2 Computational Methods
2.1 Communication
Maps: Locating Energy
Transport Networks
To identify pathways along which energy transport is facile one can
carry out nonequilibrium simulations, which have been done
extensively by all-atom MD simulations [1, 12, 89]. Simulations
in harmonic approximation via wave packet propagation [57, 58]
have also been run to examine harmonic and anharmonic contributions to energy transport. The latter approach was used, e.g., to
calculate vibrational energy transport in HbI. Those results are
plotted in Fig. 1 for the case where energy is first introduced to
one of the hemes and the flow of energy calculated over the next
few picoseconds. Anisotropic transport is observed; the cluster of
water molecules at the interface apparently serves as an energy
transport channel from one globule to the other.
More coarse-grained approaches have also been introduced,
including the local conductivity analysis of Yamato and coworkers
[13, 14], which we shall adopt below to examine a relation between
hydrogen bond dynamics and energy flow between two hydrogen
bonded residues. An alternative coarse-graining method yields a
network weighted by local energy diffusion coefficients calculated
in terms of normal modes [19]. The weights for the network are
expressed in terms of the matrix elements of the energy current
operator, S, which in harmonic approximation can be written in
terms of the Hessian matrix, H, and eigenmodes, e, of the object
[90]. The mode diffusivity, in turn, can be expressed in terms of the
matrix elements of S [90]. We break each matrix element up into
contributions from individual residues. The contribution to the
energy flux between residues A and A
0 to matrix element S αβ is [19]
S
AA0
f
g
αβ
¼
iℏ ω α þ ω β
À
Á
4V
ffiffiffiffiffiffiffiffiffiffiffi
ω α ω β
p
X
r, r 0 ∈ x, y, z
ð
Þ
X
l, l
0 ∈AA
0
e
α
l H
ll0
rr 0 R l À R l
0
ð
Þ e
β
l
0 , ð1Þ
where R l is the position of atom l and r is a coordinate (x, y, or z).
We sum the atoms l together in a given region, A, and sum atoms l
0
together in region A
0 . V is the volume of the space spanned by the
two regions. While such a volume remains somewhat ambiguous, it
cancels out in the definition of the local energy diffusivity, Eq. 2.
For mode α the energy diffusivity is a sum over the squares of
matrix elements of the heat current operator, i.e., D α /
P
β6 ¼α
S αβ
2 δ ω α À ω β
À
Á
. Considering only energy flow between residues A and A
0 , we approximate the local energy diffusivity in mode
α using the harmonic model as
D
AA0
f
g
α
¼
πV
2
3ℏ
2
ω 2
α
X
β6 ¼α
S
AA0
f
g
αβ
2 δ ω α À ω β
À
Á
:
ð2Þ
40
Korey M. Reid and David M. Leitner
2.1 Communication
Maps: Locating Energy
Transport Networks
To identify pathways along which energy transport is facile one can
carry out nonequilibrium simulations, which have been done
extensively by all-atom MD simulations [1, 12, 89]. Simulations
in harmonic approximation via wave packet propagation [57, 58]
have also been run to examine harmonic and anharmonic contributions to energy transport. The latter approach was used, e.g., to
calculate vibrational energy transport in HbI. Those results are
plotted in Fig. 1 for the case where energy is first introduced to
one of the hemes and the flow of energy calculated over the next
few picoseconds. Anisotropic transport is observed; the cluster of
water molecules at the interface apparently serves as an energy
transport channel from one globule to the other.
More coarse-grained approaches have also been introduced,
including the local conductivity analysis of Yamato and coworkers
[13, 14], which we shall adopt below to examine a relation between
hydrogen bond dynamics and energy flow between two hydrogen
bonded residues. An alternative coarse-graining method yields a
network weighted by local energy diffusion coefficients calculated
in terms of normal modes [19]. The weights for the network are
expressed in terms of the matrix elements of the energy current
operator, S, which in harmonic approximation can be written in
terms of the Hessian matrix, H, and eigenmodes, e, of the object
[90]. The mode diffusivity, in turn, can be expressed in terms of the
matrix elements of S [90]. We break each matrix element up into
contributions from individual residues. The contribution to the
energy flux between residues A and A
0 to matrix element S αβ is [19]
S
AA0
f
g
αβ
¼
iℏ ω α þ ω β
À
Á
4V
ffiffiffiffiffiffiffiffiffiffiffi
ω α ω β
p
X
r, r 0 ∈ x, y, z
ð
Þ
X
l, l
0 ∈AA
0
e
α
l H
ll0
rr 0 R l À R l
0
ð
Þ e
β
l
0 , ð1Þ
where R l is the position of atom l and r is a coordinate (x, y, or z).
We sum the atoms l together in a given region, A, and sum atoms l
0
together in region A
0 . V is the volume of the space spanned by the
two regions. While such a volume remains somewhat ambiguous, it
cancels out in the definition of the local energy diffusivity, Eq. 2.
For mode α the energy diffusivity is a sum over the squares of
matrix elements of the heat current operator, i.e., D α /
P
β6 ¼α
S αβ
2 δ ω α À ω β
À
Á
. Considering only energy flow between residues A and A
0 , we approximate the local energy diffusivity in mode
α using the harmonic model as
D
AA0
f
g
α
¼
πV
2
3ℏ
2
ω 2
α
X
β6 ¼α
S
AA0
f
g
αβ
2 δ ω α À ω β
À
Á
:
ð2Þ
40
Korey M. Reid and David M. Leitner