Non-equal Time Rules In Free Energy Simulation

The efficiency of alchemical free energy simulation with staging strategy is improved by adaptively manipulating the significance of each ensemble followed by importance sampling.The Optimum Bennett Acceptance Ratio(OBAR)method introduced in this work with explicit consideration of the statistical inefficiency in each ensemble outperforms the traditional equal time rule which is used in standard applications of alchemical transformation with window sampling regime in the sense of minimizing the total variance of free energy estimate.The Time Derivative of total Variance(TDV)is proposed for the OBAR criterion which is linearly dependent on the variance.As TDV could be used to determine the sampling efficiency,it could also be used to determine the importance of each intermediate.TDV is more sensitive to the phase space overlap than the overlap matrix.The performance of OBAR workflow is demonstrated for the solvation of several small molecules and a protein-ligand binding system.Following the previously proposed equilibrate-state sampling-based adaptive sampling regime,its nonequilibrium extension Optimum Crooks' Equation(OCE)is developed in the current work.The nonequilibrium OCE method differs from the OBAR one in two aspects.The first one is the addition of the pulling time in nonequilibrium transformation,and the second one is that the nonequilibrium framework uses transformations between neighboring states by definition.The efficiency of NonEquilibrium Work(NEW)stratification is improved by adaptively manipulating the significance of each nonequilibrium realization followed by importance sampling.As is exhibited in the equilibrium case,the nonequilibrium extension outperforms the simple equal time rule used in nonequilibrium stratification in the sense of minimizing the total variance of free energy estimate.The speedup of this non-equal time rule is more than 1-fold.The TDV proposed for the OBAR criterion is extended to determine the importance of each nonequilibrium transformation which is linearly dependent on the variance.The TDV in nonequilibrium case gives a totally different importance rank with standard errors of free energy differences and OBAR TDV due to the duration of nonequilibrium pulling is added into the OCE equation.The performance of OCE workflow is demonstrated in the solvation of several small molecules with a series of lambda increments and relaxation times between successive perturbations.Such a nonequilibrium adaptive sampling regime in alchemical transformation is unprecedented.Following the previously proposed equilibrium and nonequilibrium adaptive alchemical free energy simulation methods OBAR and OCE,the time derivative of variance guided adaptive sampling method is extended to the configurational space,falling in the term of Steered MD(SMD).By minimizing the variance of the free energy differences along the pathway in an optimized way,a new type of adaptive SMD(ASMD)is introduced.As exhibits in the alchemical case,this adaptive sampling method outperforms the traditional equal-time SMD in nonequilibrium stratification.Also,the method gives a much more efficient calculation of the potential of mean force than the selection criterion based ASMD scheme,which is proven to be more efficient than traditional SMD.The variance-linearly-dependent minus time derivative of the overall variance proposed for OBAR and OCE criterion is extended to determine the importance rank of the nonequilibrium pulling in the configurational space.It is shown that the importance rank given by the standard deviation of the free energy difference is corrected by the simulation time in the importance rank from the time derivative of the variance.The OCE workflow is periodicity-of-collective-variable(CV)dependent while ASMD is not.The performance is demonstrated in a dihedral flipping case and two distance pulling cases,accounting for periodic and non-periodic CVs,respectively.Finally,by combining the nonequilibrium enhanced sampling methods in the alchemical and configurational space,a new multi-dimensional nonequilibrium pulling framework is introduced.It is then applied in the indirect construction of the free energy landscape under multi-scale quantum mechanical(QM)Hamiltonians.The benchmark system is a dihedral flipping case.The method works in a two-step fashion.The system is simulated at the semi-empirical quantum mechanical(SQM)level with nonequilibrium free energy simulations and the free energy profile is constructed.Then the SQM-to-QM correction is obtained with bidirectional nonequilibrium pulling and bidirectional reweighting.As the method relies on nonequilibrium transformation,the result could be converged with sufficient slow pulling speeds.Further,as no parametric approximation is used,the method is generally applicable.The indirect result is in good agreement with the direct free energy profile.Efficiency comparison indicates that the indirect method is about 20-fold faster than the direct method.Therefore,the current work focuses on the development of a series of non-equal time rules in free energy simulation.The GOBAR method provides the optimal simulation time distribution protocol in equilibrium alchemical free energy calculation,while the GOCE method provides the optimal time distribution rule in nonequilibrium free energy simulations.The nonequilibrium pulling methods in the alchemical and configurational spaces are combined to introduce a general multi-dimensional protocol for indirect construction of free energy landscapes at QM levels.