Mathematical basis of affinity

3.1.1 Mathematical basis of affinity

The combination of antigen (Ag) and antibody (Ab) to form a complex is a reversible reaction: K

Ab + Ag

a Ab · Ag K d

where K a and K d are the association and dissociation constants, respectively. The law of mass action can be applied to this reaction with the affinity being given by the equilibrium constant K. As mentioned at the beginning of the section, although the law of mass action is used in general here, strictly it applies only to homogeneous systems such as monoclonal antibodies, from hybridomas or plasmacytomas, reacting with monovalent haptens. In the real world, antibodies in serum are polyclonal and therefore heterogeneous.

By the law of mass action: K

K= a [ = Ab Ag ⋅ ] K d [ Ag Ab ][ ]

where K, K a and K d are as defined above, [Ab · Ag] is moles of immune complex (products), [Ab] is moles of antibody (reactant) and [Ag] is moles of antigen (reactant). Remember that Ag refers to monovalent antigenic determinants and Ab to independent antibody-combining sites. As can be seen the amount of complex formed is proportional to the value of K. As each reactant is expressed in mol/l the overall units of K are litre/mol.

If K is determined with respect to total antibody-binding sites, Ab t , the following form of the Langmuir adsorption isotherm may be derived from the mass action equation:

Ab + Ag ↔Ab · Ag. By the law of mass action:

[ Ab Ag ⋅ ]

= K ( equilibrium constant ), [ Ab Ag ][ ]

therefore [Ab · Ag] = K[Ab][Ag].

Make Ab t = [Ab · Ag] + [Ab],

72 C H A P T E R 3: Antibody interactions with antigens 72 C H A P T E R 3: Antibody interactions with antigens

Substituting (2) in (1): [Ab · Ag] = K[Ag]([Ab t ] – [Ab · Ag]), therefore [Ab · Ag] = K[Ag][Ab t ] – K[Ag][Ab · Ag] [Ab · Ag] + K[Ag][Ab · Ag] = K[Ag][Ab t ] [Ab · Ag](1 + K[Ag]) = K[Ag][Ab t ]

[ Ab Ag ⋅ ]

K Ag = [ ]

Ab t + [ ] 1 K Ag [ ] Make [Ab · Ag] = b, then (3) may be rewritten as:

b [ Ab t ] As b = [Ab · Ag] = bound Ag concentration and [Ag] = free antigen concentration, then a plot

of 1/b against 1/[Ag] can be extrapolated to obtain [Ab t ]. This is the total of antibody-combining sites. Note: The concentrations expressed in the above equations are molar equivalents; hence, for example, when [Ab · Ag] = bound Ag concentration, this is a molar equivalence and not a weight equivalence.

The determination of the total antibody-combining sites, although theoretically straightfor- ward, is often difficult to determine experimentally. With a monoclonal antibody of high affinity there will be no problem obtaining a straight line for the Langmuir plot experimentally; thus, the extrapolation back to the ordinate is valid and accurate. However, with a heterogeneous anti- serum, there can be considerable deviation from linearity. This is often the case with a small amount that is predominantly composed of low-affinity antibody. Under these conditions the

3.1DETERMINATION OF ANTIBODY AFFINITY

Langmuir plot can curve upwards as it nears the ordinate. In these situations, it is essential to have the maximum possible number of points along the linear part of the curve so that the best estimate of the intercept of this linear portion on the ordinate can be made. It is possible to cir- cumvent some of these practical problems by using an independent method for determining the total antibody, e.g. quantitative precipitation or elution from immunoadsorbants. Unfortunately these alternatives also present their own problems.

To return to the mass action equation: = [ Ab Ag ⋅ K ]

[ Ab Ag ][ ] If increasing amounts of antigen are reacted with a fixed amount of antibody, a point is

reached where half the antibody-combining sites are occupied by antigen. At this point [Ab] = [Ab · Ag], therefore in the mass action equation,

= [ Ab Ag ⋅ K ]

[ Ab Ag ][ ]

1 K = [ Ag ]

In other words, affinity is equal to the reciprocal of the free antigen concentration when half the antibody sites are occupied by antigen.

The value of K can be obtained from the plot in Fig. 3.1 by calculating the value of 50% Ab t and reading off from the graph the value of K. In experimental systems this value can also be obtained from a plot of the logarithmic trans- formation of the Sip’s equation as follows. If:

1/ 6 Fig. 3.1 Langmuir plot of reciprocal of bound (1/

b) versus reciprocal of 4 free antigen (1/ c or 1/Ag) of a serum,

from a baboon immunized with

2 human chorionic gonadotrophin.

1/Ab The regression coefficient, r = 0.98. The value for Ab

10 t 0 obtained by extrapolation 1 2 3 4 K 5 6 × 10 to infinite antigen concentration is

1/c

3.21 × 10 –11 mmol/l. Antibody affinity K = 4.3 × 10 10 l/mol.

74 C H A P T E R 3: Antibody interactions with antigens 74 C H A P T E R 3: Antibody interactions with antigens

log r

= a ⋅ log K + a

⋅ log[ Ag ]

n r For IgG, n = 2, therefore:

[ Ab t ] moles of antibody =

2 and

b r = [ Ab t ]/ 2

where b = [Ab · Ag]. Substituting for n and r in (5):

log b

= a log log[ ⋅ K + a ⋅ Ag ].

[ Ab t ] − b If:

log b is plotted against log[Ag] [ Ab t ] − b

when: log b

= 0 [ Ab t ] − b

then: K = 1

[ Ag ] The heterogeneity index, a, is a measure of the number of different molecular species of anti-

body. It can range from a value of 0 to 1. Low values represent a large degree of heterogeneity, whereas monoclonal antibody should theoretically have an index of 1. These relationships have been derived on the assumption that the distribution of antibody affinities in a polyclonal popu- lation is random and symmetrical about the mean. There is now good experimental evidence to suggest that the distribution of affinities is often skewed or even bimodal.