Random Projection Approach to Motif Finding

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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
-1
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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 N20
  random sequences of length n600.
• 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.

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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.

k 7
l 15
P
ATGGCATTCAGATTC
TGCTGAT
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.

Input sequence x(i) TCAATGCACCTAT...
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Example

• l 7 (motif size) , k 4 (projection size)
• Choose projection (1,2,5,7)

Input Sequence
...TAGACATCCGACTTGCCTTACTAC...
Buckets
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.

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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
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Frequency Matrix Model from Bucket
Frequency matrix W
EM algorithm
Refined matrix W
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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 Wh
• Use EM algorithm with starting point Wh to obtain
  refined weight matrix model Wh
• For each input sequence x(i), return l tuple y(i)
  which maximizes likelihood ratio
• Pr(y(i) Wh )/ Pr(y(i) P0).
• 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
• P0 background probability distribution.
• Find value Wmax 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 lt l - d)
• Choose k large to avoid contamination by spurious
  l-mers. E gt (N (n - l 1))/ 4k 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.