Dynamical Systems in Gene Regulation Peter Schuster Institut fr - - PowerPoint PPT Presentation

Dynamical Systems in Gene Regulation

Peter Schuster

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

Jozef Stefan Institute Ljubljana, 10.02.2006

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

1. Forward and inverse problems in reaction kinetics 2. Reverse engineering – A simple example 3. A glimpse of regulation kinetics 4. Genetic and metabolic networks – MiniCellSim 5. How do model metabolisms evolve?

• 1. Forward and inverse problems in reaction kinetics

2. Reverse engineering – A simple example 3. A glimpse of regulation kinetics 4. Genetic and metabolic networks – MiniCellSim 5. How do model metabolisms evolve?

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. Forward and inverse problems in reaction kinetics

• 2. Reverse engineering – A simple example

3. A glimpse of regulation kinetics 4. Genetic and metabolic networks – MiniCellSim 5. How do model metabolisms evolve?

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. Forward and inverse problems in reaction kinetics 2. Reverse engineering – A simple example

• 3. A glimpse of regulation kinetics

4. Genetic and metabolic networks – MiniCellSim 5. How do model metabolisms evolve?

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

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 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

( )

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 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

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

1. Forward and inverse problems in reaction kinetics 2. Reverse engineering – A simple example 3. A glimpse of regulation kinetics

• 4. Genetic and metabolic networks – MiniCellSim

5. How do model metabolisms evolve?

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

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

Genetic regulatory network Metabolic network

Proposal of a new name: Genetic and metabolic network

Time t Concentration xi (t)

Sequences

Vienna RNA Package

Structures and kinetic parameters Stoichiometric equations

SBML – systems biology markup language

Kinetic differential equations

ODE Integration by means of CVODE

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

The model regulatory gene in MiniCellSim

The model structural gene in MiniCellSim

1. Forward and inverse problems in reaction kinetics 2. Reverse engineering – A simple example 3. A glimpse of regulation kinetics 4. Genetic and metabolic networks – MiniCellSim

• 5. How do model metabolisms evolve?

Evolutionary time: 0000 Number of genes 12 : + 06 structural 06 regulatory Number of interactions 15 : + + 04 inhibitory + 10 activating 1 self-activating

A genabolic network formed from a genotype of n = 200 nucleotides

100 1000 10000 1e+05 5 10 15 20 TF00 TF01 TF02 TF03 SP04 TF05 SP06 SP07 SP08 SP09 TF10 SP11

Evolutionary time 0000 , initial network : Intracellular time Stationary state Intracellular time scale Evolutionary time scale [generations]: 0000 initial network

Evolution of a genabolic network:

Initial genome: random sequence of length n = 200, AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit

Number of genes: total / structural genes regulatory genes

Evolution of a genabolic network:

Initial genome: random sequence of length n = 200, AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit Recorded events: (i) Loss of a gene through corruption of the start signal “TA” (analogue of the “TATA Box”), (ii) creation of a gene, (iii) change in the edges through mutation-induced changes in the affinities of translation products to the binding sites, and (iv) change in the class of genes (tf sp).

Statistics of one thousand generations Total number of genes: 11.67 2.69 Regulatory genes: 5.97 2.22 Structural genes: 5.70 2.17

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