  Protein aggregation via liquid-liquid phase separation (LLPS) is ubiquitous
in nature and intimately connects to many human diseases. Although it is widely
known that the addition of salt has crucial impacts on the LLPS of protein,
full understanding of the salt effect remains an outstanding challenge. Here,
we develop a molecular theory which systematically incorporates the
self-consistent field theory for charged macromolecules into the solution
thermodynamics. The electrostatic interaction, hydrophobicity, ion solvation
and translational entropy are included in a unified framework. Our theory fully
captures the long-standing puzzles of the non-monotonic salt concentration
dependence and the specific ion effect. We find that proteins show salting-out
at low salt concentrations due to ionic screening. The solubility follows the
inverse Hofmeister series. In the high salt concentration regime, protein
remains salting-out for small ions but turns to salting-in for larger ions,
accompanied by the reversal of the Hofmeister series. We reveal that the
solubility at high salt concentrations is determined by the competition between
the solvation energy and translational entropy of ion. Furthermore, we derive
an analytical criterion for determining the boundary between the salting-in and
salting-out regimes. The theoretical prediction is in quantitative agreement
with experimental results for various proteins and salt ions without any
fitting parameters.
