A confirmable fractal analysis only of the binding of antigen (or antibody) in solution to antibody (or antigen) immobilized on the biosensor surface provides a quantitative indication of the state of disorder (fractal dimension, Df.bind) and the binding rate coefficient, fchlnd, on the surface. Including the fractal dimensions for the dissociation step, Dfdiss, and dissociation rate coefficients, fediss provides a more complete picture of the analyte-receptor reactions occurring on the surface (Sadana, 1999). One may also use the numerical values for the rate coefficients for the binding and dissociation steps to classify the analyte-receptor biosensor system as, for example, moderate binding, extremely fast dissociation; moderate binding, fast dissociation; moderate binding, moderate dissociation; moderate binding, slow dissociation; fast binding, extremely fast dissociation; fast binding, fast dissociation; fast binding, moderate dissociation; or fast binding, slow dissociation.
The fractal dimension value provides a quantitative measure of the degree of heterogeneity that exists on the surface for the analyte-receptor systems. The degree of heterogeneity for the binding and dissociation phases is, in general, different for the same reaction. This indicates that the same surface exhibits two degrees of heterogeneity, one for the binding and one for the dissociation reaction. In our discussion, we gave examples wherein either a single- or a dual-fractal analysis was required to describe the binding kinetics. The dual-fractal analysis was used only when the single-fractal analysis did not provide an adequate fit. This was determined using the regression analysis provided by Sigmaplot (1993). The dissociation step was adequately described by a single-fractal analysis for all the examples presented.
In accord with the prefactor analysis for fractal aggregates (Sorenson and Roberts, 1997), quantitative (predictive) expressions were developed for the binding rate coefficient, f?},lnd, as a function of the fractal dimension for binding, Dfbind, for a single-fractal analysis, and for the dissociation rate coefficient, iediss, as a function of the fractal dimension for dissociation, Df diss, for a single-fractal analysis. The K( = kdls/kbind) values presented provide an indication of the stability, reusability, and regenerability of the biosensor. Also, depending on one's final goal, a higher or a lower K value may be beneficial for a particular analyte-receptor system.
The fractal dimensions for the binding and the dissociation phases, Df bmd and Dfdiss, respectively, are not typical independent variables (as is, for example, the analyte concentration) that may be directly manipulated. These fractal dimensions are estimated from Eqs. (7.2a) and (7.2b), and one may consider them as derived variables. The predictive relationships developed for the binding rate coefficient as a function of the fractal dimension as well as for the dissociation rate coefficient as a function of the fractal dimension are of considerable value because these relationships directly link the binding or the dissociation rate coefficients to the degree of heterogeneity that exists on the surface and provide a means by which the binding or the dissociation rate coefficients may be manipulated by changing the degree of heterogeneity that exists on the surface. Note that a change in the degree of heterogeneity on the surface would, in general, lead to changes in both the binding and the dissociation rate coefficients. Thus, this may require a little thought and manipulation. The binding and the dissociation rate coefficients are rather sensitive to their respective fractal dimensions, or the degree of heterogeneity that exists on the biosensor surface, as may be seen by the high orders of dependence. It is suggested that the fractal surface (roughness) leads to turbulence, which enhances mixing, decreases diffusional limitations, and leads to an increase in the binding rate coefficient (Martin et al., 1991).
More such studies are required to determine whether the binding and the dissociation rate coefficients are sensitive to their respective fractal dimensions. If they are, experimentalists may find it worth their effort to pay more attention to the nature of the surface as well as how it may be manipulated to control the relevant parameters and biosensor performance in desired directions. Also, in a more general sense, the treatment should also be applicable to non-biosensor applications wherein further physical insights could be obtained.
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