users.fred.netusers.fred.net/tds/lab/images/kanpur2016/lecture_7_thomas_schne… · rsequence and...

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

Information Theory in BiologyThomas D. Schneider, Ph.D.

Molecular Information Theory GroupCenter for Cancer Research

Gene Regulation and Chromosome Biology LaboratoryNational Cancer InstituteFrederick, MD 21702-1201

Information Theory: One-Minute Lesson

38

B

symbols of bits

4

2

M

2

1

11

01 00

number ofexample

10

number

H T

M=2 B=log2M

B


38

B

symbols of bits

4

2

M

2

1

11

01 00

number ofexample

10

number

H T

M=2 B=log2M

B

ATGC


38

B

symbols of bits

4

2

M

2

1

11

01 00

number ofexample

10

number

H T

M=2 B=log2M

B

El Duomo, Florence, Italy

T7 RNA polymerase + DNA

http://www.ebi.ac.uk/pdbe/entry/pdb/1qln/portfolio/?view=entry_index#ad-image-0

http://www.ebi.ac.uk/pdbe/entry/pdb/1qln/portfolio/?view=entry_index#ad-image-0

Sequence Logo

Schneider &StephensNucl. Acids Res.18: 6097-61001990

6 of 17 sites

1 ttattaatacaactcactataaggagag 2 aaatcaatacgactcactatagagggac 3 cggttaatacgactcactataggagaac 4 gaagtaatacgactcagtatagggacaa 5 taattaattgaactcactaaagggagac 6 cgcttaatacgactcactaaaggagaca

17 Bacteriophage T7 RNA polymerase binding sites

0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

Sequence Logo


6 of 17 sites



0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

2 bits/base

Sequence Logo


6 of 17 sites



0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

1 bit/base

Sequence Logo


6 of 17 sites



0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

0 bits/base

Sequence Logo


1 ttattaatacaactcactataaggagag 33.3 2 aaatcaatacgactcactatagagggac 37.4 3 cggttaatacgactcactataggagaac 34.4 4 gaagtaatacgactcagtatagggacaa 33.1 5 taattaattgaactcactaaagggagac 30.1 6 cgcttaatacgactcactaaaggagaca 29.1


Bits0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

Individual Information

Sequence Logo and Sequence Walker




Bits0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

29.1 bits

SequenceWalkerPatent5,867,402

Sequence Logo and Sequence Walker and Rsequence




Bits0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

Rsequence is the average: 35.0± 0.6 bits

Sequence Logo and Sequence Walker and Rsequence




Bits0

1

2

bit

s

5′ -21TCGA

-20G

TA

-19CT

GA

-18CG

AT

-17C

T-1

6TA

-15A

-14T

-13C

TA

-12G

C-1

1AG

-10A

-9

C-8

T-7

C-6

A-5

GC

-4

T-3

A-2

AT

-1

TA

0AG

1AG

2AG

3

GA

4CAG

5TCA

6TGCA

3′

Rsequence is the average: 35.0± 0.6 bits= “area under the logo”

Rfrequency

Information requiredto find a set of binding sites

G = # of potential binding sites

Rfrequency


G = # of potential binding sites= genome size in some cases

Rfrequency



γ = number of binding sites on genome

Rfrequency




Rfrequency = Hbefore −Hafter

Rfrequency





= log2G− log2 γ

Rfrequency





= log2G− log2 γ= − log2 γ/G

Rfrequency

2log 16/2 = 3 bits

16 positions 2 sites

16 positions 1 sitelog 16/1 = 4 bits2

Information requiredto find a set of binding sitesin a genome

RNA Splicing

DNA

intron exon

donor acceptor

RNA

Spliced RNA

RNA Splicing

Copy DNA (transcription)

exon

exonexonexon

intronexon

Donor and acceptor logos

5′

3′

exon

intron

exon

donor

acceptor

TGAC

GTCA

CTAG

CT

GGACTC

TGA

TCGA

CTAG

ACGT

TCAG

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

Rsequence and Rfrequency for Splice Acceptors

Rsequence

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

• Information at binding site sequences (area under sequence logo)• from: binding site sequences• 9.4 bits per site

Rsequence and Rfrequency for Splice Acceptors

Rsequence

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

• Information at binding site sequences (area under sequence logo)• from: binding site sequences• 9.4 bits per site

Rfrequency

acceptor

intron exon

donor

• Information needed to locate the sites• from: size of genome and number of sites (length of intron+exon)• 9.7 bits per site

Rfrequency/Rsequence = 0.97

Rsequence = Rfrequency Hypothesis

Hypothesis:The information in

binding site patternsis just sufficient

for the sites to be foundin the genome

Rsequence versus Rfrequency

Binding Site Total Pattern Information needed to Pattern InfoLocation Info

Recognizer1 Information Locate Site in Genome

= Rsequence = Rfrequency =Rsequence

Rfrequency

(bits) (bits)

Spliceosome acceptor2 9.35± 0.12 9.66 0.97± 0.01Spliceosome donor 7.92± 0.09 9.66 0.82± 0.01

Ribosome 11.0 10.6 1.0λ cI/cro 17.7± 1.6 19.3 0.9± 0.1LexA 21.5± 1.7 18.4 1.2± 0.1TrpR 23.4± 1.9 20.3 1.2± 0.1LacI 19.2± 2.8 21.9 0.9± 0.1ArgR 16.4 18.4 0.9O (λ Origin) 20.9 19.9 1.0Ara C 19.3 19.3 1.0Transcription at TATA3

3.3 ∼ 3 ∼ 1

T7 Promoter 35.4 16.5 2.11T. D. Schneider, G. D. Stormo, L. Gold, and A. Ehrenfeucht. J. Mol. Biol., 188:415-431, 1986.2R. M. Stephens and T. D. Schneider. J. Mol. Biol., 228:1124-1136, 1992.3F. E. Penotti. J Mol Biol, 213:37-52, 1990.

Rsequence versus Rfrequency - meaning

The information in the binding site pattern (Rsequence)is close to

The information needed to find the binding sites (Rfrequency)




But for a species in a stable environment:

• size of genome (G) is fixed (e. g. E. coli has 4.7× 106 bp)• number of binding sites (γ) is fixed (e. g. there are ∼50 E. coli LexA sites)

so Rfrequency = log2G/γ is fixed




But for a species in a stable environment:

• size of genome (G) is fixed (e. g. E. coli has 4.7× 106 bp)• number of binding sites (γ) is fixed (e. g. there are ∼50 E. coli LexA sites)

so Rfrequency = log2G/γ is fixed

Rsequence must evolve towards Rfrequency!

Evolution of Binding Sites

• Rfrequency is fixed relative to Rsequence


• Rfrequency is fixed relative to Rsequence• Does Rsequence evolve toward Rfrequency?



Setup a Computer Model, ‘Ev’:A population of “creatures” with




• genomes containing 4 bases (A, C, G, T)




• genomes containing 4 bases (A, C, G, T)• a defined genome size (G)




• genomes containing 4 bases (A, C, G, T)• a defined genome size (G)• predetermined binding site locations (γ)

(to fix the frequency of sites)




• genomes containing 4 bases (A, C, G, T)• a defined genome size (G)•

}

Rfrequencyis fixedpredetermined binding site locations (γ)

(to fix the frequency of sites)




• genomes containing 4 bases (A, C, G, T)• a defined genome size (G)•

}

Rfrequencyis fixedpredetermined binding site locations (γ)

(to fix the frequency of sites)• a recognizer gene encoded in the sequence:

use a weight matrix

How A Weight Matrix Works

Sequence matrix, s(b, l, j) for sequence j

base b position l

C A G G T C T G C A−3 −2 −1 0 1 2 3 4 5 6

A 0 1 0 0 0 0 0 0 0 1

C 1 0 0 0 0 1 0 0 1 0

G 0 0 1 1 0 0 0 1 0 0

T 0 0 0 0 1 0 1 0 0 0

Individual information weight matrix, Riw(b, l)

base b position l

−3 −2 −1 0 1 2 3 4 5 6

A +0.4 +1.3 −1.4 −8.8 −5.8 +1.1 +1.5 −1.8 −0.7 +0.0

C +0.6 −0.8 −2.4 −7.8 −5.5 −3.7 −1.6 −2.2 −0.5 −0.2G −0.6 −1.0 +1.6 +2.0 −6.2 +0.7 −1.1 +1.7 −0.3 +0.4

T −1.0 −0.9 −1.7 −5.8 +2.0 −3.4 −1.6 −2.2 +0.9 −0.5

How A Weight Matrix Works

Sequence matrix, s(b, l, j) for sequence j

base b position l

C A G G T C T G C A−3 −2 −1 0 1 2 3 4 5 6

A 0 1 0 0 0 0 0 0 0 1

C 1 0 0 0 0 1 0 0 1 0

G 0 0 1 1 0 0 0 1 0 0

T 0 0 0 0 1 0 1 0 0 0

Individual information weight matrix, Riw(b, l)

base b position l

−3 −2 −1 0 1 2 3 4 5 6

A +0.4 +1.3 −1.4 −8.8 −5.8 +1.1 +1.5 −1.8 −0.7 +0.0

C +0.6 −0.8 −2.4 −7.8 −5.5 −3.7 −1.6 −2.2 −0.5 −0.2G −0.6 −1.0 +1.6 +2.0 −6.2 +0.7 −1.1 +1.7 −0.3 +0.4

T −1.0 −0.9 −1.7 −5.8 +2.0 −3.4 −1.6 −2.2 +0.9 −0.5

5’ c a g g t c t g c a 3’

Sequence Walker

Unevolved Ev Creature

Unevolved Ev Creature}

“blue”geneweightmatrix:6 bpwide



Genome positions available G = 256 bases



Genome positions available G = 256 basesRfrequency = log2 256/16 = 4 bits

}

γ = 16bindingsites




}

γ = 16bindingsites

found real site




}

γ = 16bindingsites

found real sitemissed real site




}

γ = 16bindingsites

found real sitemissed real sitefound wrong site

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate

• EVALUATE each creature

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate


• translate the recognizer gene into a weight matrix

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate


• translate the recognizer gene into a weight matrix• scan the weight matrix across the genome

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate


• translate the recognizer gene into a weight matrix• scan the weight matrix across the genome• count the number of mistakes:

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate



missing a site at a right place

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate



missing a site at a right placefinding a site at a wrong place

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate




• Sort the creatures by their mistakes

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate





• REPLICATE: the best creatures areduplicated and replace the worst ones

Evolution Cycle

sort

mutate

replicate

kill

select

ion

evaluate





• REPLICATE: the best creatures areduplicated and replace the worst ones

• MUTATE all genomes randomly

Evolved Ev Creature


16 evolving binding sitesGeneration 100

Rs = -0.1 +/- 0.5 bits

0

1

2b

its

5′ -5 -4

CATG

-3 -2 -1 0 1CGAT

2 3TCGA

4CATG

5ATCG

6 7 8CATG

9 10 3′


Rs = -0.0 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

CATG

-3 -2

GATC

-1 0 1GACT

2 3CTAG

4 5ATGC

6 7TGAC 8 9 10 3′


Rs = 1.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5

AGCT

-4 -3

TACG

-2 -1 0GCTA

1AGCT

2 3TAG

4 5TGC

6 7GTCA

8GCTA

9 10

GCTA

3′


Rs = 2.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

ATCG

-3

ACG

-2 -1

TCAG

0TGCA

1GACT

2GCTA

3CTAG

4TCAG

5TGC

6 7TGCA 8 9ACTG

10

CGTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2 -1

CTAG 0TGA

1ACT

2GCTA

3CTAG

4TCAG

5AGC

6 7GTCA 8GCTA

9 10

GTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3 -2

GTAC

-1 0TCGA

1GACT

2TCA

3ATG

4CGA

5AGC

6 7TGCA 8GCAT 9 10

GCTA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2

ATCG

-1

TCGA 0TGA

1GCT

2CAT

3TGA

4CAG

5TGAC

6 7CTGA 8CTGA 9 10

GTA

3′


Rs = 3.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3 -2

ATGC

-1 0

GA

1ACT

2CTA

3CTGA

4ACG

5TGAC

6 7TGCA 8CGAT 9CTAG

10

CGTA

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GCAT

-4 -3 -2

ATCG

-1

TAG

0

GA

1GACT

2CTA

3ACTG 4CAG

5TGAC

6 7 8TGA

9TGA

10 3′


Rs = 4.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TAGC

-2

ATCG

-1

TAG

0GA

1

CT

2CTA

3 4CAG

5GAC

6 7 8 9CTGA

10

CGTA

3′


Rs = 5.0 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3

ATGC

-2

ATCG

-1

CAG

0

GA

1GACT

2TA

3CTAG 4CAG

5GAC

6 7 8TCGA

9CTGA

10

TAG

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCG

-2

ACGT

-1

CTGA

0GA

1GACT

2CTA

3 4ACG

5GTAC

6 7 8TCGA

9GTA

10

TGA

3′


Rs = 4.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GTCA

-4 -3

CATG

-2

ACGT

-1

TCGA

0GA

1GATC

2CTA

3ACTG 4CAG

5TGCA

6CTA

7 8 9CGTA

10

TGA

3′


Rs = 4.4 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4

GATC

-3

ATCG

-2

GCT

-1 0TGA

1ACT

2TCA

3ATCG

4CTAG

5GAC

6 7 8GCTA

9 10

CTGA

3′


Rs = 3.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3

ACTG

-2

ATGC

-1

TGCA 0TGA

1ACT

2TCA

3TACG

4CTAG

5GAC

6CGAT 7 8 9GCTA

10

TCGA

3′


Rs = 4.9 +/- 0.5 bits

0

1

2b

its

5′ -5

GATC

-4

AGTC

-3

TACG

-2 -1

TCGA

0

GA

1

CT

2TCA

3CTG

4TCAG

5TGAC

6AGCT

7 8GCTA

9CTGA

10

CTGA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

ACG

-2

ATCG

-1

TCAG

0

GA

1ACT

2GTCA

3CTG

4TAG

5GATC

6ATGC

7TACG

8GCTA

9TCGA

10

CTGA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2 -1 0

GA

1ACT

2TCA

3CATG

4

GA

5GTAC

6 7 8GCTA 9TCGA

10

GTA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2

GACT

-1 0

GA

1ACT

2TCA

3CATG

4

GA

5TAC

6 7 8TCA

9TCGA

10

CTGA

3′


Rs = 5.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

TGAC

-4 -3 -2

CGAT

-1

GTCA

0

GA

1ACT

2TCA

3ATCG

4

GA

5TAC

6CAGT 7 8GTCA 9CTGA

10

TGA

3′



Rs = -0.1 +/- 0.5 bits

0

1

2b

its

5′ -5 -4

CATG

-3 -2 -1 0 1CGAT

2 3TCGA

4CATG

5ATCG

6 7 8CATG

9 10 3′


Rs = -0.0 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

CATG

-3 -2

GATC

-1 0 1GACT

2 3CTAG

4 5ATGC

6 7TGAC 8 9 10 3′


Rs = 1.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5

AGCT

-4 -3

TACG

-2 -1 0GCTA

1AGCT

2 3TAG

4 5TGC

6 7GTCA

8GCTA

9 10

GCTA

3′


Rs = 2.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

ATCG

-3

ACG

-2 -1

TCAG

0TGCA

1GACT

2GCTA

3CTAG

4TCAG

5TGC

6 7TGCA 8 9ACTG

10

CGTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2 -1

CTAG 0TGA

1ACT

2GCTA

3CTAG

4TCAG

5AGC

6 7GTCA 8GCTA

9 10

GTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3 -2

GTAC

-1 0TCGA

1GACT

2TCA

3ATG

4CGA

5AGC

6 7TGCA 8GCAT 9 10

GCTA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2

ATCG

-1

TCGA 0TGA

1GCT

2CAT

3TGA

4CAG

5TGAC

6 7CTGA 8CTGA 9 10

GTA

3′


Rs = 3.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3 -2

ATGC

-1 0

GA

1ACT

2CTA

3CTGA

4ACG

5TGAC

6 7TGCA 8CGAT 9CTAG

10

CGTA

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GCAT

-4 -3 -2

ATCG

-1

TAG

0

GA

1GACT

2CTA

3ACTG 4CAG

5TGAC

6 7 8TGA

9TGA

10 3′


Rs = 4.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TAGC

-2

ATCG

-1

TAG

0GA

1

CT

2CTA

3 4CAG

5GAC

6 7 8 9CTGA

10

CGTA

3′


Rs = 5.0 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3

ATGC

-2

ATCG

-1

CAG

0

GA

1GACT

2TA

3CTAG 4CAG

5GAC

6 7 8TCGA

9CTGA

10

TAG

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCG

-2

ACGT

-1

CTGA

0GA

1GACT

2CTA

3 4ACG

5GTAC

6 7 8TCGA

9GTA

10

TGA

3′


Rs = 4.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GTCA

-4 -3

CATG

-2

ACGT

-1

TCGA

0GA

1GATC

2CTA

3ACTG 4CAG

5TGCA

6CTA

7 8 9CGTA

10

TGA

3′


Rs = 4.4 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4

GATC

-3

ATCG

-2

GCT

-1 0TGA

1ACT

2TCA

3ATCG

4CTAG

5GAC

6 7 8GCTA

9 10

CTGA

3′


Rs = 3.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3

ACTG

-2

ATGC

-1

TGCA 0TGA

1ACT

2TCA

3TACG

4CTAG

5GAC

6CGAT 7 8 9GCTA

10

TCGA

3′


Rs = 4.9 +/- 0.5 bits

0

1

2b

its

5′ -5

GATC

-4

AGTC

-3

TACG

-2 -1

TCGA

0

GA

1

CT

2TCA

3CTG

4TCAG

5TGAC

6AGCT

7 8GCTA

9CTGA

10

CTGA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

ACG

-2

ATCG

-1

TCAG

0

GA

1ACT

2GTCA

3CTG

4TAG

5GATC

6ATGC

7TACG

8GCTA

9TCGA

10

CTGA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2 -1 0

GA

1ACT

2TCA

3CATG

4

GA

5GTAC

6 7 8GCTA 9TCGA

10

GTA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2

GACT

-1 0

GA

1ACT

2TCA

3CATG

4

GA

5TAC

6 7 8TCA

9TCGA

10

CTGA

3′


Rs = 5.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

TGAC

-4 -3 -2

CGAT

-1

GTCA

0

GA

1ACT

2TCA

3ATCG

4

GA

5TAC

6CAGT 7 8GTCA 9CTGA

10

TGA

3′

selection no selection

0 500 1000 1500 2000 0

4

8

12

16

20

Generation

Mistakes of Best Organism



Rs = -0.1 +/- 0.5 bits

0

1

2b

its

5′ -5 -4

CATG

-3 -2 -1 0 1CGAT

2 3TCGA

4CATG

5ATCG

6 7 8CATG

9 10 3′


Rs = -0.0 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

CATG

-3 -2

GATC

-1 0 1GACT

2 3CTAG

4 5ATGC

6 7TGAC 8 9 10 3′


Rs = 1.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5

AGCT

-4 -3

TACG

-2 -1 0GCTA

1AGCT

2 3TAG

4 5TGC

6 7GTCA

8GCTA

9 10

GCTA

3′


Rs = 2.6 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4

ATCG

-3

ACG

-2 -1

TCAG

0TGCA

1GACT

2GCTA

3CTAG

4TCAG

5TGC

6 7TGCA 8 9ACTG

10

CGTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2 -1

CTAG 0TGA

1ACT

2GCTA

3CTAG

4TCAG

5AGC

6 7GTCA 8GCTA

9 10

GTA

3′


Rs = 2.7 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3 -2

GTAC

-1 0TCGA

1GACT

2TCA

3ATG

4CGA

5AGC

6 7TGCA 8GCAT 9 10

GCTA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2

ATCG

-1

TCGA 0TGA

1GCT

2CAT

3TGA

4CAG

5TGAC

6 7CTGA 8CTGA 9 10

GTA

3′


Rs = 3.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3 -2

ATGC

-1 0

GA

1ACT

2CTA

3CTGA

4ACG

5TGAC

6 7TGCA 8CGAT 9CTAG

10

CGTA

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GCAT

-4 -3 -2

ATCG

-1

TAG

0

GA

1GACT

2CTA

3ACTG 4CAG

5TGAC

6 7 8TGA

9TGA

10 3′


Rs = 4.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TAGC

-2

ATCG

-1

TAG

0GA

1

CT

2CTA

3 4CAG

5GAC

6 7 8 9CTGA

10

CGTA

3′


Rs = 5.0 +/- 0.5 bits

0

1

2b

its

5′ -5 -4 -3

ATGC

-2

ATCG

-1

CAG

0

GA

1GACT

2TA

3CTAG 4CAG

5GAC

6 7 8TCGA

9CTGA

10

TAG

3′


Rs = 4.5 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCG

-2

ACGT

-1

CTGA

0GA

1GACT

2CTA

3 4ACG

5GTAC

6 7 8TCGA

9GTA

10

TGA

3′


Rs = 4.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GTCA

-4 -3

CATG

-2

ACGT

-1

TCGA

0GA

1GATC

2CTA

3ACTG 4CAG

5TGCA

6CTA

7 8 9CGTA

10

TGA

3′


Rs = 4.4 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4

GATC

-3

ATCG

-2

GCT

-1 0TGA

1ACT

2TCA

3ATCG

4CTAG

5GAC

6 7 8GCTA

9 10

CTGA

3′


Rs = 3.9 +/- 0.5 bits

0

1

2

bit

s

5′ -5

GATC

-4 -3

ACTG

-2

ATGC

-1

TGCA 0TGA

1ACT

2TCA

3TACG

4CTAG

5GAC

6CGAT 7 8 9GCTA

10

TCGA

3′


Rs = 4.9 +/- 0.5 bits

0

1

2b

its

5′ -5

GATC

-4

AGTC

-3

TACG

-2 -1

TCGA

0

GA

1

CT

2TCA

3CTG

4TCAG

5TGAC

6AGCT

7 8GCTA

9CTGA

10

CTGA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

ACG

-2

ATCG

-1

TCAG

0

GA

1ACT

2GTCA

3CTG

4TAG

5GATC

6ATGC

7TACG

8GCTA

9TCGA

10

CTGA

3′


Rs = 3.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3

TCGA

-2 -1 0

GA

1ACT

2TCA

3CATG

4

GA

5GTAC

6 7 8GCTA 9TCGA

10

GTA

3′


Rs = 4.7 +/- 0.5 bits

0

1

2

bit

s

5′ -5 -4 -3 -2

GACT

-1 0

GA

1ACT

2TCA

3CATG

4

GA

5TAC

6 7 8TCA

9TCGA

10

CTGA

3′


Rs = 5.2 +/- 0.5 bits

0

1

2

bit

s

5′ -5

TGAC

-4 -3 -2

CGAT

-1

GTCA

0

GA

1ACT

2TCA

3ATCG

4

GA

5TAC

6CAGT 7 8GTCA 9CTGA

10

TGA

3′

selection no selection

0 500 1000 1500 2000 0

4

8

12

16

20

Generation

Mistakes of Best Organism

Rfrequency Rsequence

selection

no selection

0 500 1000 1500 2000 -1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

Generation

Information (bits per site)

Donor and acceptor logos

5′

3′

exon

intron

exon

donor

acceptor

TGAC

GTCA

CTAG

CT

GGACTC

TGA

TCGA

CTAG

ACGT

TCAG

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

Human Splice Junction Information Curves

Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s

Position L (in bases)

0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

https://alum.mit.edu/www/toms/papers/splice/

• The consensus sequences match . . .


https://alum.mit.edu/www/toms/papers/logo/


Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A G






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A G —






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A G — G






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A G — G T






Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

C A G — G T



• BUT the information curves (sequence conservation) differ!




Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

TCAG TACGTCGAGTACTGACTGAC

GTCA

CTAG

CT

GGACT

CTGA

TCGA

CTAG

ACGT

TCAG

AGTCATGCATCGATCGATCGATGC

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCA

TAC

GTCAG

CGAT

GACTATGCGATCTAGCTCAG

C A G — G T



• BUT the information curves (sequence conservation) differ!

• Put letters into the graph proportional to their frequency!




Donor

Sequence

Conservation→

inbitsper

base

Acceptor

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-9 481 444 497 371 0.01

-8 461 461 453 422 -0.00

-7 434 457 530 378 0.01

-6 498 433 452 416 0.00

-5 474 489 398 438 0.00

-4 492 527 446 334 0.02

-3 603 673 301 222 0.14

-2 1071 263 219 246 0.39

-1 169 85 1404 141 0.90

0 0 2 1789 8 1.95

1 8 10 6 1775 1.88

2 974 35 738 43 0.75

3 1274 152 215 149 0.68

4 128 95 1468 97 1.04

5 270 304 348 812 0.16

6 428 369 564 296 0.04

7 319 492 408 424 0.02

8 332 494 413 384 0.01

9 302 437 463 408 0.02

10 281 419 424 342 0.02

11 274 382 436 369 0.02

12 273 414 400 355 0.02

l

a

c

g

t

Rs

(l)

Rs(l

) =

Rseq

uenc

e(l)

, In

form

atio

n in

bit

s


0.0

1.0

2.0

-25 309 392 208 386 0.04

-24 282 420 216 388 0.04

-23 273 392 220 424 0.05

-22 290 411 214 397 0.04

-21 309 400 194 416 0.05

-20 248 412 203 461 0.08

-19 256 388 225 482 0.07

-18 213 442 246 455 0.08

-17 201 453 183 535 0.15

-16 193 471 199 517 0.14

-15 182 459 212 533 0.14

-14 158 476 193 619 0.21

-13 135 486 197 656 0.25

-12 122 511 174 673 0.29

-11 121 500 152 732 0.34

-10 110 483 149 775 0.37

-9 131 534 175 800 0.33

-8 156 583 192 727 0.27

-7 172 656 172 674 0.27

-6 165 653 158 726 0.30

-5 123 691 110 788 0.43

-4 127 590 103 914 0.46

-3 394 499 432 411 0.00

-2 67 1253 20 397 0.92

-1 1717 10 6 10 1.86

0 9 11 1720 4 1.87

1 449 231 871 187 0.26

2 426 308 413 590 0.04

3 424 426 419 468 0.00

4 400 487 440 410 0.00

5 424 502 347 464 0.01

6 394 527 424 392 0.01

7 441 431 485 380 0.00

TCAG TACGTCGAGTACTGACTGAC

GTCA

CTAG

CT

GGACT

CTGA

TCGA

CTAG

ACGT

TCAG

AGTCATGCATCGATCGATCGATGC

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCA

TAC

GTCAG

CGAT

GACTATGCGATCTAGCTCAG

1799 Human donor sites

0

1

2

bit

s

5′ -9

TCAG

-8 -7

TACG

-6

TCGA

-5

GTAC

-4

TGAC

-3

TGAC

-2

GTCA

-1

CTAG

0CT

G

1GACT 2CTGA

3TCGA

4CTAG

5ACGT

6TCAG 7AGTC 8ATGC 9ATCG

10

ATCG

11

ATCG

12

ATGC

3′

1744 Human acceptor sites

0

1

2

bit

s

5′ -25GATC

-24GATC

-23GACT

-22GATC

-21GAC

T

-20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8

AGCT

-7

GACT

-6

GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT 3GACT 4ATGC 5GATC 6TAGC 7TCAG

3′


• That’s how and why we invented sequence logos!



Splice Junction Sequence Logos

5′

3′

exon

intron

exon

donor

acceptor

TGAC

GTCA

CTAG

CT

GGACTC

TGA

TCGA

CTAG

ACGT

TCAG

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

• 90% of the splice junctioninformation is on the intron side




5′

3′

exon

intron

exon

donor

acceptor

TGAC

GTCA

CTAG

CT

GGACTC

TGA

TCGA

CTAG

ACGT

TCAG

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

donor acceptor

proto

GTAC

GTCA

CTAG

TCAG

CGAT

TGAC

GTCA

CTAG

CT

GGACT

GATC

GTCA

TAC

GTCAG

CGAT


• Hypothesis:donor and acceptor sites had acommon ancestor that duplicated




5′

3′

exon

intron

exon

donor

acceptor

TGAC

GTCA

CTAG

CT

GGACTC

TGA

TCGA

CTAG

ACGT

TCAG

GATC

GATC

GACT

GATC

GACT

GACT

GACT

AGCT

GACT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

AGCT

GACT

GACT

GACT

GACT

ATGCGATC

GTCATAC

GTCAG

CGAT

donor acceptor

proto

GTAC

GTCA

CTAG

TCAG

CGAT

TGAC

GTCA

CTAG

CT

GGACT

GATC

GTCA

TAC

GTCAG

CGAT


• Hypothesis:donor and acceptor sites had acommon ancestor that duplicated

• They evolved to put the informationinto the intron. This avoids affectingthe proteins.



3D Sequence Logos for tRNA Correlations

• tRNA reads RNAto make protein

https://alum.mit.edu/www/toms/papers/correlogo/ Nucleic Acids Res. 2006 34:W405-11

https://alum.mit.edu/www/toms/papers/correlogo/

http://www.ncbi.nlm.nih.gov/pubmed/16845037



• Correlations can be measured in bits!







• 3D Sequence logo







• 3D Sequence logo

• OBSERVED: tRNA stems




Sequence Walker example: rrnB P1

. *4164240 . *4164250 . *4164260 . *4164270 . 5’ g g a g c t g a a c a a t t a t t g c c c g t t t t a c a g c g t t a c g g c t t c g a 3’ 3’ c c t c g a c t t g t t a a t a a c g g g c a a a a t g t c g c a a t g c c g a a g c t 5’

Fis 12.0 bits

*4164280 . *4164290 . *4164300 . *4164310 . *4164320 . *4164330 5’ a a c g c t c g a a a a a c t g g c a g t t t t a g g c t g a t t t g g t t g a a t g t t g c g c g g t c a 3’ 3’ t t g c g a g c t t t t t g a c c g t c a a a a t c c g a c t a a a c c a a c t t a c a a c g c g c c a g t 5’

Fis 5.3 bits Fis 10.4 bits

. *4164340 . *4164350 . *4164360 . *4164370 . *4164380 5’ g a a a a t t a t t t t a a a t t t c c t c t t g t c a g g c c g g a a t a a c t c c c t a t a a t g 3’ 3’ c t t t t a a t a a a a t t t a a a g g a g a a c a g t c c g g c c t t a t t g a g g g a t a t t a c 5’

distalUP 6.6 bits p35 5.5 bits p10 8.4 bits | | | |- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | p35- ( 22) - p10 4164377 Gap 2.3 bits | | | | proximalUP 4.4 bits | | | | | | | | | |- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | proximalUP- ( 27) - p10 4164377 Gap 3.4 bits | | |- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | distalUP- ( 45) - p10 4164377 Gap 5.4 bits |- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |- - - - - - - - - |- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | distalUP- proximalUP- p35- p10 4164377 total 13.7 bits

rrnB P1

Complex Sequence Walker Example

• σ70 promoters have a −35 and a −10


• σ70 promoters have a −35 and a −10• Using information theory we discovered that

stress-response σ38 promoters do not have a −35



stress-response σ38 promoters do not have a −35• Instead, they have a −10 and two UP elements



stress-response σ38 promoters do not have a −35• Instead, they have a −10 and two UP elements• σ38 promoter talA P1 is complex!

Important Discovery1744 Human acceptor sites

0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′

• Area under a sequence logo is the total information.


0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′


• How is that related to the binding energy?


0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′



• Information gained for energy dissipated


0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′




• Isothermal efficiency


0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′




• Isothermal efficiency

• My most important discovery:

Molecules are often 70% efficient

Information of EcoRI DNA Binding

′

• EcoRI - restriction enzyme


EcoRI sites

0

1

2

bit

s5′

0

G1A 2A 3T 4T 5

C3′


• EcoRI binds DNA at 5′ GAATTC 3′


EcoRI sites

0

1

2

bit

s5′

0

G1A 2A 3T 4T 5

C3′


• EcoRI binds DNA at 5′ GAATTC 3′

• information required:

6 bases × 2 bits per base = 12 bits

Energy Dissipation by EcoRI

• Measured specific binding constant:

Kspec = 1.6× 105



Kspec = 1.6× 105

• Average energy dissipated by one molecule as it binds:

∆G◦

spec = −kBT lnKspec (joules per binding)



Kspec = 1.6× 105


∆G◦


• The Second Law of Thermodynamics as a conversion factor:

Emin = kBT ln 2 (joules per bit)



Kspec = 1.6× 105


∆G◦


• The Second Law of Thermodynamics as a conversion factor:

Emin = kBT ln 2 (joules per bit)

• Number of bits that could have been selected:

Renergy = −∆G◦/Emin

= kBT lnKspec/kBT ln 2

= log2Kspec ⇐ SO SIMPLE!

= 17.3 bits per binding

Information/Energy = Efficiency of EcoRI

′

EcoRI could have made 17.3 binary choices


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′

EcoRI could have made 17.3 binary choices. . . but it only made 12 choices.


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′


Efficiency is‘WORK’ DONE / ENERGY DISSIPATED


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′



12 bits per binding

17.3 bits per binding= 0.7

Information/Energy = Efficiency of EcoRI = 70%

EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′



12 bits per binding


The efficiency is 70%.

Information/Energy = Efficiency of EcoRI = 70%

EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′



12 bits per binding


The efficiency is 70%.

18 out of 19 DNA binding proteins give ∼70% efficiency.

Rhodopsin Shape Change

Dark State

P. Scheerer et al. Nature 455, 497-502(2008) doi:10.1038/nature07330



Dark State

hν




Dark State After Photon - Light State

hν





hν70%





hν70%

30%



Muscle Structure

https://en.wikipedia.org/wiki/Muscle

https://en.wikipedia.org/wiki/Muscle

Tom’s Model of Muscle Mechanism

54321 54321

54321 54321

1 2 3 4 5 54321

1 2 3 4 5 5432154321

+PiADP

+PiADP

+PiADP

ATP

heat

operation

before state

after statedegenerate

after stateforward

ATP

heat + ADP + Pi

70% success

heat + ADP + Pi

prime

"power stroke"

30% failure

Efficiency of Muscle

• Experiments by Kushmerick’s lab since (at least) 1969

http://dx.doi.org/10.1113/jphysiol.2007.146829 http://dx.doi.org/10.1242/jeb.052985

http://dx.doi.org/10.1113/jphysiol.2007.146829

http://dx.doi.org/10.1242/jeb.052985



• new work: 2008, 2011






• new work: 2008, 2011

•Weight lifting gives work done






• new work: 2008, 2011


• NMR coil gives ATP = energy used






• new work: 2008, 2011


• NMR coil gives ATP = energy used

• Efficiency: 0.68± 0.09




Why are molecular machines 70% efficient?

EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′

70% efficiency appears widely in biology:


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′


• DNA - protein binding


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′



• rhodopsin


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′



• rhodopsin• muscle


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′




• other systems


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′




• other systems

Why 70% efficiency?


EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′




• other systems

Why 70% efficiency?

Information theory explanation

Lock and Key

Like a key in a lockwhich has many independent pins,it takes many numbersto describe the vibrational stateof a molecular machine

G A A T T C

Gaussians

•2

x

ep(x) −x

Pin motion x has a Gaussian distribution:

p(x) =1√2πσ2

e−(x−µ)2

2σ2

µ = mean, σ = standard deviation

Gaussians

•2

x

ep(x) −x


p(x) =1√2πσ2

e−(x−µ)2

2σ2


• Gaussian distributions are generated by the sum ofmany small random variables

Gaussians

•2

x

ep(x) −x


p(x) =1√2πσ2

e−(x−µ)2

2σ2


• Gaussian distributions are generated by the sum ofmany small random variables

•Drunkard’s walk: Galton’s quincunx device!

http://www.youtube.com/watch?v=xDIyAOBa_yU

http://www.youtube.com/watch?v=xDIyAOBa_yU

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 (1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 (2)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y

p(x, y) = p(x)× p(y) (3)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y

p(x, y) = p(x)× p(y) (3)

∝ e−x2 × e−y

2

(4)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y

p(x, y) = p(x)× p(y) (3)

∝ e−x2 × e−y

2

(4)

∝ e−(x2+y2) (5)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y

credit: http://en.wikipedia.org/wiki/Pythagoras

p(x, y) = p(x)× p(y) (3)

∝ e−x2 × e−y

2

(4)

∝ e−(x2+y2) (5)

∝ e−r2

(6)

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y


p(x, y) = p(x)× p(y) (3)

∝ e−x2 × e−y

2

(4)

∝ e−(x2+y2) (5)

∝ e−r2

(6)

If p(x, y) is a constant,then r is a constant.

Two Gaussians

p(x) =1√2πσ2

e−(x−µ)2

2σ2 ∝ e−x2

(1)

p(y) =1√2πσ2

e−(y−µ)2

2σ2 ∝ e−y2

(2)

r

p(x,y)

x

y


p(x, y) = p(x)× p(y) (3)

∝ e−x2 × e−y

2

(4)

∝ e−(x2+y2) (5)

∝ e−r2

(6)

If p(x, y) is a constant,then r is a constant.

Circular distribution!

1 Dimension

Energy

States

1 dimension is too simple!

Bowls in 2 Dimensions

Spheres in 3 Dimensions

N Dimensional Sphere

Spheres tighten in high dimensions

D=1

D=2

D=4

D=8

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalized Radius

Normalized Density

Good Sphere Packing

Good packing of spheresgives a moleculethe capacityto make selections efficiently

N Dimensional Sphere Separation

Degenerate Sphere


Degenerate Sphere Forward Sphere



√Noise



√Noise √

Power



√Noise √

Power

Energy dissipated to escape the Degenerate Sphere must exceed the Noise



√Noise √

Power

Energy dissipated to escape the Degenerate Sphere must exceed the Noise√Power >

√Noise

Theoretical Isothermal Efficiency

T. D. Schneider, Nucleic Acids Research (2010) 38: 5995-6006

• For molecular states of molecules with dspace ‘parts’Py energy is dissipated for noise Ny and

Cy = dspace log2(Py/Ny + 1) ← machine capacity

�

t

P

y

=N

y

http://tinyurl.com/tomschneider/papers/emmgeo





ǫt =ln

(

Py

Ny+1

)

Py

Ny

← isothermal efficiency

�

t

P

y

=N

y






ǫt =ln

(

Py

Ny+1

)

Py

Ny


�

t

P

y

=N

y

0 1 2 3 40.00

1.00

0.69

0.55

0.46

0.40

Second Law upper bound

Isothermal Efficiency upper bound

The curve is an upper bound






ǫt =ln

(

Py

Ny+1

)

Py

Ny


�

t

P

y

=N

y

0 1 2 3 40.00

1.00

0.69

0.55

0.46

0.40

Second Law upper bound

Isothermal Efficiency upper bound

The curve is an upper bound

• If Py/Ny = 1 the efficiency is 70%!


Acknowledgements

• Mentors:• Larry Gold (graduate school mentor)• Andrej Ehrenfeucht (information theory)• Students:

• Mike Stephens (logos, splicing)• John Spouge (mathematics)• Paul C. Anagnostopoulos (Evolution model)• Bruce Shapiro and Eckart Bindewald (3D logo)• Kevin Franco (σ38)• Ding Jin (σ38)• Useful discussions with

• Jeff Strathern• Amar Klar• Kemi Abolude• Susan Lauffer• Cedric Cagliero• Amar Klar• Zhi-Ming Zheng• Mark Lewandoski• This research was supported by the Intramural Research Program of the NIH,National Cancer Institute, Center for Cancer Research.

Information theory: the mathematics of biology1744 Human acceptor sites

0

1

2

bit

s

5′ -20GA

CT

-19GAC

T

-18AGC

T

-17G

ACT

-16A

GCT

-15A

GCT

-14A

GCT

-13A

GCT

-12A

GCT

-11A

GCT

-10A

GCT

-9

AGCT

-8AGCT

-7GACT

-6GACT

-5

GACT

-4

GACT

-3

ATGC

-2

GATC

-1

GTCA 0TAC

G1T

CAG

2CGAT

3′


0

1

2

bit

s

5′ -3

TGAC

-2

GTCA

-1

CTAG

0CT

G1GACT 2C

TGA

3TCGA

4CTAG

5ACGT

3′ EcoRI sites

0

1

2

bit

s

5′

0

G

1A 2A 3T 4T 5

C3′

https://alum.mit.edu/www/toms/

https://alum.mit.edu/www/toms/

Version

version = 1.89 of kanpurtalk.tex 2016 Oct 16

users.fred.netusers.fred.net/tds/lab/images/kanpur2016/lecture_7_thomas_schne… · rsequence and...

Documents