Transcript Kein Folientitel

.
Modeling of Gene expression
:
Central Dogma of Molecular Biology
Cell processes
Transcription factors
Proteins
mRNA
Gene
Edda Klipp, Humboldt-Universität zu Berlin
Modeling of Gene Expression
Modeling of Expression of one/few genes
-
Binding of transcription factors/RNAPolymerasen,... to DNA
Effect of inhibitors/activators
Production of mRNA, proteins
Feedback or regulation by products or external regulators
Basis: Processes and interactions
Discovery of genetic networks
-
Cause of gene expression patterns or -profils
Modeling of the dynamics of artifical networks
Reverse Engineering
Search for Motifs and Clustern
Basis: Data
Edda Klipp, Humboldt-Universität zu Berlin
Direction of Investigation
known
to be predicted
Structure
Function
Protein interactions
TF bindiung
Expression of genes
Regulation
Impact of perturbations
Dynamic behavior, Bifurcations,...
:
:
Function
Expression pattern
Time courses of
concentrations, activities,….
:
Structure
Mutual influence of genes
Regulation network
:
Edda Klipp, Humboldt-Universität zu Berlin
Concept of state
The state of a system is a snapshot of the system at a given time that
contains enough information to predict the behaviour of the system for
all future times. The state of the system is described by the set of
variables that must kept track of in a model.
Different models of gene regulation have different representations of the state:
Boolean model: a state is a list containing for each gene involved, of whether
it is expressed („1“) or not expressed („0“)
Differential equation model: a list of concentrations of each chemical entity
Probabilistic model: a current probability distribution and/or a list of actual
numbers of molecules of a type
Each model defines what it means by the state of a system.
Given the current state the model predicts what state/s can occur next.
Edda Klipp, Humboldt-Universität zu Berlin
Kinetics – change of state
A
k
B
Deterministic, continuous time and state: e.g. ODE model
concentration of A decreases and concentration of B increases.
Concentration change in per time interval dt is given by
dB
kA
dt
Probabilistic, discrete time and state : transformation of a molecule of
type A into a molecule of type Sorte B. The probability of this event in a
time interval dt is given by
Pa 1,t  dt a ,t   k  a
a – number of molecules of type A
Deterministic, discrete time and state : e.g. Boolean network model
Presence (or activity) of B at time t+1 depends on presence (or activity) of
A at time t
Bt 1  f  At 
Edda Klipp, Humboldt-Universität zu Berlin
One Network, Different Models
A
A
A
B
B
transcription
+
gene a
translation
gene b
repression
activation
+
C
D
gene
protein
gene c
Directed graphs
gene d
Bayesian network
a
b
a
c
d
c
d
p(xa)
p(xb)
p(xc|xa,xb),
p(xd|xc),
V = {a,b,c,d}
E = {(a,c,+),(b,c,+),
(c,b,-),(c,d,-),(d,b,+)}
Boolean network
b
a
b
c
d
a(t+1) = a(t)
b(t+1) = (not c(t)) and d(t)
c(t+1) = a(t) and b(t)
d(t+1) = not c(t)
Edda Klipp, Humboldt-Universität zu Berlin
Directed Graphs
A directed graph G is a tuple
V - Set of vertices
E – Set of edges
Directed graphs
a
b
c
d
V = {a,b,c,d}
E = {(a,c,+),(b,c,+),
(c,b,-),(c,d,-),(d,b,+)}
V , E , with
Vertices are related to Genes (or other components of the system)
and edges correspond to their regulatory interactions.
An edge is a tuple i, j of vertices.
It is directed, if i and j can be associated with head and tail of the edge.
Label of edges and vertices can be enlarged to store information
about genes or interactions.
Then in general, an edge is a tuple
i , j , properties
properties: e.g: j activates i (+) or j inhibits i (-),
properties e.g. List of regulators and their effects on a specific egde
i , j ,k ,activation,l ,inhibitionas homodimeric protein
Usually not suited for presenting dynamics
Edda Klipp, Humboldt-Universität zu Berlin
Bayesian Network
Representation of network as directed acyclic graph G  V , E
Bayesian network
a
b
c
d
p(xa)
p(xb)
p(xc|xa,xb),
p(xd|xc),
Nodes i  V -- Genes
Edges E -- regulatory interactions.
Variables
x i, belonging to nodes i = for regulation relevant properties,
e.g. Gene expression leves or amount of active protein.


A conditional probability distribution p xi Lxi  is defined for every x i ,
with L xi  parent variables belonging to direct regulators of i.
Directed Graph G and conditional probability distribution together
p x 
Yield the joint probability distribution, which defines the Bayesian network.
The joint probability distribution can be decomposed to
p x  
 pxi Lxi 
i
Edda Klipp, Humboldt-Universität zu Berlin
Bayes‘sche Netze
Gerichteter Graph: Abhängigkeit von Wahrscheinlichkeiten:
Genexpressionslevel eines „Kindknotens“ ist abhängig von Expressionslevel der „Eltern“

Daher auch: bedingte Unabhängigkeiten: i xi ; y z

Die bedeuten, dass x i unabhängig von Variablen y ist, wenn Variablen z gegeben sind.
Zwei Graphen oder Bayes‘sche Netzwerke sind äquivalent, wenn sie den gleichen Satz von
Unabhängigkeiten bestimmen.
Äquivalente Graphen sind durch Beobachtung der Variablen x nicht unterscheidbar.
Für das Beispielnetz sind die bedingten Unabhängigkeiten
a
b
c
d
p(xa)
p(xb)
p(xc|xa,xb),
p(xd|xc),
i  x a ; xb 
ixd ; xa , xb xc 
Die gemeinsame Wahrscheinlichkeitsverteilung ist
pxa , xb , xc , xd   pxa   pxb   pxc xa , xb   pxd xc 
Edda Klipp, Humboldt-Universität zu Berlin
Boolean Models
(discrete, deterministic)
(George Boole, 1815-1864)
Each gene can assume one of two states:
expressed („1“) or not expressed („0“)
Background:
Not enough information for more detailed description
Increasing complexity and computational effort for more specific models
Replacement of continuous
functions (e.g. Hill function)
by step function
Edda Klipp, Humboldt-Universität zu Berlin
Boolean Models
Boolean network is characterized by
- the number of nodes („genes“): N
- the number of inputs per node (regulatory interactions): k
The dynamics are described by rules:
„if input value/s at time t is/are...., then output value at t+1 is....“
Boolean network have always a finite number of possible states and,
therefore, a finite number of state transitions.
A
A
B
C
B
D
Linear chain
A
A
Ring
C
D
B
Edda Klipp, Humboldt-Universität zu Berlin
B
C
Boolean Models
A
A
Boolean network
B
B
+
gene a
c
d
a(t+1) = a(t)
C
+
D
b(t+1) = (not c(t)) and d(t)
c(t+1) = a(t) and b(t)
repression
activation
b
gene b
transcription
translation
a
gene c
gene d
d(t+1) = not c(t)
gene
protein
0000
0001
0010
0011
0100
0101
0110
0111
 0001
 0101
 0000
 0000
 0001
 0101
 0000
 0000
Steady state: 0101
1000
1001
1010
1011
1100
1101
1110
1111
 1001
 1101
 1000
 1000
 1011
 1111
 1010
 1010
Cycle:
1000  1001  1101  1111  1010  1000
Edda Klipp, Humboldt-Universität zu Berlin
Beschreibung mit Differentialgleichungen
A
A
A
B
B
transcription
+
gene a
translation
gene b
repression
activation
+
C
D
gene
protein
gene c
gene d
Nur für mRNA:
db
 f b b , c , d 
dt
dc
 f c a , b , c 
dt
dd
 f d c , d 
dt
f a a   va  k a  a
f b b , c , d  
f c a , b , c  
f d c , d  
K
2.5
Vb  d nd
b d
nd
K
Vc  a  b nab
K c  a  b nab

 kb  b
 kc  c
c
2
1.5
a
1
b
0.5
d
0
0
Vd
K Ic  c
Ic  c
nc
Concentration
da
 f a a 
dt
nc
 kd  d
Edda Klipp, Humboldt-Universität zu Berlin
20
40
60
Time
80
100
Network motifs
Schematic view of network motif
detection. Network motifs are
patterns that recur much more
frequently
(A) in the real network than
(B) in an ensemble of
randomized networks. Each
node in the randomized networks
has the same number of incoming
and outgoing edges as does
the corresponding node in the
real network. Red dashed lines
indicate edges that participate in
the feedforward loop motif,
which occurs five times in the
real network.
R. Milo, …, U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science, 2002
Edda Klipp, Humboldt-Universität zu Berlin
X
Network motifs
Activation
Inhibition
X Y
X Y
Feedforward
loop
Feedback loop
Z
X Y
X Y
X Y
Z
Z
Z
X Y
Z
X Y
Z
Y
X
Z1 Z2 Z3
X1 X2 X3 Xm
Z1 Z2 Z3
R. Milo, …, U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science, 2002
Edda Klipp, Humboldt-Universität zu Berlin
Zn
Zn
Single
input
High
Density
Transcription
http://www.berkeley.edu/news/features/1999/12/09_nogales.html
Edda Klipp, Humboldt-Universität zu Berlin
Figure 6.1
Structure of Eukaryotic Promoter
(a)
RNAPII/GTF complex
TFIIA
TBP
TFIIF
TFIIB
TATA
TF binding sites
Distal promoter
module
(b)
TF binding sites
Proximal promoter
module
TCCCTGAACGG
TCCGAGAACCT
TTGCTCCGCA_
TTCCTGAGCTG
TTCGTAAGGAG
A
C
G
T
00001142020
02430110410
00120303113
53004000011
TYCSTGARCNG
TATA box
Aligned
TFBSs
Positional
Weight
Matrix
Consensus
Edda Klipp, Humboldt-Universität zu Berlin
INR
Transcription
start
DPE
Downstream
promoter
element
Transcription
DNA0  T F1 
 DNA1
DNA1  T F2 
 DNA2
...
K B1 
DNA1
,
DNA0  TF1
K Bn 
DNAn
,
DNAn1  TFn
Y
DNAx  RNAP olII
 DNAactive
DNA1
DNA0  DNA1
Y
DNAn
n
 DNAi
i 0
d
DNAaktiv  DNA0  Y
dt
n
 K Bi  TFi
Y
i 1
n j
1   K Bm  TFm
j 1 m 1
DNAaktiv  Nukleotide
 mRNA
d
mRNA  DNAaktiv  k  Nukleotide
dt
Edda Klipp, Humboldt-Universität zu Berlin
Time delay in Transcription
Transkriptionsfaktor TF-A aktiviert seine eigene Transkription
als phosphorylierter Homodimer, der an Enhancer TF-RE bindet.
Modell nach Smolen mit time delay: - schnelles Gleichgewicht von Monomer und Dimer
- Sättigungskinetik für Transkription
- Abbau von TF mit kd, basale Produktion mit Rbas
t – delay time
P
TF-A
P
TF-A
1.5
TF-A
P
TF-A
Delay,
translocation
of mRNA
+
P
TF-A
TF-RE
Log10TF-A
1
Delay,
translocation
of protein
Region of
multistability
0.5
0
0.5
1
0
tf-a
5
10
15
kf /min
2 
d TF  A 
 k f  TF  A 

t     k d  TF  A  Rbas
2
dt

 TF  A  K D 

Edda Klipp, Humboldt-Universität zu Berlin
20
25
30
Protein
Biosynthesis
Edda Klipp, Humboldt-Universität zu Berlin
Model for Elongation of a Peptid chain
Heyd A & Drew DA, Bulletin of Mathematical Biology (2003) 65, 1095–1109
[mRNA] - concentration of messenger RNA,
[mRNA0] - concentration of the mRNA–ribosome complex
[mRNAj ] - concentration of the mRNA–ribosome complex with a nascent peptide chain of length j attached.
reaction rate –kR [R][mRNA] - rate at which the mRNA–ribosome complex is formed
(rate of binding of the mRNA to the ribosome)
reaction rate kj [aj ][mRNAj-1] is the elongation rate
(rate constant times the concentrations of the amino acid to be attached,
and the mRNA–ribosome complex with the nascent chain)
Edda Klipp, Humboldt-Universität zu Berlin
Modell for Elongation of a Peptid chain
correct
aa-tRNA [A1]
incorrect
aa-tRNA [A2]
k52=0
A—EF-Tu:aa-tRNA complex. A1 - correct complex, and A2 - wrong complex.
B—open A-site on ribosome. In this configuration, the ribosome is
available to any amino acid.
C—initial binding.
D—codon recognition.
E—GTPase activation and GTP hydrolysis.
F—EF-Tu released after EF-Tu conformation change.
G—accommodation and peptide transfer.
A ready ribosome [B] initially binds (reversibly) with EF-Tu:aa-tRNA
complex [A]. This is followed by codon recognition [D]. After codon
recognition, GTPase activation and GTP hydrolysis follow successively [E].
EF-Tu then undergoes a conformation change allowing EF-Tu to be released [F].
At this point proofreading occurs. If the wrong aa-tRNA is present, it is rejected,
and the A-site is open again [B]. If the correct aa-tRNA is present, it is
accommodated and the peptide bond forms almost immediately [G]. The ribosome
then resets back to its open position [B].
Edda Klipp, Humboldt-Universität zu Berlin
Elongation model
correct
aa-tRNA [A1]
Edda Klipp, Humboldt-Universität zu Berlin
Regulation der Genexpression
am Beispiel des Lac-Operons
Jacob-Monod-Modell
Jacob, F. & Monod, J. (1961) On the Regulation of Gene Activity,
Cold Spring Harb. Symp. Quant. Biol., 26, 193-211.
Modell of Griffith
Griffith, J.S. (1971) Mathematical Neurobiology, Academic Press, London.
Keener, J. & Sneyd, J. (1998) Mathematical Physiology, Springer-Verlag, New York.
Nicolis-Prigogine-Modell
Nicolis, G. & Prigogine, I. (1977) Self-Organization in Non-Equilibrium Systems,
John Wiley & Sons, New York.
Edda Klipp, Humboldt-Universität zu Berlin
Experimentelle Fakten
Organismus: E.coli
Bildung von Tryptophansynthase ist reguliert durch ein Strukturgen.
In Abwesenheit von Tryptophan wird dieses Enzym synthetisiert.
In Anwesenheit von Tryptophan wird seine Synthese gestoppt.
Repression der Enzymsynthese: spezifisch für Enzyme des Trp-Syntheseweges
Bildung des Enzyms b-Galactosidase ist unter Kontrolle eines Strukturgens.
In Abwesenheit eines Galactosides wird kaum b-Galactosidase synthetisiert.
Sobald Galactosid da ist, wird die Syntheserate um das 10 000-fache gesteigert.
Induktion der Enzymsynthese, ebenfalls sehr spezifisch
Edda Klipp, Humboldt-Universität zu Berlin
Jacob and Monod: Allgemeines Modell, Annahmen
1. Das primäre Produkt struktureller Gene ist die “messenger RNA”. Sie ist kurzlebig und
bringt die Information zu den Ribosomen. Die “zweite Transkription” findet an den
Ribosomen statt, dabei werden Polypeptide geformt, die messenger RNA zerstört, die
Ribosomen aber für den nächsten Transkriptionszyklus erhalten.
2. Die mRNA-Synthese ist ein sequentieller, orientierter Prozess, der nur an bestimmten
Regionen der DNA, den Operatoren, beginnen kann. Manchmal kontrolliert ein Operator
die Transkription mehrerer aufeinanderfolgender struktureller Gene. Diese Gruppe heißt
dann Operon, eine “Einheit primärer Transkription”.
3. Neben strukturellen Genen gibt es regulatorische Gene. Ein regulatorisches Gen kodiert
für einen Repressor. Der Repressor hat eine Affinität zu und bindet reversibel an einen
spezifischen Operator. Diese Kombination blockiert die Transkriptions-initiation des
gesamten Operons und verhindert die Proteinsynthese.
4. Der Repressor R kann mit kleinen Molekülen (Effektoren, F) spezifisch reagieren:
R+F
R'+F'
In induzierbaren Systemen kann nur die R-Form mit dem Operator assoziieren und die
Transkription blocken. Der Effektor=Inducer inaktiviert den Repressor und ermöglicht
damit die Transkription.
In repremierbaren Systemen ist nur die R’-Form aktiv; die Transkription erfolgt in
Abwesenheit des Effektors und wird in seiner Anwesenheit unterdrückt.
Edda Klipp, Humboldt-Universität zu Berlin
Jacob-Monod-Model
Operon
RG
O
SG 2
SG 1
rn
R
F
R'
m2
m1
rn
aa
ribosomes
P1
Edda Klipp, Humboldt-Universität zu Berlin
P2
Modell von Griffith
Operon
RG
O
rn
R
lactose + E
allolactose + E
lactose + E
glucose + galactose + E
Genaktivierung
F
R'
rn
aa
ribosomes
p
Gactiv
Pm
m
keq
 Pm
Durchschnittliche Produktion von mRNA
Konzentrationsänderungen von
Permease (E1) und ß-Galactosidase (E2)
dE1
 c1M  d1E1
dt
Laktose Aufnahme
dLacex
Lacex
  0 E1
dt
k0  Lacex
Allolaktose (von Laktose,
to Glukose und Galaktose)
P2
Expressionsrate
dM
k1P m
 M0 
 k2 M
m
m
dt
keq  P
Interne Laktose (Aufnahme,
Umwandlung zu Allolaktose)
m2
m1
P1
Ginactiv+ m P
SG 2
SG 1
dE 2
 c2 M  d 2 E 2
dt
dLacin
Lacex
Lacin
  0 E1
 1E2
dt
k0  Lacex
ks  Lacin
Lacin
dP
P
 1E2
  2 E2
dt
ks  Lacin
kp  P
Edda Klipp, Humboldt-Universität zu Berlin
Modell von Griffith
k1  P m  M 0
M

m
m 
k 2  keq
 P  k2

Vereinfachungen Quasi-steady state für mRNA
M0
dE1
k1  P m 
 c1
 c1
 d1E1
m
m
dt
k2
k2  keq

P


Gleiche Enzymkonzentrationen E1  E2 , d1  d2
Keine Verzögerung in der
Umwandlung
von Laktose in Allolaktose
Lacex
dP
P
  0 E1
  2 E1
dt
k0  Lacex
kp  P
dLacin dt  0
Dimensionlose Variablen
lac  Lacex k0 p  P k p
Gleichungssystem
de
pm
 m0 
 e
m
m
d
k p
e  E1 e0
  t t0
 lac
dp
p 

  e

d
1

lac
1

p


dlac
s
 e
d
1 s
ck k
k
e02  1 0 1 t 0  0
 0k2
e0 0

2
0

k eq
k0
k
kp
kp
Edda Klipp, Humboldt-Universität zu Berlin
m0 
M0
k1
  t 0 d1
Modell von Griffith
1
Lösung der Differentialgleichungen
Parameter
    k 1
m0  0.001
  0.01
m2
Anfangsbedingungen
lac0  1.0 e0  0.01 p0  0.0
lactose
0.8
b - galactosidase
0.6
0.4
0.2
allolactose
20
0.1
lac0  0.1 e0  0.01 p0  0.0
40
60
80
100
lactose
0.08
0.06
b - galactosidase
0.04
0.02
allolactose
50
Edda Klipp, Humboldt-Universität zu Berlin
100
150
200
Catabolite Repression
CAP = Catabolite Activator Protein
CRP = cyclic AMP Receptor Protein
positive regulation factor
cAMP
CAP, active
cAMP
CAP, inactive
Aktives CAP bindet an die CAP Bindungsregion.
Glukose reguliert die Catabolitrepression durch Senkung der
freien cAMP-Konzentration.
Edda Klipp, Humboldt-Universität zu Berlin
Lac-Operon, Gene regulation and CAP protein
replication origin
CRP binding
coding region for
b-galactosidase
RNA polymerase binding
operator
-80
-60
-40
-20
20
1
40
60
+ glucose
+ lactose
repressor
+ glucose
- lactose
- glucose
- lactose
- glucose
+ lactose
CRP
CRP
repressor
RNA polymerase
transcription
Edda Klipp, Humboldt-Universität zu Berlin
80
transacetylase
galactoside-
permease
galactoside-
sidase
b
-galacto-
transacetylase
Induced
galactoside-
permease
sidase
b
galactoside-
Non-induced
-galacto-
Genotypes
1. i+,z+,y+
<0.1
<1
<1
100
100
100
2. i-,z+,y+
120
120
120
120
120
120
3. i+,z-,y+/F i-,z+,y+
2
2
2
200
250
250
4. i-,z-,y+/F i+,z+,y-
2
2
2
250
120
120
5. i-,z-,y+/F i-,z+,y+
250
250
250
200
250
250
200
200
200
200
200
200
6.
 izy
/F i-,z+,y+
Table: Production of b -galactosidase, galactoside-transacetylase and galactosidepermease by haploid and heterogenote, regulator-consitutive mutants.
b
i: regulator gene (i+: inducible; i-: constitutive). z and y: structural izy
genes for
galactosidase and galactoside permease, resp. F: sex factor of E. coli K12.
- deletion
of the Lac region.
Edda Klipp, Humboldt-Universität zu Berlin
Lac-Operon, Model of Nicolis and Prigogine
p
r
o
z
y
a
2
1
Ri
Ra
E7
8
G
F1
4
4
9
3
R – Repressor
I – Inducer
E, M – Enzyme
G – Glukose
O – Operator
Edda Klipp, Humboldt-Universität zu Berlin
M
Ii
5
6
Ie
p
r
(2)
Ra+Of
k2
k-2
(3)
(4)
(5)
(6)
(7)
(8)
Ra+nIIi
E
k6
k7
E7
F1
F1
dRa
 k1 Ri  k 1 Ra  k 2 Ra O f  k  2 Oc
dt
nI
 k 3 Ra I i
dt
M+Ii
k-5
F2
k9 k-8
 k  3 F1  k8 Ri G nG  k  8 Ra D
  k 2 R a O f  k  2 Oc
dE
 k 4 O f  k 7 E
dt
dM
 k 4 O f  k 6 M
dt
F3
k8
9
3
Oc
Of+E+M
k5
Ri+nGG
Ii+E
k-3
dIi
  n I k 3 Ra I inI  n I k 3 F1  k 5 I e M  k 5 I i M  k 9 I i E
dt
Ra+D
G+E
a
dG
  nG k 8 Ri G nG  nG k 8 Ra D  k 9 I i E
dt
O f  Oc    const .
(9)
Edda Klipp, Humboldt-Universität zu Berlin
4
4
Ra
G
dO f
M+Ie
M
k3
k4
+Of
1
Ri
Ra
k-1
y
2
8
k1
Ri
z
-
Mathematical formulation of the
Nicolis-Prigogine-Model
(1)
o
M
Ii
5
6
Ie
All or None Transitions
All - or - None Transition
3
analysis of steady state while neglecting

b- galactosidase
the catabolite repression ( k 8  k 8  k 9  0 )
sigmoidal dependence of the enzyme
I
concentration E on the external inducer e .
low value
10 6
, high value
3  10 3
,
2.5
2
1.5
1
0.5
correspond
to experimentally determined values

0
All or None Transition in dependence on the inducer,
-1
0
1
2
3
external inducer
4
5
Figure 9.2. Dependence of theb -galactosidase concentration of external inducer concentration
( I e , lactose concentration). The sigmoidal shape of the curve can be interpreted as All-Or-None
Transition: for low inducer concentrations almost no enzyme is detectable, increasing inducer
2
concentrations lead to a switch to ab -galactosidase concentration of 3.2  10 mol .
Parameters:..........
Edda Klipp, Humboldt-Universität zu Berlin
The dynamic behaviour
quantities known from experiments: R a , F1 , ,
,k 2 k,2
,k 3 k , 3
k, 5 k ,5
stochiometric coefficients in steps (3) and (9) are choosen as n I  nG  2
parameters: I e , k1 ,k 1 , k 4 , k 6 , k 7 , k 8 k, 8 ,k 9
k  k 8 D
  k 1  k 8
simplifications: 8

  k1 Ri  k 3 F1

dRa
   Ra  k 2 Ra O f  k 2   O f  k 3 Ra I i2  k 8 Ri G 2
dt
dO f
  k 2 Ra O f  k 2   O f
dt


dE
 k 4 O f  k 7 E
dt
p
r
o
y
a
-
dM
 k 4 O f  k 6 M
dt
dIi
 2k 3 Ra I i2  2k 3 F1  k 5 I e M  k 5 I i M  k 9 I i E
dt
z
2
1
Ri
E7
8
G
dG
 2k8 Ri G 2  2k  8 Ra  k9 I i E
dt
F1
Edda Klipp, Humboldt-Universität zu Berlin
3
4
4
Ra
9
M
Ii
5
6
Ie
Dynamic behaviour under catabolite repression
Of
Assuming a quasi-steady state for the active repressor R a , the free operator
, and for the
enzymes E and M, one obtains for the time dependence of the glucose concentration and of the
internal inducer:



2

dIi
k 5 I e  k 9  k 5 I i k 2 k 4 k 3 I i2  
2 k 8 Ri G  
 2 k 3 I i
 2k 3 F1 
2
dt
k3 I i  
k 7 k 2 k 8 Ri G 2    k 2 k 3 I i2  




 
2k 8 k 8 Ri G 2  
dG
2
 2k 8 Ri G 

2
dt
k3 I i  
k7


 

k I k k  k I   
k k R G    k k I   
2
9 i 2
2
8 i
4

2
3 i
2
2
3 i
For this equation system the steady state has been analysed for fixed parameters exept of varying
k 1 .
Edda Klipp, Humboldt-Universität zu Berlin
k1
min 1
0.2
F1
M
6  10 3
k8
min 1 M 2
0.03
Ie
M
91100
k9
min 1 M 1
5000
k 1
min 1
0.1
steady states
Stable Focus
0.0990.000248
0.2
6  10 3
0.003
51100
500
SF+
T

ULC ( 110 min )
SLC T
(  300 min )
0.000247SF+ULC
-5
10
2.0
SF, no LC
0.1Unstable F+
0.000248 SLC(T  1500 min )
0.000247UF+Stable Node
-5
10
Edda Klipp, Humboldt-Universität zu Berlin
Steady States, Catabolite Repression
2
Log Ii
HL
1
0
-1
-2
-3
-4
0.0000250.000050.000075 0.0001 0.0001250.000150.000175 0.0002
k- 1
Edda Klipp, Humboldt-Universität zu Berlin
Time evolution , Catabolite Repression
25
Concentrations
Parameters:
k1  0.2 min 1 ,
k 1  0.008 min 1 ,
k 2  4  10 5 min 1 M 1 ,
k  2  0.03 min 1 ,
k 3  0.2 min 1 M 2 ,
20
15
10
5
k 3  60 min 1
k 4  5  10
3
0
min
1
0
250
500
,
750
1000
Time
1250
1500
k 5  0.6 min 1 M 1 ,
k 5  0.006 min 1 M 1
k 6  k 7  3  10 6 min 1
k 8  10
8
min
k 9  5  10 min
3
2
1
1
25
,
,
M
1
M
1
20
,
,
F1  10 3 M
Ri  10 M
,
  2.002  10 3 M
glucose
k 8  0.03 min 1 M 2
Phase plane , Catabolite Repression
,
15
10
5
0
2
Edda Klipp, Humboldt-Universität zu Berlin
4
6
internal inducer
8
1750
Bakterielle Genexpression mit Reportergen gusA
Quantifizierung der Regulation der Genexpression durch ein externes
Signal, O2
Operon cytNOQP von A. brasilense codiert eine Cytochrome cbb3 Oxidase,
die bei Wachstum und Atmung eine Rolle spielt.
Die Expression ist abhängig vom Sauerstoffgehalt.
Die Expression von cytN wurde mittels der Fusion von cytN-gusA gemessen.
Modell
dX
 X  DX
dt
dS
 X  DS  DSin
dt
dP
 X  DP  kP
dt
X – Biomasse-Konzentration
S – Konzentration der Kohlenhydratquelle
Sin – Konz. der zugefütterten Kohlenhydrate
P – Konzentration des Fusionsproteins
D – Verdünnungsrate
µ - spezifische Wachstumsrate
 – spezifische Kohlenstoffverbrauchsrate
 – spezifische Expressionsrate des Fusionsproteins
k – Abbaurate des Fusionsproteins
Edda Klipp, Humboldt-Universität zu Berlin
Bakterielle Genexpression mit Reportergen gusA
Vorgegebenes
Sauerstoffprofil
Verdünnungsrate
Gus Aktivität
ß-Glucuronidase
Kohlenstoffquelle,
Hier:
Malat
als Maß für
cytN-Expression
Edda Klipp, Humboldt-Universität zu Berlin