Efficient Sampling of High-Dimensional …
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the weighted histogram analysis method (WHAM) [46], popularized by Alan Grossfield [47] or by applying the multi-state Bennett acceptance ratio (MBAR) estimator
introduced by Michael Shirts and John Chodera [48]. Often, however, this method
is not sufficient to fully sample the phase space and interesting chemistries of the
system because integration of the underlying potential of mean force becomes degenerate with higher-dimensional CVs. However, the most significant drawback of US is
that most CPU time during a simulation is spent sampling high energy, and uninteresting, regions of phase space more so than the relevant ones. A number of external
biasing methods have been developed to account for the drawbacks of US, including
targeted MD [49], energy landscape paving [27], the Wang–Landau algorithm [50],
variationally enhanced sampling [51], and many others.
Metadynamics (MetaD) originally developed by Laio and Parrinello [52, 53],
gained popularity as a robust and flexible technique to address a wide range of
problems including protein folding [54], nucleotide complexation [55], chemical
reactions [16, 56], ligand docking [57, 58], and phase transitions [59]. In MetaD,
an external bias potential in the form of a Gaussian is applied along a few choice
CVs. This enhances the fluctuations of those CVs, discouraging the system from
visiting previously visited metastable states, thereby allowing it to sample many
configurations. In the first variant of MetaD, constrained MD was used to impose
a history-dependent bias, allowing the system to diffuse through the free energy
surface [52]. In the subsequent versions, the bias is applied directly on the CVs while
the system dynamically evolves during an MD simulation [57, 60]. A drawback
of standard MetaD and its early variants was convergence of the bias potential to
the underlying FES, making it unclear when to stop a simulation. This problem
was alleviated by the introduction of well-tempered MetaD (WTMetaD), where
asymptotic convergence of the FES was proven [61]. Similar to standard MetaD,
small, repulsive, history-dependent Gaussian kernels that can adapt to the FES are
deposited on the phase space in WTMetaD. The height of the Gaussian bias decreases
over time, and in the long-time limit, the bias is shown to converge exactly to the
underlying FES [62]. Another variant that achieved convergence better than standard
MetaD was flux-tempered MetaD, proposed by Singh et al., an iterative method that
maximizes the flux of a random walker in CV space [63]. Still, simulations that apply
enhanced sampling methods which suffer from being computationally intensive as
the number of CVs that are biased exponentially increases the phase space to be
sampled.
Numerous variants of MetaD have been proposed over the years [63–68], and
herein, we briefly describe some of these methods which have been developed to
speed up convergence of the enhanced simulation. In the multiple walkers version
of MetaD, “walkers” (or replicas) share a history-dependent bias potential, allowing
each walker to concurrently explore the FES. The error in the energy landscape can
be reduced by increasing the number of walkers (note that this number depends
on degree of correlation between walkers and is thus limited) [69]. In the adaptive
Gaussian flavor of MetaD, the width of the Gaussian biases is adjusted on the fly to
better mimic the features of the underlying FES, which is advantageous while dealing
125
the weighted histogram analysis method (WHAM) [46], popularized by Alan Grossfield [47] or by applying the multi-state Bennett acceptance ratio (MBAR) estimator
introduced by Michael Shirts and John Chodera [48]. Often, however, this method
is not sufficient to fully sample the phase space and interesting chemistries of the
system because integration of the underlying potential of mean force becomes degenerate with higher-dimensional CVs. However, the most significant drawback of US is
that most CPU time during a simulation is spent sampling high energy, and uninteresting, regions of phase space more so than the relevant ones. A number of external
biasing methods have been developed to account for the drawbacks of US, including
targeted MD [49], energy landscape paving [27], the Wang–Landau algorithm [50],
variationally enhanced sampling [51], and many others.
Metadynamics (MetaD) originally developed by Laio and Parrinello [52, 53],
gained popularity as a robust and flexible technique to address a wide range of
problems including protein folding [54], nucleotide complexation [55], chemical
reactions [16, 56], ligand docking [57, 58], and phase transitions [59]. In MetaD,
an external bias potential in the form of a Gaussian is applied along a few choice
CVs. This enhances the fluctuations of those CVs, discouraging the system from
visiting previously visited metastable states, thereby allowing it to sample many
configurations. In the first variant of MetaD, constrained MD was used to impose
a history-dependent bias, allowing the system to diffuse through the free energy
surface [52]. In the subsequent versions, the bias is applied directly on the CVs while
the system dynamically evolves during an MD simulation [57, 60]. A drawback
of standard MetaD and its early variants was convergence of the bias potential to
the underlying FES, making it unclear when to stop a simulation. This problem
was alleviated by the introduction of well-tempered MetaD (WTMetaD), where
asymptotic convergence of the FES was proven [61]. Similar to standard MetaD,
small, repulsive, history-dependent Gaussian kernels that can adapt to the FES are
deposited on the phase space in WTMetaD. The height of the Gaussian bias decreases
over time, and in the long-time limit, the bias is shown to converge exactly to the
underlying FES [62]. Another variant that achieved convergence better than standard
MetaD was flux-tempered MetaD, proposed by Singh et al., an iterative method that
maximizes the flux of a random walker in CV space [63]. Still, simulations that apply
enhanced sampling methods which suffer from being computationally intensive as
the number of CVs that are biased exponentially increases the phase space to be
sampled.
Numerous variants of MetaD have been proposed over the years [63–68], and
herein, we briefly describe some of these methods which have been developed to
speed up convergence of the enhanced simulation. In the multiple walkers version
of MetaD, “walkers” (or replicas) share a history-dependent bias potential, allowing
each walker to concurrently explore the FES. The error in the energy landscape can
be reduced by increasing the number of walkers (note that this number depends
on degree of correlation between walkers and is thus limited) [69]. In the adaptive
Gaussian flavor of MetaD, the width of the Gaussian biases is adjusted on the fly to
better mimic the features of the underlying FES, which is advantageous while dealing