Biology Reference
In-Depth Information
steps as necessary, since their free
energy differences simply cumulate to give
to introduce as many intermediate
λ
.
D
GGG
=
D
+
D
+
L
+
D
G
AB
A
1
12
nB
The total free energy difference is then recovered by applying the
FEP method between each successive intermediate state and summing all
contributions:
n
Â
-
bl l
[(
HH
i
) ( ]
.
-
D
Gk T
=-
ln
e
+
1
i
AB
B
i
=
0
Note that the intermediate states — in other words, the unphysical path
linking states
A
and
B
— are completely arbitrary, since
G
AB
is a ther-
modynamic state function. Thus, the intermediate states can be chosen
such as to optimize the simulation convergence, for example, by adapt-
ing the functional form of the
∆
λ
dependence in different terms of
H
(
r
,
p
,
is
large. In particular, special care has to be taken to avoid numerical sin-
gularities when making Lennard-Jones particles appear, for example, by
using the soft core scaling method.
29
λ
),
28
or by setting smaller
λ
intervals in regions where d
H
/d
λ
4.1.2. Thermodynamic integration
Assuming that the two states
A
and
B
are linked by a coupling parameter
λ
as defined above, and that the free energy
G
is a continuous function
of
λ
, we have the identity
d
d
G
l
=- =
Ú
B
D
GGG
d
l
.
B
B
A
l
l
l
l
Using the definition of the free energy in Eq. (12); we have
d
d
G
kT
d
d
G
e
dl
dl
H
()
-
bl
H
()
.
B
Ú
=-
dd
r
p
=
l
Z
l
l
l
