  Inverted solubility--a crystal melting upon cooling--is observed in a handful
of proteins, such as carbomonoxy hemoglobin and $\gamma$D-crystallin. In human
$\gamma$D-crystallin, the phenomenon is associated with the mutation of the
23$^\mathrm{rd}$ residue, a proline, to a threonine, serine or valine. One
proposed microscopic mechanism for this effect entails an increase in
hydrophobicity upon mutagenesis. Recent crystal structures of a double mutant
that includes the P23T mutation allows for a more careful investigation of this
proposal. Here, we first measure the surface hydrophobicity of various mutant
structures of this protein and determine that it does not discernibly increase
upon the mutating the 23$^\mathrm{rd}$ residue. We then investigate the
solubility inversion regime with a schematic patchy particle model that
includes one of three models for temperature-dependent patch energies: two of
the hydrophobic effect, and a more generic description. We conclude that while
solubility inversion due to the hydrophobic effect may be possible, microscopic
evidence to support it in $\gamma$D-crystallin is weak. More generally, we find
that solubility inversion requires a fine balance between patch strengths and
the temperature-dependent contribution, which may explain why inverted
solubility is not commonly observed in proteins. In any event, we also find
that the temperature-dependent interaction has only a negligible impact on the
critical properties of the $\gamma$D-crystallin, in line with previous
experimental observations.
