Introduction

Sensitive detection systems (or sensors) are required to distinguish a wide range of substances. Sensors find application in the areas of biotechnology, physics, chemistry, medicine, aviation, oceanography, and environmental control. These sensors, or biosensors, may be utilized to monitor the analyte-receptor reactions in real time (Myszka et ah, 1997). Scheller et al. (1991) have emphasized the importance of providing a better understanding of the mode of operation of biosensors to improve their sensitivity, stability, and specificity. A particular advantage of this method is that no reactant labeling is required. However, for the binding interaction to occur, one of the components has to be bound or immobilized onto a solid surface. This often leads to mass transfer limitations and subsequent complexities. Nevertheless, the solid-phase immunoassay technique represents a convenient method for the separation and/or detection of reactants (for example, antigen) in a solution since the binding of antigen to an antibody-coated surface (or vice versa) is sensed directly and rapidly. There is a need to characterize the reactions occurring at the biosensor surface in the presence of diffusional limitations that are inevitably present in these types of systems.

The details of the association of analyte (antibody or substrate) to a receptor (antigen or enzyme) immobilized on a surface is of tremendous significance for the development of immunodiagnostic devices as well as for biosensors (Pisarchick et al, 1992). In essence, the analysis we will present is, in general, applicable to ligand-receptor and analyte-receptorless systems for biosensor and other applications (for example, membrane-surface reactions). External diffusional limitations play a role in the analysis of immunodiag-nostic assays (Bluestein et al, 1991; Eddowes, 1987/1988; Place et al, 1991; Giaver et al., 1976; Glaser, 1993; Fischer et al, 1994). The influence of diffusion in such systems has been analyzed to some extent (Place et al, 1991; Stenberg et al, 1986; Nygren and Stenberg, 1985; Stenberg and Nygren, 1982; Morton et al, 1995; Sadana and Sii, 1992a, 1992b; Sadana and Madagula, 1994; Sadana and Beelaram, 1995; Sjolander and Urbaniczky, 1991). The influence of partial (Christensen, 1997) and total (Matsuda, 1967; Elbicke et al, 1984; Edwards et al, 1995) mass transport limitations on analyte-receptor binding kinetics for biosensor applications is available. The analysis presented for partial mass transport limitation (Christensen, 1997) is applicable to simple one-to-one association as well as to the cases where there is heterogeneity of the analyte or the ligand. This applies to the different types of biosensors utilized for the detection of different analytes.

Chiu and Christpoulos (1996) emphasize that the strong and specific interaction of two complementary nucleic acid strands is the basis of hybridization assays. Syvanen et al (1986) have analyzed the hybridization of nucleic acids by affinity-based hybrid collection. In their method, a probe pair is allowed to form hybrids with the nucleic acid in solution. They state that their procedure is quantitative and has a detection limit of 0.67 attamoles. Bier et al (1997) analyzed the reversible binding of DNA oligonucleotides in solution to immobilized DNA targets using a grating coupler detector and surface plasmon resonance (SPR). These authors emphasize that the major fields of interest for hybridization analysis is for clinical diagnostics and for hygiene. The performance of these "genosensors" will be significantly enhanced if more physical insights are obtained into each step involved in the entire assay.

An optical technique that has gained increasing importance in recent years is the surface plasmon resonance (SPR) technique (Nylander et al, 1991). This is particularly so due to the development and availability of the BIACORE biosensor, which is based on the SPR method and has found increasing industrial usage. Bowles et al (1997) used the BIACORE biosensor to analyze the binding kinetics of Fab fragments of an antiparaquat antibody in solution to a paraquat analog (antigen) covalently attached at a sensor surface. Schmitt et al (1997) also utilized a modified form of the BIACORE biosensor to analyze the binding of thrombin in solution to antithrombin covalently attached to a sensor surface. The performance of SPR and other biosensors will be enhanced as more physical insights are obtained into each of these analytical procedures.

Kopelman (1988) indicates that surface diffusion-controlled reactions that occur on clusters (or islands) are expected to exhibit anomalous and fractallike kinetics. These fractal kinetics exhibit anomalous reaction orders and time-dependent rate (for example, binding) coefficients. As previously discussed, fractals are disordered systems in which the disorder is described by nonintegral dimensions (Pfeifer and Obert, 1989). These authors indicate that as long as surface irregularities show dilatational symmetry scale invariance such irregularities can be characterized by a single number, the fractal dimension. The fractal dimension is a global property and is insensitive to structural or morphological details (Pajkossy and Nyikos, 1989). Markel et al. (1991) indicate that fractals are scale self-similar mathematical objects that possess nontrivial geometrical properties. Furthermore, these authors indicate that rough surfaces, disordered layers on surfaces, and porous objects all possess fractal structure. A consequence of the fractal nature is a power-law dependence of a correlation function (in our case, the analyte-receptor complex on the surface) on a coordinate (for example, time).

Antibodies are heterogeneous, so their immobilization on a fiber-optic surface, for example, would exhibit some degree of heterogeneity. This is a good example of a disordered system, and a fractal analysis is appropriate for such systems. Furthermore, the antibody-antigen reaction on the surface is a good example of a low-dimension reaction system in which the distribution tends to be "less random" (Kopelman, 1988). A fractal analysis would provide novel physical insights into the diffusion-controlled reactions occurring at the surface.

Markel et al. (1991) indicate that fractals are widespread in nature. For example, dendrimers, a class of polymers with internal voids, possess unique properties. The stepwise buildup of six internal dendrimers into a dendrimer exhibits typical fractal (self-similar) characteristics (Gaillot et al., 1997). Fractal kinetics also have been reported in biochemical reactions such as the gating of ion channels (Liebovitch and Sullivan, 1987; Liebovitch et al., 1987), enzyme reactions (Li et al., 1990), and protein dynamics (Dewey and Bann, 1992). Li et al. emphasize that the nonintegral dimensions of the Hill coefficient used to describe the allosteric effects of proteins and enzymes is a direct consequence of the fractal property of proteins.

Strong fluctuations in fractals have not been taken into account (Markel et al, 1991). For example, strongly fluctuating fields bring about a great enhancement of Raman scattering from fractals. It would be beneficial to determine a fractal dimension for biosensor applications and to determine whether there is a change in the fractal dimension as the binding reaction proceeds on the biosensor surface. The final goal would be to determine how all of this affects the binding rate coefficient and subsequently biosensor performance. Fractal aggregate scaling relationships have been determined for both diffusion-limited processes and diffusion-limited scaling aggregation (DLCA) processes in spatial dimensions, 2, 3, 4, and 5, by Sorenson and Roberts (1997). Fractal dimension values for the kinetics of antigen-antibody binding (Sadana, 1997; Milum and Sadana, 1997) and for analyte-receptor binding (Sadana and Sutaria, 1997) for fiber-optic biosensor systems are available. In these studies the influence of the experimental parameters such as analyte concentration on the fractal dimension and on the binding rate coefficient (the prefactor in this case) were analyzed. We would like to delineate the role of surface roughness on the speed of response, specificity, stability, and sensitivity of fiber-optic and other biosensors. An initial attempt has been made to relate the influence of surface roughness (or fractal dimension) on the binding rate coefficient for fiber-optic and other biosensors (Sadana, 1998). High and fractional orders of dependence of the binding rate coefficient on the fractal dimension were obtained. We now extend these studies to other biosensor applications, including those where more than one fractal dimension is invloved at the biosensor surface—in other words, where complex binding mechanisms, as well as a change in the binding mechanism may be involved at the surface. Quantitative relationships for the binding rate coefficient as a function of the fractal dimension are obtained for different biosensor applications. The noninteger orders of dependence obtained for the binding rate coefficient on the fractal dimension further reinforces the fractal nature of these analyte-receptor binding systems.

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