Single Fractal Analysis

Binding Rate Coefficient

Havlin (1989) indicates that the diffusion of a particle (analyte [Ag]) from a homogeneous solution to a solid surface (e.g., receptor [Ab]-coated surface) on which it reacts to form a product (analyte-receptor complex, [Ag-Ab]) is given by

Here, Df bind is the fractal dimension of the surface during the binding step. Equation (7.2a) 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 bind)/2 at short time frames and p = \ 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), k} 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 febind being independent of time. In all other situations, one would expect a scaling behavior given by fcbind ~ k't ~ h with -b = p< 0. Also, the appearance of the coefficient p different from zero 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 r;r~£c. Above the characteristic length, rc, the self-similarity is lost. Above tc, the surface may be considered homogeneous since the self-similarity property disappears and regular diffusion is now present. For the present analysis, we choose tc arbitrarily and assume that the value of £c is not reached. 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.

Dissociation Rate Coefficient

We propose that a mechanism similar to the binding rate coefficient mechanism is involved (except in reverse) for the dissociation step. In this case, the dissociation takes place from a fractal surface. The diffusion of the dissociated particle (receptor [Ab] or analyte [Ag]) from the solid surface (e.g., analyte [Ag]-receptor [Ab] complex coated surface) into solution may be given, as a first approximation, by

(Analyte • Receptor)--k't(3~Dw-)/2 (t > tdiss). (7.2b)

Here Df diss is the fractal dimension of the surface for the desorption or dissociation step. t¿ISS represents the start of the dissociation step and corresponds to the highest concentration of the analyte-receptor complex on the surface. Henceforth, its concentration only decreases. Df bind may or may not be equal to Df diss. Equation (7.2b) indicates that during the dissociation step, the concentration of the product Ab • Ag(t) in the reaction Ag • Ab—>Ab + Ag on a solid fractal surface scales at short and intermediate time frames as [Ag • Ab]--tp with p = (3 - Dfdiss)/2 at short time frames and p = j at intermediate time frames. In essence, the assumptions that are applicable in the association (or binding) step are applicable for the dissociation step. Once again, 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), fediss 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 fediss being independent of time. In all other situations, one would expect a scaling behavior given by fediss~ — k't b with —b = p< 0. Once again, the appearance of the coefficient p different from zero is the consequence of two different phenomena: the heterogeneity (fractality) of the surface and the imperfect mixing (diffusion-limited) condition. Besides providing physical insights into the analyte-receptor system, the ratio K = kdiss/fcbmd is of practical importance since it may be used to help determine (and possibly enhance) the regenerability, reusability, stability, and other biosensor performance parameters.

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