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where
H
ss
contains contributions from interactions of the original system with
itself,
H
ee
accounts for interactions of the environment with itself, and
H
se
contains interactions between the system and its environment. Note that
H
ss
is
not simply the unperturbed Hessian, but has different diagonal super-elements
due to environmental contributions. The potential energy can be written as
1
(
′
T
′
V
=
r
)
H
r
(7.32)
2
1
1
T
T
T
=
r H
r
+
r H
e
+
e H
e
,
(7.33)
ss
se
ss
2
2
using
e
T
H
se
T
r
=
r
T
H
se
e
.
At equilibrium,
∂ ∂ = for all environmental
V
/
e
0
i
nodes, giving
0
=
H
r
+
H
e
,
(7.34)
es
es
which yields
−
1
e
=−
H
H
r
.
(7.35)
ee
es
Substitution of this expression into Eq. (7.33) permits us to write the potential
energy in terms of the 3
N
components of
r
as
1
T
V
=
2
r Hr
(7.36)
where
H
is a pseudo-Hessian with the same dimensionality as the unperturbed
Hessian, but which includes the environmental effects:
−
1
H
T
H
= −
H
H
H
.
(7.37)
ss
se
ee
se
Diagonalizing
H
leads to the normal modes in the presence of the perturbation,
and these can be directly compared to the modes of unperturbed system. This
approach has been used to examine conformational changes in myosin and
kinesin nucleotide-binding pockets. Zheng and Brooks (2005) employed a model
in which the binding pockets of motor proteins constituted the system, and the
remainder of the protein made up its environment. They showed that the
dynamics relevant to the myosin binding pocket are coupled to its global modes,
in agreement with hypothesized pathways between actin binding and force
generation. Ming and Wall (2006) used this method to demonstrate that substrate
allosteric proteins usually bind their substrates at sites that induce significant
perturbation in the collective dynamics.
