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Internal Noise Coherence Resonance in mesoscopic chemical oscillation systems. Zhonghuai Hou ( 侯中怀 ) Perugia , SR2008 Email: [email protected] Department of Chemical Physics Hefei National Lab for Physical Science at Microscale University of Science & Technology of China (USTC). - PowerPoint PPT Presentation
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Internal Noise Coherence Resonance in mesoscopic chemical oscillation systems
Zhonghuai Hou ( 侯中怀 )Perugia , SR2008
Email: [email protected] 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