Absolute free energy calculations by thermodynamic integration in four spatial dimensions

An optimized technique for calculating the excess chemical potential of small molecules in dense liquids and the binding affinity of molecular ligands to biomolecules is reported. In this method, a molecular species is coupled to the system of interest via a nonphysical fourth spatial dimension w through which insertion or extraction can be carried out [R. Pomes, E. Eisenmesser, C. B. Post et al., J. Chem. Phys. 111, 3387 (1999)]. Molecular simulations are used to compute the potential of mean force (PMF) acting on the solute molecule in the fourth dimension. The excess chemical potential of that molecule is obtained as the difference in the PMF between fully coupled and fully decoupled systems. The simplicity, efficiency, and generality of the method are demonstrated for the calculation of the hydration free energies of water and methanol as well as sodium, cesium, and chloride ions. A significant advantage over other methods is that the 4D-PMF approach provides a single effective and general route for decoupling all nonbonded interactions (i.e., both Lennard-Jones and Coulombic) at once for both neutral and charged solutes. Direct calculation of the mean force from thermodynamic integration is shown to be more computationally efficient than calculating the PMF from umbrella sampling. Statistical error analysis suggests a simple strategy for optimizing sampling. The detailed analysis of systematic errors arising from the truncation of Coulombic interactions in a solvent droplet of finite size leads to straightforward corrections to ionic hydration free energies.