Global topological properties of biological networks
Jan 04, 2016
Global topological properties of biological
networks
Node: protein
Edge: protein-protein interaction
Protein-Protein Interaction Network
Saccharomyces cerevisiae
E. coli metabolic network
Basic features of a network
• Degree distribution
• Clustering coefficients
• Average shortest path length
Degree of a node (k)Degree of ith node ki= number of nodes
linking with it
Degree of a node (k)kin= number of nodes linking in
kout= number of nodes linking out
Clustering Coefficient (CC)
Ci=2Ei/ki(ki-1)=2/9
ith node has ki neighbors linking with it
Ei is the actual number of links between ki neighbors
maximal number of links between ki neighbors is ki(ki-1)/2
Average shortest path length
jil
lNN
l
ij
jiij
and nodebetween length path shortest theis
)1(
2
Shortest path length
)pathshortest (2l
All pair shortest path Algorithm• Floyd Algorithm: d(k)
ij: shortest path between i,j with intermediate node’s label not higher than k
j
k
i
d(k)ij=min(d(k-1)
ij,d(k-1)ik+d(k-1)
kj)
d(k-1)ik d(k-1)
kj
d(k-1)ij
Pseudocode
• D(0)ij=Aij=adjacency matrix
• For k=1 -> N• for i=1 -> N• for j=1 -> N• D(k)
ij=min(D(k-1)ij,D(k-1)
ik+D(k-1)kj)
• Return D
Small world network
Three ways to generate networks
Random networks
• Paul Erdös & Alfréd Rényi model : Hugarian mathematicians in 1959
Paul Erdös Alfréd Rényi
1913~1996 1921~1970
Randomly connect two nodes with probability P=1/5 linking probability
N=10 number of nodes
<K>=NP=2 average degreeProbability distribution of degree k
Erdös & Rényi model
Poisson distribution
Exponential
NetworkNPkk
ke
ppCkkPk
kN
kNkNki
!
)1()( 11
Scale free network Albert-László Barabási
“Statistical mechanics of complex networks” Review of Modern Physics 74, 47-97 (2002)
Scale free Network• A new node is added and deleted randomly t
o and from the network, i.e. N is not fixed
• The new node preferably connects with other node with higher connections with m edges, i.e
P(k)~k-γ
ii
ii k
kkPP )(
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
Scale Free Network
Scale free network
Mean Field Theory
γ = 3
t
k
k
kAkP
t
k i
j j
ii
i
2)(
ii t
tmtk )(
, with initial condition mtk ii )(
)(1)(1)())((
02
2
2
2
2
2
tmk
tm
k
tmtP
k
tmtPktkP ititi
33
2
~12))((
)(
kktm
tm
k
ktkPkP
o
i
A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
degree distribution
Hierarchical Networks
kkP ~)(
kkC ~)(
!~)(
k
ekP
k
Are biological networks random, scale free or
hierarchical?
Degree distribution of PPI
P(k)~k-
1.62Scale free
32617 proteins
11855 interactions
Data from HMS-PCI, Yeast two hybrid, and TAP
data
Degree distribution of metabolic network
a: Archaeoglobus fulgidus
b: E.coli
c: C. elegans
d: Averaged over 43 organisms
Scale free !!!
Hierarchy in biological networks
Metabolic networks Protein networks
What does it mean?
Real Networks Have a Hierarchical Topology
Many highly connected small clusterscombine into
few larger but less connected clusters combine into
even larger and even less connected clusters
The degree of clustering follows:
Biological networks are hierarchical
kkP ~)(
kkC ~)(
Power law degree distribution
Power law clustering coefficient distribution
References
• Albert-László Barabási and Zoltán N. Oltvai,Network Biology: Understanding the Cells's Functional OrganizationNature Reviews Genetics 5, 101-113 (2004).
• O. Mason, and M. Verwoerd, Graph Theory and Networks in Biology, IET Syst. Biol, 1, 89-119, (2007).