Biology Reference
In-Depth Information
form of the system's Hamiltonian from the initial state (
λ =
0) to the
final state (
λ =
1).
4.1.1. Free energy perturbation
By inserting a unity factor in the form
e
+β
H
A
(
r
,
p
)
e
−β
H
A
(
r
,
p
)
into the numerator
of Eq. (12); we get
-
b
Hrp
(, )
+
b
H rp
(, )
-
b
H rp
(, )
Ú
e
e
e
dd
r
p
B
A
A
.
D
Gk T
=-
ln
AB
B
Z
A
This can be seen as a phase space average of the quantity e
−β [
HB
−
HA
]
in
state
A
:
-
b
[
HH
-
]
.
D
Gk T
=-
ln
e
BA
(13)
AB
B
A
This approach is generally attributed to Zwanzig.
23
In practice, a single sim-
ulation in the reference state
A
is performed, during which the above phase
space average is converged. The accuracy of the free energy evaluation can
be improved if one can perform a simulation in state
B
as well. In such a
case, the FEP from
A
to
B
and from
B
to
A
can be optimally combined in
a single expression using the so-called Bennett acceptance ratio
27
:
(
)
-
b
[
HH
-
]
min
1
,
e
BA
.
A
D
Gk T
=-
1
AB
B
(
)
-
b
[
HH
-
]
min
1
,
e
BA
B
The FEP method can give meaningful results only if the two states
A
and
B
overlap in phase space, meaning that configurations are sampled
in which the difference
H
B
H
A
is smaller than
k
B
T
. Often, for transfor-
mations of practical interest, this is not the case. The solution is to intro-
duce
n
intermediate states between
A
and
B
, such that the overlap
between successive states is good. The Hamiltonian
H
(
r
,
p
,
−
λ
) is made a
function of a parameter
λ
which characterizes the intermediate states,
such that
H
(
r
,
p
,
λ
Α
)
=
H
A
(
r
,
p
) and
H
(
r
,
p
,
λ
Β
)
=
H
B
(
r
,
p
). One is free
