Advanced Algorithms and Models for Computational Biology -- a machine learning approach

1. Advanced Algorithms and Models for Computational Biology-- a machine learning approach Computational Genomics II: HMM variants and Comparative Gene Finding Eric Xing Lecture 5, February 1, 2005 Reading: Chap 3, 5 DEKM book Chap 9, DTW book

2. Higher-order HMMs • The Genetic Code • 3 nucleotides make 1 amino acid • Statistical dependencies in triplets • Question: • Recognize protein-coding segments with an HMM

3. y1,...,N = i i e e i y1 x1,...,N = A C T T G x1 A ... y2 y3 y4 yN ... x2 A x3 A x4 A xN A Higher-order HMMs • Every state of the HMM emits 1 nucleotide • Transition probabilities: Probability of a state at one position, given those of 3 previous positions (triplets): P(yi | yi-1,yi-2, yi-3) • Emission probabilities: P(xi| yi) • Algorithms extend with small modifications

4. y1,...,N = i i e e i y1 x1,...,N = A C T T G Q1 Q2 Q3 x1 A ... Y1,Y2,Y3 Y2,Y3,Y4 Y3,Y4,Y5 y2 y3 y4 yN ... ... X1,X2,X3 X2,X3,X4 X3,X4,X5 R1 R2 R3 x2 A x3 A x4 A xN A ... Inference on Higher-order HMMs • Building 1st-order HMM on "mega" state • Use FB algorithm as usual • P(Q2|R)  P(Y2, Y3, Y4 |X)  P(Y3 |X)=SY2,Y4P(Y2, Y3, Y4 |X)

5. Modeling the Duration of States 1-p • Length distribution of region X: E[lX] = 1/(1-p) • Geometric distribution, with mean 1/(1-p) • (homework: derive this) • This is a significant disadvantage of HMMs • Several solutions exist for modeling different length distributions X Y p q 1-q

6. Observed Duration Time

7. Poisson Point Process • A counting process that represents the total number of occurrences of discrete events during a temporal/spatial interval • the number of occurrences in any internal of length t isPoisson distributedwith parameter lt: • the number of occurrences in disjoint intervals are independent • the duration of the interval between two consecutive occurrences has the following distribution:

8. Poisson point process m= lt Truncation is needed at both ends!

9. ... y2 y3 y4 yN y1 d1 d2 d3 dN d4 ... xd2 A xd3 A xd2 A xdN A xd1 A Generalized HMM Upon entering a state: • Choose duration d, according to probability distribution • Generate d letters according to emission probs • Take a transition to next state according to transition probs Disadvantage: Increase in complexity: Time: O(D2) Space: O(D) where D = maximum duration of state

10. Comparative Genomics

11. A pairwise comparison between human and mouse genome

12. Aligning One Locus

13. Three Pairwise Alignments

14. Example: a human/mouse ortholog

15. Paired HMM M (+1,+1) Alignments correspond 1-to-1 with sequences of states M, I, J I (+1, 0) J (0, +1) -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC-GGTCGATTTGCCCGACC IMMJMMMMMMMJJMMMMMMJMMMMMMMIIMMMMMIII

16. Let’s score the transitions s(xi, yj) M (+1,+1) Alignments correspond 1-to-1 with sequences of states M, I, J s(xi, yj) s(xi, yj) -d -d I (+1, 0) J (0, +1) -e -e -e -e -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC-GGTCGATTTGCCCGACC IMMJMMMMMMMJJMMMMMMJMMMMMMMIIMMMMMIII

17. 1 – 2 –  M P(xi, yj) 1 – 2 –  1 – 2 –      I P(xi) J P(yj)   A Pair HMM for alignments BEGIN  1 – 2 –    I M J    END

18. Gene Finding

19. Generalized HMM Gene finder

20. Generalized Pair-HMM gene finder

21. Hierarchical state transition in pHMM

22. Allowing for inserted exons

23. Acknowledgments • Serafim Batzoglou: for some of the slides adapted or modified from his lecture slides at Stanford University • Lior Pachter': for some of the slides modified from his lectures at UC Berkeley