Hidden Markov Models. Main source: Durbin et al., Biological Sequence Alignment (Cambridge, 98)

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1 Hidden Markov Models Main source: Durbin et al., Biological Sequence Alignment (Cambridge, 98) 1

2 The occasionally dishonest casino A P A (1) = P A (2) = = 1/6 P A->B = P B->A = 1/10 B P B (1)= P B (5)=0.1 P B (6) = Can we tell when the loaded die is used? 2

3 CpG islands: Example - CpG islands DNA stretches (100~1000bp) with frequent CG pairs (contiguous on same strand). Rare, appear in significant genomic parts. Problem (1): Given a short genome sequence, decide if it comes from a CpG island. 3

4 Preliminaries: Markov Chains (S, A, p) S: State set p: Initial state prob. vector {p(x 1 =s)} [alternatively, use a begin state] A: Transition prob. matrix a st = P(x i =t x i-1 =s) Assumption: X=x 1 x n is a random process with memory length 1, i.e.: s i S P(x i =s i x 1 =s 1,,x i-1 =s i-1 ) = P(x i =s i x i-1 =s i-1 ) = a s i-1,si Sequence probability: P(X) = p(x 1 ) Π i=2 L a x i-1,xi

5 Markov Models - Transition probs for non-cpg islands + Transition probs for CpG islands - A C G T A C G T A C G T A C G T

6 CpG islands: Fixed Window Problem (1): Given a short genome sequence X, decide if it comes from a CpG island. Solution: Model by a Markov chain. Let a + st : transition prob. in CpG islands, a - st : transition prob. outside CpG islands. Decide by log-likelihood ratio score: score( X ) P( X CpG island) = log P( X non CpG island) 1 bits _ score( X ) = n n + xi 1,x log2 i = 1 ax,x a i 1 i i = n log a a + x i 1 i = x,x 1 i 1,x i i 6

7 Discrimination of sequences via Markov Chains 48 CpG islands, tot length ~60K nt. Similar non-cpg. Durbin et. al, Fig

8 CpG islands the general case Problem(2): Detect CpG islands in a long DNA sequence. Naive Solution - Sliding windows: 1 k L-l, window: X k = (x k+1,,x k+l ) score: score(x k ) positive score potential CpG island Disadvantage: what is the length of the islands? How do we identify transitions? Idea: Use Markov chains as before, with additional (hidden) states 8

9 Hidden Markov Model (HMM) Alphabet of symbols Example: {A, C, G, T} path Π=π 1,,π n (sequence of states - simple Markov chain; convention: π 0 - begin, π L+1 - end) Given sequence X = (x 1,,x L ): a kl = P(π i =l π i-1 =k), e k (b) = P(x i =b π i =k) Finite set of states, capable of emitting symbols. Example: Q = {A +,C +,G +,T +,A -,C -,G -,T - } M=(Σ, Q, Θ) Θ=(A,E) A: Transition prob. a kl k,l Q E: Emission prob. e k (b) k Q, b Σ P(X,Π) = a π0,π 1 Π i=1 L e π i(x i ) a π i,πi+1 Ex.: express P(X,Π) in terms of counts 9

10 Viterbi s Decoding Algorithm (finding most probable state path) Want: path Π maximizing P(X, Π) v k (i) = prob. of most probable path ending in state k at step i. Init: v 0 (0) = 1; v k (0)=0 k>0 Step: v k (i+1)=e k (x i+1 ) max l {v l (i) a lk } End: P(X, Π * ) = max l {v l (L) a l0 } Time complexity: O(Ln 2 ) for n states, L steps Can find Π* using back pointers. 10

11 The occasionally dishonest casino (2) A B

12 The occasionally dishonest casino (2) 12

13 HMM for CpG Islands States: A + C + G + T + A - C - G - T - Symbols: A C G T A C G T Path Π=π 1,,π n : sequence of states + A C G T A C G T A C G T A C G transition prob. T

14 Posterior State Probabilities Goal: calculate P(π i =k X) Our strategy: P(X, π i =k) = = P(x 1,,x i, π i =k) P(x i+1,,x L x 1,,x i, π i =k) = P(x 1,,x i, π i =k) P(x i+1,,x L π i =k) P(π i =k X) = P(π i =k, X) / P(X) Need to compute these two terms - and P(X) 14

15 Forward Algorithm Goal: calculate P(X) = Σ Π P(X, Π) Approximation: take max path Π* from Viterbi alg. Not justified when several near maximal paths Exact alg : Forward Algorithm f k (i) = P(x 1,,x i, π i =k) Init: f 0 (0) = 1; f k (0)=0 k>0 Step: f k (i+1) = e k (x i+1 ) l f l (i) a lk End: P(X) = Σ l f l (L) a l0 15

16 Backward Algorithm b k (i) = P(x i+1, x L π i =k) init: k, b k (L) = a k0 step: b k (i) = l a kl e l (x i+1 ) b l (i+1) End: P(X) = Σ k a 0k e k (x 1 ) b k (1) 16

17 Posterior State Probabilities (2) Goal: calculate P(π i =k X) Recall: f k (i) = P(x 1,,x i, π i =k) b k (i) = P(x i+1, x L π i =k) Each can be used to compute P(X) P(X, π i =k) = = P(x 1,,x i, π i =k) P(x i+1,,x L x 1,,x i, π i =k) = P(x 1,,x i, π i =k) P(x i+1,,x L π i =k) = f k (i) b k (i) P(π i =k X) = P(π i =k, X) / P(X) 17

18 Dishonest Casino (3) Durbin et al. pp

19 Posterior Decoding Now we have P(π i =k X). How do we decode? 1. π i* =argmax k P(π i =k X) Good when interested in state at particular point path of states π 1*,.., π * L may not be legal 2. Define a function of interest g(i) on the states. Compute G(i X) = Σ k P(π i =k X) g(k) E.g.: g(i) =1 for states in S, 0 on the rest: G(i X) is posterior prob of symbol i coming from S e.g., CpG island 19 S={A +,C +,G +,T + }

20 Parameter Estimation for HMMs Log likelihood of model Input: X 1,,X n independent training sequences Goal: estimation of Θ = (A,E) (model parameters) Note: P(X 1,,X n Θ) = Π i=1 n P(X i Θ) (indep.) l(x 1,,x n Θ) = log P(X 1,,X n Θ) = i=1 n log P(X i Θ) Case 1 - Estimation When State Sequence is Known: A kl = #(occurred k l transitions) E k (b) = #(emissions of symbol b that occurred in state k) Max. Likelihood Estimators: a kl e k (b) = A kl / l A kl = E k (b) / b E k (b ) small sample, or prior knowledge correction: A kl = A kl + r kl E k(b) = E k (b) + r k (b) Dirichlet priors 20

21 Parameter Estimation in HMM Case 2: -Estimation When States are Unknown Input: X 1,,X n indep training sequences Baum-Welch alg. (1972): Expectation: guarantees convergence monotone many local optima special case of EM compute expected no. of k l state transitions: (ex.) P(π i =k, π i+1 =l X, Θ) = [1/P(x)] f k (i) a kl e l (x i+1 ) b l (i+1) A kl = j [1/P(X j )] i f kj (i) a kl e l (x j i+1) b lj (i+1) compute expected no. of symbol b appearances in state k E k (b) = j [1/P(X j )] {i x j i =b} f kj (i) b kj (i) (ex.) Maximization: re-compute new parameters from A, E using max. likelihood. repeat (1)+(2) until improvement ε 21

22 Profile HMMs (Haussler et al, 1993) Ungapped alignment AG---C of X against a profile M: e i (a) = prob. of A-AT-C observing a at position i. P(X M) = Π i=1 L AG-AA- e i (x i ), or Score(X M) --AAAC = i=1 L log [e i (x i ) / q x i] indels: AG---C background prob. insert I match states begin j M j end delete D j 22

23 Profile HMMs Gapped alignment of X against a profile M: assume e I j(a) = q a (backgroud prob.) gap of length k contributes to log-odds: log(a Mj,Ij ) + log(a Ij,Mj+1 ) + (k-1) log(a Ij,Ij ) { gap open } (gap extension) No log odds contribution from the emission insert I match states begin j M j end delete D j 23

24 Profile HMM Transition Probabilities Mi Mi+ 1 Mi Di+ 1 Mi Ii Emission probabilities M i a I i a Ii Mi+ 1 Ii Ii Ii Di+ 1 Di Di+ 1 Di Mi+ 1 Di Ii 24

25 Example Suppose we are given the aligned sequences **---* AG---C A-AT-C AG-AA- --AAAC AG---C Suppose also that the match positions are marked

26 Calculating A, E count transitions and emissions: transitions M-M M-D M-I I-M I-D I-I D-M D-D D-I **---* AG---C A-AT-C AG-AA- --AAAC AG---C A C G T A C T G emissions

27 Calculating A, E count transitions and emissions: transitions M-M M-D M-I I-M I-D I-I D-M D-D D-I **---* AG---C A-AT-C AG-AA- --AAAC AG---C emissions A C G T A C T G

28 Estimating Maximum Likelihood probabilities using fractions emissions A C G T A C T G A C G T A C T G

29 Estimating ML probabilities (contd) transitions M-M M-D M-I I-M I-D I-I D-M D-D D-I M-M M-D M-I I-M I-D I-I D-M D-D D-I

30 HMM from multiple alignment Shade: insert 30

31 .. and multiple alignment from a given HMM Align each sequence to the profile separately Right: a new sequence realigend with the model Inserts are unaligned. 31

32 Searching with Profile HMMs Compute: log P( X Model) P( X random) V jm (i): log odds of best path matching x 1,,x i to submodel up to level j, ending with x i emitted by state M j V ji (i) : same, ending with x i emitted by state I j V jd (i) : score for best path ending in D j after x i has been emitted (and x i+1 has not been emitted yet) V jm (i) = log [e Mj (x i ) / q x i] + max { V j-1m (i-1)+log(a Mj-1,Mj ), V j-1i (i-1)+log(a Ij-1,Mj ), V j-1d (i-1)+log(a Dj-1,Mj ) } V ji (i) = log [e Ij (x i ) / q x i] + max { V jm (i-1)+log(a Mj-1,Ij ), V ji (i-1)+log(a Ij,Ij ), V jd (i-1)+log(a Dj,Ij ) } V jd (i) = max {V j-1m (i)+log(a Mj-1,Dj ), V j-1i (i)+log(a Ij-1,Dj ), V j-1d (i)+log(a Dj-1,Dj ) } Π i q x i 32

33 Training a profile HMM from unaligned sequences choose length of the profile HMM, initialize parameters train the model using Baum-Welch obtain MA with the resulting profile as before. (Formulas inlc. forward/backward in handout) 33

34 An Illustrative Study HMMs in Computational Biology, Krogh, Brown, Mian, Sjoalnder, Haussler 93 Globin experiment: Heme-containing proteins, involved in the storage and transport of oxygen 625 globins from Swissprot, ave. legnth 145AA Training set: 400 sequences Built model and ran it against test globins and all Swissprot proteins (25K). 34

35 Representative globins Ron Shamir, CG 08 Alignment from Bashford et al 87 A-H: alpha helices 35

36 Realignment by trained HMM Ron Shamir, CG 08 36

37 Part of the final globin model 37

38 Score vs sequence length Negative LL. Length dependent using log odds is preferable For all Swissprot seqs of < 300AA NLL=-log(P(seq model)) 38

39 Z-score distribution Z-score: (S-E(S)) / sd(s) On ~25K proteins, cutoff 5 misses 2/628 globins with no fp 39

40 Discovering subfamilies 10 component HMM. Training set: 628 globins. Classified each sequence using the model. Generated 7 nonempty clusters, with ~560 falling into 4 biologically meaningful clusters. 40

41 Local alignment in HMM Add flanking states for regions of unaligned seq. Durbin et al. Ch

42 42