Internal Noise Coherence Resonance in mesoscopic chemical oscillation systems

Zhonghuai Hou ( 侯中怀 )Perugia ， SR2008

Email: hzhlj@ustc.edu.cnDepartment of Chemical PhysicsHefei National Lab for Physical Science at MicroscaleUniversity of Science & Technology of China (USTC)

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

Internal Noise Coherence Resonance

Stochastic Normal Form Theory as well as its applications

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

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)M. Wang, Z.Hou, H.Xin. ChemPhysChem 7 ， 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

New features In the literature (Two important papers): Hu Gang, ... (PRL 1993) External noise + Saddle Node Pikovsky/Kurths (PRL 1997) External noise + Excitability

In our work: Internal noise + Supercritical Hopf

Internal noise coherence resonance (INCR)Also: System Size Resonance (SSR)

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

Time scale separation

1, ~ 1O

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) ;

Applications of the theory

Extension to general reaction networks Control CR via noise modulation

Multiple Noise: External and Internal

Entropy Production: Scaling Law

General Reaction Network

CLE:

Control CR: Noise Modulation What really matters:

3

20

2r r

i

dr r C r dt dWVr V

d C r dt dWr V

2 22 / 2s r rr C V C 2 2

21/ 4 /c sVr

2

s cSNR r 2 24 /opt rV C

Example: Colored Noise

2/ ( 1)dX dt A B X X Y t 2/dY dt BX X Y

Model system: Brusselator

2

20

1

2

AS

00 0 iS e d

Type 1: Ornstein-Uhlenbeck (OU)

1ou ou

c c

Dt t

& 0 2 2

0

2

1ouc

DS

Type 2: Power-Limited (PL)

1pl pl

c c

Dt t

& 2 2

2

1c

plc

DS

Example: Colored Noise

OU

PL

Multiple Noise: External+Internal Model system: CO Oxidation

11 1 1

1 1

1 1

1( ) ( ) ( )

1( ) ( ) ( ) , 1

M M

j j j j jj j

M M

j j j j jj j

dXv w v w t D w t

dt V

dXv w v w t

dt V

X X

X X

' 't t t t

( ) ( ') ( ')i j ijt t t t Internal noise:

External noise:

2002 2002 2 2

11 21 11 1 22 2( )1

2 2

M v v av v aD

N

The Interplay

Internal Noise

Exte

rnal N

ois

e

Too much internal noise, no CR with external noise: SR as a collective behavior of ion-channel clusters

Entropy Production?

Macroscopic Level: Nonequilibrium Statistical Thermodynamics

0i ii iv

Ad Sp dv W j

dt T T

;vS t s t dv r ii

i

cs s

t c t

s

sJ

t

ii ir rr

cj v w

t

I. Prigogine 1970s

Entropy Production?

Mesoscopic Level: Stochastic Thermodynamics

; ln ;

0

X

e i i

S t P X t P X t

d S d S d SdSJ A

dt dt dt dt

Luo,Nicolis 1984; P.Gaspard 2004

Entropy Production?

Single Trajectory Level: Dynamic Irreversibity

U. Seifert, PRL 2005

0 1 2 1j

j j n ru u u u u u u

A Random Trajectory

Trajectory Entropy ln ;s p u

tot ms s s Total Entropy Change

0;0ln

;n

ps

p t

u

u

1;ln

;

j j

mj j j

Ws

W

u r

u r

R t u u

0|ln

|tot R

n

ps

p

u u

u u

0tots

Fluctuation Theorems !

1totse

totstot totp s p s e

Integrate FT

Detailed FT(NESS)

1BW G k T

Jarzynsky Equality

e

Probability of Second-law violation 0

is exponentially small tots

Brusselator

(X+1,Y-1)(X,Y-1)

(X-1,Y)

(X-1,Y-1)

(X+1,Y)(X,Y)

(X+1,Y+1)(X,Y+1)

Y

X

(X-1,Y+1)

(a)

92 94 96 98 100 102 104 106

197

198

199

200

201

202

203

Y

X

un=u0

(b)

FT holds

-4 0 4 8 12

0.0

0.1

0.2

0.3

0.4 b=1.9 b=2.0 b=2.1

P(

s m)

sm

(a)

-4.5 -3.0 -1.5 0.0 1.5 3.0 4.5-4.5

-3.0

-1.5

0.0

1.5

3.0

4.5

ln(P

(s m

)/P(-s

m))

sm

b=1.9 b=2.0 b=2.1

(b)

Scaling law

System Size Dependence

2 3 4 51

2

3

4 b=1.9 b=2.1

lnP

lnV2 3 4 5

1

2

3

4

log

(Vr2

)

logV

B=1.9 B=2.1

Simulation SNF Theory

,m L mc Ds s c X Y

u

2 2, ln 1 cos

2ms s

X rs c X Y

X X Vx x

0 2 2

0 0

21 cos

r

m L ms s

V rs rdr r d Vr

x x

u

1/22 22 /

2r

mr

C Vr

C

2lim 0V mr

rC

2

2lim 02V mr V

Conclusion Noise Induced Oscillation Stochastic Modeling is important Optimal System Size: Internal Noise Coheren

ce Resonance Intrinsic behavior Stochastic Normal Form Theory

Universality + Underlying mechanism Prediction: Control CR Nonequilibrium Thermodynamics: FT

Acknowledgements

Supported by: National science foundation (NSF)

Thank you