Jean-Paul BLONDEAU, Pierre ROCHER and Paul ROBEL

ER 125 Cnrs - Unit& de Recherches sur le MBtabolisme Mol&zulaire et la Physio-Patholoqie des Steroides (U 33) de 1'Institut National de la Sante et de la Recherche MBdicale, Hapital de Bicbtre, 78 avenue du G&n&al Leclerc, 94270, Bicbtre, FRANCE. Received7-17-78 ABSTRACT The relative competition ratio (RCR) is widely used to express the relative affinities of inhibitor(s) and agonist for a binding protein. The RCR is not a constant ; it depends on the concentrations of binding sites and of radioactive hormone, and on the presence of nonsaturable binding component(s). According to the assay conditions used, equating the RCR value to the ratio Ka/Ki of the equilibrium association constants of agonist and inhibitor can lead to large errors. In the case of homogeneous non-interacting binding sites, simple correction factors permit one to calculate the ratio Ka/Ki from the measured RCR value. Calculations are given for the eventual contribution of nonsaturable binding components. Corrections can be unnecessary under well defined experimental conditions , where the bound fraction of hormone in absence of competitor is reduced by using a large dilution of binding protein and/or an increased concentration of radioactive hormone. INTRODUCTION The theoretical basis of enzyme-substrate competitive inhibition kinetics is well established, as are the resulting plots of Lineweaver and Burk, Eadie, or Dixon f1,2,3). The latter allows one to calculate the inhibition constants and to define the competitive or noncompetitive type of inhiVotie 32, Nun&r 5





bition. Simple transposition of these plots to the quantification of steroid-protein interactions

(such as plasma bin-

ding proteins, steroid receptors, or steroid antibodies used for steroid radioimmunoassays), has been widely used (4,5). However, the conditions which allow the use of the equations established for these enzyme kinetics are not met in the field of steroid-protein interactions, since the total ligand and inhibitor concentrations cannot be equated to the concentrations of unbound ligand and inhibitor (6). Hence it has been demonstrated that the classical plots are no longer linear (7). The respective affinities of agonists and competitive inhibitors for a binding protein are best expressed by the direct determination of their equilibrium affinity constants Ka and Ki. The direct measurement of Ki is often impossible, either because the radioactive inhibitor is not available, or because Ki is too small for a direct measurement. For these reasons, and also for reasons of convenience

(such as a

limited amount of biological preparation), empirical modes of representation are widely used ; they give qualitative information concerning the respective affinities of antagonists compared to the affinity of the agonist, by investigating the inhibition of the binding of the radioactive agonist by the unlabeled

inhibitor(s). Several procedures have been

used : -(l)- The measurement of the decreased binding of radioactive agonist by single concentrations of unlabeled competitors. When these concentrations are well chosen, the inhibitors can be classified in order to decreasing affinities,

but their equilibrium association constants cannot be measured. -(2)- The measurement of the concentration(s) of inhibitor(s) which decrease(s) by half the binding of a given concentration of radioactive agonist. It is quite unjustifiable to equate this concentration of inhibitor to the inverse of Ki. -(3)- The measurement of the concentrations of unlabeled inhibitor and agonist which reduce by half the binding of a given concentration of radioactive agonist. The ratio of these concentrations has been called the relative 50 % competition ratio (RCR 50), or relative potency (8,9) or relative binding affinity (10). The parameters which relate the RCR 50 to the ratio Ka/Ki will be analysed and introduced in a correction formula which allows one to calculate the ratio Ka/Ki, from the value of RCR, without previous knowledge of Ka. From the analysis of the above parameters, experimental conditions, under which the ratio Ka/Ki can be approximated by the RCR 50, are deduced. The influence of nonspecific binding components

(i.e. of low affinity and nonsaturable under

the experimental conditions used) is examined. Synthetic data have been generated to illustrate the usefulness of the correction formula proposed. Theoretical aspects The following nomenclature, taken from a preceding publication (II), will be used : Ka = agonist equilibrium association constant, Ki = inhibitor equilibrium association constant, Bo = concentration of bound radioactive agonist in the absence of a competitor, Ba = concentration of bound radioactive agonist in the presence of an unlabeled agonist, Bi = concentration of bound

radioactive agonist in the presence of an unlabeled inhibitor (when Ba = Bi, both are called B), B'a = concentration of bound unlabeled agonist, B'i = concentration of bound unlabeled inhibitor, U'a = concentration of unbound unlabeled agonist, U'i = concentration of unbound unlabeled inhibitor, T = total concentration of radioactive agonist in the absence or presence of a competitor, Ta = total concentration of unlabeled agonist, Ti = total concentration of unlabeled inhibitor, N = total concentration of binding sites, kna = association constant times the number of agonist binding sites for low affinity nonsaturable components, kni = association constant times the number of inhibitor binding sites for low affinity nonsaturable components.. A simple model is used fox the theoretical approach : the binding protein is assumed to have homogeneous noninteracting binding sites ; the inhibition is competitive, i.e. the inhibitor(s) and the agonist compete for the same site ; the measurements are made at binding equilibrium, which means that only Ka and Ki monitor the interaction between the binding protein, the agonist, and the inhibitor, once theis respective concentrations are defined (this absolute prerequisite may be difficult to fulfiJ when the association rate constants of agonist and inhibitor are very different). Rodbaxd and Lewafd have described a relationship between Xi./Ka and Ta/Ti (8). = $$ f13. A simifar equation, derived from Ekin's previous fOrmulatiOn, has been proposed by Koxenman (10). Equation 111 can be rewritten E


T _ B


Relationship between the RCR and the ratio of association constants of agonist and inhibitor The above equation shows the relationship between the ratio of association constants and the ratio of the concentrations of inhibitor and of agonist which produce the same concentration of bound radioactive agonist, i.e. the RCR, hence : Ka (RCR)T - B C3f. The value of Ka/Ki calculated from the Ki= T-B (RCR) will be called the "corrected RCR". If Bo is the concentration of bound radioactive agonist in the absence of competitor, and if the concentrations Ta and Ti are chosen so that the bound radioactive agonist is decreased by 50 %, B will be equal to Bo/2. The following relationship is obtained between the ratio of association 50)T - 0.5 Bo f4f. constants and RCR 50 : g = (RCR T - 0.5 Eo




There is no theoretical reason for selecting RCR 50 for the calculation of Ka/Ki. Its practical interest lies in the fact that, due to the characteristics of semi-logarithmic competition binding plots, the precision of the measurement of the Ti/T.a ratio is better in the Bo/2 region of the curve than at the extremities. Equation [31 can be rewritten in order to focus on the difference (A) between the ratio of association constants and the RCR : xKa - (RCR) = A = ! (2 - 1) [51. Contribution of nonspecific binding Ti-BF Xa Equation [2] can be rearranged 's=Ta_BTa' T The expression Ta/T is characteristic of isotopic dilution and therefore BTa/T =B'a. From the equations representative of partial equilibrium reactions for the radioactive agonist in absence and in presence of agonist and inhibitor, and from conservation reactions, which allow to derive equations 111 and 121, it is easy to show that Ba = Bi = B implies that B'a = B'i. Therefore equation 121 becomes : Ka Ti - B'i U'i n = Ta - B'a = ~'a. When low affinity large capacity binding components are present together with the specific protein, the concentration of unbound and specifically bound radioactive agonist can be calculated with the use of correction factors (11). These factors make use of the product kna (association constant times the number of binding sites of the agonist for the low affinity nonsaturable component(s) and of the product kni, similarly defined for the inhibitor. In the presence of nonsaturable binding components the conservation equations have to be rewritten : Ta = B'a + U'a (kna + 1) ; Ti = B'i. + U'i (kni + 1). These equations mean that part of the binding is not saturable, and that this nonsaturable binding is proportional to the unbound ligand concentration, according to the rela- B'i tionship BNS = kn U (11). Therefore U'i = Tikni + 1 and U'a = 'Eni F'f. This shows that, in the presence of nonKa saturable binding, the relationship between hi and s Ta becoTiT_B Ka mes pi = Ta T-B . EtF 1 t 161. In this equation, B is the corrected saturable binding, according to the formula : B = (Bl - B2) T T _ B2, where Bl is the uncorrected value of Ba or Bi, whereas B is the binding observed in the presence of a very large con?entration of unlabeled agonist in the same competition experiment (11). It can be seen that when kna 2 kni, no correction for nonsaturable binding is necessary.

The measurement of kni requires the availability of a sadioactive inhibitor, or of another sufficiently sensitive method of measurement. For example, equilibrium dialysis in the presence of a large concentration of inhibitor can be used. However, if it is possible to use very dilute solutions of the protein mixture, so that the nonspecific binding will be greatly reduced, then the ratio kna + l/kni + 1 will approach 1. When the affinity of the inhibitor for the specific protein is small, it can be impossible to measure its binding to nonspecific components in conditions where the specific binding is negligible, with the use of inhibitor alone. A practical approach is to add enough agonist to prevent the binding of inhibitor to the specific component. In such conditions, even a tracer amount of inhibitor can be used for the measurement of kni. Conditions under which the ratio Xa/Ki can be approximated by the RCR From the equation ES], d = $ (2

- l), it follows that

Ka the RCR will approximate the ratio ~i whenever d is small. This will be the case when the association constants of the agonist and of the inhibitor are similar. A is also small when the

ratio g is small. This condition is met when :

-(l)- Ta and Ti are large. But in this case, the preeision of the measurement of $$ is poor, since B is measured at the lower end of the competition curvesI where the slopes are almost horizontal

(as shown on Fig. 2). -(2)- T is large

and/or the number of binding sites N is smallr i.e. a diluted solution of the binding protein is used. Examples of


retical curves are given in Fig. la, showing the influence of the above mentioned parameters on the deviation of the Ka. RCR 50 from the theoretical value of pi

b /
















I 5






40 I


N 8 I 1 15 Ka/Ki



15 Ka/Ki


: Dependence of RCR 50 upon T, N, Ka/Ki, kna and kni. It is expressed as the percent deviation (D) of the RCR 50 from the theoretical value of Ka/Ki. left : kna = kni = 0. --D as a function of Ka/Ki ; Ka = 1, N = 5, T = 1. D as a function of the concentration of binding sites N ; Ka = 1, Ki = 0.1, T = 1. -*-•- D as a function of the total concentration of radioactive agonist T ; Ka = 1, Ki = 0.1, N = 5. zh;,; f ssia_fytion of Ka/Ki ; Ka = 1, N = 5, T = 1.

--- kna = 1 ; kni = 2 ---*- kna = 1 ; kni = 0.5. Dependence of RCR 50 upon nonspecific binding In the case of steroid hormone receptor binding studies, crude preparations containing large amounts of nonsaturable low affinity binding components are often used. These components interfere with the determination of RCR. An example is given in Fig. lb, where the percent deviation of RCR 50 is plotted vs the variation of KafKi, in absence or in presence of varied nonspecific binding of agonist and inhibitor.

Comparison of corrected and uncorrected RCR 50 - Method : _--___ A Monte-Carlo evaluation of the correction factors was made by a method similar to the one described by Atkins et al. (IZ). The following binding constants were expressed in arbitrary units : Ka = 1, Ri = 0.1, N = 5, kna = kni = 0. These constants were used to construct 4 theoretical displacement curves, both for agonist and inhibktor. Each of them corresponded to a given value of T : 0.1, 5, 10 and 50, from which 4 values of h were calculated by application of equation f5fl The values of BafBo and Bi/Bo vasied between I and 0, whereas Ta and Ti varied from 0.1 to 10 . Normally distributed random errors of known means and standard deviations were introduced in theoretical B values, and were used to generate series of synthetic displacement curves (13). Two types of errors were used : - (11 - Normally distributed errors of standard deviation proportional to B. The theoretical values of B were multiplied by random numbers of mean = 1, and of standard deviation s1 = 0.01, 52 = 0.02, or s3 = 0.03. - 421 - Normally distributed errors of constant standard deviation : random numbers of mean = 0, and of standard devias2 and s , chosen so that the standard deviation is tions s the sam&*for the v il rious values of d, were added to the theoretical values of B. Fifty synthetic displacement curves for both agonist and inhibitor were constructed for each type of error, each level. of error and each value of h. For each curve, the RCR 50 was corrected according to equation [31. The accuracy (A) of R&R 50 determination was expressed by the percentage deviation of each series mean from the theoretical value of XaJRi. The precision (P) of RCR 50 determinations was expressed as the standard error of the mean of accuracy values. - Result5 : __-___L The RCR 50 is calculated from a plot of B/Be vs total concentrations of agonist and inhibitor (Fig. 2). The best choice of the plot is beyond the scope of the present work. Although with appropriate weighting, or truncation to the central portion of the curve as a simplistic approach to weighting, the logit-log or polynomial regression methods will possibly give smaller standard error of the estimate than linear interpolation, this was found to give very accep-




Log N = 5,


1 10




Fig. 2 : A representative curve for the determination of RCR 50 by linear interpolation. Svnthetic values of Ba and Bi were generated in a system of normally distributed errors of constant standard deviation. The following parameters were used : Ka = 1, Ki = 0.1,

T = 1, Bo = 0.807, s = 0.03. Binding of agonist. Binding of inhibitor.

table results when enough experimental points are located near the 0.5 ordinate. With the two types of errors in B used, the accuracy of corrected RCR 50 values is remarkably constant whatever the value of A, and very close to the theoretical value of Ka/Ki (Fig. 3). Inversely, as could be deduced from equation C'S],the RCR 50 varies in proportion to A. For the largest values of A the uncorrected RCR 50 has a slightly smaller standard error than the corrected RCR 50 However this small loss of precision is largely compensated for by the very marked improvement in accuracy.

CONCLUSION The usual empirical expression of the relative affinities of steroids for a single binding site is the relative competitive ratio (RCR 50). It has been shown already that this value is not only related to the ligand and inhibitor association



q-20 -30 &q


G ~

Fig. 3 : Accuracy and precision of RCR SO and corrected RCR SO. The accuracy (AI and precision IP) were calculated for 50 series of synthetic data , as described in the text, for each value of A. Accuracy is expressed as the percentage deviation from the theoretical value of Ka/Ki. Precision is expressed as the standard error of the mean of accuracy values. a and b : normally distributed errors of constant standard deviation, c and d : normally distributed errors of standard deviations proportional to B ---a--- corrected RCR SO --ouncorrected RCR 50.


and that it depends on the concentrations of bind-

ing sites and of radioactive hormone f8,9,lO]. However, the basic principles underlying the significance of RCR are still. often ignored. Our effoxt was oriented towards a simple explanation of the relationship between the RCR SO and the ratio of association constants. When Ki < Ka, RCR SO underestimates systematically the ratio Ka/Ki. When Ki :,Ka, overestimations of still larger amplitude between RCR SO and Ka/Ki arise. W&never

Ki is very different from Ka, we show

that RCR 50 is an acceptable approximation of Ka/Ki when the fraction bound, Be/T, is small : tne latter condition is met when a dilute solution of binding protein is used, with the limitation that the measurement of B should remain accurate (9). The use of dilute solutions also has the advantage, when crude biological preparations are used, of reducing the influence of “nonspecific“ binding. Another possibility for reducing the fraction bound is to increase T , which has the disadvantage of increasing the eventual contribution of "nonspecific" binding. We show that this type of binding influences significantl.y the values of RCR 50, and that the correction factors must be modified, by including the contribution of nonsaturable binding of agonist and inhibitor. When Be/T, i.e. the fraction of radioactive agonist bound in the absence of competitor, is < 0.05, the difference between RCR and Ka/Ki is < 10 % of Ka/Ki, and therefore RCR is an acceptable approximation of Ka/Ki. When Be/T is 3 0.05, Ka/Ki should be calculated with the use of the correction factors proposed. The Monte Carlo experiments described show that this correction increases considerably the accuracy of determination of Xa/Xi, without a noticeable change in precision. Several theoretical treatments have been proposed for the determination of Xi, whenever its direct measurement is impossible

(7,141. They all require previous knowledge of Xa,

whereas the approach proposed in this work permits a determination of KafXi, even if Xa is not known (9).




The determination of RCR at B/Be = 0.5 has no particular theoretical advantage but offers the practical advantage of greater precision in the determination of Ta and Ti. This means that, if no experimental points are available in the region of the 0.5 ordinate, for example in the case of weak inhibitors, the approximation of Ha/Xi is still feasible, by use of RCR 70 or RCR 80 (2

= 0.7 or 0.81, and application of

the correction formula. It is also possible, using the most convenient parts of the competition curvesI to make several determinations of RCR, each one to be corrected by equation [31, so that a mean value of Ka/Ki can be established. Whenever large variations are observed, the simple model of one category of noninteracting binding sites should be questioned. Acknowledgments We thank H. Mullis, C. Barrier, F. Boussacl A. Atger and S. Bellorini for the preparation of this manuscript.

REFERENCES Lineweaver, H. and Burk, D. J. AMER. CHEM. SOC. 56, 658 (1934). 2. Eadie, C.S. J. BIOL. CHEM., 146, 85 (1942). Dixon, M. BIOCHEM. J. 55, 16ml953f. 43: Scatchard, G. ANN. N.YFACAD. SCI. 51, 660 (1949). 5. Edsall, J.T. and Wyman, J. Biophysical Chemistry, Vol. II pp. 591-662 (1958) Academic Press, New York. 6. Baulieu, E.E. and Raynaud, J.P. PROGR. BIOCHEM. PHARMACOL. 2, 46 (1969). Best-Belpomme‘ M. and Dessen, P. BIOCHIMIE E I‘ 11 (19731. :: Rodbard, D. and Lewald, J.E. ACTA ~N~~~IN~L. 64 suppf. 147, 79 (1970). 9. Rodbard, D. In Receptors for reproductive hormones (O'Malley, l3.W. and Means, A-K., eds) pp. 289-326, (1973) Plenum Press, New York. 10. Korenman, S.G., Tufchinsky, D. and Eaton, L.W. ACTA ENDOCRINOL. fi4, suppl. 147, 291 (1970). 11. Blondeau, Y.P. and Rebel, P. EUR. J. BIOCHEM. 55, 375 (1975). 1.




12. Atkins, G.L. and Nimno, I.A. BIOCHEM. J., 149, 775 (1975). au calm1 13. Pelletier, P. Techniques numdriques appliqxs scientifique, pp. 290-299, Masson et Cie, Paris (1971). 14. Goertz, G., Longchamps, J., CrBpy, P., Judas, 0. and Jayle, M.F. BIOCHIM. BIOPHYS. ACTA 451, 287 (1976).

Competitive inhibition of specific steroid-protein binding: practical use of relative competition ratios for the derivation of equilibrium inhibition constants.

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