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10.6 Belfast tutorial: Metadynamics
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computed at every step) the two torsions are computed only every 10 steps. PLUMED automatically detect which
variables are used at every step, leading to better performance when complicated and computationally expensive
variables are monitored - this is not the case here, since the two torsions are very fast to compute. Note that here
the work always increase with time and never gets lower which is somewhat surprising if you tink that we are moving
in another metastable state. One would expect this to bend and give a signal of approaching a minimum like before.
Nevertheless consider what you we are doing: we are constraining the system in one specific conformation and this
is completely unnatural for a system at 300 kelvin so, even for this small system adopting a specific conformation in
which all the heavy atoms are in a precise position is rather unrealistic. This means that this state is an high free
energy state.
10.6
Belfast tutorial: Metadynamics
10.6.1
Aims
The aim of this tutorial is to introduce the users to running a metadynamics simulation with PLUMED. We will set
up a simple simulation of alanine dipeptide in vacuum, analyze the output, and estimate free energies from the
simulation. We will also learn how to run a well-tempered metadynamics simulation and detect issues related to a
bad choice of collective variables.
10.6.2
Summary of theory
In metadynamics, an external history-dependent bias potential is constructed in the space of a few selected degrees
of freedom ~s(q), generally called collective variables (CVs) [22]. This potential is built as a sum of Gaussians
deposited along the trajectory in the CVs space:
d
(si − si (q(kτ)))2
V (~s,t) = ∑ W (kτ) exp − ∑
2σi2
i=1
kτ<t
!
.
where τ is the Gaussian deposition stride, σi the width of the Gaussian for the ith CV, and W (kτ) the height of the
Gaussian. The effect of the metadynamics bias potential is to push the system away from local minima into visiting
new regions of the phase space. Furthermore, in the long time limit, the bias potential converges to minus the free
energy as a function of the CVs:
V (~s,t → ∞) = −F(~s) +C.
In standard metadynamics, Gaussians of constant height are added for the entire course of a simulation. As a
result, the system is eventually pushed to explore high free-energy regions and the estimate of the free energy
calculated from the bias potential oscillates around the real value. In well-tempered metadynamics [24], the height
of the Gaussian is decreased with simulation time according to:
V (~s(q(kτ)), kτ)
W (kτ) = W0 exp −
,
kB ∆T
where W0 is an initial Gaussian height, ∆T an input parameter with the dimension of a temperature, and kB the
Boltzmann constant. With this rescaling of the Gaussian height, the bias potential smoothly converges in the long
time limit, but it does not fully compensate the underlying free energy:
V (~s,t → ∞) = −
∆T
F(~s) +C.
T + ∆T
where T is the temperature of the system. In the long time limit, the CVs thus sample an ensemble at a temperature
T + ∆T which is higher than the system temperature T . The parameter ∆T can be chosen to regulate the extent of
free-energy exploration: ∆T = 0 corresponds to standard molecular dynamics, ∆T → ∞ to standard metadynamics.
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