School of InformationUniversity of Michigan

SI 614Network subgraphs (motifs)

Biological networks

Lecture 11

Instructor: Lada Adamic

Outline

motifs motif detection (software & Pajek) review of network characteristics

used to compare model with real-world network one more: degree assortativity

biological networks types characteristics hierarchical modularity model

Schematic view of network motif detection

Motifs can overlap in the network

http://mavisto.ipk-gatersleben.de/frequency_concepts.html

motif matches in the target graph

motif to be foundgraph

Examples of network motifs (3 nodes)

Feed forward loop Found in neural networks Seems to be used to neutralize

“biological noise”

Single-Input Module e.g. gene control networks

All 3 node motifs

Examples of network motifs (4 nodes)

Parallel paths Found in neural networks Food webs

W

X Y

Z

4 node subgraphs (computational expense increases with the size of the graph!)

Network motif detection

Some motifs will occur more often in real world networks than random networks

Technique: construct many random graphs with the same number of nodes

and edges (same node degree distribution?) count the number of motifs in those graphs calculate the Z score: the probability that the given number of

motifs in the real world network could have occurred by chance

Software available: http://www.weizmann.ac.il/mcb/UriAlon/

What the Z score means

mean number of times the motifappeared in the random graph

# of times motif

appeared in random graph

zx=x - x

x

standard deviationthe probability observing a Z

score of 2 is 0.02275

In the context of motifs:

Z > 0, motif occurs more often

than for random graphs

Z < 0, motif occurs less often

than in random graphs

|Z| > 1.65, only a 5% chance of

random occurence

Finding classes on graphs based on their motif “profiles”

Finding motifs (cliques and subgraphs) in Pajek

Create a second network that is the subgraph you are looking for e.g. an undirected triad

*Vertices 3

1 "v1"

2 "v2"

3 "v3"

*Arcs

*Edges

2 3 1

1 2 1

1 3 1

finding motifs with Pajek

Use the two drop down menus in the ‘networks’ list to specify two networks:

Then run Nets>Fragment (1 in 2)>Find under Net>Fragment (1 in 2)>Options

can select ‘induced’ subnetwork containing only overlapping fragments

in

finding motifs with Pajek (cont’d)

Now we have just the triads:

Creates a hierarchy object with the membership of each triad listed

Comparing network models with the real thing

check for structural similarity between the artificial network (the model) and the real world network degree distribution assortativity

do high degree nodes connect to other high degree nodes? average shortest path

dependence on size of network clustering coefficient

compare to a randomized version conserving node degree dependence on node degree dependence on size of network

motif profile

How can we randomize a network whilepreserving the degree distribution?

Stub reconnection algorithm (M. E. Newman, et al, 2001, also known in mathematical literature since 1960s)

Break every edge in two “edge stubs”AB to A B

Randomly reconnect stubs Problems:

Leads to multiple edges Cannot be modified to preserve additional topological

properties

Local rewiring algorithm

Randomly select and rewire two edges (Maslov, Sneppen, 2002, also known in mathematical literature since 1960s)

Repeat many times Preserves both the number of upstream and downstream

neighbors of each node

Conserving additional low-level topological properties

In addition to ki one may also conserve: The exact numbers of loops or other motifs The size and numbers of components: Internet – all nodes have

to be connected to each other

Metropolis algorithm: two edges are rewired based on E=(Nactual-Ndesired)2/Ndesired

If E0 rewiring step is always accepted If E>0 rewiring step is accepted with p=exp(-E/T)

Assortativity

Social networks are assortative: the gregarious people associate with other gregarious people the loners associate with other loners

The Internet is disassortative:

Assortative:

hubs connect to hubs

Random Disassortative:

hubs are in the

periphery

Correlation profile of a network

Detects preferences in linking of nodes to each other based on their connectivity

Measure N(k0,k1) – the number of edges between nodes with connectivities k0 and k1

Compare it to Nr(k0,k1) – the same property in a properly randomized network

Very noise-tolerant with respect to both false positives and negatives

Correlation profiles give complex networks unique identities

InternetProtein interactions

slide by Sergei Maslov

2D picture

Correlation profiles give complex networks unique identities

InternetProtein interactions

Sergei Maslov: 2D histogram

Correlation profiles -cont’d

Pastor-Satorras and Vespignani: 2D plot

average degree

of the node’s neighbors

degree of node

Correlation profiles -cont’d

Newman: single number

-0.189

internet degree correlation coefficient

The Pearson correlation coefficient of nodes on each

side on an edge

Other examples of assortative mixing

Assortativity is not limited to degree-degree correlations other attributes social networks: race, income, gender, age food webs: herbivores, carnivores internet: high level connectivity providers, ISPs, consumers

Tendency of like individuals to associate: ‘homophily’ Scott Feld paper

Biological networks

In biological systems nodes and edges can represent different things nodes

protein, gene, chemical edges

mass transfer, regulation

Can construct bipartite or tripartite networks: e.g. genes and proteins

GENOME

PROTEOME

METABOLISM

bio-chemical reactions

protein-protein interactions

protein-gene interactions

slide after Reka Albert

Cellular processes form networks on many levels

metabolic reaction networks (tri-partite)

slide after Reka Albert

Node types: metabolites (substrates or products), open rectangles metabolite-enzyme complexes (black rectangles) enzymes (open ovals)

Edges substrate to complex or complex to product symmetrical edges

regulatory networks

nodes: genes, proteins

edges: translation

regulation: activating

inhibitingslide after Reka Albert

the yeast two-hybrid method

Activation and binding domains are separated and each attached to a different protein

If the proteins interact, the two domains will be brought together and activate the transcription of a reporter gene

Can do simultaneous genome-wide experiments

slide after Reka Albert

Resulting interaction network

slide after Reka Albert

Properties and problems of resulting networks

Properties giant component exists power law distribution with an

exponential cutoff longer path length than

randomized higher incidence of short loops

than randomized

Problems false positives false negatives only 20% overlap between

different studies

Implications

Robustness resilient to random breakdowns mutations in hubs can be

deadly

Evolution most connected hubs

conserved across organisms (important)

gene duplication hypothesis new gene still has same output

protein, but no selection pressure because the original gene is still present. So some interactions can be added or dropped

leads to scale free topology

Metabolic networks: how to represent them

Can consider the one-mode projection of substrate interactions (undirected)

slide after Reka Albert

Metabolic networks are scale-free

In the bi-partite graph: the probability that

a given substrate participates in k reactions is k

indegree: = 2.2

outdegree: = 2.2

(a) A. fulgidus (Archae) (b) E. coli (Bacterium) (c) C. elegans (Eukaryote), (d) averaged over 43 organisms

Modularity

No modularity

Modularity

Hierarchical modularity

E. Ravasz et al., Science 297, 1551 -1555 (2002) (Pajek!)

How do we know that metabolic networks are modular?

clustering decreases with degree as C(k)~ k-1

randomized networks (which preserve the power law degree distribution) have a clustering coefficient independent of degree

How do we know that metabolic networks are modular?

clustering coefficient is the same across metabolic networks in different species with the same substrate

corresponding randomized scale free network:C(N) ~ N-0.75 (simulation, no analytical result)

bacteria

archaea (extreme-environment single cell organisms)

eukaryotes (plants, animals, fungi, protists)

scale free network of the same size

review: what would the clustering coefficient of a random network be

assume average degree of node is k probability of one neighbor linking to another is ~ k/N scales as N-1

Constructing a hierarchically modular network

RSMOB model Start from a fully

connected cluster of nodes

Create 4 identical replicas of the cluster, linking the outside nodes of the replicas to the center node of the original (N = 25 nodes)

This process can repeated indefinitely

(initial number of nodes can be different than 5)

Properties of the hierarchically modular model

RSMOB model Power law exponent = 2.26 (in agreement with real

world metabolic networks) C ≈ 0.6, independent of network size (also

comparable with observed real-world values) C(k) ≈ k-1, as in real world network

How to test for hierarchically arranged modules in real world networks perform hierarchical clustering on the topological overlap

map (we’ll cover hierarchical clustering in a few weeks…) can be done with Pajek

Topological overlap

A: Network consisting of nested modules B: Topological overlap matrix

hierarchical

clustering

Hubs may act within a module, or connect modules

Party hub: simultaneous interactions tends to be within the same

module

Date hub: sequential interactions connect different modules

Han et al, Nature 443, 88 (2004)

slide after Reka Albert

some matching motifs frequently overlap (e.g. feed forward loop)

Zhang et al, J. Biol 4, 6 (2005)