A Free Energy Analysis Algorithm Along Transition Path And Its Applications In Transmembrane Ion Permeation

In order to overcome the limitation of time scales of molecular dynamics simulation,a series of free energy calculation methods have emerged and are widely used in biological macromolecular simulation systems.Selecting proper reaction coordinates is the cornerstone of the success of free energy calculations.When the reaction coordinates are not properly selected,it often causes serious sampling errors,which leads to the deviation of the free energy calculation results from the correct microscopic mechanism.This paper will point out the source of such errors and design proper reaction coordinates for simulation systems.The biological reaction process can be expressed as the transition of the microscopic states from one local equilibrium state to another.According to the principle of minimum action,the transition paths are distributed around the minimum free energy path,which is essentially one-dimensional.However,when the transition paths are complex,we need multi-dimensional sampling to calculate multi-dimensional free energy landscape based on the conventional umbrella sampling free energy calculation method,which is inconvenient in practical applications.In this paper,we propose a free energy analysis method---the WEighted Least Squares method(Welsam).This method decouples the selection of potential energy functions in umbrella sampling from the bin segmentation in free energy analysis process,so it can be applied more flexibly in complex free energy calculations under the premise of ensuring the optimal statistical error.Based on this,we propose a free energy analysis method along the transition path: flexibly select multi-dimensional umbrella potential energy functions and sample along the one-dimensional transition path,and then obtain the one-dimensional Potential of Mean Force(PMF)along the transition path using Welsam.The application of the algorithm can not only greatly reduce the sampling cost,but also reduce the difficulty of finding proper reaction coordinates.Investigating the mechanism of ion permeation through membranes has broad implications for understanding biological basis and drug design.We design new one-dimensional reaction coordinates for the two popular transmembrane permeation mechanisms,namely the water-chain-assisted mechanism and the dehydration mechanism,and then apply the free energy calculation method along transition path to calculate free energy curves for transmembrane permeation of ions.First,by projecting the sample data obtained by umbrella sampling onto the phase plane/space composed of important parameters,we can intuitively ensure that the sample data obtained from the equilibrium state to the transition state of the system are continuous and sufficient.Then,we project the relaxation trajectories released from the transition state onto the phase plot.The relaxation trajectories are almost distributed around the minimum free energy path,so we can obtain the information of the minimum free energy path.Finally,we flexibly segment bins and calculate the one-dimensional PMF curve along the minimum free energy path in the phase diagram using Welsam.The simulation results show that the accuracy and precise of the PMF curve have greatly improved and also the exact energy barriers of water chain formation and break in ion permeation process have been obtained.Based on the new reaction coordinates and the new free energy calculation strategy,we systematically study the effect of membrane thickness and ion type on the selection of membrane permeation mechanisms and concluded that monovalent cations permeation through lipid systems with chain lengths of 14 to 24 carbon atoms prefer the water-chain-assisted mechanism.Monovalent anions permeation through lipid systems with chain lengths less than 18 carbon atoms employ the water-chain-assisted mechanism,while permeation through lipid systems with chain lengths greater than 18 carbon atoms prefer the dehydration mechanism or other more complex membrane permeation mechanisms.In summary,the main contribution of this paper is to propose a new free energy analysis method Welsam and the free energy calculation algorithm along transition path.Using this method,the sampling cost can be greatly reduced and the accuracy and precise of the PMF curves can also be improved.We successfully apply it to the study of transmembrane ion permeation mechanisms and explore the effect of membrane thickness and ion type on the selection of membrane permeation mechanisms.