Kopelman (1988) has 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 (for example, 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 a form given by h = k'rb.
In general, ki 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 obtained 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.
The random fluctuations in a two-state process in ligand-binding kinetics has been analyzed (Di Cera, 1991). The stochastic approach can be used as a means to explain the variable binding rate coefficient. The simplest way to model these fluctuations is to assume that the binding rate coefficient, k] (t) is the sum of its deterministic value (invariant) and the fluctuation (z(t)). This 2(t) is a random function with a zero mean. The decreasing and increasing binding rate coefficients can be assumed to exhibit an exponential form (Cuypers et al., 1987):
Here, ft and k10 are constants.
Sadana and Madagula (1993) analyzed the influence of a decreasing and an increasing binding rate coefficient on the antigen concentration when the antibody is immobilized on the surface. These authors noted that for an increasing binding rate coefficient (after a brief time interval), as time increases, the concentration of the antigen near the surface decreases, as expected for the cases when lateral interactions are present or absent. The diffusion-limited binding kinetics of antigen (or antibody or substrate) in solution to antibody (or antigen or enzyme) immobilized on a biosensor surface has been analyzed within a fractal framework (Sadana, 1997; Milum and Sadana, 1997). Furthermore, experimental data presented for the binding of HIV virus (antigen) to the antibody anti-HIV immobilized on a surface displays a characteristic ordered "disorder" (Anderson, 1993). This indicates the possibility of a fractal-like surface. It is obvious that such a biosensor system (wherein either the antigen or the antibody is attached to the surface) along with its different complexities (heterogeneities on the surface and in solution, diffusion-coupled reaction, time-varying adsorption or binding rate coefficients, etc.) can be characterized as a fractal system. Sadana (1995) has analyzed the diffusion of reactants toward fractal surfaces and earlier Havlin (1989) has briefly reviewed and discussed these results.
Havlin (1989) indicates that the diffusion of a particle (antibody [Ab]) from a homogeneous solution to a solid surface (for example, antigen [Ag]-coated surface) where it reacts to form a product (antibody-antigen complex, Ab • Ag) is given by
Here, Df is the fractal dimension of the surface. Equation (6.2a) indicates that the concentration of the product Ab • Ag(t) in a reaction Ab + Ag-» Ab • Ag on a solid fractal surface scales at short and intermediate time frames as Ab • Ag ~ tp, with the coefficient p = (3 - Df)/2 at short time frames and p = x/2 at intermediate time frames. This equation is associated with the short-term diffusional properties of a random walk on a fractal surface. Note that in a perfectly stirred kinetics on a regular (nonfractal) structure (or surface), kj is a constant; that is, it is independent of time. In other words, the limit of regular structures (or surfaces) and the absence of diffusion-limited kinetics leads to kj being independent of time. In all other situations, one would expect a scaling behavior given by ki ~ k't " h with -b = p< 0. Also, the appearance of the coefficient p different from p = 0 is the consequence of two different phenomena—the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited) condition.
Havlin indicates that the crossover value may be determined by r2~tc. Above the characteristic length rc, the self-similarity is lost. Above £c, the surface may be considered homogeneous since the self-similarity property disappears and regular diffusion is now present. For the present analysis, we chose £c arbitrarily. One may consider our analysis as an intermediate "heuristic" approach in that in the future one may also be able to develop an autonomous (and not time-dependent) model of diffusion-controlled kinetics.
We can extend the preceding single-fractal analysis to include two fractal dimensions. At present, the time (t = ti) at which the first fractal dimension "changes" to the second fractal dimension is arbitrary and empirical. For the most part, it is dictated by the data analyzed and the experience gained by handling a single-fractal analysis. A smoother curve is obtained in the transition region if care is taken to select the correct number of points for the two regions. In this case, the product (antibody-antigen complex, Ab • Ag)
concentration on the biosensor surface is given by
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