Kopelman (1988) indicates that surface diffusion-controlled reactions that occur on clusters or islands are expected to exhibit anomalous and fractal-like kinetics. These fractal kinetics exhibit anomalous reaction orders and time-dependent rate (for example, binding) coefficients. Mandelbrot (1975, 1983) initially introduced fractals, or self-similar objects that exhibit dilatational symmetry. The word fractal was taken from the Latin word fractus, meaning broken. Fractals have details on all scales; therefore euclidean geometry and classical calculus are insufficient for their description; fractal geometry is required. Markel et al. (1991) indicate that fractals are widespread in nature. They indicate that the products of a wide class of diffusion-controlled aggregation reactions in solutions and in gases may be labeled as fractals. Thus, rough surfaces, disordered layers on surfaces, and porous objects (such as heterogeneous catalysts) possess fractal structure. Furthermore, gels, soot and smoke, and most macromolecules are fractals.

Fractals are disordered systems; the disorder is described by nonintegral dimensions (Pfeifer and Obert, 1989). As long as surface irregularities show scale invariance—that is, dilatational symmetry—they can be characterized by a single number, the fractal dimension. This means that the surface exhibits self-similarity over certain-length scales. In other words, the structure exhibited at the scale of the basic building blocks is reproduced at the level of larger and larger conglomerates. Fractals possess nontrivial geometrical properties; in other words, they are geometrical structures with noninteger dimensions. A consequence of the fractal nature is a power-law dependence of a correlation function (in our case, the analyte-receptor complex on the biosensor or cell surface) on a coordinate (for example, time).

The repeating shape or form does not have to be identical. An increase in the disorder on the surface leads to higher values of the fractal dimensions. For example, a very ordered "assembly" of objects along a straight line should yield a fractal dimension of 1 (ideally). If there is some disorder or degree of heterogeneity along this straight line, a slightly higher value of the fractal dimension will be found. If there are holes along this straight line, the fractal dimension will be less than 1. Similarly, if the assembly of objects under consideration are very organized on a surface, the fractal dimension is close to 2 or exactly equal to 2 (ideally). A fractal dimension value different from 2 provides a quantitative measure of how far the surface is from an ideal or homogeneous surface exhibiting a fractal dimension of 2. Thinking along the same lines, we may have two-dimensional surfaces exhibiting fractal dimensions greater than or less than 2. We may consider the fractal dimension (loosely) as a "space-filling" ability of a system. Thus the highest value of the fractal dimension exhibited is 3, since we are restricted to three-dimensional space.

In a review of the heterogeneity of materials and multifractality, Lee and Lee (1996) note that the fractal approach provides a convenient means to quantitatively represent the different structures and morphologies at the reaction interface. The authors analyzed simulations of Eley-Rideal diffusion-limited reactions on different objects. The primary advantage is that this permits the development of a predictive approach in the field of catalysis. Lee and Lee emphasize using the fractal approach to develop optimal structures, noting that today's sensors tend to be costly, cumbersome, and specialized (Service, 1997). Service indicates that it would be helpful to develop new sensors that are based on dirt-cheap starting materials. Such sensors could then be effectively used as low-cost detectors for medical diagnostics, industrial monitoring, and environmental testing.

Avnir et al. (1998) emphasize that the power law utilized in describing the fractal nature of systems very appropriately condenses the complex nature of the system being analyzed. Furthermore, it provides a simple picture of the correlation between the system structure and the dynamics of its formation. This type of information is particularly relevant in the study of analyte-receptor binding reactions occurring on surfaces. In analyzing the optical amplification of ligand-receptor binding using liquid crystals, Gupta et al. (1998) schematically show the change in the surface heterogeneity (or the fractal dimension) as avidin or IgG molecules in solution bind to ligands attached to self-assembled monolayers of molecules supported on a gold film. Their schematic indicates that the surface roughness increases on the binding of the analyte (Av or IgG) in solution to the ligands on the surface.

Fractal kinetics exhibit anomalous reaction orders and time-dependent rate (for example, binding) coefficients. These are unlike "regular" reaction kinetics, which exhibit integer orders of reaction, such as zero, first, second, etc. The time-dependent adsorption rate coefficients observed experimentally, as indicated above, may also be due to nonidealities or heterogeneity on the surface. Antibodies are heterogeneous and their immobilization on a fiberoptic surface, for example, will definitely exhibit a degree of heterogeneity. This is a good example of a disordered system, and a fractal analysis is appropriate for such systems. In addition, 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), and a fractal analysis would provide novel physical insights into the diffusion-controlled reactions occurring at the surface.

Matuishita (1989) indicates that the irreversible aggregation of small particles occurs in many natural processes, such as polymer science, material science, and immunology. These aggregation processes frequently result in the formation of complex materials that can be described by fractals (Mandelbrot, 1983). Daccord (1989) emphasizes that when too many parameters are involved in a reaction, the fractal dimension for reactivity may be a useful global parameter. Since biosensor performance is constrained by chemical binding kinetics, equilibrium, and mass transport of the analyte to the biosensor surface, it behooves one to pay particular care to the design of such systems and to explore new avenues by which further insight or knowledge may be obtained in these systems. Fractal analysis is one such avenue by which one may obtain physical clarification of the diffusion-controlled reactions at the surface.

Havlin (1989), in a brief discussion of the diffusion of reactants on and toward fractal surfaces, indicates that although diffusion toward fractal surfaces has been studied experimentally more extensively than diffusion on fractal surfaces (owing to the number of applications, such as catalytic reactions), diffusion toward fractal surfaces has been analyzed theoretically much less. Some studies are available, however. For example, Giona (1992), reporting on first-order reaction-diffusion kinetics in complex fractal media, emphasizes that the exploration of the temporal nature of the diffusion-limited reaction on the surface could play an important role in understanding the reaction kinetics as well as the reaction itself. We now examine some typical (adsorption) studies where fractal dimension values have been obtained.

Adsorption of molecules of different diameters on a solid surface exhibit fractal characteristics (Avnir et ah, 1983, 1984; Van Damme and Fripiat, 1985). The number of molecules of A, nA, adsorbed on a surface may be given by

where dAea is the effective molecular diameter, and Djads is the fractal dimension for adsorption and lies between a value of 2 and 3. Another method of determining the fractal dimension for adsorption studies (Demertzis and Pomonis, 1997) uses particles of adsorbent of varying size (diameter) d onto which a single molecule is adsorbed. Then, the number of species adsorbed N per unit mass of the particles is given by

Nitrogen is typically the material adsorbed, and the adsorbent, for example, may be natural rocks (quartz, feldspar) and various coals. Here D is the fractal dimension for adsorption. The number 3 corresponds to the three-dimensional space in which the system is embedded.

Fractal kinetics also have been reported in such biochemical reactions as the gating of ion channels (Liebovitch et al, 1987; Liebovitch and Sullivan, 1987), enzyme reactions (Li et al, 1990), and protein dynamics (Dewey and Bann, 1992). Li et al. establish that the nonintegral dimensions of the Hill coefficient, used to describe the allosteric effects of proteins and enzymes, are a direct consequence of the fractal properties of proteins as biological macromolecules composed of amino acid residues whose branches form fractals. The substrate molecules "randomly walk" on the enzyme surface until they "hit," or react on, an active site. For a better physical understanding of reaction at interfaces, fractal analysis may be used to model the behavior of diffusion-limited antigen-antibody or, in general, analyte-receptor binding kinetics on biosensor surfaces.

Let us now look at some other examples available in the biology, biotechnology, and biomedical literature that exhibit fractal characteristics. Proteins have a hierarchical structure; during protein folding subdomains are initially formed. These subdomains then combine to yield domains, which eventually combine with other domains to produce the final active structure of the protein. This process involves many similar (though not identical) repeating biochemical units. Even in the complex protein structure there is a repeating pattern. This repeating pattern and the characteristic heterogeneity of the protein structure could be aptly described by fractals (Sadana and Vo-Dinh, 2001). It would seem appropriate to represent the different folding stages using a fractal analysis. The fractal nature is also associated with DNA, the gene frequency of which determines the protein structure.

Repeating patterns are also present in signals emanating from biological systems such as those traced by ECGs (electrocardiograms) and EEGs (electoencephalograms), as well as in the basic structures of some human organs such as the lungs and in the way that arteries divide and subdivide (Zamir, 1999). Furthermore, allometric scaling laws, including the metabolic reactions, have been analyzed by West et al. (1997), who indicate that these laws are characteristic of all organisms. For example, the authors were able to describe the 3/4 power law for metabolic reactions using a model of transport of essential materials through space-filling fractal networks of branching tubes. Note once again that a characteristic feature of fractals is the self-similarity at different levels of scale. Self-similarity implies that the features of a structure or process look alike at different levels of length or time.

Goldberger et al (1990) have indicated that when the heart rate (beats per minute) of a healthy individual is recorded for three, thirty, and three hundred minutes, the quick erratic fluctuations seem to vary in a manner similar to the slower fluctuations. This indicates a self-similarity. Note that self-similarity implies that the features of a structure look alike at different scales of length or time. This self-similarity of a process at different scales of time can be characterized with a fractal dimension: A higher value of the fractal dimension indicates a higher level of heterogeneity or state of disorder.

Pfeifer and Avnir (1983) refer to the fractal dimension as the hidden symmetry of irregular (self-similar) surfaces. In trying to determine whether there were systematic trends in the fractal dimension as the size of the protein molecule was changed, Goetze and Brickmann (1992) analyzed the self-similarity of protein surfaces and found that the fractal dimension of a protein surface increases when the size of the protein molecule is increased. Apparently, larger molecules are "rougher" (to molecular partners) than smaller molecules. Feder (1988) has defined the fractal dimension (surface dimension) as a local surface property and has attempted to associate high receptor selectivity with high values of the fractal dimension. Pfeifer et al (1984, 1985) indicate there is a balance involved for surface fractal dimension values greater than 2. Although such values would promote the transport of the analyte in solution to the surface, they would hinder the transport along the surface.

Consider the binding of an analyte of fractal nature (such as a protein or a macromolecule) in solution to the receptor immobilized on a biosensor surface. It is not unreasonable to assume that the receptor, like the analyte, would exhibit fractal characteristics. Proteins are also known to adsorb on "receptorless" surfaces. However, these surfaces themselves may or may not exhibit fractal characteristics. Low-dimension fractals have been observed for analyte-receptorless systems, for example, during the computer-simulated aggregation of ferritin (Stenberg and Nygren, 1991), the adsorption of ferritin on a quartz surface (Nygren, 1993), and polymer adsorption (Douglas et al., 1993). Note that the antibody is not fractal with binding sites on randomly distributed branches, but has only one or two binding sites on well-defined and unique parts of the molecule.

For ligand-receptor systems, it is recognized that the population of receptors for a given ligand may be represented by several subpopulations with different affinities (Lord et al., 1977; Eriksson et al, 1978; Agarwal and Phillipe, 1977; Barnett et al, 1978). Jose (1985) has developed a model for ligand-binding systems at equilibrium and has analyzed the influence of heterogeneity, cross-reactivity, and site-site interactions on this system. Sitesite interactions are themselves a source of affinity heterogeneity (Jose and Larralde, 1982), and their binding to different types of ligands may effectively be described by fractal systems. Swalen et al (1987) have indicated that the control of the structural organization of molecules at an interface are a key to understanding the reactions at interfaces and for the design of advanced materials. The characterization of a solid surface for antibody-antigen, ligand-receptor, and analyte-receptorless binding is of importance (Ebersole et al, 1990). Losche et al (1993) have analyzed the influence of surface chemistry on the adsorption of protein layers on aqueous interfaces and structural organization. Axelrod and Wang (1994) have indicated the importance of reduction of dimensionality kinetics, wherein reaction between ligands and cell-surface receptors can be enhanced by nonspecific adsorption followed by two-dimensional diffusion to a cell-surface receptor.

In their analysis of the quenching of fluorescein-conjugated lipids, Ahlers et al. (1992) indicate (1) the binding of lipid-bound haptens in biomembrane models and (2) the formation of two-dimensional protein domains. These authors emphasize that the basis of drug delivery strategies and immunoassays is the specific recognition of cell membrane epitopes by antibodies or specific sections of antibodies. They further indicate that proteins self-organize into two-dimensional crystals at the interface (lipid monolayer), for example during the high-affinity binding of antibodies to lipid-bound haptens. This self-organization of proteins into two-dimensional crystals at the surface is characteristic of fractal aggregation and formation.

Hsieh and Thompson (1994) indicate that in addition to other factors, ligand-receptor (binding and dissociation) kinetics depends on (1) the receptor density, (2) the diffusion coefficient if the ligand is bivalent or multivalent for the receptor, (3) whether the ligand induces receptor clustering, (4) and the influence of receptor clustering (Berg and Purcell, 1977; Dembo and Goldstein, 1978; Kaufmann andlain, 1991; Goldstein et al, 1989). Factors (3) and (4) lead to heterogeneities on the surface and would contribute toward a fractal surface at the reaction interface leading to fractal kinetics.

Baish and Jain (2000) have recently advocated utilizing fractal principles in cancer study and its treatment. They indicate, for example, that the tumor vessels yield fractal dimensions of 1.89 ± 0.04, while normal arteries and veins yield fractal dimensions of 1.70 ± 0.03. They emphasize the potential of fractal analysis in both treatment delivery and the diagnosis of cancer. Furthermore, these same authors (along with Losa, 1995; Cross, 1997; Coffey, 1998) indicate the widespread applications of fractals in pathology.

We now describe a typical example where fractal properties of both the analyte and the receptor are exhibited. Peng et al (1992) analyzed the nucleotide sequences in DNA using an n-step Markov chain, noting the presence of long-range correlations in nucleotide sequences. This indicated to them the presence of scale-free (fractal) phenomena. In hybridization reactions on biosensor surfaces, the analyte is typically a DNA in solution, and the receptor is a complementary DNA that is immobilized on the biosensor surface. In this case both the DNA in solution and the complementary DNA immobilized on the surface would seem to exhibit fractal characteristics. If the DNA immobilized on the biosensor surface is not complementary to the DNA in solution, effective binding does not take place.

One reason for analyzing antigen-antibody (or, in general, analyte-receptor) binding data is to provide a better physical understanding of the underlying mechanisms. We will illustrate by analyzing fractal dimensions for marine particles. This is not a directly related example, but the basic principles for using the fractal analysis should be the same. In his analysis of the correlation of fractal dimension of marine particles with ocean depth, Risovic (1998) indicates that the average fractal dimension of marine particles/aggregates changes from 2.9 ± 0.1 just beneath the surface to 2.0 + 0.1 at 800 m down. This correlates with a decrease in the turbulent energy dissipation rate with depth. His results indicate that there is a domination of shear coagulation for depths less than (or equal to) 400 m (fractal dimension = 2.7 ± 0.3) and coagulation due to a differential sedimentation rate at greater depths (fractal dimension — 2.1 ± 0.3).

Rice (1994), in his review of Kaandrop's (1994) text on fractal modeling, emphasizes that one should be able to relate the rules by which the fractal structures are generated to the underlying processes by which these structures develop. This then provides fundamental insights into the basic mechanisms involved in our case for the analyte-receptor binding process. It would be worthwhile to develop a relationship between surface roughness (characterized by a fractal dimension) and the rate of binding. This is in view of the different (statistical) fractal growth laws that are prevalent in the literature. These laws include invasion percolation, kinetic gelation, and diffusion-limited aggregation (DLA) (Viscek, 1989). These laws (or models) permit the computer simulation of the shape and the growth of natural processes. For example, in the DLA model introduced by Witten and Sander (1981), a randomly diffusing particle (seed) collides with a surface and stops. Another particle (from far away) diffuses to the surface and arrives at a site close (adjacent) to the first particle and stops. Another particle follows, and so on. In this way clusters are generated and exhibit the randon branching and open structures that are self-similar in nature.

The analyte or, in general, the receptor has to be immobilized or adsorbed to the surface. Heterogeneity of adsorption is a more realistic picture of the actual situation and should be carefully examined to determine its influence on external mass transfer limitations and on the ultimate analytical procedure. Heterogeneity in the covalent attachment of the antibody (or receptor) to the surface probably can be accounted for and needs to be considered in the analysis.

Heterogeneity may arise due to several different factors. For instance, antibodies, especially polyclonal antibodies, possess an inherent heterogeneity in that the antibodies in a particular sample are not identical. Furthermore, different sites on the antibody may become covalently bound to the surface. As a result, especially in large antibodies, steric factors will play a significant role in determining the Ag/Ab (or, in general, the receptor/analyte) ratio. It would be helpful to make the influence of heterogeneity on the kinetics of antibody-antigen interactions more quantitative. We are now implicitly indicating and associating heterogeneity on the surface with a fractal dimension [the Kopelman approach (1988)], with changes in the heterogeneity on the surface leading to changes in the fractal dimension.

Note that antigen-antibody binding is unlike reactions in which the reactant reacts with the active site on the surface and the product is released. In this sense the catalytic surface exhibits an unchanging fractal surface to the reactant in the absence of fouling and other complications. In the case of the antigen-antibody binding, the biosensor surface exhibits a changing fractal surface to the antigen or antibody (analyte) in solution. This occurs since as each binding reaction takes place, smaller and smaller amounts of binding sites are available on the biosensor surface to which the analyte may bound. This is in accord with Le Brecque's (1992) comment that the active sites on a surface may themselves form a fractal surface. Furthermore, the inclusion of nonspecific binding sites on the surface would increase the fractal dimension of the surface.

In general, a log-log plot of the distribution of molecules M(r) as a function of the radial distance (r) from a given molecule is required to demonstrate fractal-like behavior (Nygren, 1993). This plot should be close to a straight line. The slope of the log M(r) versus log(r) plot determines the fractal dimension. This is the classical definition and means of demonstrating fractal behavior.

One way of introducing heterogeneity into the analysis is to consider a time-dependent rate coefficient. Classical reaction kinetics is sometimes unsatisfactory when the reactants are spatially constrained on the microscopic level by walls, phase boundaries, or force fields (Kopelman, 1988). The types of 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 these types of reactions the rate coefficient exhibits a form given by k = k'rb, 0<b<l (£ > 1). (4.3)

Note that Eq. (4.3) fails in short time frames. In general, k depends on time, whereas k' = k(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 time dependence of the adsorption rate coefficient, ku may be due to a mathematical poisoning that is created through self-ordering (Kopelman, 1988). Kopelman emphasizes that since Eq. (4.3) fails in short time frames, the equation may be rewritten as

The range of b chosen is 0 to 1, as indicated by Kopelman. It is possible that for the reactions occurring at the interface, the values of b may be greater than 1 for antibody-antigen reactions.

The random fluctuations on a two-state process in ligand-binding kinetics can be analyzed (Di Cera, 1991). The stochastic approach can be used as a means to explain the variable adsorption rate coefficient. The simplest way to model these fluctuations is to assume that the adsorption rate coefficient, fci(i), is the sum of its deterministic value (invariant) and the fluctuation, z(t). This z{t) is a random function with a zero mean. The decreasing and increasing adsorption rate coefficients can be assumed to exhibit an exponential form (Cuypers et ah, 1987) as follows.

For A + A type of reactions, Kopelman (1988) indicates that b= 1 - {d,J2) (Kopelman, 1986; Klymko and Kopelman, 1982, 1983), where ds is the spectral (or random-walk occurrence) dimension defined by

Here, p is the probability of the random walker returning to its origin after time, t. Kopelman (1988) emphasizes that for the whole class of random fractals, all in embedded euclidean dimensions (two, three, or higher), ds is always « 1.33 (Kopelman, 1986; Alexander and Orbach, 1982). Then b equals 0.33 for A + A reactions. The self-ordering effect is much more prominent for the two-reactant case (A + B), which is closer to our case.

For the diffusion-limited case, Kopelman (1986) indicates that the reaction order, n, is given by fe1=fe'1(t+l)b, t> 0.

Then, a ds value of 4/3 yields a value of 5/2 for n. Kopelman (1988) emphasizes that, semantically, any binary reaction kinetics with b > 0 or n > 2 may be referred to as fractal-like kinetics. As b increases from 0 to 1, n increases slowly at first but more rapidly as b-> 1. For b equal to 0.25, 0.5, and 0.75, n equals 2.33, 3, and 5, respectively.

Reactions such as antibody-antigen interactions on a fiber-optic surface will be diffusion controlled and may be expected to occur on clusters or islands (indicating some measure of heterogeneity at the reaction surface). This leads to anomalous reaction orders and time-dependent adsorption (or binding) rate coefficients. It appears that the nonrandomness of the reactant distributions in low dimensions leads to an apparent "disguise" in the reaction kinetics. This disguise in the diffusion-controlled reaction kinetics is manifested through changes in both the reaction coefficient and the order of the reaction. Examples of reaction-disguised and deactivation-disguised kinetics due to diffusion are available in the literature (Malhotra and Sadana, 1989; Sadana, 1988; Sadana and Henley, 1987).

It would be of interest to obtain a characteristic value for the fractal parameter b (or perhaps a range for the fractal parameter b) for fiber-optic systems involving antibody-antigen interactions. This would be of tremendous help in analyzing these systems, in addition to providing novel physical insights into the reactions occurring at the interface. Techniques for obtaining values of fractal parameters from reaction systems are available, though they may have to be modified for fiber-optic biosensor systems. The discovery of ways to relate the fractal parameter as a measure of heterogeneity at the reaction interface would facilitate the manipulation of the interface reaction in desired directions.

Kopelman (1988) emphasizes that in a classical reaction system the distribution stays uniformly random, and in a fractal-like reaction system the distribution tends to be less random; that is, it is actually more ordered. Also, initial conditions that are usually of little importance in "re-randomizing" classical kinetics may become more important in fractal kinetics. One may wish to examine the effect of fractal-like systems of gaussian and other distributions. Finally, fractal kinetics are not the only way to obtain time-dependent adsorption rate coefficients in antibody-antigen (or, more generally, protein) interactions. As indicated in Eqs. (4.5a) and (4.5b), the influence of decreasing and increasing adsorption rate coefficients on external diffusion-limited kinetics may be analyzed.

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