Multi-seed lossless filtration Gregory Kucherov Laurent Noé LORIA/INRIA, Nancy, France Mikhail Roytberg Institute of Mathematical Problems in Biology, Puschino, Russia CPM (Istanbul) July 5-7, 2004

Text filtration: general principle potential matches

Text filtration: general principle potential matches

Text filtration: general principle lossless and lossy filters true match

Filtration applied to sequence comparison potential similarities

Filtration applied to sequence alignment potential similarities

Filtration applied to sequence alignment true similarities

Gapless similarities. Hamming distance.  Similarities are defined through Hamming distance GCTACGACTTCGAGCTGC...CTCAGCTATGACCTCGAGCGGCCTATCTA...

Gapless similarities. Hamming distance.  Similarities are defined through Hamming distance

Gapless similarities. Hamming distance.  Similarities are defined through Hamming distance  ( m, k )-problem, ( m, k )-instances m k

Gapless similarities. Hamming distance.  Similarities are defined through Hamming distance  ( m, k )-problem, ( m, k )-instances  This work: lossless filtering m k

Filtering 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 #### ######### (1) (m,k)

Superposition 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 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) ###-##

Example: (18,3)-problem ###-##

Spaced 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 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) Family F solves the (18,3)-problem ##-#-#### ###---#--##-# F

##-##-##### ###-####--## ###-##---#-### ##----####-### ###---#-#-##-## ###-#-#-#-----### Example: (18.3)-problem (cont) ##-#-#### ###---#--##-# ###-##---#-### ###---#--##-# w=7 w=9

#### ###-## ##-##-##### ###-####--## ###-##---#-### ##----####-### ###---#-#-##-## ###-#-#-#-----### Comparative selectivity ##-#-#### ###---#--##-# w=4 ~ w=5 ~ w=7 ~ w=9 ~ 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 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

Results  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

Measuring the efficiency 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 T F (m,k)1  T F (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

Measuring the efficiency of a family (cont) Using a similar DP technique we can compute, within the same time complexity bound:  the number U F (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 of seed families 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/contraction Burkhard&Karkkainen : the only two solutions of weight 12 solving the (50,5)-problem: ###-#--###-#--###-# #-#-#---#-----#-#-#---#-----#-#-#---#

Seed 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

Seed expansion/contraction Burkhard&Karkkainen : the only two solutions of weight 12 solving the (50,5)-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 = { #-#-#---#, #-#---#-# } = { #--#--#-----#, #--#-----#--# } the only solution of weight 12 of the (25,2)-problem

Seed expansion/contraction (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 ##-#-#### ###---#--##-# ##-#-#### ###---#--##-# #-#---#---#-#-#-# #-#-# #-----#-#-# (18,3) (36,7)

Periodic seeds Iterating short seeds with good properties into longer seeds ###-#--###-#--###-# ###-#--

Cyclic problem Lemma: If a seed Q solves a cyclic (m,k)-problem, then the seed Q i =[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 case Cyclic ( 11,3 )-problem Linear ( 25,3 )-problem ###-#--#--- ###-#--#---###-#--# #--#---###-#--#---###

Extension to multi-seed case Cyclic ( 11,3 )-problem Linear ( 25,3 )-problem ###-#--#--- ###-#--#---###-#--# #--#---###-#--#---###

Asymptotic 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  Fix a number of errors k.  For each seed (seed family) Q there exists m Q s.t.  mm Q, Q solves the (m,k)-problem  For a class of seeds , Q is an optimal seed in  iff Q realizes the minimal m Q over all seeds of  Lemma: Let n be an integer and r=n/3. For every k2, seed # n-r -# r is optimal among seeds of weight n with one joker.

Heuristic 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 reorder select

Example: (25,2)-problem

Example: (25,3)-problem

Application of lossless filtering: oligo 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)-problem

Experiment  This filter has been applied to the rice EST database ( 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 questions  Combinatorial structure of optimal seed families  Efficient design algorithm

Questions ? ? ?