inverse methods in systems biology · inverse methods in systems biology ... inverse bifurcation:...

24th TBI Winterseminar, Bled 2009

Inverse Methods in Systems Biology

James Lu

Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Austrian Academy of Sciences

James Lu Inverse Methods in Systems Biology

Systems Biology

◮ Cells are complex systems


Systems Biology


◮ In the face of continuously changing environments and itsstate, cells need to respond appropriately:


Systems Biology


◮ In the face of continuously changing environments and itsstate, cells need to respond appropriately:

◮ inputs: nutrients, repellants, heat shock, DNA damage, ...

◮ responses: movement, growth, protein production, death


Systems Biology

◮ Complex networks of genes and pathways are involved inmediating the various inputs to the appropriate responses

Signal transduction network: Hanahan and Weinberg, Cell (2000)


Forward Problems

◮ Consider ODE models: y(t) = f (y(t),q)


Forward Problems

◮ Consider ODE models: y(t) = f (y(t),q)

◮ The Forward analysis may include:◮ numerical integration for a set of given parameters◮ sensitivity analysis◮ bifurcation analysis

reactions→ dydt = f (y,q)→ {y(t), bifurcations}

5000 10000 15000 20000

0.2

0.4

0.6

0.8

1

1.2

Signal S = 0.095

A

5000 10000 15000 20000

0.2

0.4

0.6

0.8

1

1.2

Signal S = 1

A

5000 10000 15000 20000

0.2

0.4

0.6

0.8

1

1.2

Signal S = 0.01

A

5000 10000 15000 20000

0.2

0.4

0.6

0.8

1

1.2

Signal S = 0.087

A

0.05 0.06 0.07 0.08 0.09 0.1 0.110

0.2

0.4

0.6

0.8

1

1.2

H

LP

LP

Continuation parameter tG‘S

Sta

te c

ompo

nent

tG‘A

[t]

LPC

LPC


Inverse Problems

◮ In Inverse Problems, one looks for the causes of observedand desired effects

reactions← dydt = f (y,q)← {y(t), bifurcations}


Inverse Problems



◮ Parameter identification from time-course data


Inverse Problems



◮ Parameter identification from time-course data◮ Inverse bifurcation: infer model mechanisms that can

achieve the desired bifurcation behaviors

◮ maximize the parameter domain for oscillations◮ place bifurcation points at the desired locations


Inverse Problems






◮ Inverse problems are typically ill-posed :

◮ solutions may not exist or unique◮ the solution may not depend on the data in a continuous

manner


Inverse Problems






◮ Inverse problems are typically ill-posed :

◮ solutions may not exist or unique◮ the solution may not depend on the data in a continuous

manner

◮ To numerically tackle inverse problems, regularizationstrategies are needed


Inverse Problems

◮ Forward operator equation:

F(q) = y


Inverse Problems


F(q) = y

◮ Inverse problem: determining q from y


Inverse Problems


F(q) = y

◮ Inverse problem: determining q from y◮ Typically ill-posed (in the sense of Hadamard):

◮ non-uniqueness;◮ instability of inversion


Inverse Problems


F(q) = y



◮ Variational regularization: add penalty term

minq‖F(q)− y‖+ µR(q)


Inverse Problems


F(q) = y





◮ While stabilizing ill-posed problems, regularization bringsbias to the solution


Inverse Problems


F(q) = y





◮ While stabilizing ill-posed problems, regularization bringsbias to the solution

◮ For biological problems, usually want to find solutions thatare sparse, i.e., having as few non-zero entries aspossible: Ockam’s razor


Sparsity-Promoting Regularization

◮ As the regularization term, consider smoothed functionalsR

n→ R: lp,ε(q) = ∑i(q2i + ε)p/2




n→ R: lp,ε(q) = ∑i(q2i + ε)p/2

◮ Convex only within the box {q : |qi|<√

ε, 0 < i≤ n}

−0.2 −0.1 0 0.1 0.2 0.30

0.02

0.04

0.06

0.08

0.1

x

l 2(x)

−0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

x

l 0.1,

1e−

4(x)




n→ R: lp,ε(q) = ∑i(q2i + ε)p/2

◮ Convex only within the box {q : |qi|<√

ε, 0 < i≤ n}◮ Recent applications of sparse solutions using non-convex

penalty:◮ Exact reconstruction of sparse signals via nonconvex

minimization, R. Chartrand (2007)◮ Compressive sensing using l1 re-weighting, E. Candes, S.

P. Boyd, M. Wakin et al. (2007)◮ Log-det heuristic for matrix rank minimization with

applications to Hankel and Euclidean distance matrices, M.Fazel, H. Hindi and S. P Boyd (2003)


Inverse Bifurcation: the G1/S module of cell cycle

M. Swat, A. Kel, H. Herzel, Bioinformatics (2004)



0 0.5 1 1.5 2 2.5 3 3.5 4

0

2

4

6

8

10

12

14

SN1

SN2

TC

G0/G

1 phase

Fm

(x10−3)

E2F

1

S phase

(CyclinD/cdk4,6 activation threshold)




0 0.5 1 1.5 2 2.5 3 3.5 4

0

2

4

6

8

10

12

14

SN1

SN2

TC

G0/G

1 phase

Fm

(x10−3)

E2F

1

S phase

(CyclinD/cdk4,6 activation threshold)


◮ Which of the interactions play important roles in controllingthe geometry of the bifurcation diagram?



◮ Map various bifurcation phenotypes to parameter sets



◮ Map various bifurcation phenotypes to parameter sets◮ Consider the following 3 modes of transformations of the

nominal bifurcation diagram:



◮ Map various bifurcation phenotypes to parameter sets◮ Consider the following 3 modes of transformations of the

nominal bifurcation diagram:

◮ Inverse bifurcation problem: from conditions on bifurcationdiagrams, infer the governing mechanisms


Inverse Bifurcation: effect of sparsity-promiting penalty

0 2 4 6 80

2

4

6

8

SN

SN

SN

Fm

E2F

1

Mammalian G1/S transition

−15 −10 −5 0

k43k67k76k23k28k89k98

aJ15J18J68J13J63

Km9phipRB

phiCycDiphiAp1

phipRBpphipRBppphiCycEi

phiCycEaJ65J62J61

Km1k61k16k25

Km4J11

k3k1

k34J12

Km2phiCycDa

phiE2F1k2kp

% change

Par

amet

er li

st

(a) l0.1,10−4 regularization


Inverse Bifurcation: effect of sparsity-promiting penalty

0 2 4 6 80

2

4

6

8

SN

SN

SN

Fm

E2F

1


−15 −10 −5 0

k43k67k76k23k28k89k98

aJ15J18J68J13J63

Km9phipRB

phiCycDiphiAp1

phipRBpphipRBppphiCycEi

phiCycEaJ65J62J61

Km1k61k16k25

Km4J11k3k1

k34J12

Km2phiCycDa

phiE2F1k2kp

% change

Par

amet

er li

st

(b) l0.1,10−4 regularization

0 2 4 6 80

2

4

6

8

SN

SN

SN

Fm

E2F

1


−4 −3 −2 −1 0 1 2 3

k43k67k76k28k89k98

aJ15J18J68J13

Km9phipRB

phipRBppphiCycEi

phiCycEaJ63J61k23J65

Km1J62

phipRBpk61k16k25

Km4phiCycDi

k1J11k2

k34phiCycDa

k3J12

phiAp1phiE2F1

Km2kp

% change

Par

amet

er li

st

(c) l2 regularization


Inverse Bifurcation: identified module

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

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

Elongating SN1 nose kp ↓ 14.3% k34 ↑ 31.7% φAP-1 ↓ 20.9%Km2 ↑ 6.4% φE2F1 ↑ 7.3%

Moving SN1,2 to 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

ddt

[pRB] = k1[E2F1]

Km1+[E2F1]

J11

J11+[pRB]

J61

J61+[pRBp]− k16[pRB][CycDa]+ k61[pRBp]−φpRB[pRB],

ddt

[E2F1] = kp + k2a2 +[E2F1]2

Km22 +[E2F1]2

J12

J12+[pRB]

J62

J62+[pRBp]− φE2F1[E2F1]

ddt

[CycDi] = −k34[CycDi][CycDa]

Km4+[CycDa]+ · · ·

ddt

[CycDa] = k34[CycDi][CycDa]

Km4+[CycDa]+ · · ·


Application: Circadian Rhythm Model

◮ Circadian rhythm underlies the 24 hr activity cycle of manyorganisms




◮ Endogenous oscillator entrainable by 24 hr light-dark cycles




◮ Endogenous oscillator entrainable by 24 hr light-dark cycles

◮ Circadian model for Arabidopsis thaliana proposed by Locke etal, Molecular Systems Biology (2005)



ODE system



◮ Simulation of model with nominal parameters under 12 hrlight-dark cycle

cLm

cYm

one23

20 40 60 80 100

0.2

0.4

0.6

0.8

1.0

1.2



◮ Evidence suggests GIGANTEA could be gene Y

Phase misfit with experimental data

J. CW Locke et al., Molecular Systems Biology (2005)



◮ Evidence suggests GIGANTEA could be gene Y

Phase misfit with experimental data

J. CW Locke et al., Molecular Systems Biology (2005)

◮ Inverse problem: vary parameters so that Y mRNA peaksat ZT7 rather than ZT11



◮ The peak time t for the level of Y mRNA, satisfies

y(t) = f (y,q)fYmRNA(y(t),q) = 0





◮ Goal: obtain a solution that peaks at time t∗ by varying asfew parameters from the published values, q∗, as possible





◮ Goal: obtain a solution that peaks at time t∗ by varying asfew parameters from the published values, q∗, as possible

◮ Minimization of the objective:

‖t(q)− t∗‖2 + µ lp,ε (q−q∗)→minq



◮ Y mRNA solution profile for the original parameter set

0 10 20 30 40 50 60 70

0.0

0.1

0.2

0.3

0.4

Time HhrL

Y mRNA Level



◮ Y mRNA solution profile for the identified parameter set

0 10 20 30 40 50 60 70

0.0

0.1

0.2

0.3

0.4

Time HhrL

Y mRNA Level



◮ Y mRNA solution profile for the identified parameter set◮ 1 out of 54 parameters is identified

0 10 20 30 40 50 60 70

0.0

0.1

0.2

0.3

0.4

Time HhrL

Y mRNA Level



◮ Identified parameter: g6, Hill-constant in repression ofgene Y by LHY


Inverse Eigenvalue Problems

◮ We have seen inverse bifurcation problems inferring theunderlying mechanisms controlling the qualitative aspectsof dynamics




◮ One may also wish to probe the possibility of differentclasses of behaviors; e.g., transition of a bistable switch toan oscillator




◮ One may also wish to probe the possibility of differentclasses of behaviors; e.g., transition of a bistable switch toan oscillator

◮ Infer mechanisms governing the spectrum of thedynamical system: inverse eigenvalue problems



◮ For the dynamical system y(t) = f (y,q), many bifurcationsof equilibrium correspond to various conditions oneigenvalues of df (y,q)

dy



◮ For the dynamical system y(t) = f (y,q), many bifurcationsof equilibrium correspond to various conditions oneigenvalues of df (y,q)

dy

◮ Inverse eigenvalue problems: identify the possible modelmechanisms bringing about the desired change in thespectrum

(d) Hopf (e) BT (f) Fold-Hopf (g) Hopf-Hopf



◮ Hybrid solution algorithm:◮ Lift-and-Project (LP)



◮ Hybrid solution algorithm:◮ Lift-and-Project (LP)◮ Quasi-Newton (QN)



◮ Hybrid solution algorithm:◮ Lift-and-Project (LP)◮ Quasi-Newton (QN)

◮ Least square formulations with sparsity regularization:

LP : J(q) = ‖A(q)−Aproj‖2F + µ lp,ε(q−q∗)QN : J(q) = ∑

i

|λi(q)−λ di |2 + µ lp,ε(q−q∗)


Transition of a Bistable Switch to an Oscillator

◮ Consider a model for GATA transcription factors



◮ Consider a model for GATA transcription factors◮ Scenario: duplication of a gene A→ A1, A2




◮ Subsequent loss of the activating domain




◮ Subsequent loss of the activating domain

◮ Can oscillations emerge via a few additional mutations?



◮ Evolutionary scenario




2000 4000 6000 8000 10000

0.25

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0.75

1

1.25

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1.75

Solution for initial parameters

rMrA2rA1mMmA2mA1MA2cA2A1cA1

2000 4000 6000 8000 10000

0.5

1

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2

2.5

3

3.5

Solution for identified parameters





2000 4000 6000 8000 10000

0.25

0.5

0.75

1

1.25

1.5

1.75

Solution for initial parameters


2000 4000 6000 8000 10000

0.5

1

1.5

2

2.5

3

3.5

Solution for identified parameters


◮ Identified reactions:


Conclusions

◮ Many inverse problems arise from the modelling andanalysis of biological systems


Conclusions


◮ inverse bifurcation problems: reverse engineering,matching model to data


Conclusions



◮ inverse eigenvalue problems: model-building, exploration ofpossible behaviors


Conclusions



◮ inverse eigenvalue problems: model-building, exploration ofpossible behaviors

◮ Sparsity-promoting regularization can be effective indrawing useful insights from biological models


Acknowledgements

CollaboratorsRICAM: Heinz Engl, Philipp Kuegler, Stefan Mueller, Clemens ZarzerUniversity of Vienna: Peter Schuster, Christoph Flamm, Rainer MachneKeio University : Douglas Murray

Funding provided by Vienna Science and Technology Fund (WWTF) isgratefully acknowledged

Project: MA05, MA07


inverse methods in systems biology · inverse methods in systems biology ... inverse bifurcation:...

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