Reprinted from Paystotocican REvIEws
Vol. 23, No. 3, July, 1948

THE NATURE OF THE FORCES BETWEEN ANTIGEN AND
ANTIBODY AND OF THE PRECIPITATION REACTION

LINUS PAULING, DAN H. CAMPBELL anp DAVID PRESSMAN

Contribution no. 985 from the Gates and Crellin Laboratories of Chemistry, California Institute
of Technology, Pasadena

In one of his lectures on immunochemistry at the University of California in
the summer of 1904 Svante Arrhenius said (1) that Ehrlich and other investi-
gators, because of incomplete knowledge of the phenomenon of chemical equi-
librium, had been led to invent artificial hypotheses in order to explain their
observations in the field of immunology. Since that time, and especially during
the last few years, workers in this field have made greater and greater use of the
concepts and methods of physical chemistry, and in consequence many pre-
viously puzzling observations have been reasonably interpreted.

Another branch of chemistry which is of importance to immunology is modern
structural chemistry, which deals with the detailed structure of molecules and
with the nature of interatomic and intermolecular interactions (2). Our present
knowledge of this subject, in large part won during the past dozen years, is now
so firmly founded and so extensive that it can be confidently used as the basis
for a more penetrating interpretation of immunological observations than would
be provided by the observations alone.

In this paper we present, after a brief historical introduction, a discussion of
the nature of the specific forces between antigen and antibody and of the precipi-
tation reaction from the point of view of modern chemistry. Only the simpler
aspects of the phenomena are discussed; such complicating factors as the réles
of complement, lipids, etc., in the reactions are disregarded in our discussion.

The history of the precipitation reaction began in 1897, when Rudolf Kraus (3)
reported the results of his work with anticholera and antityphoid sera. His
observations were soon verified and extended by Nicolle, Tchistovich, Bordet,
Myers and other workers, who prepared precipitating antisera against’ a great
number of antigens of varied nature. We shall not review this early work here,
nor the later studies of the methods of preparing antisera and carrying out the
precipitation reaction, since these topics and others dealing with special phases
of the reaction have been very well covered in earlier reviews (4, 5, 6).

Two most important advances in the attack on the problem of the nature of
immunological reactions were the discovery that the specifie precipitate contains
both antigen and antibody (7) and the discovery that antibodies, which give
antisera their characteristic properties, are proteins. The verification of these
facts was provided by the work of many investigators over a score of years.
This work, which is summarized in Marrack’s monograph (6, chap. II), culmi-
nated in the preparation of purified antibody by Felton and Bailey (8), Heidel-
berger and collaborators (9), and others, and the determination of its properties,
including amino-acid composition and molecular weight, which show that it is
very closely related to normal serum globulin (6, chap. IT).

203
204 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

The work of Landsteiner (10) and other investigators on artificial conjugated
antigens provided a great body of qualitative information on the specificity of
antibodies, which, together with the experimental results for natural antigens,
led to the independent proposal by Breinl and Haurowitz (11), Alexander (12),
and Mudd (13) in 1930-32 of the theory of structural complementariness of
antigen and specific antibody. The framework theory of precipitation was
then developed by Marrack (6) and Heidelberger (14). These and other theories
are discussed in some detail in the following sections of this paper.

A new period in the study of the precipitation reaction was initiated by the
careful quantitative studies of Heidelberger and his collaborators (15) who deter-
mined the amounts of antibody and antigen in precipitates, and the similar work
of Haurowitz (16) and others. Very recently, in order to test certain aspects
of his detailed theory of the structure of antibodies (17), Pauling and his col-
laborators have carried out many quantitative experiments on the precipitation
of antisera by polyhaptenic simple substances (18, 19, 20), a phenomenon first
observed by Landsteiner and Van der Scheer (21).

THE NATURE OF THE SPECIFIC FORCES BETWEEN ANTIGEN MOLECULES AND
ANTIBODY MOLECULES. The detailed information which has been gathered in
recent years regarding the nature of the chemical bonds which hold atoms to-
gether into stable molecules has been summarized in monographs (2, 22). In-
stead of interacting strongly with one another, with interaction energy of 20
kilocalories per mole or more, to produce a chemical bond, two atoms may
interact more weakly. The nature of these weak interactions is now well under-
stood, and a brief discussion of it is given in the following paragraphs. The
properties of antigen-antibody systems, especially the reversibility of complex
formation, are such as to indicate that the antigen-antibody attraction is due
to these weaker interactions and not to the formation of ordinary chemical bonds.

The weak interactions between two molecules may be classified as electronic
van der Waals attraction, Coulomb attraction, attraction of electric dipoles or
multipoles, hydrogen-bond formation, etc. The forces increase rapidly in mag-
nitude ds the molecules approach one another more and more closely, and the
attraction between the molecules reaches its maximum when the molecules are
as close together as they can come. The molecular property which determines
the distance of closest approach of two molecules is the electronic spatial exten-
sion of the atoms in the molecules. It is possible to assign to each atom a
van der Waals radius, which describes its effective size with respect to inter-
molecular interactions. These radii vary in value from 1.2 A for hydrogen
through 1.4-1.6 A for light atoms (fluorine, oxygen, nitrogen, carbon) to 1.8-
2.2 A for heavy atoms (chlorine, sulfur, bromine, iodine, etc.). The shape of a
molecule can be predicted by locating the atoms within the molecule with use
of bond distances and bond angles and then circumscribing about each atom a
spherical surface corresponding to its van der Waals radius. This shape deter-
mines the ways in which the molecule can be packed together with other mole-
cules (2, sec. 24).

The most general force of intermolecular attraction, which operates between
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 205

every pair of molecules, is electronic van der Waals atiraction. This type of
electronic interaction between molecules was first recognized by London (23).
A molecule (of methane, for example) which has no permanent average elec-
tric dipole moment may have an instantaneous electric dipole moment, as the
center of charge of the electrons, in their rapid motion in the molecule, swings
to one side or the other of the center of charge of the nuclei. This instantaneous
dipole moment produces an instantaneous electric field, by which any other
molecule in the neighborhood would be polarized; the electrons of the second
molecule would move relative to its nuclei in such a way as to give rise to a
force of attraction toward the first molecule.

This electronic van der Waals attraction operates between every atom in a
molecule and every atom in other molecules in the near neighborhood. The
force increases very rapidly with decreasing interatomic distance, being inversely
proportional to the seventh power of the interatomic distance. Hence the
electronic van der Waals attraction between two molecules in contact is due
practically entirely to interactions of pairs of atoms (in the two molecules) which
are themselves in contact; and the magnitude of the attraction is determined by
the number of pairs of atoms which can be brought into contact. In conse-
quence, two molecules which can bring large portions of their surfaces into close-
fitting juxtaposition will in general show much stronger mutual] attraction than
two molecules with less extensive complementariness of surface topography.

Other types of molecular interactions result from the possession of a perma-
nent electric charge, electric dipole moment, or electric moment of higher order
by one or both of the interacting molecules. The effects of these charges and
moments have been classified in various ways, as ion-ion forces, dipole-dipole
forces, forces of electronic polarization of one molecule in the dipole field of
another, ete. All electrostatic interactions are very much smaller in water than
in a medium of low dielectric constant, and it ean be shown by calculation, mak-
ing use of known values of the effective dielectric constant of water for charges
a given distance apart (24), that in general these electric forces are of minor
importance, except when an isolated or essentially isolated electric charge is
involved. The electrostatic attraction of a positive group such as a substituted
ammonium ion and a negative group such as a carboxy! ion becomes significantly
strong, with bond energy 5 kilocalories per mole or more, if the structure of the
molecules containing the groups is such that they can come into juxtaposition.

A type of intermolecular attractive force which ranks in importance with the
electronic van der Waals attraction and the attraction of oppositely charged
groups is that associated with the structural feature called the hydrogen bond.
The importance and generality of occurrence of the hydrogen bond were first
pointed out in 1920 by Latimer and Rodebush (25) and summaries of the proper-
ties of the bond are given in the monographs quoted above. A hydrogen bond
results from the attraction of a hydrogen atom attached to one electronegative
atom for an unshared electron pair of another electronegative atom. The
strength of a hydrogen bond depends on the electronegativity of the two atoms
which are bonded together by hydrogen; fluorine, oxygen, and nitrogen, the
206 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

most electronegative of all atoms, are the atoms which form the strongest hydro-
gen bonds. The energy of a hydrogen bond between two of these atoms is of
the order of magnitude of 5 keal. per mole. This is so large as to have a very
important effect on the intermolecular interactions of molecules capable of
forming hydrogen bonds and on the properties of the substances consisting of
these molecules.

In synthesizing our knowledge of intermolecular forces and of immunological
phenomena into a definite picture of the antigen-antibody bond the immuno-
logical property of greatest. significance is the specificity of the combining power
of antibody for the immunizing antigen.

The forces of van der Waals attraction, hydrogen-bond formation, and inter-
action of electrically charged groups are in themselves not specific; each atom
of a molecule attracts every other atom of another molecule by van der Waals
attraction, each hydrogen atom attached to an electronegative atom attracts
every other electronegative atom with an unshared electron pair which comes
near it, and each electrically charged group attracts every other oppositely
charged group in its neighborhood. The van der Waals repulsive forces which
determine the van der Waals radii of atoms also are not specific; each atom in a
molecule repels every other atom of another molecule, holding it at a distance
corresponding to the sum of the pertinent van der Waals radii. We see, how-
ever, that specificity can arise in the interaction of large molecules as a result
of the shapes of the molecules. Two large molecules may have such spatial
configurations that the surface of one cannot be brought into contact with the
surface of the other except at a few isolated points. In such a case the total
electronic van der Waals attraction between the two molecules would be small,
because only the pairs of atoms near these few isolated points of contact would
contribute appreciably to this interaction, and, moreover, the distribution of
hydrogen-bond forming groups and of positively and negatively charged groups
of two molecules might be such that only a small fraction of these groups could
be brought into effective interaction with one another for any position and
orientation of one molecule with respect to the other; the energy of attraction —
of these two molecules would then be small. If, on the other hand, the
two molecules possessed such mutually complementary configurations that the
surface of one conformed closely to the surface of the other, if, moreover, the
electrically charged groups of one molecule and those of the other were so
located that oppositely charged groups were brought close together as the mole-
cules came into conformation with one another, and if the hydrogen-bond
forming groups were also so placed as to form the maximum number of hydrogen
bonds, the total energy of interaction would be very great, and the two molecules
would attract one another very strongly. We see that this strong attraction
might be highly specific in the case of large molecules which could bring large
areas of their surfaces into close contact. A molecule would hence show strong
attraction for another molecule which possessed complete complementariness
in surface configuration and distribution of active electrically charged and
hydrogen-bond forming groups, somewhat weaker attraction for those molecules
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 207

with approximate but not complete complementariness to it, and only very weak
attraction for all other molecules.

This specificity through complementariness of structure of the two inter-
acting molecules would be more or less complete, depending on the greater or
smaller surface area of the two molecules involved in the interaction. It may
be emphasized that this explanation of specificity as due to a complementariness
in structure which permits non-specific intermolecular forces to come into fuller
operation than would be possible for non-complementary structures is the only
explanation which the present knowledge of molecular structure and inter-
molecular forces provides.

This theory of structural complementariness of antigen and antibody was
first. suggested, in Jess detailed form than above, by Breinl and Haurowitz (11),
Alexander (12), and Mudd (13). A detailed discussion of the structure of
antibodies and of a postulated method of their formation has been presented
by Pauling (17), who has also reviewed the evidence supporting the theory of
complementariness.

It was suggested by Brein] and Haurowitz and by Mudd that the effect of an
antigen in determining the structure of an antibody might involve the ordering
of the amino-acid residues in the polypeptide chains in a way different from that
in the normal globulin. Rothen and Landsteiner (26) then pointed out that
the possibility of different ways of folding the same polypeptide chain is worth
considering, and this postulate was amplified by Pauling (17), who assumed
that all antibody molecules contain the same polypeptide chains as normal
globulin, and differ from normal globulin only in the configuration of the chains.
This assumption was made because it permits the formulation of a simple
proposed mechanism of manufacture of specific antibodies. An antibody mole-
cule, capable of existing in any one of a great number of configurations with
nearly the same energy, is synthesized, except for the final folding step, in the
same way as normal globulin. If no foreign substance is present, the chain
then folds into a stable configuration, characteristic of normal globulin; but if
an antigen molecule is present, the chain folds into a configuration stable in the
presence of the antigen, that is, into a configuration complementary to that
of a portion of the surface of the antigen molecule. This explanation of the
ability of an animal to form antibodies with considerable specificity for an
apparently unlimited number of different antigens (27), as shown especially
by the work of Landsteiner (10), is compatible with the principles of structural
chemistry and thermodynamics as well as with the immunological evidence.

To illustrate the way in which the complementariness theory accounts for
many reported observations we shall mention only one point, taken from the
great body of results on azoproteins obtained by Landsteiner. He observed a
pronounced cross reaction between an azoprotein made from m-aminobenzoic
acid and an antiserum to an azoprotein made from 4-chloro-3-aminobenzoic
acid and a different protein, but no reaction with the haptenic groups reversed.
The explanation of this is that the antibody to the 3-azo-4-chlorobenzoic acid
group conforms closely to this haptenie group, allowing either this group or the
‘208 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

-3-azobenzoic acid group, which differs in the replacement of the chlorine atom
by a smaller atom, hydrogen, to fit into the complementary cavity in the anti-
body; but the 3-azo-4-chlorobenzoic acid group cannot fit into a cavity designed
for the smaller haptenic group, and so the reverse cross reaction does not occur.
In a quantitative extension of Landsteiner’s work on hapten inhibition of pre-
cipitation reactions of simple polyhaptenic substances (10), Pauling and col-
laborators (28) have recently reported a great deal of evidence in support of the
complementariness theory. They interpreted their results on the inhibition of
the precipitation reaction between dyes containing p-azophenylarsonic acid
groups and antisera to hapten-homologous azoproteins to obtain numerical
values of the strength of the bonds formed by these antibodies with over twenty-
five different haptens. The observed correlation between the bond strengths
and the structure of the haptens is that which would be expected from the
complementariness theory.

This theory is not greatly different from some earlier proposals, such as
Ehrlich’s lock-and-key analogy (29), but it differs greatly from others. For
example, Buchner (30) considered that antigen molecules are split up and
incorporated into the antibody molecules, thus imparting specificity to them.
This theory or a closely related theory has been supported by many people,
including Burnet (31), who proposed a mechanism for the manufacture of anti-
bodies in the image of the antigens: the antigens act as templates for the manu-
facture by the body of specific enzymes, which then serve as the molds for the
production of antibodies similar to the original. antigens. Until recently there
has been no suggestion as to why antibodies similar in structure to an antigen
should combine specifically with it. Recently, however, Jordan (32) has stated
that a strong attraction would occur between such identical or nearly identical
molecules because of the quantum-mechanical resonance phenomenon; this has
been denied by Pauling and Delbriick (33), who pointed out that the resonance
energy would be so small as to be ineffective. Chemical evidence against the
identity of antibodies and specific antigens has been presented by many authors,
of whom the most recent are Haurowitz, Vardar, and Schwerin (34).

Forty years ago there was under way a keen controversy between Ehrlich and
Bordet, and their respective supporters, as to whether the bonds between anti-
bodies and antigens are chemical bonds or are physical forces of the sort pro-
ducing surface phenomena such as adsorption. The modern point of view
resolves this argument, but not in favor of either side; in fact, as in recent years
an understanding has been obtained of the forces responsible for surface phe-
nomena it has been found that these forces are the same as those which are
operative in chemical reactions, so that the old distinction between chemical
and physical forces has lost most of its meaning.

THE NATURE OF THE PRECIPITATE. Under suitable conditions (salt. concen-
tration, antibody-antigen ratio, etc.) the first stage of combination of antibody
and antigen, which may make itself evident in change in toxicity or other
properties of the antigen, is followed by precipitation. There has been much
discussion as to whether or not this second stage is specific, like the first stage,
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 209

or is non-specific. Direct experimental evidence on this point, while not con-
clusive, favors the view that the reaction is specific. The most pertinent
observations are those on the agglutination of mixed cellular antigens by mixed
antisera; Topley and collaborators (35) noted the formation of separate clumps
by the different cells, whereas Abramson (86) observed mixed clumping. Hooker
and Boyd (37) found that mixed human and chicken erythrocytes gave separate
clumps under some conditions and mixed clumps under other conditions. The
fact that separate clumping is observed at all strongly favors the concept of a
specific second stage, since mixed clumping might result from mechanical inter-
twining of specific clumps, whereas separate clumping would hardly be expected
to result from non-specific interaction. Heidelberger has pointed out that in
those cases where mixed clumping takes place the cells used were either very
large or of greatly different sizes (88).

A reasonable theory of agglutination and precipitation, the framework theory
(lattice theory), was proposed in 1934 by Marrack (6), and has received strong
support from the theoretical considerations and experiments of Heidelberger
(9, 14, 15) and Pauling (18, 19, 20, 28) and their collaborators.

It is clear that, after we have accepted a mechanism for the specific attachment
of antibody molecules to a cellular antigen, the simplest possible explanation
of the agglutination of the cells is that it results from the same mechanism; if
an antibody molecule had the power of specific attachment to two cells, it could
form specific bonds with the two cells and thus hold them together, and the
repetition of this process would lead to the formation of larger and larger clumps.
Specific precipitation of antibodies and molecular antigens would result from
the same mechanism if both antibody molecules and antigen molecules were
multivalent (capable of forming two or more antigen-antibody bonds); larger
and larger complexes A—B, A~-B—A, A--B-—-A—B, etc., would form until the
aggregates became macroscopic in size. The evidence supporting the framework
theory has been reviewed by Marrack (6), Heidelberger (14), and Pauling (17);
some of it, including that provided by recent work, is presented in the following
section in connection with a discussion of the valence of antibody molecules.

The first of the theories of non-specific precipitation is the theory of neutraliza-
tion of electrical charges. This theory was supported by many early investi-
gators, who were attracted by the analogy with the well-known phenomenon of
the mutual precipitation of oppositely charged colloids. Teague and Field (39)
investigated the charges of agglutinins and bacteria and concluded that the
former are positively and the latter negatively charged; it is now known, how-
ever, as the result of the application of improved experimental methods, that
under ordinary conditions (normal hydrogen-ion and salt concentrations) anti-
body molecules and most antigens are negatively charged, and the theory of
neutralization has in consequence been abandoned.

(The failure of precipitation or agglutination to occur in antigen-antibody

1 We have adopted the name ‘‘framework theory”’ instead of “lattice theory’? because of

our belief that the framework of antibody-antigen precipitates does not usually have the
regularity of structure which would be indicated by use of the latter expression.
210 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

systems with low salt concentration is, indeed, attributed to the electrostatic
repulsion of the negatively-charged complexes in solution, which prevents the
formation of large aggregates; agglutination or precipitation may occur in the
presence of salt, the cations of which neutralize the negative charges of the
complexes.)

Another theory of non-specific precipitation which was proposed soon after
the discovery of the precipitation reaction is that the reaction results from the
formation of a hydrophobic colloid, which precipitates in the presence of electro-
lytes. This theory has been revived recently by Eagle (40) and by Hooker and
Boyd (87). Eagle’s suggestion that the polar groups of the antigen which are
assumed to be responsible for its solubility are masked by a layer of antibody
molecules, which themselves turn their polar groups inward and present only
non-polar groups toward the solvent, has been discussed by Marrack (6), who
has marshalled some arguments against it. An important argument is that
particles can be agglutinated by an amount of antibody very much smaller than
the amount required to coat their surface; the most recently reported experi-
ments of this sort (41) indicate that azoerythrocytes can be agglutinated by
less than 0.02 per cent as much antibody as would cover their surface with a
layer 3.5 A thick.

Hooker and Boyd (37) have presented several arguments in support of the
thesis that ‘“... particles grow to visible size by the indiscriminate and non-
specific accretion of other related or unrelated, small or large, aggregates whose
primary nuclei are molecules or particles of antigen coated with antibody-
globulin.” - As additional evidence for this concept and against the framework
theory Boyd and Hooker (42) reported their failure to inhibit the agglutination
of erythrocytes by use of an excess of hemagglutinin. In our opinion the fact
that inhibition of agglutination of particles (43) as well as of precipitation of
molecular antigens (44) by excess antibody has been observed gives strong sup-
port to the framework theory. The failure of inhibition to occur under ordinary
circumstances may be due to the difficulty in saturating the multivalent anti-
gens, especially cellular antigens with thousands of combining groups, as is
indicated by the theories of Hershey (45) and Pauling (17). In particular, the
experiments of Heidelberger and Kabat (43) on the agglutination by untreated
pneumococci of pneumococci coated with antibody and then thoroughly washed
are most easily explained by the framework theory. Hooker and Boyd (46, 47)
have recently proposed the theory that precipitation of antibody by polyhaptenic
dyes may result from the action of the dye molecules in pulling the antibody
molecules to which they are bonded so tightly together as to prevent the solvent
from reaching the polar groups. This theory seems to be incompatible with our
observation (18) that in general the dyes of smaller molecular size, which accord-
ing to Boyd’s theory should pull the molecules more closely together, in fact
precipitate less completely than those of larger molecular size.

COMPOSITION OF ANTIBODY-ANTIGEN PRECIPITATES AND VALENCE OF ANTIBODY.
An essential requirement for agglutination or precipitation according to the
framework theory is that both antigen and antibody be multivalent. The
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 211

experimental observations which indicate multivalence of antigens and of
agglutinins and precipitins have been summarized by Marrack (6), Heidelberger
(14), and Pauling (17).

The most straightforward evidence for the necessary multivalence of antigen
is given by experiments on the reactions of antibodies with simple substances
of known structure. Subsequent to Landsteiner’s discovery (10) that simple
haptens inhibit the precipitation and agglutination reactions by forming soluble
complexes with antibody, it was found by Landsteiner and Van der Scheer (21)
that simple substances containing two or more haptenic groups form precipitates
with hapten-homologous antibodies. We have shown that of the twenty-seven
simple substances containing phenylarsonic acid groups which were tested with
antisera made by injecting rabbits with azoprotein made from p-arsanilic acid
each of those (twenty in number) which contained two or more of the haptenic
groups gave the precipitation reaction, whereas none of the monohaptenic
substances formed precipitates. These facts support the framework theory
strongly.

(The failure to obtain precipitates with some polyhaptenic substances reported
by Hooker and Boyd (46) and Boyd (47) may have been due to their failure to
work under conditions favorable to precipitation. We have obtained precipi-
tates with some of the same substances, and have observed that substances
which give precipitates with strong antisera may fail to do so with weak antisera.)

Experiments which have been reported on the number of haptenic groups per
azoprotein molecule necessary for precipitation with hapten-homologous anti-
body (48, 49) and some of our unpublished results indicate that a few groups are
needed, but so far they have not been precise enough to distinguish between 1
and 2 as the minimum.

Direct proof of the bivalence of diphtheria antitoxin is given by the studies
of the antitoxin in presence of an excess of toxin with use of the ultracentrifuge,
which showed that the complexes ToxAntitox and Tox:Antitox exist in the
solution (50).

The fact that slides can be coated with alternate unimolecular layers of anti-
gen and antibody in specific combination (51, 52) indicates effective bivalence
of antibody molecules as well as antigen molecules.

It has long been known that in antigen-antibody precipitates molecules of
antibody are present in larger numbers than those of antigen, the antibody-
antigen molecular ratio being considerably greater than unity for nearly all
systems (6, p. 161; 53). This was shown convincingly by the accurate quantita-
tive investigations of Heidelberger and his collaborators (54). A simple explana-
tion of this fact, which does not follow directly from the framework theory in its
original form (6, 14), is given by the theory as modified by Pauling (17), who
made the assumption that antibodies in general are at the most bivalent. (This
assumption was made because his proposed structural theory of the process of
formation of antibodies is such as to make unlikely the occurrence of anti-
bodies of higher valence.) The maximum valence N of antigens toward homolo-
gous antibodies is assumed to be determined by the sizes and shapes of the
212 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

antigen and antibody molecules, being equal to the number of antibody mole-
cules which, when bonded to the antigen molecule, can be packed around it.
If the antigen and antibody molecules were spheres of the same size, this number
would be N = 12; for smaller antigen molecules it would be smaller, and for
larger ones it would be larger.

The predicted antibody-antigen molecular ratio for small antigen molecules
would be N/2 at the equivalence zone, with the maximum valences of both
antibody and antigen effective; the limiting values of the ratio for antigen excess
would be 1, and for antibody excess MN — 1. For very large antigens the ex-
pected ratio at the equivalence zone would be less than N/2. These predictions
are in reasonably good agreement with Heidelberger’s observations (54, 17),
which correspond to the following values of N: ovalbumin and R-salt-azobi-
phenylazoovalbumin (molecular weight 40,000-46,000), N = 6; serum albumin
(m.w. 67,000), V = 6 to 8; thyroglobulin (m.w. 700,000), NV = 30 to 40. Data
of other investigators (55, 56, 57, 58, 59, 60, 61) correspond to similar values of
N, with assumed bivalence of antigen.

If the valence of the antigen were known, measurement of the antibody-
antigen molecular ratio could be interpreted to give the valence of antibody
molecules. The only reliable experiments of this sort which have been reported
so far are those of Pauling, Pressman, and Ikeda (20) with simple antigens of
known structure. They found that the dihaptenic antigens 2-methyl-4,6-
di(p-azophenylarsonic acid)phenol and 2-methyl-4,6-di(p-azobenzene(p-azo-
phenylarsonic acid))phenol gave with antisera homologous to the p-azophenyl-
arsonic acid group precipitates with the same molecular ratio throughout the
range of relative concentrations from antibody excess to antigen excess. This
is what would be expected if both antibody and dihaptenic antigen were bivalent,
the predicted molecular ratio under all conditions then being 1, corresponding
to the structure —A—-B—A—B—A—B—A—B— for the precipitate. The ob-
served molecular ratio (average of 119 analyses) was 0.75. The same independ-
ence of molecular ratio on relative concentration was found also for the four
trihaptenic and tetrahaptenic substances studied; this was interpreted as result-
ing from the effective bivalence of these molecules also, as the result of their small
size in comparison with the antibody molecules. The average molecular ratios
found for the trihaptenic and tetrahaptenic substances, 0.85,and 0.83, respec-
tively, are only slightly less than unity. These results, with assumed effective
bivalence of the antigens, indicate for antibody molecules the effective valence 2.3.

Substantiating evidence for the multivalence of precipitating antibodies is
provided by the observations which have been interpreted as resulting from
the presence in antisera of antibodies with a valence of one. Heidelberger
and Kendall (62) demonstrated the presence in rabbit antisera of univalent
antibodies, which are able to combine specifically with antigen but are not
able, in the absence of multivalent antibodies, to form precipitates. These
univalent antibodies also occur in considerable amount in horse antisera; they
seem to be produced in large amount by the first injections of ovalbumin into
horses, precipitating (multivalent) antibody being formed only on repeated
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 213

injection (63, 64). It is probable that univalent antibody confers on horse
antitoxins the peculiar properties which they show, in particular the pronounced
prezone (region of antitoxin excess in which precipitation does not occur). The
change in properties of antisera on heat treatment (65, 66, 67) or treatment
with formaldehyde (9) or other denaturing agent is probably due to the con-
version of bivalent antibody to univalent antibody by destruction of one of the
combining regions.

QUANTITATIVE THEORIES OF THE PRECIPITATION REACTION. Although the
quantitative physico-chemical treatment of immunological phenomena was
begun early in the present century, by Arrhenius and Madsen (1, 68), it is only
during the last decade that significant progress has been made. The reason
for the delay is not far to seek—it lies in the fact that the mathematical treat-
ment of numerical data of low accuracy has little significance so long as a sound
qualitative understanding of the phenomenon has not been developed. We
may mention in illustration Arrhenius’ discussion (1, p. 147) of the formula
C = K B?'8 which he found to express the relation between the amount C of
agglutinin bound by bacterial cells and the amount B of free agglutinin; Arrhe-
nius interpreted this equation (whose validity for the complex system we would
ascribe to the accidental distribution of the heterogeneous antiserum) as show-
ing that the agglutinin molecules are divided between two solvents, one within
and one without the bacterial cells, and that three molecules of the bound
agglutinin are formed from two of the free substance.

Recent theories are of two kinds: those based on thermodynamic equilibrium
among the reacting substances, and those based on the rates of reactions under
non-equilibrium conditions. There has been considerable discussion as to
whether or not immunological reactions are reversible—whether, for example,
an antigen-antibody precipitate is soluble, and is in equilibrium with free antigen
and antibody in solution. We know from general principles, however, that,
given time enough, every system reaches equilibrium, and every material is
more or less soluble; the questions of interest deal rather with such quantitative
points as the length of time required for the system to reach equilibrium, and
the magnitude of the solute concentrations in equilibrium with the precipitate.

That the precipitation reaction in some cases reaches equilibrium in the hours
or days usually allowed it is shown by various experiments on solution of the
precipitate by salt (69), acid (70), alkali (71), and excess antigen, even after
ageing for several months (72), including experiments in which there was varia-
tion in the method of approaching equilibrium (19).

Experiments on the Danysz phenomenon (73) and other related experiments
indicate that a long time—many days—is needed for equilibrium to be ap-
proached for reactions involving change in composition of antigen-antibody
precipitates.

The first quantitative theory which we shall discuss, that of Heidelberger
and Kendall (14), was based on consideration of the rate of antigen-antibody
combination under non-equilibrium conditions. The authors assumed that
antigen A and antibody B first react completely and rapidly to form the com-
214 L. PAULING, D. H. CAMPBELL AND D. PRESSMAN

plex AB, which uses up all the A (B being assumed present in excess). There
then occur two competing slow reactions:

AB + AB — AB:

The rates of formation of AB. (total number of units @) and A,B. (total number
of units 8) are

da _

= KIBITAB]
and

dp _ ? 2

wt K'{AB]

in accordance with the laws of chemical kinetics. It is assumed arbitrarily
that K = K’ and that the reactions are not reversible; by integration over the
course of the reaction until the solution is exhausted of AB there is obtained to
represent the composition of the precipitate, which consists of all the ABs and
A2B, formed, the equation

y Ra
a 2h Bo (1)
in which

y = milligrams of antibody precipitated

a = milligrams of antigen added

bo = total milligrams of antibody

R = antibody/antigen weight ratio at equivalence point

This equation for the composition of the precipitate, which corresponds to
change from 2F at large antibody excess to R at the equivalence point, has been
shown (15) to be in satisfactory agreement with the excellent experimental data
obtained by the authors. In view of the arbitrary and unlikely assumptions
originally used for its derivation, it is gratifying that Kendall himself (74) has
recently derived the equation in another way, and that we have found (unpub-
lished work) that the equation is obtained as an approximation from general con-
siderations of chemical equilibrium when the assumption is made that the ratio
may vary between the limits 2R and # and an expansion is made in powers of a/b.

Kendall’s derivation is essentially the following. (He considers also some
more general cases.)

Let antibody and antigen both be bivalent. For By and A» molecules of
antibody and antigen, respectively, there are 2B) and 2A) combining groups.
Assume, for antibody excess, that all of the 2A, antigen groups are bonded to
antibody groups, and that they are distributed at random among the 2B,
antibody groups, without regard to whether or not the antibody molecule is
already bonded at the other end. Since the chance that an antibody group is
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 215

bound is 2Ao/2Bo, the chance that it is free is 1—Ao/Bo, and the fraction of
antibody molecules free at. both ends is (1—Ao/By)?._ The fraction not free at
both ends is 1—(1—Ao/Bo)? = 2A0/Bo—(Ao/Bo)*, and the number not free at both
ends is this multiplied by the total number, Bo. If it be assumed that all anti-
body molecules not free at both ends are carried down in the precipitate with
the antigen molecules the molecular ratio for the precipitate becomes

B pp

Avp = — Ao/Bo (2)

and, introducing the ratio 2 of molecular weights, the weight ratio is found to be

Rie

=2R—-RS (3)
bo
This is identical with equation 1.

Similar considerations have also been used by Ghosh (75) for the derivation
of related equations.

An involved theory of antigen-antibody equilibria based in part on probability
considerations has been extensively developed by Hershey (45). The theory,
in common with others based on multivalent antigen and antibody, is in qualita-
tive and rough quantitative accord with experiment.

The only theory of the precipitation reaction which, following the program
begun by Arrhenius, has been developed by straightforward application of the
principles of chemical equilibrium is that of Pauling, Pressman, Campbell and
Ikeda (19). This theory applies only to relatively simple systems, namely,
those composed of bivalent antigen and bivalent antibody, univalent hapten,
certain soluble complexes, and precipitate with invariant composition AB.

In order to show the nature of the treatment, we present here the derivation
of the equation for the amount of precipitate formed in absence of hapten,
generalized over the earlier treatment by consideration also of the complex AB».

Let the molecular species A, B, AB, A.B, and ABz in solution be in equilibrium
with each other and with solid AB,». We represent the concentrations of the
five solutes by the symbols

[A] =a
[B] =8
[AB] = s
[A,B] = 4
[ABy] = b

The quantities a, 8, a, and } are variable, whereas s is constant for a given sys-
tem with precipitate present; it is the solubility of the precipitate. The equi-
librium expressions for the three reactions

A+B = AB
A+ AB = A.B
B+ AB = AB,
216 L. PAULING, D. H. CAMPBELL AND D, PRESSMAN

are respectively

8

a _

oe (5)
b —_— Mu

3s K (6)

For simplicity we have assumed that each of the bonds in the complex A—B—A
has the same strength as the bond in AB; this is probably a good approximation,
in view of the fact that the two bonding regions are probably far apart on the
large antibody molecules. (The theory can be carried through without this
assumption.) The constant K corresponds to equilibrium for one antibody
valence and one antigen valence, and the factor 4 is an entropy factor or sym-
metry factor. We use K” rather than K for the second bond in AB, because
steric repulsion between the two antibody molecules attached to the same
small antigen would be expected to decrease the stability of this complex.

The expressions for the total amounts of antigen and antibody in the system
(per unit volume of solution) are

ABpp +8 +a + 2a+ bd = Atota (7)
and

ABy +8+8 +a+ 2b = Brora (8)
Subtracting equation 8 from 7 we obtain

a—B+a—b = Aria — Broa

Eliminating 8, a, and 6 with the use of equations 4, 5, and 6 we obtain

TT ot

a—- + Ksa —

4Ka 4Keo = A total - Beotat

This quadratic equation in @ gives on solution

1

= 201 + Ks) { (A totat —_ Brotat + [s(1 + K"s)(1 + Ks)/K

a

+ (Atotat — Brota)? (9)

(The positive rather than the negative sign before the radical is seen to be cor-
rect by the consideration of limiting cases.) From equation 7 we find on elimi-
nating a and 5 the expression

2
K"'s

AByp = Atctaa — § — (1 + Ks)a — Fe

(10)

 
ANTIGEN, ANTIBODY AND PRECIPITATION REACTION 217

Equations 9 and 10 give the solution to our problem; from 9 the value of a@ is
to be found in terms of Arora and Biota: and the parameters of the system s, K,
and K”, and this on substitution in 10 gives the amount of precipitate.

It is shown in the original paper how this equation (with K” = 0) accounts
for many observed properties of antigen-antibody systems.

Future progress which may be anticipated involves the extension of this
straightforward thermodynamic treatment to include the case of variable com-
position and randomness of structure of the solid phase. This will require the
development of satisfactory approximate expressions for the free energy of such
a solid phase, by application of the methods of statistical mechanics on the basis
of sound structural concepts. This problem is not an easy one; but fortunately
promising methods for attacking it have been developed in recent years by able
theoretical physicists interested in the problem of the stability of alloys with
greater or smaller degree of randomness of atomic arrangement, and we may be
confident that great progress will soon be made in the formulation of a satis-
factory quantitative theory of the precipitation reaction.

SUMMARY

The forces responsible for combination and attraction of antigen and antibody
molecules may be classified as electronic van der Waals attraction, Coulomb
attraction, attraction of electric dipoles or multipoles, formation of hydrogen
bonds, etc. The specificity of interaction of antigen and antibody molecules
arises from their structural complementariness, which permits close contact
of the molecules over sufficient area for these weak forces to co-operate in forming
a strong antigen-antibody bond.

The weight of evidence indicates that further combination of the initial
antigen-antibody complexes to form a precipitate is a specific rather than a
nonspecific reaction and is due to a continuation of the primary combination
step to form a framework structure of alternate antigen and antibody molecules.

Furthermore it appears that both precipitating antigen and precipitating
antibody must be multivalent, at least bivalent.

The more recent quantitative theories of the precipitation reaction are
discussed.

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