- Email: [email protected]
Analysis of competitive agonist-antagonist interactions by nonlinear regression
RINCIPLES
Analvsis of comDetitive agonist-antagonist interactionsbv nonlinear regression J
MJ. Imu, and
JA Angus, Professor. Department of Pharmacology. The Unwrsry
of Melbourne.
Parkv~lle. Victoria 3052, Australia
328
Nonlinear regression analysis
A computer-based exposition of the Schild analysis has been publisheda, but none of the calculations necessarily require a computer. However, the availability of comThe rigorous estimation of a dissociation constant (KJ puters allows the type of analysis to be changed from for antagonists in functional assays has been sought by linear regression, which is solvable exactly, to nonlinear regression where the solution is obtained by iterative pharmacologists using a variety of techniques ever approximations. This change allows the experimenter since the regression method of Arunlakshana and more flexibility in experimental design, and can provide Schild in 1959. Here, Michael lew and James Angus improved accuracy over the Schild method. describe a simplified global regression method with Repeated concentration-response curves cannot be reliably done in systems where the agonist induces improved accuracy compared to Schild analysis. The desensitization. At worst, some assays may only reliably method is suitable for personal computers with give one point of a concentration-response curve (for standard graphing and statistical software. The example see Ref. 4). An analytical approach was devised accuracy of the predicted pl(,values and confidence by Stone and Anguss, based on the method of Waud6, to deal with such systems, but it is equally appropriate for intervals has been tested by comparing examples of data suitable for Schild analysis. This approach, like the published data, and by mathematical (bootstrap) Schild method, is based on the fact that the relative simulations. locations of agonist concentration-response curves is predicted from the Gaddum equation. However, instead of Determination of an antagonist dissociation constant (Kb) using the relative locations (the pair-wise comparison of from functional experiments is essentially a process of control and shifted) to produce a concentration ratio, as data extraction and reduction. The most commonly used in the Schild method, the pK, estimate is adjusted to optimethod, Schild analysis’, involves the reduction of each mize the fit of the model of competitive antagonism to all agonist concentration-response curve to a single location of the concentration-response curves simultaneously; a parameter (usually the pEC,), then the conversion of global comparison of curve locations. This means that all these values to concentration ratios. From each experi- of the concentration-response curves have equal weightmental preparation, N concentration-response curves ing, and no within-tissue control is needed. Further, small yield N-l concentration ratios. These N-l data points (random) leftward shift of agonist concentration-response from all experiments are combined and the pK, estimated curves in the presence of low antagonist concentrations cannot be dealt with in a Schild analysis as they lead to using Eqn 1: logarithms of negative numbers, but they are easily log (concentration ratio -1) = PA,+ nlog[B] (1) accommodated in the new method#. Stone and Angus devised a graphical display of where n = 1 (that is the slope of the Schild plot = l), the the results from this type of analysis, and named it the interaction is taken to be competitive, and PA, = pK,, Gen- Clark plot, because of its similarity to Clark’s plot of erally, where n is not significantly different from 1, the pK, log[agonist] vs log[antagonist]7. Unlike the Schild plot, is obtained with the slope constrained to unity (the cal- the Clark plot is not involved in the determination of the culations can take several forms, depending on the experi- pK, or pA, value. Rather, the Clark plot simply displays the relationship between the experimental curve spacing mental design2). This method is thus based on the relationship between and the predictions from a competitive interaction, along the antagonist concentration and the change of location of with the pK,, value obtained from the analysis. the agonist concentration-response curve, assessed from Despite the advantages of the computer-based metha pair-wise comparison of concentration-response curves. ods, they are frequently not used. This is probably largely A minimum requirement of this analysis is that two con- because they are less widely known and understood than centration-response curves be completed in a single the Schild plot, and some computer programming is tissue to allow a concentration ratio to be calculated, but needed, or at least the use of a sophisticated interactive ideally a family of many concentration-response curves statistical environment. This article describes a modifiwith different concentrations of antagonist is required. cation of earlier methods to make the procedure Clearly, parallel time-controls are necessary to confirm simple with tools commonly used by pharmacologists: that no time-dependent changes in tissue sensitivity to the nonlinear curve fitting with standard graphing and data Michael
Research Fellow.
agonist have taken place. When the interaction between agonist and antagonist is strictly competitive, then the PA, is an estimate of Kb.There are several ways to calculate the PA, or pK, and confidence limits, depending on the experimental design and the form of the data2.
J. Lew and James A. Angus
TIPS - October
1995 (Vol. 16)
0 1995,
Elsevier Science Ltd
PRINCIPLES
analysis software available for all personal computers. An analysis that is probably very similar to that presented here has been previously useds, but was not described in detail. Theory Fractional receptor occupation by an agonist in the presence of antagonist is predicted by the Gaddum equation:
[*I WI pq= Pl+~a(~+Pl% )
where [A] and [Bj are the agonist and antagonist concentrations, K, and Kb are their respective dissociation constants, [AR] is the concentration of occupied receptors and [R,] is the total receptor concentration. Where the agonist and antagonist interact in a simple competitive manner, any given response will occur at the same fractional receptor occupancy so:
WI )=[A’l+~a(l+[B’lJ,)
PI [A]+K$+[B]/&
log,,
(2)
[Al
Fig.1.Concentration-response curves for agonist A in the presence of four different concentrations of antagonist B. A, A’, A”, and A”’ Indicate pEC_%values for agonist A in the presence of B, B’, B”, and B”’ concentrations of antagomst B, respectively. I
1
Equation 8 indicates that (at a fixed response level) any concentration of antagonist will have a single corresponding concentration of agonist. Thus it is possible to write:
[Alec (3)
CBlI-Kb
(9)
where c is a constant. Thus a plot of [A] vs [B] will generate a unique line for each Kty and this line can be the basis of the determination of that Kb. The formula for the line is easily derived:
cross multiply:
CA1 = @I + Kb)c
(10)
(4)
+[A][A’]+[A]Q(l+[B’]/
K,)=
~~I~~‘l+~A’JKa(~+~~I~Kb)
(5)
rearrange, multiply both sides by Kb/ K,
This is a straight line where c is the slope and Kb is the intercept divided by the slope, This estimate of Kb is, in practice, unreliable because it is the ratio of two derived parameters. However, a better estimate of the antagonist affinity can be obtained by using log[A] and performing a nonlinear fit to obtain the pK,. Thus: -log[A] = -log([B] + Kb) - 1og c
(11)
pEC,, = -log([B] t 1O-pKb)- log c
(12)
(6)
[A’1 = [B’]+&
[*I *[B]+&
(7)
Eqn 12 fitted to a plot of agonist pEC, vs antagonist concentration (linear scale) provides the pK, estimate directly as a fitted parameter. The constant -log c is the difference between the antagonist pKb and the agonist control curve PEC50.
This can be extended to any number centrations (for example, B-B”‘):
[*I
_
[Bl+Kb-
of antagonist
[*‘I
= [A”]
[A”‘]
[B’]+&
[B”]+K,
= [B”‘]+&
con-
“‘(8)
It should be noted that [B] can be zero in this formulation, so the control curve is not treated differently to any other curve (Fig. 1).
Calculations Determination of PK, values require quantitation of three aspects of concentration-response curves: maximum, slope and location. These can be obtained in any convenient, unbiased way, but now are most often obtained from a curve-fitting routine. Most computerbased plotting programs include this facility, although they do not always include appropriate curve functions for concentration-response curve fitting, so ‘custom’ or
TIPS - October
1995 (Vol. 16)
329
PRINCIPLES ‘user-defined’ formulae have to be entered. The common sigmoid curve can be generated with several different formulations, but the one used in this model is:
E=a+
1 +
b e-d(c+logLW)
(13)
where E is the response, a is the resting level of response, b is the response range, c is the pEC, d determines both slope and curvature, and e is the base of the natural logarithm. For a competitive interaction, the slopes and maxima of the agonist concentration-response curves should be unaffected by the antagonist, and these are tested as they would be before performing a Schild analysis using an analysis of variance (ANOVA). The methods of Waud6 and of Stone and Angus5 test whether curves are parallel by comparing the goodness-of-fit with the curves constrained to a single slope and with individual slopes for each curve. This method is illustrated with the first data set in this article, where insufficient replicates are available for an ANOVA of slope parameters. Estimation of pK, The location parameters obtained from the concentration-response curve fitting procedure are used to obtain the pK, estimate and confidence interval by nonlinear regression of the plot of pEC,, vs [B] using Eqn 12. Tests for whether the concentration-response curve spacing is consistent with a simple competitive interaction are provided by a comparison of the goodness-of-fit of Eqn 12 with two further equations that allow the concentration-response curve spacings to deviate from the predictions of a simple competitive interaction. First the molecularity of the antagonist-receptor interaction is allowed to vary. This is a ‘power departure’ equivalent to allowing the slope of a Mild plot to vary from unity, and is obtained by allowing the exponent of [B] to vary from unity in the following equation: pEC, = -log( [B]”+ 10-pKb)-log c
(14)
A second type of deviation from simple competitive kinetics was testeds, a’quadratic departure’ equivalent to a nonlinear Schild plot. This departure is allowed in the following equation: pEC,=-log([B](l
+ n]B]/lO-PKb)+ 10-p”}-logc
(15)
The significance of these deviations from competitive kinetics is tested by comparing the goodness-of-fit of Eqn 12 with those of Eqns 14 and 15. Solution of Eqn 12 provides the estimate of pK, directly, but confidence intervals of a fitted parameter in a nonlinear curve fit cannot be obtained explicitly as they can for a linear regression. Rather, they are estimated in most curve-fitting programs from a co-variance matrix in a way
3 3 0
TiPS - October
1995 (Vol. 16)
PRINCIPLES that assumes both that the parameters are normally distributed and are independent of each other. Derivation of parameter variances and their validity is discussed to
Table. Bootstrapping of control and 10 nM methysergide Data
a widely varying degree in the documentation of personal computer graphing and data analysis programmes (for review see Ref. 9). The statistical method of data
data 10 nM methysergide
Control pEC, values
Tissue no.
Tissue no.
Realdata Pseudodatal Pseudodata Pseudodata Pseudodata Pseudodata 5 Pseudodata Pseudodata 7 Pseudodata Pseudodata Pseudodata 10
pEC, values
1
2
3
4
5
6
7
1
2
3
4
5
6
7
6.61 6.72 7.35 6.81 6.71 6.61 6.61 7.36 7.35 6.61 6.81
7.36 6.84 6.61 7.35 6.72 6.71 6.71 6.61 6.84 6.84 6.71
7.35 6.72 6.71 6.71 6.81 6.72 7 35 6.81 6.61 6.61 7.35
6.71 7.36 6.81 7.36 7.36 7.36 6.81 6.71 6.72 7.35 6.61
6.81 7.36 7.36 6.72 7.36 7.35 7.35 7.36 6.71 7.36 6.84
6.72 7.35 7.36 6.81 6.81 7.36 6.61 6.84 6.81 7.35 7.36
6.84 6.81 6.81 6.72 6.71 7.35 6.84 6.71 6.72 6.84 6.84
5.97 6.47 7.10 6.82 6.82 5.97 5.97 6.69 7.10 6.82 7.10
7.25 6.47 6.44 7.10 7.25 7.25 7.25 5.97 6.44 6.47 5.97
7.10 6.82 6.69 6.44 6.47 7.25 6.69 7.25 7.25 6.44 6.82
6.82 6.82 6.44 6.82 7.10 5.97 6.47 7.25 5.97 6.47 6.69
6.44 6.44 7.25 6.82 5.97 6.69 6.82 6.44 6.44 7.10 7.25
6.47 6.69 6.47 6.69 6.82 6.44 7.25 5.97 6.44 5.97 6.47
6.69 6.69 6.47 6.82 6.44 5.97 7.10 7.10 7.25 6.47 6.82
TiPS - October
8.23 8.34 8.38 8.19 8.21 8.29 8.13 8.07 8.20 8.22 8.26
1995 (Vol. 16)
3 3 1
PRINCIPLES bootstrapping (Boxes 1 and 2) provides a way of obtaining real confidence intervals for the parameters in nonlinear curve fittinglo, and can be used to provide confidence interval for the pK, estimate. However, it is shown
that the parameters in Eqn 12 do appear to be normally distributed and are fairly independent of each other, so the confidence interval obtained from the co-variance matrix is likely to be adequate.
Table 1. Regression analysis of the effects of tubocurarine end-plate
depolarization
on succinylcholine-induced
Common slope Tubocurarine (PM) Succinylcholine (pEC,) Slopeparameter Sum of squared deviations Degrees of freedom
0
6.168
0.1 5.858 -5.296 0.418 13-5=8
0.2 5.628
and data for the effects of methysergide
[RI(~1
log (WI+ &,I IogW
0 0 0 0 0 0 0 lo-8
10-e IO-8 It@ IV IO-8 IO-8 3.2x lo-8 3.2x IO-8 3.2x 10-s 3.2x lo-8 3.2x lo-8 3.2x 10-e 3.2x IO" IO-7 IO-7 1o-7 IO-7 IO" 1o-7 10-7 3.2x lo-7 3.2x 10-7 3.2x lo-7 3.2x lo-7 3.2x lo-7 3.2x lo-7 3.2x ID-7 104 104 104 104 104 10"
-8.23 -8.23 -8.23 -8.23 -8.23 -8.23 -8.23 -7.80 -7.80 -7.80 -7.80 -7.80 -7.80 -7.80 -7.43 -7.43 -7.43 -7.43 -7.43 -7.43 -7.43 -6.98 -6.98 -6.98 -6.98 -6.98 -6.98 6.98 -649 -6.49 -6.49 -6.49 -6.49 -6.49 -6.49 -6.00 -6.00 -6.00 -6.00 -6.00 -6.00
-6.61 -7.36 -7.35 -6.71 -6.81 -6.72 -684 -5.97 -7.25 -7.10 -6.82 -6.44 -6.47 -6.69 -6.36 -6.37 -6.72 -6.54 -6.43 -5.98 -6.06 -5.07 -4.95 6.15 -5.88 -6.00 -5.33 -6.01 -4.90 -5.22 -5.70 -5.40 -5.05 -4.84 -5.26 -4.31 4.80 -5.30 -4.57 4.47 -4.47
2
TiPS -October
1995 (Vol. 16)
:.I68 -5.210
(B) on 5-HT (Aj-induced
Average
Calculated
IogW
MAI
-6.92
-6.98 -6.98 -6.98 -6.98 -6.98 -6.98 -6.98 -6.55 -6.55 6.55 -6.55 -6.55 -6.55 6.55 6.18 -6.18 -6.18 -6.18 -6.18 -6.18 -6.18 -5.73 -5.73 -5.73 -5.73 -5.73 -5.73 -5.73 -5.24 -5.24 -5.24 -5.24 -5.24 -5.24 -5.24 -4.75 -4.75 -4.75 -4.75 -4.75 -4.75
-6.67
-6.35
-5.63
-5.19
-4.66
2 X sed; two times the standard error of the difference. Data from Ref.
3 3
Individual slope 0.4 5.431
Table 2. Calculations artery
11.
of the motor
0.1 5.868 -6.093 0.256 13-8=5
contractions
0.2 5.620 -4.864
0.4 5.428 -5.138
in the canine coronary
Difference
-0.37 0.38 0.37 -0.27 -0.17 -0.26 -0.14 -0.59 0.69 0.55 0.27 -0.11 -0.08 0.14 0.18 0.19 0.54 0.37 0.25 -0.19 -0.12 -0.66 -0.78 0.42 0.15 0.27 -0.40 0.28 -0.35 -0.03 0.45 0.16 -0.20 -0.40 0.02 -0.44 0.06 0.56 -0.18 a.28 -0.27
2 x sed
0.23
0.33
0.20
0.38
0.23
0.29
PRINCIPLES Examples Two sets of previously published data are used in this paper to illustrate the method, and to allow comparison of the pK, estimates obtained by earlier methods. First, the experiments by Goldfine that were used both by Waudsr6 and by Stone and Angus5 to illustrate the use of their respective methods of analysis. These data show the effect of tubocurarine on succinylcholine-induced depolarization of the motor end-plate of the guinea-pig lumbrical muscle. Second, data for the interaction between 5-HT and methysergide in the canine coronary artery are taken from a previously published study” with analysis by the method of Stone and Angus. The data are provided in Tables 1 and 2. Succinylcholine and tubocurarine Because the response measured is a depolarization, there is no need to have a fitted estimate of the initial value, so a = 0 in Eqn 13. The concentration-response curves are incomplete, with no upper plateau, but the control curve has a slight inflection that allows the sigmoid curve fit to converge with an upper plateau at 9.323 mV. Because the other curves have too few data points to constrain the logistical fit, a maximum was imposed by sub stituting 9.323 for b in Eqn 13. Curves were fitted with common slopes (i.e. the same d for all curves) and with individual slopes (Fig. 2; Table 1). The goodness-of-fit of the two approaches were compared using the F statistic*z?
6
0 -7
-6.5
I
-5
Fig. 2. Effects of tubocurarine (0. 0 PM; 0, 0.1 PM; W, 0.2 )LM, 0.4 pM) on succinylcholine-induced depolarizations. Data are fitted with logistic curve fits. Solid curves are constrained to a common slope, and dashed curves have individual slopes.
1x10-’
2x10-’ Tubocurarine
(16)
where SS is the sum of squared deviations, df is the degrees of freedom (number of points minus the number of parameters), subscript 1 refers to the simpler model (that is, common slopes) and subscript 2 refers to the more complex model. These data give
-5.5 (lOgloM)
0,
0
E= (=I -S%)/(dfr -df2) SS2 I df2
-6
Succinylcholine
3x10-’
4x10-’
5x10-’
(M)
Fig. 3. Effect of increasing concentrations of tubocurarine on succinylcholine pEC, value. Agonist potency data analysed using nonlinear regression (Eqn 12 in text) to yield the pK, estimate.
s
-5.4
-
-5.6
-
-5.8
-
-6.0
-
9
F3.5
=
(0.418-0.256)/(8-5) 0.25615
=l o5
which does not reach significance (F,,, = 5.41 to give p = 0.05), so it is concluded that the individual slopes do not significantly improve the fit, and the curves can be considered parallel. The pK,, is now estimated using the pEC,, values obtained from the curve fits. Using Eqn 12 and the pEC, values from the common slope fit, pK, = 7.061 + 0.037 was obtained (Fig. 3). Neither Eqns 14 nor 15 significantly changed the curve, with n in Eqn 14 being not different from unity (1.016 + 0.12) and in Eqn 15 being not different from zero (a.001 f 0.046). Therefore, the data are consistent with a simple competitive interaction between succinylcholine and tubocurarine. The results of this analysis are displayed in a Clark plot (Fig. 4). The points on the Clark plot are the mean
Y :: L $ =z g 5 .E :: s
-6.2 -7.2
I
-7.0
I
-6.8
hh
I
-6.6
I
-6.4
I
-6.2
(PI + KJ
Fig. 4. Clark plot displaying the effect of tubocurarine on succinylcholine pEC, value.
logEC,, values from the agonist concentration-response curves at each concentration of antagonist plotted against log([B]+K,) where Kb (lo-7.061)is from the pK, estimate obtained from the regression using Eqn 12. In Fig. 4 the points are the individual logEC,, values
Tip.5 -October
1995 (Vol. 16)
3 3
3
z
120
.: 3 E 8
100
z
80 60
s E:
40
g
20 0 -10
-9
-8
-7
-6
-5
-4
-3
5-HT (log,,, M) Fig. 5. Concentration-response curves to 5HT in the presence of methysergide (0, 0 nM; 0. 10 nM; 0, 32 nM; W, 100 nM; LL 320 nM; A, 1000 nM). The curves are averaged from the sigmoid regressions (Eqn 13 in text) of the individual curves. Data from Ref. 11.
7.5 7.0 f
6.5-
?J 8
6.0-
YQ,
5.5-
% ri,
5.04.5 4.0
! I 0
I 2x10-’
I 6x10-’
I 4x10-’
Methysergide
I 1x10-‘j
I 8x10-’
(M)
Fig. 6. Effect of methysergide on 5-HT pEC, value. Agonist potency data analysed using nonlinear regression (Eqn 12 in text) to yield the pK, estimate. Data from Ref. 11.
-4.5-5.o-5.5z z
-6.O-
s”
-6.5-7.0-7.5 -8.5
I
-8.0
I
-7.5
b-&o
1
-7.0
I
-6.5
I
-6.0
I
-5.5
(PI+&,)
Fig. 7. Clark plot displaying the effects of methysergide (0, 0 nM; 0.10 nM; q, 32 nbt; n , 100 nM; LL 320 nM; A, 1000 nM) on 5-HT pEC, value. Data from Ref. 11.
rather than mean values, because no replicates were performed. The line represents the ideal interaction between the agonist and antagonist (it is not a regression line), and allows an easy visual comparison of
3 3 4
TiPS - October
1995 (Vol. 16)
the concentration-response curve spacings with the ideal for simple competitivity. The affinity of tubocurarine obtained from this analysis is similar to those obtained in earlier studies5r6, although these studies estimated the Kb rather than the pi& The Kb and its estimated standard error is easily obtained with this analysis using Eqn 11 rather than Eqn 12.This gave a I$, = 86.8 f 7.3 no compared to 80.3 + 8.9 IIMand 83.0 f 4.0 IIMobtained by Waud6 and by Stone and Anguss, respectively. There is some preference to use the pK, and its estimated standard error because antagonist affinities scale geometrically and the variance should be symmetrically distributed around the pK, rather than the Kb. 5-HT and methysergide data These experiments were conducted with a design of one concentration-response curve per tissue, and seven replicate tissues per antagonist concentration giving a total of 42 concentration-response curves in the overall data set”. The data are expressed as a percentage of each curve’s own maximum, so no test of maxima is possible here. When the concentration-response curves were fitted using Eqn 13 with II= 0 and b = 100, there was no significant difference in the slope parameters with antagonist concentration (one-way ANOVA). However, the fit of the concentration-response curves was significantly improved by allowing b to not be a fixed value. This gave a predicted maximum of about 120% for each of the curves with no significant differences in either predicted maxima or slope parameters (one-way ANOVAs Fig. 5). One concentration- response curve (1 JLMmethysergide) had to be excluded from the analysis because the curve fitting failed to converge due to the lack of any inflection in the data points. The regression of location versus antagonist concentration (Eqn 12) had a sum of squared deviations (SSD) of 5.25 and gave the pKb = 8.227 f 0.163 (Fig. 6). The fit of the data was not significantly improved using Eqn 14 (SSD = 4.95, F,,, = 2.32) or Eqn 15 (SSD = 5.11, F,,, = 1.07), so it can be concluded that the interaction meets the criteria for competitivity. The results of this analysis (Table 2) are displayed in a Clark plot (Fig. 7). Because in this case there are replicates at each antagonist concentration the points are the mean logEC,, values, and the error bars provide an estimate of the confidence band around the line, obtained as two times the standard error of the differences between the observed logEC,, values and the predicted logEC, values. The results of this analysis of these data are essentially the same as those of the original analysis”. The curves were parallel and a pK, of 8.237 (95% confidence interval 7.90, 8.55) was obtained, similar to 7.89 (7.69,8.27) in the original analysis. The slight difference in pK, is probably due to the fact that in the original analysis used sigmoid curves were constrained to parallelism, whereas here the concentration-response curves were allowed to have independent slopes, although there were no significant differences in slope.
PRINCIPLES pK, variance estimations Motulsky and Ransnas12suggested that the estimated standard error values given by nonlinear regression programs underestimate the real standard errors because the uncertainties of the parameters may not be symmetrical, and significant correlation may exist between the fitted parameters (meaning that an error in one parameter can be partly compensated for by an adjustment to the other). As this is how the confidence intervals for pK,, are obtained by the earlier analyses, bootstrapping simulations were performed*0 on the 5-HT and methysergide data to provide an estimate of the standard deviation. The bootstrapping consisted of randomly resampling the pEC,, values obtained from the concentrationresponse curve fits (that is, the negative of the log[A] values in Table 2). For each concentration of antagonist, seven new pseudodata points (or six when log[B] = -6) are selected randomly from the seven real pEC, values at each antagonist concentration. The sampling is performed with replacement so that it is possible for a single real pEC,, value to be represented in the new pseudodata once, several times, or not at all. This sampling was performed 1000 times to yield 1000 sets of 41 data points. Regression using Eqn 13 was performed on each of these sets to give a sample of 1000 pK, values. The variance of this set of pK, values is considered to be a good estimate of the variance of the pi’¶meter obtained from the original curve fit of the single real data set. This bootstrapping yielded a 95% confidence band for the pK, of (7.96,&X44). The 95% confidence interval obtained from the initial curve fit analysis was (7.90,8.56), estimated using the formula: pKb@
-f0.0.5SE c
PKb
s
PKbpre,j + t0.05SE
where PK+~, is the estimated pK, SE is the estimated standard error, and to,osis the value of the two-tailed f-distribution with the appropriate degrees of freedom (that is, 40). This range encloses all but nine (that is, 99%) of the estimated pKb values from the bootstrapped data, and so appears to be conservative (Fig. 8). The estimated standard error from the nonlinear regression probably performed well because both parameters appear from the bootstrapping to be normally distributed and the correlation between them was not particularly strong (95% confidence interval for the correlation coefficient of 0.554 to 0.634).
Comparison with Schild estimation of pK, Brazenor and Angus constructed Schild plots of their data in addition to the Clark plots for comparison. They used the mean pEC, value for each concentration of antagonist as the data, and calculated a single concentration ratio for each non-zero concentration of antagonist. This allowed construction of a Schild plot and estimation of a pA, and pKty but no confidence interval for the estimates. Kenakin has suggested that such experiments can yield confidence intervals for pKb values
8.8 . 8.6
I--
l
-----
_-----
_ ----------_T
I
95% confidence interval
8.0 -
------_________ 7.8
t I
I
I
-1.5
-1.4
-1.3 -log,,
c ;_*_%.- _ c _ .*. . . I I I l
-1.2
-1.1
_
t
-1.0
c
Fig. 8. Correlation of the curve fit parameters pK, and -log c from 1000 bootstrap re-sampled pseudodata sets. Dotted lines indicate the 95% confidence interval for the p&from the nonlinear curve fit of the original data (from Ref. 11).
obtained from the Schild regression if bootstrapping is usedis. Therefore, analysis of the bootstrapped pseudodata generated above with a Schild regression to yield 1000 pKb estimates was carried out. In 89 of the pseudodata sets the log(concentration ratio -1) at the lowest concentration of antagonist was undefined because the concentration ratio was less than one. These data points were discarded, but a pKb was calculated from the remaining four data points of the Schild regression. The average pK, from this Schild analysis was significantly different from that obtained from the nonlinear regressions analysis above, 8.086 + 0.005. Further, the distribution of the pKbestimates obtained by Schild analysis was significantly wider than those obtained with the nonlinear curve fitting (95% confidence interval of variance = 0.026to 0.030for Schild, and 0.014 to 0.016 for nonlinear regression, Fig. 9). Both of these differences come about through the uneven usage of the pEC, values in the Schild adySiS. The lower average pKbiS the result of the fact that underestimation of the control pEC, changes the final pKb estimate more than overestimation by the same margin (see Box 3). This widens the distribution of the pKbvalues, but this is also the result of the fact that the control pEC, is used in all of the concentration ratios. Thus, noise from the control curve is present in each log (concentration ratio -1) value in the Schild method, whereas it is only in one of the points used to determine the pKbby the nonlinear regression method.
Concluding remarks The nonlinear regression method for pKb determination has several advantages over other methods. First, it does not require the within-tissue control concentration-response curves that are required for the standard Schild regression methods. This allows the experimenter to determine pKbvalues for antagonists in systems where
TiPS - October
1995 (Vol
16)
335
PRINCIPLES a 200 -
150
150-
E: a 100 0
E 0
50
50-
810
812
814
,
8.6
1
8:8
~6
0 7:6
7:8
8:0
8.‘2
8.4
8:6
PK,
Fig. 9. Histograms of pK, obtained from the 100 bootstrap re-sampled pseudodata sets. a: values obtained using Eqn 12; b: values obtained from Schild analysis.
1 4 -1.0
,
I
-0.4
-0.3
I I
I
I
-0.2
-0.1 Error
1 0.0
I
I
I
I
0.1
0.2
0.3
0.4
in control pECsO
Fig. Graph of error in p&obtained by nonlinear regression(O) and Schild (0) analysis of data consisting of ideal pEC,values for an agonist in the presence of a competitive antagonist at 5 concentrations (1, 3. 10, 30 and 100 times the pK,) with the control pEC, systematically changed from the ideal value.
3 3 6
TiPS - October
1995 (Vol. 16)
PRINCIPLES
Table 3. Comparison of the steps in detemdnation
of p&, by global nonlinear regression and by Schild regression
step
Global nonlinear regression
Schild regression
Agonist concentration-response curve data
Test for parallelism and maximum Calculate pEC,s
Test for parallelism and maximum Calculate pEC,s
Fit model of competitive interaction
Plot pEC, vs [B] Apply regression of Eqn 1'2
Calculate concentration ratios Plot log(concentration ratio-l) vs log[B] Apply linear regression
Test for deviation from competitivity
Compare goodness-of-fit 14 and 15 with F-test
Test for deviation from linearity and from slope = 1
Determine pK,
Fitted parameter in Eqn 12
Constrain regression to slope = 1a Determine x intercepta
Determine confidence intervals
pK,+ ttimes the estimated standard error from the nonlinear regression
Confidence interval for x intercepta
Display
Clark plot
Schild plot
of Eqns 12,
Vifferent procedures are applicable for the Schild method depending on the experimental design and the data (see Ref. 2). 6; antagonist. _ the agonist concentration-response curve cannot be repeated within tissue because of desensitization or timedependent changes in either tissue response range or sensitivity. Second, statistical balance, or even weighting, is achieved because each pEC, estimate is treated equally in the estimation of the p&,. Bootstrapping of data to allow Schild analysis without within-tissue controls is possible, but the confidence intervals obtained more easily from the nonlinear regression method appear to be adequate. The advantages of the present nonlinear regression method over the previously published computer-based methods5,6 are primarily those of convenience. Both previous methods required the use of computer programming and simultaneous curve-fitting routines, and have not become widely used. The elimination of simultaneous curve-fitting is obtained at the expense of the distillation of each concentration-response curve into a single location parameter before the model of a competitive interaction is applied. This could be argued to decrease the accuracy of the fit, but it does not appear to seriously alter the pK, estimate obtained. Elimination of the need for simultaneous curve-fitting makes this
computer-based method easily accomplished using the many graphing and data analysis packages for personal computers that allow nonlinear curve-fitting. The method is easy to use (see Table 3), and appears equally as applicable as the !%hild method for estimating antagonist dissociation constants in functional experiments. Selected references 1 Arunlakshana, 0. and Schild, H. 0. (1959)Br. J
Phurmucol. 14,48-58 2 Mackay, D. (1978)J. Pharm. Phurmacol. 30,312-313 3 Waud, D. R. and Parker, R. B.(1971)J.Pharmacol.Exp. Tkr. 177,13-24 4 Angus, J. A., Black, J. W. and Stone, M. (1980)Br. J. Pharmacol. 68, 413-423 5 Stone, M. and Angus, J. A. (1978)J. Phurmacol.Exp. l’her. 207,705-718 6 Waud, D. R. (1975) in Methods in Pharmacology(Daniel, E. E. and Paton, M., eds), pp. 471-506,Plenum 7 Stone, M. (1980)1. Pharm. Pharmucol.32,81-86 8 Trist, D. G. and Leff, P. (1985)Agents Actions 16,22%226 9 Press, W. H. et al. (1988) Numerical Recipes: The Art of Scientific Computing Cambridge University Press 10 Efron, B. and Tibshirani, R. (1986)Stat. Sci. 1,54-77 11 Brazenor, R. M. and Angus, J. A. (1981)1. Phnacol. Exp. ‘Iher. 218, 530-536 12 Motulsky, H. J. and Ransnas, L. A. (1987)FASEB J 1,365-374 13 Kenakin, T. P. (1993) Pharmacologic Analysis of Drug-Receptor Interaction Raven Press
Debate The Debate section is an occasional areas of pharmacology.
feature that allows discussion
of highly topical issues in all
Suggestions for topics for the Debate section are welcomed and may be addressed directly to the Editor, either in writing or by telephone. Priority is given to topics of broad pharmacological interest. Trends in Pharmacological Sciences 68 Hills Road, Cambridge, UK CB2 1LA
TiPS - October
1995 (Vol. 16)
3 3 7
Analvsis of comDetitive agonist-antagonist interactionsbv nonlinear regression J
MJ. Imu, and
JA Angus, Professor. Department of Pharmacology. The Unwrsry
of Melbourne.
Parkv~lle. Victoria 3052, Australia
328
Nonlinear regression analysis
A computer-based exposition of the Schild analysis has been publisheda, but none of the calculations necessarily require a computer. However, the availability of comThe rigorous estimation of a dissociation constant (KJ puters allows the type of analysis to be changed from for antagonists in functional assays has been sought by linear regression, which is solvable exactly, to nonlinear regression where the solution is obtained by iterative pharmacologists using a variety of techniques ever approximations. This change allows the experimenter since the regression method of Arunlakshana and more flexibility in experimental design, and can provide Schild in 1959. Here, Michael lew and James Angus improved accuracy over the Schild method. describe a simplified global regression method with Repeated concentration-response curves cannot be reliably done in systems where the agonist induces improved accuracy compared to Schild analysis. The desensitization. At worst, some assays may only reliably method is suitable for personal computers with give one point of a concentration-response curve (for standard graphing and statistical software. The example see Ref. 4). An analytical approach was devised accuracy of the predicted pl(,values and confidence by Stone and Anguss, based on the method of Waud6, to deal with such systems, but it is equally appropriate for intervals has been tested by comparing examples of data suitable for Schild analysis. This approach, like the published data, and by mathematical (bootstrap) Schild method, is based on the fact that the relative simulations. locations of agonist concentration-response curves is predicted from the Gaddum equation. However, instead of Determination of an antagonist dissociation constant (Kb) using the relative locations (the pair-wise comparison of from functional experiments is essentially a process of control and shifted) to produce a concentration ratio, as data extraction and reduction. The most commonly used in the Schild method, the pK, estimate is adjusted to optimethod, Schild analysis’, involves the reduction of each mize the fit of the model of competitive antagonism to all agonist concentration-response curve to a single location of the concentration-response curves simultaneously; a parameter (usually the pEC,), then the conversion of global comparison of curve locations. This means that all these values to concentration ratios. From each experi- of the concentration-response curves have equal weightmental preparation, N concentration-response curves ing, and no within-tissue control is needed. Further, small yield N-l concentration ratios. These N-l data points (random) leftward shift of agonist concentration-response from all experiments are combined and the pK, estimated curves in the presence of low antagonist concentrations cannot be dealt with in a Schild analysis as they lead to using Eqn 1: logarithms of negative numbers, but they are easily log (concentration ratio -1) = PA,+ nlog[B] (1) accommodated in the new method#. Stone and Angus devised a graphical display of where n = 1 (that is the slope of the Schild plot = l), the the results from this type of analysis, and named it the interaction is taken to be competitive, and PA, = pK,, Gen- Clark plot, because of its similarity to Clark’s plot of erally, where n is not significantly different from 1, the pK, log[agonist] vs log[antagonist]7. Unlike the Schild plot, is obtained with the slope constrained to unity (the cal- the Clark plot is not involved in the determination of the culations can take several forms, depending on the experi- pK, or pA, value. Rather, the Clark plot simply displays the relationship between the experimental curve spacing mental design2). This method is thus based on the relationship between and the predictions from a competitive interaction, along the antagonist concentration and the change of location of with the pK,, value obtained from the analysis. the agonist concentration-response curve, assessed from Despite the advantages of the computer-based metha pair-wise comparison of concentration-response curves. ods, they are frequently not used. This is probably largely A minimum requirement of this analysis is that two con- because they are less widely known and understood than centration-response curves be completed in a single the Schild plot, and some computer programming is tissue to allow a concentration ratio to be calculated, but needed, or at least the use of a sophisticated interactive ideally a family of many concentration-response curves statistical environment. This article describes a modifiwith different concentrations of antagonist is required. cation of earlier methods to make the procedure Clearly, parallel time-controls are necessary to confirm simple with tools commonly used by pharmacologists: that no time-dependent changes in tissue sensitivity to the nonlinear curve fitting with standard graphing and data Michael
Research Fellow.
agonist have taken place. When the interaction between agonist and antagonist is strictly competitive, then the PA, is an estimate of Kb.There are several ways to calculate the PA, or pK, and confidence limits, depending on the experimental design and the form of the data2.
J. Lew and James A. Angus
TIPS - October
1995 (Vol. 16)
0 1995,
Elsevier Science Ltd
PRINCIPLES
analysis software available for all personal computers. An analysis that is probably very similar to that presented here has been previously useds, but was not described in detail. Theory Fractional receptor occupation by an agonist in the presence of antagonist is predicted by the Gaddum equation:
[*I WI pq= Pl+~a(~+Pl% )
where [A] and [Bj are the agonist and antagonist concentrations, K, and Kb are their respective dissociation constants, [AR] is the concentration of occupied receptors and [R,] is the total receptor concentration. Where the agonist and antagonist interact in a simple competitive manner, any given response will occur at the same fractional receptor occupancy so:
WI )=[A’l+~a(l+[B’lJ,)
PI [A]+K$+[B]/&
log,,
(2)
[Al
Fig.1.Concentration-response curves for agonist A in the presence of four different concentrations of antagonist B. A, A’, A”, and A”’ Indicate pEC_%values for agonist A in the presence of B, B’, B”, and B”’ concentrations of antagomst B, respectively. I
1
Equation 8 indicates that (at a fixed response level) any concentration of antagonist will have a single corresponding concentration of agonist. Thus it is possible to write:
[Alec (3)
CBlI-Kb
(9)
where c is a constant. Thus a plot of [A] vs [B] will generate a unique line for each Kty and this line can be the basis of the determination of that Kb. The formula for the line is easily derived:
cross multiply:
CA1 = @I + Kb)c
(10)
(4)
+[A][A’]+[A]Q(l+[B’]/
K,)=
~~I~~‘l+~A’JKa(~+~~I~Kb)
(5)
rearrange, multiply both sides by Kb/ K,
This is a straight line where c is the slope and Kb is the intercept divided by the slope, This estimate of Kb is, in practice, unreliable because it is the ratio of two derived parameters. However, a better estimate of the antagonist affinity can be obtained by using log[A] and performing a nonlinear fit to obtain the pK,. Thus: -log[A] = -log([B] + Kb) - 1og c
(11)
pEC,, = -log([B] t 1O-pKb)- log c
(12)
(6)
[A’1 = [B’]+&
[*I *[B]+&
(7)
Eqn 12 fitted to a plot of agonist pEC, vs antagonist concentration (linear scale) provides the pK, estimate directly as a fitted parameter. The constant -log c is the difference between the antagonist pKb and the agonist control curve PEC50.
This can be extended to any number centrations (for example, B-B”‘):
[*I
_
[Bl+Kb-
of antagonist
[*‘I
= [A”]
[A”‘]
[B’]+&
[B”]+K,
= [B”‘]+&
con-
“‘(8)
It should be noted that [B] can be zero in this formulation, so the control curve is not treated differently to any other curve (Fig. 1).
Calculations Determination of PK, values require quantitation of three aspects of concentration-response curves: maximum, slope and location. These can be obtained in any convenient, unbiased way, but now are most often obtained from a curve-fitting routine. Most computerbased plotting programs include this facility, although they do not always include appropriate curve functions for concentration-response curve fitting, so ‘custom’ or
TIPS - October
1995 (Vol. 16)
329
PRINCIPLES ‘user-defined’ formulae have to be entered. The common sigmoid curve can be generated with several different formulations, but the one used in this model is:
E=a+
1 +
b e-d(c+logLW)
(13)
where E is the response, a is the resting level of response, b is the response range, c is the pEC, d determines both slope and curvature, and e is the base of the natural logarithm. For a competitive interaction, the slopes and maxima of the agonist concentration-response curves should be unaffected by the antagonist, and these are tested as they would be before performing a Schild analysis using an analysis of variance (ANOVA). The methods of Waud6 and of Stone and Angus5 test whether curves are parallel by comparing the goodness-of-fit with the curves constrained to a single slope and with individual slopes for each curve. This method is illustrated with the first data set in this article, where insufficient replicates are available for an ANOVA of slope parameters. Estimation of pK, The location parameters obtained from the concentration-response curve fitting procedure are used to obtain the pK, estimate and confidence interval by nonlinear regression of the plot of pEC,, vs [B] using Eqn 12. Tests for whether the concentration-response curve spacing is consistent with a simple competitive interaction are provided by a comparison of the goodness-of-fit of Eqn 12 with two further equations that allow the concentration-response curve spacings to deviate from the predictions of a simple competitive interaction. First the molecularity of the antagonist-receptor interaction is allowed to vary. This is a ‘power departure’ equivalent to allowing the slope of a Mild plot to vary from unity, and is obtained by allowing the exponent of [B] to vary from unity in the following equation: pEC, = -log( [B]”+ 10-pKb)-log c
(14)
A second type of deviation from simple competitive kinetics was testeds, a’quadratic departure’ equivalent to a nonlinear Schild plot. This departure is allowed in the following equation: pEC,=-log([B](l
+ n]B]/lO-PKb)+ 10-p”}-logc
(15)
The significance of these deviations from competitive kinetics is tested by comparing the goodness-of-fit of Eqn 12 with those of Eqns 14 and 15. Solution of Eqn 12 provides the estimate of pK, directly, but confidence intervals of a fitted parameter in a nonlinear curve fit cannot be obtained explicitly as they can for a linear regression. Rather, they are estimated in most curve-fitting programs from a co-variance matrix in a way
3 3 0
TiPS - October
1995 (Vol. 16)
PRINCIPLES that assumes both that the parameters are normally distributed and are independent of each other. Derivation of parameter variances and their validity is discussed to
Table. Bootstrapping of control and 10 nM methysergide Data
a widely varying degree in the documentation of personal computer graphing and data analysis programmes (for review see Ref. 9). The statistical method of data
data 10 nM methysergide
Control pEC, values
Tissue no.
Tissue no.
Realdata Pseudodatal Pseudodata Pseudodata Pseudodata Pseudodata 5 Pseudodata Pseudodata 7 Pseudodata Pseudodata Pseudodata 10
pEC, values
1
2
3
4
5
6
7
1
2
3
4
5
6
7
6.61 6.72 7.35 6.81 6.71 6.61 6.61 7.36 7.35 6.61 6.81
7.36 6.84 6.61 7.35 6.72 6.71 6.71 6.61 6.84 6.84 6.71
7.35 6.72 6.71 6.71 6.81 6.72 7 35 6.81 6.61 6.61 7.35
6.71 7.36 6.81 7.36 7.36 7.36 6.81 6.71 6.72 7.35 6.61
6.81 7.36 7.36 6.72 7.36 7.35 7.35 7.36 6.71 7.36 6.84
6.72 7.35 7.36 6.81 6.81 7.36 6.61 6.84 6.81 7.35 7.36
6.84 6.81 6.81 6.72 6.71 7.35 6.84 6.71 6.72 6.84 6.84
5.97 6.47 7.10 6.82 6.82 5.97 5.97 6.69 7.10 6.82 7.10
7.25 6.47 6.44 7.10 7.25 7.25 7.25 5.97 6.44 6.47 5.97
7.10 6.82 6.69 6.44 6.47 7.25 6.69 7.25 7.25 6.44 6.82
6.82 6.82 6.44 6.82 7.10 5.97 6.47 7.25 5.97 6.47 6.69
6.44 6.44 7.25 6.82 5.97 6.69 6.82 6.44 6.44 7.10 7.25
6.47 6.69 6.47 6.69 6.82 6.44 7.25 5.97 6.44 5.97 6.47
6.69 6.69 6.47 6.82 6.44 5.97 7.10 7.10 7.25 6.47 6.82
TiPS - October
8.23 8.34 8.38 8.19 8.21 8.29 8.13 8.07 8.20 8.22 8.26
1995 (Vol. 16)
3 3 1
PRINCIPLES bootstrapping (Boxes 1 and 2) provides a way of obtaining real confidence intervals for the parameters in nonlinear curve fittinglo, and can be used to provide confidence interval for the pK, estimate. However, it is shown
that the parameters in Eqn 12 do appear to be normally distributed and are fairly independent of each other, so the confidence interval obtained from the co-variance matrix is likely to be adequate.
Table 1. Regression analysis of the effects of tubocurarine end-plate
depolarization
on succinylcholine-induced
Common slope Tubocurarine (PM) Succinylcholine (pEC,) Slopeparameter Sum of squared deviations Degrees of freedom
0
6.168
0.1 5.858 -5.296 0.418 13-5=8
0.2 5.628
and data for the effects of methysergide
[RI(~1
log (WI+ &,I IogW
0 0 0 0 0 0 0 lo-8
10-e IO-8 It@ IV IO-8 IO-8 3.2x lo-8 3.2x IO-8 3.2x 10-s 3.2x lo-8 3.2x lo-8 3.2x 10-e 3.2x IO" IO-7 IO-7 1o-7 IO-7 IO" 1o-7 10-7 3.2x lo-7 3.2x 10-7 3.2x lo-7 3.2x lo-7 3.2x lo-7 3.2x lo-7 3.2x ID-7 104 104 104 104 104 10"
-8.23 -8.23 -8.23 -8.23 -8.23 -8.23 -8.23 -7.80 -7.80 -7.80 -7.80 -7.80 -7.80 -7.80 -7.43 -7.43 -7.43 -7.43 -7.43 -7.43 -7.43 -6.98 -6.98 -6.98 -6.98 -6.98 -6.98 6.98 -649 -6.49 -6.49 -6.49 -6.49 -6.49 -6.49 -6.00 -6.00 -6.00 -6.00 -6.00 -6.00
-6.61 -7.36 -7.35 -6.71 -6.81 -6.72 -684 -5.97 -7.25 -7.10 -6.82 -6.44 -6.47 -6.69 -6.36 -6.37 -6.72 -6.54 -6.43 -5.98 -6.06 -5.07 -4.95 6.15 -5.88 -6.00 -5.33 -6.01 -4.90 -5.22 -5.70 -5.40 -5.05 -4.84 -5.26 -4.31 4.80 -5.30 -4.57 4.47 -4.47
2
TiPS -October
1995 (Vol. 16)
:.I68 -5.210
(B) on 5-HT (Aj-induced
Average
Calculated
IogW
MAI
-6.92
-6.98 -6.98 -6.98 -6.98 -6.98 -6.98 -6.98 -6.55 -6.55 6.55 -6.55 -6.55 -6.55 6.55 6.18 -6.18 -6.18 -6.18 -6.18 -6.18 -6.18 -5.73 -5.73 -5.73 -5.73 -5.73 -5.73 -5.73 -5.24 -5.24 -5.24 -5.24 -5.24 -5.24 -5.24 -4.75 -4.75 -4.75 -4.75 -4.75 -4.75
-6.67
-6.35
-5.63
-5.19
-4.66
2 X sed; two times the standard error of the difference. Data from Ref.
3 3
Individual slope 0.4 5.431
Table 2. Calculations artery
11.
of the motor
0.1 5.868 -6.093 0.256 13-8=5
contractions
0.2 5.620 -4.864
0.4 5.428 -5.138
in the canine coronary
Difference
-0.37 0.38 0.37 -0.27 -0.17 -0.26 -0.14 -0.59 0.69 0.55 0.27 -0.11 -0.08 0.14 0.18 0.19 0.54 0.37 0.25 -0.19 -0.12 -0.66 -0.78 0.42 0.15 0.27 -0.40 0.28 -0.35 -0.03 0.45 0.16 -0.20 -0.40 0.02 -0.44 0.06 0.56 -0.18 a.28 -0.27
2 x sed
0.23
0.33
0.20
0.38
0.23
0.29
PRINCIPLES Examples Two sets of previously published data are used in this paper to illustrate the method, and to allow comparison of the pK, estimates obtained by earlier methods. First, the experiments by Goldfine that were used both by Waudsr6 and by Stone and Angus5 to illustrate the use of their respective methods of analysis. These data show the effect of tubocurarine on succinylcholine-induced depolarization of the motor end-plate of the guinea-pig lumbrical muscle. Second, data for the interaction between 5-HT and methysergide in the canine coronary artery are taken from a previously published study” with analysis by the method of Stone and Angus. The data are provided in Tables 1 and 2. Succinylcholine and tubocurarine Because the response measured is a depolarization, there is no need to have a fitted estimate of the initial value, so a = 0 in Eqn 13. The concentration-response curves are incomplete, with no upper plateau, but the control curve has a slight inflection that allows the sigmoid curve fit to converge with an upper plateau at 9.323 mV. Because the other curves have too few data points to constrain the logistical fit, a maximum was imposed by sub stituting 9.323 for b in Eqn 13. Curves were fitted with common slopes (i.e. the same d for all curves) and with individual slopes (Fig. 2; Table 1). The goodness-of-fit of the two approaches were compared using the F statistic*z?
6
0 -7
-6.5
I
-5
Fig. 2. Effects of tubocurarine (0. 0 PM; 0, 0.1 PM; W, 0.2 )LM, 0.4 pM) on succinylcholine-induced depolarizations. Data are fitted with logistic curve fits. Solid curves are constrained to a common slope, and dashed curves have individual slopes.
1x10-’
2x10-’ Tubocurarine
(16)
where SS is the sum of squared deviations, df is the degrees of freedom (number of points minus the number of parameters), subscript 1 refers to the simpler model (that is, common slopes) and subscript 2 refers to the more complex model. These data give
-5.5 (lOgloM)
0,
0
E= (=I -S%)/(dfr -df2) SS2 I df2
-6
Succinylcholine
3x10-’
4x10-’
5x10-’
(M)
Fig. 3. Effect of increasing concentrations of tubocurarine on succinylcholine pEC, value. Agonist potency data analysed using nonlinear regression (Eqn 12 in text) to yield the pK, estimate.
s
-5.4
-
-5.6
-
-5.8
-
-6.0
-
9
F3.5
=
(0.418-0.256)/(8-5) 0.25615
=l o5
which does not reach significance (F,,, = 5.41 to give p = 0.05), so it is concluded that the individual slopes do not significantly improve the fit, and the curves can be considered parallel. The pK,, is now estimated using the pEC,, values obtained from the curve fits. Using Eqn 12 and the pEC, values from the common slope fit, pK, = 7.061 + 0.037 was obtained (Fig. 3). Neither Eqns 14 nor 15 significantly changed the curve, with n in Eqn 14 being not different from unity (1.016 + 0.12) and in Eqn 15 being not different from zero (a.001 f 0.046). Therefore, the data are consistent with a simple competitive interaction between succinylcholine and tubocurarine. The results of this analysis are displayed in a Clark plot (Fig. 4). The points on the Clark plot are the mean
Y :: L $ =z g 5 .E :: s
-6.2 -7.2
I
-7.0
I
-6.8
hh
I
-6.6
I
-6.4
I
-6.2
(PI + KJ
Fig. 4. Clark plot displaying the effect of tubocurarine on succinylcholine pEC, value.
logEC,, values from the agonist concentration-response curves at each concentration of antagonist plotted against log([B]+K,) where Kb (lo-7.061)is from the pK, estimate obtained from the regression using Eqn 12. In Fig. 4 the points are the individual logEC,, values
Tip.5 -October
1995 (Vol. 16)
3 3
3
z
120
.: 3 E 8
100
z
80 60
s E:
40
g
20 0 -10
-9
-8
-7
-6
-5
-4
-3
5-HT (log,,, M) Fig. 5. Concentration-response curves to 5HT in the presence of methysergide (0, 0 nM; 0. 10 nM; 0, 32 nM; W, 100 nM; LL 320 nM; A, 1000 nM). The curves are averaged from the sigmoid regressions (Eqn 13 in text) of the individual curves. Data from Ref. 11.
7.5 7.0 f
6.5-
?J 8
6.0-
YQ,
5.5-
% ri,
5.04.5 4.0
! I 0
I 2x10-’
I 6x10-’
I 4x10-’
Methysergide
I 1x10-‘j
I 8x10-’
(M)
Fig. 6. Effect of methysergide on 5-HT pEC, value. Agonist potency data analysed using nonlinear regression (Eqn 12 in text) to yield the pK, estimate. Data from Ref. 11.
-4.5-5.o-5.5z z
-6.O-
s”
-6.5-7.0-7.5 -8.5
I
-8.0
I
-7.5
b-&o
1
-7.0
I
-6.5
I
-6.0
I
-5.5
(PI+&,)
Fig. 7. Clark plot displaying the effects of methysergide (0, 0 nM; 0.10 nM; q, 32 nbt; n , 100 nM; LL 320 nM; A, 1000 nM) on 5-HT pEC, value. Data from Ref. 11.
rather than mean values, because no replicates were performed. The line represents the ideal interaction between the agonist and antagonist (it is not a regression line), and allows an easy visual comparison of
3 3 4
TiPS - October
1995 (Vol. 16)
the concentration-response curve spacings with the ideal for simple competitivity. The affinity of tubocurarine obtained from this analysis is similar to those obtained in earlier studies5r6, although these studies estimated the Kb rather than the pi& The Kb and its estimated standard error is easily obtained with this analysis using Eqn 11 rather than Eqn 12.This gave a I$, = 86.8 f 7.3 no compared to 80.3 + 8.9 IIMand 83.0 f 4.0 IIMobtained by Waud6 and by Stone and Anguss, respectively. There is some preference to use the pK, and its estimated standard error because antagonist affinities scale geometrically and the variance should be symmetrically distributed around the pK, rather than the Kb. 5-HT and methysergide data These experiments were conducted with a design of one concentration-response curve per tissue, and seven replicate tissues per antagonist concentration giving a total of 42 concentration-response curves in the overall data set”. The data are expressed as a percentage of each curve’s own maximum, so no test of maxima is possible here. When the concentration-response curves were fitted using Eqn 13 with II= 0 and b = 100, there was no significant difference in the slope parameters with antagonist concentration (one-way ANOVA). However, the fit of the concentration-response curves was significantly improved by allowing b to not be a fixed value. This gave a predicted maximum of about 120% for each of the curves with no significant differences in either predicted maxima or slope parameters (one-way ANOVAs Fig. 5). One concentration- response curve (1 JLMmethysergide) had to be excluded from the analysis because the curve fitting failed to converge due to the lack of any inflection in the data points. The regression of location versus antagonist concentration (Eqn 12) had a sum of squared deviations (SSD) of 5.25 and gave the pKb = 8.227 f 0.163 (Fig. 6). The fit of the data was not significantly improved using Eqn 14 (SSD = 4.95, F,,, = 2.32) or Eqn 15 (SSD = 5.11, F,,, = 1.07), so it can be concluded that the interaction meets the criteria for competitivity. The results of this analysis (Table 2) are displayed in a Clark plot (Fig. 7). Because in this case there are replicates at each antagonist concentration the points are the mean logEC,, values, and the error bars provide an estimate of the confidence band around the line, obtained as two times the standard error of the differences between the observed logEC,, values and the predicted logEC, values. The results of this analysis of these data are essentially the same as those of the original analysis”. The curves were parallel and a pK, of 8.237 (95% confidence interval 7.90, 8.55) was obtained, similar to 7.89 (7.69,8.27) in the original analysis. The slight difference in pK, is probably due to the fact that in the original analysis used sigmoid curves were constrained to parallelism, whereas here the concentration-response curves were allowed to have independent slopes, although there were no significant differences in slope.
PRINCIPLES pK, variance estimations Motulsky and Ransnas12suggested that the estimated standard error values given by nonlinear regression programs underestimate the real standard errors because the uncertainties of the parameters may not be symmetrical, and significant correlation may exist between the fitted parameters (meaning that an error in one parameter can be partly compensated for by an adjustment to the other). As this is how the confidence intervals for pK,, are obtained by the earlier analyses, bootstrapping simulations were performed*0 on the 5-HT and methysergide data to provide an estimate of the standard deviation. The bootstrapping consisted of randomly resampling the pEC,, values obtained from the concentrationresponse curve fits (that is, the negative of the log[A] values in Table 2). For each concentration of antagonist, seven new pseudodata points (or six when log[B] = -6) are selected randomly from the seven real pEC, values at each antagonist concentration. The sampling is performed with replacement so that it is possible for a single real pEC,, value to be represented in the new pseudodata once, several times, or not at all. This sampling was performed 1000 times to yield 1000 sets of 41 data points. Regression using Eqn 13 was performed on each of these sets to give a sample of 1000 pK, values. The variance of this set of pK, values is considered to be a good estimate of the variance of the pi’¶meter obtained from the original curve fit of the single real data set. This bootstrapping yielded a 95% confidence band for the pK, of (7.96,&X44). The 95% confidence interval obtained from the initial curve fit analysis was (7.90,8.56), estimated using the formula: pKb@
-f0.0.5SE c
PKb
s
PKbpre,j + t0.05SE
where PK+~, is the estimated pK, SE is the estimated standard error, and to,osis the value of the two-tailed f-distribution with the appropriate degrees of freedom (that is, 40). This range encloses all but nine (that is, 99%) of the estimated pKb values from the bootstrapped data, and so appears to be conservative (Fig. 8). The estimated standard error from the nonlinear regression probably performed well because both parameters appear from the bootstrapping to be normally distributed and the correlation between them was not particularly strong (95% confidence interval for the correlation coefficient of 0.554 to 0.634).
Comparison with Schild estimation of pK, Brazenor and Angus constructed Schild plots of their data in addition to the Clark plots for comparison. They used the mean pEC, value for each concentration of antagonist as the data, and calculated a single concentration ratio for each non-zero concentration of antagonist. This allowed construction of a Schild plot and estimation of a pA, and pKty but no confidence interval for the estimates. Kenakin has suggested that such experiments can yield confidence intervals for pKb values
8.8 . 8.6
I--
l
-----
_-----
_ ----------_T
I
95% confidence interval
8.0 -
------_________ 7.8
t I
I
I
-1.5
-1.4
-1.3 -log,,
c ;_*_%.- _ c _ .*. . . I I I l
-1.2
-1.1
_
t
-1.0
c
Fig. 8. Correlation of the curve fit parameters pK, and -log c from 1000 bootstrap re-sampled pseudodata sets. Dotted lines indicate the 95% confidence interval for the p&from the nonlinear curve fit of the original data (from Ref. 11).
obtained from the Schild regression if bootstrapping is usedis. Therefore, analysis of the bootstrapped pseudodata generated above with a Schild regression to yield 1000 pKb estimates was carried out. In 89 of the pseudodata sets the log(concentration ratio -1) at the lowest concentration of antagonist was undefined because the concentration ratio was less than one. These data points were discarded, but a pKb was calculated from the remaining four data points of the Schild regression. The average pK, from this Schild analysis was significantly different from that obtained from the nonlinear regressions analysis above, 8.086 + 0.005. Further, the distribution of the pKbestimates obtained by Schild analysis was significantly wider than those obtained with the nonlinear curve fitting (95% confidence interval of variance = 0.026to 0.030for Schild, and 0.014 to 0.016 for nonlinear regression, Fig. 9). Both of these differences come about through the uneven usage of the pEC, values in the Schild adySiS. The lower average pKbiS the result of the fact that underestimation of the control pEC, changes the final pKb estimate more than overestimation by the same margin (see Box 3). This widens the distribution of the pKbvalues, but this is also the result of the fact that the control pEC, is used in all of the concentration ratios. Thus, noise from the control curve is present in each log (concentration ratio -1) value in the Schild method, whereas it is only in one of the points used to determine the pKbby the nonlinear regression method.
Concluding remarks The nonlinear regression method for pKb determination has several advantages over other methods. First, it does not require the within-tissue control concentration-response curves that are required for the standard Schild regression methods. This allows the experimenter to determine pKbvalues for antagonists in systems where
TiPS - October
1995 (Vol
16)
335
PRINCIPLES a 200 -
150
150-
E: a 100 0
E 0
50
50-
810
812
814
,
8.6
1
8:8
~6
0 7:6
7:8
8:0
8.‘2
8.4
8:6
PK,
Fig. 9. Histograms of pK, obtained from the 100 bootstrap re-sampled pseudodata sets. a: values obtained using Eqn 12; b: values obtained from Schild analysis.
1 4 -1.0
,
I
-0.4
-0.3
I I
I
I
-0.2
-0.1 Error
1 0.0
I
I
I
I
0.1
0.2
0.3
0.4
in control pECsO
Fig. Graph of error in p&obtained by nonlinear regression(O) and Schild (0) analysis of data consisting of ideal pEC,values for an agonist in the presence of a competitive antagonist at 5 concentrations (1, 3. 10, 30 and 100 times the pK,) with the control pEC, systematically changed from the ideal value.
3 3 6
TiPS - October
1995 (Vol. 16)
PRINCIPLES
Table 3. Comparison of the steps in detemdnation
of p&, by global nonlinear regression and by Schild regression
step
Global nonlinear regression
Schild regression
Agonist concentration-response curve data
Test for parallelism and maximum Calculate pEC,s
Test for parallelism and maximum Calculate pEC,s
Fit model of competitive interaction
Plot pEC, vs [B] Apply regression of Eqn 1'2
Calculate concentration ratios Plot log(concentration ratio-l) vs log[B] Apply linear regression
Test for deviation from competitivity
Compare goodness-of-fit 14 and 15 with F-test
Test for deviation from linearity and from slope = 1
Determine pK,
Fitted parameter in Eqn 12
Constrain regression to slope = 1a Determine x intercepta
Determine confidence intervals
pK,+ ttimes the estimated standard error from the nonlinear regression
Confidence interval for x intercepta
Display
Clark plot
Schild plot
of Eqns 12,
Vifferent procedures are applicable for the Schild method depending on the experimental design and the data (see Ref. 2). 6; antagonist. _ the agonist concentration-response curve cannot be repeated within tissue because of desensitization or timedependent changes in either tissue response range or sensitivity. Second, statistical balance, or even weighting, is achieved because each pEC, estimate is treated equally in the estimation of the p&,. Bootstrapping of data to allow Schild analysis without within-tissue controls is possible, but the confidence intervals obtained more easily from the nonlinear regression method appear to be adequate. The advantages of the present nonlinear regression method over the previously published computer-based methods5,6 are primarily those of convenience. Both previous methods required the use of computer programming and simultaneous curve-fitting routines, and have not become widely used. The elimination of simultaneous curve-fitting is obtained at the expense of the distillation of each concentration-response curve into a single location parameter before the model of a competitive interaction is applied. This could be argued to decrease the accuracy of the fit, but it does not appear to seriously alter the pK, estimate obtained. Elimination of the need for simultaneous curve-fitting makes this
computer-based method easily accomplished using the many graphing and data analysis packages for personal computers that allow nonlinear curve-fitting. The method is easy to use (see Table 3), and appears equally as applicable as the !%hild method for estimating antagonist dissociation constants in functional experiments. Selected references 1 Arunlakshana, 0. and Schild, H. 0. (1959)Br. J
Phurmucol. 14,48-58 2 Mackay, D. (1978)J. Pharm. Phurmacol. 30,312-313 3 Waud, D. R. and Parker, R. B.(1971)J.Pharmacol.Exp. Tkr. 177,13-24 4 Angus, J. A., Black, J. W. and Stone, M. (1980)Br. J. Pharmacol. 68, 413-423 5 Stone, M. and Angus, J. A. (1978)J. Phurmacol.Exp. l’her. 207,705-718 6 Waud, D. R. (1975) in Methods in Pharmacology(Daniel, E. E. and Paton, M., eds), pp. 471-506,Plenum 7 Stone, M. (1980)1. Pharm. Pharmucol.32,81-86 8 Trist, D. G. and Leff, P. (1985)Agents Actions 16,22%226 9 Press, W. H. et al. (1988) Numerical Recipes: The Art of Scientific Computing Cambridge University Press 10 Efron, B. and Tibshirani, R. (1986)Stat. Sci. 1,54-77 11 Brazenor, R. M. and Angus, J. A. (1981)1. Phnacol. Exp. ‘Iher. 218, 530-536 12 Motulsky, H. J. and Ransnas, L. A. (1987)FASEB J 1,365-374 13 Kenakin, T. P. (1993) Pharmacologic Analysis of Drug-Receptor Interaction Raven Press
Debate The Debate section is an occasional areas of pharmacology.
feature that allows discussion
of highly topical issues in all
Suggestions for topics for the Debate section are welcomed and may be addressed directly to the Editor, either in writing or by telephone. Priority is given to topics of broad pharmacological interest. Trends in Pharmacological Sciences 68 Hills Road, Cambridge, UK CB2 1LA
TiPS - October
1995 (Vol. 16)
3 3 7