Reaction coordinates and transition pathways of rare events via forward flux sampling

Abstract

A new approach is developed for identifying suitable reaction coordinates to describe the progression of rare events in complex systems. The method is based on the forward flux sampling (FFS) technique and standard least-square estimation (LSE) and it is denoted as FFS-LSE. The FFS algorithm generates trajectories for the transition between stable states as chains of partially connected paths, which can then be used to obtain "on-the-fly" estimates for the committor probability to the final region, p(B). These p(B) data are then used to screen a set of candidate collective properties for an optimal order parameter (i.e., reaction coordinate) that depends on a few relevant variables. LSE is used to find the coefficients of the proposed reaction coordinate model and an analysis of variance is used to determine the significant terms in the model. The method is demonstrated for several test systems, including the folding of a lattice protein. It is shown that a simple approximation to p(B) via a model linear on energy and number of native contacts is sufficient to describe the intrinsic dynamics of the protein system and to ensure an efficient sampling of pathways. In addition, since the p(B) surface found from the FFS-LSE approach leads to the identification of the transition state ensemble, mechanistic details of the dynamics of the system can be readily obtained during a single FFS-type simulation without the need to perform additional committor simulations.

An adaptive weighted ensemble procedure for efficient computation of free energies and first passage rates

“…Several powerful path sampling methods have been introduced to explore the ensembles of paths and transition rates between functional states of biological systems, such as transition path sampling, [26][27][28][29] transition interface sampling, [30][31][32] forward-flux sampling, [33][34][35][36][37] milestoning, 38,39 and weightedensemble (WE) 40,41 sampling. These methods generally explore the complete space of conformations: transition rates between states of interest are to be obtained once the probability distribution in the complete multidimensional space reaches a stationary value -a challenging task in most applications.…”

Transition path sampling and forward flux sampling. Applications to biological systems

“…On the other hand, in applications to complex systems the efficiency and accuracy of FFS can be sensitive to: [4,5,7,34,35] (i) the choice of order parameter, and (ii) the positions of the interfaces and the extent of sampling of the interface ensembles, in particular for the first interface. Such a sensitivity of FFS is partially due to a tendency of propagating sampling errors that is larger than that of other interface-based path sampling methods.…”

“…These rates are then used to reweigh contributions to P(λ) from trajectories originating from both region A and region B. A complementary method [32] that is restricted to equilibrium systems (denoted FFS-US) entails an umbrella sampling (US) simulation following an original FFS protocol (pre-optimized for order parameter [34] and staging [35]) to sample the regions inside each window until the partial path ensemble loses any "memory" of where it originated.…”

Steady-state simulations using weighted ensemble path sampling

“…Bin construction has already received attention. 17,22,26 Further, the use of smaller bins should reduce the likelihood of significant transverse relaxation within a bin-resulting in a better and quicker estimate of steady-state bin probabilities using Eq. ͑2͒.…”

“…Computationally, various paths sampling procedures, such as transition path sampling, [6][7][8][9] transition interface sampling, [10][11][12] forward flux sampling, [13][14][15][16][17] and milestoning 18,19 have been developed to compute rate constants for systems with slow kinetics. However, to our knowledge these methods have not been applied to calculate the steady-state distributions.…”