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If it is possible, how would you estimate a compound's water solubility from data like its partition coefficient, $ \log P $, its dissociation constant, $pK_a $, its molar mass, etc.?

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I think it can be done, but first you have to have model, so you can gather your data via vcclab.org/lab/alogps -> lots of compounds -> lots of descriptors (as you mentioned) -> output logS (solubility) -> heuristic(?) model (maybe ANN would be useful?) -> then input data -> output (predicted solubility) – java_xof Jan 19 at 22:41
Moreover similar things have been done at onschallenge.wikispaces.com/Predictive+Solubility – java_xof Jan 19 at 22:47

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It could certainly be estimated from those properties, the structure, etc. A lot of companies have developed proprietary algorithms for things like that: solubility, diffusion coefficient, polarity.... (For example of a company, see "ChemAxon")

The water solubility for many chemicals is already known, though. Search for a chemical ChemSpider, and look under the "Properties" tag.

They've even got an API, in case you're interested in using water solubility in an app.

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