Genome 541 Gene regulation and epigenomics Lecture 2 Transcription factor binding using functional genomics

Transcription

1 Genome 541 Gene regulation and epigenomics Lecture 2 Transcription factor binding using functional genomics

2 I believe it is helpful to number your slides for easy reference. It's been a while since I took linear algebra and this came at me a little fast. A lot of the math presented today went over my head (I have not taken linear algebra) More background on how you go about implementing equations like this doing that would be helpful. Would appreciate a more detailed look at actually implementing neural nets with some pseudocode. It was a bit unclear at the end where exactly you get the W's from. Why are logistic functions (or Rectifier-Linear) preferred for Neural networks? Does any non-linear function work? Could you point me to the paper that reports the structures of transcription factor binding the chromatin? doi: /j.tibs

3 Linear algebra review Vector (dimension 3): scalar * vector multiplication: a 2 4 x 1 x 2 x = 2 3 ax 1 4ax 5 2 ax 3 vector * vector multiplication (dot product): x T y = x 1 x 2 x y 1 y = x 1 y 1 + x 2 y 2 + x 3 y 3 3 y 3

4 Linear algebra review Matrix (3x2): matrix * vector multiplication: Ax = apple a1,1 a 1,2 a 2,1 a 2,2 apple x1 x 2 = apple a1,1 x 1 + a 1,2 x 2 a 2,1 x 1 + a 2,2 x 2 4

5 Neural networks f 3,1 (x 2 ) = sign(w 3,1 x 2 ) x 3 x 2 f 2,1 (x 1 )=W 2,1 x 1 x 1 f 1,1 (x 0 )=W 1,1 x 0 f 1,2 (x 0 )=W 1,2 x 0 x A C C G T

6 This lecture MEME optimization ChIP-seq peak calling Motivating problem: accounting for GC and mappability bias in peak calling Method: Convex functions More ChIP-seq peak calling considerations Other functional genomics assays 6

7 Chromatin immunoprecipitation followed by sequencing (ChIP-seq) Sequence and map to reference genome 7

8 Problem: Given a ChIP-seq experiment for factor X, where does X bind?

9 Problem: Given a ChIP-seq experiment for factor X, where does X bind? Short answer: Stack up the reads in the genome; choose the tall stacks. Issues to consider: Sequencing fragment lengths Sequencing read lengths Experimental biases Mappability GC bias How to pick a threshold and assign statistical confidence? 9

10 This lecture MEME optimization ChIP-seq peak calling Motivating problem: accounting for GC and mappability bias in peak calling Method: Convex functions More ChIP-seq peak calling considerations Other functional genomics assays 10

11 ChIP-seq read counts are biased by GC content and mappability 11

12 MOSAiCS enrichment model Bound? Mappability GC content Tag counts 12

13 How can we model the background distribution of ChIP-seq reads?

14 Every predictive model is composed of a mean model and an error model E[counts j Not bound] = exp( 0 + M MAP j + GC GC j ) Pr(counts j Not bound) =? Bound? Mappability GC content Tag counts 14

15 The Poisson distribution models sequencing read counts k P (k )=e k! Mean: λ Variance: λ 15

16 The negative binomial distribution models sequencing read counts more flexibly than Poisson P (k r, p) = k + r 1 (1 p) r p k k Mean: Variance: pr 1 p pr (1 p) 2 16

17 The negative binomial distribution models the mean and variance separately Principle: Make the weakest assumptions you can afford to in your modeling choices 17

18 MOSAiCs background model Number of counts at 50-bp bin j N j NegBin(a, a/µ j ) µ j =exp( 0 + M M j + GC GC j ) Mappability at bin j GC content at bin j 18

19 How do we know if the MOSAiCS model is even optimizable?

20 This lecture MEME optimization ChIP-seq peak calling Motivating problem: accounting for GC and mappability bias in peak calling Method: Convex functions More ChIP-seq peak calling considerations Other functional genomics assays

21 Answer: The negative log likelihood is convex The class of convex functions (defined on the next slide) roughly corresponds to the set of efficiently optimizable functions. When presented with an objective function, the first step is usually to check if it is convex. 21

22 Convex functions A function f(x) is convex if it satisfies the property: f( x +(1 )y) apple f(x)+(1 )f(y) for all x, y, 0 apple apple 1. Convex functions have no local minima. 22

23 Concave functions A function f(x) is concave if -f(x) is convex. Convex functions are usually efficiently minimizable. Concave functions are usually efficiently maximizable. (A function can be neither convex nor concave.) 23

24 Examples on one variable 24 Stanford EE364a

25 Examples on one variable Convex: Non-convex: 25 Duke stat376

26 Second derivative criterion for convexity If f(x) is on one variable, f(x) is convex if and only if d 2 f dx Stanford EE364a

27 Is exp(x) convex? Proof by picture: Proof from second derivative: d exp(x) =expx dx d 2 exp(x) x 2 =expx 0 d 27 Proof from definition: exp( x +(1 )x 0 )=exp( x)exp((1 )x 0 ) exp(x)+(1 )exp(x 0 ) (Arithmetic Mean / Geometric Mean inequality)

28 Is sin(x) convex? Disproof by picture: Counterexample by second derivative: d sin(x) dx = cos(x) d 2 sin(x) dx 2 = sin(x) sin( ) = 1 < 0 Counterexample from definition: sin( )/2 + sin(0)/2 =0/2+0/2 < sin(0/2+ /2) = sin( /2) = 1 28

29 29 Is x 2 convex?

30 30 Is x 3 convex?

31 31 (ax b) 2

32 Convexity of functions on multiple variables y = f(x 1,x 2,x 3,...) Convexity criterion: f( x +(1 )y) apple f(x)+(1 )f(y) for all x, y, 0 apple apple 1. 32

33 Examples of multi-variable convex functions x 2 R n a ne function (convex and concave): a T x + b a 2 R n,b2 R Euclidian norm (convex): s X i x 2 i 33

34 Examples of multi-variable convex functions Convex if P is positive semi-definite. 34 Stanford EE364a

35 Examples of multi-variable convex functions Convex if P is positive semi-definite. 35 Stanford EE364a

36 Second derivative criterion for convexity on multiple variables Derivative of a function on multiple variables: Second derivative: r 2 f(x) = f(x) is convex if and only if: x 1 x v T r 2 f(x)v 0 for v 2 R n 36 i.e. if r 2 f(x)vis positive semi-definite. Stanford EE364a

37 Second derivative criterion for convexity on multiple variables Derivative of a function on multiple variables: Second derivative: r 2 f(x) = f(x) is convex if and only if: x 1 x v T r 2 f(x)v 0 for v 2 R n 37 i.e. if r 2 f(x)vis positive semi-definite. Stanford EE364a

38 Second derivative criterion for convexity on multiple variables Derivative of a function on multiple variables: Second derivative: r 2 f(x) = f(x) is convex if and only if: x 1 x v T r 2 f(x)v 0 for v 2 R n 38 i.e. if r 2 f(x)vis positive semi-definite. Stanford EE364a

39 Second derivative criterion for convexity on multiple variables Derivative of a function on multiple variables: Second derivative: r 2 f(x) = f(x) is convex if and only if: x 1 x v T r 2 f(x)v 0 for v 2 R n 39 i.e. if r 2 f(x)vis positive semi-definite. Stanford EE364a

40 A non-convex function f(x 1,x 2 )=x 1 x 2 = x 1 x 2 apple apple x1 x 2 Not positive semi-definite 40

41 Practical ways to establish convexity of a function Verify the definition. Verify that the second derivative is always positive semidefinite. Show that the function can be obtained from simple convex functions by operations that maintain convexity.

42 Some operations that maintain convexity 42 Stanford EE364a

43 Regularization L2 regularization: f 0 (x) =f(x)+kxk 2 43

44 Is this convex? (Ax b) 2 + x 2 44

45 What if your function is not convex? Is there a monotonic transform that makes it convex? Example: Y i x i log Y x i = X log x i i i Neither convex nor concave Concave Next best thing: split the function into convex parts. Example: f(x 1,x 2 )=x 1 x 2 45 Convex in either x1 or x2 but not both at once. Optimize each in turn. Example: EM.

46 What if your function is not convex? Is there a monotonic transform that makes it convex? Example: Y i x i log Y x i = X log x i i i Neither convex nor concave Concave Next best thing: split the function into convex parts. Example: f(x 1,x 2 )=x 1 x 2 46 Convex in either x1 or x2 but not both at once. Optimize each in turn. Example: EM.

47 Optimizing convex objectives There are general convex optimization software packages. Even when these are too slow, convex functions usually admit fast optimization specific to your problem. More on convex optimization next class. 47

48 The MOSAiCS objective is concave in β =[ 0 M GC] T F j =[1MY j GC j ] T X µ(f j, )=exp( T F j )=exp( 0 + M M j + GC GC j ) Y X X log P (N ) = log Y P (N j )= Y X = j X Nj + a +1 a log + a log 1 + N j log a N j log µ(f j, ) N j j µ(f j, ) X log(1 1/ exp(x)) X N j log µ(f j, )=N j T F j (a ne in )

49 This lecture MEME optimization ChIP-seq peak calling Motivating problem: accounting for GC and mappability bias in peak calling Method: Convex functions More ChIP-seq peak calling considerations Other functional genomics assays 49

50 The ChIP-seq protocol enriches for a particular fragment size Chromatin DNA fragments ChIP, sonication size selection sequence 50 fragment length

51 Translate reads to the inferred center of the sequencing fragment correlation between strands 51 strand shift

52 The phantom peak results from mappability islands unmappable mappable ChIP-seq measure of quality: relative strand correlation (RSC) 52 read length

53 ChIP-seq controls Input: IgG: Skip IP Use irrelevant antibody 53 Reasons for controls: - sonocation bias - CNVs - sequence composition bias

54 Problem: Different background models result in wildly different false discovery-rate estimates 54

55 Idea: Control reproducibility of peaks between biological replicates 55 Irreproducible discovery rate (IDR): Expected fraction of peaks that are not reproducible between biological replicates.

56 56 IDR can handle varying quality levels

57 This lecture MEME optimization ChIP-seq peak calling Motivating problem: accounting for GC and mappability bias in peak calling Method: Convex functions More ChIP-seq peak calling considerations Other functional genomics assays 57

58 ChIP-exo has better spatial resolution than ChIP-seq 58

59 DamID measures TF binding through a fusion protein Dam+TF fusion protein Measure methylation at GATCs DamID vs. ChIP-seq: DamID can be easier ChIP requires (specific) antibody DamID requires fusion protein DamID can t query post-transcriptional modification (histone mods) ChIP has better spacial resolution ChIP is limited by cross-linking bias DamID is limited by GATC content and Dam reactivity ChIP has better temporal resolution: Dam acts over ~24 hours 59

60 DNase-seq and ATAC-seq measure DNA accessibility 60

61 High-depth DNase-seq (DNase-DGF) measures TF binding 61

62 Paired-end DNase and ATAC-seq measure nucleosome architecture 62

63 Administrivia Homework 1 will be up later today. Due Thursday Next week: Broad (non peak-y ) functional genomics assays; Chromatin architecture. Please write 1-minute responses.