3.2 Identifying Key
Sites
for Communications
3.2.1 Markov Stochastic
Model
The Markov stochastic model coupled with GNM was used for
exploring the signal transductions of perturbations in proteins.
The affinity matrix A describes the interactions between residue
pairs connected in GNM; its generic element a ij is defined as:
a ij ¼
N ij
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
N i N j
p
ð4Þ
where N ij is the number of atom-atom contacts between residues
i and j based on a cutoff distance of 4 A ˚ , N i is the number of side
chain atoms in residue i. The density of contacts at each node i is
given by:
d i ¼
X N
j ¼1
a ij
ð5Þ
The Markov transition matrix M, whose element m ij ¼ d
À1
j a ij
determines the conditional probability of transmitting a signal from
residue j to residue i in one time step. Accordingly, the hitting time
for the transfer of a signal from residue j to i is given by
H i, j
ð Þ ¼
X N
k¼1
Γ
À1
Â
Ã
kj
À Γ
À1
Â
Ã
ij
À Γ
À1
Â
Ã
ki
À Γ
À1
Â
Ã
ii
n
o
˜
nd k ð6Þ
Fig. 3 Results for predicted hinge site in Hbs. (a) The minima of square fluctuations based on the two lowest
GNM modes predict hinges that are located at different interfaces in T-Hb, and (b) green and yellow beads
denote hinges predicted by the first and the second GNM modes, respectively. Similar results (c, d) were found
for R-Hb
Identification of Allosteric Effects in Proteins by Elastic Network Models
27