Equipartition terms in transition path ensemble: Insights from molecular dynamics simulations of alanine dipeptide

Potential Energy Weighted Reactive Flux and Total Rate of Change of Potential Energy: Theory and Illustrative Applications

Reactive flux can be largely nonzero in a nonequilibrium ensemble of trajectories and provide insightful information for reactive transitions from the reactant state to the product state. Based on the reactive flux, a theoretical framework is proposed here for two quantities, the potential energy weighted reactive flux and the total rate of change of potential energy, which are useful for the identification of the mechanism from a nonequilibrium ensemble. From such quantities, two multidimensional free-energy analogues can be derived in the subspace of collective variables and they are equivalent in the regions where the reactive flux is divergence-free. These free-energy analogues are assumed to be closely related to the free energy in the subspace of collective variables, and they are reduced in the one-dimensional case to be the ensemble average of the potential energy weighted with reactive flux intensity, which was proposed recently [Li, W. J. Phys. Chem. A 2022, DOI: 10.1021/acs.jpca.2c04130] and could be decomposed into energy components at the per-coordinate level. In the subspace of collective variables, the decomposition of the multidimensional free-energy analogues at the per-coordinate level is theoretically possible and is numerically difficult to be calculated. Interestingly, the total rate of change of potential energy is able to identify the location of the transition state ensemble or the stochastic separatrix, in addition to the locations of the reactant and product states. The total rate of change of potential energy can be decomposed at the per-coordinate level, and its components can quantify the contribution of a coordinate to the reactive transition in the subspace of collective variables. We then illustrated the main insights and objects that can be provided by the approach in the applications to a two-dimensional system with various diffusion anisotropies and the alanine peptide in vacuum in various nonequilibrium ensembles of short trajectories, from which the results were found to be consistent.

Energy Decomposition along the Reaction Coordinate: Theory and Applications to Nonequilibrium Ensembles of Trajectories

A theoretical framework is proposed for an energy decomposition scheme along the reaction coordinate, in which the ensemble average of the potential energy weighted with reactive flux intensity is decomposed into energy components at the per-coordinate level. The decomposed energy quantity is demonstrated to be closely related to the free energy along the reaction coordinate, and its connection to the emergent potential energy is provided. In the application to alanine dipeptide under vacuum, illustrative calculations were performed in three nonequilibrium ensembles of trajectories: (1) transition path ensemble sampled with transition path sampling; (2) ensemble of short trajectories initiated from configurations around the transition-state region; and (3) ensemble of short trajectories shooting from configurations in several transition paths. The energy components on each coordinate were found to be consistent among the three ensembles of trajectories, indicating a broad applicability of the approach in biomolecular studies. In addition, the free energies along an optimized reaction coordinate obtained with these nonequilibrium ensembles were largely overlapped with a reference free energy calculated from a long equilibrium trajectory.

Locating Transition Zone in Phase Space

Understanding the reaction mechanism is required for better control of chemical reactions and is usually achieved by locating transition states (TSs) along a proper one-dimensional coordinate called reaction coordinate (RC). The identification of RC can be very difficult for high-dimensional realistic systems. A number of methods have been proposed to tackle this problem. A machine learning method is developed here to incorporate the influence of velocity on the reaction process. The method is also free of the unbalanced label problem resulting from the rather low fraction of configurations near the TS and can be easily extended to large systems. It locates the transition zone in the phase space and defines the dividing surface with a high transmission coefficient. Moreover, considering that the reaction environment can not only change the reaction path but also activate the reactive mode through energy transfer, we devise two measures to quantify the influence of these two factors on the reaction process and find that solvents can assist the reaction by directly doing work along the reactive mode. Not surprisingly, there is a positive correlation between the efficiency of energy transfer into the reactive mode and the reaction rate.

Time-Lagged Flux in the Transition Path Ensemble: Flux Maximization and Relation to Transition Path Theory

The transition path ensemble is of special interest in reaction coordinate identification as it consists of reactive trajectories that start from the reactant state and end in the product one. As a theoretical framework for describing the transition path ensemble, the transition path theory has been introduced more than 10 years ago, and so far, its applications have only been illustrated in several low-dimensional systems. Given the transition path ensemble, expressions for calculating flux, current (a vector field), and principal curves are derived here in the space of collective variables from the transition path theory, and they are applicable to time series obtained from molecular dynamics simulations of high-dimensional systems, i.e., the position coordinates as a function of time in the transition path ensemble. The connection of the transition path theory is made to a density-weighted average flux, a quantity proposed in a previous work to appraise the relevance of a coordinate to the reaction coordinate [Li, W. J. Chem. Phys. 2022, 156, 054117]. Most importantly, as an extension of the existing quantities, time-lagged quantities such as flux and current are also proposed. The main insights and objects provided by these time-lagged quantities are illustrated in the application to the alanine peptide in vacuum.