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