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This word “fractal” was first coined by Mandelbrot, it is from the Latin word ‘fractus’ and it means broken, used to describe the objects that are too irregular to fit into a traditional geometrical setting. Any fractal has fine structures i.e. details at all scales. Many fractals may have some degree of self similarity; they are made up of parts that resemble the whole structure in some way. Sometimes the resemblance could be weaker than strict geometrical similarity, say; the self-similarity may be approximate or similar. It was suggested by Falconer that the fractals can be regarded as a set F with the following characteristics:
- F has a fine structure, meaning details on arbitrarily small scales
- F is too irregular to be described in geometrical languages, both locally and globally
- F has some form of self similarity, which could be approximate or statistical
- Generally, the fractal dimension of F is greater than its topological dimensions
- F is usually defined in simple way, perhaps recursively
Let us look at the examples available in the biotechnology and biomedical fields and exhibit fractal characteristics. Proteins are hierarchical in structure and during protein folding protein sub-domains are formed. These subdomains then combine to form domains, which eventually combine with other domains to result into the final active structure of the protein. This process involves many similar, which may not be identical though, repeating biochemical units. Even in case of complex protein structure there is a repeating pattern. This repeating pattern and the characteristics of heterogeneity of the proteins structure can be described using fractals. It seems appropriate to represent the different folding stages using a fractal analysis. The fractal nature is also very much associated with DNA, its gene frequency determines the protein structure.
Considering the binding of an analyte of fractal nature, such as protein or some macromolecule, in solution to the receptor immobilized on a biosensor surface. It is assumed 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 reflect fractal characteristics. Low dimension fractals have been observed for analyte-receptorless system, for example, during the computer based simulation of aggregation of ferritin, the adsorption of ferritin on a quartz surface and polymer adsorption. Antibody is not fractal with binding sites on randomly distributed branches, but has one or two binding sites on well defined and unique parts of the molecule.
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 such a 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.
Reason behind analyzing antigen-antibody or analyte-receptor binding data is to provide a better physical understanding of the underlying mechanisms. The analyte or 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 in order to determine its influence on external mass transfer limitations and on the ultimate analytical procedure. Heterogeneity in the covalent attachment of the antibody or the 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 example, 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. This result into, especially in large antibodies, steric factors are significant in determining the Ag/Ab ratio. This would be of advantage in making the influence of heterogeneity on the kinetics of antibody-antigen interactions more quantitative.
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