Modelling the interactions
between plurivalent macromolecules such as
antigen and antibody
Immunologists often perceive
a need for some kind of physicochemical model that can describe the interaction
of plurivalent macromolecules and particularly the equilibrium state towards
which the reaction mixture will tend to go  a model perhaps based on
something like the Law of Mass action. They need some such model so that they
can describe the properties of a serum in a compact way by quoting only a few
numbers  and they want these few numbers to have predictive value. They want
a compact description that can be derived from a few reactions in test tubes,
but can be used to predict how the antiserum will behave at a wide range of
antiserum concentrations and antigen concentrations when used, for example, in
a diagnostic test or for serotherapy.
The concentration of the
antibody and its affinity for antigen are the first two variables that spring
to mind as neat way to describe a serum. But both of these variables are
unexpectedly troublesome even to define.
Take the affinity for
example. The common simplest measure of the affinity of a combination reaction
is the association constant K_{d} customarily defined as the product of the molar
equilibrium concentrations of the products divided by the product of the molar
equilibrium concentrations of the unchanged reactants. But which products? If
the reaction is a combination between two plurivalent macromolecules there may
be a multitude of products differing in their stoichiometry and/or in their
stereochemistry. A typical antibody will have at least two combining sites,
both with the same structure. A typical protein antigen will have a valency
from 5 to 100, and these 5 to 100 sites will sometimes all have different
structures and will all react with different antibodies.
For example, if we take the
simplest kind of antibody, IgG, which has two identical binding sites, and the
simplest kind of protein antigen, (Ag), namely one with a valency of 5, having
5 nonidentical epitopes, then even some of the simplest stoichiometry can give
numerous distinguishable products that may all have very different free
energies. e.g. IgG3Ag2 has 70 possible structures, and it is likely that some
of the complex molecular species formed have millions of antigen and antibody
molecules in them.
If we stick to the letter of
the definition of an equilibrium constant then we shall have a very complex
definition. Who is going to be interested in an association constant measured
in units of litres^{1000000}mol^{1000000 }?
I could sidestep all this
complexity only by changing the title of my talk. For plurivalent let us
substitute univalent so that we have simple products formed and then let us try
again. But this serves only to uncover another difficulty, namely that
antibodies are extremely diverse in their affinity for antigens. The
association constants quoted for artificial univalent antigens lie in the range
of about 10^{4} to 10^{10} litres per mole, and I suspect that the lower limit
does not represent any biological fact, but merely the difficulty in detecting
antibody when its affinity gets too low. The problem is that there often seems
to be a continuous spectrum of affinities between antibody and nonantibody.
How low can the affinity of an lg for its antigen get before we stop calling it
antibody? This one problem makes it difficult to define either affinity or
concentration of antibody.
For example, if we react
fixed amounts of antibody with varying concentrations of free
antigen and see how the
concentration of bound antigen depends on the concentration of free
antigen, one would expect a
graph something like this, showing that the antibody eventually becomes
saturated with antigen and won't bind any more antigen:
Instead, one often gets
something like this, with no sign of saturation at all, even when the free
antigen concentration is high enough to bind effectively to the antibodies with
affinities at the lowest limits ever reported.
Looking for antibody in a
serum like this is like looking for stars in the heavens: the harder you look,
the more you see. That is why I question the common presupposition that
concentration of antibody is a useful concept. It depends too much on what you
are willing to call antibody.
So what do immunologists do
in practice? To construct a model of the antigenantibody reaction at
equilibrium inevitably depends on making some drastic simplifying assumptions.
The most common of these is one that I find just too, too drastic to be reasonable.
They say in effect 'Yes we
know that the products of the reaction are very numerous and very complex, but
let us just assume that the antigenantibody reaction is a simple bimolecular
combination giving a simple bimolecular complex, and let us find the antibody
concentration and affinity by measuring how the bound antigen concentration
varies with the free antigen concentration at constant total antibody
concentration, and then plotting a Scatchard plot (b/f v. b) or a
doublereciprocal plot (1/b v. 1/f). These should give a straight line if
the reaction is equivalent to the Langmuir adsorption isotherm, and the
gradient and intercept of the straight line should allow us to calculate the
concentration and affinity of the antibody.
In fact, these plots often
give a very curved line, and the gradient that you get depends very much on the
range of concentrations that you choose to use. With some sera you can get
almost any association constant you want, even a negative association constant
if you want.
There is no empirical or
theoretical reason why the socalled association constant obtained in this way
should be expected to be constant at all or be expected to represent any
particular characteristic of the serum.
Here is a Scatchard plot
published by Pellequer, J.L. and Van Regenmortel, M. H. V. (Molecular
Immunology 30, 955958 (1993), and another by Mukkur, T. K. S, Szewczuk, M. R.
and Schmidt, D. E. Jr., (1974) Immunochemistry 11, 913.
There have been a few heroic
efforts to construct more valid models. Goldberg's was the most sophisticated,
based on combinatorial algebra. In principle it aimed to predict the
equilibrium concentration of any particular molecular species that may be
formed, but its formulae were too complex for most immunologists, and it was
based on simplifying assumptions that were not very realistic. He assumed that
the antibody binding sites all had the same specificity and a uniform affinity
for antigen, and also that the antigenic determinants (epitopes) were all
identical. he also assumed that intraaggregate reactions yeilding cyclic
structures do not occur. In many proteins the determinants are in fact all
nonidentical. He also had to assume, along with everyone else who has
constructed a model, that the interaction of one pair of complementary sites
had no influence on the probability of the interaction of any other pair of
complementary sites.
Steensgaard et al. constructed a model based on
much the same simplifying assumptions as Goldberg's but depending more on
numerical simulation. Unfortunately his model is defined by three simultaneous
equations that are mutually contradictory, and that, I think, disqualifies them
from any serious application .
So what can we do? Enough of
the difficulties. What about some possible solutions?
The problem of choosing which
reaction products to consider in the definition of affinity is not too
difficult. We could in principle choose any particular product or products and
define the affinity of the particular reaction that creates this particular
product, whether or not there are a multitude of other reactions going on at
the same time. Let us for example use the symbol K to denote the association constant for the reaction in which one
antibody molecule binds with a particular one of its binding sites to a
particular epitope on one antigen molecule.
This at least seems to define
a constant that may be expected actually to be more or less constant, in the
sense of being fairly independent of antigen and antibody concentrations. It
doesn't, however, make it obvious how we can measure K. It may not be feasible to measure the equilibrium concentration
of just one AgAb complex among a million.
There also remains the
problem that antibodies are often extremely heterogeneous in their values of K, making it difficult to define what we
mean by antibody, and this in turn makes it
very difficult to define any
kind of average K or any antibody
concentration. What is antibody? How small can K get before we stop calling the immunoglobulin an antibody? If we
can't answer that we can't even say what we mean
by concentration or by affinity; far less devise any way of measuring them.
I suggest this problem can be tackled by measuring the concentration of each kind of antibody present not in units of moles per litre, but in units of 1/K. In other words, we aim to measure AK the dimensionless product of antibody concentration A and affinity K. For want of a better name, we could call this the dimensionless antibody concentration of the serum. If, as we would expect, there are many different kinds of antibodies present, each with its own concentration A_{i} and its own affinity K_{i} we aim to measure _{} , and we could call this the dimensionless antibody concentration of the serum. This dimensionless concentration, unlike the traditional simple concept of concentration itself, cannot take extreme values without our knowing. A serum with very low affinity could have huge values of A without our being able to demonstrate it, but if it has very high or very low values of the dimensionless concentration, this is bound to reflect itself in laboratory measurements and in the effectiveness of the antibody when it is put to practical use.
Assumptions of the present model
the antigen molecules are all identical
different sites on the one antigen or
antibody molecule react independently of one another
any two antigenic determinants are either
identical or noncrossreacting: weakly crossreacting determinants are not
considered
the various kinds of determinant on the
antigen molecule are present in equal numbers
the antigen is sufficiently multivalent
that a large proportion of each kind of determinant is free even when only a
small proportion of the antigen molecules are free
Table of symbols used in the model:
Symbol 
Definition 
Typical value 
Units 
A_{i}(K)dK 
molar concentration of
antibodies of specificity i and
affinity in the narrow range between K
and K + dK in the serum 
10^{6}×dK 
l mol^{1} 
C 
molar concentration of
antigen (bound plus free) in reaction mixture 
10^{8} 
mol l^{1} 
D 
dilution factor of the
antiserum in the reaction mixture; i.e. the volume of the reaction mixture
divided by the volum of neat serum contained in it 
100 

dK 
a notional small increment
in the value of K; its value is not
defined 

l mol^{1} 
F 
proportion of the antigen
remaining free at equilibrium 
0.5 

i 
subscript to distinguish
different types of antigenic determinant: i = 1, 2, 3 . . . for determinants
of types 1, 2, 3 . . . 
2 

K 
Association constant for
the reaction in which one antibody molecule binds with a particular one of
its binding sites to a particular epitope on one antigen molecule to give a
bimolecular complex 
10^{7} 
l mol^{1} 
M 
A constant that defines the
shape of the graph of F as a
function of D when C is constant. It is defined by the
equation M = D lnF, and
theoretically is equal to _{} . 
10 

n_{i} 
the number of determinants
of type i present on each antigen
molecule 
1 

p_{i}(K) dK ? 
proportion of antigenic
determinants of type i that are
bound directly to antibody with affinity in the narrow range between K and K + dK 
0.01 

p_{i}^{o} 
proportion of antigenic
determinants of type i that are
bound directly to antibody of any
affinity; i.e. it is the ratio of bound concentration to total concentration 
0.01 ? 

q_{i}(K)dK 
molar concentration of
bound antibody combining sites of specificity i and affinity in the range between K and K + dK 


Derivation of the model
Let an antiserum contain
antibodies of specificities 1, 2, 3, . . . i,
. . . , each present with a distribution of affinities defined by the
distribution function A_{i}(K), where A_{i}(K)dK is the serum concentration of
antibodies of specificity i and
affinity in the narrow range between K
and K + dK. By the law of mass action
we can define K in terms of the
following three molar concentrations at equilibrium.
1. The
concentration of bound antibody combining sites of specificity i and affinity in the range between K and K + dK will be a function of K
and may be written as q_{i}(K)dK.
2. The
equilibrium concentration of free antigenic determinants (not directly involved
in an antigenantibody bond) of specificity i
is their total concentration minus the concentration of determinants of
specificity i bound directly to
antibody of any affinity. In other
words, the equilibrium concentration of free antigenic determinants of
specificity i may be written _{}, where C is the total molar concentration of
antigen present (bound plus free), n_{i} is the number of determinants of type i present on each antigen molecule and p_{i}^{o} is the proportion of antigenic determinants of type i that are bound directly to antibody of
any affinity.
3. The
concentration of free antibody combining sites of specificity i and affinity in the range between K and K + dK will be their total molar concentraton minus the
concentration bound to antigenic determinants of type i, which gives _{}, where D is the
dilution factor of the antiserum in the reaction mixture, i.e. the volume of
the reaction mixture divided by the volum of neat serum contained in it.
By the law of mass action we
can relate K to these three concentrations
as follows:
_{},
_{} 
and since the number of
antibody binding sites of specificity i bound with affinity K must equal the number of antigenic
determinants of specificity i bound with affinity K, we can replace q_{i}(K)dK with Cn_{i}p_{i}(K)
and cancel out some terms that appear in the numerator and denominator to give
_{} .
Rearranging this equation
gives _{} .
Now, from our stated
assumptions, p_{i}^{o} is small, since a multivalent antigen needs only a
small proportion of its sites bound in order to achieve degrees of antigen
binding in the range that interest us, i.e. with a measable proportion of the
antigen free. We may therefore replace _{} either with 1 or,
better, with _{} to give
_{} .
Summation of both sides of
this equation over all values of i gives
_{}.
We have assumed that n_{i} is the same for all values
of i, but some small concesson may be
made towards accommodating various values of n_{i} by
defining a weighted mean n by the
formula
_{}. The last term in the above equation could perhaps be
neglected since p_{i}^{o} is small, but it may be better to replace it by a
value in the middle of its possible range. To do this, let us express the last
term as a multiple of a parameter m, defined
as _{}. This lets us express our equation as _{}.
The small parameter m, from
its definition, varies with CK and could in principle have values ranging from
0 to _{}. Rather than neglect it, it is possible to replace it by a
value near the middle of this range, namely _{}, and this lets us express our main equation as
_{} .
Integration with respect to K gives
_{}
The probability of any given
antigenic determinant of type i being
bound directly to antibody is p_{i}^{o}, and so the probability that it is free is (1  p_{i}^{o}). Invoking our stated assumption of independent
binding, we can deduce that the probability of all n_{i} sites of type i being free is _{}, and that the probability of all sites of all types on a
given antigen molecule being free is _{}and so on. If we call this expression F, i.e. the proportion
of antigen molecules remaining free, then taking natural logarithms gives
_{}.
Now, since p_{i}^{o} is small for reasons discussed earlier, we may safely
replace ln(1  p_{i}^{o} ) by the first one or two terms of its Maclaurin
expansion, to give
_{}.
It will be noticed that the
righthand side of this formula is a factor in the righthand side of equation
1, and allows us to write equation 1 in the form
_{} ........... equation 2
This is the central equation
that defines the present model in its most general form. It can be expressed in
an extremely simple form if used to describe the reaction of various quantities
of antiserum with a fixed total concentration of antigen, because in these
circumstances the lefthand side of the equation is a constant, and the
equation is reduced to
_{}
............... equation 3,
in which only one constant, M, is needed to define the curve that
relates the dilution factor D of the
serum to the proportion F of antigen
that remains free. D usually ranges
from 1 upwards, and 0 < F < 1,
so lnF is negative and M has a positive value, often in the
range 10 to 100.
Equation 2 may be interpreted
in a fairly simple way even when C is not constant. In effect it says that D log_{e}F is equal to the antibody concentration of the serum multiplied by
the mean value of _{} (averaged over all
the antibody molecules present). Now,
_{}
where _{} . . . are the mean
values of K, K^{2}_{, }K^{3}, K^{4} averaged over
all the antibody molecules in the serum. This implies that at the lowest values
of C, DlnF will be almost independent of C, and at slightly higher values it
will be approximately a linear function of C with a negative gradient. At still
higher values of C, DlnF will be approximated by a quadratic function of C and
could have a positive gradient.
Normally distributed
affinity:
If we assume that the
antibody is normally distributed with respect to lnK, we may replace A(K) in
equation 2 with the normal distribution function:
_{} ,
and we are left with a pair
of integral equations that cannot be used without a computer. If, on the other
hand, we use Sips' first distribution function (Sips, 1948), which resembles
the normal distribution function,
i.e. if _{} ,
then the lefthand side of
equation 2 becomes integrable to give
_{}
or, _{} equation
3.
The shape of the Sips
distribution may be seen below:
The following example is based on = 0.8; K_{0} = 10^{6} l mol^{1}; A_{0}D = 10^{7} l mol^{1}:
Can we explain any of the
paradoxical curves that have been observed?
First we may show that _{} is always positive
for a Sips distribution, as we might expect.
Equation 3 gives us
_{}
Since 0 < < 1, the above expression must always be positive.
But what about the dependence
of b, i.e. the molar concentration of bound antigen, on C, the molar
concentraion of total antigen?
Since F = 1  b/C, we may
substitute for F in equation 3 to get
_{}
and differentiation with
respect to C gives
_{} .
If we divide by the
exponential factor (which must be positive, and therefore must leave the sign
unchanged) we find that the sign of db/dC is the same as the sign of
_{}
The last two terms are
clearly < 0 and the first term >0. As A_{0}/CD
increases, the last term will increase linearly in magnitude, whereas the
positive term will increase exponentially. We must therefore expect that db/dC
> 0 as expected for large A_{0}/CD, but we cannot yet rule out the paradoxical
possibility that db/dC < 0 for small A_{0}/DC. The critical change of sign will
occur when
_{} .
To simplify this, let _{} so that our condition
for a change of sign of db/dC becomes _{} .
This is an equation of the
form e^{x} = 1 + kx, where k > 0. Simple graphical methods of
solution show that this equation will always have the trivial solution x = 0,
and will also have one other solution if k > 1 . Thus our condition for a
change of sign of db/dC is k > 1, i.e.
_{}
or, _{} .
This can never happen, since
0 <
< 1. Our conclusion must be that db/dC must always be positive.
Experimental test of the
model
In order to test whether the
model is realistic, the standard Popperian approach is to derive predictions
from the equations and test whether these predictions are compatible with
experimental observations. Equations 1 and 2 give two very contrasting sets of
predictions. If we carry out the antigenantibody reaction by serially diluting
the antigen and reacting it with a fixed amount of antiserum then equation 1
will predict a wide variety of possible binding curves, depending on how the
antibody is distributed over the range of possible association constants, and
on experimental variables. Some of these predicted curves are illustrated
below. For the purposes of testing the model, these curves are not very useful,
because their variation in shape would allow them to be fitted to almost any
experimental data: they are not in much danger of being disproved. In Popper's
terminology, these predictions are not risky and so do not provide a very
stringent test of the model.
By contrast, if we carry out
the reaction by serially diluting the antiserum and reacting it with a fixed
amount of antigen, then the whole of the lefthand side of equation 1 becomes a
constant for the duration of the experiment, and it reduces to the very simple
equation 2. The prediction of equation 2 is that if the experiment is done this
way, the binding curve will always have the same shape and the same gradient,
regardless of which multivalent antigen is used, or at what concentration it is
used, or what antiserum is used. This is what Popper would have called a risky
prediction: one that could readily be falsified.
This simple and constant
theoretical curve, in fact, is just what was reported by Brownstone, Mitchison
and PittRivers. Their empirical curve, which summarises the results of many
experiments, is shown below in comparison with the theoretical curve
represented by equation 2. The shape and the slope of the two curves agree
about as closely as could be expected.
This still does not show that
the theoretical model is realistic. It merely shows that the model has been
subjected to a reasonably stringent empirical test and has not so far been
contradicted.
Appendix: Glossary of
Mathematical Symbols used
Symbol 
Definition 
Typical value 
Units 
A_{i}(K)dK 
molar concentration of
antibodies of specificity i and
affinity in the narrow range between K
and K + dK in the serum 
10^{6}×dK 
l mol^{1} 
C 
molar concentration of
antigen (bound plus free) in reaction mixture 
10^{8} 
mol l^{1} 
D 
dilution factor of the
antiserum in the reaction mixture; i.e. the volume of the reaction mixture
divided by the volum of neat serum contained in it 
100 

dK 
a notional small increment
in the value of K; its value is not
defined 

l mol^{1} 
i 
subscript to distinguish
different types of antigenic determinant: i = 1, 2, 3 . . . for determinants
of types 1, 2, 3 . . . 
2 

K 
Association constant for
the reaction in which one antibody molecule binds with a particular one of
its binding sites to a particular epitope on one antigen molecule to give a
bimolecular complex 
10^{7} 
l mol^{1} 
M 
A constant that defines the
shape of the graph of F as a
function of D when C is constant. It is defined by the
equation M = D lnF, and
theoretically is equal to _{} . 
10 

n_{i} 
the number of determinants
of type i present on each antigen
molecule 
1 

p_{i}(K) dK ? 
proportion of antigenic
determinants of type i that are
bound directly to antibody with affinity in the narrow range between K and K + dK 
0.01 

p_{i}^{o} 
proportion of antigenic
determinants of type i that are
bound directly to antibody of any
affinity; i.e. it is the ratio of bound concentration to total concentration 
0.01 ? 

q_{i}(K)dK 
molar concentration of
bound antibody combining sites of specificity i and affinity in the range between K and K + dK 

