Analysis of biological networks Part III Shalev Itzkovitz Uri Alon’s group
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Transcript of Analysis of biological networks Part III Shalev Itzkovitz Uri Alon’s group
Analysis of biological networksAnalysis of biological networks
Part IIIPart III
Shalev ItzkovitzShalev Itzkovitz Uri Alon’s groupUri Alon’s group July 2005July 2005
What is a suitable random ensemble?What is a suitable random ensemble?
Subgraphs which occur many times in the networks, significantly more than in a
suitable random ensemble.
Reminder - Network motifs definitionReminder - Network motifs definition
Types of random ensemblesTypes of random ensembles
Erdos Networks
For a given network with N nodes and E edges define : p=E/N2, the
probability of an edge existing between any one of the N2 possible
directed edges.
Erdos & Renyi, 1960
N
Ek
UMAN ensemble
a canonical version. All networks have the same numbers of Mutual, Antisymetric and Null edges as the real network, Uniformly distributed.
Used in sociology, analytically solvable for subgraph distributions.
Holland & Leinhardt, american journal of sociology 1970
Antisymetric edge
Mutual edge
The configuration model
All networks preserve the same degree sequence of the real network
All networks preserve the same degree sequence of the real network, and multiple edges between two nodes are not allowed
The configuration model+no multiple edges
Bollobas, Random graphs 1985, Molloy & Reed, Random structures and algorithms 1995, Chung et.al. PNAS 1999
Maslov & Sneppen, science 2002, Newman Phys. Rev.Lett. 2002, Milo science 2002
Stubs method for generating random Stubs method for generating random networksnetworks
Problem – multiple edges between nodesSolution – “Go with the winner” algorithm
A
B D
C A
B D
C
Markov chain Monte-Carlo algorithmMarkov chain Monte-Carlo algorithm
Uniform sampling issues : ergodicity, detailed balance, mixing time
Random networks which do not preserve Random networks which do not preserve the degree sequence are not suitablethe degree sequence are not suitable
Network hub
This v-shaped subgraph appears many timeswould be a network motif when comparing with Erdos networks
It is important to filter out subgraphs which appearIt is important to filter out subgraphs which appearin high numbers only due to the degree sequencein high numbers only due to the degree sequence
More stringent ensembles
•Preserve the number of all subgraphs of sizes 3,4..,n-1 when
counting n-node subgraphs [Milo 2002]
•Can be combined with the markov chain algorithm by using
simulated annealing
•Filters out subgraphs which appear many times only because
they contain significant smaller subgraphs
Will appear many times if Is a motif
A
B D
C A
B D
C
Simulated annealing algorithmSimulated annealing algorithm
•Randomize network by making X switches
•Make switches with a metropolis probability exp(-E/T)
•E is the deviation of any characteristic of the real network you
want to preserve (# 3-node subgraphs, clustering sequence etc)
Erdos Networks
UMAN
Degree distribution
Degree sequence
Degreesequence+triads
Subgraphs in Erdos networks: Subgraphs in Erdos networks: exact solutionexact solution
122
N
k
N
kN
N
Ep
NN nodes (8) nodes (8)
EE edges (8) edges (8)
<<kk> > mean degree (1)
Subgraphs in Erdos networks: Subgraphs in Erdos networks: exact solutionexact solution
3
NPossible tripletsPossible triplets
3p Probability of forming a ffl Probability of forming a ffl given specific 3 nodesgiven specific 3 nodes
3333
3
33 ~3
kNkN
kNp
N
# nodes# nodes # edges# edges
Number of ffls does not Number of ffls does not change with network sizechange with network size!!!!!!
• The expectancy of a subgraph with n nodes and g edges is analytically solvable. Scales as N(n-g)
gng N
N
k
n
N ~))((
N
kp
Subgraphs on Erdos NetworksSubgraphs on Erdos Networks
n=3g=3
Select n nodes
place g edges
n=3, g=2, G~O(N3-2)=O(N) n=3, g=3, G~N3-3=O(1)
n=3, g=4, G~N3-4=O(N-1) n=3, g=6, G~N3-6=O(N-3)
Subgraph scaling familiesSubgraph scaling families
P(K)~K-
Natural networks often have scale-freeNatural networks often have scale-free
outdegreeoutdegree
Erdos networkErdos network Scale free networkScale free network
P(k
)P(K)~K
-
2<<3
P(K)~K-
=3
=2
Scale-free networks have hubsScale-free networks have hubs
Edge probability in the configuration modelEdge probability in the configuration model
edgesnetwork #
indegree nodeP(edge)
high edge probabilityhigh edge probability low edge probabilitylow edge probability
Edge probability in the configuration modelEdge probability in the configuration model
edgesnetwork #
indegree) (node2*outdegree) (node1P(edge)
1122
10
2*2P
Networks with E (~N) edges, and arbitrary indegree (Ri ) and outdegree (Ki ) sequences.
Subgraphs in networks that preserve degree Subgraphs in networks that preserve degree sequence: approximate solutionsequence: approximate solution
K1, R1
K2, R2
K3, R3
E
R21KP(edge1) E
R31 )1(K)edge1P(edge2
E
R )1(K,2)edge1P(edge3 32
E
RP(edge)
Subgraph scaling depends on exact Subgraph scaling depends on exact topologytopology
3332211 )1(**)1(KK
)P(subgraphE
RRRK
3
)1()1(~#
K
RRKRKKffl
Subgraph topology effectsSubgraph topology effectsIts expected numbersIts expected numbers
Subgraph scaling depends on exact Subgraph scaling depends on exact topology – as opposed to Erdos networkstopology – as opposed to Erdos networks
Example
O(1)
O(1) O(1)
Erdos Networks
γO( N)
Directed networks with power-law out-degree, compact in-degree :
P(K)~K-
Scale-free Networks ( =2.5)
2K
K
Real networks
O( 1)
O( N)>
Itzkovitz et. al., PRE 2003
Network motifs – a new extensive variableNetwork motifs – a new extensive variable
Milo et. al., science 2002
Global constraints on network structure Global constraints on network structure can create network motifscan create network motifs
• Subgraphs which appear many times in a network (more than random)
• Might stem from evolutionary constraints of selection for some function, or be a result of other global constraints
• Degree sequence is a global constraint with a profound effect on subgraph content
• Are there other global constraints which might result in network motifs?
How do geometrical constraints How do geometrical constraints influence the local structure?influence the local structure?
Examples of geometrically constrained Examples of geometrically constrained systemssystems
• Transportation networks (highways, trains)
• Internet layout
• Neuronal networks, brain layout
• Abstract spaces (www, social, gene-array data)
The neuron network of The neuron network of C. elegansC. elegans
"The abundance of triangular connections in the nervous system of C. elegans may thus simply be a consequence of the high levels of connectivity that are present within neighbourhoods“ (White et. al.)
The geometric modelThe geometric model
• N nodes arranged on d-dimensional lattice
• Connections made only to neighbors within range R
Erdos networks – every node can Erdos networks – every node can connect to every other nodeconnect to every other node
Probability of closing triangles - small
Geometric networks – every node can Geometric networks – every node can connect only to its neighborhoodconnect only to its neighborhood
Probability of closing triangles - large
All subgraphs in geometric networks All subgraphs in geometric networks scale as network sizescale as network size
N/Rd ‘sub-networks’, each one an Erdos network of size Rd
All subgraphs scale as network size
Erdos sub-network
All subgraphs scale as NAll subgraphs scale as N
The Erdos scaling laws determine the The Erdos scaling laws determine the network motifsnetwork motifs
All subgraphs with more edges than All subgraphs with more edges than nodes are motifsnodes are motifs
Motifs – scale as N in geometric networksConstant number in random networks
Not motifs – scale as N in both random and real networks
Feedbacks in neuronal network are much Feedbacks in neuronal network are much more rare than expected from geometrymore rare than expected from geometry
= 1 : 3 = 1 : 3
= 0 : 40 = 0 : 40
geometric model
C elegans neuronal network
Imposing a field changes subgraph ratiosImposing a field changes subgraph ratios
inputsoutputs
Itzkovitz et. al., PRE 2005
A simple model of geometry + directional A simple model of geometry + directional bias is not enoughbias is not enough
abundant in C elegans
Mutual edges rare in geometric networks + directional bias
The mapping of network models and The mapping of network models and resulting network motifs is not a 1-1 resulting network motifs is not a 1-1
mappingmapping
Motif set 1Motif set 1 Motif set 2Motif set 2 Motif set 3Motif set 3
Model 1Model 1 Model 2Model 2 Model 3Model 3 Model 4Model 4
X
conclusionsconclusions
•Biological networks are highly optimized systems aimed at information processing computations.
•These networks contain network motifs – subgraphs that appear significantly more than in suitable random networks.
•The hypothesized functional advantage of each network motif can be tested experimentally.
•Network motifs may be selected modules of information processing, or results of global network constraints.
•The network motif approach can be used to reverse-engineer complex biological networks, and unravel their basic computational building blocks.
AcknowledgmentsAcknowledgments
Ron MiloRon MiloNadav KashtanNadav Kashtan
Uri AlonUri Alon
More information :More information :http://www.weizmann.ac.il/mcb/UriAlon/
PapersPapers
mfinder – network motif detection softwaremfinder – network motif detection software
Collection of complex networksCollection of complex networks