A fast algorithm for approximate string matching on gene sequences

1. A fast algorithm for approximate string matching on gene sequences Zheng Liu, Xin Chen, James Borneman and Tao Jiang University of California, Riverside

2. Outline • Background and motivation • Idea and analysis for FAAST • Experimental results • Conclusion

3. Background • Approximate string matching pattern: P = p1p2…pm text: T = t1t2…tn • K-mismatch • K-difference • Applications: text processing and gene sequence analysis.

4. Motivation of FAAST • Motivation: Gene sequence acquisition • Modeled as the k-mismatch problem Primers: AAGTC CCGTA AAGTC………CCGTA TACTT………CCGTT … ACGTC………GCGTA … AAGTC………CCGTA … ACGTC………GCGTA

5. Algorithms for the k-mismatch problem • 1992, Shift-Add by Baeza-Yates and Gonnet. • 1996, BM with Shift-Add by El-Mabrouk and Crochemore. • 1993, BM extention (bad-charcter rule) by Tarhio-Ukkonen. • 1994, BM extention (good-suffix rule) by Baeza-Yates and Gonnet.

6. FAAST • Further generalization on Tarhio-Ukkonen algorithm. tj-m+1 tj-m+2 …… tj-k …tj-2 tj-1 tj p1 p2 …… pm-k … pm-2 pm-1 pm --check last k+1 tj-m+1 tj-m+2 … tj-k-x+1…tj-k …tj-2 tj-1 tj p1 p2 … pm-k-x+1 …pm-k …pm-2 pm-1 pm --check last k+x

7. Algorithm outline T: AACTGTTAACTTGCGACTAG (k=2, x=2) P:AAATCGTAAC AAATCGTAAC Χ AAATCGTAAC Χ ……… Χ AAATCGTAAC ☺-after first shift (6)

8. An example k=2, x=3, m=10, n=20 T: AACTGTTAACTTGCGACTAG P: AAATCGTAAC T: AACTGTTAACTTGCGACTAG P: AAATCGTAAC shift 1 by Tarhio-Ukkonen T: AACTGTTAACTTGCGACTAG P: AAATCGTAAC shift 6 by FAAST

9. Construction of shift table • Heuristic: Guarantee the last k+x (or y, if y ≤ k+x) aligned text characters to have at least x (or y-k , if y ≤ k+x) matches. T:AACTGTTAACTTGCGACTA [K=2,X=3] P:AAGTCGTAAC …. AAGTCGTAAC

10. Construction details • Vkx[tj-k-x+1…tj, l] :Marks the characters that match with P after shifting P by l. • dkx[tj-k-x+1…tj] : Stores the minimum distance l, s.t.Vkx[tj-k-x+1…tj, l] contains at least min[x, m-k-l] items.

11. Construction details – cont’d • P: AAGTCGTAAC (k=2, x=3, l=[1..8]),Vkx[tj-k-x+1…tj, l] and dkx[tj-k-x+1…tj]

12. Theoretical support • Correctness of FAAST • We use random string assumption • Average shift distance • Total number of character comparisons

13. Correctness of FAAST • Theorem 1. When P is aligned with tj-k-x+1…tj,we can always shift P by dkx[tj-k-x+1…tj] to the right without miss approximate occurrences of P. tj-m+1 tj-m+2 … tj-k-x +1 …tj-2 tj-1 tj p1 p2 …pi-k-x+1 …pi-2 pi-1 pi ……pm – current p1 p2 … pi-k-x+1 … pi’-k-x+1 … pi-’2 pi’-1 pi’...pm -- (i < i’)

14. Average shift distance • Lemma 1. The prob. Pkx for the last k+x chars of T to have at least x matches is: Pkx = 1- Σi=0x-1Ck+Xi(1-p)k+x-ipi • Theorem 1. The avg. shift distance of FAAST is: Ekxd = Σs=0∞s(1-Pkx)s-1Pkx = 1/Pkx

15. Average shift distance under diff x.

16. Total character comparisons • Lemma 2. The expected number of comparisons between two shifts is: Ekxc = (k+X) / (1-p) • Theorem 2. The expected total comparisons for text of length n is: TEkxc = nPkx (k+X) / (1-p)

17. Total character comparisons

18. Difference of total character comparisons under different x

19. Experimental result • A PC with 2.8GHz CPU and 1G memory • Simulated random string testing • Real DNA gene sequence data

20. Result on simulated sequences • Text: 2M bases sequence, Pattern: 39 bases, k=3.

21. Result on real sequences • Text: 150 bacteria DNA sequences, k=3 • Text: 150 fungi DNA sequences, k=3

22. Conclusion • Competitive algorithm for k-mismatch problem on gene sequence. • Time and memory increase with larger x and alphabet size.