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Modification of an Elisa-Based procedure for affinity determination: correction necessary for use with bivalent antibody
Molecular Immunology, Printed in Great Britain
Vol. 24,
No. 10, pp. 1055-1060,
0161-5890/87 $3.00 + 0.00 Pergamon Journals Ltd
1987
MODIFICATION OF AN ELISA-BASED PROCEDURE FOR AFFINITY DETERMINATION: CORRECTION NECESSARY FOR USE WITH BIVALENT ANTIBODY* FRED J. STEVENS Division of Biological and Medical Research, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (First received 31 December
1986; accepted in revised form 3 April 1987)
Abstract-A
recently described procedure for the evaluation of the affinity of monoclonal antibodies [Friguet et al., J. Immun. Meth. 77, 305-319 (1985)]uses an ELISA system to determine the quantity of free antibody present in a mixture of antigen and antibody. However, an intact IgG may bind antigen by either of two binding sites, and an IgG can bind to a solid-phase antigen whether one or two of its binding sites are free. Therefore, this procedure does not directly provide the concn of liganded binding sites, the quantity necessary for calculation of the thermodynamic association constant. A binomial probability distribution relates the fraction of liganded binding sites to the concn of unliganded, singly liganded, and doubly liganded IgG assuming that the binding of each Fab to antigen is independent. Simulated experiments were used to compare the apparent binding characteristics of bivalent IgG and monovalent Fab and to calculate apparent association constants in each case. It was found that the affinity of binding sites on intact IgG was underestimated by a factor of at least 2 and that the error was inversely related to the fraction of liganded binding sites. Binding site affinity of an antibody may be underestimated by several orders of magnitude. On the basis of binomial analysis, it is possible to convert apparent concns of bound IgG to actual concns of liganded binding site resulting in the calculation of valid association constants for intact IgG without alteration of the experimental protocol.
INTRODUCTION
Friguet et al. (1985) have recently introduced a convenient method for the evaluation of affinity constants that characterize the interactions between antibodies and macromolecular antigens. The approach uses an enzyme-linked immunosorbent assay (ELISA) to quantitate the unbound antibody present in an equilibrated reaction mixture. By using sufficient excess antigen to justify the assumption that the concn of free antigen does not change significantly at equilibrium, the ELISA provides appropriate data to allow the use of Scatchard or Klotz analyses (Scatchard, 1949; Klotz and Hunston, 1971; Klotz, 1985) to evaluate the affinity constant of the antibody. This affinity measurement technique has been used to quantitatively evaluate and compare equilibrium constants expressed by families of monoclonal antibodies for specific antigens (Portnoi et al., 1986; Avrameas, 1986; Ternynck and Avrameas 1986; Elder et al., 1987), to analyze the specificity of an anti-histamine monoclonal antibody for a series of modified antigens (Guesdon et al., 1986), and to characterize the affinity of a monoclonal antibody for a melanoma-associated glycoprotein (Vennegoor and Rumke, 1986). Djavadi-Ohaniance et al. (1986) and
*This work is supported by the U.S. Department of Energy and the Office of Health and Enviromental Research under contract No. W-31-109-Eng-38, and by PHS grant No. ROl A119590, DHHS.
Friguet et al. (1986) measured affinity constants of monoclonal antibodies for six distinct epitopes on two domains of the f12 subunit of tryptophan synthase. Changes in affinity constants occurring with ligand binding by the synthase were observed that could be interpreted as resulting from substrateinduced conformational changes in the enzyme. Harper et al. (1987) recently measured the affinity constants of a series of anti-lysozyme monoclonal antibodies. These investigators noted that the value of the affinity constant obtained for the antigen binding fragment (Fab) from one immunologlobulin (IgG) was approximately twice the value calculated for the antibody binding site in the intact bivalent IgG. The authors suggested two possible explanations for this observation. First, binding of one lysozyme molecule might result in a steric hindrance that decreases the ability of the binding site present on the second Fab to interact with antigen. Second, since the ELISA quantifies free IgG on the basis of its ability to interact with antigen adsorbed on the solid phase, the technique might not be able to distinguish between ligand-free and singly liganded IgG. Therefore, only doubly liganded IgG is scored as bound; as a result, the extent of antigen binding is underestimated. An analogous problem was addressed in a simulation of antibody: antigen interaction occurring during size-exclusion gel chromatography (Stevens, 1986). For the simulated chromatogram, the elution behavior of a mixture of interacting mono- and bivalent components is determined by the particle-
1055
FRED J. STEVENS
1056
size distribution within the column. Therefore, it was necessary to calculate the fractional composition of unliganded, singly liganded, and doubly liganded constituents as a function of concns and aflinity constant. This report describes the use of this approach to determine that the discrepancy observed in the apparent association constants of Fab and intact IgG may directly result from the bivalency of the IgG and that this error may approach several orders of magnitude under appropriate circumstances. A procedure is described that corrects the resulting error and provides agreement in apparent association constants, regardless if they are measured for free Fabs or intact IgGs. The correction procedure also minimizes errors in evaluation of IgG affinity that are dependent upon the affinity and concn of the reagents used. BASES OF ERROR AND CORRECTION
Friguet’s procedure uses an ELISA to estimate the quantity of unliganded antibody in an equilibrium mixture of antibody and antigen to provide the necessary data for analysis by the method of Klotz, 11 11 -=r+iK.; x b 0 a
(1)
or by the method of Scatchard, X - = a
i. I& -
K,x.
c-3
Relationships (1) and (2) are both convenient expressions of the equilibrium condition x = K,(& - x)a,
(3)
in which KS is the association constant, iOis the initial binding site concn, and x and a are the equilibrium concns of complex and antigen, respectively. In the ELISA-based affinity procedure, a sample of the equilibrated antibody-antigen mixture is transferred to antigen-coated wells or tubes, and the quantity of antibody free to bind to the immobilized ligand is subsequently determined by exposure to an appropriate enzyme-linked anti-immunoglobulin antibody. Color development relative to that obtained for a corresponding sample of antigen-free antibody is taken as the fraction of free antibody in the equilibrated mixture. However, since both unliganded and singly liganded IgG bind to antigen adsorbed on the solid phase, the fraction of apparent “free” IgG is not the same as the fractional free binding sites or Fab. The problem then becomes one of converting the apparent degree of IgG binding to the true degree of Fab binding determined by the association constant. Rodbard and Weiss (1973) developed a probabilistic formulation of the effect of antibody bivalence in their quantitative analysis of solid-phase immunoassays, A similar problem was addressed in the chromatography simulation (Stevens, 1986) in which, given the concn of liganded Fab, it was necessary to
determine the concns of unliganded, singly liganded and doubly liganded antigen. In the case in which ELISA is used to assay free antibody in the equilibrium mixture of antigen and IgG, iff = .x/i0 is the fraction of bound Fab, then f can be interpreted as the probability that a randomly selected binding site or Fab is bound and 1 -f as the probability that the site is free. Accordingly, the fraction of the IgG population in each of the three categories is unliganded:
(1 -f)’
singly liganded: 2f(l -f) doubly liganded: f 2,
(4) (5) (6)
on the basis of a standard binomial distribution and assuming that the binding of antigen to an Fab is independent of the bound or free status of the second Fab. Of the three categories, only the doubly liganded antibody does not react with the solid-phase ELISA antigen. Therefore, from equation (6), the fraction of free antibody measured is equivalent to P?l=l-S2
(7)
f=Jr-m
(8)
and
is the corrected fraction of liganded binding sites in terms of the apparent fractional free IgG. The affinity assay fo~ulated by Friguet et al. (1985) conveniently measures “bound” IgG relative to the total IgG present, i.e. A,-Ai I v’=Aop
(9)
where A, is the ELISA absorbance obtained in the absence of antigen and A, is the corresponding absorbance obtained in the presence of antigen. Since the quantity represented by equation (9) is the doubly liganded fraction, equation (6), at any antigen concn the corrected fraction of experimentally observed liganded binding sites is
fF=J;;.
(~0)
RESULTS AND DISCUSSION
Figure 1 illustrates the discrepancy between observed and actual degrees of binding. The relative magnitude of the underestimated binding is most substantial at low degrees of fractional binding, suggesting that high antigen concns and/or high affinities are necessary for the more accurate results although the measured affinity of the IgG cannot be better than one-half the magnitude of the Fab affinity. At high levels of binding, the rate of change of observed binding is twice that of actual binding, accounting for the minimal 2-fold difference in measured affinity for Fab and IgG. Figure 2 shows computer-generated binding curves for IgG and Fab at three different aflinities. Doublereciprocal plots [equation (1); Lineweaver and Burk,
1057
Correction for ELISA affinity analysis
O.O~.,.,,,.,,,,,..,,,,‘,.,,‘I 0.0
0.2
0.4
0.6
0.8
1.0
FRACTION LIGANDED (ESTIMATED)
Fig. 1. Dependence of error on extent of binding. The apparent fractional binding of IgG [equation (6)] and Fab is plotted as a function of the actual fractional liganding of binding sites. The concn of complex was calculated by equation (3); the concn of antigen was not assumed to be constant through the reaction.
1934)] referred to as Klotz plots in Friguet et al. (1985), are used to represent the binding calculated on the basis of equation (3). Figure 2(A) shows linear dependence of (l/x) throughout the antigen concn range when the binding component is the monovalent Fab. Binding curves for IgG at the same molar concn of binding site (1O-9 M) are shown in Fig. 2(B) and are clearly different than those seen for free Fab. The deviation increases with decreased affinity of the binding site. Decreased affinity for a given antigen concn is equivalent to a shift to the left in the plot shown in Fig. 1. The antigen concn dependence of the IgG binding curves is not an artifact of the double-reciprocal formulation. In Fig. 3, the data shown in Fig. 2 are replotted as Scatchard plots [equation (2)]. Whereas the simulated Fab binding curves are linear throughout the antigen concn range, the maximum slope of each IgG binding curve occurs at antigen concns near saturation of the available binding sites. The slope of the tangential line at this segment corresponds to an estimated affinity one-half the value of the Fab intrinsic affinity. However, the x-axis intercept of the tangential line correctly estimates the concn of available binding sites. As the concn of antigen is decreased, the diminishing slope of the binding curve results in progressively underestimated K, and a correspondingly overestimated binding site concn. At sufficiently low antigen concns, an inflection point is reached in the binding curve and both parameters become indeterminant. The substantial curvature of the IgG binding plots shown in Figs 2 and 3 need not be evident in experiments restricted to a relatively narrow range of antigen concn. A simulated experiment is depicted in Fig. 4 for an antibody with binding site affinity of K, = 5.0 x 107M-i, concn= 1.0 x 10m9h4, and a series of antigens at concns between 30 and 600 x lo-* M. Panel (A) compares the Klotz plots obtained if the binding sites are presented as
Fabs (open circles) or as IgGs (closed circles). Data were obtained by a simple computer progam that calculated the concn of bound sites and the distribution [equations (4H6)] of bound sites; uniform noise (10%) was added by a random number process in order to test the correction procedure with experimentally realistic data. In this example, a regression line fitted to the IgG data results in an apparent affinity of 2.4 x lO’M_i. Even though this value is slightly less than one-half the value of the affinity of the binding site, the correlation coefficient of the regression line is 0.9986. Panel (B) represents the plot obtained for “corrected” IgG data. Regression analyses of both the Fab data and the corrected IgG data result in measured affinities of 5.11 x 10’ M-’ and 5.03 x 10’ M-l, respectively. In cases of significant antigen excess, as recommended by Friguet et al. (1985), the fractional liganded binding sites can be expressed f= (xl&) = &a,/(1
+&a,),
(11)
in which a, is the initial free antigen concn and remains essentially unchanged by complex formation. In these circumstances, fis effectively independent of
(4 80
I
0
I’
10
40
,,A’;) x 10”
M)
@I Fig. 2. Double-reciprocal plots [equation (I)] for Fab (A) and IgG (B). Binding site concn was set at 1 x 10e9 M; antigen concn ranged from 600 x IO-’ M to 30 x lo-* M. Binding site Ka values were 2 x 108M -I (-----); 5 x lO’M_’ (----); and 1 x 107M-’ (-). Note that the antigen concn is the actual equilibrium concn; it is not assumed that n = %. For each discrete antigen concn, the complex concn, x, was determined on the basis of equation (3).
1058
FRED J. STEVENS
relative values of the IgG affinities would accurately correspond to those of the Fabs. However, because of the exponential relationship between f* and v [equation (lo)], f * approaches zero faster than v; therefore, the ratio Kapp/Kac, + 0 for small v. Under these circumstances very small differences in the intrinsic binding site affinity can appear as differences of one or more orders of magnitude. Further, if experiments to compare antibodies were conducted with different ranges of antigen concn, then apparent ranking of IgG affinities may not correspond to the actual ranking of binding site affinities.
‘...
‘,.
‘1.. ‘1..
‘1.. ‘1.. ‘\..
‘1.. --__
--__ -_ --__
‘1.. 1.~ --_ ‘\ 8
2 x
(‘,
10'"i4-' )
(4
2
6
x
10
(: 10’06M-’ )
PI Fig. 3. Scatchard plots [equation (2)] for Fab (A) and IgG (B). Data replotted here are those of Fig. 2. the initial antibody concn. However, unless K,a, $ 1, the fraction f is dependent upon K, and a,,. Since this condition is not assured by a, $ io, the error in measurement of antibody affinity resulting from the bivalence of IgG may be dependent upon the affinity of the given antibody and the antigen concns used in the determination. Figure 5 demonstrates both effects. The ratio of apparent affinity to actual affinity is plotted as a function of (the logarithm of) K,. Simulated experiments were obtained with three 20-fold antigen concn ranges representing different but overlapping segments of the binding curves depicted in Figs 2 and 3. The magnitude of the underestimation of affinity constant increases with decreasing affinity of the binding site. Therefore, the apparent ratios of antibody affinities based on measurements with intact IgGs are not necessarily accurate. The affinities determined by correcting the IgG data presented in Fig. 5 were all consistent with the designated Fab affinity, indicating that the K,-dependent error is removed by the correction algorithm. In effect, Fig. 5 illustrates the basis for the need of the correction procedure described in this report. If the Friguet procedure systematically underestimates the affinity of all IgGs by a factor of 2, then in almost all practical experimental situations, a correction of the data would not result in a significant difference in the information content of the experiment since the
I.
I”
i
1.08-
5
. .
la-
.
1.04
:
(J-9 Fig. 4. Application of the correction procedure. Emulated experimental data points were determined as described in the legend to Fig. 2. Panel (A): the Klotz plot compares the binding of Fabs (open circles) to apparent binding of IgGs. Panel (B): corrected IgG plot. At each point, the corrected value of liganded binding sites is the square root of the apparent fractional liganded IgG [equation (IO)]. Antibody concn was set at 1 x 10m9 M; antigen concns ranged from 30 to 600 x IO-’ M. The concn of bound IgG or Fab at each antigen concn was determined in triplicate with a noise level of *lo%.
Correction for ELISA affinity analysis
LOGKm) Fig. 5. Ratio of apparent IgG K, to actual as a function of the actual K, value. Apparent affinity constants for IgG were determined in simulated experiments. Binding site concn was set at 1 x 10M9M. Three overlapping ranges of antigen concns were used: 60@-30 x lo-’ M; 200-10 x lo-‘M; and 60-3 x IOm8M. Note that although the most dilute sample of the most dilute antigen series (i.e. 3 x 10e8 M) was 30-fold greater in concn than antibody to assure appropriateness of an experimental assumption of constant antigen concn, this assumption was not made in the calculation of the apparent K,.
Consider the specific example summarized in Table 1, in which portions of the curves depicted in Fig. 5 are enumerated. A 4-fold difference in Fab affinity (U-1.35 x 10’ MO’) results in a 4.4-fold range in
apparent IgG affinities (2.62-0.59 x IO7Mlw-‘)when the measurements are made with antigen concns spanning 600-30 x lo-’ M. However, if the experiment had been performed with antigen concns IO-fold more dilute, the apparent IgG range would have been 1.91-0.0153 x lO’M_‘, a span of more than two orders of magnitude. For example, an Fab of affinity 1.35 x lo7 M-’ appears as an IgG of affinity 1.53 x 10sM-‘, whereas if the Fab had a slightly lower affinity of 1.315 x 10’ M-‘, the calculated affinity of the IgG would be only 2 x lo3 M-l. The affinity determination method described by Friguet et al. (1985) has several features that should lend it to convenient estimation of antibody equilibrium constants. Since it is necessary that the presence of the solid-phase antigen does not perturb the equilibrium established by solution-phase antibody and antigen, the incubation step in the ELISA Table 1, Dependence of apparent IgG affinity(&,J on Fab affinity (iu,,) and antigen conen f&p: IgG” ( x lo-’ M) k;,,: Fab” Ab B C - (x 10-7M) 5.400 2.620 2.460 1.910 2,700 1.270 1.120 0.624 1.350 0.595 0.456 0.0153 I.340 0.590 0.451 0.0110 1.330 0.586 0.446 0.0067 1.320 0.580 0.442 0.0024 1.315 0.577 0.439 0.0002 1.310 0.575 0.437 (- Y “Bindine site concn: 1 x 10m9M. ‘Antigenconcn ranges: A (600-30 x IO-* M); B (ZOO-IO x IO-’ M); c (6@-3x 10-EM). ‘Apparent negative affinity.
1059
must be short compared to the dissociation half of the antibody-antigen complex. Therefore, this technique may prove best suited for characterization of highaffinity antibodies. Otherwise, a competitive assay results with associated complications (e.g. Dower and Segal, 1981; Nygren et al., 1985; Stevens et al., 1986). In effect, the linear standard curve inferred from the maximal ELISA response in the absence of solutionphase antigen would not pertain for the antigenequilibrated sample. However, the protocol discussed in this report provides a simple and useful correction for IgG affinity determinations that are intended for quantitative comparisons of antibody characteristics by eliminating the error that results directly from the bivalent nature of the IgG. For many experiments, precise correction of calculated values serves little purpose, but in all cases an awareness of potential error of several orders of magnitude is important. To avoid the non-evident error in cases in which the correction is not applied, it is recommended that binding curve data for either Klotz or Scatchard analyses be obtained at two or more ranges of antigen concn for a given antibody concn. Observed consistency of KaPPmay be interpreted to verify that the analysis is undertaken with antigen concns sufficient to assure a maximum Z&,/K,, ratio and reasonable invariance of estimated K,,, . ~c~~ow~e~ge~enz-I would like to thank Dr M. Schiffer for critical reading and discussion of this manuscript. REFERENCES Avrameas S. (1986) Natural autoreactive B-cells and autoantibodies-the know thyself of the immune system. Ads fnst. Pasteur 137D, 150-I 56. Djavadi-Ohanian~ L., Friguet B. and Goldberg M. E. (1986) Conformational effects of ligand binding on the & subunit of Escherichia coli tryDtophan synthase analyzed with monoclonal antibodies. %io&emist1$25, 2502-&OS. Dower S. K. and Segal D. M. (1981) Clq binding to antibody-coated cells: predictions from a single multivalent binding modei. Molec. Immun. 18, 823-829. Elder J. H., McGee J. S., Munson M., Houghten R. A., Kloetzer W., Bittle J. L. and Grant C. K. (1987) Localization of neutralizing regions of the envelope gene of feline leukemia virus by using anti-synthetic peptide antibodies. J. Viral. 61, S-15. Friguet B., Chaffotte A. F., Djavadi-Ohaniance L. and Goldbere M. E. (19851 Measurements of the true affinitv constant-in sol&on of antigen-antibody complexes b; enz~e-links imm~osor~nt assay. J. face. Meth. 77, 305-319. Friguet B., Djavadi-Ohaniance L. and Goldberg M. E. (1986) Conformational changes induced by domain assembly within the /I2 subunit of Escherichia co/i tryptophan synthase analysed with monoclonal antibodies. Eur. J. Biochem. 160, 593-597. Guesdon T. L., Chevrier D., Mazie T. C., David B. and Avrameas S. (1986) Mon~lonal anti-histamine antibody. Preparation, characterization, and application to enzyme immunoassay of histamine. J. Immun. h4eth. 87, 69-78. Harper M., Lema F., Boulot G. and Poljak R. J. (1987) Antigen specificity and cross-reactivity of monoclonal anti-lysozyme antibodies. Molec. Immun. 24, 97-108. Klotz 1. M. (1985) Ligand-receptor interaction: facts and fantasies. Q. Rev. Biophys. 18, 227-259.
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FRED J. STEVENS
Klotz I. M. and Hunston D. L. (1971) Properties of graphical representation of multiple classes of binding &es. Biochemistry 10,306553069.. Lineweaver H. and Burk D. (1934) The determination of enzyme dissociation constants. i. Am. them. Sot. 56, 658666. Nygren H., Czerkinsky C. and Stenberg M. (1985) Dissociation of antibodies bound to surface-immobilized antigen. J. Immun. Meth. 85, 87-95. Portnoi D., Freitas A., Holmberg D., Bandeira A. and Coutinho A. (1986) Immunocompetent autoreactive B lymphocytes are activated cycling cells in normal mice. J. exp. Med. 164,25-35. Rodbard D. and Weiss G. H. (1973) Mathematical theory of immunoradiometric (labeled antibody) assays. Analyt. Biochem. 52, l&44.
Scatchard G. (1949) The attractions of proteins for small molecules and ions. Ann. N.Y. Acud. Sci. 51, 66CL672. Stevens F. J. (1986) Analysis of protein-protein interaction bv simulation of small-zone size-exclusion chromatography: application to an antibody-antigen association, Biochemistry 25, 981-993. Stevens F. J., Jwo J., Carperos W., Kohler H. and Schiffer M. (1986) Relationships between liquid- and solid-phase antibody association characteristics:implications for the use of competitive ELISA techniques to map the spatial location of idiotopes. J. Immun. 137, 1937-1944. Ternynck T. and Avrameas S. (1986) Murine natural monoclonal antibodies-a study of their polyspecificities and their affinities. Immun. Rev. 94, 99 112. Vennegoor C. and Rumke P. (1986) Circulating melanomaassociated antigen detected by monoclonal-antibody NKI/C-3. Cancer Immun. Immunother. 23, 93-100.
Vol. 24,
No. 10, pp. 1055-1060,
0161-5890/87 $3.00 + 0.00 Pergamon Journals Ltd
1987
MODIFICATION OF AN ELISA-BASED PROCEDURE FOR AFFINITY DETERMINATION: CORRECTION NECESSARY FOR USE WITH BIVALENT ANTIBODY* FRED J. STEVENS Division of Biological and Medical Research, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (First received 31 December
1986; accepted in revised form 3 April 1987)
Abstract-A
recently described procedure for the evaluation of the affinity of monoclonal antibodies [Friguet et al., J. Immun. Meth. 77, 305-319 (1985)]uses an ELISA system to determine the quantity of free antibody present in a mixture of antigen and antibody. However, an intact IgG may bind antigen by either of two binding sites, and an IgG can bind to a solid-phase antigen whether one or two of its binding sites are free. Therefore, this procedure does not directly provide the concn of liganded binding sites, the quantity necessary for calculation of the thermodynamic association constant. A binomial probability distribution relates the fraction of liganded binding sites to the concn of unliganded, singly liganded, and doubly liganded IgG assuming that the binding of each Fab to antigen is independent. Simulated experiments were used to compare the apparent binding characteristics of bivalent IgG and monovalent Fab and to calculate apparent association constants in each case. It was found that the affinity of binding sites on intact IgG was underestimated by a factor of at least 2 and that the error was inversely related to the fraction of liganded binding sites. Binding site affinity of an antibody may be underestimated by several orders of magnitude. On the basis of binomial analysis, it is possible to convert apparent concns of bound IgG to actual concns of liganded binding site resulting in the calculation of valid association constants for intact IgG without alteration of the experimental protocol.
INTRODUCTION
Friguet et al. (1985) have recently introduced a convenient method for the evaluation of affinity constants that characterize the interactions between antibodies and macromolecular antigens. The approach uses an enzyme-linked immunosorbent assay (ELISA) to quantitate the unbound antibody present in an equilibrated reaction mixture. By using sufficient excess antigen to justify the assumption that the concn of free antigen does not change significantly at equilibrium, the ELISA provides appropriate data to allow the use of Scatchard or Klotz analyses (Scatchard, 1949; Klotz and Hunston, 1971; Klotz, 1985) to evaluate the affinity constant of the antibody. This affinity measurement technique has been used to quantitatively evaluate and compare equilibrium constants expressed by families of monoclonal antibodies for specific antigens (Portnoi et al., 1986; Avrameas, 1986; Ternynck and Avrameas 1986; Elder et al., 1987), to analyze the specificity of an anti-histamine monoclonal antibody for a series of modified antigens (Guesdon et al., 1986), and to characterize the affinity of a monoclonal antibody for a melanoma-associated glycoprotein (Vennegoor and Rumke, 1986). Djavadi-Ohaniance et al. (1986) and
*This work is supported by the U.S. Department of Energy and the Office of Health and Enviromental Research under contract No. W-31-109-Eng-38, and by PHS grant No. ROl A119590, DHHS.
Friguet et al. (1986) measured affinity constants of monoclonal antibodies for six distinct epitopes on two domains of the f12 subunit of tryptophan synthase. Changes in affinity constants occurring with ligand binding by the synthase were observed that could be interpreted as resulting from substrateinduced conformational changes in the enzyme. Harper et al. (1987) recently measured the affinity constants of a series of anti-lysozyme monoclonal antibodies. These investigators noted that the value of the affinity constant obtained for the antigen binding fragment (Fab) from one immunologlobulin (IgG) was approximately twice the value calculated for the antibody binding site in the intact bivalent IgG. The authors suggested two possible explanations for this observation. First, binding of one lysozyme molecule might result in a steric hindrance that decreases the ability of the binding site present on the second Fab to interact with antigen. Second, since the ELISA quantifies free IgG on the basis of its ability to interact with antigen adsorbed on the solid phase, the technique might not be able to distinguish between ligand-free and singly liganded IgG. Therefore, only doubly liganded IgG is scored as bound; as a result, the extent of antigen binding is underestimated. An analogous problem was addressed in a simulation of antibody: antigen interaction occurring during size-exclusion gel chromatography (Stevens, 1986). For the simulated chromatogram, the elution behavior of a mixture of interacting mono- and bivalent components is determined by the particle-
1055
FRED J. STEVENS
1056
size distribution within the column. Therefore, it was necessary to calculate the fractional composition of unliganded, singly liganded, and doubly liganded constituents as a function of concns and aflinity constant. This report describes the use of this approach to determine that the discrepancy observed in the apparent association constants of Fab and intact IgG may directly result from the bivalency of the IgG and that this error may approach several orders of magnitude under appropriate circumstances. A procedure is described that corrects the resulting error and provides agreement in apparent association constants, regardless if they are measured for free Fabs or intact IgGs. The correction procedure also minimizes errors in evaluation of IgG affinity that are dependent upon the affinity and concn of the reagents used. BASES OF ERROR AND CORRECTION
Friguet’s procedure uses an ELISA to estimate the quantity of unliganded antibody in an equilibrium mixture of antibody and antigen to provide the necessary data for analysis by the method of Klotz, 11 11 -=r+iK.; x b 0 a
(1)
or by the method of Scatchard, X - = a
i. I& -
K,x.
c-3
Relationships (1) and (2) are both convenient expressions of the equilibrium condition x = K,(& - x)a,
(3)
in which KS is the association constant, iOis the initial binding site concn, and x and a are the equilibrium concns of complex and antigen, respectively. In the ELISA-based affinity procedure, a sample of the equilibrated antibody-antigen mixture is transferred to antigen-coated wells or tubes, and the quantity of antibody free to bind to the immobilized ligand is subsequently determined by exposure to an appropriate enzyme-linked anti-immunoglobulin antibody. Color development relative to that obtained for a corresponding sample of antigen-free antibody is taken as the fraction of free antibody in the equilibrated mixture. However, since both unliganded and singly liganded IgG bind to antigen adsorbed on the solid phase, the fraction of apparent “free” IgG is not the same as the fractional free binding sites or Fab. The problem then becomes one of converting the apparent degree of IgG binding to the true degree of Fab binding determined by the association constant. Rodbard and Weiss (1973) developed a probabilistic formulation of the effect of antibody bivalence in their quantitative analysis of solid-phase immunoassays, A similar problem was addressed in the chromatography simulation (Stevens, 1986) in which, given the concn of liganded Fab, it was necessary to
determine the concns of unliganded, singly liganded and doubly liganded antigen. In the case in which ELISA is used to assay free antibody in the equilibrium mixture of antigen and IgG, iff = .x/i0 is the fraction of bound Fab, then f can be interpreted as the probability that a randomly selected binding site or Fab is bound and 1 -f as the probability that the site is free. Accordingly, the fraction of the IgG population in each of the three categories is unliganded:
(1 -f)’
singly liganded: 2f(l -f) doubly liganded: f 2,
(4) (5) (6)
on the basis of a standard binomial distribution and assuming that the binding of antigen to an Fab is independent of the bound or free status of the second Fab. Of the three categories, only the doubly liganded antibody does not react with the solid-phase ELISA antigen. Therefore, from equation (6), the fraction of free antibody measured is equivalent to P?l=l-S2
(7)
f=Jr-m
(8)
and
is the corrected fraction of liganded binding sites in terms of the apparent fractional free IgG. The affinity assay fo~ulated by Friguet et al. (1985) conveniently measures “bound” IgG relative to the total IgG present, i.e. A,-Ai I v’=Aop
(9)
where A, is the ELISA absorbance obtained in the absence of antigen and A, is the corresponding absorbance obtained in the presence of antigen. Since the quantity represented by equation (9) is the doubly liganded fraction, equation (6), at any antigen concn the corrected fraction of experimentally observed liganded binding sites is
fF=J;;.
(~0)
RESULTS AND DISCUSSION
Figure 1 illustrates the discrepancy between observed and actual degrees of binding. The relative magnitude of the underestimated binding is most substantial at low degrees of fractional binding, suggesting that high antigen concns and/or high affinities are necessary for the more accurate results although the measured affinity of the IgG cannot be better than one-half the magnitude of the Fab affinity. At high levels of binding, the rate of change of observed binding is twice that of actual binding, accounting for the minimal 2-fold difference in measured affinity for Fab and IgG. Figure 2 shows computer-generated binding curves for IgG and Fab at three different aflinities. Doublereciprocal plots [equation (1); Lineweaver and Burk,
1057
Correction for ELISA affinity analysis
O.O~.,.,,,.,,,,,..,,,,‘,.,,‘I 0.0
0.2
0.4
0.6
0.8
1.0
FRACTION LIGANDED (ESTIMATED)
Fig. 1. Dependence of error on extent of binding. The apparent fractional binding of IgG [equation (6)] and Fab is plotted as a function of the actual fractional liganding of binding sites. The concn of complex was calculated by equation (3); the concn of antigen was not assumed to be constant through the reaction.
1934)] referred to as Klotz plots in Friguet et al. (1985), are used to represent the binding calculated on the basis of equation (3). Figure 2(A) shows linear dependence of (l/x) throughout the antigen concn range when the binding component is the monovalent Fab. Binding curves for IgG at the same molar concn of binding site (1O-9 M) are shown in Fig. 2(B) and are clearly different than those seen for free Fab. The deviation increases with decreased affinity of the binding site. Decreased affinity for a given antigen concn is equivalent to a shift to the left in the plot shown in Fig. 1. The antigen concn dependence of the IgG binding curves is not an artifact of the double-reciprocal formulation. In Fig. 3, the data shown in Fig. 2 are replotted as Scatchard plots [equation (2)]. Whereas the simulated Fab binding curves are linear throughout the antigen concn range, the maximum slope of each IgG binding curve occurs at antigen concns near saturation of the available binding sites. The slope of the tangential line at this segment corresponds to an estimated affinity one-half the value of the Fab intrinsic affinity. However, the x-axis intercept of the tangential line correctly estimates the concn of available binding sites. As the concn of antigen is decreased, the diminishing slope of the binding curve results in progressively underestimated K, and a correspondingly overestimated binding site concn. At sufficiently low antigen concns, an inflection point is reached in the binding curve and both parameters become indeterminant. The substantial curvature of the IgG binding plots shown in Figs 2 and 3 need not be evident in experiments restricted to a relatively narrow range of antigen concn. A simulated experiment is depicted in Fig. 4 for an antibody with binding site affinity of K, = 5.0 x 107M-i, concn= 1.0 x 10m9h4, and a series of antigens at concns between 30 and 600 x lo-* M. Panel (A) compares the Klotz plots obtained if the binding sites are presented as
Fabs (open circles) or as IgGs (closed circles). Data were obtained by a simple computer progam that calculated the concn of bound sites and the distribution [equations (4H6)] of bound sites; uniform noise (10%) was added by a random number process in order to test the correction procedure with experimentally realistic data. In this example, a regression line fitted to the IgG data results in an apparent affinity of 2.4 x lO’M_i. Even though this value is slightly less than one-half the value of the affinity of the binding site, the correlation coefficient of the regression line is 0.9986. Panel (B) represents the plot obtained for “corrected” IgG data. Regression analyses of both the Fab data and the corrected IgG data result in measured affinities of 5.11 x 10’ M-’ and 5.03 x 10’ M-l, respectively. In cases of significant antigen excess, as recommended by Friguet et al. (1985), the fractional liganded binding sites can be expressed f= (xl&) = &a,/(1
+&a,),
(11)
in which a, is the initial free antigen concn and remains essentially unchanged by complex formation. In these circumstances, fis effectively independent of
(4 80
I
0
I’
10
40
,,A’;) x 10”
M)
@I Fig. 2. Double-reciprocal plots [equation (I)] for Fab (A) and IgG (B). Binding site concn was set at 1 x 10e9 M; antigen concn ranged from 600 x IO-’ M to 30 x lo-* M. Binding site Ka values were 2 x 108M -I (-----); 5 x lO’M_’ (----); and 1 x 107M-’ (-). Note that the antigen concn is the actual equilibrium concn; it is not assumed that n = %. For each discrete antigen concn, the complex concn, x, was determined on the basis of equation (3).
1058
FRED J. STEVENS
relative values of the IgG affinities would accurately correspond to those of the Fabs. However, because of the exponential relationship between f* and v [equation (lo)], f * approaches zero faster than v; therefore, the ratio Kapp/Kac, + 0 for small v. Under these circumstances very small differences in the intrinsic binding site affinity can appear as differences of one or more orders of magnitude. Further, if experiments to compare antibodies were conducted with different ranges of antigen concn, then apparent ranking of IgG affinities may not correspond to the actual ranking of binding site affinities.
‘...
‘,.
‘1.. ‘1..
‘1.. ‘1.. ‘\..
‘1.. --__
--__ -_ --__
‘1.. 1.~ --_ ‘\ 8
2 x
(‘,
10'"i4-' )
(4
2
6
x
10
(: 10’06M-’ )
PI Fig. 3. Scatchard plots [equation (2)] for Fab (A) and IgG (B). Data replotted here are those of Fig. 2. the initial antibody concn. However, unless K,a, $ 1, the fraction f is dependent upon K, and a,,. Since this condition is not assured by a, $ io, the error in measurement of antibody affinity resulting from the bivalence of IgG may be dependent upon the affinity of the given antibody and the antigen concns used in the determination. Figure 5 demonstrates both effects. The ratio of apparent affinity to actual affinity is plotted as a function of (the logarithm of) K,. Simulated experiments were obtained with three 20-fold antigen concn ranges representing different but overlapping segments of the binding curves depicted in Figs 2 and 3. The magnitude of the underestimation of affinity constant increases with decreasing affinity of the binding site. Therefore, the apparent ratios of antibody affinities based on measurements with intact IgGs are not necessarily accurate. The affinities determined by correcting the IgG data presented in Fig. 5 were all consistent with the designated Fab affinity, indicating that the K,-dependent error is removed by the correction algorithm. In effect, Fig. 5 illustrates the basis for the need of the correction procedure described in this report. If the Friguet procedure systematically underestimates the affinity of all IgGs by a factor of 2, then in almost all practical experimental situations, a correction of the data would not result in a significant difference in the information content of the experiment since the
I.
I”
i
1.08-
5
. .
la-
.
1.04
:
(J-9 Fig. 4. Application of the correction procedure. Emulated experimental data points were determined as described in the legend to Fig. 2. Panel (A): the Klotz plot compares the binding of Fabs (open circles) to apparent binding of IgGs. Panel (B): corrected IgG plot. At each point, the corrected value of liganded binding sites is the square root of the apparent fractional liganded IgG [equation (IO)]. Antibody concn was set at 1 x 10m9 M; antigen concns ranged from 30 to 600 x IO-’ M. The concn of bound IgG or Fab at each antigen concn was determined in triplicate with a noise level of *lo%.
Correction for ELISA affinity analysis
LOGKm) Fig. 5. Ratio of apparent IgG K, to actual as a function of the actual K, value. Apparent affinity constants for IgG were determined in simulated experiments. Binding site concn was set at 1 x 10M9M. Three overlapping ranges of antigen concns were used: 60@-30 x lo-’ M; 200-10 x lo-‘M; and 60-3 x IOm8M. Note that although the most dilute sample of the most dilute antigen series (i.e. 3 x 10e8 M) was 30-fold greater in concn than antibody to assure appropriateness of an experimental assumption of constant antigen concn, this assumption was not made in the calculation of the apparent K,.
Consider the specific example summarized in Table 1, in which portions of the curves depicted in Fig. 5 are enumerated. A 4-fold difference in Fab affinity (U-1.35 x 10’ MO’) results in a 4.4-fold range in
apparent IgG affinities (2.62-0.59 x IO7Mlw-‘)when the measurements are made with antigen concns spanning 600-30 x lo-’ M. However, if the experiment had been performed with antigen concns IO-fold more dilute, the apparent IgG range would have been 1.91-0.0153 x lO’M_‘, a span of more than two orders of magnitude. For example, an Fab of affinity 1.35 x lo7 M-’ appears as an IgG of affinity 1.53 x 10sM-‘, whereas if the Fab had a slightly lower affinity of 1.315 x 10’ M-‘, the calculated affinity of the IgG would be only 2 x lo3 M-l. The affinity determination method described by Friguet et al. (1985) has several features that should lend it to convenient estimation of antibody equilibrium constants. Since it is necessary that the presence of the solid-phase antigen does not perturb the equilibrium established by solution-phase antibody and antigen, the incubation step in the ELISA Table 1, Dependence of apparent IgG affinity(&,J on Fab affinity (iu,,) and antigen conen f&p: IgG” ( x lo-’ M) k;,,: Fab” Ab B C - (x 10-7M) 5.400 2.620 2.460 1.910 2,700 1.270 1.120 0.624 1.350 0.595 0.456 0.0153 I.340 0.590 0.451 0.0110 1.330 0.586 0.446 0.0067 1.320 0.580 0.442 0.0024 1.315 0.577 0.439 0.0002 1.310 0.575 0.437 (- Y “Bindine site concn: 1 x 10m9M. ‘Antigenconcn ranges: A (600-30 x IO-* M); B (ZOO-IO x IO-’ M); c (6@-3x 10-EM). ‘Apparent negative affinity.
1059
must be short compared to the dissociation half of the antibody-antigen complex. Therefore, this technique may prove best suited for characterization of highaffinity antibodies. Otherwise, a competitive assay results with associated complications (e.g. Dower and Segal, 1981; Nygren et al., 1985; Stevens et al., 1986). In effect, the linear standard curve inferred from the maximal ELISA response in the absence of solutionphase antigen would not pertain for the antigenequilibrated sample. However, the protocol discussed in this report provides a simple and useful correction for IgG affinity determinations that are intended for quantitative comparisons of antibody characteristics by eliminating the error that results directly from the bivalent nature of the IgG. For many experiments, precise correction of calculated values serves little purpose, but in all cases an awareness of potential error of several orders of magnitude is important. To avoid the non-evident error in cases in which the correction is not applied, it is recommended that binding curve data for either Klotz or Scatchard analyses be obtained at two or more ranges of antigen concn for a given antibody concn. Observed consistency of KaPPmay be interpreted to verify that the analysis is undertaken with antigen concns sufficient to assure a maximum Z&,/K,, ratio and reasonable invariance of estimated K,,, . ~c~~ow~e~ge~enz-I would like to thank Dr M. Schiffer for critical reading and discussion of this manuscript. REFERENCES Avrameas S. (1986) Natural autoreactive B-cells and autoantibodies-the know thyself of the immune system. Ads fnst. Pasteur 137D, 150-I 56. Djavadi-Ohanian~ L., Friguet B. and Goldberg M. E. (1986) Conformational effects of ligand binding on the & subunit of Escherichia coli tryDtophan synthase analyzed with monoclonal antibodies. %io&emist1$25, 2502-&OS. Dower S. K. and Segal D. M. (1981) Clq binding to antibody-coated cells: predictions from a single multivalent binding modei. Molec. Immun. 18, 823-829. Elder J. H., McGee J. S., Munson M., Houghten R. A., Kloetzer W., Bittle J. L. and Grant C. K. (1987) Localization of neutralizing regions of the envelope gene of feline leukemia virus by using anti-synthetic peptide antibodies. J. Viral. 61, S-15. Friguet B., Chaffotte A. F., Djavadi-Ohaniance L. and Goldbere M. E. (19851 Measurements of the true affinitv constant-in sol&on of antigen-antibody complexes b; enz~e-links imm~osor~nt assay. J. face. Meth. 77, 305-319. Friguet B., Djavadi-Ohaniance L. and Goldberg M. E. (1986) Conformational changes induced by domain assembly within the /I2 subunit of Escherichia co/i tryptophan synthase analysed with monoclonal antibodies. Eur. J. Biochem. 160, 593-597. Guesdon T. L., Chevrier D., Mazie T. C., David B. and Avrameas S. (1986) Mon~lonal anti-histamine antibody. Preparation, characterization, and application to enzyme immunoassay of histamine. J. Immun. h4eth. 87, 69-78. Harper M., Lema F., Boulot G. and Poljak R. J. (1987) Antigen specificity and cross-reactivity of monoclonal anti-lysozyme antibodies. Molec. Immun. 24, 97-108. Klotz 1. M. (1985) Ligand-receptor interaction: facts and fantasies. Q. Rev. Biophys. 18, 227-259.
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FRED J. STEVENS
Klotz I. M. and Hunston D. L. (1971) Properties of graphical representation of multiple classes of binding &es. Biochemistry 10,306553069.. Lineweaver H. and Burk D. (1934) The determination of enzyme dissociation constants. i. Am. them. Sot. 56, 658666. Nygren H., Czerkinsky C. and Stenberg M. (1985) Dissociation of antibodies bound to surface-immobilized antigen. J. Immun. Meth. 85, 87-95. Portnoi D., Freitas A., Holmberg D., Bandeira A. and Coutinho A. (1986) Immunocompetent autoreactive B lymphocytes are activated cycling cells in normal mice. J. exp. Med. 164,25-35. Rodbard D. and Weiss G. H. (1973) Mathematical theory of immunoradiometric (labeled antibody) assays. Analyt. Biochem. 52, l&44.
Scatchard G. (1949) The attractions of proteins for small molecules and ions. Ann. N.Y. Acud. Sci. 51, 66CL672. Stevens F. J. (1986) Analysis of protein-protein interaction bv simulation of small-zone size-exclusion chromatography: application to an antibody-antigen association, Biochemistry 25, 981-993. Stevens F. J., Jwo J., Carperos W., Kohler H. and Schiffer M. (1986) Relationships between liquid- and solid-phase antibody association characteristics:implications for the use of competitive ELISA techniques to map the spatial location of idiotopes. J. Immun. 137, 1937-1944. Ternynck T. and Avrameas S. (1986) Murine natural monoclonal antibodies-a study of their polyspecificities and their affinities. Immun. Rev. 94, 99 112. Vennegoor C. and Rumke P. (1986) Circulating melanomaassociated antigen detected by monoclonal-antibody NKI/C-3. Cancer Immun. Immunother. 23, 93-100.