the Lac operon continued

1
the Lac operon continued math modeling, computer
simulation
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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
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Wong et al. (1997)
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discussion of Modeling network dynamics the
lac operon, a case study Vilar JMG, Guet CC,
Leibler S (2003) J. Cell Biology 161 471-476.
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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.
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All-or-none induction of Lac operon in
indi- vidual cells (expts of Novick Weiner,
1957)
Fig 1 of Vilar et al. (2003)
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cellular level
molecular level
population level
Fig 2 of Vilar et al. (2003)
WHICH LEVEL OF DESCRIPTION?
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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
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Vilar et al. claim that there are 2 stable
solutions and hysteresis
induced
Zss
(steady state)
uninduced
Iex
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Stochastic simulations (used Gillespie
algorithm - to be discussed later)
Fig 3 of Vilar et al. (2003)
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Tutorial on BerkeleyMadonna (a
dynamics simulation software) free download
from http//www.berkeleymadonna.com
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kinetic equations
dX/dt v1 - v2 dE/dt v3 - v4
note X Y constant C
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k1 10
k1 1
k1 8
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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
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we now turn to another gene regulatory network .
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Bacteriophage l
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Attractive, but nasty.
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Lysis vs
Lysogeny
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THE SIMPLE PICTURE OF THE GENETIC SWITCH
cI vs Cro
cI l repressor
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The l chromosome
DNA replication genes
cI
cro
N
cII
cIII
Recombination genes
Q
lysis genes
att site
cos ends
Tail genes
Head genes
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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
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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
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early protein N binds nut sites () allowing
continued transcription as
shown.
PL

cI
cro
N
cII
cIII

PR
O
xis
P
int
Q
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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
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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
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next 6 slides this slide were downloaded from
(reference here later)
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SUMMARY OF
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SUMMARY OF
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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.