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Expression of antibody activity measured by ELISA. Anti-ssDNA antibody activity characterized by the shape of the dose-response curve
Journal oflmmunologicalMethods, 62 (1983) 315-323 Elsevier
315
JIM02750
Expression of Antibody Activity Measured by ELISA. Anti-ssDNA Antibody Activity Characterized by the Shape of the Dose-Response Curve Marianne Gripenberg ~'* and Gustaf Gripenberg 2 l Department of Bacteriology and Immunology, Umversi O" of Helsinkt and Fourth Department of Medicine, Unieersitv Central Hospital of Helsinki, and • Institute of Mathematics, Helsinki University of Technolo~', Finland (Received 8 November 1982, accepted 11 April 1983)
To demonstrate the presence or absence of antibodies, results derived from a single serum dilution in an ELISA are sufficient. However, qualitative differences in antibodies are reflected by the shape of dose-response curves. A method based on approximating the absorbance value by a polynomial p ( x ) = a ~ x + a 2 x 2. where l / x is the dilution factor, was used to characterize the dose-response curves in an ELISA for anti-ssDNA antibodies. The parameters used are E = al and A = - a ] / a 2. It can be argued that E gives an estimate of the effective amount of antibodies in the serum and that A is essentially a function of the reaction constants between antibody and antigen. Key words: ELISA - - anti-ssDNA anttbodies
Introduction Titration has been the traditional serological method for measuring antibodies. The advent of new assays such as enzyme-linked immunosorbent assays (ELISA) has led to reconsideration of ways of expressing the amount of antibody activity (De Savigny and Voller, 1980; Malvano et al., 1982). The ELISA does not depend on an endpoint titration, and allows testing at a single serum dilution (Voller et al., 1978). A variety of ways of expressing ELISA results has been advocated, but no single method satisfies the requirements of an ideal serological report (De Savigny and Voller, 1980). Both specific antibody concentration and avidity of the antibody influence the results of antibody estimations in general and those obtained by ELISAs in particular (Butler et al., 1978). Assay conditions in the ELISA affect the shape of the dilution curve (Malvano et al., 1982). Even with identical assay conditions the shape * Address for correspondence: M. Gripenberg, MD, Dept. of Bacteriology and Immunology, University of Helsinki, Haartmanink. 3, SF-00290 Helsinki 29, Finland. 0022-1759/83/$03.00 '~' 1983 Elsevier Science Publishers B.V.
316 of the dose-response curves of different serum samples varies in some ELISAs (Ahlstedt et al., 1978; Gripenberg et al., 1979). It has recently, been shown that it is possible to estimate amounts of high avidity and total antibody by analyzing the shape of the titration curves (Lehtonen and Viljanen, 1982). Antibodies reacting with D N A are a heterogeneous group of auto-antibodies characteristically present in systemic lupus erythematosus (SLE) (Barnett, 1979). The possible pathogenic role of anti-DNA antibodies and their diagnostic and prognostic value in SLE are still not settled. Using an ELISA for class-specific anti-single stranded (ss) D N A antibodies we have found that the dose-response curves between different patients' sera vary in both shape and slope (Gripenberg et al., 1978). The aim of the present study was to obtain qualitative and quantitative data about the antibody population present in different sera by expressing the results in terms of 2 numbers characterizing the shape of the dose-response curve.
Materials and Methods
Sera Serum samples from patients with various connective tissue diseases were used in the studies. Anti-ssDNA E L I S A Antibodies against denatured (ss) D N A of IgG and IgM class were demonstrated by ELISA as previously described (Gripenberg et al., 1978). Briefly. flat-bottom microtitre plates (Linbro ~/Titertek, Flow Laboratories, Hamden, CN) were incubated with 200 t~l of a solution of ssDNA (1 ~ g / m l ) in phosphate buffered saline (PBS), pH 7.2. After 3 h at 37°C the plates were washed, 150 /~1 of the serum samples diluted in PBS containing 0.05% Tween 20 (PBS-T) were added and the plates were incubated for 2 h at 37°C. After thorough washing, 150 ~1 of a 1 : 500 PBS-T dilution of a heavy chain specific anti-human IgG or IgM alkaline phosphatase conjugate (Orion Diagnostica, Helsinki) was added and the plates left at room temperature overnight. Anti-ssDNA antibody activity was then demonstrated by degradation of Sigma 104 '~ phosphatase substrate (Sigma Chemical Co., St. Louis, MO) (1 m g / m l of a 1 M diethanolamine buffer, p H 9.7, containing 0.5 mM MgCI~). The enzyme reaction was allowed to proceed for 1 h at 37°C and the absorbance values were recorded with a Titertek Multiskan photometer (Flow Laboratories, Eflab Oy, Helsinki). All samples were run in duplicate and pooled positive and negative reference sera were included in each assay. The results of the anti-ssDNA ELISA were expressed as units, i.e., the absorbance value obtained at a serum dilution of 1 : 80 calculated as a percentage of the absorbance value given by the positive reference serum at the same dilution run on the same occasion. Calculation of results The method of expressing results used depends upon approximating the function y(x), that is the absorbance obtained in ELISA when the serum dilution factor is
317 I / x , by a polynomial p ( x ) = a l x + azx 2, using the least squares method, with equal weights at the discrete measurement values of x. Only absorbance values between 2.0 and 0.1 were taken into account. It was found that the relative error in a~ obtained from 2 parallel determinations was reduced by approximating the function
xj y*(xj)=xf'fo
y(s) ds
by the second degree polynomial instead of using the values y(xj). To obtain y(s), linear interpolation was used. The same object of minimizing the relative error determined the choice of a second and not a third or higher degree polynomial, a, and a 2 m a y be obtained as linear combinations of the values y(xj), e.g., in the case when x j+ ~ = 0.5xj and there are m measurements. a i=
x/i~-~ Cm. i.jY(X,)
j=l
where the coefficients Cm,i.j are given in the appendix. The parameters used to characterize the dose-response curve are E
= al
and A = -a]/a
2
where ~, is the mean of the values a~ determined from the parallel measurements.
Interpretation of E and A Symbols are defined as follows: s = a m o u n t of antigen binding sites; b i = a m o u n t of antibodies of type i (i = 1. . . . . n) in the serum; x = 1/dilution factor; zij = a m o u n t of antibodies of type i b o u n d at j binding sites; k~j = reaction constant for a n t i b o d y i binding at j sites; m~ = n u m b e r of binding sites of antibodies of type i. F r o m the law of mass action,
(
zij(x ) s -
~ m~1jz,j(x) )-j( xb i - m=~lzi,(x ) )-I = k,j. i=lj= j=
Assuming that the absorbance due to b o u n d antibodies of type i is a, times the total a m o u n t of b o u n d antibodies of type i, we see that the total absorbance will be
mI y(x) = E
i=1
E
j=l
318
It is possible to show will be a decreasing prozone (Gripenberg approximately equal E ~
oqb i i=l
that for certain choices of the above constants the function y(x) function of x for certain sufficiently large values of x. i.e.. a et al.. 1979). Since y(x) = y'(0)x + 2 ly"(0)x2 + .... E will be to y'(0), i.e..
(m)(
,
__~ k i j s J
,j
I
m t
1 .
I
+ ~ k@ j j=l
Thus E gives an estimate of the 'effective' a m o u n t of antibody in the serum as a weighted sum of antibody concentrations with weights that are increasing functions of the binding constants. The n u m b e r A will be approximately equal to - 2 y ' ( 0 ) e / y " ( 0 ) and if n = m I = 1, then this expression will be als(l + ktls), which is an increasing function of k~. In some cases when n > 1, however, y"(x) become~ positive for certain values of x and hence the possibility that the A value calculated is negative cannot be excluded. Results
Fig. 1 shows some typical dose-response curves for anti-ssDNA antibodies in different patients' sera. Table I gives their titres, i.e., the highest dilutions giving TABLE I A N T I - s s D N A A N T I B O D Y A C T I V I T Y G I V E N AS T I T R E , C A L C U L A T E D F R O M A S I N G L E S E R U M D I L U T I O N A N D E X P R E S S E D IN T E R M S O F T H E P A R A M E T E R S E A N D A A: I g G a n t i - s s D N A a n t i b o d i e s Sera 1 c
2 3 4 5 6
Titre a :320 :160 :640 :2560 :320 :640
U b
E
A
29 37 83 240 36 155
33 19 84 525 40 121
1.8 2.3 5.1 5.9 7.7 7.6
B: IgM a n t i - s s D N A a n t i b o d i e s Sera
Titre
U
E
A
1
1:160 1:320 1:160 1:1280 1:1280 1:1280
21 68 39 116 180 195
20 34 30 124 194 195
1.4 2.0 3.4 5.8 8.2 9.6
2 3 4 5 6
Highest d i l u t i o n giving an a b s o r b a n c e value _> 0.1. U n i t value c a l c u l a t e d from a single serum d i l u t i o n in relation to a positive serum. The n u m b e r s c o r r e s p o n d to the n u m b e r s in Fig. 1.
4
1
I
t
1:20
I
1:40
I
I
1:60
1:160 SERUM
I
1:320
I
1:640
1:1260
1
1:2560
DILUTION
B
2.0
1
I
1:20
I
1:4D
1
I
I
1:160
I:80
SERUM
1:320
I
1:640
1
1:128D
DILUTION
Fig. I. A: dose-response curves of 6 different sera obtained by ELISA for IgG antibodies against ssDNA. The numbers of the sera correspond to those in Table IA. B: dose-response curves of 6 different sera obtained by ELISA for IgM antibodies against ssDNA. The numbers of the sera correspond to those in Table IB.
320 2.0_
]1 i
/2 ,1
/.3
/ / ./ .7
////
/ / /
1.5_
/
/
/
/
/ /
/
/
//.
/
/.
/
/
j j , 2
/ J
1,0_
0.5_
//Y I I
I
:A . . . .
I
I
i m
I
SERUM
DILUTION
Fig. 2. Dose-response curves of 3 different sera in the l g G - a n t i - s s D N A ELISA. The d a s h e d line~, depict the functions Ex. For serum 1. the value of E is 194 and that of A 6.8, for serum 2 the ~alues are 81 and 4.7 and for serum 3 the 3' are 47 and - 0 . 9 respectively.
absorbance values > 0.1, the unit values calculated from a single serum dilution and the corresponding numerical values of the parameters E and A. Since the ~cale is logarithmic, horizontal translation of curves such as those in Fig. 1 leaves the number A unchanged while E is multiplied by a factor corresponding to the translation. If, on the other hand, all the absorbance values are multiplied by a factor, the values E and A will be multiplied by the same factor. A non-logarithmic scale gives dose-response curves in the ELISA like those shown in Fig. 2. The dashed lines in Fig. 2 are the graphs of the functions Ex that are approximately tangent to the curves y(x) at x = 0. Calculation of the values U and E from approximately 300 sera does not show that the relative errors in the results obtained from parallel measurements are different for U and E. It was also found, as would be expected, that low values of A (or rather high values of l / A , since sometimes A was negative) are associated with low values of E but the converse did not hold.
Discussion An inherent advantage of the enzyme-linked immunosorbent assay is the possibility of quantitating specific antibody from a single serum dilution. However, anti-
321 body affinity and avidity differ between sera giving different kinds of dose-response curves (De Savigny and Voller, 1980). Thus useful information about antibodies may be lost if ELISA results are derived from a single serum dilution (Lehtonen and Eerola, 1982). The method we suggest for the characterization of the dose-response curve relies on approximating the dilution curve by a second degree polynomial. Lehtonen and Viljanen (1982) have used the function kx l/Inx~ for this purpose. One benefit of using a polynomial lies in the simplicity of the calculation, since one has only to take a linear combination of the absorbance values. In the ELISA, the actual amounts of the reagents are unknown, which means that the results are at best expressed as relative antibody activity. The parameter E, which is closely correlated with the unit values calculated from a single serum dilution, may be taken as an estimate of the 'effective' amount of antibody in the serum. It can be argued that this parameter is approximately a weighted sum of antibody concentrations, with weights dependent on the affinity and avidity of the different antibody classes present in the serum. There is an advantage in using the parameter E when testing sera with high antibody activity since the unit value cannot discriminate between sera giving absorbance values > 2.0 at the serum dilution used. Antibody populations directed against D N A are notoriously heterogeneous (Barnett, 1979) and to have found varying dose-response curves in the anti-ssDNA ELISA is not surprising. Different methods have been used to measure the avidity of anti-DNA antibodies in sera, but none is completely satisfactory (Leon et al., 1977; Devens et al., 1978; Griffiths et al., 1978). We do not claim that the parameter A calculated from the dose-response curve is proportional to the antibody avidity or affinity, but have reason to believe that high values of A (or rather, low values of l/A, taking into account the possibility that A is negative) are associated with antibody populations of high affinity and avidity. There are widely divergent opinions about the role of antibody avidity in the pathogenesis of SLE, possibly because of variations in methodology (Barnett, 1979). Qualitative differences in anti-DNA antibodies are, however, evidently associated with disease activity in SLE (Pearson and Lightfoot, 1981). The value of determining E and A parameters for anti-ssDNA antibodies in following-up patients with SLE will be investigated in a separate study.
322
Appendix T h e coefficients Cm. ~. used to calculate a~: j
1 2 3
m 2
3
4
5
6
7
8
-0.5 3.0
- 0.3762376 0.6930693 4.118812
-0.3415237 0.6017845 2.293205
-0.3317236 0.5810315 2.219561
- 0.3290626 0.5759245 2.200752
- 0.3283647 0.5746463 2.195957
- 0.3281856 0.5743258 2.194744
3.738641
2.463203 2.854678
2.442412 1.988634 1.96228
2.4371 1.984312 1.399088
2.435755
1.259115
0.9085206
4 5 6 7
1.983217 1.398316
8
0.7709885
T h e coefficients Cm. 2., used to calculate a2: j
m
1 2 3 4 5
2
3
4
5
6
7
8
1.5 - 3.0
1.306931 0.1188119 - 5.465347
1.251887 0.2635552 - 2.849691 - 5.369938
1.236237 0.2966969 - 2.732084 - 3.486189 - 4.252656
1.231974 0.3048794 - 2.701949 --3.452878 - 2.946068
1.230854 0.3069304 - 2.694254 - 3.444355 2.939133
1.230566 0.3074452 - 2.692305 - 3.442194 - 2.937374
--2.98199
- 2.12013 - 1.936715
- 2.11889 - 1.395063 - 1.195401
6 7 8
Acknowledgements We thank Ms. Anna Liisa Ruuska for excellent technical assistance and Finska L~ikares~illskapet for financial support.
References Ahlstedt, S., B. Carlsson, L.~,. H a n s s o n , B. Kaijser, I. M a t t s b y - B a l t z e r a n d S. Sohl-,~kerlund, 1978, Scand. J. l m m u n o l . 8 (Suppl. 7), 119. Barnett, E.V., 1979, J. I m m u n o l . M e t h o d s 27. 1. Butler, J.E., T.L. Feldbush, P.L. M c G i v e r n a n d N. Stewart, 1978, I m m u n o c h e m i s t r y 15, 131. D e Savigny, D. a n d A. Voller, 1980, J. I m m u n o a s s a y 1, 105. D e v e n s , B., D. Chia a n d E.V. Barnett, 1978, J. I m m u n o l . M e t h o d s 19, 187. Griffiths, G . D . , I. Olsen a n d M.W. Steward, 1978, J. I m m u n o l . M e t h o d s 21, 143. G r i p e n b e r g , M., E. Linder, P. K u r k i a n d E. Engvall, 1978, Scand. J. I m m u n o l . 7, 15 I. G r i p e n b e r g , M., A. Nissinen, E. V~iis~inen a n d E. Linder, 1979, J. Clin. Microbiol. 10, 279.
323 Lehtonen, O.-P. and E. Eerola, 1982, J. lmmunol. Methods, 54, 233. Lehtonen, O.-P. and M.K. Viljanen, 1982, Int. J. Bio-Med. Comput. 13, 471. Leon, S.A., A. Green, G.E. Ehrlich, M. Poland and B. Shapiro, 1977, Arthritis Rheum. 20, 23. Malvano, R., A. Boniolo, M. Dovis and M. Zannino, 1982, J. Immunol. Methods 48, 51. Pearson, L. and R.W. Lightfoot Jr., 1981, J. Immunol. 126, 16. Voller, A., A. Bartlett and D.E. Bidwell, 1978, J. Clin. Pathol. 31,507.
315
JIM02750
Expression of Antibody Activity Measured by ELISA. Anti-ssDNA Antibody Activity Characterized by the Shape of the Dose-Response Curve Marianne Gripenberg ~'* and Gustaf Gripenberg 2 l Department of Bacteriology and Immunology, Umversi O" of Helsinkt and Fourth Department of Medicine, Unieersitv Central Hospital of Helsinki, and • Institute of Mathematics, Helsinki University of Technolo~', Finland (Received 8 November 1982, accepted 11 April 1983)
To demonstrate the presence or absence of antibodies, results derived from a single serum dilution in an ELISA are sufficient. However, qualitative differences in antibodies are reflected by the shape of dose-response curves. A method based on approximating the absorbance value by a polynomial p ( x ) = a ~ x + a 2 x 2. where l / x is the dilution factor, was used to characterize the dose-response curves in an ELISA for anti-ssDNA antibodies. The parameters used are E = al and A = - a ] / a 2. It can be argued that E gives an estimate of the effective amount of antibodies in the serum and that A is essentially a function of the reaction constants between antibody and antigen. Key words: ELISA - - anti-ssDNA anttbodies
Introduction Titration has been the traditional serological method for measuring antibodies. The advent of new assays such as enzyme-linked immunosorbent assays (ELISA) has led to reconsideration of ways of expressing the amount of antibody activity (De Savigny and Voller, 1980; Malvano et al., 1982). The ELISA does not depend on an endpoint titration, and allows testing at a single serum dilution (Voller et al., 1978). A variety of ways of expressing ELISA results has been advocated, but no single method satisfies the requirements of an ideal serological report (De Savigny and Voller, 1980). Both specific antibody concentration and avidity of the antibody influence the results of antibody estimations in general and those obtained by ELISAs in particular (Butler et al., 1978). Assay conditions in the ELISA affect the shape of the dilution curve (Malvano et al., 1982). Even with identical assay conditions the shape * Address for correspondence: M. Gripenberg, MD, Dept. of Bacteriology and Immunology, University of Helsinki, Haartmanink. 3, SF-00290 Helsinki 29, Finland. 0022-1759/83/$03.00 '~' 1983 Elsevier Science Publishers B.V.
316 of the dose-response curves of different serum samples varies in some ELISAs (Ahlstedt et al., 1978; Gripenberg et al., 1979). It has recently, been shown that it is possible to estimate amounts of high avidity and total antibody by analyzing the shape of the titration curves (Lehtonen and Viljanen, 1982). Antibodies reacting with D N A are a heterogeneous group of auto-antibodies characteristically present in systemic lupus erythematosus (SLE) (Barnett, 1979). The possible pathogenic role of anti-DNA antibodies and their diagnostic and prognostic value in SLE are still not settled. Using an ELISA for class-specific anti-single stranded (ss) D N A antibodies we have found that the dose-response curves between different patients' sera vary in both shape and slope (Gripenberg et al., 1978). The aim of the present study was to obtain qualitative and quantitative data about the antibody population present in different sera by expressing the results in terms of 2 numbers characterizing the shape of the dose-response curve.
Materials and Methods
Sera Serum samples from patients with various connective tissue diseases were used in the studies. Anti-ssDNA E L I S A Antibodies against denatured (ss) D N A of IgG and IgM class were demonstrated by ELISA as previously described (Gripenberg et al., 1978). Briefly. flat-bottom microtitre plates (Linbro ~/Titertek, Flow Laboratories, Hamden, CN) were incubated with 200 t~l of a solution of ssDNA (1 ~ g / m l ) in phosphate buffered saline (PBS), pH 7.2. After 3 h at 37°C the plates were washed, 150 /~1 of the serum samples diluted in PBS containing 0.05% Tween 20 (PBS-T) were added and the plates were incubated for 2 h at 37°C. After thorough washing, 150 ~1 of a 1 : 500 PBS-T dilution of a heavy chain specific anti-human IgG or IgM alkaline phosphatase conjugate (Orion Diagnostica, Helsinki) was added and the plates left at room temperature overnight. Anti-ssDNA antibody activity was then demonstrated by degradation of Sigma 104 '~ phosphatase substrate (Sigma Chemical Co., St. Louis, MO) (1 m g / m l of a 1 M diethanolamine buffer, p H 9.7, containing 0.5 mM MgCI~). The enzyme reaction was allowed to proceed for 1 h at 37°C and the absorbance values were recorded with a Titertek Multiskan photometer (Flow Laboratories, Eflab Oy, Helsinki). All samples were run in duplicate and pooled positive and negative reference sera were included in each assay. The results of the anti-ssDNA ELISA were expressed as units, i.e., the absorbance value obtained at a serum dilution of 1 : 80 calculated as a percentage of the absorbance value given by the positive reference serum at the same dilution run on the same occasion. Calculation of results The method of expressing results used depends upon approximating the function y(x), that is the absorbance obtained in ELISA when the serum dilution factor is
317 I / x , by a polynomial p ( x ) = a l x + azx 2, using the least squares method, with equal weights at the discrete measurement values of x. Only absorbance values between 2.0 and 0.1 were taken into account. It was found that the relative error in a~ obtained from 2 parallel determinations was reduced by approximating the function
xj y*(xj)=xf'fo
y(s) ds
by the second degree polynomial instead of using the values y(xj). To obtain y(s), linear interpolation was used. The same object of minimizing the relative error determined the choice of a second and not a third or higher degree polynomial, a, and a 2 m a y be obtained as linear combinations of the values y(xj), e.g., in the case when x j+ ~ = 0.5xj and there are m measurements. a i=
x/i~-~ Cm. i.jY(X,)
j=l
where the coefficients Cm,i.j are given in the appendix. The parameters used to characterize the dose-response curve are E
= al
and A = -a]/a
2
where ~, is the mean of the values a~ determined from the parallel measurements.
Interpretation of E and A Symbols are defined as follows: s = a m o u n t of antigen binding sites; b i = a m o u n t of antibodies of type i (i = 1. . . . . n) in the serum; x = 1/dilution factor; zij = a m o u n t of antibodies of type i b o u n d at j binding sites; k~j = reaction constant for a n t i b o d y i binding at j sites; m~ = n u m b e r of binding sites of antibodies of type i. F r o m the law of mass action,
(
zij(x ) s -
~ m~1jz,j(x) )-j( xb i - m=~lzi,(x ) )-I = k,j. i=lj= j=
Assuming that the absorbance due to b o u n d antibodies of type i is a, times the total a m o u n t of b o u n d antibodies of type i, we see that the total absorbance will be
mI y(x) = E
i=1
E
j=l
318
It is possible to show will be a decreasing prozone (Gripenberg approximately equal E ~
oqb i i=l
that for certain choices of the above constants the function y(x) function of x for certain sufficiently large values of x. i.e.. a et al.. 1979). Since y(x) = y'(0)x + 2 ly"(0)x2 + .... E will be to y'(0), i.e..
(m)(
,
__~ k i j s J
,j
I
m t
1 .
I
+ ~ k@ j j=l
Thus E gives an estimate of the 'effective' a m o u n t of antibody in the serum as a weighted sum of antibody concentrations with weights that are increasing functions of the binding constants. The n u m b e r A will be approximately equal to - 2 y ' ( 0 ) e / y " ( 0 ) and if n = m I = 1, then this expression will be als(l + ktls), which is an increasing function of k~. In some cases when n > 1, however, y"(x) become~ positive for certain values of x and hence the possibility that the A value calculated is negative cannot be excluded. Results
Fig. 1 shows some typical dose-response curves for anti-ssDNA antibodies in different patients' sera. Table I gives their titres, i.e., the highest dilutions giving TABLE I A N T I - s s D N A A N T I B O D Y A C T I V I T Y G I V E N AS T I T R E , C A L C U L A T E D F R O M A S I N G L E S E R U M D I L U T I O N A N D E X P R E S S E D IN T E R M S O F T H E P A R A M E T E R S E A N D A A: I g G a n t i - s s D N A a n t i b o d i e s Sera 1 c
2 3 4 5 6
Titre a :320 :160 :640 :2560 :320 :640
U b
E
A
29 37 83 240 36 155
33 19 84 525 40 121
1.8 2.3 5.1 5.9 7.7 7.6
B: IgM a n t i - s s D N A a n t i b o d i e s Sera
Titre
U
E
A
1
1:160 1:320 1:160 1:1280 1:1280 1:1280
21 68 39 116 180 195
20 34 30 124 194 195
1.4 2.0 3.4 5.8 8.2 9.6
2 3 4 5 6
Highest d i l u t i o n giving an a b s o r b a n c e value _> 0.1. U n i t value c a l c u l a t e d from a single serum d i l u t i o n in relation to a positive serum. The n u m b e r s c o r r e s p o n d to the n u m b e r s in Fig. 1.
4
1
I
t
1:20
I
1:40
I
I
1:60
1:160 SERUM
I
1:320
I
1:640
1:1260
1
1:2560
DILUTION
B
2.0
1
I
1:20
I
1:4D
1
I
I
1:160
I:80
SERUM
1:320
I
1:640
1
1:128D
DILUTION
Fig. I. A: dose-response curves of 6 different sera obtained by ELISA for IgG antibodies against ssDNA. The numbers of the sera correspond to those in Table IA. B: dose-response curves of 6 different sera obtained by ELISA for IgM antibodies against ssDNA. The numbers of the sera correspond to those in Table IB.
320 2.0_
]1 i
/2 ,1
/.3
/ / ./ .7
////
/ / /
1.5_
/
/
/
/
/ /
/
/
//.
/
/.
/
/
j j , 2
/ J
1,0_
0.5_
//Y I I
I
:A . . . .
I
I
i m
I
SERUM
DILUTION
Fig. 2. Dose-response curves of 3 different sera in the l g G - a n t i - s s D N A ELISA. The d a s h e d line~, depict the functions Ex. For serum 1. the value of E is 194 and that of A 6.8, for serum 2 the ~alues are 81 and 4.7 and for serum 3 the 3' are 47 and - 0 . 9 respectively.
absorbance values > 0.1, the unit values calculated from a single serum dilution and the corresponding numerical values of the parameters E and A. Since the ~cale is logarithmic, horizontal translation of curves such as those in Fig. 1 leaves the number A unchanged while E is multiplied by a factor corresponding to the translation. If, on the other hand, all the absorbance values are multiplied by a factor, the values E and A will be multiplied by the same factor. A non-logarithmic scale gives dose-response curves in the ELISA like those shown in Fig. 2. The dashed lines in Fig. 2 are the graphs of the functions Ex that are approximately tangent to the curves y(x) at x = 0. Calculation of the values U and E from approximately 300 sera does not show that the relative errors in the results obtained from parallel measurements are different for U and E. It was also found, as would be expected, that low values of A (or rather high values of l / A , since sometimes A was negative) are associated with low values of E but the converse did not hold.
Discussion An inherent advantage of the enzyme-linked immunosorbent assay is the possibility of quantitating specific antibody from a single serum dilution. However, anti-
321 body affinity and avidity differ between sera giving different kinds of dose-response curves (De Savigny and Voller, 1980). Thus useful information about antibodies may be lost if ELISA results are derived from a single serum dilution (Lehtonen and Eerola, 1982). The method we suggest for the characterization of the dose-response curve relies on approximating the dilution curve by a second degree polynomial. Lehtonen and Viljanen (1982) have used the function kx l/Inx~ for this purpose. One benefit of using a polynomial lies in the simplicity of the calculation, since one has only to take a linear combination of the absorbance values. In the ELISA, the actual amounts of the reagents are unknown, which means that the results are at best expressed as relative antibody activity. The parameter E, which is closely correlated with the unit values calculated from a single serum dilution, may be taken as an estimate of the 'effective' amount of antibody in the serum. It can be argued that this parameter is approximately a weighted sum of antibody concentrations, with weights dependent on the affinity and avidity of the different antibody classes present in the serum. There is an advantage in using the parameter E when testing sera with high antibody activity since the unit value cannot discriminate between sera giving absorbance values > 2.0 at the serum dilution used. Antibody populations directed against D N A are notoriously heterogeneous (Barnett, 1979) and to have found varying dose-response curves in the anti-ssDNA ELISA is not surprising. Different methods have been used to measure the avidity of anti-DNA antibodies in sera, but none is completely satisfactory (Leon et al., 1977; Devens et al., 1978; Griffiths et al., 1978). We do not claim that the parameter A calculated from the dose-response curve is proportional to the antibody avidity or affinity, but have reason to believe that high values of A (or rather, low values of l/A, taking into account the possibility that A is negative) are associated with antibody populations of high affinity and avidity. There are widely divergent opinions about the role of antibody avidity in the pathogenesis of SLE, possibly because of variations in methodology (Barnett, 1979). Qualitative differences in anti-DNA antibodies are, however, evidently associated with disease activity in SLE (Pearson and Lightfoot, 1981). The value of determining E and A parameters for anti-ssDNA antibodies in following-up patients with SLE will be investigated in a separate study.
322
Appendix T h e coefficients Cm. ~. used to calculate a~: j
1 2 3
m 2
3
4
5
6
7
8
-0.5 3.0
- 0.3762376 0.6930693 4.118812
-0.3415237 0.6017845 2.293205
-0.3317236 0.5810315 2.219561
- 0.3290626 0.5759245 2.200752
- 0.3283647 0.5746463 2.195957
- 0.3281856 0.5743258 2.194744
3.738641
2.463203 2.854678
2.442412 1.988634 1.96228
2.4371 1.984312 1.399088
2.435755
1.259115
0.9085206
4 5 6 7
1.983217 1.398316
8
0.7709885
T h e coefficients Cm. 2., used to calculate a2: j
m
1 2 3 4 5
2
3
4
5
6
7
8
1.5 - 3.0
1.306931 0.1188119 - 5.465347
1.251887 0.2635552 - 2.849691 - 5.369938
1.236237 0.2966969 - 2.732084 - 3.486189 - 4.252656
1.231974 0.3048794 - 2.701949 --3.452878 - 2.946068
1.230854 0.3069304 - 2.694254 - 3.444355 2.939133
1.230566 0.3074452 - 2.692305 - 3.442194 - 2.937374
--2.98199
- 2.12013 - 1.936715
- 2.11889 - 1.395063 - 1.195401
6 7 8
Acknowledgements We thank Ms. Anna Liisa Ruuska for excellent technical assistance and Finska L~ikares~illskapet for financial support.
References Ahlstedt, S., B. Carlsson, L.~,. H a n s s o n , B. Kaijser, I. M a t t s b y - B a l t z e r a n d S. Sohl-,~kerlund, 1978, Scand. J. l m m u n o l . 8 (Suppl. 7), 119. Barnett, E.V., 1979, J. I m m u n o l . M e t h o d s 27. 1. Butler, J.E., T.L. Feldbush, P.L. M c G i v e r n a n d N. Stewart, 1978, I m m u n o c h e m i s t r y 15, 131. D e Savigny, D. a n d A. Voller, 1980, J. I m m u n o a s s a y 1, 105. D e v e n s , B., D. Chia a n d E.V. Barnett, 1978, J. I m m u n o l . M e t h o d s 19, 187. Griffiths, G . D . , I. Olsen a n d M.W. Steward, 1978, J. I m m u n o l . M e t h o d s 21, 143. G r i p e n b e r g , M., E. Linder, P. K u r k i a n d E. Engvall, 1978, Scand. J. I m m u n o l . 7, 15 I. G r i p e n b e r g , M., A. Nissinen, E. V~iis~inen a n d E. Linder, 1979, J. Clin. Microbiol. 10, 279.
323 Lehtonen, O.-P. and E. Eerola, 1982, J. lmmunol. Methods, 54, 233. Lehtonen, O.-P. and M.K. Viljanen, 1982, Int. J. Bio-Med. Comput. 13, 471. Leon, S.A., A. Green, G.E. Ehrlich, M. Poland and B. Shapiro, 1977, Arthritis Rheum. 20, 23. Malvano, R., A. Boniolo, M. Dovis and M. Zannino, 1982, J. Immunol. Methods 48, 51. Pearson, L. and R.W. Lightfoot Jr., 1981, J. Immunol. 126, 16. Voller, A., A. Bartlett and D.E. Bidwell, 1978, J. Clin. Pathol. 31,507.