Metadynamics

Introduction

Metadynamics defines a class of flat-histogram methods useful for molecular dynamics simulations. Within metadynamics, a history-dependent bias is accrued through the periodic application of elemental Gaussian biases to the collective variables (CVs) of interest. The form of the individual biases is

\[g(\vec{\xi},\vec{s}_i) = W_i e^{-\frac{\left[\vec{\xi}-\vec{s}_i\right]^2}{2\sigma_i^2}}\;.\]

Here \(\vec{xi}\) is the collective variable, \(\vec{s}_i\) is the location of the \(i^{th}\) hill, \(W_i\) is the weight and \(\sigma_i\) the width of the hill. This bias acts to push the system away from previously visited states. As this bias accrues to the height of nearby features in the free energy surface, the system’s trajectory will begin to explore an expanded area in CV space. After a sufficient number of biases have been applied, the total applied bias within a given region will begin to oscillate around the negative of the free energy in that region.

The free energy surface (FES) at any time \(t\) may be reconstructed by summing over all applied biases thusly

\[\begin{split}F(\xi,t) = -V(\vec{\xi}) = -\sum_{i<n(t)} W_i e^{-\frac{\left[\vec{\xi}-\vec{s}_i\right]^2}{2\sigma_i^2}}\;.\end{split}\]

Here, \(n(t)\) refers to the number of biases applied before time \(t\). The time-dependent FES can be used to determine whether or not a simulation has reached a converged FES by computing block averages of the applied bias (see [Singh2012]).

Metadynamics can be applied to unbounded regions of CV space, or to bounded regions. For the case of bounded, non-periodic CVs, boundary corrections must be applied which alter the structure of the hills \(g(\vec{\xi},\vec{s}_i)\) (see [McGovern2013])

Metadynamics exists in many flavors, in which \(W_i\) and \(\sigma_i\) are altered in a time- or trajectory- dependent fashion. These include Well-Tempered Metadynamics, Transition-Tempered Metadynamics and Metadynamics with Adaptive Gaussians. Each of these methods has advantages and drawbacks. Currently, SSAGES includes only standard Metadynamics with fixed-shape hills and no tempering algorithm. Details on how to use these algorithms within SSAGES are given in the following sections.

Options & Parameters

Metadynamics is selected by defining type" : "Metadynamics" as the method in the JSON input file. It supports the following options:

widths
array of doubles (length: number of CVs). This array defines the width of the Gaussians being deposited over time in each CV dimension.
height
double. This value defines the height of the Gaussians being deposited over time, in units of energy used by the MD engine.
hill_frequency
double This value defines the frequency in iterations with which the Gaussians are deposited.

Note

The Metadynamics method does not yet support CV bounds.

Note

The Metadynamics method does not yet support restarts.

Warning

Metadynamics will run for the duration specified in the input file under MDSteps. It is the user’s responsibility to ensure that enough time is given for Metadynamics to obtain an satisfactory representation of the free energy surface of interest. A simple way to prevent Metadynamics from terminating before convergence is to define a very large number of MDSteps` and to periodically check the output file for convergence.

Example Input

"method" : {
        "type" : "Metadynamics",
        "widths" : [0.1, 0.1],
        "height" : 0.1,
        "hill_frequency" : 1
}

Output

The output of Metadynamics is stored in a file called “hills.out”. This file contains the location, width and height of each Gaussian that has been deposited over time. Each time Gaussian is deposited, a new line is written to the file in the following format:

center1 center2 ... width1 width2 ... height

The centers denote the locations in CV space where each Gaussian has been deposited; these are listed in the order that the CVs appear in the SSAGES JSON input file. The widths denote the corresponding Gaussian widths for each CV dimension. The height is the Gaussian height, which should match the parameter height defined in the JSON input file.

Note

Although the widths and height of the Gaussian currently do not change in time, future additions to the Metadynamics method will allow for adaptive Gaussians.

Example MATLAB scripts are provided in the Examples/User/Meta directory. These scripts sum the Gaussians and generate a free energy surface from the “hills.out” file.

Tutorial

Two Metadynamics examples are included in the Examples/User/Meta directory. In the first example, Metadynamics is used to sample the free energy surface of a two-dimensional particle undergoing Langevin dynamics. This example is found in the Single_Atom folder and requires LAMMPS. The files included are described below:

  • in.LAMMPS_Meta_Test - LAMMPS input file describing the Langevin particle

    and underlying free energy surface to be sampled. The free energy surface consists of two Gaussian wells at (0.98, 0.98) and (-0.98, -0.98) respectively, and one Gaussian barrier at the origin.

  • Meta.json - SSAGES JSON input file specifying Metadynamics and CVs to be sampled. In this case the CVs are the x and y coordinates of the particle.

  • analysis.m - MATLAB script that analyzes the output of the Metadynamics method.

  • Movie.m - MATLAB script that generates a movie of the free energy surface estimate over time.

To run this example:

  1. Either copy or create a symbolic link to the SSAGES executable in the examples directory.
ln -s /path/to/SSAGES/build/ssages
  1. Run the example by issuing the command below. Please note that in this example, two walkers are used to explore the system more efficiently. If you would like to use more walkers (1 processor per walker), simply include more drivers in the Meta.json input file.
mpirun -np 2 ./ssages Meta.json
  1. After the run is complete use the provided analysis.m script to generate a representation of the underlying free energy surface.

Developer

Hythem Sidky.

References Cited

[Singh2012]Singh, Sadanand, Manan Chopra, and Juan J. de Pablo. “Density of states–based molecular simulations.” Annual review of chemical and biomolecular engineering 3 (2012): 369-394.
[McGovern2013]McGovern, Michael, and Juan de Pablo. “A boundary correction algorithm for metadynamics in multiple dimensions.” The Journal of chemical physics 139.8 (2013): 084102.