CS5263 Bioinformatics Probabilistic modeling approaches for motif finding

Motif representation Collection of exact words –{ACGTTAC, ACGCTAC, AGGTGAC, …} Consensus sequence (with wild cards) –{AcGTgTtAC} –{ASGTKTKAC} S=C/G, K=G/T (IUPAC code) Position specific weight matrices (PWMs)

Sequence Logo A C G T I

Finding Motifs

Classification of approaches Combinatorial search –Based on enumeration of words and computing word similarities –Analogy to DP for sequence alignment Probabilistic modeling –Construct models to distinguish motifs vs non- motifs –Analogy to HMM for sequence alignment

Combinatorial motif finding Given a set of 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 ) W* = argmin( d(W, S) )

Exhaustive searches 1. Pattern-driven algorithm: For W = AA…A to TT…T (4 K possibilities) Find d( W, S ) Report W* = argmin( d(W, S) ) Running time: O( K N 4 K ) (where N =  i |x i |) Guaranteed to find the optimal solution.

Exhaustive searches 2. Sample-driven algorithm: For W = a K-long 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 )

WEEDER: algorithm sketch A list containing all eligible nodes: with at most α mismatches to P For each node, remember #mismatches accumulated (e), and bit vector (B) for seq occ, e.g. [ ] Bit OR all B’s to get seq occurrence for P Suppose #occ >= m –Pattern still valid Now add a letter ACGTTACGTT Current pattern P, |P| < K (e, B) # mismatches Seq occ

WEEDER: algorithm sketch Simple extension: no branches. –No change to B –e may increase by 1 or no change –Drop node if e > α Branches: replace a node with its child nodes –Drop if e > α –B may change Re-do Bit OR using all B’s Try a different char if #occ < m Report P when |P| = K ACGTTAACGTTA Current pattern P (e, B)

Probabilistic modeling approaches for motif finding

Probabilistic modeling approaches A motif model –Usually a PWM –M = (P ij ), i = 1..4, j = 1..k, k: motif length A background model –Usually the distribution of base frequencies in the genome (or other selected subsets of sequences) –B = (b i ), i = 1..4 A word can be generated by M or B

Expectation-Maximization For any word W,  P(W | M) = P W[1] 1 P W[2] 2 …P W[K] K  P(W | B) = b W[1] b W[2] …b W[K] Let = P(M), i.e., the probability for any word to be generated by M. Then P(B) = 1 - Can compute the posterior probability P(M|W) and P(B|W)  P(M|W) ~ P(W|M) *  P(B|W) ~ P(W|B) * (1- )

Expectation-Maximization Initialize: Randomly assign each word to M or B Let Z xy = 1 if position y in sequence x is a motif, and 0 otherwise Estimate parameters M,, B Iterate until converge: E-step: Z xy = P(M | X[y..y+k-1]) for all x and y M-step: re-estimate M, given Z (B usually fixed)

Expectation-Maximization E-step: Z xy = P(M | X[y..y+k-1]) for all x and y M-step: re-estimate M, given Z Initialize E-step M-step probability position

MEME Multiple EM for Motif Elicitation Bailey and Elkan, UCSD Multiple starting points Multiple modes: ZOOPS, OOPS, TCM

Gibbs Sampling Another very useful technique for estimating missing parameters EM is deterministic –Often trapped by local optima Gibbs sampling: stochastic behavior to avoid local optima

Gibbs sampling Initialize: Randomly assign each word to M or B Let Z xy = 1 if position y in sequence x is a motif, and 0 otherwise Estimate parameters M, B, Iterate: Randomly remove a sequence X* from S Recalculate model parameters using S \ X* Compute Z x*y for X* Sample a y* from Z x*y. Let Z x*y = 1 for y = y* and 0 otherwise

Gibbs Sampling Gibbs sampling: sample one position according to probability –Update prediction of one training sequence at a time Viterbi: always take the highest EM: take weighted average Sampling Simultaneously update predictions of all sequences position probability

Better background model Repeat DNA can be confused as motif –Especially low-complexity CACACA… AAAAA, etc. Solution: more elaborate background model –Higher-order Markov 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)

Gibbs sampling motif finders Gibbs Sampler –First appeared as: Larence et.al. Science 262(5131): –Continually developed and updated. webpagewebpage –The newest version: Thompson et. al. Nucleic Acids Res. 35 (s2):W232- W237 AlignACE –Hughes et al., J. of Mol Bio, ;296(5): Hughes et al., J. of Mol Bio, ;296(5): –Allow don’t care positions –Additional tools to scan motifs on new seqs, and to compare and group motifs BioProspector, X. Liu et. al. PSB 2001, an improvement of AlignACE –Liu, Brutlag and Liu. Pac Symp Biocomput. 2001;: Liu, Brutlag and Liu. Pac Symp Biocomput. 2001;: –Allow two-block motifs –Consider higher-order markov models

Limits of Motif Finders 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 0 gene ???

Challenging problem (k, d)-motif challenge problem Many algorithms fail at (15, 4)-motif for n = 20 and L = 600 Combinatorial algorithms usually work better on challenge problem –However, they are usually designed to find (k, d)-motifs –Performance in real data varies k d mutations n = 20 L = 600

(15, 4)-motif Information content: 11.7 bits ~ 6mers. Expected occurrence 1 per 3k bp

Actual Results by MEME llr = 163 E-value = 3.2e+005 llr = 177 E-value = 1.5e+006 llr = 88 E-value = 2.5e+005

Motif finding in practice Now we’ve found some good looking motifs –This is probably the easiest step What to do next? –Are they real? –How do we find more instances in the rest of the genome? –What are their functional meaning? Motifs => regulatory networks

How to make sense of the motifs? Each program usually reports a number of motifs (tens to hundreds) –Many motifs are variations of each other –Each program also report some different ones Each program has its own way of scoring motifs –Best scored motifs often not interesting –AAAAAAAA –ACACACAC –TATATATAT

How to make sense of the motifs? Combine results from different algorithms usually helpful –Ones that appeared multiple times are probably more interesting Except simple repeats like AAAAA or ATATATATA –Cluster motifs into groups. Compare with known motifs in database –TRANSFAC –JASPAR –YPD (yeast promoter database)

Strategies to improve results How to tell real motifs (functional) from noises? Statistical test of significance. –Enrichment in target sequences vs background sequences Target set T Background set B Assumed to contain a common motif, P Assumed to not contain P, or with very low frequency Ideal case: every sequence in T has P, no sequence in B has P

Statistical test for significance If n / N >> m / M –P is enriched (over-represented) in T –Statistical significance? If we randomly draw N sequences from (B+T), how likely we will see at least n sequences having P? Target set T Background set + target set B + T N M P P appeared in n sequences P appeared in m sequences

Hypergeometric distribution A box with M balls (seqs), of which m are red (with motifs), and the rest are blue (without motifs). –Red ball: sequences with motifs –Blue ball: sequences without motifs We randomly draw N balls (seqs) from the box What’s the probability we’ll see n red balls? # of choices to have n red balls Total # of choices to draw N balls

Cumulative hypergeometric test for motif significance We are interested in: if we randomly pick m balls, how likely that we’ll see at least n red balls? Null hypothesis: our selection is random. Alternative hypothesis: our selection favored red balls. When prob is small, we reject the null hypothesis. Equivalent: we accept the alternative hypothesis (The number of red balls is larger than expected).

Example Yeast genome has 6000 genes Select 50 genes believed to be co-regulated by a common TF Found a motif from the promoter seqs of these 50 genes The motif appears in 20 of these 50 genes In the rest of the genome, 100 genes have this motif M = 6000, N = 50, m = = 120, n = 20 Intuitively: –m/M = 120/6000=1/50. (1 out 50 genes has the motif) –N = 50, would expect only 1 gene in the target set to have the motif –20-fold enrichment P-value = cHyperGeom(20; 6000, 50, 120) = 6 x This motif is significantly enriched in the set of genes

ROC curve for motif significance Motif is usually a PWM Any word will have a score –Typical scoring function: Log (P(W | M) / P(W | B)) –W: a word. –M: a PWM. –B: background model To determine whether motif M occurred in a sequence, a cutoff has to be decided –Different cutoffs give different # of occurrences –Stringent cutoff: low occurrence in both + and - sequences –Loose cutoff: high occurrence in both + and - sequences –It may be better to look at a range of cutoffs

ROC curve for motif significance With different score cutoff, will have different m and n Assume you want to use P to classify T and B Sensitivity: n / N Specificity: (M-N-m+n) / (M-N) False Positive Rate = 1 – specificity: (m – n) / (M-N) With decreasing cutoff, sensitivity , FPR  Target set T Background set + target set B + T N M P Appeared in n sequences Appeared in m sequences Given a score cutoff

ROC curve for motif significance ROC-AUC: area under curve. 1: the best. 0.5: random. Motif 1 is more enriched in motif 2. 1-specificity sensitivity Motif 1 Motif 2 Random A good cutoff Highest cutoff. No motif can pass the cutoff. Sensitivity = 0. specificity = 1. Lowest cutoff. Every sequence has the motif. Sensitivity = 1. specificity =

Other strategies Cross-validation –Randomly divide sequences into 10 sets, hold 1 set for test. –Do motif finding on 9 sets. Does the motif also appear in the testing set? Phylogenetic conservation information –Does a motif also appears in the homologous genes of another species? –Strongest evidence –However, will not be able to find species-specific ones

Other strategies Finding motif modules –Will two motifs always appear in the same gene? Location preference –Some motifs appear to be in certain location E.g., within bp upstream to transcription start –If a detect motif has strong positional bias, may be a sign of its function Evidence from other types of data sources –Do the genes having the motif always have similar activities (gene expression levels) across different conditions? –Interact with the same set of proteins? –Similar functions? –etc.

To search for new instances Usually many false positives Score cutoff is critical Can estimate a score cutoff from the “true” binding sites Motif finding Scoring function A set of scores for the “true” sites. Take mean - std as a cutoff. (or a cutoff such that the majority of “true” sites can be predicted). Log (P(W | M) / P(W | B))

To search for new instances Use other information, such as positional biases of motifs to restrict the regions that a motif may appear Use gene expression data to help: the genes having the true motif should have similar activities –Risk of circular reasoning: most likely this is how you get the initial sequences to do motif finding Phylogenetic conservation is the key

References D’haeseleer P (2006) What are DNA sequence motifs? NATURE BIOTECHNOLOGY, 24 (4): D’haeseleer P (2006) How does DNA sequence motif discovery work? NATURE BIOTECHNOLOGY, 24 (8): MacIsaac KD, Fraenkel E (2006) Practical strategies for discovering regulatory DNA sequence motifs. PLoS Comput Biol 2(4): e36 Lawrence CE et. al. (1993) Detecting Subtle Sequence Signals: A Gibbs Sampling Strategy for Multiple Alignment, Science, 262(5131):