On the mechanism of the quantitative precipitin reaction

Immunochemistry. Pergamon Press 1966. Vol. 3, pp. 419-424. Printed in Great Britain

COMMUNICATION

TO THE

EDITORS

On the mechanism of the quantitative precipitin reaction (Received 7 February 1966) Abstract--The types of complexes which constitute antigen-antibody precipitate and supernatant have been computed from a theoretical interpretation of results of quantitative precipitin analyses. It was found that precipitates in the antibody and antigen excess regions are composed primarily of small complexes. For the system under study in the precipitates in far antibody excess the principal complex consists of three antibody molecules and one antigen molecule; in the precipitates in far antigen excess the most abundant complex is composed of four antigen molecules and three antibody molecules. The composition of complexes near equivalence has not yet been determined. The theory seems to have withstood its first test. It has predicted properly the experimentally observed supematant composition (antibody excess, equivalence and antigen excess).

SOME of the basic questions concerning the mechanism of the precipitin reaction are: (a) What are the types of antigen-antibody complexes which constitute precipitate? (b) What are the concentrations of each of these complexes? and (c) What are the values of the equilibrium constants for the reactions of a given system? We have recently developed mathematical theories which describe hapten-antibody binding and antigen-antibody precipitation. ~I-4~ We have now completed a computer program for the analysis of experimental precipitin data according to.<8~ In this communication we present the first results of computational analyses. While the present results are only approximate and do not answer the above questions completely, they do give partial answers and they provide some insight concerning the mechanism of the precipitin reaction. Most important, the results of computational analyses suggest experiments which need to be done, even the concentrations of antigen and antibody which should be used in a given system to test the computational results and to answer completely the questions which are posed. Before presenting the computational results, we state again the assumptions which underly the mathematical theory.~3~ T h e theory was developed to describe the reactions between homogeneous multivalent (f-valent) antigen and homogeneous 2-valent antibody. The reactions are assumed to be governed by a mass action law. Equilibrium constants (Ko) describe the reactions between the i th antigen site (i = 1 , . . . f) and thejth antibody site (j = 1, 2). Antigen and antibody molecules are assumed to combine to form branched chain aggregates; the occurrence of cyclical complexes is excluded. The type of aggregate and the concentration of each is calculated from maximizing the entropy of the system. Equilibrium constants are assumed to be independent of the state of aggregation of the complex. Since there is no evidence to the contrary and it considerably simplifies the computation, the reactivities of the two sites of antibody molecules are taken to be identical. This leads to a description of the system in terms of f equilibrium constants K~ (rather than 2f equilibrium constants K~j). As shown (Ref. 3, page 225), criteria are needed to describe either the complexes which precipitate or the complexes which remain soluble. Since there is no way at the present time to 419

420

C o m m u n i c a t i o n to t h e Editors

measure the types of complexes which precipitate--indeed this is one of the questions which we wish to answer--we must specify the types of complexes which remain soluble. Experimental evidence indicates that in the region of antibody excess only free antibody is present in supernatants, that at equivalence all antigen and antibody is in the precipitate, and that in the region of antigen excess some free antigen and several types of soluble antigen-antibody complexes may be found in the supernatants, depending upon the system and the extent of antigen excess. The types of soluble complexes which have been observed in antigen excess are antigen-antibody, antigen-antibody-antigen and antigenantibody-antigen-antibody-antigen. ~5 The computer program was written as previously outlined (Ref. 3, pages 224-226). The weight of precipitates was computed by means of the equations in the footnote of Ref. 3, page 225. In the computer program we have specified that free antigen, free antibody and each of the above complexes may be found soluble in each supernatant, i.e. in antibody excess, at equivalence, as well as in antigen excess. This was done for two reasons: (1) the experimental observations were confined to the usual type of supernatant analysis (addition of antigen and antibody); the type of soluble complex in supernatant was not determined; and (2) lack of specification of the types of complexes in particular supernatants leads immediately to a test of the theory: the theory would be shown to be wrong if significant concentrations of free antibody were computed to be in the supernatants in the equivalence or antigen excess regions, or if significant concentrations of free antigen or of antigen excess soluble complexes were computed to be present in the supernatants in the regions of antibody excess or equivalence. We have analysed several sets of precipitin data. The result of one analysis, the reactions between egg albumin and the water-soluble antibody globulin fractions of rabbit antiserum (7~ are reported here. Experimental and computed precipitin data are given in Table 1. Since the experimentally observed antigen valence (precipitin data extrapolated to zero antigen concentration) is three, the description of complexes is given by triplets of integers (al, a2, a3}. The triplets are related to the structure of complexes as follows: al is the number of antigen molecules in the complex having one reacted site, a2 is the number of antigen molecules in the complex having two reacted sites, and a3 is the number with three sites reacted; the number of antibody molecules in each complex is uniquely determined by the numbers as (Ref. 3, page 214). To the right of the colunm containing the triplets {al, a2, as} are drawn pictures of representative complexes corresponding to the triplets; antigen molecules are represented by circles and antibody molecules by bars. Several structures may be represented by a given triplet. Thus, to the triplet f

{0, 0, 4} there correspond the two complexes

I ~ I I

--O--O--O--O--

and

I

--0 --O --0 --

I

O

"

Com-

/,,

plexes are listed in order of increasing values of antigen-antibody ratio. Complexes which contribute approx. 0.5% or more to some of the precipitates are included in Table 1. The results in Table 1 reveal the following: (i) there is reasonable agreement between the values of observed and computed weights of precipitate for all but

Precipitin Reaction

421

two points*; (ii) there is reasonable agreement between the experimentally observed and the computed supernatant composition: no free antigen is computed to be present in the supernatants in antibody excess and virtually no free antibody is computed to be present in the supernatants in antigen excess. The equivalence zone, however, is computed to be in the region of maximum precipitation where some free antigen was detected in the supernatants. Antigen-rich soluble complexes are computed to be present only in the region of antigen excess, where they have been observed. It would appear, therefore, that the theory has withstood its first test, namely, that it can predict supernatant composition. (iii) A most interesting result is that in the antibody and antigen excess regions a large fraction of precipitate is computed to be present in the form of very small complexes, six or fewer antigen molecules per complex. This is contrary to the usual idea (lattice or framework hypothesis) where it is assumed that precipitation implies the formation of a relatively few very large aggregates. From the results in Table 1 we can give some answers to the questions initially posed: (a) in antigen excess and antibody excess precipitate appears to be composed primarily of small complexes. The predominant complex in the precipitates in far antibody excess contains f antibody molecules (three for the present system) and one antigen molecule. In far antigen excess the predominant complex in the precipitates is the insoluble complex for which the antigen-antibody ratio is smallest. For the present system {2, 1, 0) is the largest soluble complex in antigen excess supernatants; (2, 2, 0 } is the predominant complex in the far antigen excess precipitates. As the equivalence zone is approached, the complexes become too large for determination by the present method of computation. (b) The concentrations of the complexes which constitute precipitate are uniquely specified by the equations (Ref. 3, page 225). To determine these concentrations, only the total amounts of antigen and antibody in each tube and the concentrations of soluble complexes need to be known. Should the critical point (as defined by equation (29) Ref. 3) be reached or exceeded at any particular antigen concentration, then only the total amount of precipitate can be computed for that point, not the concentration of individual complexes. (c) T h e equilibrium constants for the system under study are given in Table 1. We have generally observed that the sharper the antigen excess inhibition zone, the higher are the values of the equilibrium constants. The experiments which are suggested by these results are almost obvious; precipitin data using purified (nearly 100% precipitable antibody) are needed, as well as weight average molecular weight measurements of supernatants. The supernatant composition of all tubes can be computed from precipitin analyses. The types of complexes and the concentrations of each in the supernatants can be calculated from molecular weight measurements. Comparison of predicted and experimentally observed supernatant compositions will constitute the critical test of the theory. We have made a number of simplifying assumptions in the present analysis. We know, for instance, that heterogeneity of antibodies exists and we have the theory to treat the effect of heterogeneity on the precipitin reaction. ~4~ The theory to * Agreement could be improved by making a finer grid least squares search but it would be costly in computer time and probably yield no further basic information.

T A B L E 1, A N A L Y S I S OF Q U A N T I T A T I V E P R E C I P I T I N DATA : T Y P E S O F C O M P L E X E S A N D A M O U N T S O F EACH I N P R E C I P I T A T E S A N D I N S U P E R N A T A N T S @ b~

Antigen (mg) Supernatant test

0"013

0"031 0"062 Ab excess-

Observed precipitate (rag) C o m p u t e d precipitate (mg) C o m p u t e d weight of complexes in precipitates (mg): {ab as, as } picture

O"144 0"143

0-338 0"336

{2,

2,

O}

{2,

3,

O}

{2,

4,

O)

{3,

2,

1}

{2,

5,

O}

0"131 0.150 ~- no Ab ÷ no Ag 0-793 0"931 0"918 0"757 0"922 0"950

0"181

0"243 0"581 -Ag excess 0-169 0"976

0-088 0"106

0"056 0"025

--

0"037

0.073

0"017

f

--

0"032

0.017

0"002

0--0--0--0

I

--

0"026

0"004

0--0--0--0

--

0"007

0.001

--

0"020

0.001

--

0'009

--

O--O--O--O

o

0

0--0

I

3,

1}

{2,

6,

O}

0--0--0--0

I

0--0--0--0

I

0--0--0 0--0--0--0

I

0"016

0--0--0--0 0--0--0--0

{3,

4,

1}

I o 1

0"010

0--0--0 0 --0 --0 --0

{2,

7,

W

0

0--0--0

{3,

0"931

0"718 0"981

o-o-o-o

0"606 0-599

0-087

I

O}

0 0--0--0--0

I

0.012

o

Txax.~ 1. (Cont.) {ab a2, as }

{3,

5,

1}

{0,

0,

7}

{0,

O,

6}

{0,

0,

5)

picture O--O--O--O I ~ ? 0 --O --0 I I I I --o--o--o--o t

0"011

0.005

0"002

0.005

0"027

0"012

0.010

0.034

0-016

--0 --0 --0 -I t

I

I

I

I

--0--0--0--0

-o-oi I I I -o-o-o-~

I I

--

ft.

--O--

t-t

~,,'%

O,

4}

{0,

0,

3}

{0,

O,

2}

{0,

O,

1}

I

I

I

I

0.001

0.018

0.044

0.021

--

--0--0--0--

0.005

0.037

0.060

0.028

--

0"023

0.077

0"087

0.042

--

0.114

0.181

0.141

0.068

0.001

--0--0--0--0--

I

I

I

m

O

-o-o-I I I

--O--

m

m

C o m p u t e d w e i g h t of c o m p l e x e s in s u p e r n a t a n t s (rag): Free Ag

o

.

.

.

.

.

.

.

0"002

0"183

0"517

{2,

0,

0}

o-o

.

.

.

.

.

.

.

0"032

0"819

1 "009

[2,

1,

0}

o-o-o

.

.

.

.

.

.

.

0.038

0"273

0-147

{1,

0,

0}

0-001

0"002

0-033

Free Ab

o. -

.

. 0"700

.

. 0"495

. 0"264

. 0-130

. 0.009

--

--

--

* E x p e r i m e n t a l data are for egg a l b u m i n - - w a t e r - s o l u b l e g l o b u l i n fraction of a n t i s e r u m ; 0"8 m g a n t i b o d y p e r tube. K1 = 5"5 x 10 t°, K2 = 3.2 x 109, K~ = 3"2 × 108 (l/mole).

~o

424

Communication to the Editors

allow for dependence of sites and for cyclical complexes can undoubtedly be developed. It is our opinion, however, that such elaborate treatments are best postponed until adequate experimental precipitin and supernatant composition data are available.

Department of Microbiology University of Southern California School of Medicine Los Angeles, California 90033

FREDERICK ALADJEM* M. T. PALMITERJ" F u - W u CHANG

REFERENCES 1 BOWMANJ. D. and ALADJEMF., J. theor. Biol. 4, 242 (1963). 2 WONG J. and ALADJEMF., J. theor. Biol. 8, 41 (1965). a PALMITERM. T. and ALADJEMF., J. theor. Biol. 5, 211 (1963). 4 ALADJEMF. and PALMITERM. T., J. theor. Biol. 8, 8 (1965). 5 SINGER S. J., J. cell. comp. Physiol. 50, Suppl. 1, 51 (1957). 6 ALADJEMF. and LIEBERMANM., J. lmmunol. 69, 117 (1952). * This work was supported by NIH Grant AI 0322 and by Research Career Development Award 2K3-GM 4817-06. t Predoctoral Trainee, Training Grant 2T1 AI 157-06.