A fractal analysis of the binding of antigen (or antibody or, in general, analyte) in solution to the antibody (or antigen or, in general, receptor) immobilized on the biosensor surface provides a quantitative indication of the state of the disorder (fractal dimension, Df) and the binding rate coefficient, k, on the surface. The fractal dimension provides a quantitative measure of the degree of heterogeneity that exists on the surface for the antibody-antigen (or, in general, analyte-receptor) systems. In our discussion, we gave examples wherein either a single- or a dual-fractal analysis was required to adequately describe the binding kinetics. The dual-fractal analysis was used only when the single-fractal analysis did not provide an adequate fit. This was done by the regression analysis provided by Sigmaplot (1993). The examples analyzed (1) the role of analyte concentration, (2) the effect of different surfaces, (3) the influence of regeneration, and (4) the effect of flow rate on binding rate coefficients and fractal dimensions during analyte-receptor binding in different biosensor systems.
In accord with the prefactor analysis for fractal aggregates (Sorenson and Roberts, 1997), quantitative (predictive) expressions were developed for the binding rate coefficient, k, as a function of the fractal dimension, Df, for a single-fractal analysis and for the binding rate coefficients, k1 and k2, as a function of the fractal dimensions, Df, and Df2, for a dual-fractal analysis. Predictive expressions were also developed for the binding rate coefficient and the fractal dimension as a function of the analyte (antigen or antibody) concentration in solution.
The fractal dimension, Df, is not a classical independent variable such as analyte (antigen or antibody) concentration. Nevertheless, the expressions obtained for the binding rate coefficient for a single-fractal analysis as a function of the fractal dimension indicate the high sensitivity of the binding rate coefficient on the fractal dimension. This is clearly brought out by the high order and fractional dependence of the binding rate coefficient on the fractal dimension. For example, in the case of the binding of premixed samples of 5 mM ATP and different concentrations of GroEL (13 to 129 nM) to GroES immobilized on a Ni+ + -NTA surface, the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, is 4.94. This emphasizes the importance of the extent of heterogeneity on the biosensor surface and its impact on the binding rate coefficient.
Note that the data analysis in itself does not provide any evidence for surface roughness or heterogeneity, and the existence of surface roughness or heterogeneity assumed may not be correct. Furthermore, there is deviation in the data that may be minimized by providing a correction for the depletion of the antigen (or antibody or analyte) in the vicinity of the surface (imperfect mixing) and by using a four-parameter equation. Other predictive expressions developed for the binding rate coefficient and for the fractal dimension as a function of the analyte (antigen or antibody) concentration in solution provide a means by which these parameters may be controlled.
In general, the binding rate coefficient increases as the fractal dimension increases. An increase in the binding rate coefficient value should lead to enhanced sensitivity and to a decrease in the response time of the biosensor. Both of these aspects would be beneficial in biosensor construction. If for a selective (or multiple) reaction system, an increase in the Df value leads to an increase in the binding rate coefficient (of interest), this would enhance selectivity. Stability is a more complex issue, and one might intuitively anticipate that a distribution of heterogeneity of the receptor on the biosensor surface would lead to a more stable biosensor. This is especially true if the receptor has a tendency to inactivate or lose its binding capacity to the analyte in solution. Similar behavior has been observed for the deactivation of enzymes wherein a distribution of activation energies for deactivation (as compared to a single activation energy or deactivation) leads to a more stable enzyme (Malhotra and Sadana, 1987).
Whenever a distribution exists, it should be precisely determined, especially if different distributions are known to exist (Malhotra and Sadana, 1990). This would help characterize the distribution present on the surface and would influence the temporal nature of the binding rate coefficient on the surface. The present analysis only makes quantitative the extent of heterogeneity that exists on the surface, with no attempt at determining the qualitative nature of the distribution that exists on the surface. Much more detailed and precise data are required before any such attempt may be made. Finally, another parameter that is only rarely considered in the biosensor literature is robustness. This may be defined as insensitivity to measurement errors as far as biosensor performance is concerned. At this point, it is difficult to see how the binding rate coefficient and the fractal dimension would affect robustness.
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