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Random Projection Approach to Motif Finding Adapted from http://genome.ucsd.edu/classes/be202/ppt/FindingSignals- RandomProjections.ppt

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daf-19 Binding Sites in C. elegans (Peter Swoboda) GTTGTCATGGTGAC GTTTCCATGGAAAC GCTACCATGGCAAC GTTACCATAGTAAC GTTTCCATGGTAAC che-2 daf-19 osm-1 osm-6 F02D8.3 -150

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Algorithmic Techniques MEME (Expectation Maximization) GibbsDNA (Gibbs Sampling) CONSENUS (greedy multiple alignment) WINNOWER (Clique finding in graphs) SP-STAR (Sum of pairs scoring) MITRA (Mismatch trees to prune exhaustive search space)

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The (l,d) Planted Motif Problem (Sagot 1998, Pevzner & Sze 2000) Generate a random length l consensus sequence C. Generate 20 instances, each differing from C by d random mutations. Plant one at a random position in each of N=20 random sequences of length n=600. Can you find the planted instances?

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Planted Motifs AGTTATCGCGGCACAGGCTCCTTCTTTATAGCC ATGATAGCATCAACCTAACCCTAGATATGGGAT TTTTGGGATATATCGCCCCTACACTGGATGACT GGATATACATGAACACGGTGGGAAAACCCTGAC Each instance differs from ACAGGATCA by 2 mutations Remaining sequence random

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Random Projection Algorithm Buhler and Tompa (2001) Guiding principle: Some instances of a motif agree on a subset of positions. Use information from multiple motif instances to construct model. ATGCGTC...ccATCCGACca......ttATGAGGCtc......ctATAAGTCgc......tcATGTGACac... (7,2) motif x(1) x(2) x(5) x(8) =M

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k-Projections Choose k positions in string of length l. Concatenate nucleotides at chosen k positions to form k-tuple. In l-dimensional Hamming space, projection onto k dimensional subspace. ATGGCATTCAGATTC TGCTGAT l = 15 k = 7 P P = (2, 4, 5, 7, 11, 12, 13)

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Random Projection Algorithm Choose a projection by selecting k positions uniformly at random. For each l-tuple in input sequences, hash into bucket based on letters at k selected positions. Recover motif from bucket containing multiple l-tuples. Bucket TGCT TGCACCT Input sequence x(i): …TCAATGCACCTAT...

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Example l = 7 (motif size), k = 4 (projection size) Choose projection (1,2,5,7) GCTC...TAGACATCCGACTTGCCTTACTAC... Buckets Input Sequence ATGC ATCCGAC GCCTTAC

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Hashing and Buckets Hash function h(x) obtained from k positions of projection. Buckets are labeled by values of h(x). Enriched buckets: contain at least s l-tuples, for some parameter s. ATTCCATCGCTC ATGC

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Motif Refinement How do we recover the motif from the sequences in the enriched buckets? k nucleotides are known from hash value of bucket. Use information in other l-k positions as starting point for local refinement scheme, e.g. EM or Gibbs sampler Local refinement algorithm ATGCGTC Candidate motif ATGC ATCCGAC ATGAGGC ATAAGTC ATGTGAC

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Frequency Matrix Model from Bucket Frequency matrix W ATGC ATCCGAC ATGAGGC ATAAGTC ATGTGAC Refined matrix W* EM algorithm

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Motif Finding as Global Optimization Scoring function (Hamming distance, likelihood ratio, etc.) Many existing algorithms (MEME, GibbsDNA) are good local optimization routines. Random projection is a procedure for finding good starting points.

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EM Motif Refinement For each bucket h containing more than s sequences, form weight matrix W h Use EM algorithm with starting point W h to obtain refined weight matrix model W h * For each input sequence x(i), return l tuple y(i) which maximizes likelihood ratio: Pr(y(i) | W h * )/ Pr(y(i) | P 0 ). T = {y(1), y(2), …, y(N)} C( T ) = consensus string

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Expectation Maximization (EM) S = { x(1), …, x(N)} : set of input sequences Given: W = An initial probabilistic motif model P 0 = background probability distribution. Find value W max that maximizes likelihood ratio: EM is local optimization scheme. Requires starting value W

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A Single Iteration Choose a random k-projection. Hash each l-mer x in input sequence into bucket labelled by h(x). From each bucket B with at least s sequences, form weight matrix model, and perform EM/Gibbs sampler refinement. Candidate motif is the best one found from refinement of all enriched buckets.

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What is the best motif? Compute score S for each motif: –Generate W, an initial PSSM from the returned l-mers {y(1), y(2), …, y(N)} Return motif with maximal score

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Parameter Selection Projection size k Choose k small so several motif instances hash to same bucket. (k < l - d) Choose k large to avoid contamination by spurious l-mers. E > (N (n - l + 1))/ 4 k Bucket threshold s: (s = 3, s = 4)

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How Many Iterations? Planted bucket : bucket with hash value h(M), where M is motif. Choose m = number of iterations, such that Pr(planted bucket contains ≥ s sequences in at least one of m iterations) ≥ 0.95. Probability is readily computable since iterations form a sequence of independent Bernoulli trials.

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Examples K = set of nt. in motif instances. P = set of nt. in positions predicted by algorithm.

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