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7.2.6.
Langevin dynamics
The equations of motion that are most commonly adopted in ENM studies do not
generally account for frictional forces (see Eq. (7.6)). Nonetheless, biomolecules
exist in viscous environment, and viscous drag may alter their normal modes of
motion. It is therefore useful to have a technique for calculating normal modes
of motion in the presence of damping forces. Perhaps the simplest way to
introduce viscous drag is through the Langevin equation:
ɺɺ
ζ
Mq
=− − +
Uq
q
ξ
( )
t
.
(7.38)
Here
ζ
is a velocity-dependent damping term and
ξ
(t) is a time-dependent vector
of random forces, also called white noise, which satisfies the conditions
i
t
ξ =
( )
0
(7.39)
′
′
ξ
( )
t
ξ
(
t
)
=
2
ζ δ
(
t
−
t
)
k T
,
(7.40)
i
j
ij
B
In mass-weighted coordinates, Eq. (7.38) becomes
,
(7.41)
ɺɺ
r
=− − +
Kr
Fr
ɺ
R
t
( )
with
K
as defined earlier,
F
=
M
-1/2
ζ
M
-1/2
is the mass-weighted friction matrix,
and
R
=
M
-1/2
ξ
.
Defining the 6
N
× 6
N
matrix (Miller
et al.
2008)
0
I
A
=
,
(7.42)
− −
K
F
in which
I
is the 3
N
× 3
N
identity matrix, Eq. (7.41) may be re-written as
ɺ
ɺɺ
r
r
0
=
A
+
,
(7.43)
ɺ
r
r
R
t
( )
and the normal modes can be solved analytically by diagonalizing
A
. The first
3
N
components of the eigenvectors of
A
provide the displacements along the
normal modes, and the last 3
N
components correspond to the mode velocities.
The eigenvalues of
A
are complex; their imaginary parts are the oscillatory
frequencies of the modes, and their real parts are the exponential decay constants
of their amplitudes. This approach has been used by Miller
et al.
(2008) to
estimate the fractional free energy loss in the myosin power stroke.
In the limit of strong friction, all of the modes are over-damped and the
system obeys Brownian dynamics. Hinsen
et al.
(2000) demonstrated that the
