Clustering of protein networks: Graph theory and terminology Scale-free architecture Modularity Robustness Reading: Barabasi and Oltvai 2004, Milo et al. 2002 Lecturer: Trey Ideker
Clustering of protein networks:
Graph theory and terminologyScale-free architecture
ModularityRobustness
Reading: Barabasi and Oltvai 2004, Milo et al. 2002Lecturer: Trey Ideker
Yeast protein-protein interaction network
What are its network properties?
Graphs
• Graph G=(V,E) is a set of vertices V and edges E
• A subgraph G’ of G is induced by some V’ V and E’ E
• Graph properties:– Node degree– Directed vs. undirected– Loops– Paths– Cyclic vs. acyclic– Simple vs. multigraph– Complete– Connected– Bipartite
Paths
A path is a sequence {x1, x2,…, xn} such that (x1,x2), (x2,x3), …, (xn-1,xn) are edges of the graph.
A closed path xn=x1 on a graph is called a graph cycle or circuit.
Network measures
• Degree ki
The number of edges involving node i
• Degree distribution P(k)The probability (frequency) of nodes of degree k
• Mean path lengthThe avg. shortest path between all node pairs
• Network Diameter“The longest shortest path”
How do the above definitions differ between undirected and directed networks?
Clustering coefficient
12
2
kk
nkn
C III
The combination “k choose 2”
# edges between node I’s neighbors
# of neighbors of I
The density of the network surrounding node I, characterized as the number of triangles through I.Related to network modularity
C(k) = avg. clustering coefficient for nodes of degree k
Directionality and Degree
What is the clustering coefficient of A in either case?
WHAT DOES SCALE FREE
REALLY MEAN, ANYWAY?
P(k) is probability of each degree k
For scale free: P(k) ~ k
What happens for
small vs. large ?
Generating random networks• Erdos-Renyi
Start with N nodes and connect each pair with equal probability p
• Scale-freeAdd nodes incrementally. New nodes connect to each existing node I with probability proportional to its degree:
J
J
I
k
k
Scale-free networks have small avg. path lengths ~ log (log N)– this is called the ‘small world’ effect
How do scale-free networks arise in evolution?
Both are well-explained by gene duplication. When a protein duplicates, it initially retains all of its previous interactions. This process drives network hubs to get even bigger.
Due to 2 basic mechanisms:
(1)Network Growth
(2)Preferential attachment
Neither network produces modular
structure
C(k) is avg. cluster coefficient of each degree k
Hierarchical networks
This class of random networks are generated based on replicating a four node “module”.
The amazing result from that paper
% E
ssen
tial
P(k
)
k k
Robustness
• Complex systems, from the cell to the Internet, can be amazingly resilient to component failure
• Network topology plays an important role in this robustness
• Even if ~80% of nodes fail, the remaining ~20% still maintain network connectivity
• This also leads to attack vulnerability if hubs are selectively targeted
• In yeast, only ~20% of proteins are lethal when deleted, and are 5 times more likely to have degree k>15 than k<5.
Network Motifs (Milo et al.)
• Motifs are “patterns of interconnections occurring in complex networks.”
• That is, connected subgraphs of a particular isomorphic topology
• The approach queries the network for small motifs (e.g., of < 5 nodes) that occur much more frequently than would be expected in random networks
• Significant motifs have been found in a variety of biological networks and, for instance, correspond to feed-forward and feed-back loops that are well known in circuit design and other engineering fields.
• Pioneered by Uri Alon and colleagues
Motif searches in 3 different contexts
All 3-node directed subgraphs
What is the frequency of each in the network?
Outline of the Approach
• Search network to identify all possible n-node connected subgraphs (here n=3 or 4)
• Get # occurrences of each subgraph type
• The significance for each type is determined using permutation testing, in which the above process is repeated for many randomized networks (preserving node degrees– why?)
• Use random distributions to compute a p-value for each subgraph type. The “network motifs” are subgraphs with p < 0.001
Schematic view of network motif detection
Networks are randomized preserving node degree
Concentration of feedforward motif:
Mean+/-SD of 400 subnetworks
(Num. appearances of motif divided byall 3 node connected subgraphs)
Transcriptional network results
Neural networks
Food webs
World Wide Web
Electronic circuits
Interesting questions
• Which networks have motifs in common?• Which networks have completely distinct motifs versus
the others?• Does this tell us anything about the design constraints
on each network?• E.g., the feedforward loop may function to activate
output only if the input signal is persistent (i.e., reject noisy or transient signals) and to allow rapid deactivation when the input turns off
• E.g., food webs evolve to allow flow of energy from top to bottom (?!**!???), whereas transcriptional networks evolve to process information