Chemical potential is a fundamental property for determining thermo-dynamic equilibria involving exchanges of molecules such as between two phases of molecular systems.Previously we developed the FMAP method for calculating ex-cess chemical potentials according to Widom insertion [1-2].Intermolecular interac-tion energies were expressed as correlation functions and evaluated via fast Fourier transform.Here we extend this method to calculate liquid-liquid phase equilibria of macromolecular solutions [3].Chemical potentials are calculated by FMAP over a wide range of molecular densities and the coexistence line between low-and high-density phases is identified by the Maxwell equal-area rule.When benchmarked on Lennard-Jones fluids,our method produces an accurate phase diagram at a frac-tion of the computational cost of the current best method.Importantly,the gain in computational speed increases dramatically as the molecules become more complex,potentially reaching many orders of magnitude in speedup for atomistically represent-ed proteins.We demonstrate the power of FMAP by reporting the first results for the liquid?liquid coexistence curve of γⅡ-crystallin represented at the all-atom level.Our method may thus open the door to accurate determination of phase equilibria for protein mixtures,which underlie the regulation of a variety of cellular functions [4].