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Prof. Yechiam Yemini (YY)

Computer Science DepartmentColumbia University

Chapter 8: The Topology of BiologicalNetworks

8.1 Introduction

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Overview A gallery of networks Scale-free network models

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A Gallery of Networks

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Introduction Network abstractions

Node: biological object Edge: interaction between nodes

Regulatory networks Node: genes; edge: regulatory interaction

Metabolic networks Node: metabolite; edge: reaction

Protein networks Node: protein; edge: interaction Node: module; edge: interaction Node: complex; edge: sharing a protein Node: residue; edge: folding neighbors

What Can Network Abstractions Teach Us About Biological Systems?

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The E.Coli Regulatory NetworkNode= TFsEdge= Regulatory interaction

Hierarchical structure and modules in theEscherichia coli regulatory network.

Hong-Wu Ma , Jan Buer, and An-Ping Zeng

http://www.biomedcentral.com/1471-2105/5/199

MODULAR VIEW

HIERARCHICAL VIEW

UNORGANIZED VIEW

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Regulatory Network Of E.ColiE.coli: 105 TFs affect 749 genes7 TFs regulate >0.5 genesConnectivity distribution

Egress: follows a power-law Ingress: follows exponential (Shen-Orr).

Martinez-Antonio, Collado-Vides,Curr Opin Microbiol 6, 482 (2003)

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Yeast Regulation

The colour scheme depicts functional category: orange, mitotic cell cycle; pink, budding and filamentformation; green, amino acid metabolism; yellow, nitrogen and sulphur utilization; blue, C-compoundand carbohydrate utilization; red, TFs; grey, unspecific or several functional categories.

http://www.biochemj.org/bj/381/0001/bj3810001.htm

Node= TFsEdge= Regulatory interaction Charting gene regulatory networks:

strategies, challenges and perspectivesGong-Hong WEI, De-Pei LIU1 and Chih-Chuan LIANG ; Biochem J. 2004 (381)

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Yeast Regulatory NetworkSergei Maslov

http://www.cmth.bnl.gov/~maslov/rockefeller_2002_networks.ppt

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Metabolic NetworkNode= MetabolitesEdge= Reaction

Ravasz et al…Science Vol 297, 2002

E-Coli

Human

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Signaling Networks

http://www.cs.tau.ac.il/~spike/ www.bioscience.org/1998/v3/d/malumbre/fig2.jpg

MAPK signaling pathway

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Yeast P2P Interaction Network

http://www.macdevcenter.com/pub/a/mac/2004/08/20/bioinformatics.html

http://www.imb-jena.de/tsb/yeast.html

Node= proteinsEdge= interaction

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Yeast P2P Domain Interaction Network

(A) Yeast SH3 domain protein-protein network; proteins are colored according to their k-core value (6-core = black, 5-core =cyan, 4-core = blue, 3-core = red, 2- core = green, 1-core = yellow), identifying subnets in which each protein has at least kinteractions. By definition, lower core numbers encompass all higher core numbers (e.g. 4-core subgraph includes 4-core, 5-core and 6-core). The 6-core subgraph is highlighted in red and depicted in (B).

http://www.utoronto.ca/boonelab/proteomics.htmNode= domainEdge= interaction

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A Network of Protein Complexes

Red, cell cycle;dark green, signalling;dark blue, transcription, DNA maintenance, chromatin structure;pink, protein and RNA transport;orange, RNA metabolism; light green, protein synthesis;brown, cell polarity and structure;violet, intermediate and energy metabolism;light blue, membrane biogenesis and traffic.

Lowe panel is an example of a complex (yeast TAP-C212) linked to two other complexes(yeast TAP-C77 and TAP-C110) by shared components.

http://www.genomenewsnetwork.org/articles/01_02/Yeast_proteins_image1.shtml

Node=complexEdge=shared proteinsColor=role

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Key Question:Are Biological Networks Random?

Or do they reflect hidden organizational principles?

First answer: biological network are random Are organized through scale-free random evolution Barabasi group: Jeong et al. Nature 407, 651-654 (2000).

Second answer: regulatory networks are not randomAre organized from statistically-significant motifsUri Alon’s group: Shen-Orr et al. Nature Gen. 31, 64 (2002)

Both can be correctContradiction is only seeming

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Statistical Topology Features

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Random Networks (Erdos Renyi, 1959)

G(n,p) a graph on n nodes where an edge has probability p Toss a coin with probability p to select an edge Average degree d=p(n-1)~pn Probability of k edges (m=n(n-1)/2): p(k)= pk(1-p)[m-k] ~ (dk/k!)exp(-d)

G(n,p(n)) has a property F, if p(G(n,p(n)∈F)1 when n∞

Main result: many properties F have threshold behavior There exists p*(n) such that

if p(n)/p*(n)>1 p(G(n,p(n)∈F)1 and if p(n)/p*(n)<1 then p(G(n,p(n)∈F)0

Example: F=connectivity p*(n)=(1/n)ln(n) As p(n) increases towards p*(n) the graph grows a giant component

mk

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Topology Measures of ER Randomness

C(k)=fraction of clique filled

Degree distribution Clustering Path length

Poisson

p(k)~ dk/k!

C=p L~ ln N

L

C

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ER Does Not Model Many Real-World Networks

Watts-Strogatz (98) many real-networks have:(A) high degree of clustering (cliquishness) and(B) short average length (small-world separation)

Network C Crand L NWWW 0.1078 0.00023 3.1 153127

Internet 0.18-0.3 0.001 3.7-3.76 3015-6209

Actors 0.79 0.00027 3.65 225226

Coauthors 0.43 0.00018 5.9 52909

Metabolic 0.32 0.026 2.9 282

Foodweb 0.22 0.06 2.43 134

C. elegan 0.28 0.05 2.65 282

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Watts-Strogatz Small World Networks

L=100 d=49.51 C=0.67 L=14 d=11.1 C=0.63 L=5 d=4.46 C=0.01

Start with a deterministic k-regular ring

Rewire connectionwith probability p

p

Converges to Random network

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But Many Real Networks HavePower-Law Degree Distribution

P(k) = k - γA: actors γ =2.3B: WWW γ =2.67C: power grid γ =4

Faloutsos & Faloutsos 99Internet graph

AS graph: γ =2.1Routers: γ =2.48

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Scale Free Networks Scale free= power-law degree distribution: p(k) = k - γ

Why is a power law “scale free”? If k is scaled by a factor α p(ak)/p(k)=α−γ regardless of k Contrast with ER: p(k)=γk/k! => p(αk)/p(k)=γ(α−1)k(k!/(αk)!)

Topological features of SF nets γ=2 hub-and-spoke topology 2<γ<3 small number of hubs γ>3 network is dispersed

Topology measures: L~ln(lnN)) for 2<γ<3 C(k) constant P(k) ~k-3

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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Scale Free vs. Random

Poisson distribution

Exponential Network

Power-law distribution

Scale-free Network

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Scale Free Network Examples (Barabasi 01)

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How Do Scale Free Networks Rise?

Evolution through preferential attachment: A new node connects to node i with probability:

where ki is the degree of i jj

ii k

kkΣ

=π )(

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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Global Topology Features

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Characterizing Metabolic Networks

Jeong et al, "The large-scale organization of metabolic networks", Nature 407 651 (2000)

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Metabolic Nets Have Power Law Distribution

H. Jeong et al, Nature, 407 651 (2000)

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Clustering

Metabolic networks Protein networks

E. Ravasz et al., Science, 2002

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The P53 Tumor Supressor NetworkVogelstein et al, Nature 2000

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node failurefc

0 1Fraction of removed nodes, f

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Robustness: SF Nets Are Robust WRT Failures

Maintain connectivity and topological features through lossS- fraction of nodes in largest connected component

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S

0 1fFailures

Albert, Jeong, Barabási Nature 406, 378 (2000)

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How About Targeted Attacks? SF networks are sensitive to attacks on hubs

1

S

0 1ffc

Disease analysis; drug design…

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Robustness of The Yeast Protein Network

Node Failure:Red: lethalGreen: robustYellow: unknown

H. Jeong et al., Nature, 2001

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Robustness of The Yeast Protein Network

Highly connected proteins are more essential (lethal)...

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Biological Networks

Evolution through duplication may explain γ<2

Approx. Exponent γNetwork

1.6Gene functional interactions1.4-1.7Yeast Gene Expression Net1.7, 2.2E.coli Metabolic Net

1.5, 1.6, 1.7, 2.5Yeast Protein-Protein NetBIOLOGICAL

2.1-2.3Phone calls4Power-grid

2.3Actors3Citations

2.1 (in), 2.5 (out)InternetNON-BIOLOGICAL

Chung, Dewey, Lu, Galas, D.J., Journal of Computational Biology(2003)

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Gene Duplication Networks

0.001

0.01

0.1

1

1 10 100

log k

log

P(k

)

Scale Free + Small world

Pastor-Satorras, Smith & R. V. Sole, “Evolving protein interaction networks through geneduplication”, Santa Fe Institute Working Paper 02-02-008, 2002

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Is The Metabolic Network of E-Coli “Small”?

Masanori Arita. PNAS 101 (6): 1543-1547

Fell, D. A. & Wagner, A. (2000) Nat. Biotechnol. 18.Wagner, A. & Fell, D. A. (2001) Proc. R. Soc. London Ser. B 268,.

Ma, H.-W. & Zeng, A.-P. (2003) Bioinformatics 19, 270–277.

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Is The Metabolic Network “Small”?

Filled bars: the direction ofreactions is considered, AL = 8.4

Open bars: all reactions areconsidered reversible, AL = 8.0

L≈ 8, much larger than that of arandom graph

Masanori Arita. PNAS 101 (6): 1543-1547

Considered more detailed structural modelFocus on carbon metabolism

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Yeast Regulatory Network

Luscombe NM, Babu MM, Yu H, Snyder M, Teichmann SA & Gerstein M (2004)Genomic analysis of regulatory network dynamics

reveals large topological changes.Nature 431: 308-312.

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Very complex network 3420 genes, 142 TFs 7074 regulatory interactions

Simplify using graph-theoretic statistics: Global topological measures Local network motifs

Target Genes

Transcription Factors

Comprehensive Dataset Available

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Global Topology Measures Connectivity:

Ingress degree: 2.1 – each gene is regulated by ~ 2TFsEgress degree 49.8 – each TF targets ~ 50 genes

Degree distribution: power law (scale free) Clustering coefficient: 0.11 (low local density)

Clustering coefficient

4 neighbours

1 existing link

6 possible links

= 1/6 = 0.17

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Partition Network Into Activity SubnetsActive subnet computations: Start with active genes Compute TFs that influence them Compute closure of TFs that influence current graph

1,385Stress response

1,715DNA damage

1,876Diauxic shift

876Sporulation

437Cell cycle

No. genesCellular Activity

Switching from glucose to lactose

Cell transforms into spores

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Cell cycle Sporulation Diauxic shift DNA damage Stress

Activity Subnets

Binary stateMulti-stage activities

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Do Global Topology Features Vary By Activity?

Literature: Network topologies are perceived to be invariantScale-free, small-world, and clusteredDifferent biological networks and genomes

Random expectation: Sample different size sub-networksfrom complete network and calculate topological measures

path length clustering coefficient outgoing degreeincoming degree

random network size

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Outgoing degree

“Binary conditions” greater connectivity

“Multi-stage conditions” lower connectivity

Binary:Quick, large-scale turnover of genes

Multi-stage:Controlled, ticking

over of genes at different stages

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Incoming degree

“Binary conditions”smaller connectivityless complex TF combinations

“Multi-stage conditions”larger connectivitymore complex TF combinations

BinaryMulti-stage

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Path length

“Binary conditions” shorter path-length “faster”, direct action

“Multi-stage” conditions longer path-length “slower”, indirect action

BinaryMulti-stage

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Clustering coefficient

“Binary conditions”smaller coefficientsless TF-TF inter-regulation

“Multi-stage conditions” larger coefficients more TF-TF inter-regulation

BinaryMulti-stage

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Literature: motif usage is well conserved for regulatory networksacross different organisms [Alon]

Random expectation: sample sub-nets for motif occurrence

single input motif multiple input motif feed-forward loop

random network size

Do Local Topology Features Vary By Activity?

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Final Notes

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Challenges & Opportunities

Improved understanding of network evolutionEvolutionary models, selection…

Modularity Network-sequence relationships Network-structure (folding) relationships Applications: drug-design…