Nonlinear Dynamics in Mesoscopic Chemical Systems

Zhonghuai Hou ( 侯中怀 )

Department of Chemical PhysicsHefei National Lab of Physical Science at Microscale

University of Science & Technology of China

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

Sub-cellular reactions

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

Heterogeneous catalysis

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

Mesoscopic Reaction SystemsN, V(Small)

Molecular Fluctuation

22 1 1orX X

X V N

Nonlinear Chemical Dynamics?

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)

Stochastic Chemical Kinetics chemical reactions are essentially

stochastic, discrete processes

1 2

1 2

( , )( 1, ) ( 1) ( 1, )

( ) ( , )

P X tk AP X t k X P X t

tk A k X P X t

1 1

2 2 ( 1)1 1

k A k A

k X k XX X X

1 2 1 1 2 2

( )( ) ( )

dX tk A k X k A t k X t

dt

Discrete Brownian Motion of X :

Prob. Evolution: Master equation

Sample Trajectory: Langevin equation

1

2

k

kA X

( )

( , )

X t

P X t

stochastic state variable

probability distribution

Chemical Langevin equation (CLE)N Species, M reaction channels, well-stirred in V

Reaction j: j X X v Rate

: ( ) jw VX

1 2

1 1

( ( ))( ( ) ) ( ( )) 1 ( )

M Mji i

ji ji jj j

w td X t V w tt

dt V VV

XX

Molecular fluctuation (Internal noise) 1 V

Deterministic kinetics for V Each channel contributes independently

to internal noise:

Fast numerical simulation

( ) ( ') ( ')i j ijt t t t

The Brusselator

Deterministic bifurcation

21 1 1 2

22 1 1 2

( ) (1 )

( )

F A B X X X

F BX X X

X

X

4

1( ) ( )j jj

dX dt F v w X X

1 2 /S SX A X B A Fixed Point:

Hopf bifurcation:2 1cB B A

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

Co

nce

ntr

atio

n X

1

Control parameter B

Hopf Bifurcation

B=1.9 Stale focus

A=1

Noise Induced Oscillation Stochastic dynamics

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

4

1

1: ( ) ( ) ( )j j jj

CLE dX F dt v w dW tV

X X

( ) ( ') ( ')j k kjdW t dW t t t dt ( ) 0jdW t

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

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)

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)

Analytical study

4

1

1: ( ) ( ) ( )j j jj

CLE dX F dt v w dW tV

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

2 2 6 2 42

2

2 4

p i s s s

s s

C r H r r V

Vr SNR H r V

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

6 2 4/ 4 /sSNR H r V 2 24 /opt rV C

2 2 2 (00)1 2( ) / 2j j jj

w

System Dependent

Internal Noise Coherent Resonance for Mesoscopic Chemical oscillations: a Fundamental Study. Z. Hou, … ChemPhysChem 7, 1520(2006)

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

Further questions

Acknowledgements

Supported by: National science foundation (NSF) Fok Yin Dong education foundation

Thank you