pattern formation in nonlinear reaction-diffusion systems
TRANSCRIPT
Pattern formation in nonlinear reaction-diffusion systems
Animal coat patterns
Stripes along the dorsoventral axis
Stripes along the AP axis
cos(x) cos(2x)
x is the normalized distance along the body axis
cos(51x)
Phenomenological models (1)
Allan Turing (1952): Reaction-diffusion model for animal skin color patterns
Turing mechanism
• Patterned state of a reaction-diffusion system arises as an instability of a spatially uniform solution
• Furthermore, this can happen when the system is linearly stable in the absence of diffusion
• The remarkable thing is that diffusion, a mechanism that works against spatial nonuniformities, destabilizes a spatially uniform state
• This, of course, is possible due to nonlinear kinetics
Phenomenological models (2)
"This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of the greatest importance in the present state of knowledge."
Nonlinear kinetics and diffusionproduction
1 1 1 1 2
production
2 2 2 1 2
21 1 1
1 1 1 220,
22 2 2
2 2 1 220,
( , ) ( , )Chemistry:
( , ) ( , )
Coupled equations & B.C.:
( , ) 0
( , ) 0
L
L
C C x t R C C
C C x t R C C
C C CD R C C
t x x
C C CD R C C
t x x
1 2/ / ( , )i i i i i ix x dxC dx D dC dx D dC dx R C C dx
Spatially uniform steady state
1 1 2
1 1 2
' '1 1 1 2 2 2
' 2 '' '1 1
11 1 12 2 1 2
( , ) 0Nonlinear algebraic system
( , ) 0
Small inhomogeneous perturbations:
( , ) ( , ) , ( , ) ( , )
Substitute and Taylor expand nonlinear terms
R C C
R C C
C x t C x t C C x t C x t C
C Ca C a C D
t x
1 2
'1
0,
' ' '' '2 2 2
21 1 22 2 2 2
0,
,
0
0
iij
L
L
j C C
C
x Ra
C C Ca C a C D
t x x
C
Dynamics in a linearized problem
' ' '
11 12 1' 1 1
21 22 21 1
1'
2
0( , )
0( , )
Solve by separation of variables: cos( )
t xx
t
C MC DC
a a DC x t CC M D
a a DC x t C
C qx e
Characteristic equation (1)
11 1
21 1
21 11 1 12 2 1 1
2 21 1 22 2 2 2
( , ) ' cos( ) substitute into linearized equation
( , )
cos( ) cos( ) cos( ) cos( )
cos( ) cos( ) cos( )
t
t t t t
t t t
C x t CC qx e
C x t C
qx e a qx e a qx e D q qx e
qx e a qx e a qx e D
2
21 11 1 12 2 1 1
22 21 1 22 2 2 2
This defines as a function of wavenumber
and parameters of the problem
For stability, Re( )should be negative f
cos
o
( )
r all
tq qx e
a a D q
a a D q
q
Characteristic equation (2)
211 11 12
2221 2 22
1
2
2 21 11 2 22 12 21
2 2 2 211 22 2 1 1 11 2 22 12 21
0Look for nontrivial solution:
0
0
d 00
0
0
et
D q a D q a a a
a a
D q a a
a D
q D D D q a D q a a
q
a
a
Very small and very large wavenumbers are stable
2 2 2 211 22 2 1 1 11 2 22 1
211 22 11 2
2 21
2 12 21
1.If the system is stable with respect to uniform perturbations, 0 :
0
Both eigenvalues are
2. Perturbations with v
negati
ery lar
ve
0
q
a a a a a
a a q D D D q a D q a
a
a a
2,
2 221 11 12 1
1,2 1,22 221 2 22 2
ge wavenumbers:
0
0
Both eigenvalues are negative
i jDq a
D q a a D qD q
a D q a D q
Thus, the system is linearly stable with respect to perturbations with both very large and very small wavenumbers.
Dispersion relation: leading eigenvalue as a function of the wavenumber
2q
2max ( )q
uniform perturbations large wavenumbers
Perturbations with very small and large wavenumbers decay. What happens at intermediate wavenumbers?
Condition for instability (1)
1 2
211 22 11 22 12 21 1 2
11 2 11 22 12 2 21
1)System is stable in the absence of diffusion ( = =0):
0 has roots, , 0
0
2) Condition for diffusion-induced instability.
One of these conditions
0,
D D
a a a a a a
a a a aa a
2 2 2
11 1 22 2
211 22
1
2
1
12 2
2
must be violated:
0 always true ( )
The only way to generate instability is
(
throu
Why
g
0
?
h
)
( ) 0
H q a D q a D q
a a
a a
H q
q D D
Condition for instability (2)
2 2 2 211 1 22 2 12 21
0,from stability of a lumped system2 4 2
1 2 1 22 2 11 11 22 1
2
2 21
( ) : a quadradic form in
The only way to generate instability is through
) 0
(
( ) 0H q
H q a D q a D q a a q
H q D D q D a D a q a a a a
2( )H q
2q
2minq
• When is this form negative?• Minimum must be negative• What is the minimum?
Condition for instability (3)
2 4 21 2 1 22 2 11 11 22 12 21
2 222 11min min
2 1
2
1 22 2 111 2 22 11 22 1111 22 12 21
2 1 2 1
2
1 22 2 1111 22 12 21
1 2
( )
1For instability, ( ) 0
2
04 2
04
H q D D q D a D a q a a a a
a aq H q
D D
D a D aD D a a a aa a a a
D D D D
D a D aa a a a
D D
Condition for instability (4)
1
2
1 22 2 1111 22 12 21
1 2
1 22 2 11 1 2 11 22 12 21
22 2 11
1 22
11 22
11 22 12 21
2 11
0
3.
1. 0
2
04
2 0
. 0Summary:
0
D a D aa a a a
D D
D a D a D D a a a
D a
a a
a a a
a
D a
D a
a
D a
Spatially uniform problem
Diffusion-induced instability
Physical interpretation (1)
1 2
11 22 11 22
11 22 12 21 11 22 12 21
1 2
22
112 2 11
111 1
1 ,
22 11 12 2
1. 0 at least one of or is negative.
2. 0
3. 0
0 promotes its own production ( )
Sinc
0
ACTIVATOR
take <0
e 0a d ,
n 0
C C
a a a a
a a a a a a a a
D a D a
Ra C
C
a a a a
a
a
1 0.This implies a pattern of a Jacobian:
+ + + -
- - + -or
Physical interpretation (2)
1 21 22 2 11 11 22
11 22
+ -Activator/Inhibitor
+ -
0 /( 0) 0
D DD a D a a a
a a
A I
Short-range activation and long-range inhibition
• Localized activator increases its own production and production of inhibitor
• Diffusible inhibitor prevents the spread of autoactivation
2 21 2Activator Inhibitor
11 22
D Dl l
a a A I
A(x)I(x)
Effect of finite size (1)
2q
2max ( )q
2 22 11min
2 1
1
2
a aq
D D
0,
All wavenumbers are allowed in an infinite system.
In the finite system, the spectrum is discrete:
From 0 , 0,1, ,L
C nq n
x L
Two systems of different size, L1<L2
2q
2max ( )q
2 22 11min
2 1
1
2
a aq
D D
2q
2max ( )q
2 22 11min
2 1
1
2
a aq
D D
, 0,1, ,n
q nL
L1 L2
1 11
( )q LL
First nonzero wavenumber
1 22
( )q LL
22
2q
L
2nd nonzero wavenumber
A finite system is linearly stable with respect to all perturbations
below a critical length
An Experimental Design Method Leading to Chemical Turing Patterns
Science 8 May 2009: Vol. 324. no. 5928, pp. 772 - 775
Judit Horváth,1 István Szalai,2 Patrick De Kepper1
Chemical reaction-diffusion patterns serve as prototypes for pattern formation in living systems, but only two isothermal single-phase reaction systems have produced sustained stationary reaction-diffusion patterns so far. We designed an experimental method to search for additional systems on the basis of three steps: (i) generate spatial bistability by operating autoactivated reactions in open spatial reactors; (ii) use an independent negative-feedback species to produce spatiotemporal oscillations; and (iii) induce a space-scale separation of the activatory and inhibitory processes with a low-mobility complexing agent. We successfully applied this method to a hydrogen-ion autoactivated reaction, the thiourea-iodate-sulfite (TuIS) reaction…
Spatially periodic steady state
4 mm
spacetime (0-100 mins) space
space
Science 8 May 2009: Vol. 324. no. 5928, pp. 772 - 775
Turing patterns in experiments
4 mm
Science 8 May 2009: Vol. 324. no. 5928, pp. 772 - 775
Questions
• What is the fate of unstable modes at long times?
• Numerical solution of the model in the unstable regime
• Turing mechanism can indeed generate spatial patterns in solution chemistry
• Does the same mechanism work in biology, in generating animal coat and other patterns?