In this paper, we describe a general procedure to evaluate the thermodynamics of the interaction between polypeptides and hydrophobic ligands in the presence of aquo-organic solvent mixtures. These studies address experimental requirements for the determination of the linear free energy relationships, derivation of partition coefficients or other extrathermodynamic parameters such as contact areas, or assessment of the conformational changes that may occur when polypeptides or proteins interact with immobilized nonpolar ligands. Not unexpectedly from thermodynamic arguments, the trends and magnitudes of free energy parameters, such as the enthalpy of association, as previously derived in many studies from gradient elution reversed-phase high-performance liquid chromatographic (RP-HPLC) measurements are often different from the data for the same parameters derived from equilibrium binding or microcalorimetric determinations. To reconcile these divergencies and to more closely examine the thermodynamic basis of the interaction of polypeptides with nonpolar ligands, the dependency of the logarithmic capacity factor, ln k', on temperature, T, for several polypeptides (bombesin, beta-endorphin, glucagon) have been investigated using a n-butylsilica and acetonitrile-water or methanol-water mixtures of defined solvent compositions. With low-pH, acetonitrile-water mixtures, the van't Hoff plots, i.e., the plots of ln k' versus 1/T, were nonlinear over the range of T = 278-358 K, although within a narrow temperature range, e.g., from T = 278-308 K, the experimental data for these polypeptides could be approximated by a linear relationship. This nonclassical van't Hoff behavior was associated with interactive processes that involved temperature-dependent enthalpic, entropic, and heat capacity changes. In contrast, with low-pH, methanol-water mixtures, the van't Hoff plots showed dependencies that were essentially linear over the range of T = 278-358 K. The slopes of the van't Hoff plots with acetonitrile-water and methanol-water mixtures at a defined T value and solvent composition were significantly larger than those found for the corresponding experiments carried out under gradient elution RP-HPLC conditions. From these plots of ln k' versus 1/T, the changes in the apparent enthalpy of association (delta H++assoc) and the apparent entropy of association (delta S++assoc) for the interaction of these polypeptides with the solvated n-butyl ligands at different T and solvent compositions have been determined. For these polypeptides, both delta H++assoc and delta S++assoc exhibited linear dependencies on the volume fraction, phi, of the organic solvent over a narrow range of T, but the slopes of these plots were dependent on the T range examined. The dependencies of the slope term, S, and the intercept term, ln ko, derived from the plots of ln k' versus phi as a function of T, have also been investigated. A new relationship linking the S values with delta H++assoc and delta S++assoc as a function of T and phi has been derived...

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An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3-D large-scale gravity inversion problems. The 3-D Gauss-Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately 2 orders of magnitude, and the memory requirement is reduced by N′ λ times compared with the traditional GLQ method, where N′ λ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3-D large-scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hr and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3-D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. Three-dimensional density distributions of the Imbrium and Serenitatis basins are obtained, and high-density bodies are found at the depths 10-60 km, likely indicating a significant uplift of the high-density mantle beneath the two mascon basins.

The solubilities of betulin in methanol, ethanol, 1-propanol, 1-butanol, 1-pentanol, 1-hexanol, methyl formate, ethyl formate, methyl acetate, ethyl acetate, chloroform, dichloromethane, cyclohexane, and acetone were measured at T = (278.2, 288.2, 298.2, and 308.2) K. The solubilities of betulin in selected organic solvents increase with temperature. A three-parameter equation was used to correlate the experimental data. These data of solubility can be used to guide the processes of crystallization in industry.