Second Order Reactions

The modeling of the specific binding of an antigen in solution to an antibody immobilized on the surface is a two-step process. The elementary steps involved in the reaction scheme are shown in Fig. 8.5. (Sadana and Chen, 1996).

•: Antigen (Ag) 0: Binding site —Antibody (Ab) with two binding sites

FIGURE 8.5 Elementary steps involved in the binding of the antigen in solution to the antibody covalently attached to the fiber-optic surface for the second-order reaction. Tq and Qq are the total concentration of the specific binding sites on the antibody and of the nonspecific binding sites on the fiber-optic surface, respectively, r® and Q" are the concentration of the antibody bound to one antigen due to specific binding and of the filled nonspecific binding sites on the fiber-optic surface, respectively. r| is the concentration of the antibody bound to two antigens due to specific binding (Sadana and Chen, 1996).

The rate of specific binding of a single antigen by an antibody is given by ,jp

= ¡^(r* - n - r2) - lel.n + kt2r2 - klcsT\, (8.7a)

where Tg is the total concentration of the antibody sites on the surface, r* is the surface concentration of antibodies bound to a single antigen at any time t, and r2 is the surface concentration of the antibody that binds two antigens. The rate at which the antibody specifically binds two antigens is given by

For initial binding kinetics, after some simplification, we obtain from Eq. (8.7a) (Sadana and Sii, 1991)

The second-order dependence on antigen concentration is not surprising since two molecules of the antigen can bind to two binding sites on the same antibody molecule. On integrating the Eq. (8.7c), we obtain the concentration of the antigen bound due to specific binding. This bound concentration is given by

Co Co Jo

The rate of nonspecific binding of the antigen to the surface is given by dQn

where QJJ is the total concentration of the nonspecific binding sites on the fiber-optic surface and Q" is the concentration of the filled nonspecific binding sites on the surface. In the initial regime, Q" ii" an<^ fef*csQo > fe^Q". Then, the total rate of antigen bound by specific and nonspecific binding is given by dt

On substituting the total rate of antigen bound into Eq. (8.2c), we obtain the boundary condition for the second-order reaction at x = 0, in dimensionless form, as

The Damkohler number, Da, now is equal to kff^Lc" ~l/D, and a = a eg""1. Here, n — 1 is the order of reaction.

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