Lecture 3: Models of gene regulation
DNA
Replication
RNA ProteinTranscription Translation
Bacterial growth on sugars
maxmax
Batch reactor:
( ) bacterialgrowth
1substrateconsumption
Substrate metabolised by a Michaelis Menten enzyme:
( ) , enzyme concentrationM
dCS C
dtdS dC
dt Y dt
SS
S K
How does the cell know which enzymes to express?
hours
[conc.]
glucose lactose
Bacterial growth on two substrates:
Two things to notice:
1.Glucose metabolized before lactose
2.Lag between glucose and lactose growth phases
[Monod, Thesis 1942]
• Monod realized this was operating like a genetic “switch”– Genes required for lactose metabolism
turned off in presence of glucose– But turned on in absence of glucose and
presence of lactose
Input/Ouput relation: Lactose
+ -
Glucose+ - -- + -
crp
lacI lacZ lacY lacA
Three main components:
1. Genes: encode protein sequence
2. Promoters: RNAP binding sites
3. Operators: Transcription factor binding sites
lacZ
The lac Operon
The Prokaryotic Promoter
NNNNTTGACANNNNNNNNNNNNNNNNNTATAATNNN
-35 -10
17 bp
• The promoter is a binding site for the protein RNA polymerase, responsible for transcription
DNA
Replication
RNA ProteinTranscription Translation
Transcription factor (protein that repress or activate)Rate usually depends on transcription factor
Gene regulation functions(rate of transcription as a function of factors or regulators)
21
12
2
1
21
(
( )
(
,
)
)n
repression n n
n
n
n
activation n n
nn
n
n
KV R A
K RThe ef
V R R A
fect of two regulators
R
RV R A
K R
K R
K
K R
Dynamics: single regulated gene
deg, 1 2
model of protein conc.:
accumulation=- degradation+ synthesis
( , ,...)pp p
A
dCk C AV R R
dt
Single regulated gene(U= protein conc or expression level)
1
.
Steady state: . ( )( )
Piecewise linear model: ( )
0,Steady state:
/ ,
n
n n
n n
s sn n
s
dU KkU A
dt K R
AK AU U H K R
k K R k
dUkU AH K R
dtR K
UA k R K
Graphical analysis of steady states and stability
sU
, repressor concentrationRR K
U
Mutual inhibition network of transcription factors
11 1 1 1 2
22 2 2 2 1
( )
( )
dRk R AH R
dtdR
k R A H Rdt
R1R2
After eliminating the mRNA variables
Nullclines
1R
2R
2 2/A k
2 1R
2R
1
1 1/A k
1 0dR
dt2 0
dR
dt
Phase plane
2R
2 2/A k
2
1
1 1/A k1R
1 1 2 2 1 23 Steady states: ( / ,0), (0, / ), ( , )A k A k
Domain of bistability
For both genes: maximal expression level should be able to repress the repressor
1 22 1
1 2
, .A A
k k
1 1A k
2 2A k
2
1
HW: Sketch the phase planes for the 4 different regimes in the model
Expression of both
R1 off
Mutual repression circuit
Toggling the switch(transients: IPTG affects R1, temp affects R2)
Parametric dependenceof steady states
(GFP- flourescence protein)
Parametric dependenceof steady statesindividual cells
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