Oscillating Gene Expressions in Feed-back Networks: Basics and applications to p53, Hes1 and NF-?B.

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Oscillating Gene Expressions in Feed-back
Networks Basics and applications to p53, Hes1
and NF-?B.
Mogens H. Jensen, Niels Bohr Institute, Copenhagen

• Three systems with oscillatory gene expressions
• Hes1-mRNA protein network (Hirata et al (2002))
• P53-mdm2 after DNA damage (Oren et al (2000))
• Transcription factor NF-?B (Hoffmann et al
  (2002))
• ? Identify the simplest feed-back loop.
• ? Stationary regimes and oscillating regimes !
• Time period of 2-3
  hours not circadian !

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• 2. After DNA-damage Sharp increase in p53.
• p53 induces production of mdm2 - binds to
  mdm2
• 3. Modelled by delay equations of two elements
  oscillating behavior if dissociation constant
  between p53, mdm2 decreases
• 4. Protein Hes1 (Notch signaling) represesses its
  own mRNA, oscillations with period of 2 hours
• Hes1 delayed by 20 mins to mRNA
• Modeled by delay equations with two variables.
• Results of importance for somite segmentation
  (current project)
• 5. Transcription factor NF-?B and I?B feed-back
  loop
• We reduce 26-dimensional dynamics to three
  variables
• 6. Oscillations possible without delay
• ? If I?B is exposed to saturated degredation
• ? If external stimuli (IKK) oscillates
  chaotic response (current project)
• 7. We discuss many basic properties of
  oscillating dynamical systems
• along in the presentation.

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Collaborators

• Guido Tiana, Milan University
• Sandeep Krishna, post doc, NBI
• Kim Sneppen, NBI
• Simone Pigolotti, post doc, NBI
• We write low-dim. dynamical equations of
• biological sub/key-networks
• Oscillations possible
• 1) Delay 2) Saturated (enzymatic)
  degradation

(Input from Eric Siggia)
Many relations to the work of Albert Goldbeter
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Typical Oscillating data Experiments on Hes1

• Serum treatment, t0
• 2-hour period
• Protein oscillations delayed 15min

(Hirata et al, Science 2002)
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5) in vivo dynamical experiment
Haupt, Maya, Kazaz Oren, Nature, 1997 BarOr,
Maya, Segel, Alon, Levine, Oren, PNAS, 2000
Haupt, Maya, Kazaz Oren, Nature, 1997 BarOr,
Maya, Segel, Alon, Levine, Oren, PNAS, 2000

• peak in p53 concentration after 1 hour lasting
  for 1 hour
• peak in mdm2 concentration partially overlapping
  with that of p53
• another peak in p53 after 4/5 hours

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(Nelson et al, Science 2004)
Not circadian time-scales !
(Hoffmann et al, Science 2002)
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p53-mdm2 response network
DNA damage
PARP
ATM
NBS
BLS
FAS
BCL2
BRCA1
E1B
BAX
ARF
p53
mdm2
Apoptosys
p21
p16
MYC
RB
Cyclin D
Growth stop
E2F
CDK4
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Experimental knowledge
1) The crystal structure of some fragments of the
protein

• tetramer built of 4393 residues
• has a DNA binding domain and a domain which
  binds to mdm2

Cho, Gorina, Jeffrey Pavletich, Science,
1994 Kussie et al., Science, 1996
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2) 45-50 of all tumors display mutations in p53
gene, in the DNA binding region.
mdm2 binding
DNA binding
Greenblatt et al., Cancer Res. 1994
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3) In vivo experiment an increase in p53
concentration causes growth stop and apoptosis
4) Knock-out experiments the concentration
of p53 is regulated by a feedback loop
DNA
constant rate
p53
concentration of p53 is kept low
p53
mdm2
mdm2
Oliner et al., Nature, 1993 Kubbutat, Jones
Voudsen, Nature, 1997
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5) in vivo dynamical experiment
Haupt, Maya, Kazaz Oren, Nature, 1997 BarOr,
Maya, Segel, Alon, Levine, Oren, PNAS, 2000

• peak in p53 concentration after 1 hour lasting
  for 1 hour
• peak in mdm2 concentration partially overlapping
  with that of p53
• another peak in p53 after 4/5 hours

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6) Mutation experiments response caused by
phosphorization of SER20 in p53.
DNA
-
p53
concentration of p53 rises
-
p53
mdm2
mdm2
Experiments performed mutating SER in ASP or ALA
Gottlieb Oren, 1996
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Physical model
DNA
S
b
p
kg
k
c
pm
m
a
d

• p53-DNA and p53-mdm2 diffusion limited with ?D
  10 -2 s, much faster than other events (? 102
  s).
• "p" is the total amount of p53, "m" of mdm2 and
  "pm" of their complex,
• consequently
• the free amount of p53, namely (p-pm), is at
  equilibrium with the complex p53DNA.
• the free amounts of p53 and mdm2 are at
  equilibrium with their complex.

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dissociation constant between p53 and DNA
(O-operator site)
(can add a Hill coefficient)
Equilibrium probability that p53 is bound to
DNA
dissociation constant between p53 and mdm2
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Master Equations
More elaborate model Proposed by Bar-Or et
al, PNAS (2000)
Variant p53 is a tetramer..... p(t)?p(t)4 in
mdm2 production
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Basic stability theory
Stationary (Fixed) point
Jacobian matrix
Eigenvalues (in fixed point)
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Dynamical behavior around fixed points
Stable node
Stable complex
Double root
Saddle point unstable
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Bifurcations of fixed points

• Creation of a stable and an unstable fixed point
  Saddle-node bifurcation
• A stable fixed point turns into two stable and
  becomes itself unstable Pitch-fork
  bifurcation
• A stable fixed point becomes unstable with
  complex eigenvalues and turns into a limit-cycle
  Hopf bifurcation
• Hopf bifurcation is the important one for genetic
  oscillations
• ? Bifurcation applets

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Limit cycles
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Still to come Chaotic strange attractors
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Choice of the constants
Known... a ? 3 10 -2 s-1 Wilkinson, Cell.
Dev. Biol. 2000 b ? 10 -4 s-1 Haupt et al.,
Nature 1997 k ? 180 (in number of molecules)
Kussie et al., Science 1996 kg? 28
Balagurumooritity et al., PNAS 1995
Guessed... (from "typical" values) S ? 1
s-1 c ? 1 s-1 d ? 10-2 s-1 Unknown... k
after phosphorization
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Numerical Results
Differential Equations solved with Adams
algorithm. Stress event (phosphorilation)
simulated as change in k at time t20000.

• E.g. k180 ? 1800
• stationary values change p154 ? 858 m81 ?
  96
• no oscillation detected, no lag-time.

p(t)
m(t)
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As k is varied.....
... the stationary value p varies at most
linearly with k.
basal value
... the Eigenvalues of the linearized dynamical
matrix are either negative real numbers or
conjugated complex numbers with small imaginary
part.
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as the other parameters vary ...
e.g., the production rate S of p53
or any of the other parameters, spanning 5 orders
of magnitude about the basal value
... p changes, but no oscillations.
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Accounting for the cooperativity induced by
tetrameric structure does not change
qualitatively the results...
no cooperativity
dimer p2 ? pp
tetramer p4? pppp
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Time scales
overall response mechanism, degradation of
proteins
1h
Wilkinson, Cell Dev. Biol. 2000
degradation of mRNA
1200 s Holstege, Cell 1995
synthesis of proteins
1m
1s
10-2 s
diffusion
there is a delay in the mdm2 response to p53
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Physical model
DNA
S
b
p
kg
c
k
pm
m
a
d
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Model with delay
with ? of the order of 1200s
Note independent of volume
Just rescaling of
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Numerical solutions with delay
k k15
lag-time 3000 s
k k/15
k k/5
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Stability theory for delay equation
Expand
Assume solution
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After a looong calculation, find solutions
We see that the time delay t appears explicitly
in the expressions for the
eigenvalues
For some values of k and t Hopf bifurcation
into an oscillating state a limit cycle
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• oscillations take place when k decreases (!!!)
• the oscillatory response is sigmoidal

no oscillations
oscillations
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Robustness with respect to the parameters
c
d
kg
b
a
scale 10-2 10-1 1 10 100
3.43
3.46
3.2 3.2
3.2 3.2
3.2
11.2
2.14
9.2
3.2
2.1
1.4
no oscillations, but positive response
no or negative response

• the system is robust against a, b and c.
• a decrease of kg or d is dangerous.

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Dependence of the response with respect to the
delay
100-200 s bifurcation into an oscillating state
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Conclusions for p53/mdm2
Taking into account the delay in mdm2 production
gives

• oscillations
• sigmoidal response
• robustess with respect to all parameters except
  upon increasing the half-life of mdm2 and upon
  decreasing the dissociation constant between p53
  and DNA (? mutations in tumors)
• lag time before the peak in p53

The response is effective if the dissociation
constant p53-mdm2 decreases
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Experiments, Hes1

• H. Hirata et al. Science 298 840 (2002)
• mRNA for Hes1 is observed to oscillate with a 2
  hour period during somite segmentation
• Oscillations can be induced by stimulating
  cultured cells with serum.
• Hes1 protein represses its own transcription by
  directly binding to the promoter

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Experimental results

• Serum treatment, t0
• 2-hour period
• Protein oscillations delayed 15min

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Control experiments

• A) Presence of MG132 ? no degradation of Hes1
  protein
• B) Hes1 expressing vector ? constant high levels
  of Hes1 protein
• C) Cycloheximide ? inhibits translation
• D) vector carrying dominant negative form of Hes1
  protein

• Is a low-dimensional network

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Simplest feed-back loop

• Induced serum initiates transcription
• More hes1 mRNA
• More Hes1 protein
• Down-regulates transcription
• Less hes1 mRNA
• Less Hes1 protein
• Up-regulation of transcription

• Half-life of hes1 mRNA 24.1 ?1.7 min

• Half-life of Hes1 protein 22.3 ?3.1 min

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Theoretical model for Hes1
o
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• Dashed curve Hes1
• Solid curve mRNA
• ?rna 24.1 min
• ?hes1 22.3 min
• ? 24 min
• ? 20 R0 min-1
• ? 1/20 min-1
• KM (0.1R0)n
• n 4

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Variations with the time delay t
Oscillations very robust to changes in a, ß and k
Oscillations are sensitive to increase in t(rna)
and t(Hes1)
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Wnt gradient and Clock are coupled
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Direct observations of oscillations in nucleus
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NF-?B oscillations
I?Bß/e knocked out
The NF-?B feed-back network

• Hoffmann et al (2002)

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Hopf bifurcation (again!)
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IKK has been observed to oscillate itself
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Maybe chaotic dynamics ?
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What is chaotic dynamics ?

• Two timeseries
• Distance grows exponentially (sensitive
  dependence)
• Positive Lyapunov exponent ? defines chaotic
  dynamics

Deterministic noise !
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Strange attractors

• Non-periodic motion, sensitive dependence on
  initial conditions

Lorenz attractor
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Strange attractor of periodically forced NF-?B
system
? Deterministic noise Relevant in biology?
?1
(2-3 hour period)
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?0.08
(circadian period)
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We are currently looking for periodic/chaotic
behavior in various genetic
feed-back loops
Only transcription and complexes
(S. Pigolotti, S. Krishna)
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