3 Methods
3.1 Protein Contact
Networks Definition
Protein Contact Networks (PCNs) are graphs whose nodes
(or vertices) are the protein residues and links (or edges) between
the i-th and the j-th nodes (residues) occur if the distance between
the two residues d ij is higher than 4 and lower than 8 Å. The lower
end excludes all covalent bonds (disulfide and peptidic bonds),
which are not sensible to environment change (so to protein functionality), while the upper end gets rid of weaker non-covalent
bonds (so not significant for protein functionality).
So, the first step is to extract from the PDB file the coordinate
for all alpha-carbon atoms: with reference to Table 1, in lines with
“ATOM” as header take only coordinates (columns 31–54) for
alpha carbons (if “Atom Name”—column 13–16—is “CA”),
keeping record as well of the “Residue name” (column 18–20).
As a result, a matrix N Â 3 is reporting X, Y, and Z coordinates
of alpha carbons for all N residues, ordered according to the primary sequence. Another vector will keep trace of residue names in
the sequence.
Starting from the coordinates matrix, it is possible to build up a
distance matrix, whose generic element d ij reports the Euclidean
distance between residues i and j:
d ij ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
x i À x j
À
Á 2 þ y i À y j
2 þ z i À z
ð
Þ
2
r
ð1Þ
(x i , y i , z i ) and (x j , y j , z j ), respectively, being the cartesian coordinates of residue i and j.
At this point, it is possible to build up adjacency matrix A,
whose generic element is defined as:
A ij ¼
1 if 4 ̊ A < d ij < 8 ̊ A
0
otherwise
&
ð2Þ
The adjacency matrix A is the mathematical descriptor of
unweighted, undirected graphs, from which all main topological
descriptors can be derived (see Note 1). The adjacency matrix A can
be visualized as matrix plot (Fig. 2). It is evident that the nature of
A is sparse matrix.
3.2 Computation
of Descriptors
Network topology translates into mathematical descriptors, derived
from the adjacency matrix A. As follows, the definition of main
descriptors and their relevance in protein structure and functionality description.
1. Node degree: the degree of the i-th node (residue) k i computes
the number of links the node participates in. It can be easily
computed as the sum of elements of the i-th row or column:
Disclosing Allostery Through Protein Contact Networks
11
3.1 Protein Contact
Networks Definition
Protein Contact Networks (PCNs) are graphs whose nodes
(or vertices) are the protein residues and links (or edges) between
the i-th and the j-th nodes (residues) occur if the distance between
the two residues d ij is higher than 4 and lower than 8 Å. The lower
end excludes all covalent bonds (disulfide and peptidic bonds),
which are not sensible to environment change (so to protein functionality), while the upper end gets rid of weaker non-covalent
bonds (so not significant for protein functionality).
So, the first step is to extract from the PDB file the coordinate
for all alpha-carbon atoms: with reference to Table 1, in lines with
“ATOM” as header take only coordinates (columns 31–54) for
alpha carbons (if “Atom Name”—column 13–16—is “CA”),
keeping record as well of the “Residue name” (column 18–20).
As a result, a matrix N Â 3 is reporting X, Y, and Z coordinates
of alpha carbons for all N residues, ordered according to the primary sequence. Another vector will keep trace of residue names in
the sequence.
Starting from the coordinates matrix, it is possible to build up a
distance matrix, whose generic element d ij reports the Euclidean
distance between residues i and j:
d ij ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
x i À x j
À
Á 2 þ y i À y j
2 þ z i À z
ð
Þ
2
r
ð1Þ
(x i , y i , z i ) and (x j , y j , z j ), respectively, being the cartesian coordinates of residue i and j.
At this point, it is possible to build up adjacency matrix A,
whose generic element is defined as:
A ij ¼
1 if 4 ̊ A < d ij < 8 ̊ A
0
otherwise
&
ð2Þ
The adjacency matrix A is the mathematical descriptor of
unweighted, undirected graphs, from which all main topological
descriptors can be derived (see Note 1). The adjacency matrix A can
be visualized as matrix plot (Fig. 2). It is evident that the nature of
A is sparse matrix.
3.2 Computation
of Descriptors
Network topology translates into mathematical descriptors, derived
from the adjacency matrix A. As follows, the definition of main
descriptors and their relevance in protein structure and functionality description.
1. Node degree: the degree of the i-th node (residue) k i computes
the number of links the node participates in. It can be easily
computed as the sum of elements of the i-th row or column:
Disclosing Allostery Through Protein Contact Networks
11