The osmotic second virial coefficients, B 2 , for atomic-sized hard spheres in water are attractive (B 2 < 0) and become more attractive with increasing temperature (ΔB 2 /ΔT < 0) in the temperature range 300 K ≤ T ≤ 360 K. Thus, these hydrophobic interactions are attractive and endothermic at moderate temperatures. Hydrophobic interactions between atomic-sized hard spheres in water are more attractive than predicted by the available statistical mechanical theory. These results constitute an initial step toward detailed molecular theory of additional intermolecular interaction features, specifically, attractive interactions associated with hydrophobic solutes.protein folding | self-assembly | Pratt-Chandler theory S olvent-mediated, noncovalent interactions within biomolecular structures are decisive for their stability and functionality over an extended range of conditions (1). Hydrophobic interactions-a principal category of noncovalent interactions in water-exhibit strong and characteristic temperature dependences (2). Molecular-scale theories of hydrophobic interactions are judged by their ability to capture those temperature dependences. Defensible theories might eventually illuminate a valid explanation and might find broader utility in modeling biomolecular structure and function.The osmotic second virial coefficienthas so far been the only direct experimental check on the molecular theory of hydrophobic interactions between slightly soluble gases (A) in liquid water (3-6). Here g AA (r) is the usual radial distribution function of AA pairs at infinite dilution. Because of low solubilities for solutes of interest, the necessary experiments are challenging. The initial comparisons between the only available molecular-scale theory, the Pratt-Chandler (PC) theory (7), and direct measurements of B 2 showed poor agreement (3-5, 8).Explanations for the discrepancy have been suggested (9-11), but the underlying disagreement has persisted (12). One explanation for this discrepancy focused on the differences between the actual interactions for accessible experimental cases and the hard-sphere solute-water interactions natural for the molecular theory. In this setting, direct high-resolution determination of hydrophobic interactions for the hard-sphere models treated by the theory would be a helpful step, but that has not been accomplished so far. The case of atomic-sized hardsphere solutes has not been treated specifically, mostly because hard-sphere models are inconvenient in available molecular dynamics simulations.These problems are of basic importance because hydrophobic interactions are acknowledged as the dominant factor that drives protein folding (13,14). Hydrophobic interactions are also expected to become more favorable with increasing temperature for physiological temperatures. Hydrophobic interactions can then be described as favorable for aggregation and endothermic at moderate temperatures. This is a primary conceptual puzzle that theories of hydrophobic effects should clarify.Summaries of the substa...