Multistate λ-local-elevation umbrella-sampling (MS-λ-LEUS): method and application to the complexation of cations by...

An extension of the λ-local-elevation umbrella-sampling (λ-LEUS) scheme [ Bieler et al. J. Chem. Theory Comput. 2014 , 10 , 3006 ] is proposed to handle the multistate (MS) situation, i.e. the calculation of the relative free energies of multiple physical states based on a single simulation. The key element of the MS-λ-LEUS approach is to use a single coupling variable Λ controlling successive pairwise mutations between the states of interest in a cyclic fashion. The Λ variable is propagated dynamically as an extended-system variable, using a coordinate transformation with plateaus and a memory-based biasing potential as in λ-LEUS. Compared to other available MS schemes (one-step perturbation, enveloping distribution sampling and conventional λ-dynamics) the proposed method presents a number of important advantages, namely: (i) the physical states are visited explicitly and over finite time periods; (ii) the extent of unphysical space required to ensure transitions is kept minimal and, in particular, one-dimensional; (iii) the setup protocol solely requires the topologies of the physical states; and (iv) the method only requires limited modifications in a simulation code capable of handling two-state mutations. As an initial application, the absolute binding free energies of five alkali cations to three crown ethers in three different solvents are calculated. The results are found to reproduce qualitatively the main experimental trends and, in particular, the experimental selectivity of 18C6 for K(+) in water and methanol, which is interpreted in terms of opposing trends along the cation series between the solvation free energy of the cation and the direct electrostatic interactions within the complex.