from reaction networks to information flow

From Reaction Networks to Information Flow

Using Modular Response Analysis to Track Information inSignalling Networks

Pascal Schulthess

Charite – Universitatsmedizin Berlin

October 15, 2010

Reaction networks → information flow?

MRA▶▶▶

Oda and Kitano (2006) Weinberg (2006)

2 / 1

Reaction networks → information flow?

B

A Ap

Bp

MRA▶▶▶ Ap

Bp

3 / 1

The dynamic behaviour of a biochemical reaction system isdetermined by

d

dt

c(t) =Nv(c(t))with

c ∈ Rm concentration vector

N ∈ Rm×n stoichiometric matrix

v ∈ Rn reaction rate vector.

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Example – System

ESE

S

P

1

3

2

d

dt

c(t) =Nv(c(t))

d

dt

��

S(t)E(t)

ES(t)P (t)

��=��

−1 0 1−1 1 01 −1 00 1 −1

��

��k+1ES(t) − k−1E(t)S(t)k+2E(t)P (t) − k−2ES(t)k+3S(t) − k−3P (t)

��

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Example – System

ESE

S

P

1

3

2

d

dt

c(t) =Nv(c(t))

d

dt

��

S(t)E(t)

ES(t)P (t)

��=��

−1 0 1−1 1 01 −1 00 1 −1

��

��k+1ES(t) − k−1E(t)S(t)k+2E(t)P (t) − k−2ES(t)k+3S(t) − k−3P (t)

��with

m0 = 26 / 1

Conservation Analysis

If conservation relations exist,one can separate N by

N = � NR

N0� ESE

S

P

(1) (2)

with

NR ∈ Rm0×n linearly independent metabolites

N0 ∈ R(m−m0)×n linearly dependent metabolites

andm0 = rank(N).

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From the separation it followsthat

N = � I⇤0�NR = ⇤NR

ESE

S

P

(1) (2)

from which the conservation relations can be determined as

� = � −⇤0 I �with

⇤ ∈ Rm×m0 link matrix

� ∈ R(m−m0)×m0 conservation matrix.

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In practice, one has to solve NT�T = 0 which is gettingnumerically expensive with increasing system size.Thus,

PNTQ = LU

where U is partitioned such that

U = � I M0 0

�from which ⇤, NR and � follow to

⇤ = � IMT � , NR = ⇤+N and � = � −MT I � ,

respectively.

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Modular Response Analysis

The unscaled elasticity coe�cient matrix

✏ = �v(c(t))�c(t)

��cgives the sensitivities of all local reaction ratesto perturbations in all species concentrations.The dependencies among species are describedby the (reduced) Jacobian matrix which isdefined as

J = N✏

JR = NR✏⇤.

B

Ap

Bp

✏

vB > 0

✏

vAp> 0

✏

vBp< 0

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After rescaling ✏� ✏ such that

✏ = ✏ c(t)v(c(t))

��c with 0 < ✏ < 1the local and global response matrices follow to

r =NR✏⇤

andR = r−1,

respectively.

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Modular Response Analysis – Example

r =��−1 2 00 −1 −31 0 −1

��B

A

C

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r =��−1 2 00 −1 −31 0 −1

��B

A

C

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r =��−1 2 00 −1 −31 0 −1

��B

A

C

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r =��−1 2 00 −1 −31 0 −1

��B

A

C

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r =��−1 2 00 −1 −31 0 −1

��B

A

C

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r =��−1 2 00 −1 −31 0 −1

��B

A

C

The influence between species in r and R can be categorised by(k, l)-th entry < 0 inactivation of k by l(k, l)-th entry > 0 activation of k by l(k, l)-th entry = 0 no (direct) influence between k by l

withk, l ∈ {1,2, ...,m0}

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Wnt model

Wnt

Dshi

Dsha

GSK3

Axin

APC

APC•Axin

APC•Axin•GSK3 APCp•Axin

p•GSK3 ß-catenin•APC

p•Axin

p•GSK3

ß-cateninp•APC

p•Axin

p•GSK3

ß-cateninp

ß-catenin•APC

TCF

ß-catenin ß-catenin•TCF

DestructionCore Cycle

TranscriptionalActivation

Wntsignaling

Non-Axin dependentproteolysis

Axin dependentproteolysis

Axin turnover

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Elasticity sampling

1. calculate ✏

2. store signs of ✏ = �v(c(t))�c(t) c(t)

v(c(t))3. sample � ✏ � randomly between 0 and 1

4. restore signs of ✏

5. calculate r for each sample

6. observe sign changes between samples

7. plot

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Local interactions of the reduced model

Dshi

GSK3

Axin

APC TCF

APCp•Axin


p•Axin

p•GSK3

ß-catenin•APC ß-catenin

ß-cateninp•APC

p•Axin

p•GSK3

ß-cateninp

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Dshi GSK3

Axin

APC TCF

APCp•Axinp•GSK3 ß-catenin•APCp•Axinp•GSK3


⇒ Input inhibits output

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Dshi GSK3

Axin

APC TCF



⇒ Input activates output

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Dshi GSK3

Axin

APC TCF




23 / 1


Dshi GSK3

Axin

APC TCF




24 / 1


Dshi GSK3

Axin

APC TCF




25 / 1


Dshi GSK3

Axin

APC TCF




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Dshi

GSK3

Axin

APC TCF

APCp•Axin


p•Axin

p•GSK3


E↵ect of Dshi on TCF varies with the use of local interactions.⇒ global interactions

R(Dshi,TCF) ≤ 0⇒ Wnt stimulus has no negative e↵ect on transcription

27 / 1

Elasticity sampling applied to 4 models

ESE

S

P

1

3

2

Figure: Example: 4 species,3 reactions

Wnt

Dshi

Dsha

GSK3

Axin

APC

APC•Axin

APC•Axin•GSK3 APCp•Axin


p•Axin

p•GSK3

ß-cateninp•APC

p•Axin

p•GSK3

ß-cateninp

ß-catenin•APC

TCF

ß-catenin ß-catenin•TCF

Figure: Wnt: 15 species,17 reactions

x1

x2

x3

x4

x5

x6

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x100x103

Figure: Schoberl: 97 species,148 reactions

Figure: Reactome: 6232 species,3652 reactions

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Sign distribution after 1000 samples

0.1

1ExampleWntMapkReactome

always zero always positive always negative positive or negative0

0.0018

Fra

ction

s

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Summary

Conservation and modular response analysis

� e↵ective algorithm based on sparse matrix operations

� transform reaction networks to interaction networks

Sampling experiments

� large definiteness (99%) of interactions

Even without knowing the reaction network in detail, itsinformation flow can be calculated.

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Acknowledgements

Group of Systems Biology of Molecular Networks

� Nils Bluthgen

� Raphaela Fritsche

� Pawel Durek

� Anja Sieber

� Nadine Schmidt

� Franziska Witzel

� Bertram Klinger

� Jorn Schmiedel

Funding

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from reaction networks to information flow

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