Protein quadratic indices of the "macromolecular pseudograph's α-carbon atom adjacency matrix". 1. Prediction of Arc repressor alanine-mutant's stability

This report describes a new set of macromolecular descriptors of relevance to protein QSAR/QSPR studies, protein's quadratic indices. These descriptors are calculated from the macromolecular pseudograph's α-carbon atom adjacency matrix. A study of the protein stability effects for a complete set of alanine substitutions in Arc repressor illustrates this approach. Quantitative Structure-Stability Relationship (QSSR) models allow discriminating between near wild-type stability and reduced-stability A-mutants. A linear discriminant function gives rise to excellent discrimination between 85.4% (35/41) and 91.67% (11/12) of near wild-type stability/reduced stability mutants in training and test series, respectively. The model's overall predictability oscillates from 80.49 until 82.93, when n varies from 2 to 10 in leave-n-out cross validation procedures. This value stabilizes around 80.49% when n was > 6. Additionally, canonical regression analysis corroborates the statistical quality of the classification model (Rcanc = 0.72, p-level <0.0001). This analysis was also used to compute biological stability canonical scores for each Arc A-mutant. On the other hand, nonlinear piecewise regression model compares favorably with respect to linear regression one on predicting the melting temperature (t _m) of the Arc A-mutants. The linear model explains almost 72% of the variance of the experimental t_m (R = 0.85 and s = 5.64) and LOO press statistics evidenced its predictive ability (q² = 0.55 and s _cv = 6.24). However, this linear regression model falls to resolve t_m predictions of Arc A-mutants in external prediction series. Therefore, the use of nonlinear piecewise models was required. The t_m values of A-mutants in training (R = 0.94) and test (R = 0.91) sets are calculated by piecewise model with a high degree of precision. A break-point value of 51.32°C characterizes two mutants' clusters and coincides perfectly with the experimental scale. For this reason, we can use the linear discriminant analysis and piecewise models in combination to classify and predict the stability of the mutants' Arc homodimers. These models also permit the interpretation of the driving forces of such a folding process. The models include protein's quadratic indices accounting for hydrophobic (z₁), bulk-steric (z₂), and electronic (z₃) features of the studied molecules. Preponderance of z₁ and z₃ over z ₂ indicates the higher importance of the hydrophobic and electronic side chain terms in the folding of the Arc dimer. In this sense, developed equations involve short-reaching (k ≤ 3), middle- reaching (3 < k ≤ 7) and far-reaching (k = 8 or greater) z_{1, 2, 3}-protein's quadratic indices. This situation points to topologic/topographic protein's backbone interactions control of the stability profile of wild-type Arc and its A-mutants. Consequently, the present approach represents a novel and very promising way to mathematical research in biology sciences.