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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
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