Genomic analysis reveals dynamic core formation in the
integrated molecular interaction network of yeast
Supplementary Material
Aswin Sai Narain Seshasayee1,+ and M. Madan Babu2, +,*
+Joint
first authors
1Centre
2MRC
for Biotechnology, Anna University, Chennai 25, India
Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, United Kingdom
* Corresponding Author:
Email: madanm@mrc-lmb.cam.ac.uk
Phone: 44-1223-402041
Fax: 44-1223-213556
1
Supplementary Material M1
M1: A guide to core densities
Networks can be decomposed into hierarchical layers called network cores, which allows us
to identify groups of nodes that are important at several levels (See Fig M1). We introduce
novel measures that allow us to identify cores that are highly populated (node core density),
and cores that are most interconnected (edge core density). We show that normalized values
of the core densities, allow us to compare networks of different sizes.
Network cores and core membership of a node: The k-core of a network is that sub-graph
that is obtained after recursive deletion of nodes with degree less than k, and all links incident
on them. Core membership of a node is the highest k-core to which the node belongs to.
1
1
1
k =3
k =2
k =1
4
5
4
5
1
4
8
k K
(node)   N
k
k
k 1
1
Hub
(highly
connected
protein)
Node and Edge Core Densities
4
3
3
k K
(edge)   E
k 1
k
n (node)  N
k K
k
 Kk
Nk 
k 1
k
n (edge)   E
k K
k
Nodes in core k
All nodes
 Kk
k 1
Ek 
Edges in core k
All edges
1
1
1
1
1
1
K  k K where, K  Maximum core value
k
1
Nodes=20, Interactions=25, K=3
Fig M1: Networks can be decomposed into network-cores, which can be characterized by
calculating the node-core density and the edge-core density values. The number on each
node denotes the degree of that node in the network. Note that degree is independent of core
membership for a node.
A k-core is defined as a sub-graph with nodes having k or more connections
Core densities: Node and edge core densities allow us to identify the most populated layer
and the most inter-connected layer in the network. Note that core membership of a node and
the degree of a node are fundamentally different measures because the former accounts for
the degree of a node’s neighbor as well.
By analyzing the core structures of various standard networks of similar size, we show that (i)
not all networks organize into network cores, and (ii) networks which are distinctly different
may have similar core structure. The standard networks include (a) scale-free network, (b)
hierarchical, modular and scale-free network, (c) erdos-renyi network, (d) star network, and
(e) fully connected network,
Note that a link between two nodes was considered as a bi-directional edge. For each of the
following networks, the various core density parameters were calculated along with the core
membership for all the nodes. The following table illustrates the above mentioned results.
2
F
Table M2: Core density values and other parameters for the various standard networks
Networks
Scale-free
Erdos-Renyi
Hierarchical,
Modular and scalefree
Fully connected
Star
Nodes
Edges
Highest core, K
Node core density
125
560
8
125
720
8
125
692
8
125
15500
248
125
248
2
(node)
5.55
7.44
8
248
2
0.69
0.93
1
1
1
7
7.95
8
248
2
0.88
0.99
1
1
1
Normalized node core
density n(node)
Edge core density
(edge)
Normalized edge core
density n(edge)
Average Degree
Average Path length
Average clustering
coefficient
3
Supplementary Material M2
M2: Degree and core membership for proteins in the molecular networks
The degree of a protein in the network refers to the number of connections it makes with the
other proteins in the network, and the core membership of a protein refers to the highest kcore to which the node belongs in the network. The following figures show that for biological
networks, degree is not the same as core membership, and appears to be two different but
inter-related properties. Note that even though there appears to be a mild positive correlation
between the normalized degree and normalized core membership, there is extensive
variability in the degree values for a particular core value, and vice-versa. This is because the
degree of a protein in the network only measures the number of connections the protein
makes, whereas the core membership of a protein takes into account the degree of a protein
and the degree of all its neighboring proteins. Thus a highly connected protein need not be a
part of a high core (see Fig. M1).
The degree and core membership for each protein was calculated, and was normalized with
respect to the maximum degree and maximum core value respectively to obtain the
normalized degree and the normalized core value. The plot below shows the relationship
between the degree and core value for the different molecular interaction networks.
Fig S2a-d: Normalized degree v/s Normalized core value for each protein in the different
interaction networks. (a) Protein interaction network (3282 proteins), (b) Transcriptional
interaction network (3459 proteins), (c) Metabolic interaction network (576 proteins) and (d)
Integrated cellular interaction network (5006 proteins). Note that there is huge variability in the
degree of proteins that populate particular cores.
4
Supplementary Material M3
5