Biomedical Engineering Reference
In-Depth Information
features. Due to space limitations, our accounts of these methods are by no means
comprehensive. Therefore, we refer readers to the many review articles and topics
on these topics for more detailed discussions [
13
,
25
,
27
,
40
,
72
,
86
,
95
,
124
].
6.1
Accelerated Molecular Dynamics
As discussed in Sect.
2
, the direct evaluation of Z
NVT
in a canonical ensemble is not
feasible using regular MD simulations. Since the microstates are sampled according
to a Boltzmann distribution in the canonical ensemble, the basic idea of enhanced
sampling methods is to escape from this distribution and sample the configurational
space in a “non-Boltzmann” way. Some examples of enhanced sampling methods
are accelerated molecular dynamics (aMD) [
49
], conformational flooding [
45
,
76
]
and hyperdynamics [
115
,
116
]. Here, we will use the aMD method as an example to
illustrate the principles behind enhanced sampling methods.
In the original aMD method [
49
], when the system's potential energy falls below
a threshold energy, E, a bias potential is added, such that the modified potential,
V
.
r
/, is related to the original potential, V.
r
/,via
V
.
r
/
D
V.
r
/
C
V.
r
/;
(25)
where V.
r
/ is the bias potential,
(
0
V.
r
/
E
V.
r
/
D
(26)
.E
V.
r
//
2
˛
C
E
V.
r
/
V.
r
/<E:
In the above equation, E is the threshold energy specified by the user, which controls
the portion of the potential surface affected by the bias. The acceleration factor ˛
determines how “aggressive” the modification to the potential surface is: the smaller
˛, the more flattened the energy surface becomes.
Under the influence of the bias potential V , the sampling in an aMD simulation
will not follow a Boltzmann distribution. Instead, the energy barriers between adja-
cent low-energy states are lowered, and the system can explore the configurational
space more efficiently. Like other enhanced sampling methods, the effect of this bias
potential must then be removed from the final result. In aMD, this is achieved by
reweighing the simulation trajectory in the calculation of the ensemble average
h
A
i
:
h
A
iD
h
A.
r
/ exp.ˇV.
r
//
i
h
;
(27)
exp.ˇV.
r
//
i
h
:::
i
represent the ensemble average in the original (unbiased)
and the aMD (biased) ensembles, respectively.
in which
h
:::
i
and
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