1. Novel size effect in mesoscopic chemical oscillation systems Zhonghuai Hou ( 侯中怀 ) Sep. 2008, Hefei Department of Chemical Physics Hefei National Lab for Physical Sciences at Microscale University of Science & Technology of China

2. Our Research Interests  Nonlinear Dynamics in Mesoscopic Chemical Systems  Dynamics of Complex Networks  Nonequilibrium Thermodynamics of Small Systems (Fluctuation Theorem)  Multiscale Modeling of Complex Systems Nonequilibrium +Nonlinear+ Complexity

3. Outline  Introduction  the question  Optimal Size Effect  Stochastic Normal Form Theory  Conclusion

4. Genetic Toggle Switch In E. Coli Nature 2000 Two or more stable states under same external constraints Reactive/Inactive bistabe CO+O2 on Pt filed tip PRL1999 Travelling/Target/Spiral/Soliton … waves PEEM Image CO Oxidation on Pt PRL 1995 Calcium Spiral Wave in Cardiac Tissues Nature 1998 Temporally Periodic Variations of Concentrations Rate Oscillation CO+O2 Nano- particle Catal.Today 2003 Synthetic transcriptional oscillator (Repressilator) Nature 2002 Stationary spatial structures in reaction-diffusion systems Cellular Pattern CO Oxidation on Pt PRL 2001 Turing Pattern BZ Reaction System PNAS 2003  Oscillation  Multistability  Patterns  Waves  Chaos Nonlinear Chemical Dynamics  far-from equilibrium, self-organized, complex, spatio-temporal structures Aperiodic/Initial condition sensitivity/strange attractor … Strange Attractor The Lorenz System Chemical turbulence CO+O2 on Pt Surface Science 2001 Collective behavior involving many molecular units Nonequilibrium Statistical Mechanics

5. Sub-cellular reactions - gene expression - ion-channel gating - calcium signaling … Heterogeneous catalysis - field emitter tips - nanostructured composite surface - small metal particles Mesoscopic Reaction System N, V (Small) Molecular Fluctuation Nonlinear Chemical Dynamics ? Chemical Oscillation Regularity Stochasticity

6. Noise Induced Pattern Transition Z.Hou, et al., PRL 81, 2854 (1998) Disorder sustained spiral waves Z.Hou, et al., PRL 89, 280601 (2002) We already know...  Noise and disorder play constructive roles in nonlinear systems Taming Chaos by Topological Disorder F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003) Y. Gong,Z.Hou, H.Xin. ChemPhysChem 6 ， 1042 （ 2005 ）； M. M.Wang, Z.Hou, H.Xin. ChemPhysChem7 ， 579( 2006); Ordering Bursting Chaos in Neuron Networks

7. Modeling of Chemical Oscillations  Macroscopic level: Deterministic, Cont. N Species, M reaction channels, well-stirred in V Reaction j: Rate: Hopf bifurcation leads to oscillation

8. Modeling of Chemical Oscillations  Mesoscopic Level: Stochastic, Discrete Master Equation Kinetic Monte Carlo Simulation (KMC) Gillespie ’ s algorithm Exactly Approximately Deterministic equation Internal Noise

9. New: Noise Induced Oscillation FFT  A model system: The Brusselator DeterministicStochastic Noisy Oscillation

10. Optimal System Size Optimal System size for mesoscopic chemical oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004) Best performance

11. Seems to be common …  Internal Noise Stochastic Resonance in a Circadian Clock System J.Chem.Phys. 119, 11508(2003)  Optimal Particle Size for Rate Oscillation in CO Oxidation on Nanometer-Sized Palladium(Pd) Particles J.Phys.Chem.B 108, 17796(2004)  Internal Noise Stochastic Resonance of synthetic gene network Chem.Phys.Lett. 401,307(2005)  Effects of Internal Noise for rate oscillations during CO oxidation on platinum(Pt) surfaces J.Chem.Phys. 122, 134708(2005)  System size bi-resonance for intracellular calcium signaling ChemPhysChem 5, 1041(2004)  Double-System-Size resonance for spiking activity of coupled HH neurons ChemPhysChem 5, 1602(2004) ? Common mechanism Analytical Study

12. Analytical study  Main idea Fact: all happens close to the HB Question: common features near HB? Answer: normal form on center manifold

13. Analytical study  Stochastic Normal Form

14. Analytical study  Stochastic Averaging

15. Analytical study( … )  Probability distribution of r Fokker- Planck equation Stationary distribution Most probable radius Noise induced oscillation

16. Analytical study( … )  Auto-correlation function

17. Analytical study( … )  Power spectrum and SNR Optimal system size:

18. Analytical study( … ) Universal near HB System Dependent ChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J. Phys. 9, 403(Nov. 2007) ;

19. System Size Bi-resonance System Size Biresonance for Intracellular Calcium Signaling J. Zhang, Z.H.Hou, H.W.Xin, ChemPhysChem 5, 1041(2004)

20. Effects of Internal Noise for Rate Oscillation During CO Oxidation Z.Hou, T. Rao, H.Xin. J.Chem.Phys. 122 ， 134708(2005) System Size Bi-resonance

21. N 个 HH 神经元的耦合体系 …… SSS S N Log(S) Coupled Neuron Network: ChemPhysChem 5 ， 1062(2004) Coupled Calcium Oscillation: Phye.Rev.E 74 031901(2006) Double System Size Resonance

22. Summary  In mesoscopic chemical systems, molecular fluctuations can induce oscillation even outside the deterministic oscillatory region  Optimal system size exists, where the noise- induced oscillation shows the best performance, characterized by a maximal SNR, a trade off between strength and regularity  Based on stochastic normal form, analytical studies show rather good agreements with the simulation results, uncovering the mechanism of NIO and OSS  System Bi-Resonance and Double System Size Resonance are observed

23. Acknowledgements Supported by: National science foundation (NSF) Prof. Yubing Gong (龚玉兵 鲁东大学) Prof. Jiqian Zhang (张季谦 安徽师范大学) Maosheng Wang (汪茂胜 安徽师范大学) Zhiwei Wang (王志伟 天津科技大学) Tiejun Xiao (肖铁军) Juan Ma(马娟) Ting Rao(饶汀)