genome evolution:

Genome Evolution. Amos Tanay 2010

Genome evolution:

Lecture 11: Transcription factor binding sites


Sequence specific transcription factors

• Sequence specific transcription factors (TFs) are a critical part of any gene activation or gene repression machinary

• TFs include a DNA binding domain that recognize specifically “regulatory elements” in the genome.

• The TF-DNA duplex is then used to target larger transcriptional structure to the genomic locus.


Sequence specificity is represented using consensus sequences or weight matrices

• The specificity of the TF binding is central to the understanding of the regulatory relations it can form.

• We are therefore interested in defining the DNA motifs that can be recognize by each TF.• A simple representation of the binding motif is the consensus site, usually derived by

studying a set of confirmed TF targets and identifying a (partial) consensus. Degeneracy can be introduced into the consensus by using N letters (matching any nucleotide) or IUPAC characters (erpresenting pairs of nucleotides, for exampe W=[A|T], S=[C|G]

• A more flexible representation is using weight matrices (PWM/PSSM):

• PWMs are frequently plotted using motif logos, in which the height of the character correspond to its probability, scaled by the position entropy

ACGCGTACGCGAACGCATTCGCGATAGCGT

123456

A60%20%0020%40%

C080%0100%00

G00100%080%0

T40%000060%


TF binding energy is approximated by weight matrices

Leu3 data (Liu and Clarke, JMB 2002)

We can interpret weight matrices as energy functions:

])[log(][

][)(

iiii

iii

spsw

swsE

This linear approximation is reasonable for most TFs.


• s

TF binding affinity is kinetically important, with possible functional implications

Kalir et al. Science 2001

Ume6

ChIP ranges

11.5

5.5

Av

era

ge

PW

M e

ne

rgy

Stronger binding

Strong

er prediction

Tanay. Genome Res 2006


An epigenetic signature of promoters (Heinzman et al., 2007)

TFs are present at only a fraction of their optimal sequence tragets. Binding is regulated by co-factors, nucleosomes and histone modifications


An epigenetic signature of enhancers (Heinzman et al., 2007)

TFs are present at only a fraction of their optimal sequence tragets. Binding is regulated by co-factors, nucleosomes and histone modifications


TFs are present at only a fraction of their optimal sequence tragets. Binding is combinatorially regulated by co-factors, nucleosomes and histone modifications

Barski et al. Cell 2007

Active

Inactive


Specific proteins are identifying enhancersHere are studies of p300 binding in the developing mouse brain

(visel et al. 2009)


TFBSs are clustered in promoters or in “sequence modules”

• The distribution of binding sites in the genome is non uniform• In small genomes, most sites are in promoters, and there is a bias toward

nucleosome free region near the TSS• In larger genomes (fly) we observe CRM (cis-regulatory-modules) which are

frequently away from the TSS. These represent enhancers.• A single binding site, without the context of other co-sites, is unlikely to represent a

functional loci


Constructing a weight matrix from aligned TFBSs is trivial

• This is done by counting (or “voting”)• Several databases (e.g., TRANSFAC, JASPAR) contain matrices

that were constructed from a set of curated and validated binding site

• Validated site: usually using “promoter bashing” – testing reported constructs with and without the putative site

Transfac 7.0/11.3 have 400/830 different PWMs, based on more than 11,000 papers

However, there are no real different 830 matrices outthere – the real binding repertoire in nature is still somewhat unclear


Probabilistic interpretation of weight matrices and a generative model

• One can think of a weight matrix as a probabilistic model for binding sites:

• This is the site independent model, defining a probability space over k-mers• Given a set of aligned k-mers, we know that the ML motif model is derived by voting (a set of independent multinomial variables – like the dice case)

• Now assume we are given a set of sequences that are supposed to include binding sites (one for each), but that we don’t know where the binding sites are.• In other words the position of the binding site is a hidden variable h.

• We introduce a background model Pb that describes the sequence outside of the binding site (usually a d-order Markov model)

• Given complete data we can write down the likelihood of a sequence s as:

k

ii imPmP

1

])[()(

k

ibackiback

S

ibackback

ildilsilsPilsPsPlsP

idisisPsP

1

||

1

]))1..[|][(/])[(()()|,(

]))1..[|][()(


• Inference of the binding site location posterior:

• Note that only k-factors should be computed for each location (Pb(s) is constant))

Using EM to discover PWMs de-novo

i

isPlsPslP )|,(/)|,(),|( 111

• Inference of the binding site location posterior:

• Note that only k factors should be computed for each location (Pb(s) is constant))

• Starting with an initial motif model, we can apply a standard EM:

E:

j Sl

jji cilsslPcP

||..0

1 )],[(),|()( M:

i

isPlsPslP )|,(/)|,(),|( 111

• As always with the EM, initializing to reasonable PWM would be critical

Following Baily and Elkan, MEME 1995


• If we assume some of the sequences may lack a binding site, this should be incorporated into the model:

Allowing false positive sequences

k

ibackiback ildilsilsPilsPsPhitPlsP

1

]))1..[|][(/])[(()(*)()|,(

hitl

s

• This is sometime called the ZOOPS model (Zero or one positions)

• In Bayesian terms: – Probability of sequence hit P(hit | S)– Probability of hit at position l = Pr(l|S)

• We can consider the PWM parameters as variables in the model• Learning the parameters is then equivalent to inference


Using Gibbs sampling to discover PWMs de-novo

hitl

s

• We can use Bibbs sampling to sample the hidden sites and estimate the PWM

hitl

s

hitl

s

• This is done by estimating the PWM from all locations except for the one we sample, and computing the hit probabilities as shown before

• Note that we are working with the MAP (Maximum a-posteriori) to do the sampling:

),,..,,,..,|( 111 SlllllP niij

Gibbs: Lawrence et al. Science 1993

),|,..,,,..,(

)|(111maxarg

SllllL

lPniiMAP

MAPj

• But this can be shown to approximate:

),,..,,,..,|( 111 niij lllllP


Generalizing PWMs to allow site dependencies: mixture of PWMs and Trees

Barash et al., RECOMB 2003

k

iback

back

ildilsilsP

lllsPsPlsP

1

])1..[|][(

)|]..[()()|,(

Mixture of PWMs

Tree motif

We only change the motif component of the likelihood model

Learning the model can become more difficult

This is because computing the ML model parameter from complete data may be challenging


Discriminative scores for motifs

• So far we used a generative probabilistic model to learn PWMs• The model was designed to generate the data from parameters• We assumed that TFBSs are distributed differently than some fixed background

model

• If our background model is wrong, we will get the wrong motifs..

• A different scoring approach try to maximize the discriminative power of the motif model.

• We will not go here into the details of discriminative vs. generative models, but we shall exemplify the discriminative approach for PWMs.

Lousy discriminator High specificity discriminator High sensitivity discriminator


Hypergeometric scores and thresholding PWMs

||

||

||||

)|(|

B

n

kB

An

k

A

kBAP

PWM score threshold

Nu

mb

er

of

seq

ue

nce

s

Positive

True positive

For a discriminative score, we need to decide on both the PWM model and the threshold.

Hyper geometric probability

(sum for j>=k is the hg p-value)


Exhaustive k-mer search

• A very common strategy for motif finding is to do exhustive k-mer search.

• Given a set of hits and a set of non hits, we will compute the number of occurrences of each k-mer in the two sets and report all cases that have a discriminative score higher than some threshold

• Since k-mers either match or do not match, there is no issue with the threshold

• For DNA, we will typically scan k=5-8. • This can be done efficiently using a map/hash:

– Iterate on short sequence windows (of the desired k length)

– For each window, mark the appearance of the k-mer in a table

– Avoid double counting using a second map

• It is easy to generalize such exhaustive approaches to include gaps or other types of degeneracy.


Refining k-mers to PWMs using heuristic “EM”

• K-mer scan is an excellent intial step for finding refined weight matrices. For example, we can use them to initialize an EM.

• If we want to find a weight matrix, but want to stick to the discriminative setting, we can heuristically use and “EM-like” algorithm:

– Start with a k-mer seed– Add uniform prior to generate a PWM– Compute the optimal PWM threshold (maximal hyper-geometric score)– Restimate the PWM by voting from all PWM true positives

• Consider additional PWM positions• Bound the position entropies to avoid over-fitting

– Repeat two last steps until fail to improve score

• There are of course no guarantees for improving the scores, but empirically this approach works very well.


High density arrays quantify TF binding preferences and identify binding sites in high throughput

Harbison et al., Nature 2004

• Using microarrays (high resolution tiling arrays) we can now map binding sites in a genome-wide fashion for any genome

• The problem is shifting from identifying binding sites to understanding their function and determining how sequences define them


Direct measurements of the in-vitro binding affinity of 8-mers and DNA binding domains (here just a library of homeodomains, from Berger et al. 2008)


Profiling binding affinity to the entire k-mer spectrum provide direct quantification of in-vitro affinity (Badis et al., 2009)

Heatmap of 2D hierarchical agglomerative clustering analysis of 4740 ungapped 8-mers over 104 nonredundant TFs, with both 8- mers and proteins clustered using averaged E-score from thetwo different array designs.

8-mers

104 TFs


If only biology was that simple…

Discrete and deterministic “binding sites” in yeast as identified by Young, Fraenkel and colleuges

In fact, binding is rarely deterministic and discrete, and simple wiring is something you should treat with extreme caution.


PWM regression exploits variable levels of binding affinity to robustly recover binding preferences.

-16.5

-14.5

-12.5

-2 2 6

ChIP log(binding ratio)

-15

-13

-11

-2 2 6


PW

M s

eq

ue

nc

e e

ne

rgy

r = 0.42 = 0.20

2

8

14

-2 2 6


PW

M s

eq

ue

nc

e e

ne

rgy

r = 0.42 = 0.28

r = 0.42 = 0.26

ABF1GCN4 MBP1

PW

M s

eq

ue

nc

e e

ne

rgy

r = 0.21 = 0.72

r = 0.28 = 0.8

r = 0.11 = 0.74

Correlation between PWM predicted binding and ChIP experiments spans high, medium and low affinity sites

)),|((maxarg svsFspearman

Motif regression optimizes the PWM given the overall correlation of the predicted binding energies and the measured ChIP values vs

Tanay, GR 2006


TFBS evolution: purifying selection and conservation

Similar function

Neutral evolution

Disrupted function

Low ratepurifying selection

TF1

TF2

Altered function

Low ratepurifying selection

TF1

CACGCGTACACGCGTT

TF1

CACGAGTTCACGCGTT

CACACGTTCACGCGTT Altered affinity

Rate?Selection?

TF1

CACACGTTCACGCGTT


Kellis et al., 2003

Binding sites conservation


Binding sites conservation: heuristic motif identification

Kellis et al., 2003


Analyzing k-mer evolutionary dynamics

• Instead of trying to identify conserved motifs try to infer the evolutionary rate of substitution between pairs of k-mers

• Start from a multiple alignment and reconstruct ancestral sequences (assuming site independence, or even max parsimony)

• Now estimate the number of substitution between pairs of 8-mers, compare this number to the number expected by the background model

• Do it for a lot of sequence, so that statistics on the difference between observed and expected substitutions can be derived


Saccharomyces TFBS Selection Network

Arcs: 1nt substitutionRate Selectio

nNormal

Low

neutral

negative

arc

not enough stat

Nodes: octamers

conserved @ 2SD

conserved @ 3SD

node

otherwise

conservation

Inter-island organization in the Reb1 cluster: selection hints toward multi modality of Reb1

Tanay et al., 2004


Leu3 selection network

log delta affinity

0.3

0.2

0.1

03210-1-2-3-4-5

High Affinity (Kd < 60)

Meidum Affinity (400 > Kd > 60)

High rate subs.

Substitution changing high affinity to high

affinity motifs

Substitution changing high affinity to low affinity motifs

Altered affinity

Rate?Selection?

Motif 1 Motif 2

TF1


A simple transcriptional code and its evolutionary implications

AAATTTAATTTTAAAATT

GATGAGGATGCGGATGAT

CACGTGCACTTG

ACGCGTTCGCGTACGCGT

All th

e re

st

TGACTGTGAGTGTGACTT

TF1

TF2

TF3

TF4TF5


The Halpren-Bruno model for selection on affinity

The basic notion here is of the relations between sequence, binding and function/fitness

Sequence

Binding energy

Function )(

)(

EF

SE

We argued that E(S) can be approximated by a PWM

F(E) is a completely different story, for example:Is there any function at all to low affinity binding sites?Is there a difference between very high affinity and plain strong binding sites?Are all appearances of the site subject to the same fitness landscape?



Ns

s

e

e2

2

1

1

According to Kimura’s theory, an allele with

fitness s and a homogeneous population would fixate with probability:

NsNs

s

ab

NsNs

s

ab

e

s

e

ef

e

s

e

ef

22

2

22

2

1

2

1

1

1

2

1

1

Assuming slow mutation rate (which allow us to assume a homogenous population) and motifs a and b with relative fitness s the fixation probabilities (chance of fixation given that mutation occurred!) are:

NsNs

NsNs

Nsbaab ee

e

s

e

e

sff 2

2

22

2 1

1

2

1

1

2/

If p represent the mutation probability, and the stationary distribution, and if we assume the process as a whole is reversible then:

Ns

ba

ab

aba

bab

ababa

babab ef

f

p

p

fp

fp 21

bab

aba

aba

bab

ab

pppp

f

1

ln

We work on deriving the substitution rate at each position of the binding site, given its observed stationary frequency. We are assuming that the fitness of the site is defined by multiplying the fitnesses of each locus. This means fitness is generally linear in the binding energy!

bab

aba

aba

bab

abab

pppp

pcr

1

ln

(Halpern and Bruno, MBE 1998)

1,1 ssfitness



Moses et al., 2003

The HB model is limited for the study of general sequences.When restricting the analysis to relatively specific sites, HB is not completely off


• The entire genome should behave like a mixture of background sequance and functional loci:

• So we can try and recover Q(E) and therefore F(E) from the maximum likelihood parameters fitting an empirical W(E)

Testing the general binding energy – fitness correspondence

• While E(S) is approximated by a PWM, F(E) is unlikely to be linear

• Assume that the background probability of a motif a is P0(a). In detailed balance, and assuming the fitness of a at functional sites is F(a), the stationary distribution at sites can be shown to be:

Mustonen and Lassig, PNAS 2005

)(2)()( aNFo eaPaQ

• If we collapse all sites with binding energy E (and hence the same F(a)=F(E(a))

)(2)()( ENFo eEPEQ

)()()1()( EQEPEW o Inferred F(E), is shown in Orange

Expected and observed energy distribution in E.Coli CRP sites (left) and background (right)

Comparison of CRP energies in E.coli and S. typhimurium

(Hwa and Gerland, 2000-)


S. cerevisiae S. mikitaeSimulation(Neutral, context aware)

High affinity

Low affinity

ΔEΔE....

ΔEΔE....

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5

KS statistics

More tests for possible conservation of low binding energy sites


More tests for possible conservation of low binding energy sites

Tanay, GR 2006

Binding site conservation

Conservation of totalenergy

0

5

10

15

20

0 50 1000

5

10

15

20

0 50 100

0

5

10

15

20

0 50 100

Reb1

Ume6

binding energy percentile

Co

nse

rvat

ion

sco

re

Cbf1 Gcn4Mbp1

binding energy percentile binding energy percentile

0

10

20

30

40

50

60

0 50 100

Co

nse

rvat

ion

sco

re

0

5

10

15

20

0 50 100 binding energy percentile

binding energy percentile


Algorithms for discovering conserved sites:

• Assuming PWM models– Search for loci that behave as predicted by the model (P(s|TFBS)/P(s|

back) > Threshold)

– Search for genomic regions with surprisingly many conserved binding sites

• Search for an ML PWM– Search for motifs and conserved sites in aligned sequences

• Assume the phylogeny, alignment, look for a PWM that will optimize the likelihood of the data

– Search for motifs and conserved sites in unaligned sequences• Assume the phylogeny, look for an ML PWM

• Search for a general ML evolutionary model (many PWMs)– Search for a set of PWMs/Motif model that will maximize the likelihood

of the data


An evolutionary model

Substitution probability = background

P(si>si’|si-1)

Phylo Tree Neutral model Motif (PWM?) TFBS evo model

*(fixation)

Substitution probability = HB model


The PhyME/PhyloGibbs model (Sinha, Blanchette, Tompa 2004, Sidharthan,Siggia,Nimwegen 2005, based on evo model by Siggia and collegues)

our presentation is a bit generalized and adapted..

• Earlier/Other similar approaches using EM• Evolutionary model:

– A phylogeny– Neutral independence model (simple tree, probabilities on branches)– A PWM-induced fixation probabilities that are proportional to the matrix weights:

• Recall the single species generative model

• The generalization to alignments is similar, but should consider a phylogeny at each locus

paxxxw

paxxxwkilpaxx

ix

ikxpax

iiikxpax

ii

i

ii

ii

][1

][))(,|Pr(

k

ibackiback ildilsilsPilsPsPlsP

1

]))1..[|][(/])[(()()|,(

Back Motif Back


Phylogenetic generative model

paxxxw

paxxxwkilpaxx

ix

ikxpax

iiikxpax

ii

i

ii

ii

][1

][))(,|Pr(

1 2 3 4

ii xpaxii noilpaxx ))(,|Pr(

i

i wilsilwls ),),(|Pr())(Pr(),|,Pr(log

)|Pr()|(maxarg wSSwLw

Joint probability – product over independent trees

Inference – for loci modes (l(i)) and for ancestral sequences (hierachicaly as in Ex 3)

Learning – find ML PWM matrix

),),(|Pr(),,|)(Pr( wsilxwsil i

Prior of having a motif/background

Prob of emitting the alignment given the mode

TFBS


EM for the PWM parameters

hl i j

ij

ij ilwpaxxilwShlwwQ

,

))(,,|Pr())(Pr(log)',|,Pr()'|(

Key point: the positions in the tree are independent once we decide on their “mode” (background or a certain positions in a binding site)

Complete data: the mode of each position (l(i)). The ancestral sequences (h)

Joint probability:

i

i wilsilwls ),),(|Pr())(Pr(),|,Pr(

The EM target:

We can use the usual trick and decompose this into a sum of independent terms, one for each PWM position. For position k we have:

i jk

ij

ij

h

wkilpaxxwskilhkilwskil ),)(,|Pr(log)',,)(|Pr())(log(Pr()',|)(Pr(

Complete data!

Joint probabilityPosterior of the missing data

i j xx xykypax

ikxpaxipaj

ij

ipaj

ij i

paj

i

ii

yw

xwkilwsxxwSkilkil

, !

])[1log(

])[log())(,',|,Pr()',|)(Pr())(log(Pr(


EM for the PWM parameters

i j xx xykypax

ikxpaxipaj

ij

ipaj

ij i

paj

i

ii

yw

xwkilwsxxwSkilkil

, !

])[1log(

])[log())(,',|,Pr()',|)(Pr())(log(Pr(

In simple words:•we are optimizing a weighted sum of log(x*w) or log(1-xw)

•The coefficient for the mutations paxj -> xj at PWM positions k is the average number of times we observe it given the previous parameter set w’

The optimization problem is solved using Lagrange multipliers (to satisfy the constraint on the wk). But because each parameter appears in terms of two forms (log(x*w) and log(1-x*w), the solution is not as trivial as the dice case.

Average number of times of we observed the variables

Log of the optimized parameter


Example for intra-site epistasis

ACGCGT

AAACGT ACGCGT

AAACGTACGCGT

“DoubleLoss”

Assume the following TFBS alignment (where ACGCGT is the motif)

According to the simple TFBS evo model, we should “pay” twice for the loss of the ACGCGT site, since each of the two mutations would be multiplied by a very low fixation probability

ACGCGT

AAACGT

ACGCGT

“Loss”

AAGCGT

“Neutrality”

The more realistic scenario involve one mutation that was under pressure, but then neutrality

Multiple possible trajectories can have different loss/gain/neutrality dynamics

Affinity

Fitness

PWM

Assuming Affinity=PWM=Fitness gives the Halpren Burno model, or the frequency selection approximation

Affinity

Fitness

PWM

When fitness(PWM) is non linear, we have epistasis which means problems for the simple loci-independent model


An evolutionary model (2)

To fully capture the effect of binding sites, we need to study a Markov model over the entire sequence.

Instantaneous rates: background * selection on TFBS changes

Mutation rate = background

P(si>si’|si-1)

Phylo Tree Neutral model TF Target Sets Selection factors(per TF)

*selection factor *selection factor


Controlling context effects: approximately independent blocks

•The sequence is decomposed into intervals, each containing one or more overlapping binding sites, or sequences that are one mutation away from a binding site. Such intervals are called epistatic intervals

•The joint probability is written as a product over epistatic intervals

•For each epistatic interval we should compute exp(Qt)(from, to), where the rate matrix is large (4d) when d is the interval size

•This is approximated by (exp(Qt/N))N


•Finding the optimal selection factor is solved by non-linear optimization of the likelihood

•Looking for target set is done using a greedy algorithm.

•The resulted target sets and their selection factors are analogous to motifs/PWMs with some additional evolutionary parameter indicating the strength of selection (or the sufficiency of the motif to determine a functional site)

•As shown to the left, similar motifs are sometime separated by significant selection factors, suggesting functional partitioning.

Learning model parameters

genome evolution:

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