Download RNA Protein - Technion moodle

Lecture 3:
Models of gene regulation
DNA
Transcription
Replication
Protein
RNA
Translation
Bacterial growth on sugars
Batch reactor:
dC
  ( S )  C bacterial growth
dt
dS
1 dC

substrate consumption
dt
Y dt
Substrate metabolised by a Michaelis Menten enzyme:
max S
 (S ) 
, max enzyme concentration
S  KM
How does the cell know which enzymes to express?
Bacterial growth
on two substrates:
Two things to
notice:
1.Glucose
metabolized
before lactose
[conc.]
glucose
lactose
2.Lag between
glucose and
lactose growth
phases
hours
[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:
Glucose
+
-
Lactose
+
+
-
The lac Operon
crp
lacI
lacZ
lacY
lacA
Three main components:
1. Genes: encode protein sequence
lacZ
2. Promoters: RNAP binding sites
3. Operators: Transcription factor binding sites
The Prokaryotic Promoter
17 bp
NNNNTTGACANNNNNNNNNNNNNNNNNTATAATNNN
-35
-10
• The promoter is a binding site for the protein RNA
polymerase, responsible for transcription
DNA
Protein
RNA
Transcription
Translation
Replication
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)
n
R
Vactivation ( R)  A  n
n
K R
n
K
Vrepression ( R)  A  n
n
K R
The
effect of
two
n
1
regulators
n
R
K2
V ( R1 , R2 )  A  n
 n
n
n
K1  R1
K 2  R2
Dynamics: single regulated gene
A model of protein conc.:
accumulation=- degradation+ synthesis
dC p
 kdeg, p C p  AV ( R1 , R2 ,...)
dt
Single regulated gene
(U= protein conc or expression level)
dU
Kn
  kU  A n
.
n
dt
K R
AK n
Steady state: U s 
.
n
n
k (K  R )
n 1
A
U s  H ( K  R)
k
dU
Piecewise linear model:
  kU  AH ( K  R)
dt
 0, R  K
Steady state: U s  
 A / k, R  K
Graphical analysis
of steady states and stability
U
Us
RK
R, repressor concentration
Mutual inhibition network of
transcription factors
R2
R1
dR1
  k1 R1  A1 H (1  R2 )
dt
dR2
  k2 R2  A2 H ( 2  R1 )
dt
After eliminating the mRNA variables
Nullclines
R2
dR2
0
dt
R2
dR1
0
dt
A2 / k2
1
2
R1
A1 / k1
R1
Phase plane
R2
A2 / k2
1
2 A / k
1
1
R1
3 Steady states: ( A1 / k1 , 0), (0, A2 / k2 ), (1 , 2 )
Domain of bistability
A1 k1
 A1

A2
  2 ,  1  .
k2
 k1

Expression of both
2
R1 off
1
A2 k2
For both genes: maximal expression level
should be able to repress the repressor
HW: Sketch the phase planes for the 4 different regimes in the model
Mutual repression circuit
Toggling the switch
(transients: IPTG affects R1, temp affects R2)
Parametric dependence
of steady states
(GFP- flourescence protein)
Parametric dependence
of steady states
individual cells