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Randomized Algorithms for Motif Finding [1] Ch 12.2

Randomized Algorithms for Motif Finding [1] Ch 12.2. l = 8. DNA. cctgatagacgctatctggctatcc a G gtac T t aggtcctctgtgcgaatctatgcgtttccaaccat agtactggtgtacatttgat C c A tacgt acaccggcaacctgaaacaaacgctcagaaccagaagtgc aa acgt TA gt gcaccctctttcttcgtggctctggccaacgagggctgatgtataagacgaaaatttt

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Randomized Algorithms for Motif Finding [1] Ch 12.2

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  1. Randomized Algorithmsfor Motif Finding[1] Ch 12.2

  2. l= 8 DNA cctgatagacgctatctggctatccaGgtacTtaggtcctctgtgcgaatctatgcgtttccaaccat agtactggtgtacatttgatCcAtacgtacaccggcaacctgaaacaaacgctcagaaccagaagtgc aaacgtTAgtgcaccctctttcttcgtggctctggccaacgagggctgatgtataagacgaaaatttt agcctccgatgtaagtcatagctgtaactattacctgccacccctattacatcttacgtCcAtataca ctgttatacaacgcgtcatggcggggtatgcgttttggtcgtcgtacgctcgatcgttaCcgtacgGc t=5 n = 69 s s1= 26s2= 21s3= 3 s4= 56s5= 60

  3. a G g t a c T t C c A t a c g t a c g t T A g t a c g t C c A t C c g t a c g G _________________ A 3 0 1 0 3 1 1 0 C 2 4 0 0 1 4 0 0 G 0 1 4 0 0 0 3 1 T 0 0 0 5 1 0 1 4 _________________ Consensusa c g t a c g t Score= 3+4+4+5+3+4+3+4=30 Profile P l t

  4. When P is Given P by count: P by probability: Score: 4+7+4+5+3+7=30

  5. Scoring a String with P The probability of a possible consensus string: Prob(aaacct|P) = 1/2 x 7/8 x 3/8 x 5/8 x 3/8 x 7/8 = .033646 Prob(atacag|P) = 1/2 x 1/8 x 3/8 x 5/8 x 1/8 x 1/8 = .001602

  6. P-Most Probable l-mer P = Given P and a sequence ctataaaccttacatc, find the P-most probable l-mer

  7. P-Most Probable l-mer Find the Prob(a|P) of every possible 6-mer: First try: c t a t a a a c c t t a c a t c Second try: c t a t a a a c c t t a c a t c Third try: c t a t a a a c c t t a c a t c -Continue this process to evaluate every possible 6-mer

  8. P-Most Probable l-mer

  9. P-Most Probable l-mers in All Sequences • Find the P-most probable l-mer in each of the sequences. ctataaacgttacatc atagcgattcgactg cagcccagaaccct cggtataccttacatc tgcattcaatagctta tatcctttccactcac ctccaaatcctttaca ggtcatcctttatcct P=

  10. New Profile ctataaacgttacat atagcgattcgactg cagcccagaaccctg cggtgaaccttacat tgcattcaatagctt tgtcctttccactca ctccaaatcctttac ggtctacctttatcc P-Most Probable l-mers form a new profile

  11. Compare New and Old Profiles New Score: 5+5+4+6+4+7=31 Old Score: 4+7+4+5+3+7=30

  12. Greedy Profile Motif Search • Select random starting positions s • Create profile P • Compute Score • Find P-most probable l-mers • Compute new profile based on new starting positions s • Compute Score • Repeat step 4-5 as long as Score increases Cost: Polynomial for each round, and number of round can be controlled

  13. Performance • Since we choose starting positions randomly, there is little chance that our guess will be close to an optimal motif, meaning it will take a very long time to find the optimal motif. • It is unlikely that the random starting positions will lead us to the correct solution at all. • In practice, this algorithm is run many times with the hope that random starting positions will be close to the optimum solution simply by chance.

  14. Gibbs Sampling • Gibbs Sampling improves on one l-mer at a time • Thus proceeds more slowly and cautiously

  15. Gibbs Sampling: an Example Input: t = 5 sequences, motif length l = 8 1. GTAAACAATATTTATAGC 2. AAAATTTACCTCGCAAGG 3. CCGTACTGTCAAGCGTGG 4. TGAGTAAACGACGTCCCA 5. TACTTAACACCCTGTCAA

  16. Gibbs Sampling: an Example 1) Randomly choose starting positions, s=(s1,s2,s3,s4,s5) in the 5 sequences: s1=7GTAAACAATATTTATAGC s2=11AAAATTTACCTTAGAAGG s3=9CCGTACTGTCAAGCGTGG s4=4TGAGTAAACGACGTCCCA s5=1TACTTAACACCCTGTCAA

  17. Gibbs Sampling: an Example 2) Choose one of the sequences at random: Sequence 2: AAAATTTACCTTAGAAGG s1=7GTAAACAATATTTATAGC s2=11AAAATTTACCTTAGAAGG s3=9CCGTACTGTCAAGCGTGG s4=4TGAGTAAACGACGTCCCA s5=1TACTTAACACCCTGTCAA

  18. Gibbs Sampling: an Example 2) Choose one of the sequences at random: Sequence 2: AAAATTTACCTTAGAAGG s1=7GTAAACAATATTTATAGC s3=9CCGTACTGTCAAGCGTGG s4=4TGAGTAAACGACGTCCCA s5=1TACTTAACACCCTGTCAA

  19. Gibbs Sampling: an Example 3) Create profile P from l-mers in remaining 4 sequences:

  20. Gibbs Sampling: an Example 4) Calculate the prob(a|P) for every possible 8-mer in the removed sequence

  21. Gibbs Sampling: an Example 5) Create a distribution of probabilities of l-mers prob(a|P), and randomly select a new starting position based on this distribution. a) To create this distribution, divide each probability prob(a|P) by the lowest probability: Starting Position 1: prob( AAAATTTA | P ) = .000732 / .000122 = 6 Starting Position 2: prob( AAATTTAC | P ) = .000122 / .000122 = 1 Starting Position 8: prob( ACCTTAGA | P ) = .000183 / .000122 = 1.5 Ratio = 6 : 1 : 1.5

  22. Turning Ratios into Probabilities b) Define probabilities of starting positions according to computed ratios Probability (Selecting Starting Position 1): 6/(6+1+1.5)= 0.706 Probability (Selecting Starting Position 2): 1/(6+1+1.5)= 0.118 Probability (Selecting Starting Position 8): 1.5/(6+1+1.5)=0.176

  23. Gibbs Sampling: an Example c) Select the start position according to computed ratios: P(selecting starting position 1): .706 P(selecting starting position 2): .118 P(selecting starting position 8): .176

  24. Gibbs Sampling: an Example Assume we select the substring with the highest probability – then we are left with the following new substrings and starting positions. s1=7 GTAAACAATATTTATAGC s2=1 AAAATTTACCTCGCAAGG s3=9 CCGTACTGTCAAGCGTGG s4=5TGAGTAATCGACGTCCCA s5=1 TACTTCACACCCTGTCAA Earlier s1=7GTAAACAATATTTATAGC s2=11AAAATTTACCTTAGAAGG s3=9CCGTACTGTCAAGCGTGG s4=4TGAGTAAACGACGTCCCA s5=1TACTTAACACCCTGTCAA

  25. Gibbs Sampling: an Example 6) iterate the procedure again with the above starting positions until we can not improve the score any more.

  26. How Gibbs Sampling Works • Randomly choose starting positions s = (s1,...,st) • Randomly choose one of the t sequences • Create a profile P from the other t -1 sequences • For each position in the removed sequence, calculate the probability that the l-mer starting at that position was generated by P. • Choose a new starting position for the removed sequence at random based on the probabilities calculated in step 4. • Repeat steps 2-5 until there is no improvement

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