The Role of Bonded Terms in Free Energy Simulations:  1. Theoretical Analysis

The role of the bonded (bond stretching and bond angle) force-field terms in free energy simulations is examined. It is shown that the proper treatment of such terms depends on the choice of the free energy methodology (single or dual topology). Furthermore, while there are no problems in describing changes in bonded terms, care has to be used in creating or destroying them in a molecular dynamics simulation. An approach that avoids the singularity caused by a bond with a zero force constant is outlined. Changes in bond stretching or bond angle terms are shown to give rise to vibrational, Jacobian factor, and potential-of-mean-force-type (pmf) contributions. The meaning of bond stretching and bond angle bending free energy components obtained in single and dual topology simulations and their connection to these three contributions is investigated. Due to the different end states used in single and dual topology simulations, the pmf contribution is projected on different free energy components. In certain dual topology methods, vibrational and Jacobian factor contributions are not included in the free energy difference. Therefore, single free energy differences (e.g., the free energy difference between two molecules in the gas phase and in solution) often cannot be compared directly between single and dual topology methods. However, identical double free energy differences (e.g., free energy differences of solvation) are obtained in all cases. The present analysis emphasizes the importance of the details of the simulation methodology in interpreting the results for bonded terms and reconciles apparently contradictory findings in the literature.