Biomedical Engineering Reference
In-Depth Information
also known as parallel tempering [11-17]. In the REM method, replicas are run
in parallel at a sequence of temperatures ranging from the desired temperature to
a high temperature at which the replica can easily surmount the energy barriers.
From time to time, the configurations of neighboring replicas are exchanged and
this exchange is accepted by a Metropolis acceptance criterion that guarantees the
detailed balance. Thus, REM is essentially a Monte Carlo (MC) method. Because the
high temperature replica can traverse high energy barriers, there is a mechanism for
the low temperature replicas to overcome the quasi-ergodicity they would otherwise
encounter in a single temperature replica. The replicas can be generated byMC, hybrid
Monte Carlo (HMC), or MD with velocity rescaling. Okamoto and co-workers [13]
have developed a temperature rescaling scheme for coupling MD with REM — the
replica exchangemolecular dynamics (REMD)method. These large-scale simulations
cannot only study the protein folding mechanism but also provide extensive data for
force field and solvation model assessment and further improvement.
Two example small protein systems, one
α
-helix [Ace-A
5
(AAARA)
3
A-Nme] and
one
β
-hairpin (Ace-GEWTYDDATKTFTVTE-Nme), are used to illustrate the power
of the parallel algorithm in this chapter. Larger protein systems have also been studied
with the REMD method, and interested readers can refer to previous publications for
details [16, 18-22]. Here, we select these two structurally simpler systems, largely
because there are a lot of data available for comparison of solvation models and force
fields with the same peptide—one of themajor benefits of the large-scale simulations.
We will also briefly discuss the application of REMD in protein structure refinement.
This chapter is not aimed to be a complete review of the subject, but instead to pick a
few specific examples, some from our own, to illustrate the parallel REMD method
and its application in protein folding and protein structure prediction.
17.2 REMD METHOD
A brief discussion of the REM method based on molecular dynamics (REMD) is
described subsequently.We have also implemented a combination of HMCwithREM,
which is more efficient for smaller systems. The implementations are basically very
similar; therefore, for simplicity, we will only describe the MD based implementation
following Okamoto's velocity rescaling approach [13, 23].
Suppose there is a protein system (or any other molecular systems) of
N
atoms
with masses
m
k
(
k
=
1, 2,
...
,
N
), and coordinates and momenta
q
≡{
q
1
,
q
2
,
...
,
q
N
}
and
p
≡{
p
1
,
p
2
,
...
,
p
N
}
, the Hamiltonian
H( p
,
q)
of the system can be expressed as,
N
p
k
H(p
,
q)
=
2
m
k
+
V(q)
(17.1)
k
=
1
where
V(q)
is the potential energy of the
N
atom system. In the canonical ensemble
at temperature
T
, each state
x
≡
(p
,
q)
with the Hamiltonian
H(p
,
q)
is weighted by
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