Dynamical Systems in Problems of Gene Regulation Peter Schuster - - PowerPoint PPT Presentation

Dynamical Systems in Problems of Gene Regulation

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA

CAS-MPG Partner Institute for Computational Biology Shanghai, 26.10.2007

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

1. The problems of quantitative biology 2. Forward and inverse problems in reaction kinetics 3. Regulation kinetics and bifurcation analysis 4. Reverse engineering of dynamical systems 5. How to upscale from small models to cells?

• 1. The problems of quantitative biology

2. Forward and inverse problems in reaction kinetics 3. Regulation kinetics and bifurcation analysis 4. Reverse engineering of dynamical systems 5. How to upscale from small models to cells?

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).

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

1. The problems of quantitative biology

• 2. Forward and inverse problems in reaction kinetics

3. Regulation kinetics and bifurcation analysis 4. Reverse engineering of dynamical systems 5. How to upscale from small models to cells?

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

1. The problems of quantitative biology 2. Forward and inverse problems in reaction kinetics

• 3. Regulation kinetics and bifurcation analysis

4. Reverse engineering of dynamical systems 5. How to upscale from small models to cells?

Active states of gene regulation

Promotor

Repressor

RNA polymerase State : inactive state

III

Promotor

Activator Repressor

RNA polymerase State : inactive state

III

Activator binding site

Inactive states of gene regulation

synthesis degradation 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 i 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

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

P P P P Q Q Q Q Q Q j i ij

d d k k p F k p F k p F k p F k d d x x a

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

A &

: regulation Cross

2 2 1 1

= ∂ ∂ = ∂ ∂ p F p F

K D K D P P P P Q Q Q Q

P P Q Q d k d k p F k d p F k d = − − − − ∂ ∂ − − ∂ ∂ − − = ε ε ε ε ε

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

I

• A

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

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

( )( ) ( )( ) ( )( )( )( )

P Q P Q

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

= ∂ ∂ ∂ ∂ − − − − − − − − − = = − − − − ∂ ∂ − ∂ ∂ − − − − − = ⋅ − ⋅ p F p F k k k k d d d d d d k p F k k p F k d d

P P Q Q P Q P Q P Q P Q P Q P Q K K D D

ε ε ε ε ε ε ε ε

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

) ε ( ) ε ( ) ε ( ) ε ( x F x F k k k k D D d d d d ∂ ∂ ∂ ∂ − = = + + + + +

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

) ε ( ) ε ( ) ε ( ) ε ( x F x F k k k k D D d d d d ∂ ∂ ∂ ∂ − = = + + + + +

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 OneD

) ( ) )( )( )( )( )( ( 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

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

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

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

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

( )( ) ( )( ) ( )( )

ε ε ε ε ε ε − − − − ∂ ∂ − − − − − ∂ ∂ − ∂ ∂ − − − − − = ⋅ − ⋅

P Q Q P P Q Q P Q P P Q k k d d

d d p F k k d d p F k k p F k k d d

3 3 2 3 3 3 2 2 1 2 2 2 3 1 1 1 1 1

P Q P Q

2 3 1 2 3 1 3 2 1 3 2 1

p F p F p F k k k k k k D

P P P Q Q Q

∂ ∂ ∂ ∂ ∂ ∂ − =

Upscaling to more genes: n = 3

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

) ε ( ) ε ( ) ε ( ) ε (

P P 1 Q Q 1

= + + + + + D d d d d

n n

K K

1 1 2 1 2 1 2 1 −

∂ ∂ ∂ ∂ ∂ ∂ − =

n n n P n P P Q n Q Q

p F p F p F k k k k k k D K K K

Upscaling to n genes with cyclic symmetry

1. The problems of quantitative biology 2. Forward and inverse problems in reaction kinetics 3. Regulation kinetics and bifurcation analysis

• 4. Reverse engineering of dynamical systems

5. How to upscale from small models to cells?

( ) ( ) ( ) ( ) ( ) { }

s s s i s i s i m m n

p p p p p p p p p x x x p x f x ∩ Σ ≡ Σ ⊕ = × ∈ = Σ ⊂ ∈ = = = ; P P P ; P P , manifold n bifurcatio P ; , , ; , , ; ;

1 1

K K K & R

( ) ( )

( )

( )

• perator

forward , ) ( , ) ( K

s i p s i

p p p F p F p F

s

Σ ⊥

= ≡ π

( )

i i i p p

p F c p p p p p F p J

s s

) ( and

• subject t

) ( min ) ( min

upp low

≤ ≤ ≤ − =

... formulation of the inverse problem

• J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Application to simple gene systems.

AMB Algorithms for Molecular Biology 1, no.11, 2006.

The bifurcation manifold

Defininition of the forward operator F(p)

Iterative solution for min J(p)

3 , 2 , 1 = k

Switch or oscillatory behavior in Escherichia coli T.S. Gardner, C.R. Cantor, J.J. Collins. Construction of a genetic toggle switch in Escherichia coli. Nature 403:339-342, 2000. M.R. Atkinson, M.A. Savageau, T.J. Myers, A.J. Ninfa. Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli. Cell 113:597-607, 2003.

Inverse bifurcation analysis of switch or oscillatory behavior in Escherichia coli

• J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Application to simple

gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

δ δ β β α α = = = =

i i i i

h h , , ,

Inverse bifurcation analysis of the repressilator model

• S. Müller, J. Hofbauer, L. Endler, C. Flamm, S. Widder, P. Schuster. A generalized

model of the repressilator. J. Math. Biol. 53:905-937, 2006.

Inverse bifurcation analysis of the repressilator model

• J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Application to simple

gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

[ ] [ ] [ ] [ ]

pRB pRB ] E2F1 [ E2F1 pRB

pRB 11 11 1 1

φ − + + = J J K k dt d

m

[ ] [ ] [ ] [ ]

E2F1 pRB ] E2F1 [ E2F1 E2F1

E2F1 12 12 2 2 2 2 2 1

φ − + + + + = J J K a k k dt d

m P

[ ] [ ] [ ] [ ]

AP1 pRB' ] p [ E2F1 AP1

AP1 11 65 15 15 25

φ − + + + = J J RB J J k F dt d

m

A simple dynamical cell cycle model J.J. Tyson, A. Csikasz-Nagy, B. Novak. The dynamics of cell cycle regulation. Bioessays 24:1095-1109, 2002

A simple dynamical cell cycle model J.J. Tyson, A. Csikasz-Nagy, B. Novak. The dynamics of cell cycle regulation. Bioessays 24:1095-1109, 2002

Inverse bifurcation analysis of a dynamical cell cycle model

• J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Application to simple

gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

1. The problems of quantitative biology 2. Forward and inverse problems in reaction kinetics 3. Regulation kinetics and bifurcation analysis 4. Reverse engineering of dynamical systems

• 5. How to upscale from small models to cells?

Suitable systems for upscaling:

• 1. Linear systems via large eigenvalue problems
• 2. Cascades
• 3. Cyclic systems in case of high symmetry
• 4. Sufficiently simple networks ???

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: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften 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, 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