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the Lac operon continued

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Title: the Lac operon continued


1
the Lac operon continued math modeling, computer
simulation
2
RNAP
lacZ
lacY
lacI
cap
P
O
cAMP/CAP
b-galactosidase
Repressor (R)
lactose permease
Lactose (L) (external)
allolactose
Lactose (L)
glucose
RL
cAMP
galactose
R
ZY
glucose (G) (external)
R
ZY
L
G
See Wong et al. (1997) in Week-3 Readings
3
Wong et al. (1997)
4
discussion of Modeling network dynamics the
lac operon, a case study Vilar JMG, Guet CC,
Leibler S (2003) J. Cell Biology 161 471-476.
5
Assumptions of standard approach of
modeling networks of biochemical reactions
1. Cell is a well-stirred reactor (homogeneous)
2. Dynamics can be described by ODEs (large
numbers of molecules)
In fact, 1. Cells are heterogeneous and
compartmentalized 2. Some molecules exist
in very few numbers.
6
All-or-none induction of Lac operon in
indi- vidual cells (expts of Novick Weiner,
1957)
Fig 1 of Vilar et al. (2003)
7
cellular level
molecular level
population level
Fig 2 of Vilar et al. (2003)
WHICH LEVEL OF DESCRIPTION?
8
The Model
dY/dt f1(I) - a1Y dYf/dt b1Y -
a2Yf dI/dt f2(Iex) - f3(I)Yf b2Iex
- a3I dZ/dt gf1(I) - a3Z
Yf
f1 c1 c2I c3I2 (for low I
increases monotonically and saturates at high
I) f2, f3 hyperbolic functions note Y, Yf, I
not affected by Z basically a 3-dimensional system
Y
Iex
I
Z
9
Vilar et al. claim that there are 2 stable
solutions and hysteresis
induced
Zss
(steady state)
uninduced
Iex
10
Stochastic simulations (used Gillespie
algorithm - to be discussed later)
Fig 3 of Vilar et al. (2003)
11
Tutorial on BerkeleyMadonna (a
dynamics simulation software) free download
from http//www.berkeleymadonna.com
12
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13
kinetic equations
dX/dt v1 - v2 dE/dt v3 - v4
note X Y constant C
14
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15
k1 10
k1 1
k1 8
16
Assignment Show that an S-shaped steady state
curve exists for certain
sets of parameters for the slightly modified
model below (note
additional term in v1). Note Treat A as a
parameter. Use A in
your diagram for the S-shaped curve as shown
below.
reaction rates
3
4
E
v1 k1EY/(Km1Y) k1bAY/(Km1bY)
Ess
A
(steady state)
v2 k2X/(Km2X)
1
Y
X
v3 k3X ko
A
v4 k4E
2
17
we now turn to another gene regulatory network .
18
Bacteriophage l
19
Attractive, but nasty.
20
Lysis vs
Lysogeny
21
THE SIMPLE PICTURE OF THE GENETIC SWITCH
cI vs Cro
cI l repressor
22
The l chromosome
DNA replication genes
cI
cro
N
cII
cIII
Recombination genes
Q
lysis genes
att site
cos ends
Tail genes
Head genes
23
patterns of gene expression
N, cro genes ON
very early
N, cro, Recomb, DNA rep genes ON
early
int, cI genes ON
late
lysogeny
lysis
24
very early host RNA polymerase transcribes
phage genes N and cro
from left promoter (PL) and right promoter
(PR), respectively
transcription stopped at termination sites
shown corresponding
proteins are produced
PL
stop
cI
cro
N
cII
cIII
O
xis
PR
P
stop
int
Q
25
early protein N binds nut sites () allowing
continued transcription as
shown.
PL

cI
cro
N
cII
cIII

PR
O
xis
P
int
Q
26
late lytic protein Q binds Qut site ()
allowing transcription of late
lysis, head, and tail genes as shown. Q
anti-terminates a small
RNA begun at promoter PR located just to the
right of gene Q. Cro
binding at sites shown (o ) inhibits further
cI and cro transcription.
OL
cI
cro
N
cII
cIII
OR
O
xis
P
int
Q
PR

lysis
tail
head
27
late lysogenic protein cII directs
transcription of two genes (int and
cI) for finally
establishing lysogeny.
PRE
cI
cro
N
cII
Pint
cIII
O
xis
P
int
Q
28
next 6 slides this slide were downloaded from
(reference here later)
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33
SUMMARY OF
34
SUMMARY OF
35
Next required reading Arkin A, Ross J, McAdams
HH (1998) Stochastic kinetic analysis
of developmental pathway bifurcation in phage
l-infected Escherichia coli cells,
Genetics 149 1633-1648.
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