dynamical systems in problems of gene regulation - tbipks/presentation/shanghai-07_lt1.pdf ·...

Dynamical Systems in Problems of Gene Regulation

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austriaand

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 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 or 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 + 9 1013 bacterial (prokaryotic) cells,

and 200 eukaryotic cell types

The spatial structure of the bacterium Escherichia coli

General conditionsInitial conditions

: T , p , pH , I , ...:

... ... S , u

Boundary conditionsboundary normal unit vector

Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =

Timet

Con

cent

ratio

n

( )x tSolution curves: xi(t)

Kinetic differential equations

);(2 kxfxDtx

+∇=∂∂

),,(;),,(;);( 11 mn kkkxxxkxftdxd

KK ===

Reaction diffusion equations

),(ˆ trgxuux S =∇⋅=

∂∂

Parameter setm,,2,1j;),I,Hp,p,T(j KK =k

The forward problem of chemical reaction kinetics (Level I)


: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =

Timet

Con

cent

ratio

n

( )x tSolution curves: xi(t)


);(2 kxfxDtx

+∇=∂∂


KK ===



∂∂

Parameter setmjIHppTkj ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G

The forward problem of biochemical reaction kinetics (Level I)

The inverse problem of biochemical reaction kinetics (Level I) Time

tCo

ncen

tratio

n

Data from measurements

(t ); = 1, 2, ... , x j Nj

xi (t )j


);(2 kxfxDtx

+∇=∂∂


KK ===



: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =


∂∂

Parameter setmjIHppTk j ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G


: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =


);(f2 kxxDtx

+∇=∂∂

),,(;),,(;);(f 11 mn kkkxxxkxtdxd

KK ===



∂∂

Parameter setmjIHppTk j ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G

Bifurcation analysis

( , ; )k ki j k

kj

ki

x t( )

time xn

xm

P

xn

xm

P

Pxn

xm

P

The forward problem of bifurcation analysis (Level II)

The inverse problem of bifurcationanalysis (Level II)


);(2 kxfxDtx

+∇=∂∂


KK ===



: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =


∂∂

Parameter setmjIHppTkj ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G

Bifurcation pattern

( , ; )k ki j k

k1

k2

P2xn

xm

P1

x

x

P

x

x

P

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

=

+=

+=

ji

pKKpF

pKp

pF

jji

j

jji

Gene regulatory binding functions

2P22

P2

2

1P21

P1

1

2Q212

Q2

2

1Q121

Q1

1

)(

)(

pdqkdt

dp

pdqkdtdp

qdpFkdt

dq

qdpFkdtdq

−=

−=

−=

−=

2211

2211

021

]P[,]P[,]Q[,]Q[

.const]G[]G[

ppqq

g

=======

2,1,

)(:Repression

)(:Activation

n

n

n

=

+=

+=

ji

pKKpF

pKp

pF

jji

j

jji

P2

Q2

P2

Q2

2P1

Q1

P1

Q1

1

1222122111

,

)(,0))((:pointsStationary

ddkk

ddkk

pFppFFp

==

==−

ϑϑ

ϑϑϑ

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

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

∂∂

∂∂

∂∂

∂∂

−−

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂

==

P

P

P

P

QQ

QQ

Q

Q

j

iij

dd

kk

pFk

pFk

pFk

pFk

dd

xxa

2

1

2

1

2

22

1

22

2

11

1

11

2

1

00

00

00

A&

0:regulation Cross2

2

1

1 =∂∂

=∂∂

pF

pF

KD

KD

PP

PP

QQ

QQ

PPQQ

dkdk

pFkd

pFkd

=

−−−−∂∂

−−

∂∂

−−

=

εε

ε

ε

ε

22

11

1

222

2

111

0000

00

00

I-A

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

KKDDDK

DKKD PQPQPP

QPPQ ⋅−⋅=⋅=⋅ KD QQhenceand

( )( )

( )( )

( )( )( )( ) 0

PQPQ

1

2

2

121212211

2221

22

12

1111

=∂∂

∂∂

−−−−−−−−−=

=−−−−

∂∂

−

∂∂

−−−−−=⋅−⋅

pF

pFkkkkdddd

ddkpFk

kpFkdd

PPQQPQPQ

PQPQ

PQPQ

KKDD

εεεε

εε

εε

1

2

2

1P2

P1

Q2

Q1

P2

P1

Q2

Q1 0)ε()ε()ε()ε(

xF

xFkkkkD

Ddddd

∂∂

∂∂

−=

=+++++

1

2

2

1P2

P1

Q2

Q1

P2

P1

Q2

Q1 0)ε()ε()ε()ε(

xF

xFkkkkD

Ddddd

∂∂

∂∂

−=

=+++++

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

2P2

P1

Q2

Q1

P2

P1

P2

Q2

P1

Q2

P2

Q1

P1

Q1

Q2

Q1

Hopf

P2

P1

Q2

Q1OneD

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

ddddddddddddddddD

ddddD

+++++++++

=

−=

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)

11;2,1,

)(:teIntermedia

)(:Repression

)(:Activation

n2321

m

n

n

n

−≤≤=

++++=

+=

+=

nmji

pppp

pF

pKKpF

pKp

pF

jjj

jji

jji

j

jji

Kκκκ

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

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

( )( )

( )( )

( )( )εε

εε

εε

−−−−∂∂

−

−−−−∂∂

−

∂∂

−−−−−

=⋅−⋅

PQQP

PQQP

QPPQ

kkdd

ddpFkk

ddpFkk

pFkkdd

332

333

221

222

3

11111

0

0

0

PQPQ

2

3

1

2

3

1321321 p

FpF

pFkkkkkkD PPPQQQ

∂∂

∂∂

∂∂

−=

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

0 1e+07 2e+07 3e+07 4e+07 5e+070

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Proteins

0 1e+07 2e+07 3e+07 4e+07 5e+070

0.02

0.04

0.06

0.08

0 1e+07 2e+07 3e+07 4e+07 5e+070

0.1

0.2

0.3

0.4

0.5

0.6

0.7

mRNAs

0 1e+07 2e+07 3e+07 4e+07 5e+070

0.05

0.1

0.15

0.2

0.25

0.3

The repressilator limit cycle

0 2e+08 4e+08 6e+08 8e+080

0.2

0.4

0.6

0.8

1

Proteins

0 2e+08 4e+08 6e+08 8e+080

0.05

0.1

0.15

0.2

0.25

0.3

0 2e+08 4e+08 6e+08 8e+080

0.2

0.4

0.6

0.8

1

mRNAs

0 2e+08 4e+08 6e+08 8e+080

0.05

0.1

0.15

0.2

0.25

0.3

The repressilator heteroclinic orbit

1 100 10000 1e+06 1e+080

0.2

0.4

0.6

0.8

1

Proteins

1 100 10000 1e+06 1e+080

0.05

0.1

0.15

0.2

0.25

0.3

1 100 10000 1e+06 1e+080

0.2

0.4

0.6

0.8

1

mRNAs

1 100 10000 1e+06 1e+080

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

P2P3

Stable heteroclinic orbit

Unstable heteroclinic orbit

1

1

2

2

2<0

2>0

2=0

Bifurcation from limit cycleto stable heteroclinic orbit

at

The repressilator heteroclinic orbit

0)ε()ε()ε()ε( PP1

QQ1 =+++++ Ddddd nn KK

11

212121

−∂∂

∂∂

∂∂

−=n

n

n

Pn

PPQn

QQ

pF

pF

pFkkkkkkD KKK

Upscaling to n genes with cyclic symmetry

( ) ( ) ( )

( ) ( ) { }sssisisi

mmn

ppppp

ppppxxxpxfx

∩Σ≡Σ⊕=×∈=

Σ

⊂∈===

;PPP;PP,

manifoldn bifurcatio

P;,,;,,;; 11

K

KK& R

( ) ( ) ( )( ) operator forward,)(,)( Ksipsi pppFpFpFsΣ⊥=≡ π

( )i

iipp

pFc

ppp

ppFpJss

)(0and

osubject t

)(min)(min

upplow

≤

≤≤

−= ... 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.

δδββαα ==== iiii hh ,,,

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


[ ] [ ][ ] [ ]pRBpRB]E2F1[

E2F1pRB pRB11

11

11 φ−

++=

JJ

Kk

dtd

m

[ ] [ ][ ] [ ]E2F1pRB]E2F1[

E2F1E2F1 E2F112

1222

2

22

1 φ−++

++=

JJ

Kakk

dtd

mP

[ ] [ ] [ ] [ ]AP1pRB']p[

E2F1AP1 AP111

65

15

1525 φ−

+++=

JJ

RBJJkF

dtd

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


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 NationalbankProject No. Nat-7813

European Commission: Contracts No. 98-0189, 12835 (NEST)

Austrian Genome Research Program – GEN-AU: BioinformaticsNetwork (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

dynamical systems in problems of gene regulation - tbipks/presentation/shanghai-07_lt1.pdf ·...

Documents