Potential distribution theory of alchemical transfer

We present an analytical description of the Alchemical Transfer Method (ATM) for molecular binding using the Potential Distribution Theory (PDT) formalism. ATM models the binding free energy by mapping the bound and unbound states of the complex by translating the ligand coordinates. PDT relates the free energy and the probability densities of the perturbation energy along the alchemical path to the probability density at the initial state, which is the unbound state of the complex in the case of a binding process. Hence, the ATM probability density of the transfer energy at the unbound state is first related by a convolution operation of the probability densities for coupling the ligand to the solvent and coupling it to the solvated receptor-for which analytical descriptions are available-with parameters obtained from maximum likelihood analysis of data from double-decoupling alchemical calculations. PDT is then used to extend this analytical description along the alchemical transfer pathway. We tested the theory on the alchemical binding of five guests to the tetramethyl octa-acid host from the SAMPL8 benchmark set. In each case, the probability densities of the perturbation energy for transfer along the alchemical transfer pathway obtained from numerical calculations match those predicted from the theory and double-decoupling simulations. The work provides a solid theoretical foundation for alchemical transfer, offers physical insights on the form of the probability densities observed in alchemical transfer calculations, and confirms the conceptual and numerical equivalence between the alchemical transfer and double-decoupling processes.