Thermodynamic Integration Based on Molecular Dynamics Simulation Combined with 3D-RISM Theory

Recently, we proposed a combination of molecular dynamics (MD) simulation with the threedimensional reference interaction site model (3D-RISM) theory, i.e. MD/3D-RISM method [1]. We extend this combination method to a thermodynamic integration method to calculate the free energy differences between different states in solution. Regarding this extension, we develop two computational tools: i.e. we introduce either (1) the reaction coordinate or (2) the coupling parameter for switching the interaction potentials. The former corresponds to the calculation of the potential of mean force (PMF) along the reaction coordinate chosen, while the latter to the free energy difference associated with the solvation of two different solutes. As to a thermodynamic integration along a reaction coordinate, we employ the algorithm proposed by Ciccotti and his coworkers, known as the blue moon ensemble [2]. In their framework, RATTLE method is employed to fix the reaction coordinate at a given value throughout the MD simulation in each window. They have shown the analytical relationship between the mean force and the constraint force, where the latter is evaluated as an ensemble average along the trajectory of the MD simulation. We have executed this scheme using the MD/3D-RISM method. We applied this method to the issue of potassium ion recognition by crown ether 18-Crown-6 (18C6) in water as one of the simplest systems for the test of our method. The separation between potassium ion and the center of mass of 18C6 was chosen as a reaction coordinate. The PMF obtained shows an existence of local maximum in the vicinity of 4 angstrom separation. When the separation approaches to smaller values than 4 angstrom, the PMF is lowered significantly. The PMF at 0.15 angstrom is lower than that at 14.5 angstrom by ca 5 kcal/mol, where the latter separation corresponds to the largest separation calculated in this study. An MD/3D-RISM thermodynamic integration along a coupling parameter was also applied to this system to evaluate the free energy difference between 18C6-K complex in water and those of uncomplexed state, i.e. at infinite separation. The result was ca 5 kcal/mol, which agreed with the aforementioned PMF difference between 0.15 angstrom and 14.5 angstrom. This agreement indicates that the effective interaction between 18C6 and K ion in aqueous solution vanishes already at the separation of 14.5 angstrom. Further analysis upon related thermodynamic quantities such as solvation free energy will also be presented.