Novel size effect in mesoscopic chemical oscillation systems

Zhonghuai Hou ( 侯中怀 )Sep. 2008, Hefei

Department of Chemical PhysicsHefei National Lab for Physical Sciences at Microscale

University of Science & Technology of China

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

Outline

Introduction the question

Optimal Size Effect

Stochastic Normal Form Theory

Conclusion

Genetic Toggle Switch

In E. ColiNature 2000

Two or more stable states under same external constraints

Reactive/Inactive bistabe

CO+O2 on Pt filed tipPRL1999

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 OscillationCO+O2 Nano-particle C

atal.Today 2003

Synthetic transcriptional oscillator (Repressilator)

Nature 2002

Stationary spatial structures in reaction-diffusion systems

Cellular PatternCO Oxidation on Pt

PRL 2001

Turing PatternBZ 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 AttractorThe Lorenz System

Chemical turbulenceCO+O2 on Pt Surface

Science 2001

Collective behavior involving many molecular

unitsMacroscopic state: ( , )tX r

Microscopic state: ,N Nq p

Nonequilibrium Statistical Mechanics

Sub-cellular reactions

- gene expression- ion-channel gating- calcium signaling … …

Heterogeneous catalysis

- field emitter tips- nanostructured composite surface- small metal particles

Mesoscopic Reaction SystemN, V(Small)

Molecular Fluctuation

22 1 1orX X

X V N

Nonlinear Chemical Dynamics? Chemical Oscillation

Regularity Stochasticity

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

Modeling of Chemical Oscillations

Macroscopic level: Deterministic, Cont.

N Species, M reaction channels, well-stirred in VReaction j:

j X X v Rate:

( ) jW VX

1

( ( ))( ( ) )

Mji

ij ij

W td X t VF

dt V

XX

Oscillation

Co

nce

ntr

atio

n

Control parameter

Hopf Bifurcation

Stale focus

Hopf bifurcation leads to oscillation

: 0

loses stabilityS S

S

X F X

X

has a pair of

pure imaginary eigenvalues

ij J F X

Modeling of Chemical Oscillations

Mesoscopic Level: Stochastic, Discrete

1

;; ;

M

j j j jj

P tW P t W P t

t

X

X ν X ν X XMaster Equation

Kinetic Monte Carlo Simulation (KMC)Gillespie’s algorithm

Exactly

( , )j

Approximately 1 2

1 1

1 ( )

M Mj ji

ij ij jj j

W WXdt

dt V V VV

X X

Chemi cal Langevi n Equati on (CLE)

V Deterministic equation

Internal Noise

New: Noise Induced Oscillation

1.4 1.6 1.8 2.0 2.2 2.4 2.60.4

0.8

1.2

1.6

2.0

2.4

2.8

Con

cent

ratio

n X

1

Control parameter B

V=1E4

Stochastic OscillationA=1, B=1.95

0.0 0.4 0.8 1.2 1.6 2.010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Frequency (Hz)

Pow

er

FFT

A model system: The Brusselator

1.4 1.6 1.8 2.0 2.2 2.4 2.60.4

0.8

1.2

1.6

2.0

2.4

B=2.2 Oscillation

Con

cent

ratio

n X

1

Control parameter B

Hopf Bifurcation

B=1.9 Stale focus

A=1DeterministicStochastic

Noisy Oscillation

Optimal System Size

:

2 :

Peak Height HSNR

Width at H

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

Best performance

Seems to be common … Internal Noise Stochastic Resonance in a Circadian Clock System J.Che

m.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

Analytical study Main idea

Fact: all happens close to the HB

Question: common features near HB?

Answer: normal form on center manifold

Analytical study

1

1: ( ) ( ) ( )

M

j j jjCLE dX F dt v w dW t

V X X

Stochastic Normal Form

3

20

1( )

1( )

r rj jj

i j jj

dr r C r dt dWV

d C r dt dWV

S

X

FJ

X

0 i

) ,1( iba

baT

01

S

S

XX

XXT

y

x

22

111

iZ x iy re

0, for 0, /( )

finite, and coupled via noiserV r C

V r

jjjj

jjjrj

w

w

)sin~cos~(

)sin~cos~(

12

21

Analytical study Stochastic Averaging

3

20

2r r

i

dr r C r dt dWVr V

d C r dt dWr V

2 2 2 (00)1 2( ) / 2 : system dependent

and are de-coupled Solvable

j j jjw

r

Analytical study(…) Probability distribution of r

2

3 2 2( , )2

2r r r

r tr C r Vr

t V

2 4

0 2

2( , )0 ( ) exp

2r

s

r C rr tr C r

t V

3 2( , )0 2 0 s

r

r tr C r Vr

r

1/ 22 2even for <0, 2 / ( 2 )s r rr C V C

Fokker-Planck

equation

Stationary distribution

Most probable radius

Noise induced

oscillation

Analytical study(…) Auto-correlation function

12 21( ) lim ( ) ( ) 2t sCorr r r t r t r e V

21

1( ) lim cos ( )cos ( ) cos( )

2tCorr t t e

2 221/ 4 /c sVr

Correl ati on Ti me:

( ) lim ( ) ( ) ( )* ( )tC x t x t Corr r Corr

Analytical study(…) Power spectrum and SNR

22

2 202 1

( ) 2 ( )( )

i srPSD C e d

2 2 4 21 0 2

22 22

2

2

p i s s s

s s c

C r H r r V

Vr SNR H r

2 2

4( )0 r

opt

CSNRV

V

Optimal system size:

Analytical study(…) 3

20

2r r

i

dr r C r dt dWVr V

d C r dt dWr V

Universalnear HB

2 22 / 2s r rr C V C

2 2

21/ 4 /c sVr , ,s cV r

2/ s cSNR H r

2 24 /opt rV C 2 2 2 (00)

1 2( ) / 2j j jjw

System Dependent

ChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J. Phys. 9, 403(Nov. 2007) ;

System Size Bi-resonance

System Size Biresonance for Intracellular Calcium SignalingJ. Zhang, Z.H.Hou, H.W.Xin, ChemPhysChem 5, 1041(2004)

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

N 个 HH 神经元的耦合体系……S S S S

N

3 41 1( ) ( ) ( ) ( 2 )i

Na i i Na K i i K L i L i i i

dUC g m h U U g n U U g U U I U U U

dt

( )(1 ) ( ) ( )i i i

i x i x i xx U x U x t

, , and 1x m n h i N

Log(S)

Coupled Neuron Network: ChemPhysChem 5 ， 1062(2004)

Coupled Calcium Oscillation: Phye.Rev.E 74 031901(2006)

Double System Size Resonance

Summary In mesoscopic chemical systems, molecular fluctua

tions 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

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 )