networks? networks are spaces generated by interaction

Transcription

1 networks? networks are spaces generated by interaction

2 DNA damage response and repair DNA metabolism DNA recombination DNA replication RNA localization and processing autophagy budding carbohydrate metabolism cell cycle cell growth and/or maintenance cell organization and biogenesis cell shape and cell size control general metabolism mating nucleolar and ribosome biogenesis protein amino acid phosphorylation/desphosphorylation protein biosynthesis protein degradation protein metabolism and modification protein transport signal transduction sporulation stress response by mass spectrometry Kss1 Bck Ste7 Dig1 Ste1 Ste11 Bem3 Rvb1 Aco1 Rvs167 Ded1 Gyp5 Ecm9 Hom6 Ilv5 Lys1 Pho8 Pre Ubp7 Ura7 Met18 Ylr3w Ubi Idh1 Apg1 Crm1 Fet3 Kap1 Kgd1 Met Pfk1 Rpn1 Car1 Hyp Ipp1 Mdh1 Cnb1 Cmp Cns1 Ecm Yhb1 Cof1 Crn1 Srv Hta1 Hhf1 Hir Htb1 Kap11 Kri1 Nap1 Nop Ret1 Rpc8 Rpo31 Spt16 Rrp13 Ylrc Cct Cys Dig Pim1 Rpa135 Rpn6 Tec1 Yer093c Ygl5w Yjr07c Yol078w Lsm8 Apa1 Gar1 Lsm Qcr Rpn1 Rpn8 Rrp Smb1 Tif6 Ygl117w Mig1 Mss116 Nop1 Nop Brx1 Ckb1 Cox6 Fet Nsa Nip7 Nmd3 Nop1 Rrp1 Sik1 Nug1 Nsa1 Noc Ypl009c Rpc19 Rpn5 Emp Kre6 Rpn9 Rpt1 Rpt Sip Arc35 Gal83 Idh Sec53 Snf1 Snf Tcp1 Smt3 Cpr1 Gph1 Pst Rod1 Sip1 Yor67c Pse1 Tem1 Cdc15 Gcd11 Mcx1 Sar1 Ybr81c Yhr033w Ynk1 Ubc6 Atp Gcn1 Los1 Pol5 Sec7 Uba1 Ybl00w Ykl056c Ypt1 Yju Cor1 Ded81 Prp19 Dss Gdi1 Mrs6 Sec Cdc53 Por1 Skp1 Ela1 Loc1 Yra1 Cic1 Glc7 Glc8 Mhp1 Reg1 Kap Ynl035c Dim1 Enp1 Nab Hrp1 Mrt Prp6 Hsl7 Rpp0 Ylr87c Fyv1 Krr1 Yjr01c Rpf Kre33 Tsr1 Tif631 Drs1 Erb1 Ydr131c Yrb Yhr197w Arf1 Arf Cmd1 Cmk Ede1 Myo Myo Myo5 Pgm Vas1 Vps13 Ils1 Sod1 Hch1 She3 Dmc1 Est1 Puf6 Rrp1 Cbf5 Dbp7 Hsh9 Pet56 Pdi1 Pwp1 Yer077c Yjl9c Ykl01c Fpr1 Hom3 Gsp1 Gsp Srm1 Rna1 Mog1 Rse1 Car Ssz1 Ydr31c Sah1 Lsm7 Pat1 Ade5 Dhh1 Lsm Prp Met30 Rub1 Sis1 Tef Met Met31 Nop13 Ebp Imd Imd3 Rrp5 Rad Rad1 Sod Ubc1 Adk1 Yhr115c Ynl311c Ypt6 Ric1 Cdc11 Cdc3 Cdc Cdc1 Tif63 Ctf13 Hrt1 Rtt1 Sec7 Guf1 Yll03c Tps1 Ubp15 Rpc0 Rpa190 Sgn1 Ygr50c Spt Cka1 Cka Ygr05w His Gpi16 Cdc Bem San1 Gbp Hpr1 Sub Thp Mft1 Rlr1 Rho Ubc1 Ula1 Clb Cdc8 Dia Cks1 Hat1 Vps1 Ypt Ypt5 Ypt3 Ypt31 Ypt7 Gpa Ymr09c Grr1 Pfk Pfs Rpt3 Ygl00c Rpn Snp1 Bcy1 Tpk3 Tpk1 Sti1 Ypt53 Cce1 Rnr3 Cmk1 Cpr6 Qns1 Trr1 Ptc Ydr7w Gin Ydr071c Mas1 Mas Gis Prb1 Sxm1 Ecm1 Lhp1 Cdc13 Clu1 Tif Gnd1 Kip3 Hef3 Inp5 Sap190 Nta1 Hsp Vma Tfp1 Mge1 Ptc5 Yal07w Spo13 Ydr19c Yhr1w Ydr306c Yku80 Msu1 Ylr35w Yol18c Oye Bur Cln1 Clb5 Clb3 Yer138c Ydr170wa Nup85 Hem15 Ymr09c Cbp3 Nup8 Seh1 Pre1 Pup Pup3 Ylr199c Pre5 Pre Pre3 Pre6 Pre7 Scl1 Ykl06c Pre9 Pre8 Rsp5 Rvs161 Bop Sgt1 Cdc Yil007c Yta6 Ygr086c Ypl00c Top Cdh1 Cct3 Cat5 Yjl068c Gyp6 Trs10 Trs130 Rpb5 Rpa3 Yck1 Yck Ypr015c Bud0 Ydr1c Prp3 Rsm5 Afg Fyv Ypl013c Rsm Yjl1w Ybl0w Nog1 Ydr036c Htb Mam33 Ygl068w Cdc3 Swm1 Cdc5 Mcd1 Smc1 Gsy1 Fpr3 Sds Fin1 Gpa1 Hex3 Ssk1 Hrr5 Prp3 Prp Ymr6c Apc1 Nop58 Ygr15w Kap95 Cbp She Mlc1 Nuf1 Myo3 Faa Rnq1 Ydr79w Ado1 Lsm1 Ygr173w Ydr15w Ykl078w Ylr097c Adh Imd Pet17 Tis11 Ytm1 Yrb1 Gal3 Cdc39 Mms1 Pma Yar009c Ygp1 Ylr035ca Ylr6c Adr1 Mkt1 Dur1 Ecm33 Hsp1 Sec6 Qri8 Ahp1 Dop1 Sry1 Rvb Npl3 Pub1 Fun1 Ste3 Ssf1 Aac3 Kre31 Rli1 Srp1 Sup5 Ygr090w Tpt1 Apc Mak5 Ynl116w Dbf Mob1 Ess1 Tfg1 Rpo1 Tom1 Spt5 Hxt6 Rpb3 Ynl53w Isu1 Nfs1 Lys1 Fol Tal1 Cct5 Cct6 Cft1 Hgh1 Mer1 Lhs1 Rpn3 Rpn7 Rpt5 Nas6 Arp Rpt Rpn11 Cpr3 Cdc33 Dut1 Ygr066c Thi Tpk Ybr08c Yck3 Ygr15c Ssn8 Sno Ygr111w Yjl07c Yil113w Slt Ynl60c Asi3 Ypl170w Pma1 Ynl7c Cdc1 Mcr1 Dpm1 Atp7 Atp5 Fur1 Hms1 Ydr53c Atp3 Spe3 Aut Spc7 Ydr9w Irr1 Smc3 Faa1 Cyr1 Dbf0 Ala1 Egd Axl1 Rpb Dbp8 Cpa Rnr1 Rnr Ydl086w Pdc6 Pdc5 Prs5 Dia Bud3 Grx Ykr038c Yml036w Isw Isw1 Kns1 Msg5 Fus3 Msh1 Pph Cdc55 Ppe1 Rts1 Tpd3 Ygr161c Hta Ygl11c Rpb11 Tap Pro1 Arc1 Fum1 Rad51 Mlh1 Tos3 Ykr096w Yak1 Dnm1 Vps1 Ylr70w Ypl7c Mpc5 Yhr11c Pyc Ybl8w Oye3 Ycr079w Vid31 Cdc60 Pro Pyc1 Ypl1c Ykl15c Emg1 Yer030w Ynl099c Siw1 Ypk Pet11 Ygr016w Yel03c Yor15w Yor0w Rad3 Dun1 Far1 Gpd1 Gpd Sen1 Ste0 Sec6 Ylr368w Rad6 Ach1 Adh Bio3 Erg0 Yhr076w Ymr318c Rhr Ydr36c Ras Ras1 Acc1 Gfa1 Isa Ngl Rpa9 Rpc5 Rpo6 Tbs1 Yfl0c Rpc3 Sen15 Egd1 Erg13 Erg6 Frs Gnd Grx1 Aat Afr1 Lro1 Met6 Ntf Prm Rsn1 Scp160 Ses1 Snu13 Ths1 Vma5 Wtm1 Ybr05c Dog1 Has1 Prp8 Sap185 Sit Ylr386w Yhr1wa Ydl05c Yal09c Yhr009c Apt1 Aro1 Ydr18w Ylr38w Erg1 Cdc7 Bfr Bir1 Nut1 Thi3 Ylr31c Ylr331c Bms1 Cdc6 Ctf Dbp Lst Mcm Mcm3 Spb1 Ssf Yblc Pph3 Ybl06w Ynl01c Prp11 Cop1 Nan1 Rex Shm Ssk Bck1 Gal7 Kic1 Kin Mkk Smk1 Ydr1w Ylr187w Gdb1 Rad50 Ubr1 Yor173w Ykl161c Ynl056w Yol05w Fun30 Fun31 Aac1 Apg17 Cvt9 Ppx1 REP1 Sec18 Tyr1 Yhr00w Ynl08w Yor086c Grx3 Idp Pho81 Sec3 Yor073w Dog Msk1 Rgd1 Vma Aip1 Arp7 Mse1 Ydr39c Ypr115w Rad5 Ynl13c Cdc9 Dbp9 Pol30 Yor378w Sld Trl1 Lif1 Dnl Mec3 Mak16 Ydr198c Ygl16c Nop15 Mlh3 Pso Mgm1 Rad Pex15 Mre11 Vma8 Xrs Rad7 Elc1 Rfa Ubc13 Aro9 Mms Rad18 Whi Csr Tdp1 Shp1 Yen1 Fpr Lcd1 Adh5 Mag1 Ai1 Hho1 Hht1 Mph1 Msh Yor155c Mus81 Anc1 Cdc16 Mes1 Mms Nhp Rad53 Rpc Yer078c Ntg1 Rfc Tif3 Rad Rfc3 Rfc5 Ylr13w Asf1 Ptc Swi Tbf1 Ymr135c Yta7 Rad59 Bem Hor Ilv Opy1 Pgm1 Ptc3 Rad5 Rad9 Rfa1 Rfa3 Hxt7 Yjr11w Brr Rfc Rom Vac8 Sml1 Adh3 Nat1 Dpb11 Hpr5 Phr1 Msd1 Pol Rho5 Rad16 Htz1 Pdx1 Rad30 Map Ycl0w Efd1 Rfc1 Rhc18 Imd1 Set1 Bre Trf Mtr Ydl175c Yil079c Ypl16c Ddc1 Suv3 Rad17 Mgt1 Rip1 Sof1 Ira Rad1 Rad Rad8 Ccl1 Kin8 Lsc1 Tfb3 Ybr18w Ald5 Bul1 Lsb1 Ykr018c Ylr39c Sir3 Gas1 Sir1 Ubp8 Sir Blm3 Sir Yfl006w Yku70 Pex19 Arc15 Arc18 Arc19 Arc0 Arp3 Puf3 Fhl1 Fkh1 Fkh Gcn3 Hmo1 Ceg1 Ckb Fyv8 Gcd Gcd7 Mbp1 Net1 Sec Sin3 Sui Sui3 Ubp1 Ure Ygr017w Ymr1w Ino80 Lap Ams1 Pib1 Pph1 Prk1 Abp1 Akl1 Dis3 Eaf3 Fip1 Nam8 Nup1 Nup Nup60 Pap1 Pct1 Reb1 Rnt1 Rrp Rrp3 Rrp6 Rtt3 Sif Snu56 Sto1 Tra1 Ume1 Ypr090w Fal1 Gcd1 Gcd6 Ubc Qcr7 Ufd Ydl0c Ybr01c Yer083c Ygl00c Ydr00c Yfr008w Yol05w Gac1 Pob3 Ycr030c Yor056c Ypr093c Rpb9 Rgp1 Hts1 Apm3 Apl5 Las17 Bzz1 Gal Pep1 Sla1 Sqt1 Vma6 Vrp1 Yhm Ynr065c Pds1 Pkh Ygr033c Cef1 Clf1 Snt309 Ygr05w Sat Pho85 Ydr516c Ygr165w Num1 Sef1 Skt5 Syf1 Ygl081w Nip1 Gsy Ylr016c Smc Ynl09w Ypl150w Aro Cbk1 Sgt Ssd1 Gip Hal5 Itr Kin8 Ynr07w Ksp1 Chs1 Pri Yhr186c Lem3 Nmd5 Yhr199c Ylr36w Ppq1 Yhl0c Nha1 Ybl09w Ykr017c Trx Usa1 Pex6 Chk1 Ctr1 Ssk Bud7 Mae1 Bfa1 Nup53 Ybr063c Dpb Lst8 Pho86 Srp5 Ykr051w Ylr71w Yml00w Ypr003c Rrp3 Slc1 Tfc7 Ygr66w Nog Met16 Slx1 Ybt1 Ynl00w Bnr1 Boi Kcs1 Nth1 Yfr017c Yil08w Cyk3 Sok1 Stu1 Svl3 Caf Ent Ybl09w Osh7 Sap1 Pex7 Fzo1 New1 Rrp9 Rtf1 Spt7 Ate1 Epl1 Mpt1 Vid1 Arp Esa1 Tuf1 Sdf1 Yel06c Snt1 Trm3 Yil11w Ylr09c Ymr155w Yrf1-3 Zds Wtm Swd3 Sfp1 Sgs1 Ydr316w Bud9 Dak Kin1 Ynl18c Q003 Bub3 Q009 Rpf1 Ypd1 Vps35 Ybrw Yil137c Sec8 Yplw Bud3 Cin8 Hir1 Mak11 Mck1 Pnt1 Yil5c Msi1 Set3 Pkh1 Prp6 Cdc5 Ypl113c Sap155 Eap1 Ybr187w Ycr076c Ykr007w Btn Hxt5 Swe1 Ahc1 Kel1 Tep1 Mlp Tup1 Cyc8 Ssy5 Sph1 Ydl156w Rpl3b Bcp1 Ypl08w Rpn13 Pol Yglc Noc3 Yhl035c Hot1 Ydr116c Ykl08c Cna1 Ygr63c Ctk1 Cdc37 Hrb1 Elp Elp3 Jip1 Iki3 Zms1 Hat Hif1 Hog1 Rck Pac11 Ybl06c Dyn Pbs Pps1 Ade13 Ppz Yor05c Pwp Ygrc Sec13 Sec31 Nup133 Yhl039w Las1 Yor83w Ycl039w Ydr55c Vid8 Fyv Vid30 Yer066ca Ygr3c Erg Yjr061w Kkq8 Yjl069c Arp Dip Dip5 Mum Yil055c Bmh Cln Gcn Ynl13c Tfc Kel Bud1 Pac Ydr9c Ygr130c Nsr1 Sul Ydr67c Ubp9 Yol111c Ygl131c Yor353c Ctk3 Stb3 Fap1 Mek1 Pcl9 Psr1 Ser1 Ufd Tsl1 Dsk Hmf1 Ydr09w Npl Rad3 Ylr7c Exo70 Cin5 Ymr91w Vps33 Ypl36c Hym1 Yfr039c Ltp1 Mot1 Mob Pcl6 Rpg1 Top1 Yfr011c Yol087c Tif35 Yor7w Aut Apg Ptc1 Ris1 Apg7 Ape3 Yml07c Cpa1 Prp8 Rpd3 Ydr66c Ymr093w Rok1 Are Bre1 Yhr19c Ypl055c Ime Pep3 Ymr086w Sum1 Lsb3 Yfr0c Prp1 Yor0w Yjl05w Ygr80c Elm1 Scd5 Ydr1w Ptp Ydl063c Dps1 Msn5 Gal11 Prp31 Skm1 Hmg Swd1 Ygr067c Yir003w Rlp Gpi15 Yhr06c Bub Ism1 Ylr15c Dcp Crz1 Dcp1 Sfb3 Tgl1 Ybr5w Sgm1 Mds3 Pin Sas Yel015w Ynl07w Rpm Cdc5 Ltv1 Yor15c Mec1 Rad7 Yhr196w Tps Scs Mms Esc Gdh Arl3 Chd1 Mlc Prs3 Ira1 Rpa1 Yer067w Yhr087w Ura3 Isa1 Ygr150c Ygr198w Red1 Rim11 Gcr Yjr08w Cki1 Pmd1 Prs Yer160c Yjr07w Ybr09w Ybr67w Ydr339c Pmc1 Ydr365c Ynr05c Ycr087w Ydrc Yfr003c Yfr016c Cap Cap1 Ylr7w Are1 Msc3 Grs1 Swi1 Kex Tao3 Ccr Cdc36 Yer08w Dbp Osh3 Ptp3 Swi5 Stb Faa Hfi1 Ygr00c Tel1 Ybr80c Ybr139w Aah1 Ycr001w Ypr13w Yhr5w Ssq1 Mub1 Spb Ubr Sth1 Ysc8 Scp1 Sps1 Apm1 Shs1 Gcn5 Ada Taf60 Sgf9 Hap Cis1 Ykl1c Ypr085c Hap5 Ypl166w Nop16 Ynl063w Grh1 Hap3 Mdh Exg1 Yil8w Med Zrg17 Psr Ssl Rad55 Rck1 Rim15 Rlf Ygl060w Ski8 Ski Ski3 Sln1 Taf90 Prp0 Ykl099c Yml093w Ykr060w Yor15c Lcp5 Caf0 Cdc0 Mad3 Cse Ime Sdh Ydr37c Ykr06c Ste Ybl036c Ydl193w Ydr8c Ygr05w Ino Msh3 Ygl0w Yll09w Yjr1w Rgr1 Yfl030w Pcl7 Efb1 Kgd Krs1 Pmi0 Lpd1 Arg1 Trp5 Ara1 Thr Sac6 Ape Pab1 Mdh3 Acs Hom Ydl1w Bat1 Gcy1 Ade6 Cys3 Ade3 Ura1 Rho Msn Van1 Mnn9 Bni1 Cdc7 Yjr09w Trp3 Imh1 Pdx3 Thi1 Ade17 Ymr15c Ktr3 Nop1 Mdj1 Cvt19 Dld3 Cdc13 Sui1 Arg Rho1 Yfr0c Ilv3 Apg5 Ymr315w Ymrc Afg3 Tfg Bgl Cbp6 Psd1 Oac1 Pet9 Ayr1 Pro3 Scw Msh6 Yilc Ymr33w Pda1 Nbp Ppz1 Snu66 Sks1 Nup15 Ctf19 Ypl181w Gsf Spc Glt1 Spc5 Ymr196w Rmt Yjr070c Sen Asc1 Trp Rcl1 Apl Ylr38w Apl Pib Rlp7 Apm Rad6 Yjl7c Sec1 Erg7 Gpt Ynl181w Ydl0w Om5 Ret Yer09w Ydr398w Sgd1 Pox1 Npr Yer18w Vps8 Ydr33c Vid Yor17w Cac Ydl113c Ynl1w Sac1 Wbp1 Hsm3 Rpl5 Nup10 Vps1 Crc1 Thick blue lines represent previously known interactions. Thin orange lines represent new interactions. Dig1 Hyp Mdh1 Ecm Cc Sec53 Prp6 Sod1 Cks1 Rpt3 Rpn Sti1 Pre3 Rvb Cct6 Slt Emg1 Acc1 Ydr1w Aac1 Mse1 Dbp9 Rad59 Hxt7 Mgt1 Srp5 Caf Nup133 Yol087c Van1

3 a systems biology hairball Drosophila protein interaction map Giot et al, Science, 30, (003) total: 0,05 interactions 7,08 proteins (of 13,656 coding loci) high confidence:,780 interactions,679 proteins

4 obtaining protein-protein interaction data on a global scale [1/] Gal transcriptional activation domain Gal DNA-binding domain AD BD Gal binding site (GAL1 UAS) a gene RNA transcript fuse a protein X to the Gal DNA-binding domain: BAIT X Y AD fuse a protein Y to the Gal transcriptional activation domain: PREY BD

5 obtaining protein-protein interaction data on a global scale [/] yeast two-hybrid assay Prey X Y AD RNA transcript Bait BD Gal binding site Reporter gene lacz (beta-galactosidase)

6 bummer in YPD 11 (0%) 81 5 in YPD Uetz et al data set: 817 (yeast) genes 69 interactions Ito et al data set: 797 (yeast) genes 81 interactions Uetz et al Ito et al 90% of known interactions not found... of newly discovered interactions, 50% are estimated to be likely biologically relevant... problems:! PCR introduces mutations that may abolish interaction! fusion constructs may affect proper folding needed for interaction! interaction may require activation of either bait or prey protein or both! stochastic gene expression may generate false positives! differences in experimental procedure (reporters, selection strategies)

7 identifying protein complexes [1/] Affinity purification Separation by size Identification by mass spectrometry Antibody B B Affinity tag m/z baiting with a tagged protein, whose tag can be immunoprecipitated Gavin et al: 3 complexes involving a total of 1,0 proteins 91% of the complexes had at least one protein of unknown function overlap with YH was 7% (ouch?); overlap with YPD was 56% (compare with 13% from YH) Ho et al: 93 complexes involving a total of 1,578 proteins

8 identifying protein complexes [/] known complexes (whose component was a bait) were missed problems:! adding a tag might interfere with complex formation (adding a tag turned even out to be lethal at times)! difficult to detect transient interactions, small proteins! wrong conditions! nonspecific binding! error rates of 30% in repeated purification experiments YH yields information about pairwise interactions. Affinity purification yields the composition of a complex, but not the contact map.

9 Now what?

10 the zoo biological networks molecular (metabolic, signaling, transcription, polymerization, binding, ), transport (blood vessels, stolons), neural, ecological, epidemiological,... social networks friendship, business, information (knowledge) networks citation, www, inference, thesaurus,... technological networks powergrid, transportation, internet,... conformational networks in physics free energy minima and saddle points in glasses and polymer conformations (protein and RNA folding),...

11 systems biology does not have a good mapsmathics yet

12 while waiting for Godot what does a network look like? (structure) is it normal? (expectation) what do looks tell us about behavior? (dynamics) how did its looks come about? (construction)

13 incomplete coverage structure: networks as graphs from structure to dynamics: networks that host dynamical processes from structure to dynamics to structure: networks that host dynamical processes that change the network structure

14 if seeing is believing,... What can you see?

15 the answer requires an analytic approach What does a network look like, when you actually can t look at it? M.E.J.Newman, U.Michigan

16 Which vertex, when removed, would prove most crucial to the networks connectivity? a meaningless question

17 What percentage of vertices, when removed, would substantially affect the networks connectivity? a much better question

18 networks as graphs a graph is defined by a set of vertices and a set of edges 1 8 undirected graph

19 networks as graphs a graph is defined by a set of vertices and a set of edges 1 8 directed graph

20 networks as graphs different vertex and edge types

21 degree (of a vertex): number of edges connected to a vertex. (not always the same as the number of neighbors.) the concept of degree

22 degree distribution : fraction (probability) of vertices in the network that have degree the degree distribution

23 the concept of a geodesic geodesic path (between two vertices): shortest path through network from one vertex to another

24 the concept of a geodesic geodesic path (between two vertices): shortest path through network from one vertex to another (not unique!)

25 take a...breadth-first 1 d=1 d=0 d=1 d=1 9 8 d= 7 d= d= d= d=infinity this procedure assumes all edges have equal weight. what if that is not the case? read up on Dijkstra s algorithm.

26 the diameter: the maximum of all the minima diameter (of network): length of longest geodesic path between any two vertices

27 the average path length mean geodesic distance in the network (average shortest path) geodesic distance between i and j number of unordered pairs (i,j), including all i=j 5 3 harmonic mean geodesic distance

28 the degree distribution the concept of a degree distribution... doesn t make much sense here makes a lot more sense here

29 the degree distribution of a random network random graph: take some number n of vertices and connect each pair of them with probability p or don t connect them with probability 1-p. n= p=0.3 p=0.5 p=0.7 degree distribution: with (average degree) exact in the limit of large n and fixed

30 old friends: binomial, normal, Poisson binomial normal Poisson n=0, p=0.5 n=0, p=0.5 n=00, p=0.05 mean variance

31 not a random degree distribution...

32 power law a tail of two worlds

33 UDP UMP ADP ATP UMP ATP UTP ADP ATP ADP UDP Mg + Mg + substrate graph reaction graph CTP NH + orthop ATP ADP UTP CTP NH + Mg + Orthophosphate UMP UDP UTP metabolic pathway (a hypergraph) CTP

34 ranked by degree ranked by mean path length glutamate 51 glutamate.6 pyruvate 9 pyruvate.59 CoA 9 CoA.69 -oxoglutarate 7 glutamine.77 glutamine acetylcoa.86 aspartate 0 oxoisovalerate.88 acetyl CoA 17 aspartate.91 phosphoribosylpp 16 -oxoglutarate.99 tetrahydrofolate 15 phosphoribosylpp 3. succinate 1 anthranilate 3. 3-phosphoglycerate 13 chorismate 3.13 serine 13 valine 3.1 oxoisovalerate 1 3-phosphoglycerate 3.15

35 a tail of two worlds power law a long tail of values far above the mean

36 a tail of two worlds power law P(k) P(k) k 1 0 1,000 k

37 you ve got the power Why is a power law interesting? 1. it is scale-free

38 a system is scale-free when it lacks a defining intrinsic size-scale scale-free means self-similar a scale transformation......leaves the form of the function invariant power laws are the only solution to this functional equation

39 measuring the tail of a power law is tricky. two ways out: increase histogram bin sizes exponentially (lose differences within bin) plot the cumulative distribution (not a direct visualization of the degree distribution) yak! tail exponent

40 1M random numbers sampled from with 1.5 (a) histogram 0 (b) log-log of the sample samples samples ouch! x x samples logarithmic binning (x) (c) x s amples with value > x (d) cumulative distribution x

41 the of and to a in is that was it for on with he be I by as at you are his had not this have from but which she they or an her were a useful consideration: cumulative distributions and rank plots the r th most frequent word has frequency k = r words occur with frequency k or more = the cumulative distribution is proportional to r traditional Zipf plot rank r slope is -1/b flip word frequency k slope is -b 0 0 word frequency k 0 0 rank r (word frequencies in Moby Dick )

42 fraction of vertices with degree larger than k degree k Uetz, P. et al. (000) Nature 03, Jeong, H. et al. (000) Nature 07, Ito, T. et al. (001) Proc. Natl. Acad. Sci. U.S.A. 98,

43 power-law distributions or power-law tails are very common 6 (a) (b) (c) citations word frequency web hits (e) (d) (f) (g) 6 (h) earthquake magnitude telephone calls received books sold (i) intensity peak intensity crater diameter in km (j) (k) (l) net worth in US dollars name frequency 3 5 population of city 7

44 more of the same 0 truncated power law? power-law tail exponential (a) collaborations in mathematics - (b) citations -3 (e) power grid (c) World-Wide Web (d) Internet (f) protein interactions 1 seems a power law not a power law source: M.E.J.Newman, The Structure and Function of Complex Networks, SIAM Review, 5/,

45 important, but not the kind of powerlaws we are focusing on here! Allometric scaling: 3/ power law of energy consumption versus body mass holds over nearly 30 (!) orders of magnitude more power to you