Transition Path Sampling - Transition Path Ensemble

Transition Path Ensemble

TPS focuses on the most interesting part of the simulation, the transition. For example, an initially unfolded protein will vibrate for a long time in an open-string configuration before undergoing a transition and fold on itself. The aim of the method is to reproduce precisely those folding moments.

Consider in general a system with two stable states A and B. The system will spend a long time in those states and occasionally jump from one to the other. There are many ways in which the transition can take place. Once a probability is assigned to each of the many pathways, one can construct a Monte Carlo random walk in the path space of the transition trajectories, and thus generate the ensemble of all transition paths. All the relevant information can then be extracted from the ensemble, such as the reaction mechanism, the transition states, and the rate constants.

Given an initial path, TPS provides some algorithms to perturb that path and create a new one. As in all Monte Carlo walks, the new path will then be accepted or rejected in order to have the correct path probability. The procedure is iterated and the ensemble is gradually sampled.

A powerful and efficient algorithm is the so-called shooting move. Consider the case of a classical many-body system described by coordinates r and momenta p. Molecular dynamics generates a path as a set of (r_t, p_t) at discrete times t in where T is the length of the path. For a transition from A to B, (r₀, p₀) is in A, and (r_T, p_T) is in B. One of the path times is chosen at random, the momenta p are modified slightly into p + δp, where δp is a random perturbation consistent with system constraints, e.g. conservation of energy and linear and angular momentum. A new trajectory is then simulated from this point, both backward and forward in time until one of the states is reached. Being in a transition region, this will not take long. If the new path still connects A to B it is accepted, otherwise it is rejected and the procedure starts again.