Biology Reference
In-Depth Information
Using Newton's equation, one gets the corresponding acceleration
a
i
:
d
d
V
r
ii
.
-=
ma
(6)
i
Starting from given initial positions and after assigning random velocities,
one can propagate the system by numerical integration of Eq. (6). Such
an integrator can be obtained by considering a Taylor expansion of the
position with respect to small time increments
δ
t
:
1
2
2
rt
(
+= +
d
t
)
rt
()
vt t
()
d
+
at t
()
d
(7)
i
i
i
i
1
2
2
.
rt
(
-= -
d
t
)
rt
()
vt t
()
d
+
at t
()
d
(8)
i
i
i
i
Summing the two equations, we obtain the so-called Verlet algorithm:
2
.
rt
(
+=
d
t
)
2
rt
( )
- -+
rt
(
d
t
)
at t
( )
d
(9)
i
i
i
i
Thus, knowledge of the system at times
t
−δ
t
and
t
allows us to obtain
the positions at time
t
+ δ
t
. Velocities can be subsequently computed
based on the positions:
rt
(
+- -
d
t
)
rt
(
d
t
)
.
i
i
vt
()
=
(10)
i
2
t
Note that these average velocities are computed for time
t
, whereas the
positions in Eq. (9) were obtained for time
t
+ δ
t
. This inaccuracy can be
accounted for in other implementations.
Repeating this procedure, one can simulate the complete time evolu-
tion of the system, as represented in Fig. 2. More sophisticated algorithms
have been developed following similar principles, like the velocity Verlet,
leap-frog, and Beeman algorithms.
A general protocol to run an MD simulation starting from a given
molecular structure is presented in Fig. 3. The initial coordinates
