Biomedical Engineering Reference
In-Depth Information
that a certain microstate can be visited in a canonical ensemble. The partition
function is a sum of the Boltzmann factors from all microstates, and is used as a
normalization factor to give the formula of P.
r
;
p
/
exp.
ˇH.
r
;
p
//
Q
NVT
P.
r
;
p
/
D
:
(7)
In practice, because the Hamiltonian is a sum of the kinetic energy, which
only depends on
p
, and the potential energy, which only depends on
r
,the
partition function Q can be expressed as a product of the kinetic (ideal gas)
contribution Q
i
NVT
and the potential (excess) contribution Q
e
NVT
[
5
]. The former
can be integrated analytically, leaving the latter our main target of calculation. In
reality, instead of Q
e
NVT
, we often use the configurational partition function
X
Z
NVT
D
exp.
ˇV.
r
//:
(8)
r
Correspondingly, the probability of visiting a configurational microstate
r
is
exp.
ˇV.
r
//
Z
NVT
P.
r
/
D
:
(9)
With (
9
), we can now calculate the ensemble average of any observable A:
X
X
A.
r
/ exp.
ˇV.
r
//
Z
NVT
h
A
i
NVT
D
A.
r
/ P.
r
/
D
:
(10)
r
r
One goal of an MD simulation is to generate the proper phase space distribution
according to (
9
), from which the ensemble average of various observables can be
calculated using (
10
). A somewhat subtle point is that once we have generated the
correct phase space distribution, we will be able to calculate an observable A as a
time average from a simulation trajectory,
A
obs
Dh
A
i
time
:
(11)
The equivalence of (
11
)and(
5
) relies on the so-called “ergodic assumption.”
Interested readers can find more on this topic in the topic by Allen and Tildesley [
5
].
Before we move on to the next section, where we will discuss how to generate
the desired phase space distribution in a MD simulation, we should say a few more
words about (
8
). As the partition function Z
NVT
contains all the information about
the microstates of a system, it's very tempting to evaluate it directly according to
(
8
). However, this remains a daunting task for most biomolecular systems. The
reason is that there are too many microstates for a typical biomolecular system,
which makes the direct evaluation of (
8
) unfeasible. To expedite the sampling of the
configurational space, various enhanced sampling methods have been developed,
and we will discuss some of these methods in Sect.
6
.
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