Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/35

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1 Efficient Algorithms for Regular Expression Constrained Sequence Alignment Yun-Sheng Chung, Chin Lung Lu, and Chuan Yi Tang Department of Computer Science National Tsing Hua University, Taiwan Department of Biological Science and Technology National Chiao Tung University, Taiwan 17th Annual Symposium on Combinatorial Pattern Matching Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

2 Regular Expression Constraints PROSITE contains biologically important sites and motifs. PROSITE motifs can be represented by regular expressions. P-loop motif: [AG]-x()-G-K-[ST] Regular expression: (A + G)ΣΣΣΣGK(S + T) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

3 Regular Expression Constraints In an alignment, it is reasonable to expect functional sites to be aligned together. Introduced by Arslan (CPM 005). T G F P S V G K T K D D A T - F - S V A - - K D D D G K S A T G F P S V G K T K D D A T F S V A K D D D G K S A (G + A)ΣΣΣΣGK(S + T): the P-loop motif. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

4 Outline Related works Weighted automata Arslan s algorithm Our algorithms Conclusion and future works Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

5 Related Works Tang et al. (CSB 00, JBCB 00) introduced the Constrained Multiple Sequence Alignment (CMSA) problem Constraint: a sequence of characters Example: Sequences from RNase family share the conserved sequence of residues H, K, H. In this case the constraint is H, K, H. W W A Q H K P H C H A Q K P Y H C Chin et al. (CSB 00, JBCB 005) then proposed a more effieicnt algorithm for pairwise CSA, along with a -approximation algorithm for CMSA. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 5/5

6 Related Works Tsai et al. (Bioinformatics 00) generalized the definition of constraints. Constraint: Sequence of strings allowing mismatches P 1 = AGCC, P = CG ɛ = 0.5 MuSiC: a web tool A G C C C G A G U C C G Lu and Huang (Bioinformatics 005) gave a more space efficient algorithm MuSiC-ME Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 6/5

7 RECSA Arslan (CPM 005) introduced the regular expression constrained sequence alignment (RECSA) problem. Constraint: Regular expression R. R = (G + A)ΣΣΣΣGK(S + T) T T F S V G A F K P D S D V D G G K K T S K D D A A In this current work, we give more time and space efficient algorithms. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 7/5

8 Weighted Automata: Topology Regular expression constraint R is converted into an ɛ-free NFA A. The topology of a weighted automaton M is essentially the same as the topology of the product machine of A. R = a; Σ- = Σ {-}: (q 0, q 0 ) (q 1, q 0 ) (a, a) a q 0 q 1 (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 8/5

9 Weighted Automata Let δ be the transition function of A. Then δ M, the transition function of M, is defined as δ M ((p, q), (a, b)) = { δ(p, a) δ(q, b) if (p, q) (F F ) {(q 0, q 0 )} (δ(p, a) δ(q, b)) {(p, q)} otherwise (q 0, q 0 ) (a, a) (q 1, q 0 ) a q 0 q 1 (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 9/5

10 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ t c a A = a t c ] (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 10/5

11 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ t c a A = a t c ] (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 11/5

12 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ t c a A = a t c ] (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

13 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ t c a A = a t c ] (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

14 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ t c a A = a t c ] (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

15 Weighted Automata and Alignments S 1 [1..] = tca, S [1..] = atc, R = a [ ] t c a A = a t c score: 1 (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 15/5

16 Weighted Automata and Alignments A reaches the final state if and only if A is a feasible constrained alignment [ ] t c a A = score: 0 a t c (q 0, q 0 ) (q 1, q 0 ) (a, a) (q 0, q 1 ) (q 1, q 1 ) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 16/5

17 Weighted Automata: Scores S 1 [1..] = tca, S [1..] = atc, R = a [ ] t c a A 1 = A = a t c [ t ] c a a t c M, : (match: scored 1; other: scored 0) A 1 A 1 (a, a) 1 A Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 17/5

18 Weighted Automata: Computation Let a = S 1 [i 1 ] and b = S [i ]: ɛ a t ɛ M 0,0 M 0,1 M 0, t M 1,0 M 1,1 M 1, c M,0 M,1 M, S [i ] M i1 1,i 1 M i1 1,i S 1 [i 1 ] M i1,i 1 M i1,i = max{m (a,b) i 1 1,i 1, M (a,-) i 1 1,i, M (-,b) i 1,i 1 } W i1,i (p, q) = max{w (a,b) (a,-) i 1 1,i 1 (p, q), W i 1 1,i (p, q), W (-,b) i 1,i 1 (p, q)} Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 18/5

19 Weighted Automata: Insertion S 1 [1..] = tca, S [1..] = atca Additional edit operation:, with score 0 M, M (-,a), A 1 A 1 (a, a) A 1 [ a (-, ] a) (a, a) A 1 A 1 [ a ] Σ- A Σ- 1 [ a ] The computation can be done by finding in M, all the arcs that are labeled with (or ). Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 19/5

20 Arslan s Algorithm Let r be the size of the transition function of M. In the algorithm by Arslan, the whole automaton is constructed and searched for each (i 1, i ), hence taking O(r) time and space. Total time and space: O(rn ) and O(rn), repectively. Let V be the set of states in A. a c #arcs in A = Θ( Σ V ) in the worst case. a t r = #arcs in M, which is Θ( Σ V ) in the worst case. c g Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 0/5

21 Time and Space Complexity Worst case complexity for Arslan s algorithm: O( Σ V n ) time and O( Σ V n) space. The space complexity is for the computation of the optimal score only. It is not mentioned how to reconstruct the optimal alignment without affecting the space complexity. In this work we propose two algorithms. Worst case time: O( V n ) and O( V log V n ), respectively. Space: O( V n), with the optimal alignment reconstructed. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

22 The First Algorithm It is more efficient to separate the transitions from the scores. (a, a) 1 L, q 0 q 1 q 0 q 1 1 Worst case space complexity: O( Σ V n) O( V n) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

23 Search in A, not M V V Consider the computation of M (a,a), from M,. (a, a) (a, a) 1 a q 0 q 1 L, q 0 q 1 q 0 q 1 1 temp q 0 q 1 q 0 q 1 L (a,a), q 0 q 1 q 0 q Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

24 The First Algorithm Let L be a temporary V V table initialized to be all. for p V do for p V with T [p, S 1 [i 1 ]; p] = 1 do for q V do L[p, q ] max{l[p, q ], L i1 1,i 1[p, q ] + γ(s 1 [i 1 ], S [i ])}; for q V do for q V with T [q, S [i ]; q] = 1 do for p V do L (S 1[i 1 ],S [i ]) i 1 1,i 1 [p, q] max{l (S 1[i 1 ],S [i ]) i 1 1,i 1 [p, q], L[p, q ]}; The insertion and deletion cases are similar. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

25 Reconstruct the Alignment If we know which substrings of S 1 and S are aligned to satisfy the constraint in an optimal constrained alignment, then the optimal alignment can be reconstructed easily in O(n) space. R S R S 1 The indices of such substrings can be found by doing some additional bookkeeping during the computation without affecting the space complexity mentioned previously. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 5/5

26 Reconstruct the Alignment M,0 0 A (a, a) A [ a a ] M (a,a),0 1 (a, a) A = [ t ] c, A [ a ] = [ t ] c a a 1 A [ a ] start: (, 1); end: (, 1) Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 6/5

27 The Second Algorithm The second algorithm requires V = O(log n). Let T [p, a] be the bit vector (T [p, a; ],..., T [p, a; p V ]). Under the assumption, T [p, a] can be stored in a single word. We take L (S 1[i 1 ],-) i 1 1,i [p, q] as an example. The other two cases are similar. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 7/5

28 The Second Algorithm The arcs shown are all transitions in δ labeled with S 1 [i 1 ]. V = {,,, p }. Fix q V. T [, S 1 [i 1 ]] = (1, 0, 1, 0), T [, S 1 [i 1 ]] = (0, 1, 1, 0), T [, S 1 [i 1 ]] = (1, 0, 0, 1), T [p, S 1 [i 1 ]] = (0, 1, 1, 1) L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] 1 1 p p p p L i1 1,i [p, q] L i1 1,i [p, q] Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 8/5

29 The Second Algorithm Y : Bit vector representing the states p in V such that L (S 1[i 1 ],-) i 1 1,i [p, q] has not yet been set. Initially: Y = (1, 1, 1, 1). T [, S 1 [i 1 ]] Y = (1, 0, 0, 1) L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] p p 1 p 1 p L i1 1,i [p, q] L i1 1,i [p, q] Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 9/5

30 The Second Algorithm Y Y T [, S 1 [i 1 ]] Y = (1, 1, 1, 1) (1, 0, 0, 1) = (0, 1, 1, 0) T [, S 1 [i 1 ]] Y = (0, 0, 1, 0) L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] p p 1 p 1 p L i1 1,i [p, q] L i1 1,i [p, q] Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 0/5

31 The Second Algorithm Y Y T [, S 1 [i 1 ]] Y = (0, 1, 1, 0) (0, 0, 1, 0) = (0, 1, 0, 0) T [p, S 1 [i 1 ]] Y = (0, 1, 0, 0) L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] p p 1 p 1 p L i1 1,i [p, q] L i1 1,i [p, q] Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 1/5

32 The Second Algorithm Y Y T [p, S 1 [i 1 ]] Y = (0, 1, 0, 0) (0, 1, 0, 0) = (0, 0, 0, 0) T [, S 1 [i 1 ]] Y = (0, 0, 0, 0) L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] p p 1 p 1 p L i1 1,i [p, q] L i1 1,i [p, q] Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

33 The Second Algorithm L (S 1[i 1 ],-) i 1 1,i [p, q] L (S 1[i 1 ],-) i 1 1,i [p, q] p p 1 p 1 p L i1 1,i [p, q] L i1 1,i [p, q] Time: O( V log V ) for each fixed q; O( V log V ) for the entire L (S 1[i 1 ],-) i 1 1,i Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

34 Conclusion and Future Works More time and space efficient algorithms for regular expression constrained sequence alignment are proposed. A multiple alignment version is important. Generalize the weighted automata structure to obtain optimal solutions: suggested by Arslan (CPM 005). Progressive heuristics: e.g., Lu and Huang s work (Bioinformatics 005). Approximation algorithms: in preparation. Local alignment version would also be useful. Has a similar motivation as PHI-BLAST; may work complementarily by taking a different approach. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. /5

35 Thank you for your attention. Efficient Algorithms forregular Expression Constrained Sequence Alignment p. 5/5