1. Nonlinear Dynamics in Mesoscopic Chemical Systems Zhonghuai Hou ( 侯中怀 ) Department of Chemical Physics Hefei National Lab of Physical Science at Microscale University of Science & Technology of China

2. 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

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

4. Noise Induced Pattern Transition Z.Hou, et al., PRL 81, 2854 (1998) Disorder sustained spiral waves Z.Hou, et al., PRL 89, 280601 (2002) Noise/Disorder  Noise and disorder play constructive roles in nonlinear dynamical systems Taming Chaos by Topological Disorder F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003)

5. Stochastic Chemical Kinetics  chemical reactions are essentially stochastic, discrete processes Discrete Brownian Motion of X : Prob. Evolution: Master equation Sample Trajectory: Langevin equation stochastic state variable probability distribution

6. Chemical Langevin equation (CLE) N Species, M reaction channels, well-stirred in V Reaction j: Rate:  Molecular fluctuation (Internal noise)  Deterministic kinetics for  Each channel contributes independently to internal noise:  Fast numerical simulation

7. The Brusselator  Deterministic bifurcation Fixed Point: Hopf bifurcation:

8. Noise Induced Oscillation  Stochastic dynamics FFT

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

10. 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)

11. Analytical study  Stochastic Normal Form

12. Analytical study  Stochastic Averaging

13. Analytical study  Probability distribution of r Fokker- Planck equation Stationary distribution Most probable radius Noise induced oscillation

14. Analytical study  Auto-correlation function

15. Analytical study  Power spectrum and SNR Optimal system size:

16. Analytical study Universal near HB System Dependent Internal Noise Coherent Resonance for Mesoscopic Chemical oscillations: a Fundamental Study. Z. Hou, … ChemPhysChem 7, 1520(2006)

17. 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

18. Further questions

19. Acknowledgements Supported by: National science foundation (NSF) Fok Yin Dong education foundation Thank you