The theory is almost the same as in Chapter 8. Equations (8.1)—(8.12) apply here, as do Figs. 8.4-8.6. These are therefore not repeated here. Since we will be examining a fractal analysis, we will now provide the additional fractal material required to provide the basis for the analysis. Kopelman (1988) indicated that classical reaction kinetics is sometimes unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Such heterogeneous reactions (e.g., bioenzymatic reactions) that occur at interfaces of different phases exhibit fractal orders for elementary reactions and rate coefficients with temporal memories. In such reactions, the rate coefficient exhibits the form
In general, kt depends on time, whereas k' = ki(t = 1) does not. Kopelman indicates that in three dimensions (homogeneous space) b = 0. This is in agreement with the results in classical kinetics. Also, with vigorous stirring, the system is made homogeneous and, again, b — 0. However, for diffusion-limited reactions occurring in fractal spaces, b> 0; this yields a time-dependent rate coefficient.
Di Cera (1991) has analyzed the random fluctuations in a two-state process in ligand-binding kinetics. The stochastic approach can be used as a means to explain the variable adsorption rate coefficient. The simplest way to model these fluctuations is to assume that the adsorption rate coefficient, fei(t), is the sum of its deterministic value (invariant) and the fluctuation, z(t). This z(t) is a random function with a zero mean. The decreasing and increasing adsorption rate coefficients can be assumed to exhibit the following exponential forms (Cuypers et al, 1987):
Furthermore, experimental data presented by Anderson (1993) for the binding of HIV virus (antigen) to anti-HIV (antibody) immobilized on a surface displays a characteristic ordered disorder. This indicates the possibility of a fractal-like surface. It is obvious that the biosensor system (wherein either the antigen or the antibody is attached to the surface) along with its different complexities—including heterogeneities on the surface and in solution, diffusion-coupled reactions, time-varying adsorption rate coefficients, etc.—can be characterized by a fractal system.
Sadana and Madagula (1994) performed a theoretical analysis using fractals for the time-dependent binding of antigen in solution to antibody immobilized on a fiber-optic biosensor surface. The authors noted that an increase in the fractal parameter utilized in their studies decreased both the rate of antigen and the amount of antigen bound. They recommended obtaining or estimating a fractal parameter (or perhaps a range of such) to help characterize the antibody-antigen interactions for fiber-optic biosensor systems. Note that in this case the diffusion is in the Euclidean space surrounding the fractal surface (Giona, 1992). Sadana and Madagula did not consider the effect of nonspecific binding on the specific binding of the antigen in solution to the antibody immobilized on the fiber-optic surface. However, it is beneficial to analyze the influence of nonspecific binding on both the rate of specific binding and the amount of antigen bound to the antibody immobilized on the biosensor surface.
Equation 9.1 is associated with the short-term diffusional properties of a random walk on a fractal surface. Also, in a perfectly stirred kinetics on a regular (nonfractal) structure (or surface), fei is a constant; that is, it is independent of time. In other words, and as indicated in earlier chapters, the limit of regular structures (or surfaces) and the absence of diffusion-limited kinetics leads to hi being independent of time. In all other situations, one would expect a scaling behavior given by fci ~ k t~b with - b < 0. Also, the appearance of the coefficient — b different from b = 0 is the consequence of two different phenomena—the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited condition).
It is of practical interest to evaluate or estimate the parameter b in Eq. (9.1) for a real surface. In a brief analysis of diffusion or reactants toward fractal surfaces, Havlin (1989) indicates that the diffusion of a particle (antibody or antigen, as the case may be) from a homogeneous solution to a solid surface (the biosensor surface) where it reacts to form a product (antibody-antigen complex) is given by (De Gennes, 1982; Pfeifer et al, 1984; Nyikos and Pajkossy, 1986)
Here, p = —b and Dj is the fractal dimension of the surface. Havlin (1989) states that the crossover value may be determined by rc~t2c. Above the characteristic length, rc, the self-similarity of the surface is lost and the surface may be considered homogeneous. Note that the product (Ab • Ag) in a reaction (Ab + Ag->Ab • Ag) on a solid fractal surface corresponds to p = (3 — Dj)/2 at short time frames and p = \ at intermediate time frames. The values of the parameters k, p, and Dj may be obtained for antigen ic^ t>tc.
antibody kinetics data where nonspecific binding is either present or absent. This may be done by a regression analysis using, for example, Sigmaplot (1993), along with Eq. (9.3), where (Ab • Ag) = ktp (Sadana and Beelaram, 1994, 1995). The fractal dimension may be obtained from the parameter p. Higher values of the fractal dimension would indicate higher degrees of disorder or heterogeneity on the surface. It is reasonable to anticipate that increasing levels of nonspecific binding would lead to higher levels of heterogeneity on the sensing surface.
Note that antigen-antibody binding is unlike reactions in which the reactant reacts with the active site on the surface and the product is released. In this sense, the catalytic surface exhibits an unchanging fractal surface to the reactant in the absence of fouling and other complications. In the case of antigen-antibody binding, the biosensor surface exhibits a changing fractal surface to the antigen or antibody (analyte) in solution. This occurs because as each binding reaction takes place, fewer sites are available on the biosensor surface to which the analyte may bind. This is in accord with Le Brecque's (1992) comment that the active sites on a surface may themselves form a fractal surface. Furthermore, the inclusion of nonspecific binding sites on the surface would increase the fractal dimension of the surface.
In general, log-log plots of the distribution of molecules, M(r), as a function of the radial distance, (r), from a given molecule are required to demonstrate fractal-like behavior (Nygren, 1993). This plot should be close to a straight line. The slope of the log M(r) versus log(r) plot determines the fractal dimension.
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