Lagrangian Approaches for a class of Matching Problems in Computational Biology

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Lagrangian Approaches for a class of Matching Problems in Computational Biology

Nicola Yanev, Rumen Andonov, Philippe Veber, Stefan Balev

To cite this version:

Nicola Yanev, Rumen Andonov, Philippe Veber, Stefan Balev. Lagrangian Approaches for a class of Matching Problems in Computational Biology. [Research Report] RR-5973, INRIA. 2006. �inria- 00090635v2�

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inria-00090635, version 2 - 7 Sep 2006

a p p o r t

d e r e c h e r c h e

SN0249-6399ISRNINRIA/RR--5973--FR+ENG

Thème BIO

Lagrangian Approaches for a class of Matching Problems in Computational Biology

Nicola Yanev and Rumen Andonov and Philippe Veber and Stefan Balev

N° 5973

31st August 2006

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Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71

NiolaYanev

∗

and RumenAndonov

†

and PhilippeVeber

†

and Stefan Balev

‡

ThèmeBIOSystèmesbiologiques

ProjetsSymbiose

Rapportdereherhe n°597331stAugust200618pages

Abstrat: Thispaperpresentseientalgorithmsforsolvingtheproblemof aligningaprotein

struture template to a query amino-aid sequene, known as protein threading problem. We

onsider the problem as a speial ase of graph mathing problem. We giveformal graphand

integerprogrammingmodelsof the problem. After studying the properties of these models, we

proposetwokindsofLagrangianrelaxationforsolvingthem. Wepresentexperimentalresultson

reallife instanesshowingtheeieny ofourapproahes.

Key-words: sequene-struturealignment,omplexity,integerprogramming,Lagrangianrelax-

ation

∗

hobymath.bas.org

†

prenom.nomirisa.fr

‡

stefan.balevuniv-lehavre.fr

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d'une lasse de problème d'appariement en bioinformatique

Résumé : Cet artile propose des algorithmes eaes pour déterminer l'alignement optimal

entreunestruture etune séqueneprotéique,problèmeonnu souslenom deprotein threading.

Nous posons e problème ommeun as partiulier d'appariement. Nousprésentonsunmodèle

formelduproblèmesouslaformed'unefamilledegraphes,etdesprogrammesennombreentiers

orrespondants. Nousétudionsdansunpremiertempslespropriétésdeesmodèles,pourensuite

proposer deuxapprohesde relaxationlagrangiennepourlarésolution. Enn, nousmontrons,à

l'aidededonnéesexpérimentalessurdesinstanesréelles,l'eaitédeesapprohes.

Mots-lés : alignement séquene-struture, omplexité, programmation en nombres entiers,

relaxationlagrangienne

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A C G C A A

A G T C T

ACG_CAA A_GTC_T

Figure1: Mathinginterpretationofsequenealignmentproblem

1 Preliminaries

Mathing is important lass of ombinatorial optimization problems with many real-life appli-

ations. Mathing problems involve hoosing a subset of edges of a graph subjet to degree

onstraintsontheverties. Manyalignmentproblemsarisinginomputationalbiologyarespeial

ases of mathing in bipartite graphs. In these problems the verties of the graph anbe nu-

leotidesofaDNAsequene,aminoaidsofaproteinsequeneorseondarystrutureelementsof

aproteinstruture. Unlikelassialmathingproblems, alignmentproblems haveintrinsiorder

onthegraphvertiesandthisimpliesextraonstraintsontheedges. Asanexample,Fig.1shows

an alignment of two sequenes as a mathing in bipartite graph. We an see that the feasible

alignmentsare1-mathingswithoutrossingedges.

Inthis paper we deal with the problem of aligning aprotein struture template to a query

proteinsequeneoflengthN^,^knownâs^protein^threading^problem^(PTP).Â^templateîsânôrdered

set of m ^seondary^struture êlements^(or ^bloks)ôf ^lengthsli^, i = 1, . . . , m^. Ân âlignment^(or

threading)isoveringofontiguoussequeneareasbythebloks. Athreadingisalledfeasibleif

theblokspreservetheirorderanddonotoverlap. A threadingisompletely determinedbythe

startingpositionsof allbloks. Forthesakeofsimpliitywewilluserelativepositions. Ifbloki

startsatthej^th^query^harater,^its^relative^position^isri=j−Pi−1

k=1lk^. ^In^this^way^the^possible

(relative)positionsofeahsegmentarebetween1andn=N+ 1−P^m

i=1li ^(see^Fig.^2(b)). ^The

setoffeasiblethreadingsis

T ={(r1, . . . , rm)|1≤r1≤ · · · ≤rm≤n}.

Proteinthreadingproblem isamathingproblemin abipartitegraph(U∪V, U×V)^,^where U ={u1, . . . , um} îs^theôrdered^set ôf^bloksând V ={v1, . . . , vn}îs ^theôrdered^set ôf^relative

positions. Thethreadingfeasibilityonditionsanberestatedintermsofmathinginthefollowing

way. AmathingM ⊆U×V ^is^feasible^if:

(i) d(u) = 1^,u∈U ^(whered(x)îs ^the^degreeôfx^). ^This^means^that êah ^blokîs âssigned^to

exatlyoneposition). Bythewaythisimpliesthattheardinalityofeahfeasiblemathing

ism^.

(ii) There are no rossing edges, or more preisely, if (ui, vj) ∈ M^, (uk, vl) ∈ M ^and i < k^,

then j ≤l^. ^This ^means^that ^the ^bloks^preserve^their ôrder ând ^do^notôverlap. ^The^last

inequalityisnotstritbeauseofusingrelativepositions.

Note that while (i) isa lassialmathing onstraint, (ii)is spei for thealignmentproblems

andmakesthemmorediult. Fig.2()showsamathingorrespondingtoafeasiblethreading.

Proposition 1. Thenumberof feasiblethreadings is|T |= ^m+n−1_m

.

Proof. Wean denetherelativepositionsasri =j−Pi−1

k=1lk+i−1^. ^In^this^ase^the^relative

positionsofthefeasiblethreadingsare relatedby

1≤r1<· · ·< rm≤m+n−1

andathreadingisdeterminedbyhoosingm^out^ofm+n−1^positions.

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(a)

abs.position 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16 1718 19 20

rel.positionblok1 1 2 3 4 5 6 7 8 9

(b)

1 2 3

1 2 3 4 5 6 7 8 9

V

U

()

Figure 2: (a) Example of alignmentof query sequene of length 20 and template ontaining 3

segmentsoflengths3,5and4. (b)Correspondenebetweenabsoluteandrelativeblokpositions.

()Amathing orrespondingtothealignmentof(a).

One of the possible ways to deal with alignment problems is to try to adapt the existing

mathing tehniquesto thenewedgeonstraintsof type(ii). Insteadof doingthisweproposea

newgraphmodel andwedevelopeientmathingalgorithmsbasedonthismodel.

Weintrodue analignment graph G= (U×V, E)^. Êah ^vertex ôf^this ^graphôrresponds^to

an edge of themathing graph. Forsimpliity wewill denote thevertiesby vij^, i = 1, . . . , m^, j = 1, . . . , n ând ^draw ^them âs ân n×m ^grid ^(see ^Fig. ^3). ^The ^verties vij^, j = 1, . . . , n ^will

be alled i^th ^layer. Â ^layer ôrresponds ^to â ^blok ând êah ^vertex ⁱⁿ â ^layerôrresponds ^to

positioningofthisblokin thequerysequene.

Oneanonnetbyedgesthepairsofvertiesof G^whih ^orrespond^to ^pairs^of ^nonrossing

edges in the mathing graph. In this ase a feasible threading is an m^-lique ⁱⁿ G^. ^A ^similar

approahisusedin[12℄. Weintrodueonlyasubsetoftheaboveedges,namelytheonesthaton-

netvertiesfromadjaentolumnsandhavethefollowingregularpattern: E={(vij, vi+1,l)|i= 1, . . . , m−1,1 ≤j ≤l ≤n}^. ^We âdd^two^more^vertiesS ând T ând êdgesônnetingS ^to âll

vertiesfrom the rstolumn and T ^to âll^verties^from ^the ^last ôlumn. ^Nowît îs êasy^to ^see

theone-to-one orrespondene betweentheset of feasiblethreadings (or mathings)and theset

ofS^-T ^pathsⁱⁿG^. ^Fig.³illustratesthisorrespondene.

Till now we gave several alternative ways to desribe the feasible alignments. Alignment

problemsinomputationalbiologyinvolvehoosingthebestofthembasedonsomesorefuntion.

Thesimplestsorefuntions assoiateweightsto theedges ofthemathinggraph. Forexample,

this istheaseofsequene alignmentproblems. Byintroduingalignmentgraphssimilarto the

above, lassialsequenealignment algorithms, suh asSmith-Waterman orNeedleman-Wunh,

anbeviewedasndingshortestS^-T ^paths. ^When^the^sore^funtions^use^struturalinformation, theproblemsaremorediultandtheshortestpathmodelannotinorporatethisinformation.

Thesorefuntionsin PTPevaluatethedegreeofompatibilitybetweenthesequeneamino

aidsandtheirpositionsinthetemplatebloks. Theinterations(orlinks)betweenthetemplate

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block position

T S

i = 1 i = 2 i = 3 i = 4 i = 5 i = 6

j = 1 j = 2 j = 3 j = 4

Figure 3: Exampleof alignmentgraph. The pathin thiklines orresponds to thethreading in

whihthepositionsofthebloksare1,2,2,3,4,4.

bloksaredesribedbytheso-alledgeneralizedontatmapgraph,whosevertiesarethebloks

andwhoseedgesonnetpairsof interatingbloks. LetL^be^the^set^of^these^edges:

L={(i, k)|i < kând^bloksi ândkînterat}

Sometimes we need to distinguish the links between adjaent bloks and the other links. Let

R={(i, k)|(i, k)∈L, k−i >1} ^be^the^set^of^remote^(or^non-loal)^links. ^The^links^from L\R

arealled loallinks. Withoutlossofgeneralityweansuppose thatallpairsofadjaentbloks

interat.

Thelinksbetweenthebloksgeneratesoreswhihdependontheblokpositions. Inthisway

asorefuntionofPTP anbepresentedbythefollowingsetsofoeients

cij^, i= 1, . . . , m^, j= 1, . . . , n^,^the^sore^of^putting ^bloki^on^position j

dijkl^,(i, k)∈L^,1≤j≤l≤n^,^the^sore^generated^by^theînteration^between^bloksiând k ^when^blokiîsôn^positionj ând^blokk îsôn^positionl^.

Theoeientscij âre^some^funtion ^(usually ^sum) ôf^the^preferenes ôfêah ^queryâminoâid

plaedin bloki^forôupyingîtsâssigned^position,âs^wellâs^the^soresôf^pairwiseinterations betweenaminoaidsbelongingtobloki^. ^Theôeientsdijkl înlude^the^soresôfinterations between pairs of amino aids belonging to bloks i ând j^. ^Loops ^(sequenes ^between âdjaent

bloks)mayalsohavesequenespei sores,inludedintheoeientsdi,j,i+1,l^.

The soreof a threading is the sum of the orresponding sore oeientsand PTP is the

optimizationproblem ofndingthethreadingof minimumsore. Iftherearenoremotelinks (if

R=∅⁾^weân^put^the^soreôeientsôn^the^vertiesând^theêdgesôf^theâlignment^graphând

PTP isequivalentto theproblem ofnding theshortestS^-T ^path. ^In^order ^to^take^the ^remote

linksinto aount,weaddtothealignmentgraphtheedges

{(vij, vkl)|(i, k)∈R, 1≤j≤l≤n}

whihwewillreferasz^-edges.

An S^-T ^pathîs^said^toâtivate^thez^-edges^that ^have^bothêndsôn^this^path. ÊahS^-T ^path

ativates exatly|R| z^-edges, ône ^forêah ^link ⁱⁿ R^. ^The ^subgraphîndued ^by ^theêdges ôf ân S^-T ^pathând^theâtivatedz^-edgesîsâlled âugmented^path. ^Thus^PTPîsêquivalent^to ^nding

theshortestaugmentedpathin thealignmentgraph(seeFig.4).

Aswewillseelater,themainadvantageofthisgraphisthatsomesimplealignmentproblems

redue to nding the shortest S^-T ^path ⁱⁿ ît ^with ^some ^pries âssoiated ^to ^the êdges ând/or

verties. The last problem an be easily solved by a trivial dynami programming algorithm

of omplexity O(mn²)^. În ôrder ^to âddress ^the ^general âse ^we ^need ^to ^represent ^this ^graph

optimisationproblem asanintegerprogrammingproblem.

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j = 2 j = 3 j = 4

i = 1 i = 2 i = 3 i = 4 i = 5 i = 6

j = 1

block position

T S

c₁₁₂₂

c2232

c3243

c₄₃₅₄

c₅₄₆₄

c1132

c3264

c4364

Figure4: Exampleofaugmentedpath. Thegeneralizedontatmapgraphisgiveninthebottom.

Thexârsôf^theS^-T ^pathâreⁱⁿ^solid^lines. ^Theâtivatedz^-arsâreⁱⁿ^dashed^lines. ^The^length

oftheaugmentedpathisequaltothesoreof thethreading(1,2,2,3,4,4)^.

2 Integer programming formulation

Letyij ^be^binary^variablesâssoiated^to^the^vertiesôfG^. yij îsôneîf^blokiîsôn^positionjând

zerootherwise. LetY ^be^the^polytope^dened^by^the^followingonstraints:

n

X

j=1

yij= 1 i= 1, . . . , m ⁽¹⁾

j

X

l=1

yil−

j

X

l=1

yi+1,l≥0 i= 1, . . . , m−1, j= 1, . . . , n−1 ⁽²⁾ yij ≥0 i= 1, . . . , m, j= 1, . . . , n ⁽³⁾

Constraints(1) ensure the feasibility ondition (i) and(2) are responsiblefor (ii). That is why

Y ∩B^mnîsêxatly^the^setôf^feasiblethreadings.

In order totakeinto aountthe interationosts, weintrodue aseond set ofbinary vari-

ables zijkl^, (i, k) ∈ L^, 1 ≤ j ≤ l ≤ n^. ^Tô âvoid âdded ^notation ^we ^will ûse ^vetor ^nota-

tion for the variables yi = (yi1, ...yin) ∈ Bⁿ ^with âssigned ôsts ci = (ci1, ...cin) ∈ Rⁿ ând zik = (zi1k1, . . . , zi1kn, zi2k2, . . . , zi2kn, . . . , zinkn) ∈ Bⁿ⁽ⁿ⁺¹⁾² ^for (i, k) ∈ L ^with âssigned ôsts dik = (di1k1, . . . , di1kn, di2k2, . . . , di2kn, . . . , dinkn)∈R

n(n+1) 2 .

Considerthe 2n×ⁿ⁽ⁿ⁺¹⁾₂ ^node-edgeînidene ^matrixôf ^the^subgraph^spanned ^by^twoînter-

atinglayersiândk^. ^The^submatrixA^′ ôntaining^the^rstn^rows^(resp. A^′′ôntaing^the^last n

rows)orrespondsto thelayeri^(resp. ^layerk^).

Nowtheproteinthreadingproblemanbedenedas

z_IP^L =v(P T P(L)) = min{

m

X

i=1

ciyi+ X

(i,k)∈L

dikzik} ⁽⁴⁾

subjetto: y= (y1, . . . , ym)∈Y, ⁽⁵⁾

yi=A^′zik (i, k)∈L ⁽⁶⁾

yk=A^′′zik (i, k)∈L ⁽⁷⁾

zik∈Bⁿ⁽ⁿ⁺¹⁾² (i, k)∈L ⁽⁸⁾

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Theshortutnotationv(.)^will ^beûsed ^for^theôptimalôbjetive^funtion ^valueôfâ^subproblem

obtainedfrom P T P(L)^with^somez ^variables^xed.

3 Complexity results

In this setion we study thestruture of thepolytopedened by (5)-(7) and zik ∈ R

n(n+1) 2

+ ^, ^as

wellastheimpatofthesetLôn^theômplexityôf^theâlgorithms^for^solving^theP T P ^problem.

Throughoutthissetion,vertexostsciâreâssumed^to^be^zero. ^Thisâssumptionîs^not^restritive

beausetheostscij ân^beâdded^todi,j,i+1,l^,l=j, . . . , n^. ^We^willônsider^theôstsdik âsn×n

matries ontainingthe oeients dijkl âbove^the ^main ^diagonal ând ârbitrary ^large ^numbers

below the main diagonal. In order to simplify the desriptions of the algorithms given in this

setionweintroduethefollowingmatrixoperations.

Denition 1. LetA ând B ^be ^two^matries ôfômpatible ^size. A·B îs^the ^matrix^produtôf A ând B ^where ^the âdditionôperation îs ^replaed ^bymin ând^the multipliation operationis replaedby+.

Denition 2. LetA ândB ^be^two^matries ôf^size n×n^. M =A⊗B îs^dened ^by M(i, j) = mini≤r≤jA(i, r) +B(i, j)

Belowwepresentfourkindsofontatgraphsthat makePTPpolynomiallysolvable.

3.1 Contat graph ontains only loal edges

Asmentionedabove,inthisasePTPredues tondingtheshortestS^-T ^pathⁱⁿ^the^alignment

graphwhihanbedonebyO(mn²)^dynamiprogrammingalgorithm. Animportantpropertyof analignmentgraphontainingonlyloaledgesisthatithasatightLPdesription.

Theorem1. Thepolytope Y îsîntegral, î.e. ît^hasônly integer-valuedverties.

Proof. Let A ^be ^the ^matrixôf ^the ôeientsⁱⁿ ^(1)-(2) ^with ôlumns ^numbered^by ^theîndies

of the variables. Onean provethat A ^is ^totaly ^unimodular ^(TU) ^by ^performing ^the^following

sequeneofTUpreservingtransformations.

fori= 1, . . . , n

deleteolumn(i, n)^(theseâreûnitôlumns)

fori= 1, . . . , m

forj=n−1, . . . ,1

pivotonaij ⁽Aîs^TUⁱ^the^matrixôbtained^byâ^pivotôperationônAîs^TU

deleteolumn(i, j)^(now^thisîsûnitôlumn)

ThenalmatrixisanunitolumnthatisTU.SineallthetransformationsareTUpreserving,

Aîs^TUândY îsîntegral.

One ouldprovethe sameassertionbyshowingthat anarbitrary feasiblesolutionto (1)-(3)

isaonvexombination ofsomeinteger-valuedvertiesof Y^. ^The^best^suh^vertex⁽ⁱⁿ^the^sense

ofanobjetivefuntion)mightbeagoodapproximatesolutiontoaproblemwhosefeasiblesetis

anintersetionofY ^with^additionalonstraints.

Let y îs ân ârbitrary^non ^-integer^solution ^to ^(1)-(3). ^Beause ôf ^(1), ⁽²⁾ ân ûnit ôw¹ f = (fsj, f(i,k)(i+1,j)) i= 1, m−1 j= 1, n ⁱⁿ Gêxist^s.t.

X

k≤j

f(i,k)(i+1,j)=yij i= 1, m−1 fsj=y_1j j= 1, n

Bythewellknownproperties ofthenetworkowpolytope,the owf ân^beêxpressedâsâ

onvexombinationof integer-valuedunit ows(pathsinG)^. ^Butêah^suh ôwôrresponds^to

1

The4indeesi, k, p, jûsed^forârs^labeling^follows^the ônvention: ^tailât^vertex(i, k)^headât^vertex(p, j)^.

Sometimesthebraketswillbedropped.