Application of Inverse Methods to Problems from Systems Biology - - PowerPoint PPT Presentation

Application of Inverse Methods to Problems from Systems Biology Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Inverse Problems: Computational Methods an Emerging Applications UCLA Conference Center at Lake Arrowhead, 11.– 16.06.2006

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

1. What is systems biology ? 2. Forward and inverse problems in modeling 3. Three examples 4. Bifurcation analysis of gene regulation 5. Analysis of a synthetic oscillator

• 1. What is systems biology ?

2. Forward and inverse problems in modeling 3. Three examples 4. Bifurcation analysis of gene regulation 5. Analysis of a synthetic oscillator

From qualitative data to quantitative modeling Genomics, proteomics, interactomics Metabolomics, functional genomics Systems biology (quantitative biology)

time

Analysis by gel electrophoresis

Jeff Rogers, Gerald F. Joyce. RNA 7:395-404, 2001

The same section of the microarray is shown in three independent hybridizations. Marked spots refer to: (1) protein disulfide isomerase related protein P5, (2) IL-8 precursor, (3) EST AA057170, and (4) vascular endothelial growth factor.

Gene expression DNA microarray representing 8613 human genes used to study transcription in the response of human fibroblasts to serum. V.R.Iyer et al., Science 283: 83-87, 1999

A pH-modulated, self-replicating peptide

Shao Yao, Indraneel Ghosh, Reena Zutshi, Jean Chmielewski. J.Am Chem.Soc. 119:10559-10560, 1997

Time t Concentration xi (t) Stoichiometric equations

SBML – systems biology markup language

Kinetic differential equations

ODE Integration

Solution curves

A + B X 2 X Y Y + X D y x k d y x k x k y y x k x k b a k x b a k b a

3 3 2 2 3 2 2 1 1

t d d t d d t d d t d d t d d = − = − − = − = = The elements of the simulation tool MiniCellSim

SBML: Bioinformatics 19:524-531, 2003; CVODE: Computers in Physics 10:138-143, 1996

1 2 3 4 5 6 7 8 9 10 11 12 Regulatory protein or RNA Enzyme Metabolite Regulatory gene Structural gene

A model genome with 12 genes

Sketch of a genetic and metabolic network

A B C D E F G H I J K L 1

Biochemical Pathways

2 3 4 5 6 7 8 9 10

The reaction network of cellular metabolism published by Boehringer-Ingelheim.

The citric acid

• r Krebs cycle

(enlarged from previous slide).

• E. coli:

Length of the Genome 4×106 Nucleotides Number of Cell Types 1 Number of Genes 4 290 Man: Length of the Genome 3×109 Nucleotides Number of Cell Types 200 Number of Genes 30 000 - 60 000

The bacteriophage lysis/lysogeny decision circuit.

• A. Arkin, J. Ross, H.H. McAdams. Genetics 149:1633-1648, 1998.

genomic DNA mRNA

Elimination of introns through splicing AAA

The gene is a stretch of DNA which after transcription and processing gives rise to a mRNA

Sex determination in Drosophila through alternative splicing The process of protein synthesis and its regulation is now understood but the notion of the gene as a stretch of DNA has become obscure. The gene is essentially associated with the sequence of unmodified amino acids in a protein, and it is determined by the nucleotide sequence as well as the dynamics of the the process eventually leading to the m-RNA that is translated.

The difficulty defining the gene Helen Pearson, Nature 441: 399-401, 2006

Stefan Bornholdt. Less is more in modeling large genetic networks. Science 310, 449-450 (2005)

1. What is systems biology ?

• 2. Forward and inverse problems in modeling

3. Three examples 4. Bifurcation analysis of gene regulation 5. Analysis of a synthetic oscillator

General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• Time

t Concentration ( ) x t Solution curves: xi(t) Kinetic differential equations ) ; (

2

k x f x D t x + ∇ = ∂ ∂

) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d

K K

= = = Reaction diffusion equations

) , ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m , , 2 , 1 j ; ) , I , H p , p , T (

j

K K = k

The forward problem of chemical reaction kinetics (Level I)

General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• Time

t Concentration ( ) x t Solution curves: xi(t) Kinetic differential equations ) ; (

2

k x f x D t x + ∇ = ∂ ∂ ) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d K K = = = Reaction diffusion equations

) , ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T kj , , 2 , 1 ; ) , , , , ; I ( G K K =

Genome: Sequence IG

The forward problem of biochemical reaction kinetics (Level I)

The inverse problem of biochemical reaction kinetics (Level I)

Time t Concentration Data from measurements (t ); = 1, 2, ... , x j N

j

xi (t )

j

Kinetic differential equations

) ; (

2

k x f x D t x + ∇ = ∂ ∂ ) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d

K K

= = = Reaction diffusion equations General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• )

, ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T k j , , 2 , 1 ; ) , , , , ; I ( G K K

=

Genome: Sequence IG

General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) (

x

) , ( t r g x S =

• Kinetic differential equations

) ; ( f

2

k x x D t x + ∇ = ∂ ∂

) , , ( ; ) , , ( ; ) ; ( f

1 1

m n

k k k x x x k x t d x d

K K

= = = Reaction diffusion equations

) , ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T k j , , 2 , 1 ; ) , , , , ; I ( G K K =

Genome: Sequence IG

Bifurcation analysis

( , ; ) k k

i j k

kj ki

x t

( )

time

xn

xm

P

xn

xm

P P

xn xm

P

The forward problem of bifurcation analysis (Level II)

The inverse problem of bifurcation analysis (Level II)

Kinetic differential equations

) ; (

2

k x f x D t x + ∇ = ∂ ∂

) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d

K K

= = = Reaction diffusion equations General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) (

x

) , ( t r g x S =

• )

, ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T kj

, , 2 , 1 ; ) , , , , ; I ( G K K

=

Genome: Sequence IG

Bifurcation pattern

( , ; ) k k

i j k

k1 k2

P2

xn xm

P1

x

x

P

x

x

P

Stock Solution [A] = a Reaction Mixture [A],[X]

A A A A A A A A A A A A A A A A A A A X X X X X X X X X X X X

Flow rate =

r

1

R- A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

Flow rate r

Stationary concentration x

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.1 0.2 0.3 0.4 0.5

rcr,1 rcr,2

Bistability Thermodynamic branch

r

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x Steady states: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

2 4 8 ) 5 12 ( ) 8 6 ( D 216

4 2 2 3 2 2 2 2 3

= + + + − + − + = a a a r a r r α α α α α α

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Polynomial discriminant of the cubic equation: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

2 4 8 ) 5 12 ( ) 8 6 ( D 216

4 2 2 3 2 2 2 2 3

= + + + − + − + = a a a r a r r α α α α α α

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Polynomial discriminant of the cubic equation: Kinetic differential equations: D < 0 r : 3 roots , 2 are positive =

• r , r , and r

r r

1 2 3 1 2

0.4 0.6 0.2 0.0 r 0.00 0.01 0.02 0.03

• 0.5

1.0 1.5 2.0 2.5 a0

Range of hysteresis as a function of the parameters a0 and

1. What is systems biology ? 2. Forward and inverse problems in modeling

• 3. Three examples

4. Bifurcation analysis of gene regulation 5. Analysis of a synthetic oscillator

The cell division cycle

Talk by Philipp Kügler

M mitosis G1 gap 1 S synthesis G2 gap 2 M mitosis

time The division cycle of eukaryotic cells

John J. Tyson. Modeling the cell division cycle: cdc2 and cyclin insteractions. Proc.Natl.Acad.Sci. 88:7328-7332, 1991.

The budding yeast cell cycle.

Katherine C. Chen, Attila Csikasz-Nagy, Bela Gyorffy, John Val, Bela Novak, John J. Tyson. Molecular Biology of the Cell 11:369-391, 2000.

Budding yeast cell cycle model

Circadian rhythms

Talk by James Lu

PER ... period protein TIM ... timeless protein

Jean-Christophe Leloup, Albert Goldbeter. Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila. BioEssays 22:84-93, 2000.

Jean-Christophe Leloup, Albert Goldbeter. Modeling the molecular regulatory mechanism

• f circadian rhythms in Drosophila.

BioEssays 22:84-93, 2000.

Jean-Christophe Leloup, Albert Goldbeter. Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila. BioEssays 22:84-93, 2000.

Jean-Christophe Leloup, Albert Goldbeter. Modeling the molecular regulatory mechanism

• f circadian rhythms in Drosophila.

BioEssays 22:84-93, 2000.

Jean-Christophe Leloup, Albert Goldbeter. Modeling the molecular regulatory mechanism

• f circadian rhythms in Drosophila.

BioEssays 22:84-93, 2000.

The immune synapse

Joined work with Gerhard Schütz (Linz), Alois Sonnleitner (Linz) and Hannes Stockinger (Wien) and the RICAM Group of Heinz Engl.

The immune synapse assembly model

Sung-Joo E. Lee, Yuko Hori, Jay T. Groves, Michael L.Dustin, Arup K. Chakraborty. Trends in Immunology 23:492-499, 2002

Immune synapse Anergy T-cell activation Amnon Altman, Noah Isakow, Gottfried Baier. Immunology Today 21:567-573, 2000

Ca2+ transport in the T-cell

1. What is systems biology ? 2. Forward and inverse problems in modeling 3. Three examples

• 4. Bifurcation analysis of gene regulation

5. Analysis of a synthetic oscillator

Basal transcription and active state of a gene

Inactive or silent state of a gene

Cross-regulation of two genes

2 , 1 , ) ( : Repression ) ( : Activation

n n n

= + = + = j i p K K p F p K p p F

j j i j j j i

Gene regulatory binding functions

2 P 2 2 P 2 2 1 P 2 1 P 1 1 2 Q 2 1 2 Q 2 2 1 Q 1 2 1 Q 1 1

) ( ) ( p d q k dt dp p d q k dt dp q d p F k dt dq q d p F k dt dq − = − = − = − =

2 2 1 1 2 2 1 1 2 1

] P [ , ] P [ , ] Q [ , ] Q [ . const ] G [ ] G [ p p q q g = = = = = = = 2 , 1 , ) ( : Repression ) ( : Activation

n n n

= + = + = j i p K K p F p K p p F

j j j j i j j j j i

P 2 Q 2 P 2 Q 2 2 P 1 Q 1 P 1 Q 1 1 1 2 2 2 1 2 2 1 1 1

, ) ( , )) ( ( : points Stationary d d k k d d k k p F p p F F p = = = = − ϑ ϑ ϑ ϑ ϑ

Qualitative analysis of cross-regulation of two genes: Stationary points

act-act act-rep rep-rep

n n n

p K K p p K p p

1 1 1 2 2 1 1 1 2 2

• r

+ = + = ϑ ϑ

Stationary protein concentrations for Hill coefficient n

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ = =

P P P P Q Q Q Q Q Q K D K D j i ij

d d k k p F k p F k p F k p F k d d P P Q Q x x a

2 1 2 1 2 2 2 1 2 2 2 1 1 1 1 1 2 1

A &

K K D D D K D K K D

P Q P Q P P Q P P Q ⋅ − ⋅ = ⋅ = ⋅

K D

Q Q hence and

Special case:

) ( and ) (

1 2 2 2 1 1

p F F p F F = =

( )

1 2 2 1 1 2 2 1 2 1

, p F p F p F p F p p ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ∂ ∂ − = Γ

Qualitative analysis of cross-regulation of two genes: Jacobian matrix

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

( )

1 2 2 1 1 2 2 1 2 1

, p F p F p F p F p p ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ∂ ∂ − = Γ

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

Eigenvalues of the Jacobian of the cross-regulatory two gene system

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

Eigenvalues of the Jacobian of the cross-regulatory two gene system

2 P 2 P 1 Q 2 Q 1 P 2 P 1 P 2 Q 2 P 1 Q 2 P 2 Q 1 P 1 Q 1 Q 2 Q 1 Hopf P 2 P 1 Q 2 Q 1 trans

) ( ) )( )( )( )( )( ( d d d d d d d d d d d d d d d d D d d d d D + + + + + + + + + = − =

Regulatory dynamics at D 0 , act.-act., n=2

Auxiliary parameter s K1,2 = 1,2 / s kQ

1,2 = 1,2 ( + s)

Regulatory dynamics at D 0 , act.-rep., n=3

Auxiliary parameter s K1,2 = 1,2 / s kQ

1,2 = 1,2 ( + s)

Regulatory dynamics at D < DHopf , act.-repr., n=3

Regulatory dynamics at D > DHopf , act.-repr., n=3

Regulatory dynamics at D 0 , rep.-rep., n=2

Auxiliary parameter s K1,2 = 1,2 / s kQ

1,2 = 1,2 ( + s)

Hill coefficient: n Act.-Act. Act.-Rep. Rep.-Rep. 1 S , E S S 2 E , B(E,P) S S , B(P1,P2) 3 E , B(E,P) S , O S , B(P1,P2) 4 E , B(E,P) S , O S , B(P1,P2)

1 1 ; 2 , 1 , ) ( : te Intermedia ) ( : Repression ) ( : Activation

n 2 3 2 1 m n n n

− ≤ ≤ = + + + + = + = + = n m j i p p p p p F p K K p F p K p p F

j j j j j i j j i j j j i

K κ κ κ

Regulatory dynamics, int.-act., m=2, n=4

Auxiliary parameter s K1,2 = 1,2 / s kQ

1,2 = 1,2 ( + s)

Auxiliary parameter s K1,2 = 1,2 / s kQ

1,2 = 1,2 ( + s)

Regulatory dynamics, rep.-int., m=2, n=4

1. What is systems biology ? 2. Forward and inverse problems in modeling 3. Three examples 4. Bifurcation analysis of gene regulation

• 5. Analysis of a synthetic oscillator

An example analyzed and simulated by MiniCellSim

The repressilator: M.B. Ellowitz, S. Leibler. A synthetic oscillatory network of transcriptional

• regulators. Nature 403:335-338, 2002

Stable stationary state Limit cycle oscillations Fading oscillations caused by a stable heteroclinic orbit Hopf bifurcation Bifurcation to May-Leonhard system Increasing inhibitor strength

1e+07 2e+07 3e+07 4e+07 5e+07 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Proteins

1e+07 2e+07 3e+07 4e+07 5e+07 0.02 0.04 0.06 0.08 1e+07 2e+07 3e+07 4e+07 5e+07 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mRNAs

1e+07 2e+07 3e+07 4e+07 5e+07 0.05 0.1 0.15 0.2 0.25 0.3

The repressilator limit cycle

2e+08 4e+08 6e+08 8e+08 0.2 0.4 0.6 0.8 1

Proteins

2e+08 4e+08 6e+08 8e+08 0.05 0.1 0.15 0.2 0.25 0.3 2e+08 4e+08 6e+08 8e+08 0.2 0.4 0.6 0.8 1

mRNAs

2e+08 4e+08 6e+08 8e+08 0.05 0.1 0.15 0.2 0.25 0.3

The repressilator heteroclinic orbit

1 100 10000 1e+06 1e+08 0.2 0.4 0.6 0.8 1

Proteins

1 100 10000 1e+06 1e+08 0.05 0.1 0.15 0.2 0.25 0.3 1 100 10000 1e+06 1e+08 0.2 0.4 0.6 0.8 1

mRNAs

1 100 10000 1e+06 1e+08 0.05 0.1 0.15 0.2 0.25 0.3

The repressilator heteroclinic orbit (logarithmic time scale)

P1 P2 P3

start start

The repressilator limit cycle

P1 P2 P2 P2 P3

Stable heteroclinic orbit Unstable heteroclinic orbit

1 1 2 2 2<0 2>0 2=0

Bifurcation from limit cycle to stable heteroclinic orbit at

The repressilator heteroclinic orbit

The bacterial cell as an example for the simplest form of autonomous life The human body: 1014 cells = 1013 eukaryotic cells + 91013 bacterial (prokaryotic) cells, and 200 eukaryotic cell types The spatial structure of the bacterium Escherichia coli

Acknowledgement of support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 09942, 10578, 11065, 13093 13887, and 14898 Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Contracts No. 98-0189, 12835 (NEST) Austrian Genome Research Program – GEN-AU Siemens AG, Austria Universität Wien and the Santa Fe Institute

Universität Wien

Coworkers

Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE Paul E. Phillipson, University of Colorado at Boulder, CO Heinz Engl, Philipp Kügler, James Lu, Stefan Müller, Josef Schicho, RICAM Linz, AT Jord Nagel, Kees Pleij, Universiteit Leiden, NL Walter Fontana, Harvard Medical School, MA Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE Ivo L.Hofacker, Christoph Flamm, Andreas Svrček-Seiler, Universität Wien, AT Kurt Grünberger, Michael Kospach , Andreas Wernitznig, Stefanie Widder, Stefan Wuchty, Universität Wien, AT Jan Cupal, Stefan Bernhart, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Hakim Tafer, Thomas Taylor, Universität Wien, AT

Universität Wien

Web-Page for further information: http://www.tbi.univie.ac.at/~pks