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