nearest-neighbor model in microarray...

Nearest-neighbor model in microarray hybridization

Enrico Carlon, KULeuven

Plon – May 9, 2011

Enrico Carlon, KULeuven () Nearest-neighbor model in microarray hybridization Plon – May 9, 2011 1 / 28

Outline

Enrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 2 / 28

Outline

1 Hybridization properties from controlled experiments (Agilent)


Outline

1 Hybridization properties from controlled experiments (Agilent)1 Breakdown of thermal equilibrium


Outline

1 Hybridization properties from controlled experiments (Agilent)1 Breakdown of thermal equilibrium2 Modeling non-equilibrium effects


Outline

1 Hybridization properties from controlled experiments (Agilent)1 Breakdown of thermal equilibrium2 Modeling non-equilibrium effects3 Exploring other arrays


Outline


2 Algorithms for biological data analysis (Affymetrix)


Outline


2 Algorithms for biological data analysis (Affymetrix)1 AffILM (available at www.bioconductor.org)


Nearest neighbor parameters ∆G37◦C (DNA/DNA)

Table of nearest neighbor parameters for the hybridization free energies ∆G37◦C

in 1M NaCl expressed in kcal/mol. The orientation is 5′-3′ for the upper strand

and 3′-5′ for the lower strand. Only 10 of the 16 parameters are independent.

AATT -1.00 AT

TA -0.88 ACTG -1.44 AG

TC -1.28

TAAT -0.58 TT

AA -1.00 TCAG -1.30 TG

AC -1.45

CAGT -1.45 CT

GA -1.28 CCGG -1.84 CG

GC -2.17

GACT -1.30 GT

CA -1.44 GCCG -2.24 GG

CC -1.84


Example of a calculation

G C A C −3’

C G T G −5’A

G

C

= 11.4 kcal/mole

T

0.881.451.451.84

2.241.30

A5’− T

3’− T

0.88

A

. . . plus boundary terms (effect of the double helix ends)


Mismatches

Base pairings can occurr also for non-complementary bases.

Here: possible structure of a mismatchbetween G (left) and A (right)G·A forms two hydrogen bonds!

Table of nearest-neighbor parameters for AG mismatches at 37◦C in 1MNaCl (unit kcal/mol)

AATG 0.14 AG

TA 0.02 CAGG 0.03 CG

GA 0.11

GACG -0.25 GG

CA -0.52 TAAG 0.42 TG

AA 0.74


Hybridization in a biological experiment

Complex system with a large set of chemical reactions

Aim

Understand/characterize hybridization in microarrays




Aim

Understand/characterize hybridization in microarraysEnrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 6 / 28



Aim

Understand/characterize hybridization in microarrays




Aim

Understand/characterize hybridization in microarraysEnrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 6 / 28

Determination of ∆G from Agilent Custom Arrays

Sufficiently simple setup to avoid “spurious” effects




5’-GTTTTCGAAGATTGGGTGGCACTGTTGTAA-3’ target in solution





3’-CAAAAGCTTCTAACCCACCGTGACAACATT-5’ perfect match






3’-CAAAAACTTCTAACCCACCGTGACAACATT-5’ 1 mismatch







3’-CAAAACCTTCTAACCCACCGTGACAACATT-5’ 1 mismatch








3’-CAAAATCTTCTAACCCACCGTGACAACATT-5’ 1 mismatch









3’-CAAAAGATTCTAACCCACCGTGACAACATT-5’ 1 mismatch










...

3’-CAAAAGCTTCTAACCCACCGTGACGACATT-5’ 1 mismatch










...


3’-CAAAAGCTTCTAACCCACCGTGACTACATT-5’ 1 mismatch










...



3’-CAAAAACTTCTCACCCACCGTGACAACATT-5’ 2 mismatches










...




3’-CAAAAACTTCTGACCCACCGTGACAACATT-5’ 2 mismatches










...





3’-CAAAAACTTCTTACCCACCGTGACAACATT-5’ 2 mismatches

...

3’-CAAAAGCTTCTAACCCACTGTGACTACATT-5’ 2 mismatches










...





3’-CAAAAACTTCTTACCCACCGTGACAACATT-5’ 2 mismatches

...

3’-CAAAAGCTTCTAACCCACTGTGACTACATT-5’ 2 mismatches

total 1006 sequences, replicated 15 times = 15K array


Intensity vs. ∆Gsol

-8 -6 -4 -2 0-∆∆G

sol (kcal/mol)

100

101

102

103

104

105

I

Experiment 1

50 pM

500 pM

5000 pM

PM

1MM

2MM

Expected: Langmuir isotherm

I =Ace∆G/RT

1 + ce∆G/RT

data for 3 differentconcentrations (c)

lines: slope = 1/RT

scatter of the data

deviations from 1/RT already

far from chemical saturation

No fitting parameters!

∆Gsol is computed from tabulated literature data (parameters measured inhybridization in solution)


Intensity vs. ∆Gsol

-∆∆G (arb. units)

I (a

rb. u

nits

)

slope 1/RT

A

Expected Langmuir behavior

Expected: Langmuir isotherm

I =Ace∆G/RT

1 + ce∆G/RT

data for 3 differentconcentrations (c)

lines: slope = 1/RT

scatter of the data

deviations from 1/RT already

far from chemical saturation

No fitting parameters!

∆Gsol is computed from tabulated literature data (parameters measured inhybridization in solution)


Fitting free energy parameters from experimental data

-8 -6 -4 -2 0∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

Experiment 1

c = 50 pM

1006 data points and58 parameters

Fit to

I = A exp(∆G/RT )


Fitting free energy parameters from experimental data

-8 -6 -4 -2 0∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

Experiment 1

c = 50 pM

T = 850K

T = 338K = 65oC

1006 data points and58 parameters

Fit to

I = A exp(∆G/RT )Two temperatures!


Identical behavior for 4 different sequences

-8 -6 -4 -2 0-∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

-8 -6 -4 -2 0

10-4

10-2

Experiment 1

c = 50 pM

Texp

Teff

-8 -6 -4 -2 0-∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

-8 -6 -4 -2 0

10-4

10-2

Experiment 2

c = 50 pM

-8 -6 -4 -2 0-∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

-8 -6 -4 -2 0

10-4

10-2

Experiment 3

c = 50 pM

-8 -6 -4 -2 0-∆∆Gµarray

(kcal/mol)10

-5

10-4

10-3

10-2

10-1

I/I PM

*

-8 -6 -4 -2 0

10-4

10-2

Experiment 4

c = 50 pM


Comparison of free energy in the array and in solution

0 1 2 3 4 5∆∆G

sol

0

1

2

3

4

5

∆∆G

µarr

ay

CC mismatches

All Experiments

∆∆G: difference between perfectmatch hybridization andhybridization with an internalmismatch.Moderate correlation betweensolution and microarray free energies


Understanding the two regimes

k2

k−2

k−1

k1

configuration 1 configuration 2

Hybridization in solution:nucleation and fast zippering

Is zippering rapid inmicroarrays?Probably not!

The simplest description of along lived partially zippedstate is through a 3 statemodel


Understanding the two regimes

k2

k−2

k−1

k1

configuration 1 configuration 2

Hybridization in solution:nucleation and fast zippering

Is zippering rapid inmicroarrays?Probably not!

The simplest description of along lived partially zippedstate is through a 3 statemodel

Four parameters

Hybridization rates: k1, k2 (forward), k−1, k−2 (reverse)

Thermodynamics k−1/k1 = e−∆G′/RT k−2/k2 = e−(∆G−∆G′)/RT


Two vs. three state model

Two state modelkinetics

dθ

dt= ck1(1− θ)− k−1θ

Equilibrium θeq =ck1

k−1 + ck1=

cK

1 + cK=

ce∆G/RT

1 + ce∆G/RT


Two vs. three state model

Two state modelkinetics

dθ

dt= ck1(1− θ)− k−1θ

Equilibrium θeq =ck1

k−1 + ck1=

cK

1 + cK=

ce∆G/RT

1 + ce∆G/RT

Threestatemodel

dθ(1)

dt= ck1(1− θ(1) − θ(2)) + k−2θ

(2)− (k−1 + k2)θ

(1)

dθ(2)

dt= k2θ

(1)− k−2θ

(2)

Equilibrium(c → 0) θ(1)eq = c

k1k−1

= ce∆G′/RT θ(2)eq = ck1k2

k−1k−2= ce∆G/RT

Assumption:

k1, k2 independent on ∆G∆G′

≈ γ∆G (they should be monotonically linked)Enrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 13 / 28

Three state model

10 20 30 40-∆G(kcal/mol)

10-8

10-6

10-4

10-2

100

θ

Langmuir isotherm

intermediateregime

dynamical saturation

(b)chemical saturation

Langmuir isotherm

intermediateregime

dynamical saturation

(b)chemical saturation

c fixed at 5.10-12

, various times

Solid line:Equilibrium solution(t → ∞)

Dashes lines:Finite time behavior


Experiments (Agilent arrays)

-7 -6 -5 -4 -3 -2 -1 0101

102

103

104

105

I

a)

-7 -6 -5 -4 -3 -2 -1 0101

102

103

104

105

I

b)

-7 -6 -5 -4 -3 -2 -1 0

∆∆G (kcal/mol)10

1

102

103

104

105

I

c)

-7 -6 -5 -4 -3 -2 -1 0


1

102

103

104

105

I

d)

t = 40min t = 3h24min

t = 17h t = 86h8min

kinetic model

Target sequenceL = 30Exper. to ≈ 86h

Temperature 65◦C

Solid line:three state modelDashed line:

Equilibrium isotherm



-7 -6 -5 -4 -3 -2 -1 0101

102

103

104

105

I

a)

-7 -6 -5 -4 -3 -2 -1 0101

102

103

104

105

I

b)

-7 -6 -5 -4 -3 -2 -1 0


1

102

103

104

105

I

c)

-7 -6 -5 -4 -3 -2 -1 0


1

102

103

104

105

I

d)

t = 40min t = 3h24min

t = 17h t = 86h8min

Target sequenceL = 25Exper. to ≈ 86h

Temperature 65◦C

Dashed line:Equilibrium isotherm



14 16 18 20 22 24 26 28 30-∆G

sol (kcal/mol)

100

102

104

106

I

50pM

500pM

5000pMequilibrium

region

non-equilibrium region

Target sequenceL = 30Exper. ≈ 17h

Temperature 55◦C

Dashed line:Three state model

Evidence of dynamicalsaturation


Experiments (High density arrays, Suzuki et al.)

15 20 25 30-∆G (kcal/mol)

10

103

105

I

1.4fM14fM140fM1.4pM14pM140pM1.4nM

140 targets andprobes with 1 or 2mismatches



15 20 25 30-∆G (kcal/mol)

10

103

105

I



Solid line:Teff ≈ 1200K



15 20 25 30-∆G (kcal/mol)

10

103

105

I




Dashed line:Texp ≈ 45C



15 20 25 30-∆G (kcal/mol)

10

103

105

I





Dynamical saturation



15 20 25 30-∆G (kcal/mol)

10

103

105

I





Dynamical saturation... or depletion?



10 15 20 25 30 35-∆G (kcal/mol)

10

103

105

I

14fM140fM1.4pM




Dynamical saturation... or depletion?Solid line: three statemodel


Experiments (Spotted array, Weckx et al.)

25 30 35 40-∆G

sol (kcal/mol)

102

103

104

105

I -

I 0 scannersaturation

slope 1/RTexp

PM1MM2MM3MM

Experiment with 4targets and probes upto three mismatches



25 30 35 40-∆G

sol (kcal/mol)

102

103

104

105

I -


slope 1/RTexp

PM1MM2MM3MM


Large range of ∆G



25 30 35 40-∆G

sol (kcal/mol)

102

103

104

105

I -


slope 1/RTexp

PM1MM2MM3MM


Large range of ∆G

Solid line:Texp ≈ 45C



25 30 35 40-∆G

sol (kcal/mol)

102

103

104

105

I -


slope 1/RTexp

PM1MM2MM3MM


Large range of ∆G

Solid line:Texp ≈ 45C

Dashed line:scanner saturation


Practical consequences

Microarrays working in a non-equilibrium regime suffer from enhancedcross-hybridization




Example: two different sequences in solution binding to the same probe atthe surface. The first is perfect match and has concentration c, the secondcarries mismatches and has concentration c′.

I = cf(∆G) + c′f(∆G′)





I = cf(∆G) + c′f(∆G′)

We wish to have f(∆G) ≫ f(∆G′). In equilibrium

feq(∆G)

feq(∆G′)= e(∆G−∆G′)/RTexp

is maximized. . .





I = cf(∆G) + c′f(∆G′)

We wish to have f(∆G) ≫ f(∆G′). In equilibrium

feq(∆G)

feq(∆G′)= e(∆G−∆G′)/RTexp

is maximized. . . while in non-equilibrium

f(∆G)

f(∆G′)= e(∆G−∆G′)/RTeff ≪

feq(∆G)

feq(∆G′)


Application: Mutant identification (Jef’s talk)

-8 -6 -4 -2 010

0

101

102

103

I

c’ = 0

baseline

(a) slope 1/RT

-8 -6 -4 -2 010

0

101

102

103

c’/c = 0.01

(b)

-8 -6 -4 -2 0

∆∆G (kcal/mol)

100

101

102

103

I

c’/c = 0.03

(c)

-8 -6 -4 -2 0

∆∆G (kcal/mol)

100

101

102

103

c’/c = 0.1

(d)

Target:Mixture of 2 sequences

Wild type at cMutant at c′

Detection limitc′/c ≈ 0.01


Back to complex biological experiments (Affy Genechip)

AffyILM: Algorithm available from BioConductor

AffyILM




AffyILM

. . . Langmuir hybridization (with Teff)




AffyILM

. . . background subtraction




AffyILM

. . . not taken into account (should affect 5% of targets)Enrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 20 / 28



AffyILM

. . .mean-field approximationEnrico Carlon, KULeuven (ITF, K.U.Leuven)Nearest-neighbor model in microarray hybridization Apr. 11, 2011 20 / 28

1st - Estimating of the Background Noise

Motivation:

Strong signals(Isp ≫ I0)⇒ Gene expressed

Weak signals (Isp ≈ I0)⇒ Weakly expressedgene or backgroundnoise?

Aim

Calculation of background intensity for each probe on array


2nd - Specific signal: extended Langmuir model

θ: fraction of bound probesc: mRNA concentrationα: fraction “free” mRNA

θ =αce∆G/RT

1 + αce∆G/RT

α =1

1 + ce∆GR/RT ′

kin

α c c(1−α)

kout

Intensity: I = Aθ∆G, ∆GR: from data in solution

Parameters: A, T , T ′, c

Scaling variablex′ = αc exp(∆G/RT )Fitted effective temperature T ≈ 700K


Modeling Hybridization in solution

~~tijt

+ + + + ....25

1i

j^

Chemical equilibrium

Target conservation

[t][ti,j ]

[tti,j ]= e∆GR(i,j)/RT

[t] +∑

i,j

[tti,j ] = c

α =[t]

c=

1

1 +∑

i,j [ti,j ]e−∆GR(i,j)/RT

≈1

1 + ce−∆GR(1,25)/RTeff


Calibration data (spike-in)

≈ 300 data points (concentrations c = 0, 0.25, 0.5 . . . 1024 pM)

Scaling collapses (4 fitting parameters) RED : PM / GREEN : MM

12

3 5678 910

1112 1314

15161 23

4567

8 910

11121314

15161 23 45678 910

11121314

15161 2

34567

8 910

1112 1314

15161 23 4

5678 910

11

1213

14

15161 23

4

5678 910

1112 1314

1516

1 2345678 910

1112 1314

1516

12

3 45678 9

10

1112 1314

15161 2

3456

78 910

1112 1314

1512

345

678 91011

1213

14

15

161 23 45678

910

11

12131415

16

1 2

34

5678 9

1011

121314

15

16

1 23 45678 910

1112 1314

15161 23 45678 910

1112 1314

15161 2

3 45678 910

1112 1314

15161 2

34567

8 910

11121314

15161 2

3 45678 910

1112 1314

15161 2

34567

8 910

1112 1314

15161 23 4567

8 910

1112 1314

1516

10−4

10−2

100

102

104

x’

100

102

104

I(c)

−I(

0)

236

71023 6710

16

2

36710

16

23

6710

16

23 67

10

162367

102

36710

162

3

67

10

16

23 6710

27

10

162

3

6710 16

23 710

162

3 671016

2

3 671016

2

3 6710

16

2

3 671016

2

3 6710

16

2

3 6710

16

2

36710

16

Ax’/(1+x’)

1024_at 12

345

678

9101112

1314

15161

2345

678

910

1112

1314

151612

34

5678

910

111213

14

151612

345

678

9101112

1314

15161

2 3

45

678

910

1112

1314

15161

2 3

45

678

910

1112

1314

15161

2 3

45678

910

1112

1314

15161

2 3

45678

9101112

1314

15161

23

45

678

910

1112

1314

15161

23

45

678

9

10

1112

1314

15161

23

45

6

78

9

101112

1314

1516

12

3

45

6

78

9

101112

1314

15161

23

45

6

78

9

10

1112

1314

15

161

23

45

678

9

101112

1314

15

161

2

3

45

6

78

9

101112

1314

15

161

2

3

45

6

78

101112

1314

15

161

3

45

678

9101112

1314

15

1613

4

57

8

9

101112

1314

15161

3

4

578

101112

13

14

15

16

10−4

10−2

100

102

104

x’

100

102

104

I(c)

−I(

0)

1

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

5712

161

2

57

12

161

2

57

12

16

1

2

57

12

161

2

57

12

161

2

5712

161

2

57

12

161

2

57 16

12

57

12

16

7

16

12 5

7

1216

Ax’/(1+x’)

37777_at

x-axis: scaling variable x′ = αc exp(∆G/RT )y-axis: background subtracted intensities


Outliers

1 23

45

6 78

9

10111213141516

1 23

45

6 78

9

10111213141516 1 2

34

5

6 78

9

10111213141516 1 2

34

5

6 78

9

10111213141516

1 23

4

5

6 78

9

10111213141516

1 23

4

5

6 78

9

10111213141516 1 2

34

5

6 78

9

10111213141516

1 23

45

6 78

9

10111213141516

1 23

4

5

6 78

9

10111213141516

1 2

3

45

6 78

9

10111213141516

1 23

45

6 78

9

10111213141516

1 23

4

5

6 78

9

10111213141516

23

4

5

67813141516

123

45

6 7

8

11121415162356

78

11

13141516

1 2

3

56

711

1215

161

234

56

7

81011

1214

1516

1 23456 7

8

9

101112131415161 2

34

5

6 78

9

10111213141516

10−4

10−2

100

102

104

x’

100

102

104

I(c)

−I(

0)

12

4

59

1013

152

45

9

1013

15

1 2

4

5 9

1013

15

12

4

5 9

1013

15

1

24

5

101315

1

24

5

9

1013

15

1

24

59

1013

15

12

45

9

1013

15

24

5

101315

1 24

5

1013

151 24

5 9

1013

1512

4

5

9

10

13

1525

13

152

5 25

9132

5

91 2

45

10

15

12

4

5 9

101315

12

4

5 9

1013

15

Ax’/(1+x’)

1091_at

probe 9

1091 at9sequence alignment (GenBank)

��

��

��

��

Affymetrix

M65066.1

BC013368.2

AL833563.1 Dat

a ba

se

3’−CTG TCC TTG GTC CG C ATG GCT CGT T−5’

3’−CTG

3’−CTG

3’−CTG

TCC

TCC

TCC

TTG

TTG

TTG

GTC

GTC

GTC

CG

CG

CG

C ATG GCT

GCT

GCT

CGT

CGT

T−5’

T−5’

T−5’

CGTAGGCT

AGGCT

Error in the annotation:the correct sequence was submitted to GenBank in 2003, after themicroarray was designed.


affyILM (available at www.bioconductor.org)

Input: Experimental microarray data (Affymetrix)Output: Estimate of Concentration per each probe sequence

20 25 30 35 40

5010

020

050

010

0020

0050

0010

000

206158_s_at − RIri−CEP−A−1a−U133A.CEL

−RT ln K

Inte

nsity

+

++

+

+

+

+

+

+

+

+

median c = 200 pM

m.a.d. c = 69 pM

Probe no. conc [pM]

1 437.24

2 466.92

3 176.57

4 200.39

5 433.10

6 153.85

7 211.93

8 154.55

9 124.68

10 35.74

11 201.57

Median 200

M.A.D. 69


Collaborators

Institute for Theoretical Physics - KULeuvenM. Baiesi, F. Berger, A. Ferrantini, K. M. Kroll, J. C. WalterJ. Hooyberghs (& Flemish Institute for Technological Research, VITO)

Institute for Theoretical Physics - Utrecht UniversityG. T. Barkema, J. Klein Wolterink, G. Mulders

Micorarray Facility - Flemish Institute for Biotechnology (VIB)P. Van Hummelen, S. Weckx

Financial support fromFWO (Flemish Research Council), KULeuven, VITO, VIB


Recent Papers

G. Mulders, G. Barkema and E. CarlonInverse Langmuir method for oligonucleotide microarray analysisBMC Bioinformatics 10, 64 (2009)

J. Hooyberghs, M. Baiesi, A. Ferrantini and E. CarlonBreakdown of thermodynamic equilibrium for DNA hybridization in microarraysPhys. Rev. E 81, 012901 (2010).

J. Hooyberghs and E. CarlonHybridisation thermodynamic parameters allow accurate detection of pointmutations with DNA microarraysBiosens. and Bioel. 26, 1692 (2010)

J. C. Walter, K. M. Kroll, J. Hooyberghs and E. CarlonNon-Equilibrium Effects in DNA Microarrays: A Multiplatform StudyJ.Phys. Chem. B (in press). DOI: 10.1021/jp2014034

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nearest-neighbor model in microarray...

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