Assignments for lecture Bioinformatics III WS 03/04. Assignment 5, return until Dec 16, 2003, 11 am. Your name: Matrikelnummer: Fachrichtung:

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1 Assignments for lecture Bioinformatics III WS 03/04 Assignment 5, return until Dec 16, 2003, 11 am Your name: Matrikelnummer: Fachrichtung: Please direct questions to: Jörg Niggemann, tel , Suggested Reading: You can find additional information on Hidden Markov Models in these two tutorials: R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. "Biological sequence analysis: Probabilistic models of proteins and nucleic acids", chapter 3: Markov chains and hidden Markov models Lawrence Rabiner. A tutorial on hidden markov models and selective applications in speech recognition. In Alex Waibel and K. F. Lee, editors, Readings in Speech Recognition. Morgan Kaufmann Publishers, An excellent introduction to the viterbi algorithm is in: This involves forecasting the weather from the state of some leaves of seaweed. It also contains a little Java applet showing the algorithm in function. Another viterbi algorithm demo:

2 (1) Simple Hidden Markov Model Consider a hidden Markov model H with the following parameters: states S = {S 1, S 2, S 3 }, alphabet A = {C, G, T}, P = S1 S2 S3 S S S p = B = C G T S S S What are all possible state sequences for the following observed sequences O, and for each case, what is Prob(O H)? a) O = C, G, T. b) O = C, T, C. (20 points)

3 (2) Gene Finding Suppose we model a stretch of DNA containing two types of regions. Region 1 contains bases A,T,C, and G with equal frequency. Region 2 has a higher frequency of G and C than of A and T. We also have some knowledge about the length of these regions and the overall length of the modelled sequence. Consider the following simple 4-state Hidden Markov Model H for this stretch of DNA: states: B (Begin), E (End), S1 (Region 1), S2 (Region 2). Transition probabilities: P = E S1 S2 B S S Emission probabilities: B= A T G C S S Note: B and E are special begin and end states, respectively. They don't emit symbols. When generating output, the HMM is initiated in state B and terminates in state E. Use this definition for the following tasks a d. (a) Propbability for known path: Simple Probability Calculation [10 points] Compute P(obs=ATGCG, path=b,s1,s1,s1,s1,s1,e ), the probability of following state path B,S1,S1,S1,S1,S1,E and emitting the sequence ATGCG.

4 (b) Overall Probability of Observation: Forward Algorithm [ 20 points] Compute P(O = "ATGCG" H), the probability of the output sequence "ATGCG" being generated by the above model H, using the forward algorithm. As a reminder, here is the formula for the forward algorithm: Pr(X) means the probability of some event X. i (state s) = Pr (from start to state s)* Pr(output symbol at stae s ) for i = 1 r [ i 1 previous state r Pr output symbol at state s Pr transition from r to s ] for i 1 where i (state s) is an intermediary result that stores the probability of observing the symbols up to point i and ending in state s at point i. Step 1: Use the forward algorithm to fill in the dynamic programming matrix: a i (s) Output Symbol at state s State A(i=1) T(i=2) G(i=3) C(i=4) G(i=5) S1 S2 Step 2: Using the table, what is the probability of the output sequence "ATGCG" being generated by the above model H: Pr(O = "ATGCG" H) =

5 c) Most Probable Path: Viterbi Algorithm [ 30 points] Which state sequence was most likely to have generated the observation sequence "ATGCG"? The Viterbi algorithm realizes the following recurrence relation: i(state s) = Pr (from start to state s)* Pr(output symbol at stae s ) for i = 1 maximum [ i 1 previous state r Pr output symbol at state s Pr transition from r to s ] for i 1 allpossibler Step 1: Use the viterbi algorithm to fill in the dynamic programming matrix: δ i (s) Output Symbol at state s State A(i=1) T(i=2) G(i=3) C(i=4) G(i=5) S1 S2 Step 2: The most probable sequence of states is: Reasoning: Note that the viterbi algorithm is quite similar to the forward algorithm. What is the difference? Why?

6 (d) Structuring the HMM [ 20 points] How does the given HMM specify the length of Regions 1 and 2 and the numbers of occurrences of Regions 1 and 2? How could this be done more explicitly, e.g., if length distributions for Regions 1 and 2, and distributions of the number of occurrences of these regions are given?