A mathematical analysis using fractals for binding interactions of estrogen receptors to different ligands on biosensor surfaces

A fractal analysis which takes into account the affect of surface heterogeneity brought about by ligand immobilization on the reaction kinetics is presented. The binding and dissociation of estrogen receptors ERα and ERβ to different ligands is analyzed within the fractal framework. The heterogeneity on the biosensor surface is made quantitative by using a single number, the fractal dimension, Df. The analysis provides physical insights into the binding of these receptors to different ligands and compounds, particularly the endocrine disrupting compounds (EDCs), which can have deleterious affects on humans and wildlife. Single- and dual-fractal models were employed to fit the ER binding data obtained from literature. Values of the binding and dissociation rate coefficient and fractal dimensions were obtained from a regression analysis provided by Corel Quattro Pro 8.0 [Corel Corporation Limited, Ottawa, Canada, 1997]. In some cases both a single- and dual-fractal model was required to completely and adequately describe the kinetics involved. For example, the binding of 40 nM ERα+17β-estradiol in solution to 300 RU of TIF2 immobilized on a SPR biosensor surface [J. Clin. Ligand Assay 18 (4) (1995) 215] requires a dual-fractal model. The dissociation, however, could be satisfactorily described using a single-fractal analysis. Values for the affinity, KD(=kd/ka) were also calculated. This provides us with some extra flexibility in designing biomolecular assays. Predictive expressions relating the binding and dissociation rate coefficients and fractal dimensions to the analyte concentration were also developed. The analysis should provide further information on the mode of action and interaction of EDCs with the ERs. This would help in the design of agents and modulators against these EDCs.