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In-Depth Information
Note that the potential energy (Eq. (7.4)) can be expressed in terms of the
changes in mass-weighted coordinates and the mass-weighted stiffness matrix
K
as V(
r
) = ½
r
T
Kr
. The solution to Eq. (7.7) is
−
i
ω
t
r
( )
t
=
a
e
.
(7.8)
From Eq. (7.7) and Eq. (7.8), we find that the coefficients
a
solve the eigenvalue
equation
Ka =
λ
a
, where
a
is a vector of displacements along a normal mode of
vibration, and the eigenvalue, λ, is the square of the normal mode frequency ω.
In most cases,
K
is not invertible, but has a well-defined number of
eigenvalues that are identically zero. This occurs because the potential energy
only depends on internal degrees of freedom and places no energetic restrictions
on rigid-body rotations and translations. With this in mind, the inverse of
K
,
when required, is replaced by the pseudo-inverse, defined as
T
v v
−
1
=
∑
k
k
K
,
(7.9)
λ
≠
0
λ
k
k
where λ
k
are the nonzero eigenvalues of
K
, and
v
k
are their associated
eigenvectors.
ENM partition function
. The system's partition function can be calculated by
integrating the potential over all possible changes in structure:
1
n
T
∫
Z
=
d r
exp
−
r Kr
(7.10)
2
k T
B
n
/ 2
−
1
1/ 2
=
(2
π
k T
)
[det(
K
)]
,
(7.11)
where
k
B
is the Boltzmann constant and
T
is the absolute temperature. As
det (
K
-1
) is simply the product of the reciprocal nonzero eigenvalues of
K
, the
lowest frequency modes contribute most to the partition function. These modes
are also of highest interest when seeking to determine the most probable
global
fluctuations of a molecule. Indeed, the low-frequency, or 'slow', modes of an
ENM are robust to variations in network topology, the level of resolution adopted
in describing the network (see for example Doruker
et al.
, 2002) and the force
field adopted in NMA, and they reflect the intrinsically accessible motions that
are endowed upon the molecule by its structure.
Mean-square fluctuations and cross-correlations.
Expectation values for
dynamical variables predicted by the ENM can be directly compared to
experimental measurements. X-ray temperature factors (B-factors) provide a
measure of the mean-square fluctuations of individual atoms. Similarly, an