Preparing for path sampling

Or: What OpenPathSampling does not do for you

While OpenPathSampling does a lot to simplify the process of performing a path sampling simulation, it doesn’t do everything for you. In particular, setting up a path sampling simulation can still be challenging.

Just as configurational Monte Carlo requires an initial configuration and a description of the ensemble (e.g., setting a temperature in an \(NVT\) simulation), path sampling requires an initial trajectory and a description of the path ensemble. In both cases, you use Monte Carlo to obtain the equilibrated ensemble. However, both describing the ensemble and creating an initial trajectory are more difficult in path sampling than the equivalent problems in configurational sampling.

This document discusses the general idea of setting up path sampling simulations. In includes a few references to specifics of OPS, but is mainly focused on the concepts. The examples section of the website includes many practical uses of these ideas.

Defining the ensemble

The nature of the ensemble depends on the specific algorithm being used. Here, we’ll discuss ensembles for TPS and TIS. In OPS, the transition network is used to efficiently create all the ensembles to be used in a simulation. For TPS, there is only one ensemble to be sampled. For TIS, there is one ensemble per interface (plus additional helper ensembles, such as the minus ensemble). In multiple state systems, TIS can involve hundreds of ensembles.

Monte Carlo methods (configurational or path sampling) draw samples from a statistical ensemble. For configurational sampling, the thermodynamic ensemble to be sampled is defined by the underlying Hamiltonian of the system (which includes the masses of the particles and the interaction energies) and the natural variables of the ensemble (e.g., for the canonical, or \(NVT\), ensemble, these are the number of particles, the volume, and the temperature). Similarly, a path ensemble will be defined by the underlying dynamics (including the underlying thermodynamic ensemble) and additional restrictions (such as state definitions).

Selecting the extra restrictions for path sampling is much more difficult that selecting the thermodynamic state to sample: whereas the temperature and density usually come from experimental considerations, the volume of phase space that defines a stable state is not an experimentally observable quantity. Typically, defining the state requires performing MD simulations in each stable state to characterize its bounds, and there is not a unique appropriate definition when studying a particular problem of interest.

In TPS, only the states need to be defined. In TIS, one must also define the interfaces.

Defining good states

In TPS, the main traits defining the path ensemble are the state definitions. The essential point is that states should be stable: a trajectory started from any point in the state should be expected to take a long time before visiting any other state.

The topic of state definitions has been included in several review articles. For more, see:

In replica exchange TIS, the role of the minus move in decorrelating trajectories should also be considered. Ideally, a single minus move trajectory should be just long enough to fully decorrelate (lose all memory) within the state. This puts restrictions on both the volume of the state as well as that of the innermost interface.

Defining good interfaces

Much of the literature on TIS uses the expression that “interfaces foliate phase space.” In the OPS picture, where interfaces are volumes instead of surfaces, this means that inner interfaces (closer to the state) must be fully contained by outer interfaces. In other words, for every volume in the ordered set of the state, interface 0, interface 1, etc.; any point in phase space that is inside interface \(i\) is also inside interface \(i+1\). Importantly, this includes the state: the innermost interface must entirely encapsulate the state. Note that OPS does not check this; it requires user attention.

Interface placement in TIS is essential to its efficiency. If interfaces are too far apart, the statistics in the estimate for \(P_A(\lambda_{i+1}|\lambda_i)\) (see TIS Analysis) are poor. If interfaces are too close together, then to span the transition, we’ll need more interfaces. Since each interface represents an ensemble that must be sampled to convergence, this can badly hurt the efficiency. Typically, we aim for about 20-30% overlap between successive interfaces.

There are approaches to iteratively optimize the location of interfaces. These have been implemented for OPS, and will be added to the core code in the future.

Getting an initial trajectory

Even with configurational sampling, initial conditions are not always completely trivial. A PDB entry might be missing hydrogens, and probably needs to be solvated (and energy-minimized). MC simulations of a liquid might be started on a lattice, and then melted during an equilibration phase. However, obtaining initial conditions for path sampling of rare events is much more difficult. Path sampling makes it possible to generate many rare trajectories once a first rare trajectory is given as input. But how can you get that first rare trajectory?

The answer is, “any way that works.” Some approaches include running long MD, or using a ratcheting approach (such as, or similar to, adaptive multilevel splitting or forward flux sampling) to get natural trajectories that go further and further toward the products.

Another important idea is that the initial trajectory does not need to be a valid physical trajectory from the dynamics of this path ensemble. For example, it can come from a run at a different temperature. A non-physical path can be equilibrated to a physical path by performing an equilibration path sampling simulation before performing the production simulation.

Note that a very bad initial trajectory could equilibrate to a nonsensical region of trajectory space, just like a bad configuration could equilibrate to a nonsensical metastable state in configurational Monte Carlo. Selecting a “nearly” physical path is important.

One approach that has shown promise is to use metadynamics to obtain a first transition. After the first transition, metadynamics usually will not have significantly altered the potential energy surface near the barrier, so this trajectory is likely to equilibrate to a good initial trajectory.

Generating with OPS

There are a few approaches within OPS for generating initial trajectories. One approach is to use run with a higher temperature. This is used in the alanine dipeptide examples, such as the notebooks for Flexible Length TPS on Alanine Dipeptide. This can work in practical cases for transitions that are not too rare. However, here the dynamics is often altered dramatically, and so special attention is needed. One option is to do a quick committor analysis on the high temperature trajectories, close to the expected transition. A configuration with a non-zero committor can be used as a valid good initial trajectory.

Another approach is to use the FullBootstrapping approach, which starts from a snapshot and rachets up through the path ensembles for a given transition. This will sample each ensemble until the trajectory satisfies the next ensemble, then it switches to the new ensemble and samples that. The process continues until the all ensembles in the transition have trajectories. It is essentially a version of adaptive multilevel splitting that has been discretized along the progress parameter. This approach sounds promising, but in reality is very dependent on the quality of the order parameter. It is very efficient some simple models, which is why we use it for the examples like the Multiple State TIS on a Toy Model notebooks. In complicated systems, it may fail.

Loading from a file

In addition to creating a trajectory from scratch using OPS, you can also load a trajectory from an existing file. This enables you to get an initial trajectory using whatever tool you’re already familiar with (e.g., metadynamics with PLUMED). For example, to load a Gromacs XTC file:

from import ops_load_trajectory
traj = ops_load_trajectory("trajectory_file.xtc", top="conf.gro")

The top argument must be specified as an explicit keyword (using =). It is required by MDTraj, which is used internally to load files. Since the MDTraj trajectory object does not have velocities, OPS will default to giving zero velocity to all atoms. You can still equilibrate the trajectory by doing two-way shooting with thermalized velocities (see the two-way shooting example from the ops_additional_examples repository.)

If your input trajectory file also has velocities, you can assign the correct velocities to the OPS trajectory by using the trajectory_from_mdtraj() method. For example, with a Gromacs TRR file:

import mdtraj as md
from import trajectory_from_mdtraj
mdt = md.load("trajectory_file.trr", top="conf.gro")
# next two lines create numpy array for velocities
trr = md.formats.TRRTrajectoryFile("trajectory_file.trr")
vel = trr._read(n_frames=len(mdt), atom_indices=None, get_velocities=True)[5]
# combine `mdt` and `vel` in an OPS trajectory
traj = trajectory_from_mdtraj(mdt, velocities=vel)

Note that the tricks to get the velocities from the file will depend on what kind of file you’re using. In this particular example, we use part of MDTraj’s private API. The important thing is that the result should be a Numpy array with shape (n_frames, n_atoms, n_spatial) and with values in units of nm/ps.