THE PRECIPITIN REACTION BETWEEN TYPE III PNEU- MOCOCCUS POLYSACCHARIDE AND HOMOLOGOUS ANTIBODY Ill. A QUANTITATIVE STUDY AND A THEORY OF THE REACTION MECHANISM" . By MICHAEL HEIDELBERGER, Pu.D., anv FORREST E. KENDALL, Ps.D. (From the Department of Medicine, College of Physicians and Surgeons, Columbia University, and the Presbyterian Hospital, New York) (Received for publication, December 27, 1934) In the first paper of this series (1) it was concluded as a first approxi- mation that under a standard set of conditions the entire course of the precipitin reaction between the specific polysaccharide of Type III pneumococcus and homologous purified antibody could be quantita- tively accounted for by three simple equations. The mass law was believed to hold for these equations, the more so as the reactions were found to be reversible. Studies of the theoretical factors involved have since been continued under more varied conditions, and the present paper describes experiments which have necessitated modifica- tion of the conclusions originally drawn. EXPERIMENTAL The quantitative precipitin determinations were made as in previous papers (2-4), except that the technique was modified as described below in order to study the effect of varying a given set of conditions. In general, precipitates were analyzed, rather than supernatants, as had been done in (1). Much of the serum used was obtained through the kindness of Dr. William H. Park, to whom the writers again wish to express their gratitude. Unless otherwise stated, antibody solu- tions were prepared according to Felton (6). The specific poly- * The work reported in this communication was carried out under the Harkness Research Fund of the Presbyterian Hospital], New York. . 563 564 QUANTITATIVE PRECIPITIN STUDY saccharide of Type III pneumococcus is referred to throughout as § Il. In view of the temperature effects shown in the preceding paper (7) it was found necessary to work at a single temperature in studying the reaction mechanism, and experiments were accordingly run either at 0° or at 37°C. The differences between the amounts of antibody nitrogen precipi- tated when the reaction mixture is kept in the cold throughout (0°) and when the precipitation and centrifugation are carried out at 37° are strikingly shown in Tables I and JI, which summarize, respectively, the results of the addition of increasing amounts of S ITI to 1 cc. portions of serum or antibody solution and of serial additions of small amounts of S III to antibody solutions under those conditions. la. Addition of Increasing Amounts of S III to Antibody at 37° and at 0°.— Experiments corresponding to those in the earlier work (1) have now been made with the temperature constant throughout. Duplicate tubes were set up at 37° using 1.0 cc. of antibody Solution B 61 and varying amounts of S III in a total volume of 4cc. The tubes were incubated for 2 hours at 37° and then centrifuged at that temperature. The precipitate was washed twice with cold saline and analyzed for nitrogen. The results are given in Table I, Column 2, and are represented by the circles along Curve C, Fig. 1. The point connected with the curve by the dotted line indicates the maximum amount of nitrogen specifically precipitable (0.4 mg. $ ITI) from this solution at 37°. A similar experiment was run with antibody Solution B 62 in which the tubes were set up at 0° and centri- fuged in the cold instead of at 37°. The amount of S ITI combined in the region of excess S III was found by determining the amount left in the supernatant by the method described in a previous paper (3), except that the determinations were run at 0° in the B 62 experiments and entirely at 37° in the B 61 series. Aliquot portions of the supernatants containing a suitable amount of S IIT were set up with another 1.0 cc. portion of the antibody solution used in the experiment, adding saline to bring the volume to4cc. ‘The precipitates were analyzed after 24 hours. The amount of S III in the B 61 aliquots was read off from the experi- mental points along Curve C, Fig. 1 (Column 2, Table I). A similar curve was constructed for B 62. The results are given in Table III. 1b. Serial Experiments at 0° and at 37°.—In these experiments successive small portions of the antibody were precipitated. In the 0° experiment the reagents were chilled in an ice bath. Duplicate 5.0 cc. portions of antibody Solution B 61 were measured into Wassermann tubes and mixed with 0.5 cc. of a 1 to 10,000 solution of S III. The tubes wete kept in the ice box overnight and were then centrifuged in the cold. The supernatants from the duplicate tubes were mixed. M. HEIDELBERGER AND F. E. KENDALL 565 and 5.0 cc, samples set up with 0.5 cc. of S III as before. This procedure was repeated until the antibody was exhausted. In the case of antibody Solution 2 va oh -- p— SPN NS 5 Mg. N pptd. per cc. J ead TS 0.02 0.06 0.10 O44 0.18 Mg. SII per ce. Text-Fic, 1 B 62, 0.02 mg. portions of S IIT were added, The precipitates were washed and analyzed as in the preceding section. The 37° experiment was carried out in the same way except that the solutions were mixed at room temperature. The tubes were allowed to stand for 2 hours at 37° and were then centrifuged in a small angle centrifuge at 37°. TABLE I Addition of Increasing Amounts of S III to 1.0 Ce. of Serum or Antibody 5 Antibody Solution B 61, 37° Horse Serum 607, 0* Antibody Solution B 62, 0* Horse Serum 607, 37° Antibody Solution B 62, 37° saveeenes ees seeces [6] It will be noted that the volume factors cancel, so that the amount of antibody precipitated depends only on the relative amounts of antibody and S III present and not on their concentration. This treatment of the problem involves only the formation of compounds having ratios between R and 2 R, where R is the ratio of antibody to S ITI in the equivalence point compound. The experimental data show that compounds hav- ing ratios greater than 2 R may be formed, for at 0° in the presence of a large excess of antibody ratios greater than 4 R are encountered. By extending the process used for the calculation of the second step to stage [3] and beyond, it is possible to calculate the amount of antibody precipitated by a given amount of S III when the ratio varies between R and 3 R and also between R and 4R. The calculations are complicated, as step [3] involves the bimolecular formation of five compounds, that is, AsS, AsSe, AaSe, A.S3, and A,S,, and extension of the process to 4 R results in 20 compounds. In this calculation it is assumed that the ratio in which any two products are formed is unaffected by the other competing reactions, The expression thus calculated for antibody precipitated in the range R to 3 R is: 2zA-— BY, Units of antibody pptd. = A — ATA -B) + Al] ve denetesvoes * Copies of the derivation will be furnished on request. The same formulas apply in the region of excess S, and in their derivation S and A are interchanged. , In the above calculations the simplifying assumption was made that the composition of the precipitate at the equivalence point is repre- sented by the molecular formula AS. It will now be shown that this assumption is not necessary, and that if the antibody nitrogen: S III ratio in the precipitate varies between the value found at the equiva- lence point and one twice as great when a large excess of antibody is present, the reaction follows the same course regardless of the molecu- lar composition at the equivalence point. Tf the compound at this point be taken as AS., formed as a result of a series of bimolecular reactions between A and S, making up the first step of the reaction, the 582 QUANTITATIVE PRECIPITIN STUDY course of the reaction as far as AoS,, which has double the N:S ratio, proceeds similarly to [2]: ASa+ A PAS. Qe. 3] AS, + ASa = AS,+ASq, In the presence of still more antibody it would proceed according to [3], and both reactions would be calculated as were these steps. H, on the other hand, the equivalence point compound be taken as A.S, the course of the reaction between A,S and A,S requires two successive steps, as in (2] and [3]: AS A BAS cece eteee eee 19] AS + A:S 72 A)S-A:S and AS+A AS AS-A:S + A = ASe A AS: + AS EASA foo. cece. [10} AS + AS: = AS-AS: AUSa ++ AS: = AjS:- Ae and is therefore calculated in the same way as the reaction between AS and A;S. The expression derived for the reaction between the limits AgS and A,S is: . . 2(A — 2B)¢ Units antibody N pptd. = A — K- BAB) + A OB woe (hh] * Copies of the derivation will be furnished on request. In Table VII, Columns 1, 2, and 3, a calculation of the reaction is given according to [1] and (2], and [9] and [10], respectively, and it is evident that the differences are small. In making this calculation and in applying the derived equations to the experimental data it is necessary to convert units of antibody and S$ IIT into milligrams. This may be done by assuming that 1 mg. of antibody nitrogen equals 1 unit, that the number of milligrams of antibody nitrogen precipitated at the equivalence point equals A, and that the ratio of A to S IT at this point is equalto R. It follows that in equations [6] and [7} B = A and B = RS at the equivalence point. Equation [6] then becomes: mg. antibody N pptd. = 2RS — RS. {6a} Equation [7] becomes « 2(A — RS) mg. antibody N pptd. = A — Nh RSE sete ee ee tans [7a). M. HEIDELBERGER AND F. E. KENDALL 583 and since in equation [11] 2 B = A at the equivalence point and 2 B = RS, this equation becomes: ~— 4 mg. antibody N pptd. = A ~ 2(A ~ RS) ;..fl ta] (5 [= ra In order to permit comparisons to be made between antibody solutions con- taining different amounts of antibody and having different equivalence point ratios it was found convenient to reduce the amounts of S ITI and N precipitated to percentages of the quantities precipitated at the equivalence point. To convert {6 a} into an expression involving percentages, use is made of the relationship A = RSaq. at the equivalence point. Dividing all terms of [6 a] by RSeq. (RS)! St Npptd. 4 RS _ (RSa)? | Npptd. — 5 S _ Sea! RS.q. RS.g. A A Sea A RS.a. A Multiplying each side of the equation by 100, eS Per cent A pptd. = 2 X %S ~— Too ccc [60] 2(100 — %S)* Per cent A pptd. = 100 — 100 [100 = 95)" 1005] Toa A Per cent A pptd. = 100 — 2(400 — 78)" .. .[Lt3] (100 - #8) [ (100 - ze + (100 — %5) | The percentages of A precipitated by increasing percentages of S III, calculated according to these equations, are given in Table VII. These data are shown graphically in Fig. 3, in which Curve A is calculated according to [7 5} and Curve B according to [6 8]. In Table VIII the data on Serum 607 in Table I are calculated in terms of percentage of the total precipitated at the equivalence point, and the percentage of S III used and antibody N precipitated are plotted on Fig. 3. It will be seen that the circles representing the values found at 0° lie very close to the curve (A) for the reaction in which the N : S ratio varies from R to 3 R (equation [7 8] ) while the 37° values are very near those calculated for the R to 2R reaction (equation [6 6], Curve B). Thus the course of the reaction at 0° appears to be determined by a greater complexity of the reactions occurring after the initial A and S combination than is indicated by the data for the reaction at 37°. 584 QUANTITATIVE PRECIPITIN STUDY If, instead of using the equivalence point as the basis of the calcula- tion, the ratio at the beginning of the equivalence zone (from the region of excess antibody) be used, the course of the reaction follows the two stage expression [6 a] very closely in all but one of the anti- TABLE VII Calculated Percentage of Antibody Precipitated Total S TIT added Antibody N precipitated Calculated according to..... Equation 65 Equation 116 Equation 76 Ratio himits.........6.c000e Rand2R Raud2R Rand3R per cent per cent per cent per cent 10 19.0 19.4 27.5 20 36.0 37.2 50.0 30 51.0 53.4 67.8 40 64.0 67.6 80.9 50 75.0 79.5 90.0 69 84.0 88.7 95.6 70 $1.0 95.1 98.5 8a 96.0 93.7 99.7 90 99.0 99,88 99.98 100 100.0 100.9 100.0 TABLE VII Antibody N Precipitated by S fT from Serum 607 Expressed as Percentage of Quantity Precipitated at Equivalence Point Reaction at 0° Reaction at 37° $ TT used Antibod; Antibod: Antibod Antibod: ntibody | {Antibody ntibody tbody Sul N pp N pptd. si N pptd. WN pptd. mg, per cent mg. per cent per cent me. per cent 0.02 14.5 0.62 42.8 15.8 0.42 31.6 0.04 29.0 1.03 71.0 31.5 0.74 55.6 0.06 43.5 1.25 86.2 9.075 54.4 1.35 93.4 59.1 1.16 87,2 0.09 65.2 1.40 96.6 70.9 1.23 92.5 0.10 72.5 1.43 98.6 body solutions studied in sufficient detail, regardless of the tempera- ture at which the reaction is carried out.’ The theoretical amounts * The exception, BX, is one of the solutions studied in the beginning of the work (1), in which aliquot portions of supernatant were analyzed instead of entire precipitates. It was, moreover, an exceedingly concentrated solution. M. HEIDELBERGER AND F. E. KENDALL 585 of antibody nitrogen precipitated by varying quantities of S III from different antibody solutions according to equation [6 a] were calculated with the aid of the experimental values for R given in Table VI for the ratio at the beginning of the equivalence zone, A being nitrogen precipi- tated at this point. A comparison is given in Table IX of the calcu- lated and experimental values for nitrogen precipitated. In a previous paper (3) it was shown that the antibody nitrogen precipitated by S III from Solution B 31 in the region of excess anti- body followed the empirical equation, N = 18.6S — 605% It will be noted that this equation is in the same form as [6 a], so that the theoretical significance of the two constants is now clear, for 18.6 = IR and 60 = & The results of the serial experiments (Table II) also conform to equation [6]. In order to make the comparison with other data, the result from each successive addition of S III was calculated to the 1.0 cc, basis. The ratio of total antibody nitrogen precipitated to total S ITI used was taken as R and the total antibody nitrogen per _ 1.0 cc. of antibody solution as A in the equation [6a]. Curves A and B, Fig. 1, were calculated with the aid of these values. The circles along the curves represent the actual experimental data. In the region of excess S III, by interchange of S and A (page 581), equation {6 a] becomes: (RIA . = ORA — mg. S III precipitated = 2 R’A ToS ‘ccc 5 in which R’ = A at the equivalence point. Thus, as in equation [6], the amount of S precipitated depends only on the relative amounts of S and A present. Expressions similar to [12] may also be found corresponding to [64], [7a], and [74] by inter- changing A and S. In making calculations according to these equa- ‘ T tions 100 per cent A was taken as rote een, or the amount which would combine with the S III present to form the equiva- lence point compound. The points obtained from the data in Table III are plotted as crosses in Fig. 3, using the per cent of S III pre- 586 QUANTITATIVE PRECIPITIN STUDY cipitated as ordinates and the per cent of A as abscissae. It will be seen that the values fall quite close to the theoretical values for a three step reaction. If, however, instead of the ratio of N : S at the 7 ee 4 90}—— - po a do A COO LZ y Fer cent N (S II) pptd. ° iif 10 20 30 40 30 60 70 80 90 100 For ciecles: per cent SII precipitated For crosses and triangles: per cent antibody N precipitated TEXT-FicG, 3 equivalence point, the ratio be taken as 7.5, at which point an an- alytically determinable amount of S III first appears in the superna- tant, the reaction follows the two stage mechanism, according to equa- tion [6 4]. This procedure is analogous to that used in the region of M. HEIDELBERGER AND F. E. KENDALL 587 excess antibody. The points derived in this way are plotted as triangles in Fig. 3. The three lowest points are partially in the inhibi- tion zone and could scarcely be expected to conform closely: to: the curves. Similar considerations apply to antibody Solution B 62 at 0°, for which the data are also given in Table III. TABLE IX Comparison of Experimental Daia wiih Values Calculated According to: sag RIS? N Precipitated = 2 RS — —— A Antibody No....... BVa B 36 B61 B62 B 62 Serum 607 | Serum 607 Temperature, °C... 37,0 37,0 37, 37 0,0 37, 37 0,9 37, 37 re 13.6 12.4 11.4 (17) 12 (15) (11) Al vcceecereceeees 4.08 1.86 1.71 (1.23) 1.20 (1.42) (1.31) N pptd. N pptd. N pptd. N pptd. N pptd. N pptd. | N pptd. 7? 3 ~~ <2 3 ?s S ITI used Ce ee g us} g a gla l2)/3/2i3lel3i2l3ié)3|2la me. mg. | mg. | mg. | sg. | meg. | ang. | mg. | mg. | meg. | meg. | og. | og. | amg. | meg. 0.01 0,36] 0.32 0.02 0.50! 0.46] 0.45] 0.43) 0.57/ 0.59] 0.44) 0.43] 0.62] 0.54] 0.42] 0.40 0.03 0.78 0.81 0.04 0.79] 0.79! 1.03] 0.95} 0.74]. 0.73 0.05 1,22] 1.25] 1.03] 1.03] 0.97] 0.95) 1.07} 1.11 0.06 1.08] 1.09) 0.96] 1.01] 1.25} 1.23 0.075 1.41] 1.40 1.35} 1.36) 1.26F 1.13 0.08 1.29} 1.34 0.09 1.40] 1.42] 1.23} 1.23 0.10 2.24] 2.27] 1,66] 1.65} 1.54] 1.52 0.12 1.68] 1.64] | 0.20 3.62] 3.62 0.25 3.87] 3.96 Rand A values in parentheses deduced from nearest actual determunation. A relationship useful in its application to unknown sera may be derived from expression [6 a]. If both sides of the equation be divided N R? by S, the resulting equation, 5 = 2R ~ x 5S, is that of a straight line. ‘Thus, if the values of the ratio found in the region of excess antibody are plotted as ordinates against the amounts of § III added as abscissae, a straight line is obtained. The intercept on the 588 QUANTITATIVE PRECIPITIN STUDY R: y axis gives the value of 2R, while the slope is A? from which A, the amount of nitrogen precipitated at the beginning of the equivalence zone, may be calculated. Line F, Fig. 2 (page 576), illustrates this TABLE X Calculation of Precipitated Antibody N from N:S III Ratio at Two Points by : . N R Means of Linear Relation 77 2R - a 5 Antibody No........ BVa B 36 B 61 B 62 Serum 607 Serum 607 Temperature, °C... 37,0 37,0 37, 37 0,0 0,0 37, 37 S Til added at 2 points used, wg....] 0.10,0.20* | 0.05,0.10" ) 0.05, 0.10*| 0.03,0.05* | 0.04, 0.075" | 0.04, 0.075* Values given 21.9 by line drawn 2R 26.8 24.6 23.4 33 34.8 : through the 2 [ R! points 7 43.5 80 80 233 224 85.5 N pptd. N pptd. N pptd. N pptd. N pptd. N pptd. yg yg yg yg yg a . . a . Z ‘ 2 2 5 2 3 3 5 2 3 4 a elale/sj2lslelei2[s| els mg. meg. mg. mg. mg mg. mg. {| mg. | mg. mg. meg. mE. Awses-.. 4.08 | 4.131 1.86 | 1.89 | 1.717 1.78 1.17 1.35 1 Amt. § II mg. 0.01 0.36 | 0.31 0.02 @.50 | 0.46 |0.45 | 0.44/0.57 | 0.57)0.62 | 0.61 | 0.42 | 0.40 0.03 0.78* 0.04 0.79 | 0.81 1.03* 0.74* 0.05 1.22 § 1.23 | 1.03* 0.97 1.07* 0.06 1.08 | 1.12 1.25 | 1.28 0.075 1.41 | 1.40 1.35* 1,16* 0.08 1.29 | 1.36 0.09 1.40 | 1.32 | 4.23 | 1.28 0.10 2,24* 1.66" 1.54" 0.12 1.68 | 1.66 0.20 3,62* 0.25 3.87 | 3.98 * These points are also marked with an asterisk in the N pptd. columns below. procedure in the case of antibody Solution B 61, the circles being the experimentally found ratios in Column 3, Table I. The calculated ratios at the beginning of the equivalence zone given in Column M. HEIDELBERGER AND F. E. KENDALL 589 3 of Table VI were obtained in this way and are probably more accurate than the observed ratios because the experimental errors in the determination of the individual points are averaged in this method. This linear relationship makes it possible to characterize an unknown Type III antipneumococcus serum or antibody solution in the region of excess antibody by two analyses, in duplicate. If the ratio of antibody N to S$ II precipitated be determined for two different amounts of S$ IIT in the region of excess antibody and a straight line be drawn through the two points so obtained, the intercept on the y axis = 2 R and the 2 R slope = A: With the R and A values at the beginning of the equivalence zone calculated in this way the amount of antibody nitro- A gen precipitated by any quantity of S III Jess than R may be cal- culated with a fair degree of accuracy. In choosing the amounts of S III to be used in the determination of these points it is best to pre- cipitate more than 50 per cent of the antibody, since above this level experimental errors in the determination of nitrogen have a smaller effect on the . ratio. The application of this procedure to several antibody solutions is illustrated in Table X. It will be seen that there is in general good agreement between the observed and calculated points, but it is better, of course, to have three points with which to determine the position of the line. In Fig. 2 and in making calculations in the region of excess antibody it is assumed that all of the antibody present is precipitated at the beginning of the equivalence zone. The data in Reference 1 and in Tables I and III show that this is actually not the case, and that the amount of antibody precipitated usually increases as S III is increased in the equivalence zone, often reaching its maximum only when S III is present in appreciable excess. In different sera the amount of additional antibody nitrogen precipitated in this way varies from a few hundredths to one- or two-tenths of a milligram. This behavior appears due to the varying amounts of the relatively easily dissociable antibody occurring in different sera, and renders necessary for the complete description of the behavior of a serum in the precipitin re- action a separate determination of the maximum amount of specifically precipitable nitrogen (3, 4, 7). 590 QUANTITATIVE PRECIPITIN STUDY In the region of excess S III the behavior of a serum as far as the beginning of the inhibition zone may be characterized by the deter- mination of the A and S III precipitated at two points, since in this Spptd. _ oR! (R'P A A Total S region the linear relation applies ifR’ be taken as the= ratio at the end of the equivalence zone at which S III appears in excess and A be taken as the amount of antibody precipitated. In the inhibition zone, in which large amounts of 5 III are present and the amount of precipitate has begun to diminish, this equation is no longer applicable and it is necessary to determine the dissociation constant of the soluble compound AS, according to the method in- dicated in Table V, Reference 1. The determination of two prop- erly spaced points should be sufficient to establish the dissociation constant and permit the calculation of other points in this range. SUMMARY AND CONCLUSION The precipitin reaction between the specific polysaccharide of Type III pneumococcus and homologous antibody formed in the horse can be accounted for quantitatively by assuming the chemical combination of the components in a bimolecular reaction, followed by a series of competing bimolecular reactions which depend upon the relative pro- portions of the components. These reactions would lead to the for- mation of larger and larger aggregates until precipitation ultimately occurred. The mathematical formulation of this theory on the basis of the mass law is described. The derived expressions are shown to be in accord with the experimental findings and the constants used in these expressions are shown to have definite significance. In spite of the wide variation in the properties of individual sera these ex- pressions permit the complete description of the behavior of an un- known serum with S III without an unduly burdensome number of analyses. The quantitative theory presented has been found applicable to other instances of the precipitin reaction, as will be shown in sub- sequent papers. BIBLIOGRAPHY Nos. 1-6 are the same as in the accompanying paper (7). 7. Heidelberger, M., and Kendall, F. E., J. Exp. Med., 1935, 61, 559. 14, 15. 16. 17. 18. 19. 20. 21. 22, 23. 24. 25. 26, M. HEIDELBERGER AND F. E. KENDALL 591 . Heidelberger, M., and Kendall, F. E., J. Exp. Med., 1933, 57, 373. . Breinl, F., and Haurowitz, F., Z. physiol. Chem., 1930, 192, 45, 10. 11. 12. 13. 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