Modeling Periodic Oscillations and Designing Bio-sensor for Genetic Networks

1. Modeling Periodic Oscillations and Designing Bio-sensor for Genetic Networks L. Chen, T.Kobayashi, K.Aihara

2. 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

3. Rhythms in Living Organisms Periodic oscillation Circadian Rhythms, etc. Period : Seconds ? Years Drosophila, Neurospora, mammal, etc Jumping Dynamics ?Phage, etc

4. 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)

6. 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 )

7. Problems: robustness of repressilator The experimental data revealed that the repressilator was subject to stochastic fluctuations of its components. It was observed to have irregular phase, irregular amplitude. Moreover, Only 40% of repressilators successfully oscillated. These are experimental data shown in Elowitz et al, Nature, 2000. This indicates that a repressilator is not good enough and that other designs of an artificial genetic oscillator are required. The experimental data revealed that the repressilator was subject to stochastic fluctuations of its components. It was observed to have irregular phase, irregular amplitude. Moreover, Only 40% of repressilators successfully oscillated. These are experimental data shown in Elowitz et al, Nature, 2000. This indicates that a repressilator is not good enough and that other designs of an artificial genetic oscillator are required.

8. 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)

9. A Two-Gene Model

10. Singular Perturbation Slow Subsystem for Scale t Fast Subsystem for Scale t

11. Relaxation Oscillation

12. Relaxation Oscillator

13. Two-Gene System with Delay

14. Effectiveness of Time Delay Enlarge the stability region of oscillation Influence the period length

15. Delay Enhances Oscillation

16. Numerical Simulation(A limit cycle with and without delay)

17. Time Evolution for e=0.01

18. Limit Cycles with Parameter Variations

19. Robustness of Oscillation with Delay for noise

20. A Three-Gene Model

21. Relaxation Oscillation for Three-Gene System

22. Mathematical Model

23. 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

24. Bio-Sensor This relation does not necessarily hold for the coupling model because it generally does not satisfy the required conditions. However, this bifurcation analysis motivated by the result show that it seems to hold for the coupling model. Here, the blue regions are parameter regions in which the reduced model oscillate, and the region enclosed by broken lines are parameter regions in which the original model oscillate where the degradation rates of proteins are set the same as that of mRNAs. These analysis suggests us that we should set the time-scale of gene expression as large as possible and also that we should makes degradation rates of proteins as equal to that of mRNA as possible. This relation does not necessarily hold for the coupling model because it generally does not satisfy the required conditions. However, this bifurcation analysis motivated by the result show that it seems to hold for the coupling model. Here, the blue regions are parameter regions in which the reduced model oscillate, and the region enclosed by broken lines are parameter regions in which the original model oscillate where the degradation rates of proteins are set the same as that of mRNAs. These analysis suggests us that we should set the time-scale of gene expression as large as possible and also that we should makes degradation rates of proteins as equal to that of mRNA as possible.

25. Relaxation-based sensor

26. Two-Gene Model (A)

27. 12min change of Arc ? 120 min change of Cro

28. Numerical Simulation

29. Synchronization of Oscillators

30. 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