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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
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 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
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 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)
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 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
Kinetic differential equations
);(2 kxfxDtx
+∇=∂∂
),,(;),,(;);( 11 mn kkkxxxkxftdxd
KK ===
Reaction diffusion equations
General conditionsInitial conditions
: T , p , pH , I , ...:
... ... S , u
Boundary conditionsboundary normal unit vector
Dirichlet
Neumann
:
:
:
)0(x
),( trgx S =
),(ˆ trgxuux S =∇⋅=
∂∂
Parameter setmjIHppTk j ,,2,1;),,,,;I( G KK =
Gen
ome:
Seq
uenc
e I G
General conditionsInitial conditions
: T , p , pH , I , ...:
... ... S , u
Boundary conditionsboundary normal unit vector
Dirichlet
Neumann
:
:
:
)0(x
),( trgx S =
Kinetic differential equations
);(f2 kxxDtx
+∇=∂∂
),,(;),,(;);(f 11 mn kkkxxxkxtdxd
KK ===
Reaction diffusion equations
),(ˆ trgxuux S =∇⋅=
∂∂
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)
Kinetic differential equations
);(2 kxfxDtx
+∇=∂∂
),,(;),,(;);( 11 mn kkkxxxkxftdxd
KK ===
Reaction diffusion equations
General conditionsInitial conditions
: T , p , pH , I , ...:
... ... S , u
Boundary conditionsboundary normal unit vector
Dirichlet
Neumann
:
:
:
)0(x
),( trgx S =
),(ˆ trgxuux 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
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?
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
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
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
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
+++++++++
=
−=
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
( )( )
( )( )
( )( )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
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
pF
pF
pFkkkkkkD KKK
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?
( ) ( ) ( )
( ) ( ) { }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.
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
J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Application to simple gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.
[ ] [ ][ ] [ ]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
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 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
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