- Email: [email protected]
What can be measured with rast?
Journal o f Immunological Methods, 11 (1976) 197--212
197
© North-Holland Publishing Company, Amsterdam -- Printed in The Netherlands
WHAT CAN BE MEASURED WITH RAST?
PIERRE BONGRAND *, DANIEL VERVLOET **, RAYMOND DEPIEDS * and JACQUES CHARPIN ** • Laboratoire dTmmunologie, U.E.R. de Mddecine, Boulevard Jean Moulin, 13385 Marseille, Cedex 4, France, and ** Hbpital Sainte Marguerite, Clinique de Pneumophtisiologie, B.P. 29, 13274 Marseille Cedex 2, France
(Received 24 December 1975, accepted 1 January 1976)
A theoretical study of the basic principles involved in Radioallergosorbent test (RAST) showed that: 1) When a given serum is tested, the significance of the numerical value obtained with RAST depends upon the serum assayed and the allergosorbent preparation, in a rather unpredictable way. Three factors can be measured: a) The percentage of specific IgE antibodies among all allergen-specific antibodies; b)The specific IgE antibody level; c)The product of the specific IgE antibody level and its mean affinity constant. 2) Simple graphical techniques allow a straightforward determination of all these factors if four dilutions of each serum are assayed at the same time. The results are expressed in two constant parameters (arbitrary IgE unit and allergosorbent capacity). It is concluded that these theoretical calculations may give a fair account of a lack of correlation between specific IgE antibody levels (as assayed with RAST) and several clinical and biological parameters. Furthermore, they provide a simple procedure which makes such tedious manipulations as specific IgE antibody purification quite unnecessary.
INTRODUCTION
R a d i o a l l e r g o s o r b e n t t e s t ( R A S T ) , first d e s c r i b e d b y Wide et al. ( 1 9 6 7 ) , is t h e b e s t t e c h n i q u e available to q u a n t i t a t e I g E - t y p e reaginic a n t i b o d i e s in various sera; such assays are useful to d i a g n o s e allergic diseases a n d e x a m i n e pat i e n t s d u r i n g d e s e n s i t i z a t i o n t r e a t m e n t {Berg a n d J o h a n s s o n , 1 9 7 1 ; J o h a n s s o n e t al., 1974}. R A S T is m a d e o f 5 s e q u e n t i a l phases: 1) t h e t e s t e d s e r u m is i n c u b a t e d w i t h a given a m o u n t o f a l l e r g o s o r b e n t (insolubilized allergen); 2) t h e n , t h e a U e r g o s o r b e n t is w a s h e d u n t i l o n l y specific anti-allergen a n t i b o d i e s r e m a i n b o u n d t o t h e solid p h a s e ; 3} t h e a l l e r g o s o r b e n t is i n c u b a t e d w i t h labelled rad i o a c t i v e a n t i - h u m a n IgE; 4) t h e a l l e r g o s o r b e n t is w a s h e d again; 5) t h e radioa c t i v i t y r e t a i n e d in t h e solid p h a s e is m e a s u r e d w i t h a s p e c t r o m e t e r . It is c u r r e n t l y a s s u m e d t h a t t h e a m o u n t o f r a d i o a c t i v e anti-IgE r e t a i n e d b y t h e a U e r g o s o r b e n t gives a fair a c c o u n t o f t h e specific IgE level in t h e a s s a y e d
198 sera (Stenius et al., 1971; ZetterstrSm and Wide, 1974). In fact, this crude assumption makes the interpretation of some experimental data rather awkward. When a given serum is diluted, the a m o u n t of counted radioactivity may decrease in the same ratio, but it is rarely the case. Thus the ratio cpm/IgE level evidently depends on some neglected factors that so far remain unaccounted for (Johansson et al., 1971; Foucard et al., 1972; Vervloet et al., 1974). These factors may be: 1) the existence of affinity constant of specific reaginic antibodies that can vary in different sera; 2) the inhibition of binding of reaginic antibodies by non-IgE allergen-specific antibodies (Fujita et al., 1974}; 3) the total number of antigen-binding sites: in fact, this factor is a constant when the allergosorbent preparation procedure is not varied; 4) possible enzymatic degradation of allergen and antibody; this remains an unproven hypothesis (Aalberse et al., 1973); 5 ) t h e presence of anti-IgE antibodies in some sera has been reported, but these do not seem to alter RAST results, and the presence of anti-heterologous gamma-globulins in 20% human sera is not a problem, provided their effect is eliminated by a judicious experimental device (Johansson et al., 1974). Furthermore, it is well known that during desensitization a patient's serum IgE level, as assayed by RAST, usually reaches a maximum after several months and decreases slowly (Lichtenstein et al., 1974). In fact, it must be noted that: a) The real variation of the specific IgE level may be much more pronounced than the variation of the a m o u n t of radioactivity measured with RAST; b) An apparent change of the IgE level may be due to a change in the affinity constant of these antibodies, or to a variation of the a m o u n t of specific non-IgE antibodies, w i t h o u t any difference in the real a m o u n t of specific IgE antibodies, which might explain a lack of correlation between the evolution of specific IgE (as measured by RAST and the total IgE level (AnfossoCapra, 1974; Berg and Johansson, 1971). This paper describes a study of the basic principles of RAST in order to determine which quantity is really measured, and to provide a simple method taking account of most equilibrium factors and using some more refined assumptions. STUDY OF THE THEORETICAL MODEL We incubate in a unit volume: a) an allergosorbent containing G identical antigenic determinants, b) an adequate dilution of a serum containing T independent specific antibody combining sites. We define r as follows:
r
=
IgE antibody a m o u n t total antibody a m o u n t
Thus: there are r T IgE antibody sites; there are (1--r)T non-IgE antibody sites.
199 When the chemical equilibrium is reached, there remains in the medium: E f-free IgE antibody sites Eb-bound IgE antibody sites J E f + Eb = rT nEf-free non-IgE antibody sites }n E f + nEb = (1--r)T n E b - b o u n d non-IgE antibody sites Eb can be measured with RAST, using arbitrary units. We shall derive theoretical formulae for these quantities, using three models that rely upon some sligthly different assumptions.
FIRST MODEL Basic a s s u m p tions
We assume that all (IgE and non-IgE) antibody combining sites are identical, and all antigenic determinants are identical. We define K as the affinity constant of the antigen--antibody reaction (K = (Eb)/(ED(G--Eb--nEb) = (nEb)/(nEf)(V--Eb--nEb)).
Thus, we obtain: (see appendix I)
Eb = rG"
(K/r)(rT--Eb) 1 + (K/r)(rT--Eb)
(1)
Significance o f R A S T
In large antibody excess (T is much larger than G), we have: If K T >> 1 (high affinity constant) E b ~ rG. The quantity actually assayed is r {when G is known) so, we obtain a quantitative determination of the importance of non-IgE blocking antibody. If K T is not much larger than 1, (moderate affinity constant): Eb ~ rG • ( K t ) / (1 + K T ) . So, the numerical result depends on b o t h K T (affinity constant multiplied b y total antibody concentration) and rT(IgE antibody level). In large antigen excess (G is much larger than T), we have: If K G >> 1 (high affinity constant) Eb ~ fT. The specific IgE antibody level in the serum is actually measured, but non-IgE blocking antibodies and affinity constant are ignored. If K G is much smaller than 1 (low affinity constant) Eb G.K.
fT.
The quantity actually measured is the product of K (affinity constant) and rT (IgE antibody level). Therefore it seems obvious that the values of RAST obtained with various sera may not be compared, since their significances may be different (for example, it is meaningless to compare rG from one serum and r T from another). Thus, to compare two sera, all three quantities r, K and T must be determined, which is possible provided that various dilutions of each serum are assayed with RAST.
200
Numerical
determination
o f all r e l e v a n t p a r a m e t e r s
K / r , r T and rG can be adjusted so as to make the experimental and theoretical (eq. 1) curves obtained coincide w h e n E b is plotted versus serum dilution. When three dilutions ( 1 / 1 , 1 I x a n d l / y ) o f a serum are assayed, let a, b and c be the results respectively obtained, then, w e have (Vervloet et al., 1975): r G = a b c ( ( 1 - - 1 / x ) c _-- (1 -- 1/y)b ( l--/ _ y _ , 1/x)a) (b - - a / x ) c 2 - - ( b 2 - - a : / x ) c - - a b ( a - - b ) / y
(2)
~G
KG = 2.5 1L"
KG=IO
"
.
KG=I
KG =0.4 KG=0.1
0.5L
Z G
I E___~b rG '
I
/i
(ant,body excess)
/ .
4, kJ.
7: iz-!: zone a
nl(c~t.~ "0.1
. ',
',
excess) i
J i 1
i G
l 10
Fig. 1. Variation o f R A S T value w i t h serum dilution. E b / r G = R A S T value (arbitrary units). T / G = serum c o n c e n t r a t i o n (arbitrary units). Curves are obtained by means o f theoretical calculations, using Model I. a) D e p e n d e n c e u p o n K • G (product of affinity constant and allergosorbent c o n c e n t r a t i o n ) ; b) occurrence o f 3 zones.
201 K r
(1 -- 1 / x ) a b - - (b - - a / x ) r G =
(rG - - a ) ( r G - - b ) ( r G - - c)
ab ( a - - b) r T = (1 -- 1 / x ) a b - - (b - - a / x ) r G
(3) (4)
It must be noted that the absolute value of r remains u n k n o w n as long as G is expressed in arbitrary units (using a given immunoadsorbent sample as a reference). Thus, when two sera (1) and (2) are compared, the ratio r l . T 1 / r2 • T2 can be calculated, but the ratio of non-IgE antibody levels (i.e. (1--rl)T 1 / ( 1 - - r 2 ) T 2 ) cannot. Such a limitation does not depend upon the particular assumptions that were needed to derive formulae (2), (3) and (4) except an identical affinity constant for IgE and non-IgE antibodies. Further, using formula (1), the decrease of E b m a y be quantified when the serum is diluted ( T / G is decreased). In fig. l a , ( E b / r G ) is plotted versus ( T / G ) for various K G values. Each curve may be divided into three parts (fig. lb): Zone 1: a n t i b o d y excess; the immunosorbent is saturated, RAST is not decreased when the serum is diluted (arbitrarily, we may define zone 1 as follows: when T / G is divided by 2, E b is divided by a number smaller than 1.25). Zone 2: transitional zone (when serum is diluted twice, Eb is divided by a number larger than 1.2 and smaller than 1.75). The extent of the transitional zone depends upon KG, but not upon r; it does not depend on blocking antibodies. Zone 3: low antibody concentration: Eb is fairly proportional to 1/serum dilution (e.g. when T/G is divided by 2, Eb is divided by a number smaller than 2 and larger than 1.75). Thus, in our model, the behaviour of Eb during serum dilution depends upon KG only. SECOND MODEL Basic a s s u m p t i o n s
We assume that all the antigenic determinants are identical. We assume that all IgE a n t i b o d y combining sites are identical, and define K e as the affinity constant. We assume that all non-IgE a n t i b o d y sites are identical, and define K n e as the affinity constant. We obtain (see appendix I) Eb
= G .
Ke(rT-- Eb) 1 + Ke(rT--Eb) + Kne[(1 --r)T--nEb]
nEb = G. -Kne((1 -- r)T -- nEb) 1 + Ke(rT--Eb) + Kne[(1 -- r)T-- nEb]
(5) (6)
An algebraic resolution of equations (5) and (6) is rather intricate, but the
202 I i
Eb
?
0
.
o
5
1 2
~
1 ; 4 ~, J-6 S e r u m cMution
1 32
_1._ 64
_IZ 128
Fig. 2. Inhibition of IgE binding by non-lgE antibodies: RAST value is plotted versus serum dilution. Curve (0): IgE only (rT = 10.G, K e = 1 / G ) C u r v e (1): IgE ( r T = 10.G, K e = I / G ) + non-IgE ((]--r)T = 1000G, K n e = 1/100G) Curve (2): lgE ( r T = 10G, K e = 1 / G ) + non-IgE ((1--r)T = 100G, K n e = 1/10G) Curve (3): IgE ( r T = 10G, K e = I / G ) + non-IgE ((1--r)T = t0G, Kne = l / G ) Curve (4): IgE ( r T = 10G, K e = I / G ) + non-lgE ( ( 1 - - r ) T = G , K n e = 10/G) Curve (5): IgE (rT = 10G, K e = 1 / G ) ÷ non-IgE ({1--r)T = G , K n e = 100/G) Curve (6): IgE (rT = 10G, K e = 1 / G ) + non-lgE ((1--r)T = 10G, K n e = 100/G).
occurrence of competition between IgE and non-IgE antibodies is examined in Appendix II. Conclusions may be summarized as follows: If (1--r)T < G: no competition. If (1--r)T ~ G; K n e / K e ~ r / ( 1 - - r ) : no competition; K n e / K e r / ( 1 - - r ) : competition. It seemed of interest to determine how the curve of IgE binding was affected by non-lgE antibodies: so, arbitrary constants were chosen in order to study a numerical example thoroughly enough. The binding of IgE antibodies ( r T = 10 G, K e = 1 / G } was plotted versus serum dilution in various cases (fig. 2) -- IgE antibodies only (curve 0) -IgE antibodies + non-IgE antibodies, with: - - ( 1 - - r ) T = 1000G, K n e = 1/10G (curve 1) --(1--r)T = 100G, K n e = 1/10G (curve 2) - - ( 1 - - r ) T = 10G, K n e = 1 / G {curve 3)
203 --(1--r)T = G, K n e = I O / G (curve 4) --(1--r)T = G, K n e = 100/G (curve 5) --(1--r)T = 10G, K n e = 100/G {curve 6) It can be seen that: a) When K n e ~ K e and (1--r)T ~ r T (curves 1, 2, 3 and 4, fig. 2), competition decreases the maximum IgE binding level; this decrease depends on: [ K n e (1--r)T]/(Ke . r T ) only if (1--r)T>~ G and rT>~ G; [ K n e (1--r)T]/(Ke • rT) and (1--r)T when (1--r)T ~ G. The shape of the IgE binding/serum dilution curve is not drastically changed by non-IgE antibodies. b) When K n e > K e {curves 5 and 6, fig. 2): The maximum IgE binding level is also decreased, as was described in (a). The shape of the IgE binding/ serum dilution curve can entirely change, since serum dilution may result in an increase of IgE binding. This increase is seen when serum is diluted to 1 / x , where x ~ [(1--r)T]/G. THIRD MODEL Experimental use of eqs. (2), (3) and (4) led to paradoxical results. Experimental E b versus dilution curves generally cannot be superimposed on theoretical curves (shown in fig. 1); in fact, experimental 'transitional zones' as defined in the first model are much wider than theoretical ones, which may be due to: a) heterogeneity of antibody sites, b) multiplicity of antigenic sites, (b) was neglected and (a) was accounted for with Sips' formula (Sips, 1948). Basic a s s u m p t i o n s
We assume that all (IgE and non-IgE) antibody combining sites are characterized by: A mean affinity constant K, an heterogeneity index a: 0 < a <~ 1. When a = 1, all antibody sites are identical; heterogeneity is larger as a is smaller. Thus, we have (see Appendix I): T _ Eb + G rG
(8)
. 1--Eb/rG]
We have devised a practical method of evaluating T / G , K G , rG and a: serial two-fold dilutions (1, 1/2, 1/4, 1/8) of a given serum are assayed by RAST. Let a l , a2, a4 and a8 be the numerical results expressed in arbitrary units: 1) The values of RAST are plotted versus dilution logarithm {fig. 3), so, experimental values t h a t are obviously erroneous may be discarded and an experimental curve can be drawn. 2) (see explanation in Appendix III): the curve of y = l o g ( x / 1 - - x ) versus log(x) is drawn {fig. 4) in rectangular coordinates (xOy); then, using the same scale, three parallel lines are drawn on a tracing paper, defined as: x = log(a1); x = log(a2);x = log(a4) (see fig. 5) Then, the tracing paper is deposited on the preceding curve (fig. 6), both
204 !
251-
20
i
J
0.5
_
i
-1.5 IC~:J(x)
..1. -0.5
4-05
i
0L
j
I
!
2
I
___J~
! ! 4 8 S e r u m dilut,on
I I-~
~ -
J_ 32
F i g . 3. V a r i a t i o n o f R A S T v a l u e w i t h s e r u m d i l u t i o n ( S e r u m A ) ~ , ( M o d e l III). x . . . . . . x, experimental data. F i g . 4. T h e o r e t i c a l c u r v e o f l o g ( x / ( l - - x ) )
theoretical curve
versus log(x).
axes Oy remaining parallel. Let A1, A2, A4 be the points where each line intersects the curve. The tracing paper is slowly moved (without any rotation) until the ordinates y l , y2, y4 of A1, A2, A4 are such that: y2--yl
-- y4 - - y 2
Let log(x1) be the first coordinate of A1, we have: x l = E b / r G 1/a = l o g ( 2 ) / ( y l -- y 2 )
Thus, rG is determined (in the same arbitrary units as E b ) and so is a. 3) 1 / K G and T / G can be calculated using two linear equations:
T = x l + 1---I x l
1,o
G
K G \1 -- x l !
T _xl 8G
a8 1 ~ xl.aS/al ]lla "-~ + ~ ~ l -- x l • a 8 / a l ]
205
Both Oy .l~rallel ..,'
YI
~
• ''~'"
I
,,
I ""'.
II
J
i
".11
r'
I ~ I l
i
I i
I i I i
I ,
Ii /I / l / !
I
i
i
i
/
,
/I
L
',
I
:
I
I
I
I
I
,
,
Z'
I
I
I
I
I
/4
I
I
/
,/ I
i/.
I
( I_
......... I x /~A1 IY1
,
i
,, 2
'
I
J"
-li
1.5
/
i
, ..
log(x1)
.."
tY
"...
[
.......
I' ......
I
..........
I ...........
1i Y2
X
Fig. 5. Graphical determination of rG and 1/a (Appendix III), lines drawn of a tracing paper. Fig. 6. Graphical determination of rG and 1/a (Appendix III). As a proof o f the validity of the m e t h o d , the theoretical curve of Eb versus serum dilution can be drawn and compared to the experimental one (fig. 3); in m o s t cases, the agreement between theoretical and experimental curves is satisfactory (see figs. 3, 7, 8 and 9). Numerical examples Numerical results obtained with four different sera are shown in figs. 3, 7, 8, 9. Experimental and theoretical curves are in good agreement, which proves the validity of our graphical approximation technique. The results m a y be summarized as follows: Serum A B C D
RAST (undiluted) 24 28 22 44
rT 792 583 35.2 6490
KG 0.0625 0.73 1.86 0.016
206
! i
!
25 U
9°.L. "
I
,
,
25-
20
i
x\\\
x×
20
15F-
~.
~
,\
i
~,
,
!
I L_
I . . - - ~
1 2-
I
1 4
I.
1 1__ 8 16 Serum (511ui,On
.
I
_1_ 32
~ _ _ ~ _ _
1 -2
-L_
__
-I.
--
~.
1 1 1 4 ~ ~ Serum tl~Lutlon
A-l--
--L
1 32
1 64
--
, theoretical curve
Fig. 7. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m B) (Model III). × . . . . . . x, e x p e r i m e n t a l data. Fig. 8. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m C). ( M o d e l III). × . . . . . . x, e x p e r i m e n t a l d a t a .
__
-
, theoretical curve
It can be noticed t h a t serum t~, compared to serum A, contains t w e n t y times less antibodies whose affinity constant is thirty times higher, whereas the undiluted RAST values are similar. DISCUSSION
Three models were studied: Model 1, IgE and non-IgE antibodies are perfectly homogenous and there is only one affinity constant. Only this model yields algebraically solvable equations, but it is unable to give a fair account of: a) Quantitative experimental data concerning the evolution of inhibition of IgE binding by non-IgE antibodies when a serum is diluted, b) Experimental width of the transitional zone where IgE binding is significantly decreased when serum is diluted, but the ratio IgE binding/serum concentration depends on serum dilution.
207
50L i 40~
~li~ ~ •
302-
L
'~,
1
l
2
i
!
1
!
i
!
4 8 16 Serum ddubon
32
Fig. 9. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m D ) {Model III). x . . . . . . ×, e x p e r i m e n t a l d a t a .
, theoretical curve
Model 2 was studied in order to take account of (a): two affinity constants were defined. Model 3 was studied in order to take account of b: Sips' equation was used, to describe antibody heterogeneity. Such a study enabled us to answer two main questions: What does R A S T assay ? The significance of the quantity actually measured by RAST depends upon 5 parameters: Ke--IgE antibody affinity constant; Kne-non-IgE antibody affinity constant; T-total a n t i b o d y site concentration; r = IgE antib o d y / t o t a l antibody; G-antigenic site concentration. Let z be the quantity measured by RAST; four cases may be examined: 1) Large antibody excess: (T>> G) Ke • rT RAST measures G . . . . 1 + K e . - r T + Kne--(i ~ r ) T 1.1) high affinity constant (Ke- rT>> 1), z=G.
1/ l + K n e ( 1 - - r ) Ke. r
208 RAST gives an account of the competition between non-IgE and IgE antibodies; this competition depends only on K n e ( 1 - - r } / ( K e • r}. RAST value is n o t markedly decreased when the serum is diluted. 1 . 2 ) l o w a f f i n i t y c o n s t a n t ( K e • r T ~ 1) z=G
•
Ke. rT 1 + Kne(1 -- r)T
1.2.1} K n e (1--r)T ~ 1 ; z = G • K e • r T : RAST measures the product of affinity constant and antibody concentration. When the serum is diluted, z / s e rum concentration is a constant. 1.2.2) K n e (1--r)T ~ 1;z depends on K e • r T a n d K n e (1--r}T, z/serum conconcentration is not a constant when the serum is diluted. 2) Large antigen excess (T ~ G)
No competition can occur between IgE and non-IgE antibodies, z = f T . [ K e • G / ( 1 + K E G ) ] , z/serum concentration is a constant when the serum is diluted. 2 . 1 ) h i g h a f f i n i t y c o n s t a n t ( K e • G >> 1), z = r T , the IgE antibody concen-
tration is measured. 2 . 2 ) l o w a f f i n i t y c o n s t a n t ( K e G ~ 1}, z = G • K e • r T (as 1.2.1). Thus, when various sera are assayed with RAST, the significance of the amounts of retained radioactivity may be altogether different, and the comparison of the numerical results is meaningless. Many authors have been disappointed b y the lack of correlation between the variations of specific IgE antibodies (assayed with RAST) and several clinical or biological parameters, during desensitization treatment. So far, no satisfactory explanation of this fact has been provided. A possible interpretation is that RAST yields a numerical result whose significance m a y vary when different samples are tested. Perhaps some better correlation could be found if such quantities as: r, ratio of IgE reaginic antibodies to total (including non-IgE, possibly blocking) antibodies; r T , total a m o u n t of specific IgE antibodies; K, average affinity constant, were related to the variations of clinical and biological parameters. For example, it might be speculated that K / r determines the 'antigenic threshold level' which causes clinical trouble, and r T - - the maximum intensity of this trouble. Therefore we must answer a second question: H o w is it p o s s i b l e t o d e t e r mine the most important equilibrium factors?
The above formulae provide a simple w a y to study thoroughly a given serum: RAST may be performed with pure serum, and 1/2, 1/4 and 1/8 dilutions, rG, K G a n d r T can be calculated using a graphical technique. It must be pointed o u t that our main assumption (identity of antigen determinants) is subject to criticism when usual allergens are considered, which may alter the real significance of parameters K / r , r T and rG. Further experimental data will determine whether the calculation of these parameters may be o f clinical and biological value.
209 ACKNOWLEDGEMENT T h i s w o r k was s u p p o r t e d b y a g r a n t f r o m t h e ' I n s t i t u t N a t i o n a l de la Sant~ et de la R e c h e r c h e M~dicale', S e c t i o n 'Allerg~nes a t m o s p h ~ r i q u e s ' .
REFERENCES Aalberse, R.C., E.E. Reerink-Brongers and E. Vermeulen, 1973, Int. Arch. Allergy 45, 46. Anfosso-Capra, F., D. Vervloet, P. Autran, J. Aubert and J. Charpin, 1974, Acta AllergoIogica 29, 79. Berg, T., S.G.O. Johansson, 1971a, Int. Arch. Allergy 41,452. Berg, T. and S.G.O. Johansson, 1971b, Int. Arch. Allergy 41,434. Foucard, T., S.G.O. Johansson, H. Bennich and T. Berg, 1972, Int. Arch. Allergy 43,360. Fujita, Y, J.I. Wypich, K. Whicher, R.E. Reisman and C.E. Arbesman, 1974, J. Allergy Clin. Immunol. 53, 80, abst 40. Johansson, S.G.O., H. Bennich and T. Berg, 1971, Int. Arch. Allergy 41,443. Johansson, S.G.O., T. Berg, H. Bennich and T. Foucard, 1974, Ailergology, eds. ¥. Yamamura, O.L. Frick, Y. Horuchi, S. Kishimoto, T. Miyamoto, P. Naranjo, A. De Weck (Excerpta Medica, Amsterdam) p. 25. Johansson, S.G.O., A.C.M.L. Miller, N. Mullah, B.O. Overell, E.C. Tees and A. Wheeler, 1974, Clin. Allergy 4,255. Lichtenstein, M.L., P.S. Norman and K. Ishizaka, 1974, Allergology, eds. Y. Yamamura, O.L. Frick, Y. Horuehi, S. Kishimoto, T. Miyamoto, P. Naranjo and A. De Weck (Excerpta Medica, Amsterdam) p. 61. Sips, R, 1948, J. Chem; Phys. 16,490. Vervloet, D., P. Bongrand and J. Charpin, 1975, C.R. Acad. Sci. (in press). Vervloet, D., Y. Fujita, J.I. Wypych, R.E. Reisman and C.E. Arbesman, 1974, Clin. Allergy 4, 359. Wide, L., H. Bennich S.G.O. Johansson, 1967, Lancit ii, 1105. Zetterstr6m, O. and L. Wide, 1974, Clin. Allergy 4,273
APPENDIX I First, we d e f i n e o u r p a r a m e t e r s : E f : free IgE a n t i b o d y sites E b : b o u n d I g E a n t i b o d y sites n E f : free n o n - I g E a n t i b o d y sites n E b : b o u n d n o n - I g E a n t i b o d y sites
T: t o t a l a n t i b o d y c o n c e n t r a t i o n r = IgE antibody/total antibody G = a n t i g e n i c site c o n c e n t r a t i o n K e = IgE a n t i b o d y a f f i n i t y c o n s t a n t Kne = non-IgE antibody affinity constant. A p p l y i n g t h e law o f m a s s a c t i o n : Ke =
Eb ( r T - - E b )( G - - E b - - n E b )
1.1
210 nEb Kne = [(1--r)T--nEb] (G -- E-b-- nEbi
1.2
So: Eb = Ke(rT--
1.3
E b ) ( G -- E b -- n E b )
nEb = Kne[ (1--r)T--
1.4
nEb ] ( G -- Eb -- nEb )
The following equation is self-obvious: 1.5
G--Eb--nEb=G--Eb--nEb
Adding 1.3, 1.4 and 1.5, we have: E b + n E b + G - - E b -- n E b = 1 + K e ( r T - -
Eb) + Kne[(1--r)T--
nEb)]
×
(G -- Eb -- nEb) G -- Eb --nEb
= G/1 + Ke(rT--Eb)
+ Kne[(1--r)T--nEb)]
1.1 and 1.2 yield: G • K e ( ~ : T _ -" E b ) Eb ........... 1 + Ke(rT--Eb) + Kne[(1--r)T--nEb] nEb =
'
G "--Kne[(1------r-)T ----nEb] . . . . . . 1 + Ke(rT--Eb) + [(1--r)T--nEb]
If K e = K n e = K , dividing E b by n E b , we get: Eb nEb
rT--Eb
(1--r)T--nEb
Eb +rT--Eb nEb + (1--r)T--nEb
r 1--r
So, E b may be written as: E b = r(; • - - ( K / r ) - ( r . T Eb ) I + (K/r)(rT-Eb)
The last equation may be replaced with an empirical equation derived from Sips' formula (Sips, 1948), using a slight modification: Eb
rG
[(K/r)(rT--
. . . . . . . . . . G
r G
Eb)] a
1 + [(K/r)(rT--Eb)]
1.6 ~
+ _!_.
=
rG
KG
\1 - - E b / r G l
where 'a' is defined as the heterogeneity index. Only a comparison of theoretical curves and experimental data will prove the validity of 1.6.
211 APPENDIX II When a serum contains IgE only, we have, using the first mode]: E ' b = G . - Ke-(rT---- E--'b-)--1 + Ke(rT-- E'b)
II.1
When IgE and non-IgE antibodies are contained in the same serum: Ke(rT -- Eb ) E b = G " T :+-Ke(r-T - - El)) + K n e [ ( l - r - - ) T - - - - n--Eb
II.2
Kne[ ( l--r)T - - n E b ] nEb = G" T -UKe(rT ~ E b ) -~Kne[(l-r)T--- n--Eb-]]
II.3
IgE antibody binding is markedly reduced by non-lgE antibodies (Eb' > Eb) only if: Kne [(1--r)T -- nEb] is larger, or at least of the same order of magnitude as 1 + Ke(rT -- Eb), which may be expressed as:
Kne[(1--r)T -- nEb] >~ 1 + Ke(rT -- Eb)
II.4
II.3 yields:
nEb ~ Kne[(1--r)T--nEb] G
.....
so, as nEb < (1 -- r)T, II.4 and II.3 yields: (1 -- r)T >~ G. Thus, competition cannot occur if (1 -- r)T < < G. If IgE antibody binding is markedly reduced by non-lgE antibodies, Eb must be significantly smaller than fT. For example we may assume that: Eb ~ 3rT/4, so: rT-- Eb >~rT/4
1 + Ke(rT-- Eb) > Ke • rT/4 Then, II.4 yields:
Kne[(1 -- r)T -- nEb] > KerT/4 Kne[(l -- r)T-- nEb] > 1 so: Kne(1 --r)>~ Ke. r/4 Therefore, competition cannot occur if:
Kne(1 -- r)T < < 1 or if
Kne/Ke < < r/(l -- r)
A P P E N D I X III
It is impossible to derive parameters T / G , rG, K G and 1 / a algebraically, but in most cases, the following a p p r o x i m a t i o n is valid: if a 2 / a 4 is significant-
212 ly smaller t h a n 2, I[KG [(Eb/rG)/(1 -- Eb/rG)] 1 la m u s t be significantly larger t h a n Eb/rG, so we t e n t a t i v e l y write, instead o f (8)
G KG"
(9)
(provided serum dilution is not too high). Let Ebl, Eb2, Eb4 be the concentration of bound IgE antibodies when T is decreased to T, T]2, T/4 respectively,we have:
T _ 1 ( E b l / r G t'la V gV_l :E~-~-rV]
(10)
T _ 1 ~. Eb2/rG 111~ 2 6 K G ~I -- Eb-~r G ]
(11)
1 ~ E b 4 / r V I~1 '~ KG ~-1 -- Eb4/rG]
(12)
T4G So:
log(2) = l o g ( T / G ) - - log(T/2G) = log( T/2G) -- log(T/4G)
Eb l / r G a • log(2) = log 1 - - E b l / r G
Eb2/rG = log 1 - - Eb2/rG
Eb2/rG log 1 - - Eb2/rG
Eb4/rG log 1 - - Eb4/rG
(13)
T h u s we m u s t find t h r e e n u m b e r s E b l / r G , Eb2/rG, Eb4/rG such t h a t (13) is valid and:
E b l / a l = Eb2/a2 = Eb4]a4 or:
log(Eb2/rG) -- log(Eb l / r G ) = log(a2) - - l o g ( a l ) log(Eb4/rG) -- log(Eb l / r G ) = log(a4) - - log(a1) T h e n , using (13), " a " c o u l d be readily calculated.
(14)
197
© North-Holland Publishing Company, Amsterdam -- Printed in The Netherlands
WHAT CAN BE MEASURED WITH RAST?
PIERRE BONGRAND *, DANIEL VERVLOET **, RAYMOND DEPIEDS * and JACQUES CHARPIN ** • Laboratoire dTmmunologie, U.E.R. de Mddecine, Boulevard Jean Moulin, 13385 Marseille, Cedex 4, France, and ** Hbpital Sainte Marguerite, Clinique de Pneumophtisiologie, B.P. 29, 13274 Marseille Cedex 2, France
(Received 24 December 1975, accepted 1 January 1976)
A theoretical study of the basic principles involved in Radioallergosorbent test (RAST) showed that: 1) When a given serum is tested, the significance of the numerical value obtained with RAST depends upon the serum assayed and the allergosorbent preparation, in a rather unpredictable way. Three factors can be measured: a) The percentage of specific IgE antibodies among all allergen-specific antibodies; b)The specific IgE antibody level; c)The product of the specific IgE antibody level and its mean affinity constant. 2) Simple graphical techniques allow a straightforward determination of all these factors if four dilutions of each serum are assayed at the same time. The results are expressed in two constant parameters (arbitrary IgE unit and allergosorbent capacity). It is concluded that these theoretical calculations may give a fair account of a lack of correlation between specific IgE antibody levels (as assayed with RAST) and several clinical and biological parameters. Furthermore, they provide a simple procedure which makes such tedious manipulations as specific IgE antibody purification quite unnecessary.
INTRODUCTION
R a d i o a l l e r g o s o r b e n t t e s t ( R A S T ) , first d e s c r i b e d b y Wide et al. ( 1 9 6 7 ) , is t h e b e s t t e c h n i q u e available to q u a n t i t a t e I g E - t y p e reaginic a n t i b o d i e s in various sera; such assays are useful to d i a g n o s e allergic diseases a n d e x a m i n e pat i e n t s d u r i n g d e s e n s i t i z a t i o n t r e a t m e n t {Berg a n d J o h a n s s o n , 1 9 7 1 ; J o h a n s s o n e t al., 1974}. R A S T is m a d e o f 5 s e q u e n t i a l phases: 1) t h e t e s t e d s e r u m is i n c u b a t e d w i t h a given a m o u n t o f a l l e r g o s o r b e n t (insolubilized allergen); 2) t h e n , t h e a U e r g o s o r b e n t is w a s h e d u n t i l o n l y specific anti-allergen a n t i b o d i e s r e m a i n b o u n d t o t h e solid p h a s e ; 3} t h e a l l e r g o s o r b e n t is i n c u b a t e d w i t h labelled rad i o a c t i v e a n t i - h u m a n IgE; 4) t h e a l l e r g o s o r b e n t is w a s h e d again; 5) t h e radioa c t i v i t y r e t a i n e d in t h e solid p h a s e is m e a s u r e d w i t h a s p e c t r o m e t e r . It is c u r r e n t l y a s s u m e d t h a t t h e a m o u n t o f r a d i o a c t i v e anti-IgE r e t a i n e d b y t h e a U e r g o s o r b e n t gives a fair a c c o u n t o f t h e specific IgE level in t h e a s s a y e d
198 sera (Stenius et al., 1971; ZetterstrSm and Wide, 1974). In fact, this crude assumption makes the interpretation of some experimental data rather awkward. When a given serum is diluted, the a m o u n t of counted radioactivity may decrease in the same ratio, but it is rarely the case. Thus the ratio cpm/IgE level evidently depends on some neglected factors that so far remain unaccounted for (Johansson et al., 1971; Foucard et al., 1972; Vervloet et al., 1974). These factors may be: 1) the existence of affinity constant of specific reaginic antibodies that can vary in different sera; 2) the inhibition of binding of reaginic antibodies by non-IgE allergen-specific antibodies (Fujita et al., 1974}; 3) the total number of antigen-binding sites: in fact, this factor is a constant when the allergosorbent preparation procedure is not varied; 4) possible enzymatic degradation of allergen and antibody; this remains an unproven hypothesis (Aalberse et al., 1973); 5 ) t h e presence of anti-IgE antibodies in some sera has been reported, but these do not seem to alter RAST results, and the presence of anti-heterologous gamma-globulins in 20% human sera is not a problem, provided their effect is eliminated by a judicious experimental device (Johansson et al., 1974). Furthermore, it is well known that during desensitization a patient's serum IgE level, as assayed by RAST, usually reaches a maximum after several months and decreases slowly (Lichtenstein et al., 1974). In fact, it must be noted that: a) The real variation of the specific IgE level may be much more pronounced than the variation of the a m o u n t of radioactivity measured with RAST; b) An apparent change of the IgE level may be due to a change in the affinity constant of these antibodies, or to a variation of the a m o u n t of specific non-IgE antibodies, w i t h o u t any difference in the real a m o u n t of specific IgE antibodies, which might explain a lack of correlation between the evolution of specific IgE (as measured by RAST and the total IgE level (AnfossoCapra, 1974; Berg and Johansson, 1971). This paper describes a study of the basic principles of RAST in order to determine which quantity is really measured, and to provide a simple method taking account of most equilibrium factors and using some more refined assumptions. STUDY OF THE THEORETICAL MODEL We incubate in a unit volume: a) an allergosorbent containing G identical antigenic determinants, b) an adequate dilution of a serum containing T independent specific antibody combining sites. We define r as follows:
r
=
IgE antibody a m o u n t total antibody a m o u n t
Thus: there are r T IgE antibody sites; there are (1--r)T non-IgE antibody sites.
199 When the chemical equilibrium is reached, there remains in the medium: E f-free IgE antibody sites Eb-bound IgE antibody sites J E f + Eb = rT nEf-free non-IgE antibody sites }n E f + nEb = (1--r)T n E b - b o u n d non-IgE antibody sites Eb can be measured with RAST, using arbitrary units. We shall derive theoretical formulae for these quantities, using three models that rely upon some sligthly different assumptions.
FIRST MODEL Basic a s s u m p tions
We assume that all (IgE and non-IgE) antibody combining sites are identical, and all antigenic determinants are identical. We define K as the affinity constant of the antigen--antibody reaction (K = (Eb)/(ED(G--Eb--nEb) = (nEb)/(nEf)(V--Eb--nEb)).
Thus, we obtain: (see appendix I)
Eb = rG"
(K/r)(rT--Eb) 1 + (K/r)(rT--Eb)
(1)
Significance o f R A S T
In large antibody excess (T is much larger than G), we have: If K T >> 1 (high affinity constant) E b ~ rG. The quantity actually assayed is r {when G is known) so, we obtain a quantitative determination of the importance of non-IgE blocking antibody. If K T is not much larger than 1, (moderate affinity constant): Eb ~ rG • ( K t ) / (1 + K T ) . So, the numerical result depends on b o t h K T (affinity constant multiplied b y total antibody concentration) and rT(IgE antibody level). In large antigen excess (G is much larger than T), we have: If K G >> 1 (high affinity constant) Eb ~ fT. The specific IgE antibody level in the serum is actually measured, but non-IgE blocking antibodies and affinity constant are ignored. If K G is much smaller than 1 (low affinity constant) Eb G.K.
fT.
The quantity actually measured is the product of K (affinity constant) and rT (IgE antibody level). Therefore it seems obvious that the values of RAST obtained with various sera may not be compared, since their significances may be different (for example, it is meaningless to compare rG from one serum and r T from another). Thus, to compare two sera, all three quantities r, K and T must be determined, which is possible provided that various dilutions of each serum are assayed with RAST.
200
Numerical
determination
o f all r e l e v a n t p a r a m e t e r s
K / r , r T and rG can be adjusted so as to make the experimental and theoretical (eq. 1) curves obtained coincide w h e n E b is plotted versus serum dilution. When three dilutions ( 1 / 1 , 1 I x a n d l / y ) o f a serum are assayed, let a, b and c be the results respectively obtained, then, w e have (Vervloet et al., 1975): r G = a b c ( ( 1 - - 1 / x ) c _-- (1 -- 1/y)b ( l--/ _ y _ , 1/x)a) (b - - a / x ) c 2 - - ( b 2 - - a : / x ) c - - a b ( a - - b ) / y
(2)
~G
KG = 2.5 1L"
KG=IO
"
.
KG=I
KG =0.4 KG=0.1
0.5L
Z G
I E___~b rG '
I
/i
(ant,body excess)
/ .
4, kJ.
7: iz-!: zone a
nl(c~t.~ "0.1
. ',
',
excess) i
J i 1
i G
l 10
Fig. 1. Variation o f R A S T value w i t h serum dilution. E b / r G = R A S T value (arbitrary units). T / G = serum c o n c e n t r a t i o n (arbitrary units). Curves are obtained by means o f theoretical calculations, using Model I. a) D e p e n d e n c e u p o n K • G (product of affinity constant and allergosorbent c o n c e n t r a t i o n ) ; b) occurrence o f 3 zones.
201 K r
(1 -- 1 / x ) a b - - (b - - a / x ) r G =
(rG - - a ) ( r G - - b ) ( r G - - c)
ab ( a - - b) r T = (1 -- 1 / x ) a b - - (b - - a / x ) r G
(3) (4)
It must be noted that the absolute value of r remains u n k n o w n as long as G is expressed in arbitrary units (using a given immunoadsorbent sample as a reference). Thus, when two sera (1) and (2) are compared, the ratio r l . T 1 / r2 • T2 can be calculated, but the ratio of non-IgE antibody levels (i.e. (1--rl)T 1 / ( 1 - - r 2 ) T 2 ) cannot. Such a limitation does not depend upon the particular assumptions that were needed to derive formulae (2), (3) and (4) except an identical affinity constant for IgE and non-IgE antibodies. Further, using formula (1), the decrease of E b m a y be quantified when the serum is diluted ( T / G is decreased). In fig. l a , ( E b / r G ) is plotted versus ( T / G ) for various K G values. Each curve may be divided into three parts (fig. lb): Zone 1: a n t i b o d y excess; the immunosorbent is saturated, RAST is not decreased when the serum is diluted (arbitrarily, we may define zone 1 as follows: when T / G is divided by 2, E b is divided by a number smaller than 1.25). Zone 2: transitional zone (when serum is diluted twice, Eb is divided by a number larger than 1.2 and smaller than 1.75). The extent of the transitional zone depends upon KG, but not upon r; it does not depend on blocking antibodies. Zone 3: low antibody concentration: Eb is fairly proportional to 1/serum dilution (e.g. when T/G is divided by 2, Eb is divided by a number smaller than 2 and larger than 1.75). Thus, in our model, the behaviour of Eb during serum dilution depends upon KG only. SECOND MODEL Basic a s s u m p t i o n s
We assume that all the antigenic determinants are identical. We assume that all IgE a n t i b o d y combining sites are identical, and define K e as the affinity constant. We assume that all non-IgE a n t i b o d y sites are identical, and define K n e as the affinity constant. We obtain (see appendix I) Eb
= G .
Ke(rT-- Eb) 1 + Ke(rT--Eb) + Kne[(1 --r)T--nEb]
nEb = G. -Kne((1 -- r)T -- nEb) 1 + Ke(rT--Eb) + Kne[(1 -- r)T-- nEb]
(5) (6)
An algebraic resolution of equations (5) and (6) is rather intricate, but the
202 I i
Eb
?
0
.
o
5
1 2
~
1 ; 4 ~, J-6 S e r u m cMution
1 32
_1._ 64
_IZ 128
Fig. 2. Inhibition of IgE binding by non-lgE antibodies: RAST value is plotted versus serum dilution. Curve (0): IgE only (rT = 10.G, K e = 1 / G ) C u r v e (1): IgE ( r T = 10.G, K e = I / G ) + non-IgE ((]--r)T = 1000G, K n e = 1/100G) Curve (2): lgE ( r T = 10G, K e = 1 / G ) + non-IgE ((1--r)T = 100G, K n e = 1/10G) Curve (3): IgE ( r T = 10G, K e = I / G ) + non-IgE ((1--r)T = t0G, Kne = l / G ) Curve (4): IgE ( r T = 10G, K e = I / G ) + non-lgE ( ( 1 - - r ) T = G , K n e = 10/G) Curve (5): IgE (rT = 10G, K e = 1 / G ) ÷ non-IgE ({1--r)T = G , K n e = 100/G) Curve (6): IgE (rT = 10G, K e = 1 / G ) + non-lgE ((1--r)T = 10G, K n e = 100/G).
occurrence of competition between IgE and non-IgE antibodies is examined in Appendix II. Conclusions may be summarized as follows: If (1--r)T < G: no competition. If (1--r)T ~ G; K n e / K e ~ r / ( 1 - - r ) : no competition; K n e / K e r / ( 1 - - r ) : competition. It seemed of interest to determine how the curve of IgE binding was affected by non-lgE antibodies: so, arbitrary constants were chosen in order to study a numerical example thoroughly enough. The binding of IgE antibodies ( r T = 10 G, K e = 1 / G } was plotted versus serum dilution in various cases (fig. 2) -- IgE antibodies only (curve 0) -IgE antibodies + non-IgE antibodies, with: - - ( 1 - - r ) T = 1000G, K n e = 1/10G (curve 1) --(1--r)T = 100G, K n e = 1/10G (curve 2) - - ( 1 - - r ) T = 10G, K n e = 1 / G {curve 3)
203 --(1--r)T = G, K n e = I O / G (curve 4) --(1--r)T = G, K n e = 100/G (curve 5) --(1--r)T = 10G, K n e = 100/G {curve 6) It can be seen that: a) When K n e ~ K e and (1--r)T ~ r T (curves 1, 2, 3 and 4, fig. 2), competition decreases the maximum IgE binding level; this decrease depends on: [ K n e (1--r)T]/(Ke . r T ) only if (1--r)T>~ G and rT>~ G; [ K n e (1--r)T]/(Ke • rT) and (1--r)T when (1--r)T ~ G. The shape of the IgE binding/serum dilution curve is not drastically changed by non-IgE antibodies. b) When K n e > K e {curves 5 and 6, fig. 2): The maximum IgE binding level is also decreased, as was described in (a). The shape of the IgE binding/ serum dilution curve can entirely change, since serum dilution may result in an increase of IgE binding. This increase is seen when serum is diluted to 1 / x , where x ~ [(1--r)T]/G. THIRD MODEL Experimental use of eqs. (2), (3) and (4) led to paradoxical results. Experimental E b versus dilution curves generally cannot be superimposed on theoretical curves (shown in fig. 1); in fact, experimental 'transitional zones' as defined in the first model are much wider than theoretical ones, which may be due to: a) heterogeneity of antibody sites, b) multiplicity of antigenic sites, (b) was neglected and (a) was accounted for with Sips' formula (Sips, 1948). Basic a s s u m p t i o n s
We assume that all (IgE and non-IgE) antibody combining sites are characterized by: A mean affinity constant K, an heterogeneity index a: 0 < a <~ 1. When a = 1, all antibody sites are identical; heterogeneity is larger as a is smaller. Thus, we have (see Appendix I): T _ Eb + G rG
(8)
. 1--Eb/rG]
We have devised a practical method of evaluating T / G , K G , rG and a: serial two-fold dilutions (1, 1/2, 1/4, 1/8) of a given serum are assayed by RAST. Let a l , a2, a4 and a8 be the numerical results expressed in arbitrary units: 1) The values of RAST are plotted versus dilution logarithm {fig. 3), so, experimental values t h a t are obviously erroneous may be discarded and an experimental curve can be drawn. 2) (see explanation in Appendix III): the curve of y = l o g ( x / 1 - - x ) versus log(x) is drawn {fig. 4) in rectangular coordinates (xOy); then, using the same scale, three parallel lines are drawn on a tracing paper, defined as: x = log(a1); x = log(a2);x = log(a4) (see fig. 5) Then, the tracing paper is deposited on the preceding curve (fig. 6), both
204 !
251-
20
i
J
0.5
_
i
-1.5 IC~:J(x)
..1. -0.5
4-05
i
0L
j
I
!
2
I
___J~
! ! 4 8 S e r u m dilut,on
I I-~
~ -
J_ 32
F i g . 3. V a r i a t i o n o f R A S T v a l u e w i t h s e r u m d i l u t i o n ( S e r u m A ) ~ , ( M o d e l III). x . . . . . . x, experimental data. F i g . 4. T h e o r e t i c a l c u r v e o f l o g ( x / ( l - - x ) )
theoretical curve
versus log(x).
axes Oy remaining parallel. Let A1, A2, A4 be the points where each line intersects the curve. The tracing paper is slowly moved (without any rotation) until the ordinates y l , y2, y4 of A1, A2, A4 are such that: y2--yl
-- y4 - - y 2
Let log(x1) be the first coordinate of A1, we have: x l = E b / r G 1/a = l o g ( 2 ) / ( y l -- y 2 )
Thus, rG is determined (in the same arbitrary units as E b ) and so is a. 3) 1 / K G and T / G can be calculated using two linear equations:
T = x l + 1---I x l
1,o
G
K G \1 -- x l !
T _xl 8G
a8 1 ~ xl.aS/al ]lla "-~ + ~ ~ l -- x l • a 8 / a l ]
205
Both Oy .l~rallel ..,'
YI
~
• ''~'"
I
,,
I ""'.
II
J
i
".11
r'
I ~ I l
i
I i
I i I i
I ,
Ii /I / l / !
I
i
i
i
/
,
/I
L
',
I
:
I
I
I
I
I
,
,
Z'
I
I
I
I
I
/4
I
I
/
,/ I
i/.
I
( I_
......... I x /~A1 IY1
,
i
,, 2
'
I
J"
-li
1.5
/
i
, ..
log(x1)
.."
tY
"...
[
.......
I' ......
I
..........
I ...........
1i Y2
X
Fig. 5. Graphical determination of rG and 1/a (Appendix III), lines drawn of a tracing paper. Fig. 6. Graphical determination of rG and 1/a (Appendix III). As a proof o f the validity of the m e t h o d , the theoretical curve of Eb versus serum dilution can be drawn and compared to the experimental one (fig. 3); in m o s t cases, the agreement between theoretical and experimental curves is satisfactory (see figs. 3, 7, 8 and 9). Numerical examples Numerical results obtained with four different sera are shown in figs. 3, 7, 8, 9. Experimental and theoretical curves are in good agreement, which proves the validity of our graphical approximation technique. The results m a y be summarized as follows: Serum A B C D
RAST (undiluted) 24 28 22 44
rT 792 583 35.2 6490
KG 0.0625 0.73 1.86 0.016
206
! i
!
25 U
9°.L. "
I
,
,
25-
20
i
x\\\
x×
20
15F-
~.
~
,\
i
~,
,
!
I L_
I . . - - ~
1 2-
I
1 4
I.
1 1__ 8 16 Serum (511ui,On
.
I
_1_ 32
~ _ _ ~ _ _
1 -2
-L_
__
-I.
--
~.
1 1 1 4 ~ ~ Serum tl~Lutlon
A-l--
--L
1 32
1 64
--
, theoretical curve
Fig. 7. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m B) (Model III). × . . . . . . x, e x p e r i m e n t a l data. Fig. 8. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m C). ( M o d e l III). × . . . . . . x, e x p e r i m e n t a l d a t a .
__
-
, theoretical curve
It can be noticed t h a t serum t~, compared to serum A, contains t w e n t y times less antibodies whose affinity constant is thirty times higher, whereas the undiluted RAST values are similar. DISCUSSION
Three models were studied: Model 1, IgE and non-IgE antibodies are perfectly homogenous and there is only one affinity constant. Only this model yields algebraically solvable equations, but it is unable to give a fair account of: a) Quantitative experimental data concerning the evolution of inhibition of IgE binding by non-IgE antibodies when a serum is diluted, b) Experimental width of the transitional zone where IgE binding is significantly decreased when serum is diluted, but the ratio IgE binding/serum concentration depends on serum dilution.
207
50L i 40~
~li~ ~ •
302-
L
'~,
1
l
2
i
!
1
!
i
!
4 8 16 Serum ddubon
32
Fig. 9. V a r i a t i o n o f R A S T value w i t h s e r u m d i l u t i o n ( S e r u m D ) {Model III). x . . . . . . ×, e x p e r i m e n t a l d a t a .
, theoretical curve
Model 2 was studied in order to take account of (a): two affinity constants were defined. Model 3 was studied in order to take account of b: Sips' equation was used, to describe antibody heterogeneity. Such a study enabled us to answer two main questions: What does R A S T assay ? The significance of the quantity actually measured by RAST depends upon 5 parameters: Ke--IgE antibody affinity constant; Kne-non-IgE antibody affinity constant; T-total a n t i b o d y site concentration; r = IgE antib o d y / t o t a l antibody; G-antigenic site concentration. Let z be the quantity measured by RAST; four cases may be examined: 1) Large antibody excess: (T>> G) Ke • rT RAST measures G . . . . 1 + K e . - r T + Kne--(i ~ r ) T 1.1) high affinity constant (Ke- rT>> 1), z=G.
1/ l + K n e ( 1 - - r ) Ke. r
208 RAST gives an account of the competition between non-IgE and IgE antibodies; this competition depends only on K n e ( 1 - - r } / ( K e • r}. RAST value is n o t markedly decreased when the serum is diluted. 1 . 2 ) l o w a f f i n i t y c o n s t a n t ( K e • r T ~ 1) z=G
•
Ke. rT 1 + Kne(1 -- r)T
1.2.1} K n e (1--r)T ~ 1 ; z = G • K e • r T : RAST measures the product of affinity constant and antibody concentration. When the serum is diluted, z / s e rum concentration is a constant. 1.2.2) K n e (1--r)T ~ 1;z depends on K e • r T a n d K n e (1--r}T, z/serum conconcentration is not a constant when the serum is diluted. 2) Large antigen excess (T ~ G)
No competition can occur between IgE and non-IgE antibodies, z = f T . [ K e • G / ( 1 + K E G ) ] , z/serum concentration is a constant when the serum is diluted. 2 . 1 ) h i g h a f f i n i t y c o n s t a n t ( K e • G >> 1), z = r T , the IgE antibody concen-
tration is measured. 2 . 2 ) l o w a f f i n i t y c o n s t a n t ( K e G ~ 1}, z = G • K e • r T (as 1.2.1). Thus, when various sera are assayed with RAST, the significance of the amounts of retained radioactivity may be altogether different, and the comparison of the numerical results is meaningless. Many authors have been disappointed b y the lack of correlation between the variations of specific IgE antibodies (assayed with RAST) and several clinical or biological parameters, during desensitization treatment. So far, no satisfactory explanation of this fact has been provided. A possible interpretation is that RAST yields a numerical result whose significance m a y vary when different samples are tested. Perhaps some better correlation could be found if such quantities as: r, ratio of IgE reaginic antibodies to total (including non-IgE, possibly blocking) antibodies; r T , total a m o u n t of specific IgE antibodies; K, average affinity constant, were related to the variations of clinical and biological parameters. For example, it might be speculated that K / r determines the 'antigenic threshold level' which causes clinical trouble, and r T - - the maximum intensity of this trouble. Therefore we must answer a second question: H o w is it p o s s i b l e t o d e t e r mine the most important equilibrium factors?
The above formulae provide a simple w a y to study thoroughly a given serum: RAST may be performed with pure serum, and 1/2, 1/4 and 1/8 dilutions, rG, K G a n d r T can be calculated using a graphical technique. It must be pointed o u t that our main assumption (identity of antigen determinants) is subject to criticism when usual allergens are considered, which may alter the real significance of parameters K / r , r T and rG. Further experimental data will determine whether the calculation of these parameters may be o f clinical and biological value.
209 ACKNOWLEDGEMENT T h i s w o r k was s u p p o r t e d b y a g r a n t f r o m t h e ' I n s t i t u t N a t i o n a l de la Sant~ et de la R e c h e r c h e M~dicale', S e c t i o n 'Allerg~nes a t m o s p h ~ r i q u e s ' .
REFERENCES Aalberse, R.C., E.E. Reerink-Brongers and E. Vermeulen, 1973, Int. Arch. Allergy 45, 46. Anfosso-Capra, F., D. Vervloet, P. Autran, J. Aubert and J. Charpin, 1974, Acta AllergoIogica 29, 79. Berg, T., S.G.O. Johansson, 1971a, Int. Arch. Allergy 41,452. Berg, T. and S.G.O. Johansson, 1971b, Int. Arch. Allergy 41,434. Foucard, T., S.G.O. Johansson, H. Bennich and T. Berg, 1972, Int. Arch. Allergy 43,360. Fujita, Y, J.I. Wypich, K. Whicher, R.E. Reisman and C.E. Arbesman, 1974, J. Allergy Clin. Immunol. 53, 80, abst 40. Johansson, S.G.O., H. Bennich and T. Berg, 1971, Int. Arch. Allergy 41,443. Johansson, S.G.O., T. Berg, H. Bennich and T. Foucard, 1974, Ailergology, eds. ¥. Yamamura, O.L. Frick, Y. Horuchi, S. Kishimoto, T. Miyamoto, P. Naranjo, A. De Weck (Excerpta Medica, Amsterdam) p. 25. Johansson, S.G.O., A.C.M.L. Miller, N. Mullah, B.O. Overell, E.C. Tees and A. Wheeler, 1974, Clin. Allergy 4,255. Lichtenstein, M.L., P.S. Norman and K. Ishizaka, 1974, Allergology, eds. Y. Yamamura, O.L. Frick, Y. Horuehi, S. Kishimoto, T. Miyamoto, P. Naranjo and A. De Weck (Excerpta Medica, Amsterdam) p. 61. Sips, R, 1948, J. Chem; Phys. 16,490. Vervloet, D., P. Bongrand and J. Charpin, 1975, C.R. Acad. Sci. (in press). Vervloet, D., Y. Fujita, J.I. Wypych, R.E. Reisman and C.E. Arbesman, 1974, Clin. Allergy 4, 359. Wide, L., H. Bennich S.G.O. Johansson, 1967, Lancit ii, 1105. Zetterstr6m, O. and L. Wide, 1974, Clin. Allergy 4,273
APPENDIX I First, we d e f i n e o u r p a r a m e t e r s : E f : free IgE a n t i b o d y sites E b : b o u n d I g E a n t i b o d y sites n E f : free n o n - I g E a n t i b o d y sites n E b : b o u n d n o n - I g E a n t i b o d y sites
T: t o t a l a n t i b o d y c o n c e n t r a t i o n r = IgE antibody/total antibody G = a n t i g e n i c site c o n c e n t r a t i o n K e = IgE a n t i b o d y a f f i n i t y c o n s t a n t Kne = non-IgE antibody affinity constant. A p p l y i n g t h e law o f m a s s a c t i o n : Ke =
Eb ( r T - - E b )( G - - E b - - n E b )
1.1
210 nEb Kne = [(1--r)T--nEb] (G -- E-b-- nEbi
1.2
So: Eb = Ke(rT--
1.3
E b ) ( G -- E b -- n E b )
nEb = Kne[ (1--r)T--
1.4
nEb ] ( G -- Eb -- nEb )
The following equation is self-obvious: 1.5
G--Eb--nEb=G--Eb--nEb
Adding 1.3, 1.4 and 1.5, we have: E b + n E b + G - - E b -- n E b = 1 + K e ( r T - -
Eb) + Kne[(1--r)T--
nEb)]
×
(G -- Eb -- nEb) G -- Eb --nEb
= G/1 + Ke(rT--Eb)
+ Kne[(1--r)T--nEb)]
1.1 and 1.2 yield: G • K e ( ~ : T _ -" E b ) Eb ........... 1 + Ke(rT--Eb) + Kne[(1--r)T--nEb] nEb =
'
G "--Kne[(1------r-)T ----nEb] . . . . . . 1 + Ke(rT--Eb) + [(1--r)T--nEb]
If K e = K n e = K , dividing E b by n E b , we get: Eb nEb
rT--Eb
(1--r)T--nEb
Eb +rT--Eb nEb + (1--r)T--nEb
r 1--r
So, E b may be written as: E b = r(; • - - ( K / r ) - ( r . T Eb ) I + (K/r)(rT-Eb)
The last equation may be replaced with an empirical equation derived from Sips' formula (Sips, 1948), using a slight modification: Eb
rG
[(K/r)(rT--
. . . . . . . . . . G
r G
Eb)] a
1 + [(K/r)(rT--Eb)]
1.6 ~
+ _!_.
=
rG
KG
\1 - - E b / r G l
where 'a' is defined as the heterogeneity index. Only a comparison of theoretical curves and experimental data will prove the validity of 1.6.
211 APPENDIX II When a serum contains IgE only, we have, using the first mode]: E ' b = G . - Ke-(rT---- E--'b-)--1 + Ke(rT-- E'b)
II.1
When IgE and non-IgE antibodies are contained in the same serum: Ke(rT -- Eb ) E b = G " T :+-Ke(r-T - - El)) + K n e [ ( l - r - - ) T - - - - n--Eb
II.2
Kne[ ( l--r)T - - n E b ] nEb = G" T -UKe(rT ~ E b ) -~Kne[(l-r)T--- n--Eb-]]
II.3
IgE antibody binding is markedly reduced by non-lgE antibodies (Eb' > Eb) only if: Kne [(1--r)T -- nEb] is larger, or at least of the same order of magnitude as 1 + Ke(rT -- Eb), which may be expressed as:
Kne[(1--r)T -- nEb] >~ 1 + Ke(rT -- Eb)
II.4
II.3 yields:
nEb ~ Kne[(1--r)T--nEb] G
.....
so, as nEb < (1 -- r)T, II.4 and II.3 yields: (1 -- r)T >~ G. Thus, competition cannot occur if (1 -- r)T < < G. If IgE antibody binding is markedly reduced by non-lgE antibodies, Eb must be significantly smaller than fT. For example we may assume that: Eb ~ 3rT/4, so: rT-- Eb >~rT/4
1 + Ke(rT-- Eb) > Ke • rT/4 Then, II.4 yields:
Kne[(1 -- r)T -- nEb] > KerT/4 Kne[(l -- r)T-- nEb] > 1 so: Kne(1 --r)>~ Ke. r/4 Therefore, competition cannot occur if:
Kne(1 -- r)T < < 1 or if
Kne/Ke < < r/(l -- r)
A P P E N D I X III
It is impossible to derive parameters T / G , rG, K G and 1 / a algebraically, but in most cases, the following a p p r o x i m a t i o n is valid: if a 2 / a 4 is significant-
212 ly smaller t h a n 2, I[KG [(Eb/rG)/(1 -- Eb/rG)] 1 la m u s t be significantly larger t h a n Eb/rG, so we t e n t a t i v e l y write, instead o f (8)
G KG"
(9)
(provided serum dilution is not too high). Let Ebl, Eb2, Eb4 be the concentration of bound IgE antibodies when T is decreased to T, T]2, T/4 respectively,we have:
T _ 1 ( E b l / r G t'la V gV_l :E~-~-rV]
(10)
T _ 1 ~. Eb2/rG 111~ 2 6 K G ~I -- Eb-~r G ]
(11)
1 ~ E b 4 / r V I~1 '~ KG ~-1 -- Eb4/rG]
(12)
T4G So:
log(2) = l o g ( T / G ) - - log(T/2G) = log( T/2G) -- log(T/4G)
Eb l / r G a • log(2) = log 1 - - E b l / r G
Eb2/rG = log 1 - - Eb2/rG
Eb2/rG log 1 - - Eb2/rG
Eb4/rG log 1 - - Eb4/rG
(13)
T h u s we m u s t find t h r e e n u m b e r s E b l / r G , Eb2/rG, Eb4/rG such t h a t (13) is valid and:
E b l / a l = Eb2/a2 = Eb4]a4 or:
log(Eb2/rG) -- log(Eb l / r G ) = log(a2) - - l o g ( a l ) log(Eb4/rG) -- log(Eb l / r G ) = log(a4) - - log(a1) T h e n , using (13), " a " c o u l d be readily calculated.
(14)