modeling of gene expression
DESCRIPTION
Modeling of Gene expression. Central Dogma of Molecular Biology. :. Cell processes. Transcription factors. Proteins. mRNA. Gene. Modeling of Gene Expression. Modeling of Expression of one/few genes Binding of transcription factors/RNAPolymerasen,... to DNA - PowerPoint PPT PresentationTRANSCRIPT
Edda Klipp, Humboldt-Universität zu Berlin
Modeling of Gene expression.
:
Gene
mRNA
Proteins
Cell processes
Central Dogma of Molecular Biology
Transcription factors
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
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
Basis: Processes and interactions
Edda Klipp, Humboldt-Universität zu Berlin
Direction of Investigation
known to be predicted
Structure FunctionProtein interactions Expression of genesTF bindiung Regulation
Impact of perturbationsDynamic behavior,
Bifurcations,... : :
Function StructureExpression pattern Mutual influence of genesTime courses of Regulation network concentrations, activities,…. : :
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“)
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.
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
Edda Klipp, Humboldt-Universität zu Berlin
Kinetics – change of state
A Bk
Deterministic, continuous time and state: e.g. ODE modelconcentration of A decreases and concentration of B increases. Concentration change in per time interval dt is given by
Akdt
dB
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
aktadttaP ,,1a – number of molecules of type A
Deterministic, discrete time and state : e.g. Boolean network modelPresence (or activity) of B at time t+1 depends on presence (or activity) of A at time t tAftB 1
Edda Klipp, Humboldt-Universität zu Berlin
One Network, Different Models
gene a gene b
gene c gene d
C
A
D
B
AB
+
+
repression
activation
transcription
translation
gene
protein
A
a b
c d
Directed graphs
V = {a,b,c,d}
E = {(a,c,+),(b,c,+), (c,b,-),(c,d,-),(d,b,+)}
a b
c d
Boolean network
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)
a b
c d
Bayesian network
p(xa)
p(xb)
p(xc|xa,xb),
p(xd|xc),
Edda Klipp, Humboldt-Universität zu Berlin
Directed Graphs
a b
c d
Directed graphs
V = {a,b,c,d}
E = {(a,c,+),(b,c,+), (c,b,-),(c,d,-),(d,b,+)}
A directed graph G is a tuple , with V - Set of verticesE – Set of edges
Vertices are related to Genes (or other components of the system) and edges correspond to their regulatory interactions.
An edge is a tuple 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
properties: e.g: j activates i (+) or j inhibits i (-), properties e.g. List of regulators and their effects on a specific egde
EV ,
proteinc homodimeri as inhibitionactivation ,,,,, lkji
propertiesji ,,
ji ,
Usually not suited for presenting dynamics
Edda Klipp, Humboldt-Universität zu Berlin
Bayesian Network
Representation of network as directed acyclic graph
Nodes -- Genes
Edges E -- regulatory interactions.
Variables , belonging to nodes i = for regulation relevant properties,
e.g. Gene expression leves or amount of active protein.
A conditional probability distribution is defined for every ,
with parent variables belonging to direct regulators of i.
Directed Graph G and conditional probability distribution together
Yield the joint probability distribution, which defines the Bayesian network.
The joint probability distribution can be decomposed to
EVG ,
Vi
ix
ii xLxp ix ixL
xp
i
ii xLxpp x
a b
c d
Bayesian network
p(xa)
p(xb)
p(xc|xa,xb), p(xd|xc),
Edda Klipp, Humboldt-Universität zu Berlin
Bayes‘sche Netze
zy;ixi
ix
Gerichteter Graph: Abhängigkeit von Wahrscheinlichkeiten:Genexpressionslevel eines „Kindknotens“ ist abhängig von Expressionslevel der „Eltern“
Daher auch: bedingte Unabhängigkeiten:
Die bedeuten, dass unabhängig von Variablen y ist, wenn Variablen z gegeben sind.
Zwei Graphen oder Bayes‘sche Netzwerke sind äquivalent, wenn sie den gleichen Satz vonUnabhängigkeiten bestimmen.
Äquivalente Graphen sind durch Beobachtung der Variablen x nicht unterscheidbar.
Für das Beispielnetz sind die bedingten Unabhängigkeiten
Die gemeinsame Wahrscheinlichkeitsverteilung ist
ba xxi ;
cbad xxxxi ,;
cdbacbadcba xxpxxxpxpxpxxxxp ,,,,
a b
c dp(xa)
p(xb)
p(xc|xa,xb),
p(xd|xc),
Edda Klipp, Humboldt-Universität zu Berlin
Boolean Models
(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 descriptionIncreasing complexity and computational effort for more specific models
(discrete, deterministic)
Replacement of continuousfunctions (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.
B C
Linear chain
Ring
A B C D
A B
C D
A
B
A
Edda Klipp, Humboldt-Universität zu Berlin
Boolean Models
gene a gene b
gene c gene d
C
A
D
B
AB
+
+
repression
activation
transcription
translation
gene
protein
a b
c d
Boolean network
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)
0000 00010001 01010010 00000011 00000100 00010101 01010110 00000111 0000
Steady state: 0101
1000 10011001 11011010 10001011 10001100 10111101 11111110 10101111 1010
Cycle: 1000 1001 1101 1111 1010 1000
Edda Klipp, Humboldt-Universität zu Berlin
Beschreibung mit Differentialgleichungen
aft
aa
d
d
dcbft
bb ,,
d
d
cbaft
cc ,,
d
d
dcft
dd ,
d
d
akvaf aaa
bkcKdK
dVdcbf bn
Icn
b
nb
bcd
d
,,
ckbaK
baVcbaf cn
c
nc
cab
ab
,,
dkcK
Vdcf dn
Ic
dd
c
,
0 20 40 60 80 100
0
0.5
1
1.5
2
2.5
Time
Co
nce
ntr
atio
n
a
d
b
c
Nur für mRNA:
gene a gene b
gene c gene d
C
A
D
B
AB
+
+
repression
activation
transcription
translation
gene
protein
A
Edda Klipp, Humboldt-Universität zu Berlin
Network motifs
R. Milo, …, U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science, 2002
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. Eachnode in the randomized networks has the same number of incoming and outgoing edges as doesthe 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.
Edda Klipp, Humboldt-Universität zu Berlin
Network motifs
X Y
X Y
X Y Z
X Y Z
X Y Z
X Y Z
X Y Z
X
Z1 Z2 Z3 Zn
X1 X2 X3 Xm
Z1 Z2 Z3 Zn
Singleinput
HighDensity
Feedforwardloop
Feedback loop
Activation
Inhibition
R. Milo, …, U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science, 2002
X
YZ
Edda Klipp, Humboldt-Universität zu Berlin
Transcription
http://www.berkeley.edu/news/features/1999/12/09_nogales.html
Edda Klipp, Humboldt-Universität zu Berlin
Structure of Eukaryotic Promoter
(a)
Figure 6.1
TATA INR DPETFIIA
TBPTFIIF
TFIIB
RNAPII/GTF complex
TF binding sitesDistal promoter module
TF binding sitesProximal promoter module
TATA box Transcriptionstart
Downstream promoterelement
(b) TCCCTGAACGGTCCGAGAACCTTTGCTCCGCA_TTCCTGAGCTGTTCGTAAGGAG
A 00001142020C 02430110410G 00120303113T 53004000011
Aligned TFBSs
TYCSTGARCNG
Positional WeightMatrix
Consensus
Edda Klipp, Humboldt-Universität zu Berlin
Transcription
activeDNARNAPolIIDNA
DNATFDNA
DNATFDNA
x
221
110
mRNANukleotideDNA aktiv
...
NukleotidekDNAmRNAdt
daktiv
n
j
j
mmBm
n
iiBi
aktiv
TFK
TFK
Y
YDNADNAdt
d
1 1
1
0
1
n
ii
n
nn
nBn
B
DNA
DNAY
TFDNA
DNAK
DNADNA
DNAY
TFDNA
DNAK
0
1
10
1
10
11
,
,
Edda Klipp, Humboldt-Universität zu Berlin
Time delay in Transcription
TF-RE
TF-A
P
TF-ATF-AP
P
TF-ATF-AP
tf-a
Delay, translocation of mRNA
Delay, translocation of protein
+
0 5 10 15 20 25 30
1
0.5
0
0.5
1
1.5
kf /min
Lo
g 10T
F-A Region of
multistability
basdD
fRATFkt
KATF
ATFk
dt
ATFd
2
2
Transkriptionsfaktor TF-A aktiviert seine eigene Transkriptionals 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
Edda Klipp, Humboldt-Universität zu Berlin
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 j [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
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].
k52=0
incorrect aa-tRNA [A2]
correct aa-tRNA [A1]
Edda Klipp, Humboldt-Universität zu Berlin
Elongation model
correct aa-tRNA [A1]
Edda Klipp, Humboldt-Universität zu Berlin
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.
Regulation der Genexpression am Beispiel des Lac-Operons
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 -Galactosidase ist unter Kontrolle eines Strukturgens.
In Abwesenheit eines Galactosides wird kaum -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
J a c o b a n d M o n o d : A l l g e m e i n e s M o d e l l , A n n a h m e n
1 . D a s p r i m ä r e P r o d u k t s t r u k t u r e l l e r G e n e i s t d i e “ m e s s e n g e r R N A ” . S i e i s t k u r z l e b i g u n db r i n g t d i e I n f o r m a t i o n z u d e n R i b o s o m e n . D i e “ z w e i t e T r a n s k r i p t i o n ” f i n d e t a n d e nR i b o s o m e n s t a t t , d a b e i w e r d e n P o l y p e p t i d e g e f o r m t , d i e m e s s e n g e r R N A z e r s t ö r t , d i eR i b o s o m e n a b e r f ü r d e n n ä c h s t e n T r a n s k r i p t i o n s z y k l u s e r h a l t e n .
2 . D i e m R N A - S y n t h e s e i s t e i n s e q u e n t i e l l e r , o r i e n t i e r t e r P r o z e s s , d e r n u r a n b e s t i m m t e nR e g i o n e n d e r D N A , d e n O p e r a t o r e n , b e g i n n e n k a n n . M a n c h m a l k o n t r o l l i e r t e i n O p e r a t o rd i e T r a n s k r i p t i o n m e h r e r e r a u f e i n a n d e r f o l g e n d e r s t r u k t u r e l l e r G e n e . D i e s e G r u p p e h e i ß td a n n O p e r o n , e i n e “ E i n h e i t p r i m ä r e r T r a n s k r i p t i o n ” .
3 . N e b e n s t r u k t u r e l l e n G e n e n g i b t e s r e g u l a t o r i s c h e G e n e . E i n r e g u l a t o r i s c h e s G e n k o d i e r tf ü r e i n e n R e p r e s s o r . D e r R e p r e s s o r h a t e i n e A f f i n i t ä t z u u n d b i n d e t r e v e r s i b e l a n e i n e ns p e z i f i s c h e n O p e r a t o r . D i e s e K o m b i n a t i o n b l o c k i e r t d i e T r a n s k r i p t i o n s - i n i t i a t i o n d e sg e s a m t e n O p e r o n s u n d v e r h i n d e r t d i e P r o t e i n s y n t h e s e .
4 . D e r R e p r e s s o r R k a n n m i t k l e i n e n M o l e k ü l e n ( E f f e k t o r e n , F ) s p e z i f i s c h r e a g i e r e n :
I n i n d u z i e r b a r e n S y s t e m e n k a n n n u r d i e R - F o r m m i t d e m O p e r a t o r a s s o z i i e r e n u n d d i eT r a n s k r i p t i o n b l o c k e n . D e r E f f e k t o r = I n d u c e r i n a k t i v i e r t d e n R e p r e s s o r u n d e r m ö g l i c h td a m i t d i e T r a n s k r i p t i o n .I n r e p r e m i e r b a r e n S y s t e m e n i s t n u r d i e R ’ - F o r m a k t i v ; d i e T r a n s k r i p t i o n e r f o l g t i nA b w e s e n h e i t d e s E f f e k t o r s u n d w i r d i n s e i n e r A n w e s e n h e i t u n t e r d r ü c k t .
R+F R'+F'
Edda Klipp, Humboldt-Universität zu Berlin
RG O SG1 SG2
RF
R'
r n
m1 m2
r n
aa
P1 P2
ribosomes
Operon
Jacob-Monod-Model
Edda Klipp, Humboldt-Universität zu Berlin
lactose + E allolactose + E
lactose + E glucose + galactose + E
Ginactiv + m P Gactiv mmeq
m
Pk
Pp
Genaktivierung
Durchschnittliche Produktion von mRNA MkPk
PkM
dt
dMmm
eq
m
21
0
Konzentrationsänderungen von Permease (E1) und ß-Galactosidase (E2) 111
1 EdMcdt
dE 222
2 EdMcdt
dE
ex
exex
Lack
LacE
dt
dLac
010
ins
in
ex
exin
Lack
LacE
Lack
LacE
dt
dLac
21
010
Pk
PE
Lack
LacE
dt
dP
pins
in
2221
Laktose Aufnahme
Interne Laktose (Aufnahme, Umwandlung zu Allolaktose)
Allolaktose (von Laktose, to Glukose und Galaktose)
RG O SG1 SG2
RF
R'
r n
m1 m2
r n
aa
P1 P2
ribosomes
Operon
Modell von Griffith
Expressionsrate
Edda Klipp, Humboldt-Universität zu Berlin
2
0
2
1
k
M
Pk
P
k
kM
mmeq
m
Vereinfachungen Quasi-steady state für mRNA
Gleiche Enzymkonzentrationen 2121 ddEE ,11
2
11
2
01
1 EdPk
P
k
kc
k
Mc
dt
dEmm
eq
m
Pk
PE
Lack
LacE
dt
dP
pex
ex
12
010
Keine Verzögerung in der Umwandlungvon Laktose in Allolaktose
0dtdLacin
Dimensionlose Variablen 0kLaclac ex pkPp 01 eEe 0tt
ep
pm
d
demm
m
0
p
p
lac
lace
d
dp
11
s
se
d
dlac
1
Gleichungssystem
20
10120 k
kkce
00
00e
kt
0
2
pk
k0 p
eq
k
k
1
00k
Mm 10dt
Modell von Griffith
Edda Klipp, Humboldt-Universität zu Berlin
20 40 60 80 100
0.2
0.4
0.6
0.8
1lactose
allolactose
b- galactosidase
50 100 150 200
0.02
0.04
0.06
0.08
0.1lactose
allolactose
b- galactosidase
1 010.
00100 .m
2m
Lösung der Differentialgleichungen
010 .lac 0100 .e 000 .p
Parameter
Anfangsbedingungen
100 .lac 0100 .e 000 .p
Modell von Griffith
Edda Klipp, Humboldt-Universität zu Berlin
Catabolite Repression
CAP = Catabolite Activator ProteinCRP = cyclic AMP Receptor Protein
positive regulation factor
cAMP
CAP, active CAP, inactive
cAMP
Aktives CAP bindet an die CAP Bindungsregion.
Glukose reguliert die Catabolitrepression durch Senkung der freien cAMP-Konzentration.
Edda Klipp, Humboldt-Universität zu Berlin
1 20 40 60 80-20-40-60-80
operator
CRP binding RNA polymerase binding
coding region for -galactosidase
replication origin
+ glucose+ lactose
+ glucose- lactose
- glucose- lactose
- glucose+ lactose
CRP
CRP
repressor
repressor
RNA polymerase
transcription
Lac-Operon, Gene regulation and CAP protein
Edda Klipp, Humboldt-Universität zu Berlin
Non-induced InducedGenotypes
-gal
acto
-
sida
se
gala
ctos
ide-
perm
ease
gala
ctos
ide-
tran
sace
tyla
se
-gal
acto
-
sida
se
gala
ctos
ide-
perm
ease
gala
ctos
ide-
tran
sace
tyla
se
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
6. izy /F i-,z+,y+ 200 200 200 200 200 200
Table: Production of -galactosidase, galactoside-transacetylase and galactoside-permease by haploid and heterogenote, regulator-consitutive mutants.i: regulator gene (i+: inducible; i-: constitutive). z and y: structural genes for -galactosidase and galactoside permease, resp. F: sex factor of E. coli K12. izy - deletion
of the Lac region.
Edda Klipp, Humboldt-Universität zu Berlin
r p o z y a
Ri Ra
E MIeIi
F1
G
-
1
2
3
4 4
5
678
9
Lac-Operon, Model of Nicolis and Prigogine
R – RepressorI – InducerE, M – EnzymeG – GlukoseO – Operator
Edda Klipp, Humboldt-Universität zu Berlin
( 1 )
( 2 )
( 3 )
( 4 )
( 5 )
( 6 )
( 7 )
( 8 )
( 9 )
k 1
k -1
R i R a
O cR a +O fk -2
k 2
k 3
k -3
R a + n I I i F 1
k 4 +O f O f +E+M
M+I ek -5
k 5M+I i
F 2Mk 6
k 7E F 3
R a +DR i + n G Gk -8
k 8
G+EI i +Ek 9
DRkGRkFkIRk
OkORkRkRkdt
dR
an
in
ia
cfaaia
GI88133
2211
cfaf
OkORkdt
dO22
EkOkdt
dEf 74
MkOkdt
dMf 64
EIkMIkMIkFknIRkndt
dIiieI
niaI
i I955133
EIkDRknGRkndt
dGiaG
niG
G988
.constOO cf
Mathematical formulation of theNicolis-Prigogine-Model
r p o z y a
Ri Ra
E MIeIi
F1
G
-
1
2
3
4 4
5
678
9
Edda Klipp, Humboldt-Universität zu Berlin
- 1 0 1 2 3 4 5external inducer
0
0.5
1
1.5
2
2.5
3
b-esadisotcalag
All-or-None Transition
Figure 9.2. Dependence of the -galactosidase concentration of external inducer concentration
( eI , 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
concentrations lead to a switch to a -galactosidase concentration of mol21023 . .Parameters:..........
All or None Transitions
analysis of steady state while neglecting
the catabolite repression ( 0988 kkk )
sigmoidal dependence of the enzyme
concentration E on the external inducer eI .
low value 610 , high value 3103 ,
correspond to experimentally determined values
All or None Transition in dependence on the inducer,
Edda Klipp, Humboldt-Universität zu Berlin
T h e d y n a m i c b e h a v i o u r
q u a n t i t i e s k n o w n f r o m e x p e r i m e n t s : aR , 1F , , , 2k , 2k , 3k , 3k , 5k , 5k
s t o c h i o m e t r i c c o e f f i c i e n t s i n s t e p s ( 3 ) a n d ( 9 ) a r e c h o o s e n a s 2 GI nn
p a r a m e t e r s : eI , 1k , 1k , 4k , 6k , 7k , 8k , 8k , 9k
s i m p l i f i c a t i o n s : Dkk 88 81 kk 131 FkRk i
28
2322 GRkIRkOkORkR
dt
dRiiaffaa
a
ffaf
OkORkdt
dO 22
EkOkdt
dEf 74
MkOkdt
dMf 64
EIkMIkMIkFkIRkdt
dIiieia
i95513
23 22
EIkRkGRkdt
dGiai 98
28 22
r p o z y a
Ri Ra
E MIeIi
F1
G
-
1
2
3
4 4
5
678
9
Edda Klipp, Humboldt-Universität zu Berlin
Dynamic behaviour under catabolite repression
Assuming a quasi-steady state for the active repressor aR, the free operator fO , and for the
enzymes E and M, one obtains for the time dependence of the glucose concentration and of theinternal inducer:
2
322
827
2342595
1323
282
3 22ii
iie
i
ii
i
IkkGRkkk
IkkkIkkIkFk
Ik
GRkIk
dt
dI
2
322
827
23429
23
2882
82
2ii
ii
i
ii
IkkGRkkk
IkkkIk
Ik
GRkkGRk
dt
dG
For this equation system the steady state has been analysed for fixed parameters exept of varying
1k .
Edda Klipp, Humboldt-Universität zu Berlin
11
min k
M
F
1
218
min M
k
M
I e
119
min M
k
11
min k s t e a d y s t a t e s
0 . 2 3106 0 . 0 3 9 1 1 0 0 5 0 0 0 0 . 1 S t a b l e F o c u s
0 . 0 9 9 -0 . 0 0 0 2 4 8
S F +U L C ( min110T )S L C ( min300T )
0 . 0 0 0 2 4 7 -1 0 - 5
S F + U L C
0 . 2 3106 0 . 0 0 3 5 1 1 0 0 5 0 0 2 . 0 S F , n o L C
0 . 1 -0 . 0 0 0 2 4 8
U n s t a b l e F +S L C ( min1500T )
0 . 0 0 0 2 4 7 -1 0 - 5
U F + S t a b l e N o d e
Edda Klipp, Humboldt-Universität zu Berlin
0.0000250.000050.000075 0.0001 0.0001250.000150.000175 0.0002k- 1
- 4
- 3
- 2
- 1
0
1
2
goLHI iL
Steady States, Catabolite Repression
Edda Klipp, Humboldt-Universität zu Berlin
0 250 500 750 1000 1250 1500 1750Time
0
5
10
15
20
25
snoitartnecnoC
Time evolution , Catabolite Repression
0 2 4 6 8internal inducer
5
10
15
20
25
esoculg
Phase plane, Catabolite Repression
P a r a m e t e r s :1
1 min2.0 k ,1
1 min008.0 k ,
1152 min104 Mk ,
12 min03.0
k ,21
3 min2.0 Mk ,1
3 min60 k
134 min105 k ,
115 min6.0 Mk ,
115 min006.0
Mk 16
76 min103 kk ,21
8 min03.0 Mk ,118
8 min10 Mk ,
1139 min105 Mk ,
MR i 210 , MF 31 10 ,
M 310002.2
Edda Klipp, Humboldt-Universität zu Berlin
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
kPDPXdt
dP
DSDSXdt
dS
DXXdt
dX
in
X – Biomasse-KonzentrationS – Konzentration der KohlenhydratquelleSin – Konz. der zugefütterten KohlenhydrateP – Konzentration des FusionsproteinsD – Verdünnungsrateµ - spezifische Wachstumsrate – spezifische Kohlenstoffverbrauchsrate – spezifische Expressionsrate des Fusionsproteinsk – Abbaurate des Fusionsproteins
Edda Klipp, Humboldt-Universität zu Berlin
Bakterielle Genexpression mit Reportergen gusA
VorgegebenesSauerstoffprofil
Kohlenstoffquelle,Hier:Malat
Verdünnungsrate
Gus Aktivitätß-Glucuronidase
als Maß fürcytN-Expression