Other Order Reaction

For the present, no reaction mechanisms are proposed for the one-and-one-half-order reaction kinetics. Nevertheless, it is useful to display curves of with respect to time for these reaction orders.

One-and-one-half-order reaction

For the one-and-one-half-order reaction, the rate of antigen bound specifically is given by

On integrating Eq. (8.11a), we obtain the relative concentration of the antigen bound by specific binding:

Co Jo

The boundary condition in dimensionless form is

In this case, the Damkohler number is given by k\TlLclJ2/D. Similar expressions can be derived for the one-half-order case by substituting 1/2 for 3/2 where the reaction order exponent is used.

The solution for the diffusion equation [Eq. (8.1a)] for the different reaction orders may be obtained by using different but appropriate boundary conditions atx = 0, and the same initial condition at £ = 0 [Eqn. (8.2a)], and the same boundary condition at x = co. Note that since the boundary condition is nonlinear, except for the first-order reaction, and the initial condition exhibits a discontinuity, the solution to the diffusion equation is obtained by a numerical method. After obtaining the numerical solution of the diffusion equation, one can obtain the concentration of the antigen bound specifically to the antibody on the biosensor surface by numerically integrating the appropriate equations for the different reaction orders.

Different numerical techniques were considered in the solution of Eq. (8.1a). The explicit finite difference method was unsuitable due to severe restrictions placed by the stability conditions on the interval size. The Crank-Nicholson implicit finite difference method was also unsuitable since in this method very slowly decaying finite oscillations can occur in the neighborhood of discontinuities in the initial values or between the initial and boundary values. In our model, the initial condition in the neighborhood of z = 0 is discontinuous.

The technique found to be suitable was that in which the partial differential equation is reduced to a system of ordinary differential equations. Appropriate expressions for the different reaction orders can easily be obtained (Chen, 1994). Once the solution of the diffusion equation is obtained, the concentration of the antigen bound to the antibody due to specific binding can be obtained using the Hermite cubic quadrature. Chen (1994) used a computer subroutine called SDRIV2 for solving the initial value problem. This subroutine is particularly useful for solving a variety of initial value problems.

8.2.4. INFLUENCE OF NONSPECIFIC BINDING Forward Binding Rate Constant, kf

When nonspecific binding is absent (a = 0), an increase in the forward binding rate constant, kf, leads to an increase in the specific binding of the antigen in solution to the antibody immobilized in the biosensor surface, F^g/co for first-, one-and-one-half-, and second-order reactions. Since these results can be found in the literature (Sadana and Sii, 1991, 1992), they are not repeated here. As expected, an increase in the forward binding rate constant, kf, leads to a decrease in the normalized concentration of the antigen near the surface, cs/c0, and an increase in the antigen specifically bound to the antibody on the surface, P^/to-

Figures 8.6a and 8.6b show the influence of kf on cs/c0 and on Fs4g/co for a first-order reaction when a = 0.5. Note that, as expected, an increase in the kf value leads to a decrease in the cs/c0 value and an increase in the F^/cq values.

When nonspecific binding is present (a>0), the influence of kf on the binding curve for r^g/co becomes complicated for one-and-one-half- and second-order reactions. Apparently, there is an optimum value of kf that leads

FIGURE 8.6 The influence of the forward binding rate constant, kj, for a first-order reaction when a = 0.5 on (a) the normalized antigen concentration near the fiber-optic surface, c5/c0, and (b) the amount of antigen specifically bound to the antibody immobilized on the biosensor surface, PA /C0.

to the maximum amount of antigen that can be specifically bound to an antibody immobilized on the surface. Figures 8.7a and 8.7b show that for a one-and-one-half-order reaction initially as kf increases, T^g/co increases. However, as fef increases further, /c0 begins to decrease. Similar behavior is observed for a second-order reaction, as seen in Figs. 8.7c and 8.7d. The curves in Fig. 8.7 are for a values of 0.01 and 0.1 for both the reaction orders. Note that for the first-order reaction no such complicated behavior is observed.

Time-Dependent Forward Binding Rate Constant, kj

Due to complexities and heterogeneities on the reaction surface in real-life situations, the specific binding forward rate coefficient of the antigen in solution to the antibody immobilized on the surface may exhibit a temporal

0 20 40 60 80 100 120 140 160 180 t (seconds)

- k'

= 1E7

k'

= 5E7

---k*

= 1E8

k'

= 5E8

—- k'

= 1E9

k-

= 5E9

FIGURE 8.7 The influence of the forward binding rate constant, k^, for different x values on the amount of antigen specifically bound to the antibody immobilized on the biosensor surface, rig/c0. One-and-one-half-order reaction (a) a = 0.01; (b) a = 0.1. Second-order reaction (c) a = 0.01; a = 0.1.

nature. We will examine the influence of a temporal forward specific binding rate coefficient, fef, on the normalized concentration of the antigen in solution near the biosensor surface, cs/c0, and on the amount of antigen bound specifically to the antibody immobilized on the surface, F^/co, for cases when a = 0 and when a > 0.

The decreasing and increasing specific binding rate coefficients are assumed to exhibit following the exponential forms (Sadana and Sii, 1991, Cuypers et al, 1987):

Nonspecific Binding Absent

Figure 8.8 shows the influence of a decreasing forward binding rate coefficient, kf, on the amount of normalized antigen concentration in solution near the biosensor surface, cs/c0, for first-, one-and-one-half-order, and second-order reactions when nonspecific binding is absent (a = 0). As expected, after a brief initial period, a decrease in the ks( value leads to an increase in the cs/c0 value for a first-order reaction (Fig. 8.8a). The cs/c0 value is rather sensitive to the ft value of the forward binding rate coefficient. For example, for a reaction time of 3 min, the cs/c0 value changes from about 0.25 to about 0.01 as the /i value changes from 0.02 to 0 (time-invariant forward binding rate constant). Note that the changes in cs/c0 values are almost nonexistent for the one-and-one-half-order reaction (Fig. 8.8b) and for the second-order reaction (Fig. 8.8c). This is because very little antigen in solution is bound specifically to the antibody immobilized on the surface or nonspecifically to the biosensor surface under these conditions.

Figure 8.9 shows the influence of a decreasing fcj1 on F^„/c0. As expected, a decrease in the binding rate coefficient leads to a decrease in the amount of the antigen bound specifically to the antibody on the surface. Note how sensitive F^g/c0 is to the order of the reaction. The r^g/c0 values are considerably lower (by orders of magnitude) for the one-and-one-half- and the second-order reaction when compared to the first-order reaction for the same t and ft values. Note that higher fi values lead to increasing tendencies toward earlier exhibition of "saturation type" behavior by the binding curves for the reaction orders analyzed. Since the curves for an increasing kf exhibit trends for the cs/c0 and rAg/c0 curves that are similar to a decreasing k5{, they are not presented here. (See Chen, 1994).

Nonspecific Binding Present

Figures 8.10a and 8.10b show the influence of a decreasing forward binding rate coefficient, k5{ on the cs/c0 and the r^g/c0 curves when nonspecific binding is present (a = 0.5) for a first-order reaction. As expected, an increase in the value leads to an increase in the cs/c0 value and a corresponding decrease in the T^g/co values.

Figures 8.11a-8.11d show the influence of a decreasing kf on the cs/c0 and the PAg/c0 values when nonspecific binding is present for a one-and-one-half-order reaction. Figure 8.11a shows that for an a value of 0.01 an increase in the fi value leads to an increase in the cs/c0 value, as expected. When /3 = 0, the cs/c0 value decreases continuously. However, for /i > 0, the cs/c0 curve exhibits an initial decrease followed by an increase that asymptotically approaches a value of 1 for large time t. Figure 8.11b shows that an increase in

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