We have computed values for the betweenness centrality, C B ,
for all the nodes of unliganded and liganded structures HbI and
plot the largest values for each structure in Fig. 3. The largest value
for the unliganded structure, 0.15, lies on the cluster of water
molecules at the interface, highlighting again the centrality of this
feature to the global network of energy transport in unliganded
HbI. The next largest values, 0.12, lie on the Lys30-Asp89 salt
bridge, with somewhat smaller but comparable values on nearby
residues, including Arg67, Leu84, Asp85, Asn86, Pro87, and
Asp88. The hubs of information flow on the energy transport
network as quantified by C B lie in regions critical to the stability
of the protein, and to the allosteric regulation of HbI.
Mutation studies that influence interactions between the water
cluster and the protein reveal significant effects on cooperativity
[83–85]. Mutation of Lys30 to Asp30 destabilizes the protein
altering the mechanism of cooperativity, which then involves dissociation of the two globules upon oxygen binding, and reformation
of the dimer upon dissociation [86]. The largest values of the
betweenness centrality calculated for the energy transport network
apparently identify two regions that control allostery in this protein. Both of these regions controlling allosteric regulation identified here were also identified as NBNs.
The largest values of C B calculated for the liganded protein,
also plotted in Fig. 3, are 0.16 and 0.13 for, respectively, Lys30 and
Asp89, with somewhat smaller values obtained for several residues
of the E and F helices at the interface between the proteins, including residues Tyr75, Leu77, Gln78, Asn79, Gln83, Leu84, Asp85,
Val91, Cys92, and Val93. The Lys30-Asp89 salt bridge forms the
basis of the only interglobule network for the liganded protein, and
its pivotal role in cooperativity was noted above. The residues
around Cys92 on the F helix form a hinge with residues around
Arg67 on the E helix that has been observed in time-resolved
crystallography experiments to serve as a pivot point for structural
change following ligand photolysis [141]. We find sizable values of
the betweenness centrality, 0.11 and 0.06, respectively, for Cys92
and Arg67. More details concerning network analysis of this allosteric protein can be found in Ref. [22].
2.3 Master Equation
Simulations of Energy
Dynamics
on a Network
We turn now to dynamics along a network, where we simulate
energy dynamics with a master equation, using as rate constants
the time constants obtained with the local energy diffusion coefficients, Eq. 5. In a recent study of the 36-amino acid fragment from
the villin headpiece subdomain, HP36, the results of a master
equation simulation using the rate constants obtained from communication maps were compared with results of all-atom nonequilibrium simulations. Here we summarize work carried out in that
study [17]. The master equation is,
Locating and Navigating Energy Transport Networks in Proteins
45
for all the nodes of unliganded and liganded structures HbI and
plot the largest values for each structure in Fig. 3. The largest value
for the unliganded structure, 0.15, lies on the cluster of water
molecules at the interface, highlighting again the centrality of this
feature to the global network of energy transport in unliganded
HbI. The next largest values, 0.12, lie on the Lys30-Asp89 salt
bridge, with somewhat smaller but comparable values on nearby
residues, including Arg67, Leu84, Asp85, Asn86, Pro87, and
Asp88. The hubs of information flow on the energy transport
network as quantified by C B lie in regions critical to the stability
of the protein, and to the allosteric regulation of HbI.
Mutation studies that influence interactions between the water
cluster and the protein reveal significant effects on cooperativity
[83–85]. Mutation of Lys30 to Asp30 destabilizes the protein
altering the mechanism of cooperativity, which then involves dissociation of the two globules upon oxygen binding, and reformation
of the dimer upon dissociation [86]. The largest values of the
betweenness centrality calculated for the energy transport network
apparently identify two regions that control allostery in this protein. Both of these regions controlling allosteric regulation identified here were also identified as NBNs.
The largest values of C B calculated for the liganded protein,
also plotted in Fig. 3, are 0.16 and 0.13 for, respectively, Lys30 and
Asp89, with somewhat smaller values obtained for several residues
of the E and F helices at the interface between the proteins, including residues Tyr75, Leu77, Gln78, Asn79, Gln83, Leu84, Asp85,
Val91, Cys92, and Val93. The Lys30-Asp89 salt bridge forms the
basis of the only interglobule network for the liganded protein, and
its pivotal role in cooperativity was noted above. The residues
around Cys92 on the F helix form a hinge with residues around
Arg67 on the E helix that has been observed in time-resolved
crystallography experiments to serve as a pivot point for structural
change following ligand photolysis [141]. We find sizable values of
the betweenness centrality, 0.11 and 0.06, respectively, for Cys92
and Arg67. More details concerning network analysis of this allosteric protein can be found in Ref. [22].
2.3 Master Equation
Simulations of Energy
Dynamics
on a Network
We turn now to dynamics along a network, where we simulate
energy dynamics with a master equation, using as rate constants
the time constants obtained with the local energy diffusion coefficients, Eq. 5. In a recent study of the 36-amino acid fragment from
the villin headpiece subdomain, HP36, the results of a master
equation simulation using the rate constants obtained from communication maps were compared with results of all-atom nonequilibrium simulations. Here we summarize work carried out in that
study [17]. The master equation is,
Locating and Navigating Energy Transport Networks in Proteins
45