Inverse dynamical analysis

Inverse dynamical analysis

a methodology for probing dynamics in gene regulation and

metabolism

James Lu

Inverse Problems Group

Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, Austria

Rainer Machné

Theoretical Biochemistry Group (TBI) University of Vienna

Vienna, Austria

In collaboration with: Heinz Engl (RICAM), Peter Schuster (TBI, Vienna)

Outline

◮ _{Gene regulation models: G1}/_{S module, GATA networks}

◮ Auto-regulatory feedback loops

◮ _{The Forward Problem}

◮ Dynamics and Bifurcations

◮ Inverse Bifurcation Analysis ◮ Numerical Results ◮ Biological interpretation ◮ _{Inverse Eigenvalue Analysis}

Outline

◮ _{Gene regulation models: G1}/_{S module, GATA networks}

◮ Auto-regulatory feedback loops

◮ _{The Forward Problem}

◮ Dynamics and Bifurcations

◮ Inverse Bifurcation Analysis

◮ Numerical Results

◮ Biological interpretation

◮ _{Inverse Eigenvalue Analysis}

The G

1

/

S Module of Mammalian Cell Cycle

◮ _{What can be said about the core regulatory mechanism of the}

bifurcation points?

◮ Inverse Bifurcation and Eigenvalue Analysis

from Bio

to Math and back

The Forward Problem: bottom-up modeling

interaction graph→reactions→dx_dt =f(x, α)→x(t)→bifurcations

interaction graph

compilingdecades of diploma and

PhD students’wet labwork

reaction network

Snuclear−−→Snuclear+ARNA

ARNA−−→AmRNA

AmRNA−−→AmRNA+Aprotein

Aprotein←−→Anuclear

Anuclear−−→Anuclear+ARNA

Ax−−→

1. stoichiometry: analysis

from Bio

to Math

and back

The Forward Problem:

interaction graph→reactions→dx_dt =f(x, α)→x(t)→bifurcations

equations drA dt = N· 0 @V_ab+V_aA· 0 @ A “ KA a +A ” 1 A hA +V_aS· 0 @ S “ KS a +S ” 1 A 1 A−(kexa+ ΦrA)·rA dmA dt = kexa·rA·c1/c2−ΦmA·mA dAc dt = ktla·mA+kexA·A·c1/c2−(kimA+ ΦAc)·Ac dA dt = kimA·Ac−(kexA+ ΦA)·A

dynamical systems theory:

ODEs, PDEs, Markov process, Boolean nw., Bayesian nw., Petri nets,

from Bio to Math

and back

The Forward Problem:

interaction graph→reactions→dx_dt =f(x, α)→x(t)→bifurcations

time courses: x(t) 0 150 300 450 600 750 900 1050 1200 1350 1500 time [minutes] 0 0.25 0.5 0.75 1 1.25

nuclear A [molecules / femtolitre]

low S concentration

still rarely availabe from wet labs

(steady-state paradigm)

from Bio to Math

and back

The Forward Problem:

interaction graph→reactions→dx_dt =f(x, α)→x(t)→bifurcations

time courses: x(t) 0 150 300 450 600 750 900 1050 1200 1350 1500 time [minutes] 0 11.152 22.304 33.456 44.608 55.761 66.913 78.065 89.217 100.37

nuclear A [molecules / femtolitre]

high S concentration

still rarely availabe from wet labs

(steady-state paradigm)

from Bio to Math

and back

: bifurcation phenotypes

The Forward Problem:

interaction graph→reactions→dx_dt =f(x, α)→x(t)→bifurcations

time courses: x(t) 0 150 300 450 600 750 900 1050 1200 1350 1500 time [minutes] 0 11.152 22.304 33.456 44.608 55.761 66.913 78.065 89.217 100.37

nuclear A [molecules / femtolitre]

high S concentration

still rarely availabe from wet labs

(steady-state paradigm)

stability and bifurcation analysis

w.r.t. physiological signals

≈signal/response curves from titration experiments

Feedback and Bifurcations I: hysteresis/bistability

2 saddle-node bifurcations

:

signal switches, memory effects

conjecture: multistability requires positive feedback

◮ _{S: e.g. low nutrient concentration or infection} ◮ _{S should really decrease significantly to switch-off A}

◮ ⇒avoid oscillating deficiency or infection

◮ _{Irreversibility w.r.t. S: terminal differentiation!}

Feedback and Bifurcations I: hysteresis/bistability

2 saddle-node bifurcations

:

signal switches, memory effects

conjecture: multistability requires positive feedback

◮ _{S: e.g. low nutrient concentration or infection}

◮ _{S should really decrease significantly to switch-off A}

◮ ⇒avoid oscillating deficiency or infection

◮ _{Irreversibility w.r.t. S: terminal differentiation!}

Feedback and Bifurcations I: hysteresis/bistability

2 saddle-node bifurcations

:

signal switches, memory effects

conjecture: multistability requires positive feedback

◮ _{S: e.g. low nutrient concentration or infection}

◮ _{S should really decrease significantly to switch-off A}

◮ ⇒avoid oscillating deficiency or infection

◮ _{Irreversibility w.r.t. S: terminal differentiation!}

Feedback and Bifurcations II: oscillations

Hopf bifurcations:oscillatory phenomena

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0 0.2 0.4 0.6 0.8 1 1.2 H LP LP Continuation parameter tG‘S

State component tG‘A[t]

LPC

LPC

conjecture: oscillation requires negative feedback and delay

◮ _{Life is an intrinsically rhythmic phenomenon:} ◮ metabolic oscillations, nanovibrations

◮ _{yeast respiration cycles with genome-wide effects,}

e.g. cell cycle gating!

◮ cell cycle ◮ circadian clock ◮ lunar and solar cycle

Feedback and Bifurcations II: oscillations

Hopf bifurcations:oscillatory phenomena

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0 0.2 0.4 0.6 0.8 1 1.2 H LP LP Continuation parameter tG‘S

State component tG‘A[t]

LPC

LPC

conjecture: oscillation requires negative feedback and delay

◮ _{Life is an intrinsically rhythmic phenomenon:}

◮ metabolic oscillations, nanovibrations

◮ _{yeast respiration cycles with genome-wide effects,}

e.g. cell cycle gating!

◮ cell cycle

◮ circadian clock

◮ lunar and solar cycle

S. cerevisiae and C. elegans GATA networks

secondary mesoderm (ABa) mesoderm gastrulation endoderm SKN-1 MED-1,2 END-1,3 ELT-2,7 ELT-4

◮ _{common origin in auto-activator?}

⇒coherent functional story up to mammals

◮ _{different duplication and diversification events}

◮ _{generating cascades by mutation of bindings sites and domains}

◮ _{generating competitive inhibitors by loss of activating domain}

Dosage effects: hypersensitivity / haploinsufficiency

◮ _{gene duplication:N:1→2}

◮ hypersensitivity or even irreversibility wrt to S ◮ _{related to hypersensitive immune response?}

◮ IL4→STAT6→GATA-3→GATA-3→T

h2activation

◮ _{haploinsufficiencyN:2→1}

◮ haploinsufficiency diseases known for both

GATA-2 (confirmed auto-activator) and GATA-6

Dosage effects: hypersensitivity / haploinsufficiency

◮ _{gene duplication:N:1→2}

◮ hypersensitivity or even irreversibility wrt to S

◮ _{related to hypersensitive immune response?} ◮ IL4→STAT6→GATA-3→GATA-3→T

h2activation

◮ _{haploinsufficiencyN:2→1}

◮ haploinsufficiency diseases known for both

GATA-2 (confirmed auto-activator) and GATA-6

Machné R, Lu J, Müller S, Endler L, Widder S, Flamm C, Engl H, Schuster PK, Murray DB et al. in preparation 2007 Susumu Ohno. Evolution by gene duplication. 1970

Dosage effects: hypersensitivity / haploinsufficiency

◮ _{gene duplication:N:1→2}

◮ hypersensitivity or even irreversibility wrt to S

◮ _{related to hypersensitive immune response?}

◮ IL4→STAT6→GATA-3→GATA-3→T

h2activation ◮ _{haploinsufficiencyN:2→1}

◮ haploinsufficiency diseases known for both

GATA-2 (confirmed auto-activator) and GATA-6

Machné R, Lu J, Müller S, Endler L, Widder S, Flamm C, Engl H, Schuster PK, Murray DB et al. in preparation 2007 Höfer T et al. PNAS 2002

Dosage effects: hypersensitivity / haploinsufficiency

◮ _{gene duplication:N:1→2}

◮ hypersensitivity or even irreversibility wrt to S

◮ _{related to hypersensitive immune response?}

◮ IL4→STAT6→GATA-3→GATA-3→T

h2activation

◮ _{haploinsufficiencyN:2→1}

◮ haploinsufficiency diseases known for both

GATA-2 (confirmed auto-activator) and GATA-6

Machné R, Lu J, Müller S, Endler L, Widder S, Flamm C, Engl H, Schuster PK, Murray DB et al. in preparation 2007 Rodrigues NP et al. Blood 2005, Grass JA et al. Mol Cell Biol. 2006

Inverse dynamical analysis

◮

In modelling gene regulation systems, one would like to:

◮ probethe possibility for the model to exhibit bistability or

oscillations

◮ characterizeparameter variations that can give rise to

different qualitative dynamics

◮

Given a plausible model:

◮ identifymechanisms in model that can give rise to various

bifurcation phenotypes: verify or falsify experimentally

◮ designfor desired dynamical characteristics

◮

Methods:

◮ inverse eigenvalue analysis

◮ inverse bifurcation analysis ◮ sparsity-promoting regularization

Inverse dynamical analysis

◮

In modelling gene regulation systems, one would like to:

◮ probethe possibility for the model to exhibit bistability or

oscillations

◮ characterizeparameter variations that can give rise to

different qualitative dynamics

◮

Given a plausible model:

◮ identifymechanisms in model that can give rise to various bifurcation phenotypes: verify or falsify experimentally ◮ designfor desired dynamical characteristics

◮

Methods:

◮ inverse eigenvalue analysis ◮ inverse bifurcation analysis

Inverse dynamical analysis

◮

In modelling gene regulation systems, one would like to:

◮ probethe possibility for the model to exhibit bistability or

oscillations

◮ characterizeparameter variations that can give rise to

different qualitative dynamics

◮

Given a plausible model:

◮ identifymechanisms in model that can give rise to various bifurcation phenotypes: verify or falsify experimentally ◮ designfor desired dynamical characteristics

◮

Methods:

◮ inverse eigenvalue analysis ◮ inverse bifurcation analysis

Inverse problems

◮

Forward problem:

dx dt

=

f

(

x

, α

)

→

x

(

t

)

→

bifurcations

F

(

α

) =

x

◮

Inverse problem:

dx dt

=

f

(

x

, α

)

←

x

(

t

)

←

bifurcations

◮ typically ill-posed (in the sense of Hadamard) ◮ non-uniqueness;

◮ instability of inversion

◮

Variational

regularization: add penalty term

min

α

k

F

(

α

)

−

x

k

+

µ

R

(

α

)

◮

While stabilizing ill-posed problems, regularization brings

bias to the solution

◮

For biological systems, we usually want the solution to be

sparse, i.e., having as few non-zeros

in the solution as

Inverse problems

◮

Forward problem:

dx dt

=

f

(

x

, α

)

→

x

(

t

)

→

bifurcations

F

(

α

) =

x

◮

Inverse problem:

dx dt

=

f

(

x

, α

)

←

x

(

t

)

←

bifurcations

◮ typically ill-posed (in the sense of Hadamard)

◮ non-uniqueness;

◮ instability of inversion

◮

Variational

regularization: add penalty term

min

α

k

F

(

α

)

−

x

k

+

µ

R

(

α

)

◮

While stabilizing ill-posed problems, regularization brings

bias to the solution

◮

For biological systems, we usually want the solution to be

sparse, i.e., having as few non-zeros

in the solution as

Inverse problems

◮

Forward problem:

dx dt

=

f

(

x

, α

)

→

x

(

t

)

→

bifurcations

F

(

α

) =

x

◮

Inverse problem:

dx dt

=

f

(

x

, α

)

←

x

(

t

)

←

bifurcations

◮ typically ill-posed (in the sense of Hadamard)

◮ non-uniqueness;

◮ instability of inversion

◮

Variational

regularization

: add penalty term

min

α

k

F

(

α

)

−

x

k

+

µ

R

(

α

)

◮

While stabilizing ill-posed problems, regularization brings

bias to the solution

◮

For biological systems, we usually want the solution to be

sparse, i.e., having as few non-zeros

in the solution as

Inverse problems

◮

Forward problem:

dx dt

=

f

(

x

, α

)

→

x

(

t

)

→

bifurcations

F

(

α

) =

x

◮

Inverse problem:

dx dt

=

f

(

x

, α

)

←

x

(

t

)

←

bifurcations

◮ typically ill-posed (in the sense of Hadamard)

◮ non-uniqueness;

◮ instability of inversion

◮

Variational

regularization

: add penalty term

min

α

k

F

(

α

)

−

x

k

+

µ

R

(

α

)

◮

While stabilizing ill-posed problems, regularization brings

bias to the solution

◮

For biological systems, we usually want the solution to be

sparse, i.e., having as few non-zeros

in the solution as

Inverse problems

◮

Forward problem:

dx dt

=

f

(

x

, α

)

→

x

(

t

)

→

bifurcations

F

(

α

) =

x

◮

Inverse problem:

dx dt

=

f

(

x

, α

)

←

x

(

t

)

←

bifurcations

◮ typically ill-posed (in the sense of Hadamard)

◮ non-uniqueness;

◮ instability of inversion

◮

Variational

regularization

: add penalty term

min

α

k

F

(

α

)

−

x

k

+

µ

R

(

α

)

◮

While stabilizing ill-posed problems, regularization brings

bias to the solution

◮

For biological systems, we usually want the solution to be

sparse, i.e., having as few non-zeros

in the solution as

possible

Sparsity-promoting regularization: l

p

, p

≤

1

◮

Consider (smoothed) functionals

R

n

→

R

:

l

p,ǫ

(

α

) =

P

i

(

α

2i

+

ǫ

)

p/2

Sparsity-promoting regularization: l

p

, p

≤

1

◮

Consider (smoothed) functionals

R

n

→

R

:

l

p,ǫ

(

α

) =

P

i

(

α

2i

+

ǫ

)

p/2

◮

Convex only within the box

{

α

:

|

α

_i

|

<

√

ǫ,

0

<

i

≤

n

}

−0.2 −0.1 0 0.1 0.2 0.3 0 0.02 0.04 0.06 0.08 0.1 α l2 ( α ) −0.2 −0.1 0 0.1 0.2 0.3 0 0.1 0.2 α l0.1, 1e−4 ( α )

Inverse bifurcation: mammalian G

1

/

S transition

◮

Map: bifurcation phenotypes

→

parameter sets

◮

Consider the following 3 modes of geometric

transformations of the nomimal bifurcation diagram:

Inverse bifurcation: mammalian G

1

/

S transition

◮

Map: bifurcation phenotypes

→

parameter sets

◮

Consider the following 3 modes of geometric

Inverse bifurcation: effect of sparsity-promiting penalty

0 2 4 6 8 0 2 4 6 8 SN SN SN Fm E2F1 Mammalian G 1/S transition −15 −10 −5 0 k43 k67 k76 k23 k28 k89 k98 a J15 J18 J68 J13 J63 Km9 phipRB phiCycDi phiAp1 phipRBp phipRBpp phiCycEi phiCycEa J65 J62 J61 Km1 k61 k16 k25 Km4 J11 k3 k1 k34 J12 Km2 phiCycDa phiE2F1 k2 kp % change Parameter list (a) l_0.1,10−4regularization 0 2 4 6 8 0 2 4 6 8 SN SN SN Fm E2F1 Mammalian G 1/S transition −4 −3 −2 −1 0 1 2 3 k43 k67 k76 k28 k89 k98 a J15 J18 J68 J13 Km9 phipRB phipRBpp phiCycEi phiCycEa J63 J61 k23 J65 Km1 J62 phipRBp k61 k16 k25 Km4 phiCycDi k1 J11 k2 k34 phiCycDa k3 J12 phiAp1 phiE2F1 Km2 kp % change Parameter list (b) l2regularization

Inverse bifurcation: identified module

Table:Result of hierarchical algorithm with p=0.1, ǫ=10−4

Modification Case Level j=1 Level j=2 Level j=3

Elongating SN1nose kp↓14.3% k34↑31.7% φAP-1↓20.9%

Km2↑6.4% φE2F1↑7.3% Moving SN1,2to right Km4↑269.3% J11↑191.7% k2↓39.9% kp↑17.3% φE2F1↓11.7% Km2↓10.3% Decreasing bistabiliy J11↑128.5% k1↑169.1% k2↓43.7% kp↑33.8% Km2↓21.7% φE2F1↓28.3% J12↓20.1%

Inverse bifurcation: identified module

d

dt

[

pRB

] =

k

1

[

E2F1

]

K

m1

+ [

E2F1

]

J

11

J

11

+ [

pRB

]

J

61

J

61

+ [

pRB

p

]

−

k

16

[

pRB

][

CycD

a

] +

k

61

[

pRB

p

]

−

φpRB

[

pRB

]

,

d

dt

[

E2F1

] =

k

p

+

k

2

a

2

+ [

E2F1

]

2

K

m22

+ [

E2F1

]

2

J

12

J

12

+ [

pRB

]

J

62

J

62

+ [

pRB

p

]

−

φ

E2F1

[

E2F1

]

d

dt

[

CycD

i

] =

−

k

34

[

CycD

i

]

[

CycD

_a

]

K

m4

+ [

CycD

a

]

+

· · ·

d

dt

[

CycD

a

] =

k

34

[

CycD

i

]

[

CycD

_a

]

K

m4

+ [

CycD

a

]

+

· · ·

Possible interprations: G

1

/

S module

1.

Qualitative model - arbitrary interpretations!

2.

Modification 1 + 2 : move bistability

◮ move SN

1or SN1/SN2: insensitivity to mitogen

→E2F1/pRB↔p53/Mdm2 involved in cell-cycle arrest

◮ ↓k

p:decreasing basal transcription of E2F1

→induction of differentiation markers in squamous cancer cell line (Wong et al. Oncogene 2005)

3.

Modification 3 : decrease bistability

◮ loss of threshold:continuous response to mitogen →increase stochasticity of response

◮ ↑k

p,↑J11 etc. :weaken E2F1 self-control

→yeast swi4 knock-out (≈E2F1) shows increased stochasticity in cell cycle (Ubersax JA Mol Syst Biol 2006)

4.

Compare Mod. 2, level 2 with Mod. 3, level 1

→

dynamic properties of complex systems unintuitive!

5.

Only one step in model-experiment loop!

6.

Apply to GATA evolution: based on realistic data, only

relative changes in evolutionary duplication scenarios!

Possible interprations: G

1

/

S module

1.

Qualitative model - arbitrary interpretations!

2.

Modification 1 + 2 : move bistability

◮ move SN

1or SN1/SN2: insensitivity to mitogen

→E2F1/pRB↔p53/Mdm2 involved in cell-cycle arrest

◮ ↓k

p:decreasing basal transcription of E2F1

→induction of differentiation markers in squamous cancer

cell line (Wong et al. Oncogene 2005)

3.

Modification 3 : decrease bistability

◮ loss of threshold:continuous response to mitogen →increase stochasticity of response

◮ ↑k

p,↑J11 etc. :weaken E2F1 self-control

→yeast swi4 knock-out (≈E2F1) shows increased stochasticity in cell cycle (Ubersax JA Mol Syst Biol 2006)

4.

Compare Mod. 2, level 2 with Mod. 3, level 1

→

dynamic properties of complex systems unintuitive!

5.

Only one step in model-experiment loop!

6.

Apply to GATA evolution: based on realistic data, only

relative changes in evolutionary duplication scenarios!

Possible interprations: G

1

/

S module

1.

Qualitative model - arbitrary interpretations!

2.

Modification 1 + 2 : move bistability

◮ move SN

1or SN1/SN2: insensitivity to mitogen

→E2F1/pRB↔p53/Mdm2 involved in cell-cycle arrest

◮ ↓k

p:decreasing basal transcription of E2F1

→induction of differentiation markers in squamous cancer

cell line (Wong et al. Oncogene 2005)

3.

Modification 3 : decrease bistability

◮ loss of threshold:continuous response to mitogen

→increase stochasticity of response

◮ ↑kp,↑J11 etc. :weaken E2F1 self-control

→yeast swi4 knock-out (≈E2F1) shows increased

stochasticity in cell cycle (Ubersax JA Mol Syst Biol 2006)

4.

Compare Mod. 2, level 2 with Mod. 3, level 1

→

dynamic properties of complex systems unintuitive!

5.

Only one step in model-experiment loop!

6.

Apply to GATA evolution: based on realistic data, only

relative changes in evolutionary duplication scenarios!

Possible interprations: G

1

/

S module

1.

Qualitative model - arbitrary interpretations!

2.

Modification 1 + 2 : move bistability

◮ move SN

1or SN1/SN2: insensitivity to mitogen

→E2F1/pRB↔p53/Mdm2 involved in cell-cycle arrest

◮ ↓k

p:decreasing basal transcription of E2F1

→induction of differentiation markers in squamous cancer

cell line (Wong et al. Oncogene 2005)

3.

Modification 3 : decrease bistability

◮ loss of threshold:continuous response to mitogen

→increase stochasticity of response

◮ ↑kp,↑J11 etc. :weaken E2F1 self-control

→yeast swi4 knock-out (≈E2F1) shows increased

stochasticity in cell cycle (Ubersax JA Mol Syst Biol 2006)

4.

Compare Mod. 2, level 2 with Mod. 3, level 1

→

dynamic properties of complex systems unintuitive!

5.

Only one step in model-experiment loop!

6.

Apply to GATA evolution: based on realistic data, only

relative changes in evolutionary duplication scenarios!

Possible interprations: G

1

/

S module

1.

Qualitative model - arbitrary interpretations!

2.

Modification 1 + 2 : move bistability

◮ move SN

1or SN1/SN2: insensitivity to mitogen

→E2F1/pRB↔p53/Mdm2 involved in cell-cycle arrest

◮ ↓k

p:decreasing basal transcription of E2F1

→induction of differentiation markers in squamous cancer cell line (Wong et al. Oncogene 2005)

3.

Modification 3 : decrease bistability

◮ loss of threshold:continuous response to mitogen →increase stochasticity of response

◮ ↑k

p,↑J11 etc. :weaken E2F1 self-control

→yeast swi4 knock-out (≈E2F1) shows increased stochasticity in cell cycle (Ubersax JA Mol Syst Biol 2006)

4.

Compare Mod. 2, level 2 with Mod. 3, level 1

→

dynamic properties of complex systems unintuitive!

5.

Only one step in model-experiment loop!

6.

Apply to GATA evolution: based on realistic data, only

relative changes in evolutionary duplication scenarios!

Inverse eigenvalue problems: ODE setting

Definition (IEP for Saddle-Node and Hopf bifurcations)

Denote scalars

λ

SN

=

{

0

}

or

λ

H

=

{±

ω

i

}

. With equilibrium

condition f

(x

,

α

) =

0, determine parameter values

α

such that

σ

(

_dxdf

)

⊃

λ

SN,H

.

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

(c) IEA for Saddle-Node

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

Inverse eigenvalue problems: ODE setting

◮

Hybrid solution algorithm:

◮ Lift-and-Project (LP) ◮ Quasi-Newton (QN)

◮

Least square formulations with

regularization:

LSIEP1

:

J

(

α

) =

X

i

|

λ

i

(

α

)

−

λ

di

|

2

+

µ

k

α

−

α

∗

k

_l p

,

LSIEP2

:

J

(

α

) =

k

A

(

α

)

−

Q

T

˜

(

{

λ

d_i

}

)

Q

H

k

2F

+

µ

k

α

−

α

∗

k

lp

.

Inverse eigenvalue problems: ODE setting

◮

Hybrid solution algorithm:

◮ Lift-and-Project (LP)

◮ Quasi-Newton (QN)

◮

Least square formulations with

regularization:

LSIEP1

:

J

(

α

) =

X

i

|

λ

i

(

α

)

−

λ

di

|

2

+

µ

k

α

−

α

∗

k

_l p

,

LSIEP2

:

J

(

α

) =

k

A

(

α

)

−

Q

T

˜

(

{

λ

d_i

}

)

Q

H

k

2F

+

µ

k

α

−

α

∗

k

lp

.

Inverse eigenvalue problems: ODE setting

◮

Hybrid solution algorithm:

◮ Lift-and-Project (LP)

◮ Quasi-Newton (QN)

◮

Least square formulations with

regularization

:

LSIEP1

:

J

(

α

) =

X

i

|

λ

i

(

α

)

−

λ

di

|

2

+

µ

k

α

−

α

∗

klp

,

LSIEP2

:

J

(

α

) =

k

A(

α

)

−

Q

T

˜

(

{

λ

d_i

}

)Q

H

k

2F

+

µ

k

α

−

α

∗

klp

.

Inverse eigenvalue problems: ODE setting

◮

Hybrid solution algorithm:

◮ Lift-and-Project (LP)

◮ Quasi-Newton (QN)

◮

Least square formulations with

regularization

:

LSIEP1

:

J

(

α

) =

X

i

|

λ

i

(

α

)

−

λ

di

|

2

+

µ

k

α

−

α

∗

klp

,

LSIEP2

:

J

(

α

) =

k

A(

α

)

−

Q

T

˜

(

{

λ

d_i

}

)Q

H

k

2F

+

µ

k

α

−

α

∗

klp

.

Emergence of an oscillator from bistable switch

◮ _{Time-series: the initial and identified systems}

10 20 30 40 50 0.05 0.1 0.15 0.2 0.25 0.3

Solution for initial parameters

rI rA mI mA Ic I Ac A 500 10001500200025003000 0.1 0.2 0.3 0.4 0.5

Solution for identified parameters

rI rA mI mA Ic I Ac A ◮ _{Identified reactions:}

Emergence of an oscillator from bistable switch

◮ _{Time-series: the initial and identified systems}

10 20 30 40 50 0.05 0.1 0.15 0.2 0.25 0.3

Solution for initial parameters

rI rA mI mA Ic I Ac A 500 10001500200025003000 0.1 0.2 0.3 0.4 0.5

Solution for identified parameters

rI rA mI mA Ic I Ac A ◮ _{Identified reactions:}

System tG : identified reactions tG‘v13@tDŠtG‘Ac@tDtG‘D€Ac -1®0.15 tG‘v14@tDŠtG‘Ic@tDtG‘D€Ic -1®0.046 Ac v7 A NTP v1 rA I v3 mA AA v5 v9 rA_deg v11 mA_deg v13 Ac_deg v2 rI v4 mI Ic v8 v6 v10 rI_deg v12 mI_deg v14 Ic_deg v1b v2b v1s S v2s

System tG : identified reactions tG‘v7@tDŠtG‘kin€A tG‘Ac@tD-tG‘A@tDtG‘kout€A

-1®0.9 tG‘v1@tDŠ tG‘Va tG‘A@tD2 €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ i k j j j j j j j j j tG‘Ka

€A+tG‘A@tD+tG‘Ka€A tG‘I@tD

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€tG‘Ki€A -1®0.14 y { z z z z z z z z z 2 tG‘v2@tDŠ tG‘Vi tG‘A@tD2 €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ i k j j j j j j j j

jtG‘Ka€I+tG‘A@tD+€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€tG‘Ka€I tG‘I@tD

tG‘Ki€I -1®0.25 y { z z z z z z z z z 2

tG‘v8@tDŠtG‘kin€I tG‘Ic@tD-tG‘I@tDtG‘kout€I

-1®0.42 Ac v7 A NTP v1 rA I v3 mA AA v5 v9 rA_deg v11 mA_deg v13 Ac_deg v2 rI v4 mI Ic v8 v6 v10 rI_deg v12 mI_deg v14 Ic_deg v1b v2b v1s S v2s

Conclusions and Outlook

◮

Inverse dynamical analysis

◮ Sparsity-cosntraint

→identify governing modules and parameters

◮ Highly useful in model-experiment loop

1. find crucial parameters for observed behaviour

2. propose new experiments

◮

Evolution of the bifurcation phenotype

1. duplication of auto-activator causes hypersensitivity stress 2. analyze all possible relieving mutations by

Inverse Dynamical Analysis

3. reconstruct evolutionary pathways towards complex transcription factor networks

Conclusions and Outlook

◮

Inverse dynamical analysis

◮ Sparsity-cosntraint

→identify governing modules and parameters

◮ Highly useful in model-experiment loop

1. find crucial parameters for observed behaviour

2. propose new experiments

◮

Evolution of the bifurcation phenotype

1. duplication of auto-activator causes hypersensitivity stress

2. analyze all possible relieving mutations by Inverse Dynamical Analysis

3. reconstruct evolutionary pathways towards complex

Thanks:

BIRD07

Lukas Endler, Stefan Müller Christoph Flamm, Stefanie Widder Peter Schuster, Heinz Engl Douglas Murray

Funded by: