Biomedical Engineering Reference
In-Depth Information
of ionic solvation energy [
56
]. Despite these artifacts, the PBC are generally
considered to have little impact on the equilibrium thermodynamic properties of
asystem[
5
], and are routinely used in MD simulations of biomolecules.
3.3
Equations of Motion
Now that we have discussed the various components of V and outlined the
procedure of their calculation, the next step is to derive the time evolution of
our system by integrating the equations of motion. The goal is to calculate, at
timestep n
C
1, the coordinates X
n
C
1
, velocities V
n
C
1
and forces F
n
C
1
of all atoms,
given the corresponding values of these quantities at the previous timestep n.The
Verlet algorithm [
114
], which belongs to the class of finite difference methods, is
commonly used to perform such calculations. In practice, we often use the velocity
form of the Verlet algorithm [
110
], which has improved numerical accuracy over
the original method. This algorithm contains the following equations,
t
2
M
1
F
n
;
V
n
C
2
D
V
n
C
(19)
1
X
n
C
1
D
X
n
C
tV
n
C
2
;
(20)
F
n
C
1
D
F.X
n
C
1
/;
(21)
t
2
M
1
F
n
C
1
:
V
n
C
1
D
V
n
C
2
C
(22)
1
1
2
is first calculated, followed
by the calculation of the coordinates at step n
C
1. Based on the new coordinates,
the potential energy function is evaluated and new forces are obtained. The velocity
is then advanced by another half a timestep to produce the new value at n
C
1.
A key parameter in the above equations is the timestep t, which determines
how frequently we perform the integration. Ideally, we would like to use a timestep
as large as possible to minimize the computational cost. In reality, we are often
limited to a timestep that is rather small, e.g., 1 fs (10
15
s), because the timestep
must be small enough to allow for accurate evaluation of the fastest motion in
a system, which is the vibration of the bond length between two atoms. Using
constraint methods, such as the SHAKE algorithm [
99
], we can fix the bond
lengths and increase the timestep from 1 fs to 2 fs. Even with these algorithms,
however, the timescale we can routinely access with MD is currently limited to the
submicrosecond range. Compared with most experimental techniques, the limited
timescale accessible by MD remains a bottleneck of the method.
In (
19
)-(
22
), we have used the Verlet algorithm to integrate the Newtonian
equations of motion. Since these equations conserve the total energy of a system,
the phase space distribution generated above is that of a microcanonical (NVE)
ensemble. In order to simulate other statistical ensembles, such as the canonical
As shown above, the velocity of the system at step n
C
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