Combinatorial Problems and Algorithms in Comparative Genomics. Max Alekseyev

Transcription

1 Comintoril Prolems n Algorithms in Comprtive Genomis Mx Alekseyev Dept. Computer Siene, University of South Crolin, Columi, SC, U.S.A. Algorithmi Biology L, Aemi University, Sint Petersurg, Russi 2011

2 Genome Rerrngements Mouse X hromosome Unknown nestor ~ 80 M yers go Humn X hromosome

3 Genome Rerrngements: Evolutionry Senrios Unknown nestor ~ 80 M yers go Wht is the evolutionry senrio for trnsforming one genome into the other? Wht is the orgniztion of the nestrl genome? Are there ny rerrngement hotspots in mmmlin genomes? Reversl (inversion) flips segment of hromosome

4 Genome Rerrngements: Anestrl Reonstrution Wht is the evolutionry senrio for trnsforming one genome into the other? Wht is the orgniztion of the nestrl genome? Are there ny rerrngement hotspots in mmmlin genomes?

5 Genome Rerrngements: Evolutionry Erthqukes Wht is the evolutionry senrio for trnsforming one genome into the other? Wht is the orgniztion of the nestrl genome? Are there ny rerrngement hotspots in mmmlin genomes? (ontroversy in )

6 Genome Rerrngements: Evolutionry Erthqukes Wht is the evolutionry senrio for trnsforming one genome into the other? Wht is the orgniztion of the nestrl genome? Where re the rerrngement hotspots in mmmlin genomes?

7 Rerrngement Hotspots in Tumor Genomes Rerrngements my isrupt genes n lter gene regultion. Exmple: rerrngement in leukemi yiels Philelphi hromosome: Chr 9 Chr 22 promoter ABL gene promoter BCR gene promoter BCR gene promoter -1 onogene Thousns of iniviul rerrngements hotspots known for ifferent tumors.

8 Biologil Prolem: Who re evolutionry loser to humns: mie or ogs?

9 Who is Closer to Us: Mouse or Dog?

10 Primte Roent Crnivore Split roent-rnivore split time primte-roent split primte-rnivore split

11 Primte Roent Crnivore Split roent-rnivore split time primte-roent split primte-rnivore split

12 Primte Roent vs. Primte Crnivore Split July 2007 n up new ppers supporting the primte-rnivore split April 2007 Lunter et l., PLoS CB 2007 refute Cnnrozzi et l. rguments Jnury 2007 Cnnrozzi et. l., PLoS CB 2007 rgue for the primte-rnivore split 2001 Murphy et. l., Siene 2001 set new ominnt view: the primte-roent split efore 2001 most iologists elieve in the primte-rnivore split primte-roent split primte-rnivore split

13 Reonstrution of Anestrl Genomes: Humn / Mouse / Rt

14 Reonstrution of MANY Anestrl Genomes: Cn It Be Done?

15 Algorithmi Bkgroun: Genome Rerrngements n Brekpoint Grphs

16 Unihromosoml Genomes: Reversl Distne A reversl flips segment of hromosome. For given genomes P n Q, the numer of reversls in shortest series, trnsforming one genome into the other, is lle the reversl istne etween P n Q. Hnnenhlli n Pevzner (FOCS 1995) gve polynomil-time lgorithm for omputing the reversl istne.

17 Prefix Reversls A prefix reversl flips prefix permuttion. Pnke Flipping Prolem: sort given stk (permuttion) of pnkes of ifferent sizes with the minimum numer of flips of ny numer of top pnkes.

18 Multihromosoml Genomes: Genomi Distne Genomi Distne etween two genomes is the minimum numer of reversls, trnslotions, fusions, n fissions require to trnsform one genome into the other. Hnnenhlli n Pevzner (STOC 1995) extene their lgorithm for omputing the reversl istne to omputing the genomi istne. These lgorithms were followe y mny improvements: Kpln et l. 1999, Ber et l. 2001, Tesler 2002, Ozery-Flto & Shmir 2003, Tnnier & Sgot 2004, Bergeron , et.

19 HP Theory Is Rther Complite: Is There Simpler Alterntive? HP theory is key tool in most genome rerrngement stuies. However, it is rther omplite tht mkes it iffiult to pply in omplex setups. To stuy genome rerrngements in multiple genomes, we use 2-rek rerrngements, lso known s DCJ (Ynopoulus et l., Bioinformtis 2005).

20 Simplifying HP Theory: Swith from Liner to Cirulr Chromosomes A hromosome n e represente s yle with irete re n unirete lk eges, where: re eges enoe loks n their iretions; jent loks re onnete with lk eges.

21 Reversls on Cirulr Chromosomes reversl Reversls reple two lk eges with two other lk eges

22 Fissions fission Fissions split single yle (hromosome) into two. Fissions reple two lk eges with two other lk eges.

23 Trnslotions / Fusions fusion Trnslotions/Fusions trnsform two yles (hromosomes) into single one. They lso reple two lk eges with two other lk eges.

24 2-Breks 2-rek 2-Brek reples ny pir of lk eges with nother pir forming mthing on the sme 4 verties. Reversls, trnslotions, fusions, n fissions represent ll possile types of 2-reks.

25 2-Brek Distne The 2-Brek istne ist(p,q) etween genomes P n Q is the minimum numer of 2reks require to trnsform P into Q. In ontrst to the genomi istne, the 2-rek istne is esy to ompute.

26 Two Genomes s Blk-Re n Green-Re Cyles P Q

27 Rerrnging P in the Q orer P Q

28 Brekpoint Grph = Superposition of Genome Grphs: Gluing Re Eges with the Sme Lels P Brekpoint Grph (Bfn & Pevzner, FOCS 1994) G(P,Q) Q

29 Blk-Green Cyles Blk n green eges represent perfet mthings in the rekpoint grph. Therefore, together these eges form olletion of lk-green lternting yles (where the olor of eges lternte). The numer of lk-green yles yles(p,q) in the rekpoint grph G(P,Q) plys entrl role in omputing the 2-rek istne etween P n Q.

30 Rerrngements Chnge Cyles Trnsforming genome P into genome Q y 2-reks orrespons to trnsforming the rekpoint grph G(P,Q) into the rekpoint grph G(Q,Q). G(P,Q) yles(p,q) = 2 G(P',Q) yles(p',q) = 3 G(Q,Q) trivil yles yles(q,q) = 4 = loks(p,q)

31 Trnsforming P into Q y 2-reks 2-reks P=P0 P1... P= Q G(P,Q) G(P1,Q)... G(Q,Q) yles(p,q) yles... loks(p,q) yles # of lk-green yles inrese y loks(p,q) - yles(p,q) How muh eh 2-rek n ontriute to this inrese?

32 2-Brek Distne Any 2-Brek inreses the numer of yles y t most one (Δyles 1) 1 Any non-trivil yle n e split into two yles with 2-rek (Δyles = 1) 1 Every sorting y 2-rek must inrese the numer of yles y loks(p,q) yles(p,q) The 2-Brek Distne etween genomes P n Q: ist(p,q) = loks(p,q) yles(p,q) (p. Ynopoulos et l., 2005, Bergeron et l., 2006)

33 Complex Rerrngements: Trnspositions Snkoff, PLoS CB 2006, suggeste tht rerrngement hotspots oserve in the Pevzner-Tesler nlysis my e result of omplex rerrngement opertions suh s trnspositions. Sorting y Trnspositions Prolem: Given two genomes, fin the shortest sequene of trnspositions trnsforming one genome into the other. First 1.5-pproximtion lgorithm ws given y Bfn n Pevzner, SODA Reent hievement: pproximtion lgorithm ue to Elis n Hrtmn, WABI The omplexity sttus ws unknown for long time. Only in 2011 it ws prove to e NP-omplete.

34 Complex Rerrngements: Trnspositions (с)

35 Sorting y Trnspositions Sorting y Trnspositions Prolem: Given two genomes, fin the shortest sequene of trnspositions trnsforming one genome into the other. First 1.5-pproximtion lgorithm ws given y Bfn n Pevzner (SODA 1995). Best hievement: pproximtion lgorithm ue to Elis n Hrtmn (WABI 2005). Prove to e NP-omplete y Bulteu, Fertin, n Rusu (ICALP 2011).

36 Trnspositions trnsposition Trnspositions ut off segment of one hromosome n insert it t some position in the sme or nother hromosome

37 Trnspositions Are 3-Breks 3-rek 3-Brek reples ny triple of lk eges with nother triple forming mthing on the sme 6 verties. Trnspositions re 3-Breks.

38 3-Brek Distne: Fous on O Cyles A 3-rek n inrese the numer of o yles (i.e., yles with o numer of lk eges) y t most 2 (Δyleso 2) A non-trivil o yle n e split into three o yles with 3-rek (Δyleso = 2) An even yle n e split into two o yles with 3-rek (Δyleso = 2) The 3-rek Distne etween genomes P n Q is: 3(P,Q) = ( loks(p,q) yleso(p,q) ) / 2

39 Multi-Brek Rerrngements The stnr rerrngement opertions (reversls, trnslotions, fusions, n fissions) mke 2 rekges in genome n glue the resulting piees in new orer. k-brek rerrngement opertion mkes k rekges in genome n glues the resulting piees in new orer. Rerrngements re rre evolutionry events n iologists elieve tht krek rerrngements re unlikely for k>3 n reltively rre for k=3 (t lest in the mmmlin evolution). Also, in rition iology, hromosome errtions for k>2 (initive of hromosome mge rther thn evolutionry vile vritions) my e more ommon, e.g., omplex rerrngements in irrite humn lymphoytes (Shs et l., 2004; Levy et l., 2004).

40 Multi-Brek Rerrngements We propose ext formuls for the k-rek istne etween multi-hromosoml irulr genomes s well s liner-time lgorithm for omputing it. (MA & PP, Theor. Comput. Si. 2008) The ext formuls for k(p,q) eomes omplex s k grows, e.g.: The formul for 20(P,Q) is estimte to ontin over 1,500 terms.

41 Brekpoints Are Verties in Non-trivil Cyles Brekpoints orrespon to regions in the genome tht were roken y some rerrngement(s). In the rekpoint grph, rekpoints orrespon to verties hving two neighors (while verties with just one neighor represent ommon jenies etween loks in the genomes). All verties in non-trivil yles in the rekpoint grph represent rekpoints.

42 Brekpoint Uses n Reuses Eh 2-rek uses four verties (the enpoints of the ffete eges). A vertex (rekpoint) is reuse if it is use y t lest two ifferent 2-reks (i.e., the numer of uses > 1). 1 Numer of uses:

43 Algorithmi Prolem: Reonstrution of Anestrl Genomes

44 Anestrl Genomes Reonstrution in Nutshell Given set of genomes, reonstrut genomes of their ommon nestors. The evolutionry tree of these genomes my e known or unknown.

45 Existing Tools for Anestrl Genomes Reonstrution GRAPPA: J. Tng, B. Moret et l. (2001) MGR: G. Bourque n P. Pevzner (2002) InferCARs: J. M, D. Hussler et l. (2006) EMRAE: H. Zho n G. Bourque (2007) MGRA: M. Alekseyev n P. Pevzner (2009)

46 Anestrl Genomes Reonstrution Prolem (with known phylogeny) Input: set of k genomes n phylogeneti tree T Output: genomes t the internl noes of the tree T Ojetive: minimize the totl sum of the genomi istnes long the rnhes of T NP-omplete in the simplest se of k=3. Wht mkes it hr?

47 Brekpoints Are Footprints of Rerrngements on the Groun of Genomes Input: set of k genomes n phylogeneti tree T Output: genomes t the internl noes of the phylogeneti tree T Ojetive funtion: the sum of genomi istnes long the rnhes of T (ssuming the most prsimonious rerrngement senrio) NP-omplete in the simplest se of k=3. Wht mkes it hr? BREAKPOINTS RE-USES (resulting in messy footprints )! Anestrl Genome Reonstrutions of MANY Genomes (i.e., for lrge k) my e esier to solve.

48 Solution: Multiple Brekpoint Grphs n MGRA Algorithm

49 How to Construt Brekpoint Grph for Multiple Genomes? P Q R

50 Construting Multiple Brekpoint Grph: rerrnging P in the Q orer P Q R

51 Construting Multiple Brekpoint Grph: rerrnging R in the Q orer P Q R

52 Multiple Brekpoint Grph: Still Gluing Re Eges with the Sme Lels P Q R Multiple Brekpoint Grph G(P,Q,R)

53 Prolem: Reonstrut Anestors of Six Mmmlin Genomes time Mouse Rt Dog mque Humn Chimpnzee

54 Multiple Brekpoint Grph of 6 Genomes Multiple Brekpoint Grph G(M,R,D,Q,H,C) of the Mouse, Rt, Dog, mque, Humn, n Chimpnzee genomes.

55 k=2 Genomes: Two Wys of Sorting y 2-Breks Trnsforming P into Q with lk 2-reks: 2-reks P = P0 P1... P-1 P = Q G(P,Q) G(P1,Q)... G(P,Q) = G(Q,Q) Trnsforming Q into P with green 2-reks: 2-reks Q = Q0 Q1... Q = P G(P,Q) G(P,Q1)... G(P,Q) = G(P,P) Let's omine these two wys...

56 Sorting By 2-Breks: Meet In The Mile Let X e ny genome on pth from P to Q: P = P0 P1... Pm = X = Q-m... Q1 Q0 = Q 2-Breks t the left n right hn sies of X re inepenent! Sorting By 2-Breks Prolem is equivlent to fining shortest trnsformtion of G(P,Q) into set of trivil yles G(X,X) (n ientity rekpoint grph of priori unknown genome X): G(P0,Q0) G(P1,Q0) G(P1,Q1) G(P1,Q2)... G(X,X) The lk n green 2-reks my ritrrily lternte.

57 MGRA: Trnsformtion into n Ientity Brekpoint Grph We fin trnsformtion of the multiple rekpoint grph G(P1,P2,...,Pk) with relile rerrngements (reognize from their footprints ) into some ( priori unknown!) ientity multiple rekpoint grph G(X,X,...,X): G(P1,P2,...,Pk)... G(X,X,...,X) Eh rerrngement is onsistent with the given tree T n thus is ssigne to some rnh of T. Rerrngements re pplie in ritrry orer tht ielly (when there re no extensive rekpoint re-uses) oes not ffet the result. Previously pplie rerrngements my revel footprints of new ones.

58 Brekpoints s Rerrngements' Footprints Q D H C M R In this yle there four rekpoints of type MR+DQHC. It n e trnsforme into trivil multi-yles: y two 2-reks: one in Mouse, Mouse one in Rt; Rt or y four 2-reks: one in eh of Dog, Dog mque, mque Humn, Humn n Chimpnzee; pnzee or y single 2-rek operting in the Mouse-Rt nestor (i.e., simultneously in Mouse n Rt)

59 Tree-Consistent Rerrngements Eh rnh of the given tree T efines two omplementry groups of genomes, to eh of whih the sme rerrngements my e pplie simultneously. For exmple, the rnh lele MR+DQHC efines groups {M, R} (Mouse n Rt) n {D,Q,H,C} (Dog, mque, Humn, Chimpnzee). But there re no groups like {M,C} or {R,D,H}. So, we n pply the sme rerrngement to M n R simultneously, viewing it s hppening in their ommon nestor (enote MR) long the MR+DQHC rnh.

60 When All Relile 2-Breks Are Ientifie n Unone The multiple rekpoint grph is reue rmtilly! The remining (non-trivil) omponents n e proesse mnully in the se-y-se fshion.

61 MGRA: Reonstrution of the Anestrl Genomes The resulting ientity rekpoint grph G(X,X,...,X) efines its unerlying genome X. The reverse trnsformtion is pplie to the genome X to trnsform it into eh of the originl genomes P1, P2,..., Pk. This trnsformtion trverses ll internl noes of T n thus efines the nestrl genome t every noe.

62 Reonstrute X Chromosomes The Mouse, Rt, Dog, mque, Humn, Chimpnzee genomes n their reonstrute nestors:

63 If The Evolutionry Tree Is Not Known For the set of 7 mmmlin genomes: Mouse, Rt, Dog, mque, Humn, Chimpnzee, n Opossum, the evolutionry tree T is not known. Depening on the primte roent rnivore split, three topologies re possile (only two of them re vile). However, these three topologies shre mny ommon rnhes in T (onfient rnhes). We n restrit the trnsformtion only to suh rnhes in orer to simplify the rekpoint grph, not reking n eviene for either of the topologies.

64 Resolving The Primte-Roent-Crnivore Split Controversy We reue the multiple rekpoint grph G(M,D,Q,O) (of representtives of eh fmily n n outgroup) with relile 2-reks on the onfient groups of genomes. M M M Q Q O Q D O D O D Wht woul e n eviene for one topology over the others?

65 Rerrngement Eviene For The PrimteCrnivore Split Eh of the three topologies hs n unique rnh in the tree. A single rerrngement ssigne to suh rnh woul orrespon to lest two rerrngements if this rnh is sent in the tree. M O M M 12 Q 19 Q 26 D O D O D We oserve the prevlene of rerrngements' footprints speifi to the primte rnivore split. Q

66 Aknowlegments Pvel Pevzner, UC Sn Diego Glenn Tesler, UC Sn Diego Jin M, University of Illinois t Urn-Chmpign

67 Biologil Prolem: Why n Where Genome Rerrngements Hppen?

68 Chromosome Brekge Moels Chromosome Brekge Moels speify how hromosomes re roken y rerrngements. While the ext mehnism of rerrngements is not known, suh moels try to explin s mny s possile sttistil hrteristis oserve in rel genomes. The more hrteristis re pture y moel, the etter is this moel. The hoie of moel is prtiulrly importnt in simultions tht im retion of simulte genomes whose hrteristis shoul mth those of rel genomes.

69 Testing Moels Given hrteristi oserve in rel genomes n hromosome rekge moel, we n test whether the moel explins this hrteristi. Test: Simulte genomes using the moel n hek if the simulte genomes possess the require hrteristi. As soon s new hrteristi in rel genomes is isovere, the existing moels n e teste ginst it. If they fil, this lls for new moel tht woul explin ll previously known hrteristis s well s the new one.

70 Susumu Ohno: Rerrngements our rnomly Ohno, 1970, 1973 Rnom Brekge Hypothesis: Genomi rhitetures re shpe y rerrngements tht our rnomly.

71 Rnom Brekge Moel (RBM) The rnom rekge hypothesis ws emre y iologists n hs eome e fto theory of hromosome evolution. Neu & Tylor, Pro. Nt'l A. Sienes 1984 First onvining rguments in fvor of the Rnom Brekge Moel (RBM) RBM implies tht there is no rerrngement hotspots RBM ws re-iterte in hunres of ppers

72 Frgile Brekge Moel (FBM) Pevzner & Tesler, PNAS 2003 rgue tht every evolutionry senrio for trnsforming Mouse into Humn genome must result in lrge numer of rekpoint re-uses, ontrition to the RBM. propose the Frgile Brekge Moel (FBM) tht postultes existene of rerrngement hotspots n vst rekpoint re-use FBM implies tht the humn genome is mosi of soli n frgile regions

73 Reuttl of the Reuttl Snkoff & Trinh, J. Comput. Biol. 2004, presente rguments ginst the Frgile Brekge Moel: we hve shown tht rekpoint re use of the sme mgnitue s foun in Pevzner n Tesler, 2003 my very well e rtifts in ontext where NO re use tully ourre.

74 Reuttl of the Reuttl of the Reuttl Snkoff & Trinh, J. Comput. Biol. 2004, presente rguments ginst the Frgile Brekge Moel: we hve shown tht rekpoint re use of the sme mgnitue s foun in Pevzner n Tesler, 2003 my very well e rti fts in ontext where NO re use tully ourre. Peng et l., PLoS Comput. Biol. 2006, foun n error in the Snkoff Trinh rguments. Snkoff, PLoS Comput. Biol. 2006, knowlege the error: Not only i we foist hstily oneive n inor retly exeute simultion on n overworke RECOMB on ferene progrm ommittee, ut worse nostr mxim ulp we olige tem of high powere reserhers to len up fter us!

75 All Reent Stuies Support FBM Kikut et l., Genome Res. 2007:... the Neu n Tylor hypothesis is not possile for the explntion of synteny in rt.

76 With One Influentil Exeption M et l., Genome Res. 2006: Simultions suggest tht this frequeny of rekpoint re use is pproximtely wht one woul expet if rekge ws eqully likely for every genomi position reful nlysis [of the RBM vs. FBM ontroversy] is eyon the sope of this stuy.

77 Our Contriution We reonile the eviene for limite rekpoint reuse in M et l., 2006 with the Frgile Brekge Moel n revel rmpnt ut elusive rekpoint reuse. We provie eviene for the irth n eth of the frgile regions, implying tht they move to ifferent lotions in ifferent lineges, explining why M et l., 2006, foun limite rekpoint reuse etween ifferent rnhes of the evolutionry tree. We introue the Turnover Frgile Brekge Moel (TFBM) tht ounts for the irth n eth of the frgile regions n shes light on possile reltionship etween rerrngements n Mthing Segmentl Duplitions. Duplitions TFBM points to lotions of the urrently frgile regions in the humn genome.

78 Tests vs. Moels Why iologists elieve in RBM? Beuse RBM implies the exponentil istriution of the sizes of the synteny loks oserve in rel genomes. A flw in this logi: RBM is not the only moel tht omplies with the exponentil istriution test. Why Pevzner n Tesler refute RBM? Beuse RBM oes not omply with the rekpoint reuse test: RBM implies low reuse ut rel genomes revel high reuse. FBM omplies with oth the exponentil istriution n rekpoint reuse tests. But is there test tht oth RBM n FBM fil? Test Moel RBM FBM Exponentil Brekpoint istriution reuse YES NO YES YES

79 Tests vs. Moels Why iologists elieve in RBM? Beuse RBM implies the exponentil istriution of the sizes of the synteny loks oserve in rel genomes. A flw in this logi: RBM is not the only moel tht omplies with the exponentil istriution test. Why Pevzner n Tesler refute RBM? Beuse RBM oes not omply with the rekpoint reuse test: RBM implies low reuse ut rel genomes revel high reuse. FBM omplies with oth the exponentil istriution n rekpoint reuse tests. RBM n FBM fil the Multispeies Brekpoint Reuse (MBR) test. Test Moel RBM FBM Exponentil Brekpoint istriution reuse MBR YES NO NO YES YES NO

80 Tests vs. Moels Why iologists elieve in RBM? Beuse RBM implies the exponentil istriution of the sizes of the synteny loks oserve in rel genomes. A flw in this logi: RBM is not the only moel tht omplies with the exponentil istriution test. Why Pevzner n Tesler refute RBM? Beuse RBM oes not omply with the rekpoint reuse test: RBM implies low reuse ut rel genomes revel high reuse. FBM omplies with oth the exponentil istriution n rekpoint reuse tests. TFBM psses ll three tests. Test Moel RBM FBM TFBM Exponentil Brekpoint istriution reuse MBR YES NO NO YES YES NO YES YES YES

81 Algorithmi Prolem: Brekpoint Re-use Anlysis

82 Brekpoints Are Verties in Non-trivil Cyles Brekpoints orrespon to regions in the genome tht were roken y some rerrngement(s). In the rekpoint grph, rekpoints orrespon to verties hving two neighors (while verties with just one neighor represent ommon jenies etween synteny loks). All verties in non-trivil yles in the rekpoint grph represent rekpoints.

83 Brekpoint Uses n Reuses Eh 2-rek uses four verties (the enpoints of the ffete eges). A vertex (rekpoint) is reuse if it is use y t lest two ifferent 2-reks (i.e., the numer of uses > 1). 1 Numer of uses:

84 Intr- n Inter- Reuses For n evolutionry tree with known rerrngement senrios, rekpoint is intr-reuse on some rnh if it is use y t lest two ifferent 2-reks long this rnh. Similrly, rekpoint is inter-reuse ross two rnhes if it is use on oth these rnhes.

85 Rerrngement Senrios Remin Amiguous In mmmlin evolution we know only genomes of existing speies ut o not know the nestrl genomes. While nestrl genomes n e relily reonstrute, the ext rerrngement senrios etween them remin miguous. Cn we ompute the numer of rekpoint intrn inter- reuses without knowing rerrngement senrios?

86 Numer of Intr-Reuses (Lower Boun) For rerrngement senrio etween genomes P n Q: The numer of 2-reks is t lest ist(p,q) Eh 2-rek uses 4 rekpoints The numer of rekpoints is 2 loks(p,q) Hene the totl numer of intr-reuses is: 4 ist(p,q) 2 loks(p,q)

87 Numer of Inter-Reuses (Lower Boun) For two rnhes (P,Q) n (P',Q') in the tree: Set V of the verties in non-trivil yles in G(P,Q) represents the rekpoints etween genomes P n Q Set V' of the verties in non-trivil yles in G(P',Q') represents the rekpoints etween genomes P' n Q' Hene, the numer of inter-reuses is size of the intersetion of V n V'

88 Surprising Irregulrities in Brekpoint Reuse Aross Vrious Pirs of Brnhes Sttistis of rekpoint intr- n inter-reuses etween the rnhes of the tree of six mmmlin genomes: Colors represent the istne etween pir of rnhes: re = jent rnhes; green = rnhes seprte y one other rnh; yellow = rnhes seprte y two other rnhes. Wht is surprising out this Tle?

89 Solution: Turnover Frgile Brekge Moel n Multispeie Brekpoint Reuse Test

90 Turnover Frgile Brekge Moel (TFBM) The M et l. oservtion n the sttistis of inter-reuses inites: Brekpoint inter-reuses mostly hppen ross jent rnhes of the evolutionry tree. Turnover Frgile Brekge Moel (TFBM): Frgile regions re sujet to irth n eth proess n thus hve limite lifespn.

91 Simplest TFBM: Fixe Turnover Rte for Frgile Regions TFBM(m,n,x): TFBM(m,n,x) genomes hve m frgile regions n (out of m) frgile regions re tive eh 2-rek is pplie to 2 (out of n) rnomly hosen tive frgile regions fter eh 2-rek, x tive frgile regions (out of n) ie n x new tive frgile regions (out of m-n) m-n re orn FBM is prtiulr se of TFBM with x=0 RBM is prtiulr se of TFBM with x=0 n n=m

92 Reognizing the Birth n Deth Given n evolutionry tree with known rerrngement senrios, how one woul etermine whether they followe TFBM with x = 0 (tht is, FBM/RBM) or x > 0? Compring rekpoint inter-reuse ross ifferent pirs of rnhes woul help, ut it lso epens on the rnh lengths tht my iffer signifintly ross the tree.

93 Sle Brekpoint Reuse The numer of rekpoint intr- n inter- reuses epens on the length of rnhes. To eliminte this epeneny, we efine the sle intr- n inter- reuse: We efine n expresse nlytilly: E(t) = the expete numer of intr-reuses long rnh of length t; E(t1,t2) = the expete numer of inter-reuses ross rnhes of length t1 n t2. Sle intr- n inter-reuse is the numer of reuses ivie y E(t) or E(t1,t2) respetively.

94 Sle Inter-Reuse in Colore Cells (Simulte Genomes with Vrile Brnh Length) FBM TFBM TFBM TFBM Simultions for the se when n=900 out of m=2000 frgile regions re tive n vrious turnover rte x=0..4.

95 Mesuring Reuse in the Whole Evolutionry Tree TFBM suggests tht on verge the numer rekpoint reuses r(r1,r2) for 2-reks r1 n r2 epens on the istne (in the evolutionry tree) etween them. The lrger is the istne, the smller is r(r1,r2). Our gol is to efine single mesure for the whole tree tht woul esrie this tren n llow one to test whether the rerrngement proess follow the TFBM with x>0.

96 Multispeies Brekpoint Reuse The multispeies rekpoint reuse is funtion R(L) expressing verge rekpoint reuse etween pirs of rerrngements seprte y L other rerrngements in the given tree. It n e expliitly efine s: R(L) = Σ r(r1,r2) / Σ 1 where oth sums re tken over ll pirs of rerrngements r1 n r2 t istne L in the tree.

97 Multispeies Brekpoint Reuse Test For RBM/FBM, R(L) is onstnt. For TFBM with x > 0, R(L) is eresing funtion. MBR Test: ompute R(L), n hek if it is eresing. (A stronger vrint: etermine x n hek if x>0.)

98 Multispeies Brekpoint Reuse in TFBM (theoreti urve) For TFBM with prmeters m, n, x, we erive n nlyti formul: R(L) = 8(m-n)/(mn) * ( 1 xm/(n(m-n)) )L + 8/m For smll L, R(L) is pproximte y stright line: 8/n 8x/n2 L whih oes not epen on m. Given R(L), the prmeters n n x n e etermine from the vlue n slope of R(L) t L=0.

99 Multispeies Brekpoint Reuse in TFBM (theoreti vs. empiri urve) Simultions for the se when n=160 out of m=800 frgile regions re tive n vrious turnover rte x

100 From Simulte to Rel Genomes: Complitions It is esy to ompute R(L) for simulte genomes, whose rerrngement history is efine y simultions. For rel genomes, while we n relily reonstrut the nestrl genomes, the ext evolutionry senrios etween them remin miguous.

101 From Simulte to Rel Genomes: Complitions It is esy to ompute R(L) for simulte genomes, whose rerrngement history is efine y simultions. For rel genomes, while we n relily reonstrut the nestrl genomes, the ext evolutionry senrios etween them remin miguous. We n smple rnom senrios inste.

102 Multispeies Reuse etween Mmmlin Genomes Best fit: m 4017 n 196 x 1.12

103 Implitions: How will the Humn Genome Evolve in the Next Million Yers?

104 Preition Power of TFBM Cn we etermine urrently tive regions in the humn genome H from omprison with other mmmlin genomes? RBM provies no lue FBM suggests to onsier the rekpoints etween H n ny other genome TFBM suggests to onsier the losest genome suh s the mque-humn nestor QH. QH Brekpoints in G(QH,H) re likely to e reuse in the future rerrngements of H.

105 Vlition of Preitions for the Mque-Humn Anestor (QH) Preition of frgile regions on (QH,H) se on the mouse, rt, n og genomes: Using mouse genome M s proxy: ury 34 / 552 6% Using mouse-rt-og nestor genome MRD: MRD ury 18 / % Using mque genome Q: ury 10 / 68 16% (using synteny loks lrger thn 500K)

106 Puttive Ative Frgile Regions in the Humn Genome

107 Unsolve Mystery: Wht Cuses Frgility? Zho n Bourque, Genome Res. 2009, suggeste tht frgility is promote y Mthing Segmentl Duplitions, pir of long similr regions lote within rekpoint regions flnking rerrngement. TFBM is onsistent with this hypothesis sine the similrity etween MSDs eteriortes with time, implying tht MSDs re lso sujet to irth n eth proess.

108 Aknowlegments Pvel Pevzner, UC Sn Diego Glenn Tesler, UC Sn Diego Jin M, University of Illinois t Urn-Chmpign