretical basis for studies already published (l-51, and for several manuscripts in preparation. Additionally, we believe that the principles we describe may be generally applicable to other receptor systems where distinguishing between two binding models is neccessary.
Computer simulations, data analysis and graphics were done with an Apple II computer. Curve fitting was done with the nonlinear least squares curve fitting program "CURFIT" (11) modified to fit a function of multiple independent variables. The equation which describes the two-site competitive binding model (eq [41): (100-A) x L AX L B = ~~~~~_c~--~~-~~-~~~--~~--- + _______________-____-----~ (1 + I/KIP + J/K L+K L+K a (1 + I/KIa + J/K ) ) JP Ja was derived in reference 2 and has been modified to reflect the presence of two inhibitors (12). The equation describing the allosteric model proposed and derived by Rothman and Westfall (2) (eq. f51):
1)I IU
J I ------------------- + ---------_--------I + KIFi(1 + J/KJII ) J+K Jo (1 + I/K 100 x
X
__________________-_------
L
L t Ka (1 + I/KIa + J/KJ,) we have similarly modified to r,eflectthe presence of 2 inhibitors. In this case, the term "Vmax" is set to 0.5 indicating that 50% of the delta-sites can be non-competitively inhibited. The binding equations within this term represent the fractional saturation of the mu receptor by both I and J.
NEUR.
C
369
To accurately define a surface of a function F (with J=O) requires that the amount of binding of L be measured at all possible concentractions of L and I. In practice, we approximate this surface by conducting two types of experiments. First, we measure the binding of varied concentrations of L at different fixed concentrations of I. We also conduct the converse experiments: i.e., measuring the binding of different fixed concentrations of L at various concentrations of I. In this manner, we gain information about F over the full range of L and I, which is more then either type of experiment could provide alone. In order to simulate a surface generated by the allosteric equation [51, we have fixed Ka = 3.0 nM, KI,,= 10 nM and Kia = 500 nM. Scatchard plots (13) of the binding of L in the a sence and presence of varying concentrations of I (all with J-0) are shown in Figure 1 as solid lines. Figure 2 shows the displacement of fixed concentrations of L by varying concentrations of I (all with J=O). Data has been expressed as percent inhibition of the binding in the absence of I to allow for easy visual comparison of curves. Together, these data approximate the surface defined by the allosteric equation [51 with J=O.
In order to demonstrate a redundant binding model, we analyzed the 3 nM displacement curve shown in Figure 2 according to the two site competitive model 141 with J=O. An almost perfect fit was obtained with the parameters K, = Ka = 3 nM, A = 50.5, KI,,= 5 nM and KIa = 500 nM. Figure 3 graphically illustrates how well the two binding models can generate the same data. The reason that we illustrated the possiblity of model redundancy when analysing a single curve is that this approach is commonly used in the laboratory. That is, the only data obtained is a Scatchard plot in the absence of inhibitor and a displacement curve using only a single concentration of L. Such a set of data comprises only a small part of the surface defined by a binding function. When this is the only data obtained, the chances of model redundancy are greatly increased. In principle, these two models might be distinguished if data describing the entire surface were obtained. To examine this point, we used the apparently redundant two-site competitive model (Table 1) to generate Scatchard plots and displacement curves similar to those generated for the allosteric model in Figures 1 and 2. These simulations are shown in Figure 4 and as dashed lines in Figure 1. When all of the curves generated by the two-site model are compared to those generated by the allosteric equation, a qualitative and quantitative difference is clearly apparent. For example, the allosteric model predicts linear Scatchard plots with a decrease in the apparent Bmax value while the two site model predicts curvilinear Scatchard plots 370
BOUND
FIGURE 1; Scatchard plots of the binding of L in the presence of 0, 10, 100 and 500 nM I according to the Allosteric (solid lines) and the Two-Site (dashed lines) Models. Binding parameters are given in Table 1.
4 -I
0
I
Lo6* (IN&
4
5
6
JXGURE Ir Displacement curves of I vs. .3, 3 and 30 nM L according to the Allosteric Model. Binding parameters are given units are in nM. in Table 1. Concentration
371
FIGURE h Displacement curves of I vs. 3 nM L according to the Allosteric Model ( Cl 1 and the Two-site Model ( X 1. Binding parameters are given in Table 1. Concentration units are in nM.
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-I
0
I
2 LOG
,
,
,
)
3 (INH)
4
5
6
FIGURE 4; Displacement curves of I vs. .3, 3 and 30 nM L according to the Two-site Model. Binding parameters are given in Table 1. Concentration units are in nM.
372
with no change in the apparent Bmax value. In addition, the allosteric model predicts displacement curves that overlap at low inhibitor concentrations while the two-site competitive model predicts that the entire curve shift to the right as the concentration of labeled ligand is increased. However, these differences occur in regions of the curves where accurate experimental data is difficult to obtain: i.e., the highest and lowest concentrations of I and L.
In the previous section we showed that, in theory, two models which generate identical data in 2-dimensional space (a fixed concentration of L and varied concentrations of I), can be distinguished by examining the entire binding surface. In this section we consider the hypothesis that such models may be shown to differ more dramatically by mapping their behavior when they are defined as a function of three independent variables, which maps into 4-dimensional space. In addition, it is possible that the set of parameters determined by analyzing a 3-D surface defined by experimental data may not be unique. There may exist other sets of parameter values which will fit this data equally well using the same model. We feel that the ability of the set of parameters to predict the behavior of the function in 4-D space validates not only the model but also the accuracy of the parameter values. From the practical viewpoint, evaluating a function in 4dimensional space requires an immense amount of data. Using equations [41 and 151 as examples , one would have to generate a series of Scatchard plots and displacement curves similar to those in Figures. 1-3 using all possible combinations of L, I, and J. However, such an exhaustive approach is not neccessary to distinguish between apparently redundant binding models. A simpler method is to first examine the surface defined by L and I, followed by an examination of of the surface defined by L and J. Each set of data is then analyzed according to each of the apparently redundent models. The correct model should then predict the behavior of F when both I and J are present. In practice, the concentrations of L, I and J are selected so as to maximize the difference in the curves predicted by the two models, so that they can be most easily distinguished. To illustrate this approach, we used the allosteric equation 153 to generate data for the interaction of J and L. In doing so we set I = 0, Ka = 3 nM, KJ,,= 1 nM and KJa = 100 nM. The predicted displacement curve with L = 3 nM was then analyzed according to the two-site model (Eq. 141). An almost perfect fit was obtained with A = 50.5, KJ~ = 0.5 nM, KJa = 100 nM and Ka = K = 3 nM. Using the parameters of the two-site and allosteric models summarized in Table 1, we simulated the displacement of L by J in the presence of fixed concentrations of I and chose practical 373
Ligand L I J
KU
Ka
-10 1
3 500 100
Kp
3 5 .5
%Mu
Ka
3 500 100
50.5 50.5 50.5
%Delta 49.5 49.5 49.5
The parameters used to generate the graphs in Figures l-5 according to the Allosteric (eq. [41) and Two-Site Competitive Models (eq. 151) are summarized.
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8070-
FIGURE 2 Displacement curves of J vs. 16 nM L in the presence of 0 and 50 nM I according to the Allosteric (solid lines) and Two-Site (dashed lines) Models. Binding parameters are given in Table 1. The ICSO's in the presence of 0 nM I and 50 nM I predicted by the Allosteric Model are 21.7 nM and 267 nM respectfully, while those predicted by the Two-Site Model are 39.4 nM and 159 nM respectfully. The Allosteric Model therefore predicts a 12.3 fold shift in the IC50 while the Two-Site Model predicts a 4.0 fold shift.
374
experimental conditions which would maximize the differences in Figure 5 illustrates the the curves predicted by the two models. displacement of 16 nM L by J in the absence and presence of 50 nM I according to the allosteric (solid lines) and two-site (dashed lines) models. Based upon our experience with opioid receptor analysis, these curves are different enough to be distinguished.
The question of how one tests for statistical significance between predicted and observed curves will be explored in the Discussion.
A major goal of ligand binding studies is understanding the interaction of ligands with the receptor of interest. Quantitative examination of binding data requires that the equation which relates the concentration of free ligand to the concentration of bound ligand be determined. To quantitate the binding parameters according to a competitive model, a common experimental approach is to examine a Scatchard plot of the radiolabeled ligand along with a single displacement curve. These data are just a small portion of the surface defined by a binding function of two independent variables. As shown in this paper, knowledge of such a small portion of the surface cannot distinguish between the two-site competitive model and the allosteric model recently proposed by Rothman and Westfall (l-4). In principle, we have shown that it is possible to correctly choose the appropriate binding model if one can accurately measure the amount of radiolabeled ligand bound over the full concentration range of L and I: i.e.,the entire 3-D surface. However, we demonstrated an alternate and perhaps experimentally simpler method of ruling out an inappropriate model. In this approach, one tests the the ability of a proposed model to predict data when a second inhibitor is considered. In an abstract sense, the correct model should fit the binding data when it is a function of two independent variables, as well as when binding is examined as a function of three independent variables. More simply stated, if a model can explain the interaction of I with L and J with L, then it should also predict the binding of L in the simultaneous presence of I and J. Thus, we showed that two binding models which gave exactly the same single displacement curves for I and J could easily be distinguished when the binding of L was measured in the presence of a fixed concentration I and varied concentrations of J. The approach we have taken raises two important issues when applied to experimental data. Most importantly, given a model and a set of "best fit" parameters which do not predict the behavior of the binding function in 4-dimensional space, does one reject the model or reject the parameter estimates. It is theoretically possible that there are other sets of parameters which give equally good fits and that these may lead to correct predictions in 4-D space.
375
The possibility of redundancy in the estimates of binding parameters is a complex problem. The probability of the curvefitting program converging upon the correct parameters is greatly increased when an entire surface rather than a small part of the surface is considered. Even then, we feel that the question of whether to reject the model or reject the parameters is unanswerable. The question in essence queries the basic assumption underlaying our approach to the ligand binding technique: that quantitative analysis of binding data will lead to meaningful conclusions. In the final analysis, any conclusion reached on the basis of ligand binding studies must be confirmed using different techniques. The second issue which arises in the experimental application of the methods outlined in this paper is how to test for statistical significance between observed and predicted curves. In this paper, we have not addressed this point directly since this is a computer simulation study. In the laboratory, given a model and its parameters , we generate a predicted curve using the same ligand concentrations used to generate the observed curve. We then compare the two curves by looking at indices which reflect properties of the entire curve. For example, with predicted and observed displacement curves, we compare predicted and observed ICSO's and Hill coefficients. Each of these parameters has an associated standard deviation, thus statistical significance can be assessed. However, in the final analysis, judgement and acumen are as important as statistical tests. In summary, we have illustrated in this paper the techniques used in our laboratories (l-4) to distinguish between two proposed models for mu and delta opioid receptors. We feel that these techniques are not restricted to these models and will prove to be generally applicable to other receptor systems.
This work was supported by a Pharmaceutical Manufacturers Association Research Starter Grant and a grant from McNeil Pharmaceuticals. R.W.B. is supported by a PMA Foundation Advanced Predoctoral Fellowship. R.B.R. is supported by the Pharmacology Research Associate Training Program of the NIGMS.
1.
Rothman, R.B. and Westfall, T.C. (1981). Allosteric modulation by leucine enkephalin of 3H-Naloxone binding in rat brain. Eur. J. Pharmacol. 12: 365-368.
2.
Rothman, R.B. and Westfall, T.C. (1982). Morphine allosterically modulates the binding of 3H-leucine enkephalin to a particulate fraction of rat brain. Mol. Pharmacol. a: 538-547.
3.
Rothman, R.B. and Westfall, T.C. (1982). Allosteric coupling 376
between morphine and enkephalin receptors in uitro. Mol. Pharmacol. 21: 548-557. 4.
Rothman, R.B. and Westfall, T.C. (1982). Interaction of naloxone with the opioid receptor complex in vitro. Neurochem. Res. 2: 1375-1384.
5.
Vaught, J.L., Rothman, R.B. and Westfall, T.C. (1982). Mu and delta receptors: Their role in analgesia and in the differential effects of opioid peptides on analgesia. Life Sci. 3-Q: 1443-1455.
6.
Chang, K.J. and Cautrecacas, P. (1979). Multiple opiate receptors: Enkephalins and morphine bind to receptors of different specificities. J. Biol. Chem. U: 2610-2618.
7.
Goodman, R.R., Snyder, S.H., Kuhar, M.J. and Young, W.S. (1980). Differention of delta and mu opiate receptors localization by light microscopic autoradiography. Proc. Natl. Acad. Sci. U.S.A. ljl: 6239-6243.
8.
Kosterlitz, H.W., Paterson, S.S. and Robson, L.E. (1981). Characterization of the K-subtype of the opiate receptor in the guinea pig brain. Br. J. Pharmacol. 13: 939-949.
9.
Zhang, A.Z. and Pasternak, G.W. (1981). Opiates and enkephalins: a common binding site mediates their analgesic actions in rats. Life Sci. Z&843-851.
10. Bowen, W.D., Gentleman, S., Herkenham, M. and Pert, C.B. (1981). Interconverting p and a forms of the opiate receptor in rat striatal patches. Proc. Natl. Acad. Sci. U.S.A. a: 4818-4822. 11. Bevington, P.R. (1969). pntn fnr
Lhe
Ehybsal
Sciences.
m
McGraw
*
UldErrorAnalvsis
Hill, Inc., New York.
12. Feldman, H.A. (1972). Mathematical theory of complex ligandbinding systems at equilibrium: some methods of parameter fitting. Anal. Biochem. 48: 317-338. 13. Scatchard, G. (1949). The attractions of proteins for small molecules and ions. Ann. N.Y. Acad. Sci, a: 660-672. Accepted
04.01.83
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3: 367 - 377, 1983
MULTIDIMENSIONAL ANALYSIS OF LIGAND BINDING DATA: APPLICATION TO OPIOID RECEPTORS Richard B. Rothman', Ronald W. Barrett2 and Jeffry L. Vaught' 'Clinical Neuroscience Branch , Section on Brain Bjochemistry, Bld 10, Rm 3N256, NIMH, Bethesda, Maryland 20205 and Dept. of Pharmacology and Toxicology, College of Pharmacy, Rutgers University, Piscataway, N.J. 08854. [Reprints requests to RBRI. The existence of distinct mu and delta opioid receptors is now well accepted. Most investigators favor the hypothesis that these receptors are physically distinct and that the enkephalins are only 2-10 fold selective for the delta receptor. Rothman and Westfall (Mol. Pharmacol. 21:548-557) recently challenged this hypothesis, proposing that at least some population of mu and delta receptors coexist in an opioid receptor complex and that the enkephalins are at least 100 fold selective for the delta receptor. In this paper we describe a generally applicable method we have used to design and analyze ligand binding experiments which distinguish between the two different models. The opiate binding field is remarkable for the diversity of opinion regarding the interaction of various opioids with putative subclasses of opioid receptors (l-10). Given that most investigators use similar methodology, it seems likely that this diversity mainly reflects differences in the interpretation of data. In our laboratories, we have attempted to closely analyze the binding of various opioids utilizing computerized curvefitting of experimental binding data as well as simulations of various binding models. In these studies, we have often encountered the problem of model redundancy: i.e., two binding models describe equally well the same set of data. In this paper, we describe the approach used in our laboratories to design and analyze ligand binding experiments. We believe this approach may be a generally applicable methodology for distinguishing between apparently redundant binding models. In its simplest form, the binding of a ligand to its receptor at equilibrium can be described by the function: 367
B = F(L) In this case, the amount bound (B) is defined as a function of the ligand concentration (L). A major goal of binding studies is to determine the functional form of F. For example, is L binding to one or two classes of binding sites? Clearly, the more information available about the binding of L, the easier it will be to determine the form of F. Thus, the measurement of binding at a single concentration of L provides considerably less information about F than examining the binding of L over a broad range of ligand concentrations. Likewise, F can be more precisely defined by the inclusion of additional independent variables. To fully define F requires that it be "mapped" as a function of its independent variables. When one measures the binding of L at various concentrations [Eq. 11, F maps into 2-dimensional space , with one axis being ligand concentration and the other being the observed amount of binding. However, when F is defined as a function of two independent variables, B=
F(L,I)
r21
where I is an inhibitor, the function F maps into 3-dimensional space. The three axes are ligand concentration, inhibitor concentration and observed binding. As defined by Eq.[21, the function F maps a surface of which Eq.[ll (equivilent to I=01 is but a 2-dimensional slice. Similarly, experiments conducted with a fixed concentration of I or L while varying the other generate slices of the 3-dimensional surface defined by Eq.[21, and thus provide much more information. Further, with F defined as a function of 3 independent variables: B=
F(L,I,J)
[31
where J is a second inhibitor, F maps into I-dimensional space. Eq.131 defines a series of surfaces of which Eq.[21 is but one (J=O), and thus contains considerably more information about F. The above formulation is particularly useful regarding model redundency. Two models which generate the same data when examined as a function of one independent variable may generate different surfaces when examined as a function of two independent variables and be distinguished on that basis. Alternatively, two models which may look identical as a function of two independent variables may be distinguished from each other by the inclusion of an additional variable as defined by Eq.131. As alluded to previously, several models have been proposed to explain the interaction of opioid ligands with mu and delta receptors (3-10). Most investigators favor the hypothesis that: 1. mu and delta receptors are physically distinct; 2. a two-site competitive model describes the interactions of various opioids
368
with these receptors; 3. the enkephalins are only 2-10 fold selective for the delta receptor (6,8). On the other hand, Rothman and Westfall (l-4) recently proposed that: 1. some proportion of these receptors coexist in an opioid receptor complex; 2. mu and delta receptor ligands can interact in a noncompetitive manner: 3. the enkephalins are at least 100 fold selective for the delta receptor. Using computer simulations, we examine in this paper these two models according to the principles outlined above and demonstrate that they can be distinguished with properly designed and analyzed experiments. In doing so, we hope to provide a theoretical basis for studies already published (l-51, and for several manuscripts in preparation. Additionally, we believe that the principles we describe may be generally applicable to other receptor systems where distinguishing between two binding models is neccessary.
Computer simulations, data analysis and graphics were done with an Apple II computer. Curve fitting was done with the nonlinear least squares curve fitting program "CURFIT" (11) modified to fit a function of multiple independent variables. The equation which describes the two-site competitive binding model (eq [41): (100-A) x L AX L B = ~~~~~_c~--~~-~~-~~~--~~--- + _______________-____-----~ (1 + I/KIP + J/K L+K L+K a (1 + I/KIa + J/K ) ) JP Ja was derived in reference 2 and has been modified to reflect the presence of two inhibitors (12). The equation describing the allosteric model proposed and derived by Rothman and Westfall (2) (eq. f51):
1)I IU
J I ------------------- + ---------_--------I + KIFi(1 + J/KJII ) J+K Jo (1 + I/K 100 x
X
__________________-_------
L
L t Ka (1 + I/KIa + J/KJ,) we have similarly modified to r,eflectthe presence of 2 inhibitors. In this case, the term "Vmax" is set to 0.5 indicating that 50% of the delta-sites can be non-competitively inhibited. The binding equations within this term represent the fractional saturation of the mu receptor by both I and J.
NEUR.
C
369
To accurately define a surface of a function F (with J=O) requires that the amount of binding of L be measured at all possible concentractions of L and I. In practice, we approximate this surface by conducting two types of experiments. First, we measure the binding of varied concentrations of L at different fixed concentrations of I. We also conduct the converse experiments: i.e., measuring the binding of different fixed concentrations of L at various concentrations of I. In this manner, we gain information about F over the full range of L and I, which is more then either type of experiment could provide alone. In order to simulate a surface generated by the allosteric equation [51, we have fixed Ka = 3.0 nM, KI,,= 10 nM and Kia = 500 nM. Scatchard plots (13) of the binding of L in the a sence and presence of varying concentrations of I (all with J-0) are shown in Figure 1 as solid lines. Figure 2 shows the displacement of fixed concentrations of L by varying concentrations of I (all with J=O). Data has been expressed as percent inhibition of the binding in the absence of I to allow for easy visual comparison of curves. Together, these data approximate the surface defined by the allosteric equation [51 with J=O.
In order to demonstrate a redundant binding model, we analyzed the 3 nM displacement curve shown in Figure 2 according to the two site competitive model 141 with J=O. An almost perfect fit was obtained with the parameters K, = Ka = 3 nM, A = 50.5, KI,,= 5 nM and KIa = 500 nM. Figure 3 graphically illustrates how well the two binding models can generate the same data. The reason that we illustrated the possiblity of model redundancy when analysing a single curve is that this approach is commonly used in the laboratory. That is, the only data obtained is a Scatchard plot in the absence of inhibitor and a displacement curve using only a single concentration of L. Such a set of data comprises only a small part of the surface defined by a binding function. When this is the only data obtained, the chances of model redundancy are greatly increased. In principle, these two models might be distinguished if data describing the entire surface were obtained. To examine this point, we used the apparently redundant two-site competitive model (Table 1) to generate Scatchard plots and displacement curves similar to those generated for the allosteric model in Figures 1 and 2. These simulations are shown in Figure 4 and as dashed lines in Figure 1. When all of the curves generated by the two-site model are compared to those generated by the allosteric equation, a qualitative and quantitative difference is clearly apparent. For example, the allosteric model predicts linear Scatchard plots with a decrease in the apparent Bmax value while the two site model predicts curvilinear Scatchard plots 370
BOUND
FIGURE 1; Scatchard plots of the binding of L in the presence of 0, 10, 100 and 500 nM I according to the Allosteric (solid lines) and the Two-Site (dashed lines) Models. Binding parameters are given in Table 1.
4 -I
0
I
Lo6* (IN&
4
5
6
JXGURE Ir Displacement curves of I vs. .3, 3 and 30 nM L according to the Allosteric Model. Binding parameters are given units are in nM. in Table 1. Concentration
371
FIGURE h Displacement curves of I vs. 3 nM L according to the Allosteric Model ( Cl 1 and the Two-site Model ( X 1. Binding parameters are given in Table 1. Concentration units are in nM.
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7060~ 50, 40,
iiy/
-I
0
I
2 LOG
,
,
,
)
3 (INH)
4
5
6
FIGURE 4; Displacement curves of I vs. .3, 3 and 30 nM L according to the Two-site Model. Binding parameters are given in Table 1. Concentration units are in nM.
372
with no change in the apparent Bmax value. In addition, the allosteric model predicts displacement curves that overlap at low inhibitor concentrations while the two-site competitive model predicts that the entire curve shift to the right as the concentration of labeled ligand is increased. However, these differences occur in regions of the curves where accurate experimental data is difficult to obtain: i.e., the highest and lowest concentrations of I and L.
In the previous section we showed that, in theory, two models which generate identical data in 2-dimensional space (a fixed concentration of L and varied concentrations of I), can be distinguished by examining the entire binding surface. In this section we consider the hypothesis that such models may be shown to differ more dramatically by mapping their behavior when they are defined as a function of three independent variables, which maps into 4-dimensional space. In addition, it is possible that the set of parameters determined by analyzing a 3-D surface defined by experimental data may not be unique. There may exist other sets of parameter values which will fit this data equally well using the same model. We feel that the ability of the set of parameters to predict the behavior of the function in 4-D space validates not only the model but also the accuracy of the parameter values. From the practical viewpoint, evaluating a function in 4dimensional space requires an immense amount of data. Using equations [41 and 151 as examples , one would have to generate a series of Scatchard plots and displacement curves similar to those in Figures. 1-3 using all possible combinations of L, I, and J. However, such an exhaustive approach is not neccessary to distinguish between apparently redundant binding models. A simpler method is to first examine the surface defined by L and I, followed by an examination of of the surface defined by L and J. Each set of data is then analyzed according to each of the apparently redundent models. The correct model should then predict the behavior of F when both I and J are present. In practice, the concentrations of L, I and J are selected so as to maximize the difference in the curves predicted by the two models, so that they can be most easily distinguished. To illustrate this approach, we used the allosteric equation 153 to generate data for the interaction of J and L. In doing so we set I = 0, Ka = 3 nM, KJ,,= 1 nM and KJa = 100 nM. The predicted displacement curve with L = 3 nM was then analyzed according to the two-site model (Eq. 141). An almost perfect fit was obtained with A = 50.5, KJ~ = 0.5 nM, KJa = 100 nM and Ka = K = 3 nM. Using the parameters of the two-site and allosteric models summarized in Table 1, we simulated the displacement of L by J in the presence of fixed concentrations of I and chose practical 373
Ligand L I J
KU
Ka
-10 1
3 500 100
Kp
3 5 .5
%Mu
Ka
3 500 100
50.5 50.5 50.5
%Delta 49.5 49.5 49.5
The parameters used to generate the graphs in Figures l-5 according to the Allosteric (eq. [41) and Two-Site Competitive Models (eq. 151) are summarized.
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8070-
FIGURE 2 Displacement curves of J vs. 16 nM L in the presence of 0 and 50 nM I according to the Allosteric (solid lines) and Two-Site (dashed lines) Models. Binding parameters are given in Table 1. The ICSO's in the presence of 0 nM I and 50 nM I predicted by the Allosteric Model are 21.7 nM and 267 nM respectfully, while those predicted by the Two-Site Model are 39.4 nM and 159 nM respectfully. The Allosteric Model therefore predicts a 12.3 fold shift in the IC50 while the Two-Site Model predicts a 4.0 fold shift.
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experimental conditions which would maximize the differences in Figure 5 illustrates the the curves predicted by the two models. displacement of 16 nM L by J in the absence and presence of 50 nM I according to the allosteric (solid lines) and two-site (dashed lines) models. Based upon our experience with opioid receptor analysis, these curves are different enough to be distinguished.
The question of how one tests for statistical significance between predicted and observed curves will be explored in the Discussion.
A major goal of ligand binding studies is understanding the interaction of ligands with the receptor of interest. Quantitative examination of binding data requires that the equation which relates the concentration of free ligand to the concentration of bound ligand be determined. To quantitate the binding parameters according to a competitive model, a common experimental approach is to examine a Scatchard plot of the radiolabeled ligand along with a single displacement curve. These data are just a small portion of the surface defined by a binding function of two independent variables. As shown in this paper, knowledge of such a small portion of the surface cannot distinguish between the two-site competitive model and the allosteric model recently proposed by Rothman and Westfall (l-4). In principle, we have shown that it is possible to correctly choose the appropriate binding model if one can accurately measure the amount of radiolabeled ligand bound over the full concentration range of L and I: i.e.,the entire 3-D surface. However, we demonstrated an alternate and perhaps experimentally simpler method of ruling out an inappropriate model. In this approach, one tests the the ability of a proposed model to predict data when a second inhibitor is considered. In an abstract sense, the correct model should fit the binding data when it is a function of two independent variables, as well as when binding is examined as a function of three independent variables. More simply stated, if a model can explain the interaction of I with L and J with L, then it should also predict the binding of L in the simultaneous presence of I and J. Thus, we showed that two binding models which gave exactly the same single displacement curves for I and J could easily be distinguished when the binding of L was measured in the presence of a fixed concentration I and varied concentrations of J. The approach we have taken raises two important issues when applied to experimental data. Most importantly, given a model and a set of "best fit" parameters which do not predict the behavior of the binding function in 4-dimensional space, does one reject the model or reject the parameter estimates. It is theoretically possible that there are other sets of parameters which give equally good fits and that these may lead to correct predictions in 4-D space.
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The possibility of redundancy in the estimates of binding parameters is a complex problem. The probability of the curvefitting program converging upon the correct parameters is greatly increased when an entire surface rather than a small part of the surface is considered. Even then, we feel that the question of whether to reject the model or reject the parameters is unanswerable. The question in essence queries the basic assumption underlaying our approach to the ligand binding technique: that quantitative analysis of binding data will lead to meaningful conclusions. In the final analysis, any conclusion reached on the basis of ligand binding studies must be confirmed using different techniques. The second issue which arises in the experimental application of the methods outlined in this paper is how to test for statistical significance between observed and predicted curves. In this paper, we have not addressed this point directly since this is a computer simulation study. In the laboratory, given a model and its parameters , we generate a predicted curve using the same ligand concentrations used to generate the observed curve. We then compare the two curves by looking at indices which reflect properties of the entire curve. For example, with predicted and observed displacement curves, we compare predicted and observed ICSO's and Hill coefficients. Each of these parameters has an associated standard deviation, thus statistical significance can be assessed. However, in the final analysis, judgement and acumen are as important as statistical tests. In summary, we have illustrated in this paper the techniques used in our laboratories (l-4) to distinguish between two proposed models for mu and delta opioid receptors. We feel that these techniques are not restricted to these models and will prove to be generally applicable to other receptor systems.
This work was supported by a Pharmaceutical Manufacturers Association Research Starter Grant and a grant from McNeil Pharmaceuticals. R.W.B. is supported by a PMA Foundation Advanced Predoctoral Fellowship. R.B.R. is supported by the Pharmacology Research Associate Training Program of the NIGMS.
1.
Rothman, R.B. and Westfall, T.C. (1981). Allosteric modulation by leucine enkephalin of 3H-Naloxone binding in rat brain. Eur. J. Pharmacol. 12: 365-368.
2.
Rothman, R.B. and Westfall, T.C. (1982). Morphine allosterically modulates the binding of 3H-leucine enkephalin to a particulate fraction of rat brain. Mol. Pharmacol. a: 538-547.
3.
Rothman, R.B. and Westfall, T.C. (1982). Allosteric coupling 376
between morphine and enkephalin receptors in uitro. Mol. Pharmacol. 21: 548-557. 4.
Rothman, R.B. and Westfall, T.C. (1982). Interaction of naloxone with the opioid receptor complex in vitro. Neurochem. Res. 2: 1375-1384.
5.
Vaught, J.L., Rothman, R.B. and Westfall, T.C. (1982). Mu and delta receptors: Their role in analgesia and in the differential effects of opioid peptides on analgesia. Life Sci. 3-Q: 1443-1455.
6.
Chang, K.J. and Cautrecacas, P. (1979). Multiple opiate receptors: Enkephalins and morphine bind to receptors of different specificities. J. Biol. Chem. U: 2610-2618.
7.
Goodman, R.R., Snyder, S.H., Kuhar, M.J. and Young, W.S. (1980). Differention of delta and mu opiate receptors localization by light microscopic autoradiography. Proc. Natl. Acad. Sci. U.S.A. ljl: 6239-6243.
8.
Kosterlitz, H.W., Paterson, S.S. and Robson, L.E. (1981). Characterization of the K-subtype of the opiate receptor in the guinea pig brain. Br. J. Pharmacol. 13: 939-949.
9.
Zhang, A.Z. and Pasternak, G.W. (1981). Opiates and enkephalins: a common binding site mediates their analgesic actions in rats. Life Sci. Z&843-851.
10. Bowen, W.D., Gentleman, S., Herkenham, M. and Pert, C.B. (1981). Interconverting p and a forms of the opiate receptor in rat striatal patches. Proc. Natl. Acad. Sci. U.S.A. a: 4818-4822. 11. Bevington, P.R. (1969). pntn fnr
Lhe
Ehybsal
Sciences.
m
McGraw
*
UldErrorAnalvsis
Hill, Inc., New York.
12. Feldman, H.A. (1972). Mathematical theory of complex ligandbinding systems at equilibrium: some methods of parameter fitting. Anal. Biochem. 48: 317-338. 13. Scatchard, G. (1949). The attractions of proteins for small molecules and ions. Ann. N.Y. Acad. Sci, a: 660-672. Accepted
04.01.83
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