"Flexible-Acceptor" General Solubility Equation for beyond Rule of 5 Drugs

This study describes a novel nonlinear variant of the well-known Yalkowsky general solubility equation (GSE). The modified equation can be trained with small molecules, mostly from the Lipinski Rule of 5 (Ro5) chemical space, to predict the intrinsic aqueous solubility, S0, of large molecules (MW > 800 Da) from beyond the rule of 5 (bRo5) space, to an accuracy almost equal to that of a recently described random forest regression (RFR) machine learning analysis. The new approach replaces the GSE constant factors in the intercept (0.5), the octanol-water log P (-1.0), and melting point, mp (-0.01) terms with simple exponential functions incorporating the sum descriptor, Phi+B (Kier F molecular flexibility and Abraham H-bond acceptor potential). The constants in the modified three-variable (log P, mp, Phi+B) equation were determined by partial least-squares (PLS) refinement using a smallmolecule log S-0 training set (n = 6541) of mostly druglike molecules. In this "flexible-acceptor" GSE(Phi,B) model, the coefficient of log P (normally fixed at -1.0) varies smoothly from -1.1 for rigid nonionizable molecules (Phi+B = 0) to -0.39 for typically flexible (Phi similar to 20, B similar to 6) large molecules. The intercept (traditionally fixed at +0.5) varies smoothly from +1.9 for completely inflexible small molecules to -2.2 for typically flexible large molecules. The mp coefficient (-0.007) remains practically constant, near the traditional value (-0.01) for most molecules, which suggests that the small-to-large molecule continuum is mainly solvation responsive, apparently with only minor changes in the crystal lattice contributions. For a test set of 32 large molecules (e.g., cyclosporine A, gramicidin A, leuprolide, nafarelin, oxytocin, vancomycin, and mostly natural-product-derived therapeutics used in infectious/viral diseases, in immunosuppression, and in oncology) the modified equation predicted the intrinsic solubility with a root-mean-square error of 1.10 log unit, compared to 3.0 by the traditional GSE, and 1.07 by RFR.