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ensemble of NMR structures can be used to calculate the correlations between
the displacements of atoms. The correlations between node fluctuations in the
ENM are given by
1
1
∫
n
T
〈 〉=
r r
d r
exp
−
r Kr
r r
(7.12)
i
j
i
j
Z
2
k T
B
−
1
=
k T
(
K
)
,
(7.13)
B
ij
and from Eq. (7.9) it is clear that the most significant contributions to the inter-
residue correlations are also from the slowest modes.
7.2.2.
Anisotropic network models
The most common ENMs are anisotropic network models (ANM) (Doruker
et al.
, 2000; Atilgan
et al.
, 2001; Tama & Sanejouand, 2001) that use the 3
N
mass-weighted coordinates of the nodes as generalized coordinates:
r
= (Ș
x
1
,
Ș
y
1
, Ș
z
1
, …, Ș
x
N
, Ș
y
N
, Ș
z
N
)
T
, where Ș
x
i
=
x
i
-
x
i
0
is the
x
-component of
the displacement of node
i
from its equilibrium position,
r
i
0
. In this case
the interaction matrix
K
is the 3
N ×
3
N
Hessian matrix,
H
, of mixed second
derivatives of the potential with respect to the coordinates of the residues. The
Hessian might be thought of as an
N
×
N
matrix of 3 × 3 submatrices, each of
which describes the energetic contribution from the interaction of two nodes.
The elements of
H
can be calculated from the potential energy,
1
∑
V
=
γ
(
R
−
R
0
)
2
,
(7.14)
ij
ij
ij
2
ij
where γ
ij
is the spring constant between nodes
i
and
j
,
R
ij
is their distance, and
i
R
is their equilibrium distance. The second derivatives of the potential function at
equilibrium have the general form
2
γ −
(
x
x
)(
y
−
y
)
∂
V
ij
j
i
j
i
=−
,
(7.15)
2
∂ ∂
x
y
R
i
j
ij
where
x
i
and
y
j
are the
x
- and
y
-coordinates of nodes
i
and
j
, respectively. Using
the notation
x
ij
= (
x
j
-
x
i
), and similarly for
y
ij
and
z
ij
, the off-diagonal super-
elements of
H
are
2
x
x y
x z
ij
ij
ij
ij
ij
γ
ij
H
=−
x y
y
2
y z
,
(7.16)
ij
ij
ij
ij
ij
ij
2
R
ij
x z
y z
z
2
ij
ij
ij
ij
ij
