Discovery and analysis of biochemical subnetwork hierarchies October 6, 2003 Petter Holme Department of Physics, Ume˚ a University, Ume˚ a, Sweden NORDITA, Copenhagen, Denmark Mikael Huss SANS, NADA, Royal Institute of Technology, Stockholm, Sweden
B IOCHEMICAL NETWORKS : T HE CLASSICAL VIEW http://www.tp.umu.se/ ∼ holme/ 1 Ume˚ a University, Sweden
B IOCHEMICAL PATHWAYS AS GRAPHS metabolic pathways of Borrelia burgdorferi (a bacterium) http://www.tp.umu.se/ ∼ holme/ 2 Ume˚ a University, Sweden
M OTIVATION Complexity Our work Even E. coli has a metabolism involving Earlier algorithms have been based on over 850 substances and 1500 reactions ⇒ local algorithms that may miss some large- scale features. The coarsest level of description—the Not much known about how the large graph representation—is needed, at least scale subnetwork ordering looks like. Can as a complement. the network easily be decomposed into One would like to decompose the graph autonomous subnetworks? How inde- into functional subunits. Both for con- pendent are the modules? Is it useful to ceptual and analytical purposes. talk about modules at all? The basic assumption If we find a subnetwork that is well-connected within, and sparsely connected to the out- side, then it is likely to be a relatively au- tonomously functioning subnetwork. http://www.tp.umu.se/ ∼ holme/ 3 Ume˚ a University, Sweden
B IOCHEMICAL NETWORKS AS DIRECTED BIPARTITE NETWORKS C A D B The reaction A + B ↔ C + D in a directed bipartite representation: Two types of vertices, representing substrates and chemical reactions. Arcs (arrows) between different types of vertices We denote the set of chemical substances by S and the set of reaction vertices by R . http://www.tp.umu.se/ ∼ holme/ 4 Ume˚ a University, Sweden
T HE CLUSTER DETECTION ALGORITHM Based on: M. Girvan & M. Newman, PNAS 99 (2002), pp. 7821-7826. Presented in: P. Holme, M. Huss, and H. Jeong, Bioinformatics 19 (2003), pp. 532-538. The idea Recursively delete reactions situated between densely connected regions. 1. Calculate the effective be- tweenness c B ( r ) for all reac- tion vertices. 2. Remove the reaction ver- tex with highest effective betweenness and all its in- and out-going links. 3. Save information about the current state of the net- work. http://www.tp.umu.se/ ∼ holme/ 5 Ume˚ a University, Sweden
The cluster detection algorithm (continued) Let C B be the betweenness of r with respect to the substance-vertices. C B ( r ) = ∑ σ ss ′ ( r ) s ∈ S ∑ , (1) σ ss ′ s ′ ∈ S \{ s } where σ ss ′ ( r ) is the number of shortest paths between s and s ′ that passes through r , and σ ss ′ is the total number of shortest paths between s and s ′ . The reactions we delete recursively are the one having the highest effective betweenness : c B ( r ) = C B ( r ) / k in ( r ) (2) where k in ( r ) is the in-degree (# of substrates) of the reaction r . This rescaling is sensible since all substrates needs to be present for a reaction to occur. http://www.tp.umu.se/ ∼ holme/ 6 Ume˚ a University, Sweden
S UBNETWORK HIERARCHIES 0 h 0 max h S 2 h 0 ( ) S 1 ( h 0 ) h The substrates are at the base of the tree. If a horizontal line is drawn across the tree, the vertices below are connected at that particular level of the hierarchy. Clusters that are isolated high in the hierarchy (close to the bottom of the tree) are more entangled in other pathways. http://www.tp.umu.se/ ∼ holme/ 7 Ume˚ a University, Sweden
Subnetwork hierarchies (continued) (a) (b) (a) Clusters that get isolated at the same level are more highly wired within, than to its surrounding (and therefore a candidate to a functional module). (b) Vertices that becomes isolated at the same level forms an outer shell of the cluster in question. http://www.tp.umu.se/ ∼ holme/ 8 Ume˚ a University, Sweden
E XAMPLES : Treponema pallidum (a) (b) http://www.tp.umu.se/ ∼ holme/ 9 Ume˚ a University, Sweden
L ARGE SCALE SHAPE OF THE TREES T. pallidum M. pneumoniae C. elegans h h h 0 10 20 30 40 50 0 10 20 30 0 40 80 120 160 1 1 metabolic network 0.8 0.8 We test 43 organ- S 2 S 1 / S 2 S 2 S 1 / S 2 isms of the WIT 0.6 0.6 ~ database. S 1 , , ~ ~ S ~ 2 0.4 0.4 , , S 1 / S 2 S 1 S 1 ~ ~ 0.2 0.2 0 0 whole cellular network S 1 size of the biggest cluster. 0.8 0.8 S 2 S 1 / S 2 S 2 S 1 / S 2 0.6 0.6 S 2 size of the sec- , , ~ ond biggest cluster. ~ , 0.4 0.4 S 1 , ~ S 1 ~ 0.2 0.2 0 0 0 20 40 60 80 0 20 40 60 0 50 100 150 200 250 h h h http://www.tp.umu.se/ ∼ holme/ 10 Ume˚ a University, Sweden
http://www.tp.umu.se/ ∼ holme/ Large scale shape of the trees h 1/2 / h max 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 A. fulgidus archae A. pernix M. thermoautotrophicum M. jannaschii P. furiosus P. horikoshii h 1/2 S 2 max A. aeolicus A. actinomycetemcomitans / h B. burgdorferi / S max B. subtilis 1 C. acetobutylicum max C. jejuni C. tepidum C. pneumoniae metabolic C. trachomati Synechocystis sp. D. radiodurans E. coli E. faecalis H. influenzae bacteria H. pylori M. bovis 11 M. genitalium M. leprae M. pneumoniae M. tuberculosis N. gonorrhoeae N. meningitidi P. aeruginosa S 2 h 1/2 P. gingivalis max S. pneumoniae / h R. capsulatus / R. prowazekii S max S. pyogenes 1 max M. maritima T. pallidum S. typhi whole cell Y. pestis eukaryotes C. elegans Ume˚ E. nidulans S. cerevisiae a University, Sweden O. sativa A. thaliana 0 0.05 0.1 0.15 0.2 (continued) max max S 2 / S 1
C RITERIA FOR IDENTIFYING SUBNETWORKS F. Radicchi et al. , preprint 2003 (http://arxiv.org/abs/cond-mat/0309488/): If, during the iterations of the GN algorithm, an isolated vertex set S ′ ⊂ S fulfills the following criterion it is said to be a weak community if: K in ( s ) > ∑ ∑ K out ( s ) , (3) s ∈ S ′ s ∈ S ′ and a strong community if: K in ( s ) > K out ( s ) for all s ∈ S ′ , (4) where K in ( s ) is the number of s ∈ S that are products of a reaction involving a substrate s ∈ S , and K out ( s ) is the number of s ∈ S \ S ′ that are products of a reaction involving a substrate s ∈ S . http://www.tp.umu.se/ ∼ holme/ 12 Ume˚ a University, Sweden
Criteria for identifying subnetworks (continued) These criteria works well for social networks and electronic circuits, but gives only trivial clusters for biochemical networks. Modified criteria Idea: Networks with some degree of autonomy have loops. To implement this idea, consider the subnetworks with substrate vertex set S ′ that fulfills: L ( S ′ ) � Λ | S ′ | , (5) where L ( S ′ ) is the number of vertices in S ′ that lies on an elementary cycle (a closed non- self-intersecting path) of only vertices in S ′ and length larger than three, | S ′ | is the number of vertices in S ′ , and the parameter Λ ∈ [0 , 1] is the required fraction of loop vertices. 0 . 5 < Λ � 1 gives sensible subnetworks. http://www.tp.umu.se/ ∼ holme/ 13 Ume˚ a University, Sweden
S OME DETECTED SUBNETWORKS Treponema pallidum i ii iii α hypoxanthine −D−ribose 1−phosphate substrate (a) reaction node adenosine link inosine in−flow out−flow H O 2 (b) adenine α −D−ribose 1−pyrophosphate orthophosphate deoxyadenosine ATP, ADP 2−deoxy−D−ribose 1−phosphate NADPH, NADH pyrophosphate CoA D−glucosamine 1−phosphate −acetyl−D−glucosamine 1−phosphate −acetyldihydrolipoamide acetyl−CoA dihydrolipoamide deoxyguanosine 2−deoxy−D−ribose 1−phosphate deoxyadenosine guanine guanosine −D−ribose 1−phosphate −D−ribose 1−pyrophosphate adenine adenosine hypoxanthine inosine orthophosphate H O 2 N −acetyl−D−glucosamine pyrophosphate CO 2 1−phosphate CoA guanosine guanine acetyl−CoA deoxyguanosine pyruvate, CO 2 α S D−glucosamine 1−phosphate −acetyldihydrolipoamide S α dihydrolipoamide i ii iii N http://www.tp.umu.se/ ∼ holme/ 14 Ume˚ a University, Sweden
Some detected subnetworks (continued) enzyme III Glc p (b) enzyme III Glc N −phosphohistidine enzyme III Fru enzyme III Man p p N −phosphohistidine N −phosphohistidine Mycoplasma pneumoniae enzyme III Fru HPr protein N−pros− phosphohistidine enzyme III Man HPr protein histidine enzyme III Man enzyme III Glc (a) enzyme III Scr enzyme III Scr p enzyme III Man p N −phosphohistidine N −phosphohistidine HPr protein N−pros− phosphohistidine pyruvate Glc p enzyme III N −phosphohistidine enzyme III Scr HPr protein histidine phospho enol pyruvate enzyme III Fru p N −phosphohistidine enzyme III Fru 2.7.7.7.DNA polymerase III 5.99.1.2.DNA topoisomerase I Scr p (c) enzyme III N −phosphohistidine Rep 2.7.7.7.DNA polymerase I 6.5.1.2.DNA ligase SSB open prepriming complex 5.99.1.3.DNA topoisomerase II SSB primosome complex 2.7.7.7.DNA polymerase I DNA helicase II DNA helicase II CTP 6.5.1.2.DNA ligase 2.7.7.7.DNA polymerase III Rep GTP 5.99.1.3.DNA topoisomerase II ortophosphate open prepriming complex 5.99.1.2.DNA topoisomerase I primosome complex ADP ortophosphate ATP prepriming complex UTP RNA primer−primosome complex http://www.tp.umu.se/ ∼ holme/ 15 Ume˚ a University, Sweden
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