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Setting Up a Replica Exchange Approach to Motif
Discovery in DNAJeffrey Goett
Advisor:
Professor Sengupta
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Protein Synthesis from DNA
Translation to
Proteins
TranscriptionRegulation
RNA polymerase
Binding
Proteins
geneBinding
sites
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Binding Sites
Sequence A:
code for protein
Binding protein “A”Binding Site
A - A - C - G - A - C -
T - T - G - C - T - G -
T - T - C - A - A - C - C - A -
A - A - G - T - T - G - G - T -
Sequence B:
code for protein
A - A - G - G - A - C -
T - T - C - C - T - G -
C - G - T - T - G - C - T - C -
G - C - A - A - C - G - A - G -
Binding protein “A”
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Discovering New Binding Motifs
…ATCG GCTCAG CTAG……CACT GATCAG AGTA……TTCC GCTCTG TAAC……GCTA GCTCAA ATCG…
€
A 0 .25 0 0 .75 .25
T 0 0 1 0 .25 0
C 0 .75 0 1 0 0
G 1 0 0 0 0 .75
Motif Probability Model
Motif: GCTCAG
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Modeling Motifs in Sequences
ATATCCGTA
AATCGAGAC
TCGATGTGT
CCACCTGCA
Assume:
Break into N sequences
Each sequence has one instance of motif embedded in random background
Variations of motif by point mutation, but not insertion or deletion
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Modeling Motifs in Sequences
AT ATC CGTA
A ATC GAGAC
TCG ATG TGT
CC ACC TGCA
€
p j,ρ =
A 1 0 0
T 0 .75 0
C 0 .25 .75
G 0 0 .25
The “Alignment:” Starting position of motif in each sequence
The “Motif Probability Distribution:” Probability of each letter occurring at each motif position
€
rx = {x1,x2, x3 ...xN }
€
ex : r x = {3, 2, 4, 3}
€
p j,ρ
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Scoring a Model
€
p(r x , p j ,ρ | S) =
p(S |r x ,p j ,ρ )p(
r x )p( p j ,ρ )
p(S )
€
p(r x , p j ,ρ | S) ⏐ → ⏐ log(
p(S |r x ,p j ,ρ )p( p j ,ρ )
p(S | pρ0 )
) + log(p(r x )) + log(p(S)) =
1N n j,ρ log(
ˆ p j ,ρ
pρ0 ) + constant
ρ ∈Σ
∑j=1
w
∑
€
p(S |r x , p j ,ρ ) :
“Log-likelihood” score:
ATATCCGTA
AATCGAGAC
TCGATGTGT
CCACCTGCA
p1,T p2,A p3,T
p1,A p2,G p3,A
p1,A p2,T p3,G
p1,C p2,C p3,A
pC pC pG pT pA0 0 0 0 0
pA0
pA pA pT pC pG0 0 0 0 0 pC
0
pT pC pG0 0 0 pT pG pT
0 0 0
pC pC pT pG pC pA0 0 0 0 0 0
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Example Models
A TAT CCGTA
AAT CGA GAC
TCGATG TGT
CC ACC TGCA
€
p j,ρ =
A 1 0 0
T 0 .75 0
C 0 .25 .75
G 0 0 .25
€
rx = {3, 2, 4, 3}
AT ATC CGTA
A ATC GAGAC
TCG ATG TGT
CC ACC TGCA
€
L(S |r x , p j ,ρ , p j
0) ≈ 3
€
p j,ρ =
A .25 .25 .25
T .5 0 .5
C .25 .25 .25
G 0 .5 0
€
rx = {2, 4, 7, 3}
€
L(S |r x , p j ,ρ , p j
0) ≈1.1
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The Gibbs SamplerWe want to find
€
pj, ρ
€
p( p j,ρ | S)that maximizes
€
pj, ρ
€
rx
€
L( p j,ρ ,r x | S)
€
p( p j,ρ | S) = p( p j,ρ∫ ,r x | S)d
r x
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The Gibbs Sampler
€
pj, ρ
€
p( p j,ρ ,r x | S)
€
pj, ρ
€
rx
€
pj, ρ
€
rx
€
pj, ρ
€
rx
€
pj, ρ
€
rx
€
pj, ρ
€
rx
€
pj, ρ
€
rx
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The Gibbs Sampler
Times visited
€
pj, ρ
Over time, the frequency distribution approaches
€
p( p j,ρ | S)
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Biasing our search to these areas may discover the pj,ro values which maximize faster.
If we assume areas of local maximization contribute the most during “integration” to the local maximizations of
Optimization Technique
€
p( p j,ρ | S)
€
p( p j,ρ | S)
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Multiple Gibbs Samplers
By combining results from Gibbs Samplers begun at random positions, find maximizing sooner
€
p( p j,ρ | S)
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Replica Exchange/Parallel Tempering
“Low-sensitivity” samplers which “scout out area” periodically swap with “high-sensitivity” samplers good at focused searches if swap appears promising.
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Controlling Sensitivity
€
˜ p (x i | p j,ρ ,S) = eβL(xi ,p j ,ρ |S )Adjust the relative probability of sampling an xi by adjusting a new parameter in distribution:
Small
€
β Large
€
β
Search breadth of space Focused search of region
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Testing the Sensitivity
Running on randomly generated sequences to see motifs found, different sensitivity samplers converge to different scores.
Betas
21.9.1
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Predicting Convergence Score
Measure of Similarity:
magnetization
€
m = 1N si
i=1
N
∑
“Configuration Score:” energy
Ex: m=.5
€
E = −12 Jsis j
j=1j≠ i
N
∑i=1
N
∑m=.5
E=0
m=1
E=-6J
m=0
E=2J
m=0
E=2J
m=0
E=2J
€
p ≈ e−β 0
€
p ≈ eβ 6J
€
p ≈ e−β 2J
€
p ≈ e−β 2J
€
p ≈ e−β 2J
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Alignment Analogue
m=.77
E=-5J
m=1
E=-9J
m=.77
E=-5J
m=.77
E=-5J
€
p ≈ eβ 9J
€
p ≈ eβ 5J
€
p ≈ eβ 5J
A:
B:
C:
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Test Results
L < |alphabet|w
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Test Results
L > |alphabet|w
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Test Results
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Test Results
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Hidden Motifs: Gibbs SamplerBeta = .1 Beta = .5 Beta = .9
Beta = 1.3 Beta = 1.7 Beta = 2
W=5, l=500
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Hidden Motifs: Replica Exchange
Betas
.9
.93
.961
.8
1.5
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