3.3.5. Experimental Determination of Ka (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 Kd be reciprocal of Ka:
Substituting Eq. (8) in Eq. (9):
In reciprocal form,
Therefore, from a plot of 1/[AbAg]e vs. 1/[Ag]e, Kd can be obtained experimentally, and Ka = 1/Kd. This process is shown graphically in Fig. 3.16b. Kd 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 Kd; (b) graphical
determination of Ka.
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 103 fold increase in binding energy of IgG when both valencies (combining sites) are utilized, and 107 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.