Nucleotide's bilinear indices: Novel bio-macromolecular descriptors for bioinformatics studies of nucleic acids. I. Prediction of paromomycin's affinity constant with HIV-1 Ψ-RNA packaging region

A new set of nucleotide-based bio-macromolecular descriptors are presented. This novel approach to bio-macromolecular design from a linear algebra point of view is relevant to nucleic acids quantitative structure-activity relationship (QSAR) studies. These bio-macromolecular indices are based on the calculus of bilinear maps on ℜⁿ [b_mk (over(x, -)_m, over(y, -)_m) : ℜⁿ × ℜⁿ → ℜ] in canonical basis. Nucleic acid's bilinear indices are calculated from kth power of non-stochastic and stochastic nucleotide's graph-theoretic electronic-contact matrices, M_m^k and ^s M_m^k, respectively. That is to say, the kth non-stochastic and stochastic nucleic acid's bilinear indices are calculated using M_m^k and ^s M_m^k as matrix operators of bilinear transformations. Moreover, biochemical information is codified by using different pair combinations of nucleotide-base properties as weightings (experimental molar absorption coefficient ε{lunate}₂₆₀ at 260 nm and pH=7.0, first (Δ E₁) and second (Δ E₂) single excitation energies in eV, and first (f₁) and second (f₂) oscillator strength values (of the first singlet excitation energies) of the nucleotide DNA-RNA bases. As example of this approach, an interaction study of the antibiotic paromomycin with the packaging region of the HIV-1 Ψ-RNA have been performed and it have been obtained several linear models in order to predict the interaction strength. The best linear model obtained by using non-stochastic bilinear indices explains about 91% of the variance of the experimental Log K (R=0.95 and s=0.08×10^-4 M^-1) as long as the best stochastic bilinear indices-based equation account for 93% of the Log K variance (R=0.97 and s=0.07×10^-4 M^-1). The leave-one-out (LOO) press statistics, evidenced high predictive ability of both models (q²=0.86 and s_cv=0.09×10^-4 M^-1 for non-stochastic and q²=0.91 and s_cv=0.08×10^-4 M^-1 for stochastic bilinear indices). The nucleic acid's bilinear indices-based models compared favorably with other nucleic acid's indices-based approaches reported nowadays. These models also permit the interpretation of the driving forces of the interaction process. In this sense, developed equations involve short-reaching (k≤3), middle-reaching (4<k<9), and far-reaching (k=10 or greater) nucleotide's bilinear indices. This situation points to electronic and topologic nucleotide's backbone interactions control of the stability profile of paromomycin-RNA complexes. Consequently, the present approach represents a novel and rather promising way to theoretical-biology studies.