Biology Reference
In-Depth Information
as well as in simulation, free energies are always computed relatively to a
reference state. Let state
B
be described by
H
B
and characterized
by
Z
B
. The free energy difference between two states
A
and
B
is given by
a ratio of partition functions:
Z
ZA
.
B
D
Gk T
=-
B
ln
AB
The main idea behind the methods presented below is to avoid
direct computation of the individual partition functions
Z
A
and
Z
B
by
using the fact that the variations between states
A
and
B
of interest
are often localized in relevant regions of the configuration space; else-
where, the corresponding partition functions
Z
A
and
Z
B
have a high
degree of similarity. Most approaches correspond, therefore, to refor-
mulating Eq. (12) such that common parts of
Z
A
and
Z
B
not directly
relevant to the process under investigation cancel out. A fundamental
aspect of these approaches is that they express, as we will see, the free
energy difference in terms of an ensemble average, which can
be directly measured or calculated in a simulation, unlike an absolute
free energy.
In Sec. 4.1, we present methods derived from first principles that
are exact at the statistical mechanics level. For these methods,
the quality of the results (for a given model or force field) depends
mainly on the quality of the sampling and on convergence properties.
In Secs. 4.2 and 4.3, approximate methods will be described. These
methods are not exact at the statistical mechanics level, but do show
interesting convergence properties that make them very useful in some
applications.
4.1. Exact Statistical Mechanics Methods for Free
Energy Differences
Here, we briefly derive the free energy perturbation (FEP) and ther-
modynamic integration (TI) expressions, which can be applied in MD
or in Monte Carlo simulations to calculate free energy differences.
26
In the following,
λ
is an external parameter changing the functional
