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In-Depth Information
6.5Å and 7.5Å, corresponding to the radius of the first coordination shell, is
typically selected for use in GNM.
The potential in Eq. (7.14) is a sum over pairwise potentials, each of which
depends on the difference between the instantaneous distance between two nodes
and their equilibrium separation. By assuming isotropic fluctuations, we can
separate the spatial components of each node's motion, resulting in the GNM
potential
1
∑
0
0
V
=
γ
[(
R
− ⋅ −
R
) (
R
R
)]
(7.18)
GNM
ij
ij
ij
ij
2
ij
ij
1
γ
∑
0
2
0
2
0
2
=
(
x
−
x
)
+ −
(
y
y
)
+ −
(
z
z
)
(7.19)
ij
ij
ij
ij
ij
ij
ij
2
ij
γ
∑
2
2
2
=
Γ
(
∆
x
+ +
∆
y
∆
z
)
(7.20)
ij
ij
ij
ij
ij
2
γ
T
T
T
=
(
∆
x
Γ∆
x +
∆
y
Γ∆
y +
∆
z
Γ∆
z
)
(7.21)
2
0
,
In the preceding lines we use the notation
∆ = − and similarly for
y
ij
and
z
ij
. Likewise, we used the notation
x
= (
x
1
,
x
2
,
x
3
, …,
x
N
)
T
and
similar expressions for ∆
y
and ∆
z
, where
x
i
is the
x
-component of the vector
r
i
describing the fluctuation in the position of node
i
. The Kirchhoff adjacency
matrix,
x
x
x
ij
ij
ij
Γ, has elements
−
1 if
R
≤
r
and
i
≠
j
ij
c
Γ
=
0 if
R
>
r
and
i
≠
j
.
(7.22)
−
ij
ij
c
∑
Γ
if
i
=
j
ik
k k
;
≠
i
The equations of motion separate into three identical equations, one for each
spatial coordinate, and the normal modes are obtained by diagonalizing
Γ. The
assumption of isotropy essentially reduces the system to one dimension, and
Γ
has
N
- 1 non-zero eigenvalues. Correlations between nodes can be found as
before with
−
1
∆
r
⋅
∆
r
=
3
k T
(
Γ
)
,
(7.23
)
i
j
B
ij
which is identical to Eq. (7.13) with
Γ taking the place of
K
and a factor of three
from the summation over spatial coordinates.
