Biomedical Engineering Reference
In-Depth Information
(NVT) and the isothermal-isobaric (NPT) ensembles, the Newtonian equations of
motion must be modified. Many techniques are available for this purpose [
5
,
38
],
almost all of which can be found in some commonly used MD simulation packages,
such as AMBER [
18
], CHARMM [
15
], GROMACS [
107
], NAMD [
94
], and
DESMOND [
14
]. Regardless of the details of these methods, most of them rely
on the Verlet algorithm described above to perform the integration of the modified
Newtonian equations of motion.
4
Applications of MD: A Few Examples
So far, we have concentrated on the theoretical aspects of MD: its statistical
mechanics background, the functional form of the potential energy, and how to
integrate the equations of motion. In this section, we will use a few examples
to showcase the applications of MD in biomolecular modeling. Due to space
limitations, we cannot hope to be comprehensive on this subject, hence, we refer the
readers to recent review articles for more application examples of MD [
1
,
66
,
71
,
98
].
4.1
Calculation of Water Diffusion
As discussed earlier, we can calculate various macroscopic properties of the system
from a MD simulation. One such macroscopic property often calculated is the
diffusion coefficient of water (D), which describes the mobility of the solvent
molecules in the system. It can be calculated from a simulation trajectory in one
of the two following ways:
Z
1
1
3
D
D
dt
h
v
i
.t/
v
i
.0/
i
;
(23)
0
1
6t
hj
2
D
D
r
i
.t/
r
i
.0/
j
i
;
(24)
where
v
i
.t/ and
r
i
.t/ are the velocity and coordinates of a single water molecule at
time t, respectively. Equation (
24
) is derived from (
23
) via the “Einstein relation,”
which also holds for other transport coefficients, such as the shear viscosity. These
transport coefficients can be calculated from a simulation with equations similar to
(
23
)and(
24
)[
5
].
To reduce the usage of disk space, the velocity trajectory is usually not saved in
a simulation, and (
24
) is used in the calculation of D. Here, we should emphasize
that (
24
) is only valid at long time intervals compared with the correlation time of
r
.
This calculation is performed for all the water molecules in the system and the
results are averaged to improve the statistical precision. Another commonly used
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