Geometric Suffix Tree: A New Index Structure for Protein 3-D Structures

Transcription

1 CPM 2006 Geometrc Suffx ree: A New Idex Structure for Prote 3-D Structures etsuo Shbuya Huma Geome Ceter, Isttute of Medcal Scece, Uversty of okyo

2 oday's alk Backgrouds Prote structures Suffx rees Geometrc suffx tree Geeralzato of suffx trees for dexg rote structures Exermets Coclusos

3 Prote Structure Prote A cha molecule cosstg of 20 kds of amo acds Folded to some structure 3-D structure Coordates of C atoms (backboe) C atom: he reresetatve atom of a amo acd W L V A K E

4 Backgrouds Structurally smlar rotes ted to have smlar fuctos eve f ot smlar the resdue level Structural search o a rote structure database Fuctoal aalyss for rotes wth ewly solved structures Icreasg database sze (PDB: 35,000~ etres) Sohstcated dex structure s desred! A B C It's smlar! Query: Prote structure Prote Structure Database

5 Suffx ree [Weer '73] A sohstcated dex structure for strgs Comacted tre of all the suffxes of a strg S Each leaf corresods to a suffx of S Eables effcet substrg search Suffx tree of 'mssss$' All the suffxes mssss$ ssss$ ssss$ sss$ ss$ ss$ s$ $ $ $ $ $ $ ss $ ss ss$ $ mssss$ s $ $ s $ ss$ ss$

6 Suffx ree Features Lear-tme costructo Varous atter matchg alcatos Motf fdg Reeat fdg Large-scale algmet etc. Good!...But they are ot for structures...

7 oday's oc Exted suffx trees for rote structures "Geometrc Suffx ree" based o the RMSD measures Related Work Suffx trees for rote 3-D structure PSIS [Gao et al. '05] Covert structures to alhabetcal strgs Does ot suort RMSD query A B

8 How to comare two rotes? RMSD: Root Mea Square Devato he most famous measure for rote structure comarso RMSD( A, B) R, v m a R ( b v) 2 / b 5 a 4 a 5 a b 3 b 4 a 2 b b 2 a3 Corresodece of atoms s gve

9 Problem Gve a substructure of a rote as a query a structure DB Fd all the smlar substructures.e. RMSD some gve boud d 'All' meas o false egatves/ostves Query a rote substructure Search! Prote Structure Database It's smlar! (.e. RMSD d) A B C

10 Geometrc re ree that reresets multle structures Smlar refx substructures are comacted to a edge.e. f sqrt(l) RMSD b (l: legth of the refx, b: gve boud) RMSD(P,Q ) s ot always RMSD(P +,Q + ) But sqrt(l) RMSD(P,Q ) s always sqrt(l+) RMSD(P +,Q + ) Edge formato - O() sze!, j: start / ed dces (+sequece#) R, v: the rotato matrx ad the traslato vector 2 q 2 q q 3 P[..7], R, v RMSD b/sqrt(7) q 4 q 5 q 6 q 7 q 8 q 9 P[8..], R, v Q[8..], R 2, v q 0 q P[..7]Q[8..] s smlar to Q[.]

11 Geometrc ree Geometrc tre over all the suffx substructures O() sace (though there are O( 2 ) substructures) But how to comute the RMSD cremetally? RMSD, R, ad v betwee [..6] ad q[..6] 2 q 2 q comute cremetally q 3 q 4 q 5 q 6 q 7 q 8 q RMSD, R, ad v betwee [..7] ad q[..7] q 0 q

12 RMSD Comutato raslato otmzato Ideedet from Rotato Otmzato raslate so that two cetrods comes to the same osto Rotato otmzato Rotate after traslato mmze R R q raslated vectors 2

13 Otmal Rotato for Mmzg RMSD Problem [Aru et al. '87, Schwartz et al. '87] Soluto by SVD (Sgular Value Decomosto) Comutato tme: O() Post-rocessg s requred some rare degeerate cases R VU mmze R where UΣV s the SVD of R q H 2 q 3x3 matrx

14 Icremetal RMSD Comutato heorem: he value of the RMSD, R ad v ca be comuted costat tme f we are gve the followg values whch ca be comuted cremetally! q q q q

15 Costructo Algorthm Just add avely each suffx substructures O( 2 ) tme for a strg of sze cf. O( 3 ) tme f we do ot use cremetal RMSD comutato O(k 2 ) tme for k structures of szes at most New odes

16 Search Algorthm Search for all the odes wth (refx) structures of legth l whose RMSD to the query s b/sqrt(l) + d where l: query legth b: boud used costructo Check whether RMSD d Search for substructures wth RMSD d

17 Comutato me Costructo me Lear to DB sze Due to the rote legth boud Search me Iut Query: 50 DB 37 related structures 4,79 atoms RMSD Boud:.0Å Results the boud most ofte used rote aalyss Search tme 0.39 sec About 3 tmes faster tha the ave search Reasoable boud 9 hts foud CPU:.2GHz UltraSPARC III Cu

18 Coclusos Geometrc suffx trees Suffx trees exteded for Prote 3-D structures Future work More flexble smlarty search Faster algorthms (costructo / query) Boformatcs alcatos Motf fdg / fuctoal aalyss / rote structure clusterg

19 hak you very much.

20 URMSD: Ut-Vector Root Mea Square Devato Varato of the RMSD URMSD( A, B) m R, v u R ( w v) 2 / w 7 where u w ( a+ a ) / a+ ( b + b ) / b + a b u 5 w 6 u 4 u 6 u 7 w 5 w 2 u u 2 u 3 w w 3 w 4

21 Otmzg Rotato Problem wthout traslato 2 q R mmze rotato R raslated vectors.e. ) ( 2 ) ( )} ( { q R trace q q R q Rq q q q R Let t be H

22 s for Otmzg Rotato heorem Gve: Postve defte matrx M ad orthogoal matrx Q Proerty: trace(m) > trace(qm) he... If RH s ostve defte, It s the R ad H to comute! Note: here's a degeerate case that R s a symmetrc matrx Let RVU where HU V (Sgular value decomosto) RHV V s ostve defte! SVD of H ca be comuted costat tme (as H s a 3x3 matrx) How to guaratee that R s ot a symmetrc matrx? det(r) should be (It s - case of a symmetrc matrx) If the object s o a 2-D lae, t s easy to comute the actual R by flg R. I other cases, t s dffcult to comute the actual matrx... but t's a rare case. (Heurstcally, the above fled R could be used.)

23 Sgular Value Decomosto Comutato tme: O( 3 ) (for a x matrx) matrx A Orthoormal vectors v, v2,... Orthoormalzed traslated vectors u, u2,... u u v v A σ σ σ O 2 2 2,...), (,...), ( V U V U A Postve, dagoal matrx

24 URMSD: Ut-Vector Root Mea Square Devato Varato of the RMSD URMSD( A, B) m R, v u R ( w v) 2 / w 7 where u w ( a+ a ) / a+ ( b + b ) / b + a b u 5 w 6 u 4 u 6 u 7 w 5 w 2 u u 2 u 3 w w 3 w 4