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

理论化学的三个层面 夸克 美洲豹 电子结构理论 量子力学 结果：反应个体的物理化学性质；基本参数 反应动力学 量子+统计 结果：反应势能面，反应途径，速率常数 统计行为 统计+唯象 结果：宏观量时空演化行为 －实际体系功能 夸克 美洲豹

Nonlinear Chemical Dynamics far-from equilibrium, self-organized, complex, spatio-temporal structures Oscillation Multistability Patterns Waves Chaos Aperiodic/Initial condition sensitivity/strange attractor… Strange Attractor The Lorenz System Chemical turbulence CO+O2 on Pt Surface Science 2001 Travelling/Target/Spiral/Soliton … waves PEEM Image CO Oxidation on Pt PRL 1995 Calcium Spiral Wave in Cardiac Tissues Nature 1998 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 Stationary spatial structures in reaction-diffusion systems Cellular Pattern CO Oxidation on Pt PRL 2001 Turing Pattern BZ Reaction System PNAS 2003 Temporally Periodic Variations of Concentrations Rate Oscillation CO+O2 Nano-particle Catal.Today 2003 Synthetic transcriptional oscillator (Repressilator) Nature 2002 Collective behavior involving many molecular units Nonlinear chemical dynamics is a field to study … far from equilibrium. Typical behaviors include … Oscillation Mul.. Patterns… Different from patterns, waves as non-stationary. There are a few types of waves, Chaos has now been a quite popular word to us. It lacks a clear definition, nevertheless, they share some features like … One notes that all these behaviors concerns the spatio-temporal evolution of some ‘macroscopic’ state variable, such as the concentration or total number of molecules of some reaction species, rather than the ‘microscopic’ state, given by the position and momentum of all the molecules. Nonequilibrium Statistical Mechanics

Mesoscopic Reaction System Heterogeneous catalysis - field emitter tips - nanostructured composite surface - small metal particles Sub-cellular reactions - gene expression - ion-channel gating - calcium signaling … … N, V (Small) Molecular Fluctuation Recently years, growing attention has been paid to the properties of mesoscopic systems, due to the fast development of nano-science technology and life science. For a mesoscopic reaction system, the total number of molecules N or the system size V is small. An important feature of mesoscopic system is that molecular fluctuation is large, i.e., large deviation from the average exists. Generally, the standard deviation of some macroscopic state variable is proportional to 1 divided by square root of the system size. Examples of such meso-l reaction systems include … and … . As just mentioned, a lot of nonlinear dynamic behavior are observed in these systems. Therefore, our basic question is: how the large molecular fluctuations would influence the nonlinear dynamics in these meso- reaction systems. Nonlinear Chemical Dynamics ? Chemical Oscillation Regularity Stochasticity

Why Noise/Disorder ? Noise and disorder play constructive roles in nonequilibrium systems Taming Chaos by Topological Disorder F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003) Disorder sustained spiral waves Z.Hou, et al., PRL 89, 280601 (2002) Noise Induced Pattern Transition Z.Hou, et al., PRL 81, 2854 (1998) Another main reason for us to study m-f is: it is now well-known that noise and disorder can often play rather constructive roles in nonlinear dynamic systems. There have been a large amount of literature regarding this issue, and here I will only show a few of our previous works. For example, … These results give a hint that molecular fluctuations, or internal noise may also have rather interesting, constructive effects.

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

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

New: Noise Induced Oscillation A model system: The Brusselator Stochastic Deterministic FFT Noisy Oscillation

Optimal System Size Best performance As in literature, we can use the effective SNR, defined as the H in the PS divided by the half-H width, to measure the performance of the SO. Consequently, we find that the SNR shows a clear maximum at an optimal system size, also indicating an optimal noise level, as shown in the figure. Therefore, we demonstrate an interesting effect of internal noise in mesoscopic chemical oscillation systems. Optimal System size for mesoscopic chemical oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)

Seems to be common … ? Common mechanism Analytical Study Internal Noise Stochastic Resonance in a Circadian Clock System J.Chem.Phys. 119, 11508(2003) ? Common mechanism 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) Optimal Particle Size for Rate Oscillation in CO Oxidation on Nanometer-Sized Palladium(Pd) Particles J.Phys.Chem.B 108, 17796(2004) Such phenomenon seems to be common in mesoscopic oscillation systems. For instance, we have found very similar behaviors in clock, cal..., CO.., gene.... These brings the question to us whether they share some common mechanisms. Very recently, we have been able to perform an analytical study on such behaviors, and the analytical results show rather good agreements with the numerical results. Effects of Internal Noise for rate oscillations during CO oxidation on platinum(Pt) surfaces J.Chem.Phys. 122, 134708(2005) Analytical Study Internal Noise Stochastic Resonance of synthetic gene network Chem.Phys.Lett. 401,307(2005)

Analytical study Fact: all happens close to the HB Main idea Fact: all happens close to the HB Question: common features near HB? In the following part, I will briefly outline the basic steps of the analytical study. I would not go into the details, but focus on the main idea. Since NIO and optimal size effect are observed near the HB, we believe that they must be related to some common features of the HB. From the dynamical system theory, we know the dynamics near the HB can be described by a normal form on the center manifold, which involves the evolution of the oscillation amplitude and phase angle. Therefore, we believe that ‘normal form’ is the key for analytical study. Answer: normal form on center manifold

Analytical study Stochastic Normal Form

Analytical study Stochastic Averaging

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

Analytical study(…) Auto-correlation function

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

Analytical study(…) Universal near HB System Dependent Here is a few remarks of the analytical study. Since the analysis is based on the normal form, we can conclude that NIO and Optimal size effect is a universal behavior near the HB. When the system size goes small, i.e., the internal noise goes large, the oscillation amplitude increases, while the correlation time decreases, both monotonically. The SNR shows a maximum at intermediate values of r and tau_c, therefore the best performance is a trade off between oscillation strength and regularity. Though the behavior is common near the HB, the location of the optimal size is system dependent, through the parameter Cr and eps2, which is related to the details of all the elementary reactions. (This analytical work is published very recently. ) System Dependent ChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J. Phys. 9, 403(Nov. 2007) ;

Entropy Production? Macroscopic Level: Nonequilibrium Statistical Thermodynamics I. Prigogine 1970s

Entropy Production? Mesoscopic Level: Stochastic Thermodynamics Luo,Nicolis 1984; P.Gaspard 2004

Entropy Production? Single Trajectory Level: Dynamic Irreversibity A Random Trajectory Trajectory Entropy Total Entropy Change U. Seifert, PRL 2005

Fluctuation Theorems ! Integrate FT Detailed FT(NESS)

Application to Brusselator Detailed FT holds

Application to Brusselator System Size Dependence Simulation SNF Theory

Summary Noise Induced Oscillation  Stochastic Modeling is important Optimal system size  Noise + Nonlinearity Success Analytical Study  Universality + Underlying mechanism Fluctuation Theorem  Single Trajectory Thermodynamics + Dynamic Irreversibility

Statistical physics in mesoscopic chemical systems Future directions How does fluctuation influence the properties of small chemical systems Thermodynamics Size dependence Jarzynski equality Fluctuation theorems… Dynamics Nucleation and growth Transport/relaxation Nonlinear dynamics … Here I would like to take sometime to outlook some related questions in our future studies. The basic question of us is that how …, including thermodynamics and dynamics. For instance, for small systems, ‘size’ is a new parameter, and thermodynamic functions should be size-dependent, hence new thermodynamic relations may be required. This is the main topic of the book written by Terill Hill, “Thermodynamics of small systems’. Also in recent years, the Jarzynsky equation, which relates the free-energy difference of two equilibrium states to the exponential average of irreversible work, has gained much attention, which is also relevant with some so-called FTs. It is interesting to apply these new ideas to small chemical reaction systems, where the system is modeled as stochastic Markov process. We all also interested in how dynamic processes, such as nucleation and growth, transport and relaxation processes, are affected by large molecular fluctuations. What I talked today actually mainly concern the nonlinear dynamics, that is only one small part of the whole question: statistical mechanics of mesoscopic chemical systems. I really hope that we can move forward toward these interesting and important directions in our future works. Statistical physics in mesoscopic chemical systems

Thank you Acknowledgements Supported by: National science foundation (NSF)

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