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where the components of the matrix refer to the equilibrium distance vectors, and
the diagonal super-elements satisfy
=−
∑
H
H
.
(7.17)
ii
ij
j;j
j;j≠i
≠i≠i
≠i
Diagonalization of
H
yields 3
N -
6 normal modes, each of which has a 3-vector
component for every node. The remaining 6 modes have zero eigenvalue and
correspond to rigid-body rotations and translations of the system.
The spring constants γ
ij
are the only adjustable parameters in this model, and a
variety of methods are used to select their values. Pairwise interactions are
predominantly local, and a common practice is to assign a uniform spring
constant, γ
ij
= γ, to all pairs of nodes separated by less than some cutoff distance,
and γ
ij
= 0 for all others. It has been found empirically (Eyal
et al.
, 2006) that
when the nodes are taken to be the α-carbons of a protein, a cutoff distance of
about 15Å results in residue mean-square fluctuations that correlate well with
experimental B-factors. An alternative approach (Hinsen, 1998) that agrees
comparably with experiments is to assign spring constants that decay with
distance. Recent studies (Kondrashov
et al
., 2006) show that the adoption of
stiffer force constants for the springs that connect first neighbors along the
sequence further enhances the correlation with B-factors.
7.2.3.
Gaussian network model
A simplification of the above-described model is the Gaussian network model
(GNM) (Bahar
et al.
, 1997). This model uses the assumptions that node
fluctuations are isotropic and Gaussian to reduce the interaction matrix from a
3
N ×
3
N
Hessian to an
N × N
Kirchhoff matrix. Interestingly, this model often
agrees better with experimental data than does its anisotropic counterpart,
because the underlying potential penalizes the vectorial changes
0
∆ = −
in internode distances (as opposed to penalizing the changes in the magnitudes
0
R
R
R
ij
ij
ij
ij i
− only, as in the ANM; see Eq. (7.14)).
The inherent assumption of vibrational isotropy allows GNM to predict the
size of motions and their cross-correlations, but not their directions. It also
predicts the displacements along normal coordinates (e.g., slow modes) and
permits us to define the domains engaged in concerted motions in the global
modes, but not their mechanism/direction of concerted rearrangements. Note that
the GNM uses only a single parameter, the elastic constant γ that defines the
interactions between nodes that are separated by a distance less than a cutoff
distance,
r
c
. When applied to proteins at the residue level, a value of
r
c
between
|
R
|
|
R
|
