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Gene Regulation and Microarrays

Gene Regulation and Microarrays. Finding Regulatory Motifs. Given a collection of genes with common expression, Find the TF-binding motif in common. Characteristics of Regulatory Motifs. Tiny Highly Variable ~Constant Size Because a constant-size transcription factor binds

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Gene Regulation and Microarrays

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  1. Gene Regulation and Microarrays

  2. Finding Regulatory Motifs Given a collection of genes with common expression, Find the TF-binding motif in common . . .

  3. Characteristics of Regulatory Motifs • Tiny • Highly Variable • ~Constant Size • Because a constant-size transcription factor binds • Often repeated • Low-complexity-ish

  4. Sequence Logos • Information at pos’n I, H(i) = – {letter x} freq(x, i) log2 freq(x, i) • Height of x at pos’n i, L(x, i) = freq(x, i) (2 – H(i)) • Examples: • freq(A, i) = 1; H(i) = 0; L(A, i) = 2 • A: ½; C: ¼; G: ¼; H(i) = 1.5; L(A, i) = ¼; L(not T, i) = ¼

  5. Probabilistic Motif: Mij; 1  i  W 1  j  4 Mij = Prob[ letter j, pos i ] Find best M, and positions p1,…, pN in sequences Combinatorial Motif M: m1…mW Some of the mi’s blank Find M that occurs in all si with  k differences Or, Find M with smallest total hamming dist Problem Definition Given a collection of promoter sequences s1,…, sN of genes with common expression

  6. Essentially a Multiple Local Alignment • Find “best” multiple local alignment • Alignment score defined differently in probabilistic/combinatorial cases . . .

  7. Algorithms • Combinatorial CONSENSUS, TEIRESIAS, SP-STAR, others • Probabilistic • Expectation Maximization: MEME • Gibbs Sampling: AlignACE, BioProspector

  8. Combinatorial Approaches to Motif Finding

  9. Discrete Formulations Given sequences S = {x1, …, xn} • A motif W is a consensus string w1…wK • Find motif W* with “best” match to x1, …, xn Definition of “best”: d(W, xi) = min hamming dist. between W and any word in xi d(W, S) = i d(W, xi)

  10. Approaches • Exhaustive Searches • CONSENSUS • MULTIPROFILER • TEIRESIAS, SP-STAR, WINNOWER

  11. Exhaustive Searches 1. Pattern-driven algorithm: For W = AA…A to TT…T (4K possibilities) Find d( W, S ) Report W* = argmin( d(W, S) ) Running time: O( K N 4K ) (where N = i |xi|) Advantage:Finds provably “best” motif W Disadvantage:Time

  12. Exhaustive Searches 2. Sample-driven algorithm: For W = any K-long word occurring in some xi Find d( W, S ) Report W* = argmin( d( W, S ) ) or, Report a local improvement of W* Running time: O( K N2 ) Advantage: Time Disadvantage: If the true motif is weak and does not occur in data then a random motif may score better than any instance of true motif

  13. CONSENSUS Algorithm: Cycle 1: For each word W in S (of fixed length!) For each word W’ in S Create alignment (gap free) of W, W’ Keep the C1 best alignments, A1, …, AC1 ACGGTTG , CGAACTT , GGGCTCT … ACGCCTG , AGAACTA , GGGGTGT …

  14. CONSENSUS Algorithm: Cycle t: For each word W in S For each alignment Aj from cycle t-1 Create alignment (gap free) of W, Aj Keep the Cl best alignments A1, …, ACt ACGGTTG , CGAACTT , GGGCTCT … ACGCCTG , AGAACTA , GGGGTGT … … … … ACGGCTC , AGATCTT , GGCGTCT …

  15. CONSENSUS • C1, …, Cn are user-defined heuristic constants • N is sum of sequence lengths • n is the number of sequences Running time: O(N2) + O(N C1) + O(N C2) + … + O(N Cn) = O( N2 + NCtotal) Where Ctotal = i Ci, typically O(nC), where C is a big constant

  16. MULTIPROFILER • Extended sample-driven approach Given a K-long word W, define: Nα(W) = words W’ in S s.t. d(W,W’)  α Idea: Assume W is occurrence of true motif W* Will use Nα(W) to correct “errors” in W

  17. MULTIPROFILER Assume W differs from true motif W* in at most L positions Define: A wordlet G of W is a L-long pattern with blanks, differing from W • L is smaller than the word length K Example: K = 7; L = 3 W = ACGTTGA G = --A--CG

  18. MULTIPROFILER Algorithm: For each W in S: For L = 1 to Lmax • Find the α-neighbors of W in S  Nα(W) • Find all “strong” L-long wordlets G in Na(W) • For each wordlet G, • Modify W by the wordlet G  W’ • Compute d(W’, S) Report W* = argmin d(W’, S) Step 1 above: Smaller motif-finding problem; Use exhaustive search

  19. Expectation Maximization in Motif Finding

  20. Expectation Maximization • The MM algorithm, part of MEME package uses Expectation Maximization Algorithm (sketch): • Given genomic sequences find all K-long words • Assume each word is motif or background • Find likeliest Motif Model Background Model classification of words into either Motif or Background

  21. Expectation Maximization • Given sequences x1, …, xN, • Find all k-long words X1,…, Xn • Define motif model: M = (M1,…, MK) Mi = (Mi1,…, Mi4) (assume {A, C, G, T}) where Mij = Prob[ letter j occurs in motif position i ] • Define background model: B = B1, …, B4 Bi = Prob[ letter j in background sequence ]

  22. Expectation Maximization • Define Zi1 = { 1, if Xi is motif; 0, otherwise } Zi2 = { 0, if Xi is motif; 1, otherwise } • Given a word Xi = x[1]…x[k], P[ Xi, Zi1=1 ] =  M1x[1]…Mkx[k] P[ Xi, Zi2=1 ] = (1 - ) Bx[1]…Bx[K] Let 1 = ; 2 = (1- )

  23. Expectation Maximization Define: Parameter space  = (M,B) 1: Motif; 2: Background Objective: Maximize log likelihood of model:

  24. Expectation Maximization • Maximize expected likelihood, in iteration of two steps: Expectation: Find expected value of log likelihood: Maximization: Maximize expected value over , 

  25. Expectation Maximization: E-step Expectation: Find expected value of log likelihood: where expected values of Z can be computed as follows:

  26. Expectation Maximization: M-step Maximization: Maximize expected value over  and  independently For , this is easy:

  27. Expectation Maximization: M-step • For  = (M, B), define cjk = E[ # times letter k appears in motif position j] c0k = E[ # times letter k appears in background] • cij values are calculated easily from E[Z] values It easily follows: to not allow any 0’s, add pseudocounts

  28. Initial Parameters Matter! Consider the following “artificial” example: x1, …, xN contain: • 212 patterns on {A, T}: A…A, A…AT,……, T… T • 212 patterns on {C, G}: C…C , C…CG,…… , G…G • D << 212 occurrences of 12-mer ACTGACTGACTG Some local maxima: •  ½; B = ½C, ½G; Mi = ½A, ½T, i = 1,…, 12 •  D/2k+1; B = ¼A,¼C,¼G,¼T; M1 = 100% A, M2= 100% C, M3 = 100% T, etc.

  29. Overview of EM Algorithm • Initialize parameters  = (M, B), : • Try different values of  from N-1/2 up to 1/(2K) • Repeat: • Expectation • Maximization • Until change in  = (M, B),  falls below  • Report results for several “good” 

  30. Overview of EM Algorithm • One iteration running time: O(NK) • Usually need < N iterations for convergence, and < N starting points. • Overall complexity: unclear – typically O(N2K) - O(N3K) • EM is a local optimization method • Initial parameters matter MEME: Bailey and Elkan, ISMB 1994.

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