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

1 2 3 4 5 6 7 8 9

A .97 .10 .02 .03 .10 .01 .05 .85 .03

C .01 .40 .01 .04 .05 .01 .05 .05 .03

G .01 .40 .95 .03 .40 .01 .3 .05 .03

T .01 .10 .02 .90 .45 .97 .6 .05 .91

I 1.76 0.28 1.64 1.37 0.40 1.76 0.60 1.15 1.42

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 = {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 a word in xi

d(W, S) = i d(W, xi)

W* = argmin( d(W, S) )

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|)

Guaranteed to find the optimal solution.

Exhaustive searches

2. Sample-driven algorithm:

For W = a K-long word 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 )

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. [011100010]

• Bit OR all B’s to get seq occurrence for P

• Suppose #occ >= m– Pattern still valid

• Now add a letter

ACGTT

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

ACGTTA

Current pattern P

(e, B)

Probabilistic modeling approaches

for motif finding

Probabilistic modeling approaches

• A motif model– Usually a PWM– M = (Pij), 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 = (bi), i = 1..4

• A word can be generated by M or B

Expectation-Maximization

• For any word W, P(W | M) = PW[1] 1 PW[2] 2…PW[K] K

P(W | B) = bW[1] bW[2] …bW[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 Zxy = 1 if position y in sequence x is a motif, and 0

otherwise• Estimate parameters M, , B

Iterate until converge:• E-step: Zxy = 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: Zxy = P(M | X[y..y+k-1]) for all x and y

• M-step: re-estimate M, given Z

Initialize E-step

M-step

prob

abili

ty

position

1

9

5

1

9

5

MEME

• Multiple EM for Motif Elicitation

• Bailey and Elkan, UCSD

• http://meme.sdsc.edu/

• 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 Zxy = 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 Zx*y for X*• Sample a y* from Zx*y. • Let Zx*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

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

position

prob

abili

ty

Sampling

Simultaneously update predictions of all sequences

position

prob

abili

ty

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

0th order: B = { pA, pC, pG, pT }1st order: B = { P(A|A), P(A|C), …, P(T|T) }…Kth order: B = { P(X | b1…bK); X, bi{A,C,G,T} }

Has been applied to EM and Gibbs (up to 3rd order)

Gibbs sampling motif finders• Gibbs Sampler

– First appeared as: Larence et.al. Science 262(5131):208-214. – Continually developed and updated. webpage– The newest version: Thompson et. al. Nucleic Acids Res. 35 (s2):W232-

W237 • AlignACE

– Hughes et al., J. of Mol Bio, 2000 10;296(5):1205-14. – 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;:127-38. – 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 mutationsn = 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 setT

Background setB

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 setT

Background set + target setB + TN 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

N

M

nN

mM

n

m

mNMnhypegeom ),,;(

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?

1

0

1

0

),min(

1

),,;(1

),,;(),,;(

n

i

n

i

Nm

ni

N

M

iN

mM

i

m

mNMihypogeom

mNMihypogeommNMncHypegeom

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 = 100+20 = 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 10-22

• 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 setT

Background set + target setB + TN 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

sens

itivi

ty

Motif 1Motif 2Random

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 = 0.

0

1

10

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 50-150bp 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):423-425

• D’haeseleer P (2006) How does DNA sequence motif discovery work? NATURE BIOTECHNOLOGY, 24 (8):959-961

• 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):208-214