Weighted Finite-State Transducers in Computational Biology

Transcription

1 Weighted Finite-State Transducers in Computational Biology Mehryar Mohri Courant Institute of Mathematical Sciences Joint work with Corinna Cortes (Google Research). 1

2 This Tutorial Weighted finite-state transducers and software library Pairwise alignments for bioinformatics Learning weighted transducers Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 2 2

3 Biological Motivation General idea: use sequence similarity and known properties of some proteins to predict functional and structural information for other proteins. Applications: Reconstruction of long DNA sequences, Searching and mining DNA databases, Finding frequent nucleotide patterns, Determining informative sequences. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 3 3

4 Why Weighted Finite-State Transducers? (Mohri, Pereira, and Riley, 2000) Generality Unified representation framework Single general algorithm No need to design and implement a novel algorithm for each new sequence alignment problem Just define a new alignment transducer Alignments between sets of strings, finite automata, or weighted automata (Mohri, 2003). Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 4 4

5 Why Weighted Finite-State Transducers? Efficiency Compact representation of alignment transducer Benefit of general transducer optimization algorithms General quadratic-time alignment algorithm Exploits graph representation and sparseness Most pruning ideas used in dynamic programming can be used with WFSTs as well. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 5 5

6 Why Weighted Finite-State Transducers? Flexibility Can use any weighted finite-state transducer Can create more complex alignment transducers using general transducer algorithms, e.g., rational operations Easy to modify, e.g., tailor alignment transducer to a specific problem Graphical representation helps design and understanding Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 6 6

7 Terminology Biology sequence subsequence Computer science: Computer Science string factor - substring Factor: contiguous symbols. Substring or subsequence: nonnecessarily contiguous symbols. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 7 7

8 Local Edits Insertion: ε a Deletion: a ε Substitution: a b (a b) Example: 2 insertions, 3 deletions, 1 substitution cttgɛɛac ɛ taɛgtɛc This is called an alignment. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 8 8

9 Edit-Distance - Global Alignment Let c(a, b) be the cost associated to the alignment of a with b, where a and b are in Σ {ɛ}. Problem: Find the best alignment (minimal cost) of two sequences x and y. Solution: general algorithm; Compute best path of Construct edit transducer T e using c(a, b) costs. Represent x and y by automata X and Y. X T e Y. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 9 9

10 Global Alignment - Example Example: c(a, G) = 1, c(a, T) = c(g, C) =.5, no cost for matching symbols. Representation: A G C T A C C T G C:G/0.5 G:C/0.5 T:A/0.5 A:T/0.5 G:A/1.0 A:G/1.0 T:T/0.0 G:G/0.0 C:C/0.0 A:A/0.0 0 echo A G C T farcompilestrings >X.fsm Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 10 10

11 Global Alignment - Example Program: fsmcompose X.fsm Te.fsm Y.fsm fsmbestpath -n 1 >A.fsm Graphical representation: A:A/0 0 1 G:C/0.5 2 C:C/0 3 T:T/0 4 e:g/2 5/0 e:e/0 1 A:A/0 2 G:C/0.5 3 C:C/0 T:T/0 e:g/ /0 0 e:e/0 7 A:A/0 8 G:e/2 9 e:c/2 10 C:C/0 11 T:T/0 12 e:g/2 13/0 Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 11 11

12 Local Alignment Problem: Find the best local alignment (alignment between factors) of two sequences x and y. Motivation: ignore non-coding regions (introns). Solution: general algorithm; Construct edit transducer T e Construct factor transducer T f. Represent x and y by automata X and Y. Compute X =Project 2 (X T f ) and Y =Project 1 (T 1 f Y ). Compute best path of X T e Y. using c(a, b) costs. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 12 12

13 Factor Transducer Definition: transducer mapping a string to the set of factors of that string. T:e G:e C:e A:e T:T G:G C:C A:A T:e G:e C:e A:e 0 e:e 1 e:e 2 Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 13 13

14 Factor Automaton of a String Program: fsmcompose X.fsm Tf.fsm fsmproject -2 fsmrmepsilon fsmdeterminize fsmminimize >Xp.fsm Representation: size linear in that of X. A A G G T C T G A G 5 T T C 4 C G Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 14 14

15 Local Alignment - Example Program: fsmcompose Xp.fsm Te.fsm Yp.fsm fsmbestpath -n1 >B.fsm Graphical representation: C:C/ T:e/1 2 G:G/ A:A/-1.5 4/0 X = T G T C T G A G Y = A C A C G A C T G A C ε G A matching cost: -1.5 insertion/deletion/substitution costs: +1 Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 15 15

16 Alignment with Gaps Problem: Find the best local or global alignment with gaps (consecutive sequence of deletions/ insertions) two sequences x and y. Motivation: ignore long gaps due to introns. Example: Gap penalties: A T G A ε ε G A C A ε ε G A T G ε C Constant: constant penalty Affine: w 0 + w 1 gap Convex: w 0 log gap. w 0 per gap. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 16 16

17 Gap Alignment Transducer Representation: affine gap penalties. a:b/1 a:a/0 0/0 a:e/(w0 + w1) e:a/(w0 + w1) a:a/0 a:b/1 e:a/w1 a:e/w1 1/0 Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 17 17

18 More Complex Alignments Complex alignment transducers Simpler alignment transducers can be combined using rational operations to create more complex ones. Any weighted transducer can help define an alignment. Alignment transducers can be learned from examples. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 18 18

19 Learning Alignment Transducers Problem: given a fix topology for the transducer examples., learn the transition weights from Example: (Ristad and Yianilos, 1997) learn edit costs of one-state edit-distance transducer; data: large corpus of pairs of related sequences; algorithm: use the Expectation-Maximization (EM) algorithm. T e Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 19 19

20 Alignment of Sets of Sequences Problem: Find the best alignment (minimal cost) of two sets of sequences X and Y. Note: X and Y may be finite sets of biological sequences or infinite sets, e.g., represented or modeled by HMMs. Solution: T e Construct edit transducer. A X Represent X and Y by automata and. Compute best path of A X T e A Y. A Y Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 20 20

21 Longest Common Subsequence of Automata Problem: Find the longest common subsequence of two (finite) sets of sequences X and Y each given by a finite automaton. lcs(x, Y ) = max { lcs(x, y) : x X, y Y }. Solution: Construct LCS transducer. Represent X and Y by automata and. Compute best path of T lcs A X A X T lcs A Y. A Y Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 21 21

22 Longest Common Subsequence Transducer Representation: example. e:b/0 e:a/0 b:e/0 a:e/0 b:b/-1 a:a/-1 0/0 Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 22 22

23 Summary An alignment scoring function can be defined by a WFST. A general algorithm can be used to efficiently compute the best alignments between sequences, or sets of sequences represented by finite automata. Complex alignments can be learned from examples. The FSM library can be used for the implementation and computation of alignments. Corinna Cortes and Mehryar Mohri - WFSTs in Computational Biology - Part II page 23 23