next up previous contents index
Next: Harmonic restraints Up: Biasing and analysis methods Previous: Adaptive Biasing Force   Contents   Index



The metadynamics method uses a history-dependent potential [45] that generalizes to any type of colvars the conformational flooding [46] and local elevation [47] 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:

$\displaystyle V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}}) \; = \; { \sum_{t' ...
...t(-\frac{(\xi_{i}-\xi_{i}(t'))^{2}}{2\delta_{\xi_{i}}^{2}}\right) } }\mathrm{,}$ (13.20)

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)$ . Each Gaussian functions has a width of approximately $ 2\delta_{\xi_{i}}$ along the direction of the $ i$ -th colvar.

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 ``hill'' functions. At that stage the 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:

$\displaystyle A({\mbox{\boldmath {$\xi$}}}) \; \simeq \; { -V_{\mathrm{meta}}({\mbox{\boldmath {$\xi$}}}) + K }$ (13.21)

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$ , $ \delta_{\xi_{i}}$ and $ \delta{}t$ [48]. 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: $ \delta_{\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 only mandatory keyword is the colvars option listing all the variables involved: multidimensional PMFs are obtained by the same metadynamics instance applied to all the colvars.

The parameters $ W$ and $ \delta{}t$ are specified by the keywords hillWeight and newHillFrequency, respectively. The values of these options are optimal for colvars with correlation times $ \tau_{{\mbox{\boldmath {$\xi$}}}}$ in the range of a few thousand simulation steps, typical of many biomolecular simulations:

It is the user's responsibility to either leave hillWeight and newHillFrequency at their default values, or to change them to match the specifics of each system. The parameter $ \delta_{\xi_{i}}$ is instead defined as approximately half the width of the corresponding colvar $ \xi_{i}$ (see 13.2.1).

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:

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:

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.

Well-tempered metadynamics

The following options define the configuration for the ``well-tempered'' metadynamics approach [49]:

Multiple-replicas metadynamics

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

Compatibility and post-processing

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

next up previous contents index
Next: Harmonic restraints Up: Biasing and analysis methods Previous: Adaptive Biasing Force   Contents   Index