Free Energy Calculation of Transmembrane Ion Permeation: Sample with a Single Reaction Coordinate and Analysis along Transition Path

A single reaction coordinate is designed to perform umbrella sampling for the water-chain-assist transmembrane permeation processes of ions and highly polar groups. In these permeation processes, there are three types of configuration changes including deformation of both leaflets of the lipid bilayer, formation of a water chain across the bilayer, and translocation of the ion through the bilayer. Due to the complexity of the permeation process, multiple reaction coordinates and expensive high dimensional free energy calculations have been used in previous studies. Our single reaction coordinate can be used to obtain sample data of the entire permeation process, which significantly reduces the computational cost in free energy calculations. We show that one-dimensional umbrella sampling along the new reaction coordinate can be used to obtain continuous sample data roughly along the minimum free energy path (MFEP) on the phase plane of the two components of the new reaction coordinate. Along the MFEP, a one-dimensional bin-segmentation of the sample data is then used to calculate the potential of mean force accurately by the newly developed free energy analysis method: weighted least-square analysis method (Welsam). Our work not only proposes a useful reaction coordinate to obtain sample data for water-chain assist transmembrane permeation processes but also establishes a general methodology for one-dimensional free energy calculations with bin-segmentation along the transition path in multiple dimensional phase space.