Replica exchange 3030 10 20 30 40 50
Dihedral angle number0
2
4
6
8
10
Number of barrier crossings
0 10 20 30 40 50
Dihedral angle number0
200
400
600
800
1000
Number of barrier crossings(a) (b)Fig. 7.4Comparison of hybrid Monte Carlo (a) and parallel temperingreplica exchange (b)
for C 50 H 102.
force field (MacKerellet al., 1998). Here, the conformational preferences of the molecule
can be characterized using the set of backbone dihedral angles. Since each dihedral an-
gle has three attractive basins corresponding to twogaucheand atransconformation,
the number of local minima is 3^50 ≈ 7 × 1023. Fig. 7.4 displays a histogram of the num-
ber of times each of the backbone dihedral angles crosses an energy barrier(Minary
et al., 2007). The replica exchange calculations are carried out using hybrid Monte
Carlo to evolve each of ten individual replicas. The temperatures ofthe replicas all lie
in the range 300 K to 1000 K with a distribution chosen to give an average acceptance
probability of 20%, as recommended by Rathoreet al.(2005) and Kone and Kofke
(2005). The replica exchange Monte Carlo calculations are compared to a straight hy-
brid Monte Carlo calculation on a single system at a temperature ofT= 300 K. The
figure shows a significant improvement in sampling efficiency with replicaexchange
compared to simple hybrid Monte Carlo. Although the replica exchange calculation is
ten times more expensive than straight hybrid Monte Carlo, the gainin efficiency more
than offsets this cost. It is interesting to note, however, that even for this simple sys-
tem, replica exchange does not improve the sampling uniformly over the entire chain.
Achieving more uniform sampling requires algorithms of considerable sophistication.
An example of such an approach was introduced by Zhuet al.(2002) and by Minary
et al.(2007).
It is important to note that a direct correlation exists between thenumber of repli-
cas and the acceptance probability. If sufficient computational resources are available,
then a replica exchange calculation can be set up that contains a large number of
replicas. In principle, this facilitates exchanges between neighboring replicas, thereby
increasing the average acceptance probability. However, more attempted exchanges
(and hence more computational time) are needed for high-temperature copies to per-
colate down to the low-temperature copies. Thus, the number of replicas and average
acceptance probability need to be optimized for each system, computer program, and
available computational platform. Indeed, sinceU∼N, as the system size increases,