Metadynamics

The metadynamics method uses a history-dependent potential [59] that generalizes to any type of colvars the conformational flooding [60] and local elevation [61] methods, originally formulated to use as colvars the principal components of a covariance matrix or a set of dihedral angles, respectively. The metadynamics potential on the colvars ${\mbox{\boldmath {$\xi$}}} = (\xi_{1}, \xi_{2}, \ldots, \xi_{N_{\mathrm{cv}}})$ is defined as:

$$V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}}(t)) \; = \; { \sum_{... ...\frac{(\xi_{i}(t)-\xi_{i}(t'))^{2}}{2\sigma_{\xi_{i}}^{2}}\right) } }\mathrm{,}$$ (13.22)

where $V_{\mathrm{meta}}$ is the history-dependent potential acting on the current values of the colvars ${\mbox{\boldmath {$\xi$}}}$ , and depends only parametrically on the previous values of the colvars. $V_{\mathrm{meta}}$ is constructed as a sum of $N_{\mathrm{cv}}$ -dimensional repulsive Gaussian ``hills'', whose height is a chosen energy constant $W$ , and whose centers are the previously explored configurations $\left({\mbox{\boldmath {$\xi$}}}(\delta{}t), {\mbox{\boldmath {$\xi$}}}(2\delta{}t), \ldots\right)$ .

During the simulation, the system evolves towards the nearest minimum of the ``effective'' potential of mean force $\tilde{A}({\mbox{\boldmath {$\xi$}}})$ , which is the sum of the ``real'' underlying potential of mean force $A({\mbox{\boldmath {$\xi$}}})$ and the the metadynamics potential, $V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}})$ . Therefore, at any given time the probability of observing the configuration ${\mbox{\boldmath {$\xi^{*}$}}}$ is proportional to $\exp\left(-\tilde{A}({\mbox{\boldmath {$\xi^{*}$}}})/\kappa_{\mathrm{B}}T\right)$ : this is also the probability that a new Gaussian ``hill'' is added at that configuration. If the simulation is run for a sufficiently long time, each local minimum is canceled out by the sum of the Gaussian ``hills''. At that stage the ``effective'' potential of mean force $\tilde{A}({\mbox{\boldmath {$\xi$}}})$ is constant, and $-V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}})$ is an accurate estimator of the ``real'' potential of mean force $A({\mbox{\boldmath {$\xi$}}})$ , save for an additive constant:

$$A({\mbox{\boldmath {$\xi$}}}) \; \simeq \; { -V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}}) + K }$$ (13.23)

Assuming that the set of collective variables includes all relevant degrees of freedom, the predicted error of the estimate is a simple function of the correlation times of the colvars $\tau_{\xi_{i}}$ , and of the user-defined parameters $W$ , $\sigma_{\xi_{i}}$ and $\delta{}t$ [62]. In typical applications, a good rule of thumb can be to choose the ratio $W/\delta{}t$ much smaller than $\kappa_{\mathrm{B}}T/\tau_{{\mbox{\boldmath {$\xi$}}}}$ , where $\tau_{{\mbox{\boldmath {$\xi$}}}}$ is the longest among ${\mbox{\boldmath {$\xi$}}}$ 's correlation times: $\sigma_{\xi_{i}}$ then dictates the resolution of the calculated PMF.

To enable a metadynamics calculation, a metadynamics block must be defined in the colvars configuration file. Its mandatory keywords are colvars, which lists all the variables involved, and hillWeight, which specifies the weight parameter $W$ . The parameters $\delta{}t$ and $\sigma_{\xi}$ specified by the optional keywords newHillFrequency and hillWidth:

• name: see definition of name (biasing and analysis methods)
• colvars: see definition of colvars (biasing and analysis methods)
• outputEnergy: see definition of outputEnergy (biasing and analysis methods)

• hillWeight $\langle\,$ Height of each hill (kcal/mol)$\,\rangle$
  Context: metadynamics
  Acceptable values: positive decimal
  Description: This option sets the height $W$ of the Gaussian hills that are added during this run. Lower values provide more accurate sampling of the system's degrees of freedom at the price of longer simulation times to complete a PMF calculation based on metadynamics.

• newHillFrequency $\langle\,$ Frequency of hill creation$\,\rangle$
  Context: metadynamics
  Acceptable values: positive integer
  Default value: 1000
  Description: This option sets the number of integration steps after which a new hill is added to the metadynamics potential. Its value determines the parameter $\delta{}t$ in eq. 13.23. Higher values provide more accurate sampling at the price of longer simulation times to complete a PMF calculation.

• hillWidth $\langle\,$ Relative width of a Gaussian hill with respect to the colvar's width$\,\rangle$
  Context: metadynamics
  Acceptable values: positive decimal
  Default value: $\sqrt{2\pi}/2$
  Description: The Gaussian width along each colvar, $2\sigma_{\xi_{i}}$ , is set as the product between this number and the colvar's parameter $w_{i}$ given by width (see 13.2.1); such product is printed in the standard output of VMD. The default value of this number gives a Gaussian hill function whose volume is equal to the product of $\prod_{i}w_{i}$ , the volume of one histogram bin (see 13.5.7), and $W$ . Tip: use this property to estimate the fraction of colvar space covered by the Gaussian bias within a given simulation time. When useGrids is on, the default value also gives acceptable discretization errors [44]: for smoother visualization, this parameter may be increased and the width $w_{i}$ decreased in the same proportion. Note: values smaller than 1 are not recommended.

Output files

When interpolating grids are enabled (default behavior), the PMF is written every colvarsRestartFrequency steps to the file outputName.pmf. The following two options allow to control this behavior and to visually track statistical convergence:

• writeFreeEnergyFile $\langle\,$ Periodically write the PMF for visualization$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: on
  Description: When useGrids and this option are on, the PMF is written every colvarsRestartFrequency steps.

• keepFreeEnergyFiles $\langle\,$ Keep all the PMF files$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: When writeFreeEnergyFile and this option are on, the step number is included in the file name, thus generating a series of PMF files. Activating this option can be useful to follow more closely the convergence of the simulation, by comparing PMFs separated by short times.

Note: when Gaussian hills are deposited near lowerBoundary or upperBoundary (see 13.2.1) and interpolating grids are used (default behavior), their truncation can give rise to accumulating errors. In these cases, as a measure of fault-tolerance all Gaussian hills near the boundaries are included in the output state file, and are recalculated analytically whenever the colvar falls outside the grid's boundaries. (Such measure protects the accuracy of the calculation, and can only be disabled by hardLowerBoundary or hardUpperBoundary.) To avoid gradual loss of performance and growth of the state file, either one of the following solutions is recommended:

• enabling the option expandBoundaries, so that the grid's boundaries are automatically recalculated whenever necessary; the resulting .pmf will have its abscissas expanded accordingly;
• setting lowerWall and upperWall well within the interval delimited by lowerBoundary and upperBoundary.

Performance tuning

The following options control the computational cost of metadynamics calculations, but do not affect results. Default values are chosen to minimize such cost with no loss of accuracy.

• useGrids $\langle\,$ Interpolate the hills with grids$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: on
  Description: This option discretizes all hills for improved performance, accumulating their energy and their gradients on two separate grids of equal spacing. Grids are defined by the values of lowerBoundary, upperBoundary and width for each colvar. Currently, this option is implemented for all types of variables except the non-scalar types (distanceDir or orientation). If expandBoundaries is defined in one of the colvars, grids are automatically expanded along the direction of that colvar.

• rebinGrids $\langle\,$ Recompute the grids when reading a state file$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: When restarting from a state file, the grid's parameters (boundaries and widths) saved in the state file override those in the configuration file. Enabling this option forces the grids to match those in the current configuration file.

Well-tempered metadynamics

The following options define the configuration for the ``well-tempered'' metadynamics approach [63]:

• wellTempered $\langle\,$ Perform well-tempered metadynamics$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: If enabled, this flag causes well-tempered metadynamics as described by Barducci et al.[63] to be performed, rather than standard metadynamics. The parameter biasTemperature is then required.

• biasTemperature $\langle\,$ Temperature bias for well-tempered metadynamics$\,\rangle$
  Context: metadynamics
  Acceptable values: positive decimal
  Description: When running metadynamics in the long time limit, collective variable space is sampled to a modified temperature $T+\Delta T$ . In conventional metadynamics, the temperature ``boost'' $\Delta T$ would constantly increases with time. Instead, in well-tempered metadynamics $\Delta T$ must be defined by the user via biasTemperature. The written PMF includes the scaling factor $(T+\Delta T)/\Delta T$ [63]. A careful choice of $\Delta T$ determines the sampling and convergence rate, and is hence crucial to the success of a well-tempered metadynamics simulation.

Multiple-replicas metadynamics

The following options define metadynamics calculations with more than one replica:

• multipleReplicas $\langle\,$ Multiple replicas metadynamics$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: If this option is on, multiple (independent) replica of the same system can be run at the same time, and their hills will be combined to obtain a single PMF [64]. Replicas are identified by the value of replicaID. Communication is done by files: each replica must be able to read the files created by the others, whose paths are communicated through the file replicasRegistry. This file, and the files listed in it, are read every replicaUpdateFrequency steps. Every time the colvars state file is written (colvarsRestartFrequency), the file:
  ``outputName.colvars.name.replicaID.state'' is also written, containing the state of the metadynamics bias for replicaID. In the time steps between colvarsRestartFrequency, new hills are temporarily written to the file:
  ``outputName.colvars.name.replicaID.hills'', which serves as communication buffer. These files are only required for communication, and may be deleted after a new MD run is started with a different outputName.

• replicaID $\langle\,$ Set the identifier for this replica$\,\rangle$
  Context: metadynamics
  Acceptable values: string
  Description: If multipleReplicas is on, this option sets a unique identifier for this replica. All replicas should use identical collective variable configurations, except for the value of this option.

• replicasRegistry $\langle\,$ Multiple replicas database file$\,\rangle$
  Context: metadynamics
  Acceptable values: UNIX filename
  Default value: ``name.replica_files.txt''
  Description: If multipleReplicas is on, this option sets the path to the replicas' database file.

• replicaUpdateFrequency $\langle\,$ How often hills are communicated between replicas$\,\rangle$
  Context: metadynamics
  Acceptable values: positive integer
  Default value: newHillFrequency
  Description: If multipleReplicas is on, this option sets the number of steps after which each replica (re)reads the other replicas' files. The lowest meaningful value of this number is newHillFrequency. If access to the file system is significantly affecting the simulation performance, this number can be increased, at the price of reduced synchronization between replicas. Values higher than colvarsRestartFrequency may not improve performance significantly.

• dumpPartialFreeEnergyFile $\langle\,$ Periodically write the contribution to the PMF from this replica$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: on
  Description: When multipleReplicas is on, the file outputName.pmf contains the combined PMF from all replicas, provided that useGrids is on (default). Enabling this option produces an additional file outputName.partial.pmf, which can be useful to quickly monitor the contribution of each replica to the PMF.

Compatibility and post-processing

The following options may be useful only for applications that go beyond the calculation of a PMF by metadynamics:

• name $\langle\,$ Name of this metadynamics instance$\,\rangle$
  Context: metadynamics
  Acceptable values: string
  Default value: ``meta'' + rank number
  Description: This option sets the name for this metadynamics instance. While it is not advisable to use more than one metadynamics instance within the same simulation, this allows to distinguish each instance from the others. If there is more than one metadynamics instance, the name of this bias is included in the metadynamics output file names, such as e.g. the .pmf file.

• keepHills $\langle\,$ Write each individual hill to the state file$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: When useGrids and this option are on, all hills are saved to the state file in their analytic form, alongside their grids. This makes it possible to later use exact analytic Gaussians for rebinGrids. To only keep track of the history of the added hills, writeHillsTrajectory is preferable.

• writeHillsTrajectory $\langle\,$ Write a log of new hills$\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: on
  Description: If this option is on, a logfile is written by the metadynamics bias, with the name ``outputName.colvars.$<$ name$>$ .hills.traj'', which can be useful to follow the time series of the hills. When multipleReplicas is on, its name changes to
  ``outputName.colvars.$<$ name$>$ .$<$ replicaID$>$ .hills.traj''. This file can be used to quickly visualize the positions of all added hills, in case newHillFrequency does not coincide with colvarsRestartFrequency.

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