Biomedical Engineering Reference
In-Depth Information
6.3
Free-Energy Perturbation
In the FEP approach, we start by combining (
8
)and(
29
),
P
exp.
ˇV
1
/
P
exp.
ˇV
0
/
A
D
ˇ
1
ln
(30)
P
expŒ
ˇ.V
1
V
0
/ exp.
ˇV
0
/
P
exp.
ˇV
0
/
D
ˇ
1
ln
(31)
D
ˇ
1
ln
h
expŒ
ˇ.V
1
V
0
/
i
0
:
(32)
For simplicity, we have omitted the dependence of V on
r
in the above equations,
and the summation is to be understood as being performed over all the configura-
tions
r
. The transition from (
31
)to(
32
) is made using the definition of ensemble
average in (
10
), and
h
:::
i
0
represents an ensemble average performed in state 0. If
we further define V
D
V
1
V
0
,then(
32
) can be simplified to
A
D
ˇ
1
ln
h
exp.
ˇV/
i
0
:
(33)
In essence, the above equation can be thought of as calculating the exponential of
V when the system is “frozen” at a particular configuration in state 0, and the
obtained exponentials are averaged over all configurations in state 0 to give A.
Note that we can get an equivalent formula by using the ensemble average in state 1,
A
D
ˇ
1
ln
h
exp.ˇV/
i
1
:
(34)
The above two equations, which form the basis of the FEP approach, are often
referred to as the “forward” and “reverse” calculation, respectively. Combining
both forward and reverse calculations using the Benett acceptance ratio (BAR)
method [
11
] has been demonstrated to yield the most accurate result [
106
]. Since in
many MD programs, performing calculations in both directions can be done nearly
as efficiently as performing the calculation in a single direction, combining forward
and reverse FEP has been recommended as a standard practice [
95
].
6.4
Thermodynamic Integration
In the TI approach, a parameter is used to describe the transition of the system
from state 0 to state 1. For instance, in an alchemical transformation where a ligand
is annihilated,
0
and
1
will correspond to the system with and without the ligand,
respectively. To calculate A using TI, we start by differentiating (
28
),
dA
d
D
1
ˇ
1
Z
NVT
@Z
NVT
@
:
(35)
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