Novel size effect in mesoscopic chemical oscillation systems
Zhonghuai Hou ( 侯中怀 )
Department of Chemical PhysicsHefei National Lab of Physical Science at Microscale
University of Science & Technology of China
理论化学的三个层面
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
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
反应动力学
量子 + 统计
结果:反应势能面,反应途径,速率常数
夸克 美洲豹
kA B C
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)
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)
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) ;
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
Application to Brusselator Detailed FT holds
(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)
-4 0 4 8 12
0.0
0.1
0.2
0.3
0.4
B C D
P(s
tot)
stot
-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
4.5ln
(P(A
)/P(-
A))
A
b=1.9 b=2.0 b=2.1
0 20 40 60 80
800
1200
1600
X(t
)
t
Application to Brusselator System Size Dependence
2 3 4 51
2
3
4 b=1.9 b=2.1
lnP
lnV
Thermodynamic Characterization of
dynamic Hopf bifurcation
‘ ’
2 3 4 51
2
3
4
log
(Vr2
)logV
B=1.9 B=2.1
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
Future directions
? How does fluctuation influence the properties of small chemical systems
Thermodynamics
Size dependence
Jarzynski equality
Fluctuation theorems…
DynamicsNucleation and growth
Transport/relaxation
Nonlinear dynamics …
Statistical physics in mesoscopic chemical systems
Acknowledgements
Supported by: National science foundation (NSF)
Thank you
Details