SI 614 Network subgraphs (motifs)

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SI 614Network subgraphs (motifs) Biological
networks
Lecture 11 Instructor Lada Adamic
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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

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Schematic view of network motif detection
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Motifs can overlap in the network
motif to be found
graph
motif matches in the target graph
http//mavisto.ipk-gatersleben.de/frequency_concep
ts.html
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Examples of network motifs (3 nodes)

• Feed forward loop
• Found in neural networks
• Seems to be used to neutralizebiological noise
• Single-Input Module
• e.g. gene control networks

X Y Z
X
a
b
c
d
b
c
d
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All 3 node motifs
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Examples of network motifs (4 nodes)

• Parallel paths
• Found in neural networks
• Food webs

W
X
Y
Z
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4 node subgraphs (computational expense increases
with the size of the graph!)
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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/

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What the Z score means
m mean number of times the motifappeared in
the random graph
the probability observing a Z score of 2 is
0.02275 In the context of motifs Z gt 0, motif
occurs more often than for random graphs Z lt 0,
motif occurs less often than in random
graphs Z gt 1.65, only a 5 chance of random
occurence
s standard deviation
of times motif appeared in random graph
x - mx
zx

sx
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Finding classes on graphs based on their motif
profiles
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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

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finding motifs with Pajek

• Use the two drop down menus in the networks
  list to specify two networks
• Then run NetsgtFragment (1 in 2)gtFind
• under NetgtFragment (1 in 2)gtOptions
• can select induced subnetwork containing only
  overlapping fragments

in
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finding motifs with Pajek (contd)

• Now we have just the triads
• Creates a hierarchy object with the membership of
  each triad listed

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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

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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 stubsA??B to A?
  ?B
• Randomly reconnect stubs
• Problems
• Leads to multiple edges
• Cannot be modified to preserve additional
  topological properties

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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

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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 ?E?0 rewiring step is always accepted
• If ?Egt0 rewiring step is accepted with
  pexp(-?E/T)

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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
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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

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Correlation profiles give complex networks unique
identities
2D picture
Protein interactions
Internet
slide by Sergei Maslov
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Correlation profiles give complex networks unique
identities
Sergei Maslov 2D histogram
Protein interactions
Internet
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Correlation profiles -contd

• Pastor-Satorras and Vespignani 2D plot

average degree of the nodes neighbors
degree of node
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Correlation profiles -contd

• Newman single number

-0.189
internet degree correlation coefficient The
Pearson correlation coefficient of nodes on
each side on an edge
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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

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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

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GENOME
protein-gene interactions
PROTEOME
protein-protein interactions
METABOLISM
bio-chemical reactions
slide after Reka Albert
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Cellular processes form networks on many levels

• metabolic reaction networks (tri-partite)

• 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

slide after Reka Albert
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regulatory networks
nodes genes, proteins edges translation
regulation activating inhibiting
slide after Reka Albert
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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
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Resulting interaction network
slide after Reka Albert
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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

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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

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Metabolic networks how to represent them

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

slide after Reka Albert
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Metabolic networks are scale-free

• In the bi-partite graph
• the probability that a given substrate
  participates in k reactions is k-a
• indegree a 2.2
• outdegree a 2.2

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

• No modularity
• Modularity
• Hierarchical modularity

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

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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 networkC(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
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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

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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)

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Properties of the hierarchically modular model

• RSMOB model
• Power law exponent g 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 (well cover hierarchical
  clustering in a few weeks)
• can be done with Pajek

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Topological overlap

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

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

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