**3.3.5. Experimental Determination of K _{a }(Homogeneous
Case)**

Suppose we start with an antibody concentration of [Ab]_{o }and
vary antigen concentration [Ag]. At equilibrium, the antigen concentration
will be:

Let K_{d} be reciprocal of K_{a}:

Substituting Eq. (8) in Eq. (9):

In reciprocal form,

Therefore, from a plot of 1/[AbAg]_{e} vs. 1/[Ag]_{e},
K_{d} can be obtained experimentally, and K_{a} = 1/K_{d}.
This process is shown graphically in Fig. 3.16b. K_{d} is equal
to the reciprocal concentration of free antigen necessary to occupy half
of the antigen binding sites of the antibody (Fig. 3.16a).

**Fig. 3.16. (a) Definition of K_{d}; (b) graphical
determination of K_{a}.**

**Avidity. **Multivalent binding between antibody and antigen (avidity
or functional affinity) results in a considerable increase in stability
as measured by the equilibrium constant, compared to simple monovalent
binding (affinity or intrinsic affinity, in the example of Fig. 3.17, an
arbitrarily assigned value of 104 L/mole) is used). This is sometimes referred
to as the 'bonus effect' of multivalency. Thus there may be a 10^{3}
fold increase in binding energy of IgG when both valencies (combining sites)
are utilized, and 10^{7} fold increase when IgM binds antigen in
a multivalent manner. Monovalent antigen combines with multivalent antibody
with no greater afinity than it does with monovalent antibody.