De novo identification of motifs in one species. Modified from Serafim Batzoglou s lecture notes

Transcription

1 De novo identification of motifs in one species Modified from Serafim Batzoglou s lecture notes

2 Finding Regulatory Motifs... Given a collection of genes that may be regulated by the same transcription factor, Find the TF-binding motif in common.

3 Characteristics of Regulatory Motifs TCCGGAAGC TCCGGATGC TCCGGATCT CATGGATGC CCAGGAAGT GGTGGATGC ACCGGATGC Tiny degenerate ~Constant Size Because a constant-size transcription factor binds Occurs everywhere

4 Enormous non-coding sequences??? 0 gene Given upstream regions of coregulated genes: Increasing length makes motif finding harder random motifs clutter the true ones Decreasing length makes motif finding harder true motif missing in some sequences

5 Problem Definition Given a collection of promoter sequences s 1,, s N of genes with common expression Combinatorial Motif M: m 1 m W Some of the m i s blank Find M that occurs in all s i with k differences Probabilistic Motif: M ij ; 1 i W 1 j 4 M ij = Prob[ letter i, pos j ] Find best M, and positions p 1,, p N in sequences

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

7 Probabilistic Algorithms 1. Expectation Maximization: MEME 2. Gibbs Sampling: AlignACE, BioProspector Combinatorial CONSENSUS

8 Discrete Approaches to Motif Finding

9 Discrete Formulations Given sequences S = {x 1,, x n } A motif W is a consensus string w 1 w K Find motif W * with best match to x 1,, x n Definition of best : d(w, x i ) = min hamming dist. between W and a word in x i d(w, S) = i d(w, x i )

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

11 Exhaustive Searches Pattern-driven algorithm: For W = AA A to TT T possibilities) Find d( W, S ) Report W* = argmin( d(w, S) ) Running time: O( K N 4 K ) (where N = i x i ) (4 K

12 Exhaustive Searches (2) 2. Sample-driven algorithm: For W = a K-bp word in some x i Find d( W, S ) Report W* = argmin( d( W, S ) ) OR Report a local improvement of W * Running time: O( K N 2 )

13 Exhaustive Searches (3) Problem with sample-driven approach: If: True motif does not occur in data, and True motif is weak Then, random strings may score better than any instance of true motif

14 CONSENSUS (1) Algorithm: Cycle 1: For each word W in S 1 For each word W in S 2 Create alignment (gap free) of W, W Keep the C 1 best alignments, A 1,, A C1 ACGGTTG, CGAACTT, GGGCTCT ACGCCTG, AGAACTA, GGGGTGT

15 CONSENSUS (2) Algorithm (cont d): Cycle 2: For each word W in S For each alignment A j from the previous cycle,create alignment (gap free) of W, A j Keep the C l best alignments A 1,, A cl

16 CONSENSUS (3) C 1,, C n are user-defined heuristic constants Running time: O(N 2 ) + O(N C 1 ) + O(N C 2 ) + + O(N C n ) = O( N 2 + NC total ) Where C total = i C i, typically O(nC), where C is a big constant

17 MULTIPROFILER Extended sample-driven approach Given a K-bp long word W, define: N a (W) = words W in S s.t. d(w,w ) a Idea: Assume W is occurrence of true motif W * Will use N a (W) to correct errors in W

18 MULTIPROFILER (2) Assume W differs from true motif W * in at most L positions Define: A wordlet G of W is a L-bp pattern with blanks, differing from W Example: K = 7; L = 3 W = ACGTTGA G = --A--CG

19 Algorithm: MULTIPROFILER (2) For each W in S: For L = 1 to L max Find all strong L-long wordlets G in N a (W) Modify W by the wordlet G -> W Compute d(w, S) Report W * = argmin d(w, S)

20 Expectation Maximization in Motif Finding

21 Expectation Maximization (1) The MM algorithm, part of MEME package uses Expectation Maximization Algorithm (sketch): Given genomic sequences find all K-bp words 1. Assume each word is motif or background 2. Find likeliest motif & background models, and classification of words

22 Expectation Maximization (2) Given sequences x 1,, x N, Find all k-bp words X 1,, X n Define motif model: M = (M 1,, M K ) M i = (M i1,, M i4 ) (assume {A, C, G, T}) where M ij = Prob[ motif position i is letter j ] Define background model: B = B 1,, B 4 B i = Prob[ letter j in background sequence ]

23 Expectation Maximization (3) Define Z i1 = { 1, if X i is motif; 0, otherwise } Z i2 = { 0, if X i is motif; 1, otherwise } Given a word X i = a[1] a[k], P[ X i, Z i1 =1 ] = M 1a[1] M ka[k] P[ X i, Z i2 =1 ] = (1 - ) B a[1] B a[k]

24 Expectation Maximization (4) Define: Parameter space = (M,B) Objective: Maximize log likelihood of model: log ) ( log )) ( log( ),,... ( log j j j ij n i j i ij n i n i j j i j ij n Z Z Z X P X P Z X X P

25 Expectation Maximization (5) Maximize expected likelihood, in iteration of two steps: Expectation: Find expected value of log likelihood: E[log P( X... X, Z, )] 1 n Maximization: Maximize expected value over,

26 Expectation Maximization (6): E-step Expectation: Find expected value of log likelihood: ]log [ ) ( ]log [ )],,... ( [log j j j ij n i j i ij n i n Z Z E X P E Z X X P E where expected values of Z can be computed as follows: 2 1 ) ( ) ( k k i k j i j ij X P X P Z

27 Expectation Maximization (7): M-step Maximization: Maximize expected value over and independently For, this is easy: NEW j arg m a x n E[ Z ]log j ij j i1 i1 n Z ij n

28 Expectation Maximization (8): M-step For = (M, B), define c jk = E[ # times letter k appears in motif position j] c 0k = E[ # times letter k appears in background] It easily follows: M NEW jk c jk 4 k 1 c jk B NEW k c 0k 4 c k 1 0 k to not allow any 0 s, add pseudocounts

29 Overview of EM Algorithm 1. Initialize parameters = (M, B), : Try different values of from N -1/2 up to 1/(2K) 2. Repeat: a. Expectation b. Maximization 3. Until change in = (M, B), falls below 4. Report results for several good

30 Conclusion One iteration running time: O(NK) Usually need < N iterations for convergence, and < N starting points. Overall complexity: unclear typically O(N 2 K) -O(N 3 K) EM is a local optimization method Initial parameters matter MEME: Bailey and Elkan, ISMB 1994.

31 Gibbs Sampling in Motif Finding

32 Overview of Gibbs Sampler Iteratively sample from conditional distribution when other parameters are fixed. In order to draw: ( 1 2 n ~ X, X,, X ) ( X ) i, draw X i ~ ( X i X [ i] )

33 Gibbs Sampling (1) Given: x 1,, x N, motif length K, background B, Find: Model M Locations a 1,, a N in x 1,, x N Maximizing log-odds likelihood ratio: N K M ( k, x i a k log i B( x i ) i1 k 1 a k i )

34 Gibbs Sampling (2) AlignACE: first statistical motif finder BioProspector: improved version of AlignACE Algorithm (sketch): 1. Initialization: a. Select random locations in sequences x 1,, x N b. Compute an initial model M from these locations 2. Sampling Iterations: a. Remove one sequence x i b. Recalculate model c. Pick a new location of motif in x i according to probability the location is a motif occurrence

35 Gibbs Sampling (3) Initialization: Select random locations a 1,, a N in x 1,, x N For these locations, compute M: N 1 M kj ( xa k j) i N That is, M kj is the number of occurrences of letter j in motif position k, over the total i1

36 Gibbs Sampling (4) Predictive Update: Select a sequence x = x i Remove x i, recompute model: M kj ( N 1 1) ( j B s N 1, s ( x i a s k j)) where j are pseudocounts to avoid 0s, and B = j j M

37 Gibbs Sampling (5) Sampling: For every K-long word x j,,x j+k-1 in x: Q j = Prob[ word motif ] = M(1,x j ) M(k,x j+k-1 ) P i = Prob[ word background ] B(x j ) B(x j+k-1 ) Let A j Q j x k 1 j1 / Q j P / j P j Prob 0 x Sample a random new position a i according to the probabilities A 1,, A x -k+1.

38 Gibbs Sampling (6) Running Gibbs Sampling: 1. Initialize 2. Run until convergence 3. Repeat 1,2 several times, report common motifs

39 Advantages / Disadvantages Very similar to EM Advantages: Easier to implement Less dependent on initial parameters More versatile, easier to enhance with heuristics Disadvantages: More dependent on all sequences to exhibit the motif Less systematic search of initial parameter space

40 Repeats, and a Better Background Model Repeat DNA can be confused as motif Especially low-complexity CACACA AAAAA, etc. Solution: more elaborate background model 0 th order: B = { p A, p C, p G, p T } 1 st order: B = { P(A A), P(A C),, P(T T) } K th order: B = { P(X b 1 b K ); X, b i {A,C,G,T} } Has been applied to EM and Gibbs (up to 3 rd order)

41 Applications

42 Application 1: Motifs in Yeast Group: Tavazoie et al. 1999, G. Church s lab, Harvard Data: Microarrays on 6,220 mrnas from yeast Affymetrix chips (Cho et al.) 15 time points across two cell cycles

43 Processing of Data 1. Selection of 3,000 genes Genes with most variable expression were selected 2. Clustering according to common expression K-means clustering 30 clusters, genes/cluster Clusters correlate well with known function 3. AlignACE motif finding 600-long upstream regions 50 regions/trial

44 Motifs in Periodic Clusters

45 Motifs in Non-periodic Clusters

46 Application 2: Discovery of Heat Shock Motif in C. Elegans Group: GuhaThakurta et al. 2002, C.D. Link s lab & colleagues Data: Microarrays on 11,917 genes from C. Elegans Isolated genes upregulated in heat shock

47 Processing of Data, and Results Isolated 28 genes upregulated in heat shock during 5 separate experiments Motif finding with CONSENSUS and ANNSpec on 500-long upstream regions 2 motifs found: TTCTAGAA: known heat shock factor (HSF) GGGTGTC: previously unreported Conserved in comparison with C. Briggsae Validation by in vitro mutagenesis of a GFP reporter