Biomedical Engineering Reference
In-Depth Information
Integration of the new equation of motion proceeds along the lines discussed in
Chapter 10. A straightforward algorithm has been given by Ermak and Buckholtz (1980).
The equations of motion are integrated over a time interval
t
that is sufficiently short so
that the interparticle forces remain approximately constant. The algorithm for advancing
the position
r
A
and velocity
v
A
of particleA is then similar to those discussed in Chapter 10;
we have
c
1
d
r
A
d
t
c
2
d
2
r
A
d
t
2
(
t
)
2
r
A
(
t
+
t
)
=
r
A
(
t
)
+
t
+
+
r
A
t
t
c
0
d
r
A
d
t
c
1
d
2
r
A
d
t
2
t
v
A
v
A
(
t
+
t
)
=
t
+
t
+
(23.3)
Here
r
A
(
t
),(d
r
A
/d
t
)
t
and (d
2
r
A
/d
t
2
)
t
are the instantaneous position, velocity and acceleration
vector of particleA. The acceleration is calculated from the force.
r
G
and
v
G
are random
vectors chosen from a Gaussian distribution with zero mean and standard deviations
t
2
k
B
T
m
A
1
γ
t
2
2γ
t
))
1
γ
t
(3
σ
r
=
−
−
4 exp(
−
γ
t
)
+
exp(
−
k
B
T
m
A
(1
σ
v
=
−
exp(
−
γ
t
))
(23.4)
The numerical coefficients are given by
c
0
=
exp (
−
γ
t
)
1
c
0
γ
t
−
c
1
=
1
c
1
γ
t
−
c
2
=
At low values of the friction coefficient, the dynamical aspects dominate. If the inter-
particle forces are taken to vary linearly with time between each time step, the equations
of motion can be rewritten in a form that is said to produce a more accurate simulation:
c
1
d
r
A
d
t
c
2
d
2
r
A
d
t
2
(
t
)
2
r
A
(
t
+
t
)
=
r
A
(
t
)
+
t
+
+
r
A
c
0
d
r
A
d
t
c
2
)
d
2
r
A
d
t
2
c
2
d
2
r
A
d
t
2
t
t
(23.5)
v
A
(
t
+
t
)
=
t
+
(
c
1
−
t
+
t
+
v
A
t
t
+
t
and as γ
0 we recover the
velocity Verlet
algorithm discussed in Chapter 10. For large
values of γ , the random collisions dominate and the motion becomes diffusion-like.
The same considerations apply to Langevin dynamics as to standard MD; there are
three stages to a simulation, the heating phase, the data collection stage and the optional
cooling stage. For the sake of illustration, Figure 23.2 shows the end-to-end distance in the
hydrocarbon C
20
H
42
, over a 10 ps experiment witha4ps
−
1
friction coefficient.
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