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 non-identical 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^1000000mol^-1000000 ?

I could side-step 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 10⁴ to 10¹⁰ 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 non-antibody. 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 antigen-antibody 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 antigen-antibody 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 double-reciprocal 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 so-called 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, 955-958 (1993), and another by Mukkur, T. K. S, Szewczuk, M. R. and Schmidt, D. E. Jr., (1974) Immunochemistry 11, 9-13.

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 intra-aggregate reactions yeilding cyclic structures do not occur. In many proteins the determinants are in fact all non-identical. 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 non-cross-reacting: weakly cross-reacting 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
Ai(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
ni	the number of determinants of type i present on each antigen molecule	1
pi(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
pi^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 ?
qi(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 antigen-antibody 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_ip_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 right-hand side of this formula is a factor in the right-hand 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 left-hand 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_eF 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, K²_, K³, K⁴ 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 left-hand 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; K₀ = 10⁶ l mol^-1; A₀D = 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 A₀/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 A₀/CD, but we cannot yet rule out the paradoxical possibility that db/dC < 0 for small A₀/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 antigen-antibody 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 left-hand 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 Pitt-Rivers. 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
Ai(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
ni	the number of determinants of type i present on each antigen molecule	1
pi(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
pi^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 ?
qi(K)dK	molar concentration of bound antibody combining sites of specificity i and affinity in the range between K and K + dK