Modeling Periodic Oscillations and Designing Bio-sensor for Genetic Networks

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Modeling Periodic Oscillations and Designing
Bio-sensor for Genetic Networks

• L. Chen, T.Kobayashi, K.Aihara

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Contents

• Background and Motivation
• Multiple Time Scales, Hysteresis and Delay
• A Two-Gene Model
• Singular Perturbation
• Relaxation Oscillator, Time-Delay
• Numerical Simulation
• A Three-Gene Model
• One-Gene Oscillator
• Bio-sensor
• Synchronization model

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Rhythms in Living Organisms

• Mathematical Model
• Boolean Model
• Differential Equation Model
• Hybrid Model
• Etc
• Analysis
• Computer Simulation
• Stability Analysis
• Bifurcation Analysis
• Etc

• Periodic oscillation
• Circadian Rhythms, etc.
• Period Seconds ? Years
• Drosophila, Neurospora,
• mammal, etc
• Jumping Dynamics
• ?Phage, etc

Gene Network
Mechanism ?
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Theoretical Models

• Enzymatic control process, Goodwin(1965)
• Negative feedback loop, Hastings, et al.(1977)
• Circadian oscillation in Drosophila, Goldbeter
  (1995,1998)
• Mathematical model of transcription, Keller(1995)
• Gene regulatory dynamics, Wolf et al.(1998)
• Switching dynamics, Smolen et al.(1998)
• Synthetic Toggle switch, Gardner et al. (2000)
• Repressilator, Elowitz and Leibler (2000)
• ?switch, Hasty et al. (2001)

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Synthetic genetic oscillator
Repressilator by Elowitz and Leibler
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Properties in Living Organizm and Motivation

• Time Scale Differences of Bio-reaction
• Transcription, Translation
• (sec ?
  minutes )
• Binding, Multimerization, Phosphorylation
  (msec ? sec)
• Hysteresis and Switching Dynamics
• Dynamics Change Drastically at Certain State
• Long Time Lags in Living Organism
• Unknown Chemical Modification
• (msec ? minutes )
• Transportation (sec ? Hours )

Possible Mechanism for Oscillation ?
Effectiveness for Oscillation ?
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Problems robustness of repressilator

• Irregular period
• irregular amplitude
• Failure of oscillation

Elowtiz et al., Nature, 2000
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Purpose

• Robust Oscillator delay and noise
• --- hysteresis and time scale
  differences
• Sensitive Bio-sensor
• --- bifurcation
• Bio-modules as building blocks
• --- synthetic genetic networks
• (Negative feedback, frustration system,
  competitive system)

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A Two-Gene Model
p(t-t)
p(t-t)
kp
slow
Protein p

DNA sequences
Gene-P
Gene-Q
RERQ
RERP

fast
kq degradation
Protein q
q(t)
q(t)
q(t)
q(t)
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Singular Perturbation

• Slow Subsystem for Scale t
• Fast Subsystem for Scale t

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Relaxation Oscillation
Protein q
Protein p
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Relaxation Oscillator
Theorem Assume that the system has only one real
equilibrium E(p,q). When time-scale difference
is sufficiently large, if J gt0, the system has a
periodic solution around E if J lt0, the system
has a stable equilibrium at E.
Time scale difference generates switching
dynamics and hysteresis creates oscillation.
Robustness to noise and parameters
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Two-Gene System with Delay
There is a time delay for protein p. Equilibrium
is the same as the model without time delay.
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Effectiveness of Time Delay

• Enlarge the stability region of oscillation
• Influence the period length

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Delay Enhances Oscillation
Theorem When time-scale difference is
sufficiently large, if J gt0, the system has a
periodic solution around E for any t if J lt0,
the system has a periodic solution around E for t
gt t0 . Delay enlarges the stability region of
oscillation and periodic length.
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Numerical Simulation(A limit cycle with and
without delay)
e0.01, t0.5
e0.01, t0
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Time Evolution for e0.01
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Limit Cycles with Parameter Variations
e0.001
e0.3
e0.25
e0.01
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Robustness of Oscillation with Delay for noise
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A Three-Gene Model
PR
PRM
cI
LuxR
LuxI
ai
Placlux0
ai auto-inducer
Protein p1 and p3 form a heterodimer to inhibit
gene-2. Protein p2 forms a homodimer to activate
gene-3 and inhibit gene-1
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Relaxation Oscillation for Three-Gene System
Gene Ocillator Gene Switch
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Mathematical Model
,
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Bio-sensor by Bifurcation

• Saddle bifurcation jumping dynamics
• Singular homoclinic bifurcation (Hopf
  bifurcation) equilibrium oscillation
• Canard Orbit
• There exists only in parameter ranges
  exponentially small
• in relation to e when the equilibrium moves.
• Trajectory changes exponentially from equilibrium
  to
• full size periodic orbit.
• Bio-Sensor spiking and pulse

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Bio-Sensor
?
Enlarge oscillation region
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Relaxation-based sensor
Green Input
Red proteins.
concentration
time
concentration
time
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Two-Gene Model (A)
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12min change of Arc ? 120 min change of Cro
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Numerical Simulation
cro
cI (temperature sensitive)
cI life-time 1min ?2min
cI life-time 4min ? 2min
cI
cro
mRNAcro
mRNAcI
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Synchronization of Oscillators
CELL Oscillator ai
CELL Oscillator ai
Autoinducer
ai
CELL Oscillator ai
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Conclusion

• Oscillator is constructed by time differences
• and hysteresis
• Periodic oscillation with switching dynamics
• Robust oscillator with delays and noise
• Exmples by two and three genes
• Bio-modules, e.g. switch, oscillator, bio-sensor