Information Technology Reference
In-Depth Information
Hessian is projected into the space of rigid blocks with the transformation
H
BLK
=
P
T
H
P
,
H
BLK
is diagonalized with
V
BLK
H
V
BLK
=
Λ
BLK
,
and the resulting
eigenvectors are projected back into the full
3N
-dimensional space with the
inverse projection
V
=
P
T
V
BLK
.
An example application of the RTB/BNM formalism is to virus maturation.
Tama and Brooks (2005) modeled viral capsids of different sizes and symmetries
using an ENM in which each capsomer was taken to be a rigid block. Their
analysis showed that the conformational changes that occur during viral
maturation can be largely accounted for with only a few icosahedrally symmetric
slow modes. Another example application is to molecular motors. Li and Cui
(2004) found that the conformational changes that occur in myosin and the
calcium transporter Ca
2+
-ATPase are dominated by a small number of slow
modes, suggesting that Brownian motions have an essential role in the function
of molecular motors.
7.2.5.
Treatment of perturbations
It is often interesting to compare the global dynamics of a system in the absence
and presence of some environmental perturbation applied at a given position,
such as ligand binding. In such cases the perturbation takes the form of
additional nodes, and the Hessian is calculated for the extended system, including
these additional nodes. Comparing the normal modes of the original system to
those of the perturbed system is not straightforward: Including the perturbation
provides additional degrees of freedom, so the normal modes of the perturbed
system are not necessarily orthogonal when projected into the space of the
unperturbed system. It is useful to have an effective Hessian that will account for
the influence of the perturbation without modifying the size of the system. This
can be calculated as follows.
The state vector for an
N
node system is
r
= (
r
1
, …,
r
3
N
)
T
, and the state vector
for the same system in the presence of a perturbation,
e
= (
e
1
, …,
e
3
n
)
T
, by a
system of
n
nodes is
r
′ = (
s
1
, …,
s
3
N
,
e
1
, …,
e
3
n
)
T
= (
r
T
e
T
)
T
, where the first 3
N
components refer to the original system, and the last 3
n
to the environment or
perturbing molecule. As demonstrated by Ming and Wall (2005), and by Zheng
and Brooks (2005), the Hessian of a molecule within a specific environment can
be decomposed as follows:
H
H
ss
se
H
=
,
(7.31)
T
se
H
H
ee
