Multi-seed lossless filtration

Multi-seed lossless filtrationMulti-seed lossless filtration

Gregory KucherovLaurent Noé

LORIA/INRIA, Nancy, France

Mikhail RoytbergInstitute of Mathematical Problems in Biology,

Puschino, Russia

CPM (Istanbul)July 5-7, 2004

Text filtration: general principleText filtration: general principle

potential matches


potential matches


lossless and lossy filters

true match

Filtration applied to sequence comparisonFiltration applied to sequence comparison

potential similarities

Filtration applied to sequence alignmentFiltration applied to sequence alignment

potential similarities

Filtration applied to sequence alignmentFiltration applied to sequence alignment

true similarities

GaplessGapless similarit similarities. Hamming distance.ies. Hamming distance.

Similarities are defined through Hamming distance

GCTACGACTTCGAGCTGC

...CTCAGCTATGACCTCGAGCGGCCTATCTA...





(m,k)-problem, (m,k)-instances

m

k



(m,k)-problem, (m,k)-instances This work: lossless filtering

m

k

Filtering by contiguous fragmentFiltering by contiguous fragment

PEX (Navarro&Raffinot 2002)– Searching for a contiguous pattern

PEX with errors– Searching for a contiguous pattern with l possible errors

• requires retrieval of all l-variants in the index. Efficient for– small alphabets (ADN,ARN)– relatively small l (<= 2)

m=18

k=3

11

km

####

conserved1

#########(1)

(m,k)

Superposition of two filtersSuperposition of two filters

Pevzner&Waterman 1995

Idea: combine PEX with another filter based on a regularly-spaced seed

PEX :

spaced PEX (matches occurring at every k positions).

####

#---#---#---#

#---#---#---# #---#---#---# #---#---#---# #---#---#---#

k+1

Spaced seedsSpaced seeds

Spaced seeds (spaced Q-grams)– proposed by Burkhardt & Kärkkäinen (CPM 2001) for solving (m,k)-

problems

Principle– Searching for spaced rather than contiguous patterns

– Selectivity• defined by the weight of the seed (number of #’s)

###-##

ExExaamplemple: (18,3)-problem: (18,3)-problem

###-##

###-##

###-##

###-## ###-## ###-##

Spaced seeds for sequence comparisonSpaced seeds for sequence comparison

Ma, Tromp, Li 2002 (PatternHunter)

Estimating seed sensitivity: Keich et al 2002, Buhler et al 2003, Brejova et al 2003, Choi&Zhang 2004, Choi et al 2004, Kucherov et al 2004, ...

Extended seed models: BLASTZ 2003, Brejova et al 2003, Chen&Sung 2003, Noé&Kucherov 2004, ...

This work: lossless filtration using spaced seed families (extension of Burkhard&Karkkainen 2001)

single filter based on several distinct seeds each seed detects a part of (m,k)-instances but

together they must detect all (m,k)-instances

Families of spaced seedsFamilies of spaced seeds

Independent work (lossy seed families for sequence alignment):

Li, Ma, Kisman, Tromp 2004 (PatternHunter II) Xu, Brown, Li, Ma, this conference Sun, Buhler, RECOMB 2004 (Mandala)

– every (18,3)-instance contains an occurrence of a seed of F

– all seeds of the family have the same weight 7

Example: (18.3)-problem (cont)Example: (18.3)-problem (cont)

Family F solvesthe (18,3)-problem

##-#-#######---#--##-#

F

##-##-########-####--#####-##---#-#####----####-######---#-#-##-#####-#-#-#-----###

Example: (18.3)-problem (cont)Example: (18.3)-problem (cont)

##-#-#######---#--##-#

###-##---#-###

###---#--##-# ###---#--##-#

w=7

w=9

####

###-##

##-##-########-####--#####-##---#-#####----####-######---#-#-##-#####-#-#-#-----###

Comparative selectivityComparative selectivity

##-#-#######---#--##-#

w=4 ~39. 10-4

w=5 ~9.8 10-4

w=7 ~1.2 10-4

w=9 ~0.23 10-4

Selectivity of families on Bernoulli similarities (p(match) = 1/4) estimated as the probability for one of the seeds to occur at a given position

How far should we goHow far should we go

A trivial extreme solution ... – would be to pick all seeds of weight m - k. – prohibitive cost except for very small problems

We are interested in intermediate solutions:– relatively small number of seeds (< 10) to keep the hash table of a

reasonable size,– the seed weight sufficiently large to obtain a good selectivity

kmC ~

ResultsResults

Computing properties of seed families Seed design

– Seed expansion/contraction– Periodic seeds– Seed optimality– Heuristic seed design

Experiments– Examples of designed seed families– Application to computing specific oligonucleotides

Conclusions

MeMeaasursuringing the the efficefficiency of a familyiency of a family

Optimal threshold (Burkhard&Karkkainen): minimal number of seed occurrences over all (m,k)-instances

A seed family F is lossless iff the optimal threshold TF(m,k)1

TF(m,k) can be computed by a dynamic programming algorithm in time O(m·k·2(S+1)) and space O(k·2(S+1)), where S is the maximal length of a seed from F

optimizations are possible (see the paper) the resulting space complexity is the same as in the

Burkhard&Karkkainen algorithm

MeMeaasursuringing the the efficefficiency of a family (cont)iency of a family (cont)

Using a similar DP technique we can compute, within the same time complexity bound:

the number UF(m,k) of undetected (m,k)-similarities for a (lossy) family F

the contribution of a seed of F, i.e. the number of (m,k)-similarities detected exclusively by this seed

[see the paper for details]

Design Design of seedof seed famil familiesies

Pruning exhaustive search tree (Burkhard&Karkkainen)

– Construct all solutions of weight w from solutions of weight w – 1

– Example:if ##--#--# and ##-#---# are solutions of weight w-1,

consider their «union» ##-##--# of weight w.

– Prohibitive cost: • more than a week for computing all single-seed solutions of

the (50,5)-problem• the search space blows up for multi-seed families

Seed expansion/contractionSeed expansion/contraction

Burkhard&Karkkainen: the only two solutions of weight 12 solving the (50,5)-problem:

###-#--###-#--###-#

#-#-#---#-----#-#-#---#-----#-#-#---#



###-#--###-#--###-#

#-#-#---#-----#-#-#---#-----#-#-#---#

the only solution of weight 12 of the (25,2)-problem



###-#--###-#--###-#

#-#-#---#-----#-#-#---#-----#-#-#---#

– Let be the i-regular expansion of F obtained by inserting i-1 jokers between successive positions of each seed of F

– Example:If F = { ###-# , ##-## } then

= { #-#-#---# , #-#---#-# } = { #--#--#-----# , #--#-----#--# }

Fi

F2F3

the only solution of weight 12 of the (25,2)-problem

Seed expansion/contractionSeed expansion/contraction (cont)(cont)

Lemma:

– If a family F solves an (m,k)–problem, then both F and solves the (i·m, (i+1)·k- 1)–problem

– If a family solves the (i·m,k)–problem, then its i-contraction F solves the (m, )-problem

Fi

Fi

ik

##-#-#######---#--##-#

##-#-#######---#--##-#

#-#---#---#-#-#-##-#-#-------#-----#-#-#

(18,3)

(36,7)

Periodic seedsPeriodic seeds

Iterating short seeds with good properties

into longer seeds

###-#--###-#--###-#

###-#--

Cyclic problemCyclic problem

Lemma: If a seed Q solves a cyclic (m,k)-problem, then the seed Qi=[Q,- (m-s(Q))]i solves the linear (m·(i+1)+s(Q)-1,k)-problem.

Cyclic (11,3)-problem

Linear (30,3)-problem

###-#--#---

###-#--#---###-#--#

Extension to multi-seed caseExtension to multi-seed case



###-#--#---

###-#--#---###-#--##--#---###-#--#---###

Extension to multi-seed caseExtension to multi-seed case



###-#--#---

###-#--#---###-#--# #--#---###-#--#---###

AAsymptotsymptotic optimalityic optimality

Theorem:Fix a number of errors k. Let w(m) be the maximal weight

of a seed solving the linear (m,k)-problem. Then

the fraction of the number of jokers tends to 0 but the convergence speed depends on k

seed expansion cannot provide an asymptotically optimal solution

( )

Non-asymptotic optimality Non-asymptotic optimality

Fix a number of errors k. For each seed (seed family) Q there exists mQ s.t.

mmQ, Q solves the (m,k)-problem

For a class of seeds , Q is an optimal seed in iff Q realizes the minimal mQ over all seeds of

Lemma: Let n be an integer and r=n/3. For every k2, seed #n-

r-#r is optimal among seeds of weight n with one joker.

Heuristic seed design: genetic algorithmHeuristic seed design: genetic algorithm

a population of seed families is evolving by mutating and crossing over

seed families are screened against sets of difficult (m,k)-instances

for a family that detects all difficult instances, the number of undetected similarities is computed by a DP algorithm. A family is kept if it yields a smaller number than currently known families

compute the contribution of each seed of the family. Mutate the less “valuable” seeds.

difficult(m,k)-instances

seed families

select and reorderselect

Example: (25,2)-problemExample: (25,2)-problem

Example: (25,3)-problemExample: (25,3)-problem

Application Application of lossless filtering: of lossless filtering: oligooligo design design

Specific oligonucleotides: small DNA molecules (10-50bp) that hybridize with a given target sequence and do not hybridize with the other background sequences (e.g. the rest of the genome)

Formalization: given a sequence, find all windows of length m which do not occur elsewhere within k substitution errors

Seed design: (32,5)-problemSeed design: (32,5)-problem

ExperimentExperiment

This filter has been applied to the rice EST database (100015 sequences of total size ~42 Mbp)

All 32-windows occurring elsewhere within 5 errors have been computed

The computation took slightly more than 1 hour on a P4 3GHz computer

87% of the database have been “filtered out”

Further questionsFurther questions

Combinatorial structure of optimal seed families

Efficient design algorithm

QuestionsQuestions

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Multi-seed lossless filtration