1. Transcription factor binding motifs
Prof. William Stafford Noble
GENOME 541

2. Outline Representing motifs Motif discovery
Gibbs sampling
MEME
Scanning for motif occurrences
Multiple testing correction redux

3. Motif (n): a succession of notes that has some special importance in or is characteristic of a composition

4. Motif (n): a recurring genomic sequence pattern
TCCACGGC

5. Sequence-specific transcription factors drive gene regulation

6. Motif discovery problem
Given sequences
Find motif
seq. 1
seq. 2
seq. 3
IGRGGFGEVY at position 515
LGEGCFGQVV at position 430
VGSGGFGQVY at position 682
seq. 1
seq. 2
seq. 3

7. Motif discovery problem (harder version)
Given:
a sequence or family of sequences.
Find:
the number of motifs
the width of each motif
the locations of motif occurrences

8. Why is this hard?
Input sequences are long (thousands or millions of residues).
Motif may be subtle
Instances are short.
Instances are only slightly similar.

9. The most common model of sequence motifs is the position-specific scoring matrix A C G T …

10. Log-odds score Estimate the probability of observing each amino acid.
The amino acid was generated by the foreground model (i.e., the PSSM).
The amino acid “A” is observed.
Estimate the probability of observing each amino acid.
Divide by the background probability of observing the same amino acid.
Take the log so that the scores are additive.
The amino acid was generated by the background model (i.e., randomly selected).

11. Motif logos scale letters by relative entropy
Splice site motif
pi, = probability of  in matrix position i
b = background frequency of 
CTCF binding motif

12. Gibbs sampling
Lawrence et al. “Detecting subtle sequence signals: A Gibbs sampling strategy for multiple alignment.” Science 1993

13. Alternating approach Guess an initial weight matrix
Use weight matrix to predict instances in the input sequences
Use instances to predict a PSSM
Repeat 2 & 3 until satisfied.

14. Initialization
Randomly guess an instance si from each of t input sequences {S1, ..., St}.
sequence 1
ACAGTGT
TTAGACC
GTGACCA
ACCCAGG
CAGGTTT
sequence 2
sequence 3
sequence 4
sequence 5

15. Gibbs sampler
Initially: randomly guess an instance si from each of t input sequences {S1, ..., St}.
Steps 2 & 3 (search):
Throw away an instance si: remaining (t - 1) instances define PSSM.
PSSM defines instance probability at each position of input string Si
Pick new si according to probability distribution
Return highest-scoring motif seen

16. Sampler step illustration:
ACAGTGT
TAGGCGT
ACACCGT
???????
CAGGTTT A C G T
.45
.05
.25
.65
.85
ACAGTGT
TAGGCGT
ACACCGT
ACGCCGT
CAGGTTT
sequence 4
11%
ACGCCGT:20%
ACGGCGT:52%

17. MEME
Bailey and Elkan. “Fitting a mixture model by expectation-maximization to discover motifs in biopolymers.” ISMB 1994.

18. MEME problem statement
Input: Collection of sequences.
Assumption: One TFBS per sequence
Output: High-likelihood PSSM

19. MEME likelihood model
TFBS starti
sequencei

20. The EM algorithm optimizes latent variable (missing data) models
TFBS start
position C A T G G

21. The EM algorithm optimizes latent variable (missing data) modelsX Y θ

22. The EM algorithm optimizes latent variable (missing data) modelsX Y θ
Want to optimize:

23. The EM algorithm optimizes latent variable (missing data) modelsX Y θ
Want to optimize:
Easier to optimize:
E-step:
M-step:

24. Idea #1: Compute probability distribution jointly over all sites
TFBS starts (Y) …
sequence (X)
E-step:

25. In order to efficiently compute a probability distribution over many variables, that distribution must be factorizable …
Independent:
Chain-structure: …
Tree-structure:

26. The MEME probability distribution is efficiently computable because each sequence is independent given θ θ Yi
Xi:
E-step:
Distribution for sequence i:
TFBS starti (Yi)
sequencei (Xi)

27. MEME update rule θ Yi Xi: M-step: Update rule: Position #
Sequence #
Position #
Indicator function
Update rule:

28. Alternating approach Guess an initial weight matrix
Use weight matrix to predict instances in the input sequences
Use instances to predict a weight matrix
Repeat 2 & 3 until satisfied.

29. MEME initializes EM separately using each subsequence as a starting point A C G T
One round of EM
Choose highest-likelihood initializations
run EM to convergence

30. Scanning for motifs
Grant et al. “FIMO: Scanning for occurrences of a given motif.” Bioinformatics 2011.

31. CTCF One of the most important transcription factors in human cells.
Responsible both for turning genes on and for maintaining 3D structure of the DNA.

32. Motivating question: How accurately does a PSSM predict the binding of a given transcription factor?

33. Scanning for motif occurrences
Given:
a long DNA sequence, and
TAATGTTTGTGCTGGTTTTTGTGGCATCGGGCGAGAATAGCGCGTGGTGTGAAAG
a DNA motif represented as a PSSM
Find:
occurrences of the motif in the sequence A C G T

34. Scanning for motif occurrences
– = 6.87
TAATGTTTGTGCTGGTTTTTGTGGCATCGGGCGAGAATAGCGCGTGGTGTGAAAG

35. Scanning for motif occurrences
– 3.32 – = -1.39
TAATGTTTGTGCTGGTTTTTGTGGCATCGGGCGAGAATAGCGCGTGGTGTGAAAG

36. Searching human chromosome 21 with the CTCF motif

37. Significance of scores
Motif
scanning
algorithm
26.30
Low score = not a motif occurrence
High score = motif occurrence
How high is high enough?
TTGACCAGCAGGGGGCGCCG

38. Two way to assess significance
Empirical
Randomly generate data according to the null hypothesis.
Use the resulting score distribution to estimate p-values.
Exact
Mathematically calculate all possible scores

39. CTCF empirical null distribution

40. Poor precision in the tail

41. Converting scores to p-values
Linearly rescale the matrix values to the range [0,100] and integerize.

42. Converting scores to p-values
Find the smallest value.
Subtract that value from every entry in the matrix.
All entries are now non-negative.

43. Converting scores to p-values
100 / 7 =
Find the largest value.
Divide 100 by that value.
Multiply through by the result.
All entries are now between 0 and 100.

44. Converting scores to p-values
Round to the nearest integer.

45. Converting scores to p-values… A C G T
Say that your motif has N columns. Create a matrix that has N rows and 100N columns.
The entry in row i, column j is the number of different sequences of length i that can have a score of j.

46. Converting scores to p-values… A C G T 1 1 1 1
For each value in the first column of your motif, put a 1 in the corresponding entry in the first row of the matrix.
There are only 4 possible sequences of length 1.

47. Converting scores to p-values… A C G T 1 1 1 1 1
For each value x in the second column of your motif, consider each value y in the zth column of the first row of the matrix.
Add y to the x+zth column of the matrix.

48. Converting scores to p-values… A C G T 1 1 1 1 1
For each value x in the second column of your motif, consider each value y in the zth column of the first row of the matrix.
Add y to the x+zth column of the matrix.
What values will go in row 2?
10+67, 10+39, 10+71, 10+43, 60+67, …,
These 16 values correspond to all 16 strings of length 2.

49. Converting scores to p-values… A C G T 1 1 1 1 1
In the end, the bottom row contains the scores for all possible sequences of length N.
Use these scores to compute a p-value.

50. Multiple testing correction
Noble. “How does multiple testing correction work?” Nature Biotechnology 2010.

52. Multiple testing
Say that you perform a statistical test with a 0.05 threshold, but you repeat the test on twenty different observations.
Assume that all of the observations are explainable by the null hypothesis.
What is the chance that at least one of the observations will receive a p-value less than 0.05?

53. Multiple testing
Say that you perform a statistical test with a 0.05 threshold, but you repeat the test on twenty different observations. Assuming that all of the observations are explainable by the null hypothesis, what is the chance that at least one of the observations will receive a p-value less than 0.05?
Pr(making a mistake) = 0.05
Pr(not making a mistake) = 0.95
Pr(not making any mistake) = = 0.358
Pr(making at least one mistake) = = 0.642
There is a 64.2% chance of making at least one mistake.

54. Bonferroni correction
How does it work?

55. Bonferroni correction
Divide the desired p-value threshold by the number of tests performed.
For the previous example, 0.05 / 20 =
Pr(making a mistake) =
Pr(not making a mistake) =
Pr(not making any mistake) = =
Pr(making at least one mistake) = =
Does not assume that individual tests are independent.

56. Sample problem
You have scanned both strands of the human genome with a single PSSM, yielding 6 × 109 scores.
You use dynamic programming to assign a p-value of 2.1 × to the top-scoring match.
Is this alignment significance at a 95% confidence threshold?
No, because 0.05 / 6 × 109 = 8.3 ×

57. Proof: Bonferroni adjustment controls the family-wise error rate
Note: Bonferroni adjustment does not require that the tests be independent.
Boole’s inequality
Definition of p-value
m = number of hypotheses
m0 = number of null hypotheses
⍺ = desired control
pi = ith p-value
Definition of m and m0

58. Types of errors
False positive: the algorithm indicates that this position is a binding site, but it actually is not.
False negative: the site a binding site, but the algorithm indicates that it is not.
Both types of errors are defined relative to some confidence threshold.
Typically, researchers are more concerned about false positives.

59. False discovery proportion
5 FP
13 TP
The false discovery proportion (FDP) is the percentage of target sequences above the threshold that are false positives.
The FDR is the expected value of the FDP.
In the context of motif scanning, the false discovery proportion is the percentage of sites above the threshold that are not binding sites.
33 TN
5 FN
Binding site
Non-binding site
FDP = FP / (FP + TP) = 5/18 = 27.8%

60. Family-wise error rate vs. false discovery rate
Bonferroni controls the family-wise error rate; i.e., the probability of at least one false positive among the sequences that score better than the threshold.
With FDR control, you aim to control the percentage of false positives among the sequences that score better than the threshold.

61. Controlling the FDR Order the unadjusted p-values p1  p2  …  pm.
To control FDR at level α,
Reject the null hypothesis for j = 1, …, j*.
(Benjamini & Hochberg, 1995)

62. FDR example
Rank (jα)/m p-value …
Choose the largest threshold j so that (jα)/m is less than the corresponding p-value.
Approximately 5% of the examples above the line are expected to be false positives.

63. Summary – Multiple testing correction
Selecting a significance threshold requires evaluating the cost of making a mistake.
Bonferroni correction divides the desired p-value threshold by the number of statistical tests performed.
The false discovery proportion is the percentage of false positives among the target sequences that score better than the threshold.
Use Bonferroni correction when you want to avoid making a single mistake; control the false discovery rate when you can tolerate a certain percentage of mistakes.