Nonlinear Dynamics in
Mesoscopic Chemical Systems
Zhonghuai Hou (侯中怀)
Department of Chemical Physics
Hefei National Lab of Physical Science at Microscale
University of Science & Technology of China
Nonlinear Chemical Dynamics
 far-from equilibrium, self-organized,
complex, spatio-temporal structures





Oscillation
Multistability
Patterns
Waves
Chaos
…
Temporally
Variations
of
Travelling/Target/Spiral/Soliton
Two
Stationary
or morePeriodic
spatial
stablestructures
states
under
in
Aperiodic/Initial
condition
reaction-diffusion
same Concentrations
external
waves
constraints
systems…
sensitivity/strange
attractor
Collective
behavior involving
many molecular
units
Macroscopic state: X(r, t )
Microscopic state: q N , p N 
Reactive/Inactive
Synthetic
Strange
Turing
transcriptional
Pattern
Attractor
bistabe
Calcium
Spiral
Wave
oscillator
CO+O2
BZ
The
Reaction
Lorenz
on
(Repressilator)
PtTissues
System
System
filed tip
in
Cardiac
Nature
PNAS
PRL1999
2003
2002
Nature 1998
Genetic
Chemical
Rate
Cellular
PEEM
Toggle
Oscillation
turbulence
Image
Pattern
Switch
CO+O2
COCO+O2
Oxidation
Inon
E. Pt
Coli
NanoSurface
on Pt
particle
Science
Nature
PRL
Catal.Today
1995
2001
2000
2001
Mesoscopic Reaction Systems
 Heterogeneous
catalysis
N, V
(Small)
- field emitter tips
- nanostructured
composite surface
- small metal particles
Molecular
Fluctuation
?
X2  X
X
 Sub-cellular
reactions
- gene expression
- ion-channel gating
- calcium signaling
……
2

1 or 1
V
N
Nonlinear Chemical
Dynamics
Noise/Disorder
 Noise and disorder play constructive
roles in nonlinear dynamical systems

Noise Induced
Disorder
Taming
Chaos
sustained
by
Pattern
Topological
spiral
Transition
waves
Disorder
et al.,H.PRL
81,
89,
2854
280601
(2002)(2003)
F.Z.Hou,
Qi, Z.Hou,
Xin,
PRL
91, (1998)
064102
Stochastic Chemical Kinetics
 chemical reactions are essentially
stochastic, discrete processes
k1
X (t )
stochastic state variable
k2
P( X , t )
probability distribution

X
A 

Discrete Brownian
Motion of X :
k1 A
k1 A



 X 1
X  1 X 

k2 X
k2 ( X 1)
Prob. Evolution:
Master equation
P( X , t )
 k1 AP( X  1, t )  k2 ( X  1) P( X  1, t )
t
 (k1 A  k2 X ) P ( X , t )
Sample Trajectory:
Langevin equation
dX (t )
  k1 A  k2 X   k1 A1 (t )  k2 X  2 (t )
dt
Chemical Langevin equation (CLE)
N Species, M reaction channels, well-stirred in V
Reaction j:
X  X vj
Rate:
w j ( X)  V
12
 w j ( X(t )) 
d ( X i (t ) V )
 wi ( X(t ))  1
  ji 
 ji 

  j (t )

dt
V
V j 1 
 V

j 1

M
M
 Molecular fluctuation (Internal noise)  1 V
 Deterministic kinetics for V  
 Each channel contributes independently
to internal noise: i (t ) j (t ')   ij (t  t ')
 Fast numerical simulation
The Brusselator
 Deterministic bifurcation
dX  dt  F ( X)   j 1 v j w j ( X)
4
F1 ( X)  A  (1  B) X 1  X 12 X 2
Fixed Point:
X1S  A X 2 S  B / A
Hopf bifurcation:
B  Bc  A2  1
2.4
Concentration X1
F2 ( X)  BX 1  X X 2
2
1
A=1
B=1.9
Stale focus
2.0
B=2.2
Oscillation
1.6
1.2
0.8
0.4
1.4
Hopf Bifurcation
1.6
1.8
2.0
2.2
Control parameter B
2.4
2.6
Noise Induced Oscillation
 Stochastic dynamics
1
CLE: dX   F ( X)dt 
V
v j w j ( X)dW j (t )
-2
10
Stochastic Oscillation
A=1, B=1.95
-4
FFT
10
-6
10
2.0
1.6
Power
Concentration X1
2.4
j 1
dW j (t )dWk (t ')   kj (t  t ')dt
dW j (t )  0
2.8

4
4
V=1E
1.2
-8
10
-10
10
-12
10
0.8
0.4
1.4
-14
10
-16
1.6
1.8
2.0
2.2
Control parameter B
2.4
2.6
10
0.0
0.4
0.8
1.2
Frequency (Hz)
1.6
2.0
Optimal System Size
Peak Height : H
SNR 
Width at H 2 : 
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.Chem.Phys. 119, 11508(2003)

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)

Optimal Particle Size for Rate Oscillation in CO Oxidation on
Nanometer-Sized Palladium(Pd) Particles
J.Phys.Chem.B 108, 17796(2004)

Effects of Internal Noise for rate oscillations during CO oxidation on
platinum(Pt) surfaces
J.Chem.Phys. 122, 134708(2005)

Internal Noise Stochastic Resonance of synthetic gene network
Chem.Phys.Lett. 401,307(2005)
Analytical study
 Stochastic Normal Form
CLE: dX 
1
 F ( X)dt 
V
 F
J 
 X
 
1
dr  (  r  Cr r )dt 
V
1
2
d  (0  Ci r )dt 
V
3

4
j 1
v j
w j ( X)dW j (t )
    i0



 X S (1, a  ib )
 1 0

T  
a b
 j rj dW j
X 1  X 1S 
x
1 

   T 
 y
 X 2  X 2S 

Z  x  iy  rei
 dW j
j j
V  , for   0,r0   /(Cr )
V finite, r and  coupled via noise
 rj  (~j1 cos  ~j 2 sin  ) w j
j  (~j 2 cos  ~j1 sin  ) w j
Analytical study

Stochastic Averaging





3
dr    r  Cr r 
dt 
dWr

2Vr 
V


d  0  Ci r  dt 
dW
r V
2
   j (  ) w
2
2
j1
2
j2
(00)
j
/ 2 : system dependent
r and  are de-coupled  Solvable
Analytical study

Probability distribution of r
FokkerPlanck
equation
2
 (r , t )

  r   r  Cr r 3   2 2Vr    
 2r 
t
2V
Stationary
distribution
 2 r 2  Cr r 4 
 (r , t )
 0   s (r )  C0 r exp 

2
t
 2 V

Most
probable
radius
Noise
induced
oscillation
 s (r , t )
 0   r  Cr r 3   2 2Vr  0
r


1/ 2
even for  <0, rs    2  2Cr 2 / V   (2Cr ) 


Analytical study

Auto-correlation function
Corr (r )  limt  r (t )r (t   )  rs2   2e1 21V
Corr ( )  limt  cos  (t ) cos  (t   ) 
1
cos(1 )e  2
2
C( )  limt  x(t ) x(t   )  Corr (r )* Corr ( )
Cor r el at i on Ti me:
 c  1/ 2  4Vrs 2 /  2
Analytical study

Power spectrum and SNR

PSD( )  2 C ( )e
0
 i
d
r 2
2
2
2  (  1 )
2
s
 p  1  0  Ci rs2
H  rs2 2
  2
SNR  H   4rs6V 2  4
 2 2Vrs2
Optimal system size:
4Cr
( SNR)
 0  Vopt  2 2
V
 
2rs4V  2
Analytical study

 

3
dr    r  Cr r 
dt

dWr

2Vr 
V


d  0  Ci r 2  dt 
dW
r V
rs 

 2  2Cr  2 / V  
  2C 
 c  1/ 2  4Vrs / 
2
r
2
SNR  H /   4rs6V 2 /  4
Vopt  4Cr /  2  2
 2   j ( 2j1  2j 2 ) w(00)
/2
j
Universal
near HB
V , rs , c 
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 noiseinduced 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
Thank you
Supported by:
National science foundation (NSF)
Fok Yin Dong education foundation