784
Lwus PAULING, DAVID F~ESSMAN, AND ALLAN L. GROWERG
2. The interference of alcoholic hydroxyl has been investigated, including the relative reactivity of primary, secondary and tertiarv hydroxyl.
Vol. 66
3. The method can be used indirectly for the determination of tertiary amine. WILMINGTON,
DELAWARERECEIVED FEBRUARY 17, 1944
[C~NTIUBVTION FROM THE GATESAND CRELLTN LABORATOXXES OF CHEMISTRY, CALIFORNIA INSTITUTE OF TECHNOLOGY, No. 9,561
The Serological Properties of Simple Substances. VII. A Quantitative Theory of the Inhibition by Haptens of the Precipitation of Heterogeneaus Antisera with Antigens, and Comparison with Experimental Results for Polyhaptenic Simple Substances and for Azoproteins BY LINUSP A U L I N G , DAVID PRESSMAN,
In an earlier paper in this series’ there was developed a simple physicochemical theory of the precipitation of bivalent antibody molecules and bivalent antigen molecules and the inhibition of precipitation by haptens, and it was shownlJ that the results of experiments on the Precipitation of specific antisera by polyhaptenic substances agree qualitatively but not quantitatively with the theory: in particular, the predicted linear decrease in amount of precipitate with increase in amount of hapten was observed only in the region of small inhibition. The deviation from linearity for larger amounts of hapten was attributed to the heterogeneity of the antisera, which were assumed, as is indicated by many experimental observations, to contain antibody molecules with greatly varying combining powers. We have now developed an extended quantitative theory of hapten inhibition on the assumption of an errorfunction distribution of antibody .molecules with Merent combining powers in a heterogeneous antiserum, and have found it to be in generally satisfactory quantitative agreement with experiment. The theory permits the evaluation of two constants from each hapten-inhibition experiment, an average bond-strength constant for antibody and hapten (in competition with antigen) and an effective heterogeneity index for the antiserum; values of each of these constants can be interpreted in relation to the molecular structure of the hapten and the antigen. There have previously been reported* the results of quantitative investigations of the inhibition by haptens of the precipitation of polyhaptenic simple substances by antisera made by inoculating rabbits with sheep’serum coupled either with diazotized p-arsanilic acid2 (anti-R sera) or with diazotized p-(p-amhophenylazo)phenylarsonic acid3 (anti-R’ sera). It was found that the relative inhibiting powers of various haptens are essentially the same for different
AND
ALLANL.GROSSBERG
polyhaptenic simple substances precipitated by the s a m e p l of antiserum, but are somewhat M e r ent for Merent pools of anti-R serum or of anti-R’ serum precipitated by the same polyhaptenic substance. Stit1 greater differences are observed between anti-R sera and anti-R’ sera in general. We have now carried out a comparative study of hapten, inhibition of the precipitation of a polyhaptenic simple substance (XXX) and an azo OH OH SO,H(J,JSO,H R‘
=
-NN~CZ)A~~O,H~
protein (K‘-ovalbumin) by anti-R serum and anti-R’ serum, and of a polyhaptenic substance (XI)by anti-R serum obtained by a single three OH
R~AR~ HOOOH R’
weeks’ course of inoculations, with the results described below. The data obtained in these and the earlier experiments have been analyzed by use of the new theory, and the results are discussed in relation to the molecular structure of the interacting substances.
Experimental Methods
JOURNAL., 64,3003 (1942)
Yateriak-There have been previously described the ~ ~ *haptens and methods of preparation of the a n t i ~ e r a ,the antigens,*,8~~ and the R’-ovalbumin,’ which contained 2 per cent. arsenic. The Rscletion of Antiserum with Antigen md Eapten.The-reaction mixtures were set up in triplicste, with use in each series of experiments of the amount of antigen giving the largest amount of precipitate in the absence of hapten; borate buffer was used as di1uent.l The tubes were allowed to stand one hour at room temperature and over two nights m the refrigerator, and the precipitates werr then analyzed by our standard method.’ The results of the study of the inhibition by each of 24 haptens of the precipitation of antiserum with antigen
(2) D. Pressman. D H Brown, and L Pauling, ibrd, 64,3015 (1942) (3) D P m m a n , J T Maynard, A L Grossberg, End Llnus Paulrng. l b d . 61, 728 (1943)
(4) L. Pauling, D. Pressman, I3 H Campbell, C. Ikeda,and M Ikawa. ibid., 64, 2994 (1942). (5) D Pressmao,qnd. Enp C k m , Anal. Ed., 16, 387 (1943).
(1) L. P d i o g , D Pressman D H Campbell and C. Ikedr. Tars
786
INHIBITION BY HAPTENSOF ANTIGEN PRECIPITATION OF ANTISRRA
May, 1944
TABLEI HAPTBNINHIBITION OF PRECIPITATION OF ANTI-RSERUM WITH ANTIGEN AND WITH R'-OVALBVM For antigen XXX: antigeh solution, 1.76 ml. (22 pg.); anti-R serum, 0.25 ml.;. hapten solution, 1 ml. For R'-ovolbumin: antigen solution, 1.80 ml. (140 pg.); anti-R serum, 0.20 ml.; hapten solution, 1 ml. pH of all supernates 8.068.10.
xxx
Antiga
xxx
K,'
Acid p-(p-Aminopheny1azo)1.00 phcnylamonic P: (fi-HydtoXyphenylpzo) 1.00 phenylarwnic p- Acetunbophcnylannic 1.00 9- (9-Nitrobenzoylamino)-
-
0.43 phenyhnic p- (p-Arninobenzoylamino).54 phenylarsonic p- Benzoylaminophenyl.83 arsonic 1.52 9-Nitrophenylarsonic 0.47 n-Nitrophcnylarsonic .095 o-Nitrophenylrrsonic .89 p-Iod&enpl.rsonie
p-Brornophenyluronic p-ChlorophenyIanonic 9-hftthylphenyhrsonic m-MethylphcaylPrsonic a-Methylplunylotsonic 8-Naphthylarsonic a-Naphthylotsopic
1,4-Aminonaphthylprsonic 9- Hydroxyphenylarsic p-Carboxyphcnylarsonic Phenylarsonic 9-Aminophenylarsonic m-Aminophenplarsonic o-Aminophenglarsonic
.81 .71
.so .22 ,022 .41 .061 ,088 .22 .19 ,137 .19 ,127 .027
a
R'-oval-
2.4
$*#
964'
896b
88Ob
JIO
96Sb 780
914' 834
62ab 421
766
827
sal
6i3e
440
1.6 816
768
578
844
710
580
1.2 889 1.2 719 1.6 1.2 1.2 1.5 1.2 1.0 1.6 1.0 1.2 1.5 1.0 1.5 1.6 1.6 1.5 1.5 1.0
760 640
585 291
Wbl 648
2.0 1.34
1.0 906b 872b 815'
1.6 1.34 1.2 1.07
1.0 946' 1.5 815
920'
2.1 0.98
1.2 849
.77
2.0
1.6 .71 1.2 2.06 1.0 0.48 1.2 .os9 1.0 .98 1.0 .81 1.0 .69 1.1 .so 1.0 .18 1.8 .031 1.1 .49 1.5 ,049 1.0 .079 1.8 .20 1.0 .17 1.2 ,125 0.9 .18 1.1 .125 1.6 ,035
ma
843b
nsn
494 521 682 880 885 723 79.5 89R
(4621 218 726 527 288 226 80 262 99 301 122 448 192 732 500 913 846 726 518 262 802 646 463 764 668 321 855 480 643 282 122 630 410 179 766 640 867 428 192 916 807 879
[746] 16661 437 193 813 678 414 880 469 216 661 514 291 687 662 322 769 668 463 839 &4 748 916 980 446 868 820 919 684 824 700 919
776 OOO
918 882 832 797 798 602
761 676 368
908
861
804 em 484 733 781 mi 497 864 908 004
0 The amounts are tabulated as fractions per mille of the amount precipitated in absence of hapten: for antigen XXX this was 715 pg. and for R'-ovalbumin 769 pg. Values for R'-ovalbumin include the precipitated antigen protdo. Values are averages for triplicate analyses, with mean deviation f2%; single analyses are given in brackets. These values are for hapten concentrations one-fifth of those indicated.
in which AB (pp) is the amount of precipitate, the total amount of hapten present in all sera were obtained by successive inoculationsand bleedings of several rabbits over a period of several months; for com- forms, K the equilibrium constant for combinaparison a study was also made of the effect of fifteen hap- tion of a haptenic group of the antigen and a tens on the precipitation by antigen XI of an anti-R serum complementary region of the antibody, K' the obtained after a single three-weeks' course of inoculations, corresponding constant for hapten and antibody, with the results given in Table 111. and s .the solubility of the precipitate. This equation may be rewritten2in the form Discussion
XXX and with R'-ovalbumin are given for anti-R serum in Table I and for anti-R' serum in Table 11. These anti-
In an earlier paper' it was shown that in a system containing bivalent antigen, A, homogeneous bivalent antibody, B, and univalent hapten, H, and capable of forming the soluble complexes AB, ABA, HB, HBA, and HBH and the precipitate AB the amount of precipitate, calculated according to the principles of chemical equilibrium with use of reasonable values for the pertinent equilibrium constants, falls off, in the region of the equivalence zone, in linear relation to the amount of hapten present. The slope of the curve for a system containing equivalent amounts of antigen and antibody was found to be given by the equation
Htotpl
with C and C' independent of hapten, and constant for a given antigen-antibody system; and it has been suggested2that, as an approximation, the constant C' may be neglected, and the hapten inhibition constants K' for a series of haptens be taken as roportional to the negative slopes -d'4B ( P P ) l J d .
It has been found,1~2~* however, that the experiniental points showing the dependence of the amount of precipitate on the amount of added hapten do not fall on a straight line except in the region of very low hapten concentration-with larger amounts of hapten the inhibition of precipitation is less than predicted by the theory (see Fig. 1 of ref. 2); and this deviation from linearity has been attributed to heterogeneity
LINUSP A U L I N G , DAVIDP R E S S M A N ,
786
AND
ALLANL.GROSSBERG
Vol. 66
TABLE I1 HAPTENINHIBITION OF PRECIPITATION OF ANTI-R'SERUM WITH ANTIGEN AND WIl" R'-OVALBLMRT For antigen XXX: antigen solution, 1.25 ml. (67 p g . ) ; anti-R' s e f m , 0.75 ml.; hapten solution, 1.0 ml: For R'ovalbumin: antigen solution, 1.50 ml. (175 p g . ) ; anti-R'serum, 0.50 ml.; hapten solution, 1.0 ml. pH of all supernates
xxx
8.1-8.2
Antigen
xxx
Kj
Acid
9-(p-Aminopheny1azo)-phenylarsonic 1 80 P-(P-Hydroxyphenylazo)-phenylarsonic 1 05 p- Acetaminophen ylarsonic 0 24 9-(p-Nitrobenzoy1amino)-phenylarsonic 32 p-(p-Aminobenzoy1amino)-phenylarsonic 17 9-Benzoylaminophenylarsonic 97 p-Nitrophenylarsonic 22 fn-hTitrophenylarsonic .15 o-Nitropbenylarsonic .18 9-Iodophenylnrsonic 32 p-Bromophenylarsonic 22 p-Chlorophenylarsonic .I7 P-Methylpbenylarsonic 17 m-Methylphenylarsonic ,089 o-Methylphenylarsonir 084 8-Naphthylarsonic 19 a-Naphthylarsonic .35 1,4-Aminonnphthylarsonic 32 P-Hydroxyphenylarsonic ,095 8-Carboxyphenylarsonic ,074 Phenylarsonic ,067 9-Aminophenylarsonic .074 m-Aminophenylarsonic ,054 o-Aminophenylarsonic ,029
R'-oval-
bumin
ff
K,'
I
1.9 1.20 1.8 1.02 1.7 0.20 2 0 .30 2 . (1 .38 1 8 ,29 1 8 .30 2 0 .22 2.0 ,I1 1.6 .34 2 0 19 1.5 .17 2 0 ,16 1.5 .063 2 0 ,060 1 5 1.5
17
.15 15 .19
I 5
,069 ,054
1.5 1.5
060 062
1.0 2.0
,041 ,038
1 5
12.5 25
50
Moles of hapten added, X 10, I00 200 400 12.5 25 50
Amount of precipitste with antigen XXXa
680 481 840 690 541 886 813 772 755 648 775 571 780 648 903 773
2.0 629 452 270 2.0 651 501 305 2.0 812 670 510 2.5 736 619 436 2.0 675 501 343
2.5 2.0 2.5 2.0 1.5
754 568 424 825 701 539 502 602 420 531
1.5 2 0 3.0 2 5 2 0 2 0 1.7 1 5 1.5 1 5 1.5 1.5 2.0
602 750 725 603 516 431 800 827 821 801 915 867
1.0
596
318 426 234 357 417 411 607 575 404 313 252 600 620 680 669 778
190 288 134 198 250 268 404 430 196 182 137 381 482 495 460 578 809 671
100 200 400
Amount of precipitate with R'-ovalbumino
605 487 446 521 541 588 706 483 656 694 6643' 7.f4 795 664 678 641 884 890 875 880 957 880
413 260 566 445 282 187 454 278 457 295 465 324 655 530 684 550 468 348 494 347 434 276 687 557 720 655 703 606 709 598 834 686 765 673
The amounts are tabulated as fractions per mille of the amount precipitated in absence of hapten: for antigen XXX this was 748 pg. and for R'-ovalbumin 562 pg. Values for R'-ovalbumin include t h e precipitated antigen protein. Values are averages of triplicate analyses, with mean deviation *4?&.
TABLE 111 HAPTBNINHIBITION OF PRECIPITATION OF ANTIGENXI AND ANTI-R SERUM PRODUCED BY A BRIEF COURSEOF INOCULATIONS Antigen solution, 2 ml. (40 p g . ) , anti-R serum, 2 ml.: hapten solution, 2 tnl. PH of supernates 8.16-8.18. Acid p - (P-Hydroxyphenylazo)phenylarsonic p - Acetsminophenylarsonic 9-Benzop laminophenylarsonic P-Nitrophenylarsonic m-Nitrophenylarsonic o-Nitrophenylarsonic p- Methylphenylarsonic m- Methylphenylarsonic 0-Methylphenylarsonic 1-Naphthylarsonic o-Naohthvlarsonir Phenylarsonic p - Aminophenylarsonic m-Aminophenylarsonic ~~~~
~
o- Aminophenylarsoaic
KU'
Moles of hapten added X 10' 12.5 25 50 IO!' 200 Amount of precipitate0
I
*
1.1 (2.0) 1,l5 2.5 759 729 568
1.04 1.7 629 429 1.58 2.0 795 610 520 0 . 3 9 2.5 (756) 624 22 3 5 756 '690 b67 529 .56 3 0 .i4
2
:,
.13 1 ;. , 33 2 (I
.iz
2
a
44 2 5
(263)
511 590 450 689 835 502 953 876 770 812 665 535 'it32 795 702 710 635 478
3.i
3 0 2 0
$650) ,511 (454) 816 662 523
.22
1.5
914
b1
820 686
The amounts arc tabulated as fractions per mille of the amount precipitated iii absence of hapten, which was ;3,31 pg. \'slues are averages for triplicate analyses, with mean deviation *1%; duplicate analyses are given in parentheses. * Values of 916 and 867 found for 5 and 10 X 10-9 mole of hapten, respectively; the value 8.0 for u was absumed. a
of the antiserum, which appears to contain antibodies of greatly varying coiiibiriirig powers. We have now extended the theory to cover heterogeneous antisera of a particular sort, and have
found that the extended theory agrees in a generally satisfactory way with experiment. Let us assume that the heterogeneity of the antiserum is such that it can be described by a probability distribution function which is an error function in the effective free energy of combination of hapten and antibody (in competition with antigen); that is, in the quantity In(K'/KI), where K' is the effective hapten inhibition constant of the particular antibody molecule under consideration and K6 is an average effective hapten inhibition constant. The normalized distribution function itself is (3)
and the fractional number of antibody molecules with given value of K' in a differential region is
Curves showing the distribution function for
several values of the heterogeneity index , are shown in Fig. For = most Of the antibody
'*
molecules (84%) have K' values in the range from 0.368 K6 to 2.718 K;, for u = 2 this range is from 0.135 KI, to 7-48 K;, for ,, = 3 from 0.050 K6 to 20.1 K & and for = 4 from 0.018 K6 to 54.5 Ki. The corresponding ranges of values of free energy of interaction of antibody and hapten have widths of 1200, 2400, . . ., cal. per . mole for u = 1, 2, 1
May, 1944
787
INHIBITION BY HAPTENSOF ANTIGENPRECIPITATION OF ANTISERA
with larger values of K’. The integral is easily evaluated in terms of the Gaussian probability integral H(x), for which numerical values are given in tables’; the equation then assumes the form P 211 I H(x1)) - ~1 H t d K o @ / 4 ( 1
+
+
H(4I
with
(6)
In ( 1 / H t o t a l K 6 ) and xz = U X I - u/2. The function H(x) is to be taken with the same sign as x . Curves of the calculated amounts of precipitate for u = 0, 1, 2, 3, 4, and 5 plotted against H t o t a l are shown in Fig. 2. It is seen that with increase in u there is increasing deviation from the linear relation, which holds for u = 0. The similarity in form of the theoretical and experi-4 -3 -2 -1 0 1 2 a 4 mental curves is evident from a comIn (K’/Ki). parison of Fig. 3, which shows calcuFig. 1.-Assumed distribution function for values 1, 2, 3, and4 of the lated curves for u = 1.5 and a seheterogeneity index u,plotted against In (K’/KL). quence of values of KQ,with Fig. 1 of This distribution of antibodies would occur in reference 2, in which experimental data for several an antiserum if the heterogeneity were the result haptens with the same antigen-antibody system of a very large number of independent influences, are dotted. The values of Ko’for each antigen-antibody each of which could increase or decrease the free energy of bond formation by an additive con- system have been normalized by assigning the tribution. Since this is not unreasonable, in the value 1-00 to an azo hapten or, for some systems, light of current concepts of the structure of anti- to an average of several haptens. A very satisfactory way of comparing experibodies, we may well expect this distribution to be rather closely approximated by the antiserum ment and theory is to use a logarithmic scale for from a single animal or by the combined antisera the amount of hapten. Since K6 and H t o t a l from a large number of animals.6 On the other occur in Equation 6 only as their product, the hand, the combined antisera from only a few animals or a combination of unlike fractions obtained in the course of fractionation of antibodies might not have an error-function distribution and might not correspond in their properties to the present theory; the theory could, however, easily be extended to cover such special cases. If we assume that for each kind of antibody the amount of precipitate is linear in the amount of hapten, with slope proportional to K’, the amount of precipitate formed by the heterogeneous antim m is given by the equation In (1/Rtotd K;) A B (PPI - 1 AB ( P P ) H ~ ~ ~ = 6 O us(1 - K’Htotd) e - ( l n (K’/Kh)*/@’dln (K’IK:)
XI
=
P =
(5)
The upper limit of the integral represents the value of K’ at which the hapten just inhibits completely the precipitation of the corresponding antibody: no precipitate is formed by antibody (6) It would not be surprising if the antiserum from a single rabbit inoculated with an azoprotein were t o have a different type of distribution. The antibodies produced by the rabbit would be influenced by the nature of the place of attachment of the hapten to the protein (histidine or tyrosine side-chains). The distribution function for such an antiserum might be closely approximated by the rum of two or more error functions.
0.5 1.o Amount of hapten, Hw. Fig. 2.-Theoretical cum-es showing depehdence of amount of precipitate P on amount of hapten H ~ t afor e = 0, 1, 2,3,4, and 5 and K: = 1. 0.0
(7) “Tablw of Probability Functions,” Vol. I, Federal Works Agency. Works Projects Administration, New York, 1841.
788
LINUS PAULING,
DAVIDPRESSMAN
AND ALLAN
L. GROSSBERG
Vol. 66
:l
f
0.5 1.o Amount of hapten, HtQt,l. Fig. 3.-Family of curves with u = 1.5and 'Ki = 0.25, 0.5, 1.2, and 4,showing dependence of amount of precipitate on K: for fixed u
0.0
curves showing the amount of precipitate plotted against the logarithm of H t o t a l have the same shape when they correspond to the same value of a; the &ect of changing Kb is simply to shift the curve along the log Hto-l axis. Hence only a single family of such curves, covering a range of values of a, is needed for the evaluation of u and Ko' from a set of experimental points. Such a set of for u = 0, 1, 2, 3 , 4 , and 5 , is shown in Fig. 3.
0.0
IC I I 1 I I 1 I I I I I Logarithm of amount of hapten. Fig. 5.-Comparison of theoretical hapten inhibitiotr curve (with o = 1.3) with experimental data for the inhibition by p-(P-hydroxyphenylazo)-phenylarsonic acid of the precipitation of anti-R serum and antigen XXX; the points correspond to successive two-fold increases in hapten concentration. "I
successive two-fold Fhanges in amount of hapteri. The data obtained are given in Table IV and are compared with the theory in Fig. 5, from which it is seen that the experimental points (each of which is the average of either five or six analyses, with mean deviation *1.5%) agree closely with thc indicated theoretical curve, which corresponds t u = 1.3. TABLE IV INRIBITION BY ~(P-HYDROXYPHENYLAZO)-PHENYLARS I
ACID OF PRECIPITATION OF ANTI-R SERUM AND ANPIGBN
xxx
Antigen solution, 2 ml. (6.7 X lo-* mole); antiseruin, 0.33 ml.; hapten solution, 2 . 6 i ml.; PH of supernates, 8.0; amount of precipitate in absence of hapten, 762 fig.; valrir, corrected by 17 pg., the blank for serum and buffer. .Moles of added hapten, X 1oP
2.44 4.88 9.75 19.5 39 78
156
tn n
(8) In interpreting t h e experimental ( I ~ t a11 was foiinrf m m v r n i e n t prepare and use a &et01 trmnsparent templnttv. m\erini: thr rang,' 0 I O .5 at intervals of I 1 3
-
965 950 92.5 833 707 490 277
313 625
-1.0 -0.5 0 0.5 1.0 log Hbtt.1. Pig. 4.--Curves of amount of precipitate as function of log Hto~i, for u = 0, 1, 2, 3, 4, and 5 and K i = 1. -1.5
Only two experimental points are needed to evaluate KO' and a ; the agreement of three or more points with a theoretical curve provides a test of the theory. We have found that nearly all of the sets of hapten inhibition data so far obtained, numbering about three hundred, fit the theoretical curves t o within the probable errors of the experiments, usually about +4%. This agreement is particularly striking for the eightpoint sets in Tables IV and VI of reference 2 . In order to test the theory further a set of analyses was made with special care over a range of ten
Amount of precipitated antibody, in fractions per milk of amount for zero hapten"
iec,
91 3,i 1250 1,Ft 2500 12 Each value is the average of tivv o r \ix replickate aiialy with mean clrviation * 1 .5yc
The antiserum used in this experiment was, lilic most of those which we have used, a pooled serum made by mixing the sera obtained in several bleedings of a number of rabbits (usually about six). Since the mixing of sera would be expected to produce a serum with increased heterogeneity, some experiments were carried out with antisera obtained by single bleedings of individual rabbits, which had been inoculated over a period of months. The results are given in Table V and Fig. f i . I t is seen that these antisera are heterogeneous, with = I for Rabbit 3.5 and u = 1.5 for l
