BZ and the Turing Instability
Tamas Bansagi
BZ Boot camp @ Brandeis
What are these two patterns?
Tropical fishTuring patterns ina chemical reaction
Alan Turing’s theory‘The chemical basis of morphogenesis’
Philosophical Transactions of the Royal Society of London, (Series B, No.641, Vol. 237, 37-72,1952).
vDvugdt
dv
uDvufdt
du
v
u
2
2
),(
),(
Kinetics DiffusiveTransport+
u activator
v inhibitor
Di diffusion coefficients
f , g kinetic rate equations
Reaction-Diffusion equations
- In a Reaction-Diffusion system, patterns stationary in time and periodic in space
may develop if Du≠Dv.
- In the same system, if Du=Dv≥0 u and v tend to a stable uniform steady state.
- More precisely: Du<Dv (Long range inhibition, short range activation required)
NonlinearityConsider:
Alan Turing’s theory
Chemical pre-patterning through diffusion driven instability.
Fo
rma
tion
an
d d
eve
lop
me
nt o
f em
bry
o
Early stageMorphogenesis (development of pattern and form)
Positional information template
Cell differentiation, migration, shape change
Turing patterns in experimentLiving systems:
• Difficult to identify pre-patterning species (morphogens)
• Mechanisms are very complicated Chemical systems:
• Relatively easy to identify species• Mechanisms tend to be simpler • Seemed easier to find/design systems supporting Turing patterns
Turing patterns in experimentLiving systems:
• Difficult to identify pre-patterning species (morphogens)
• Mechanisms are very complicated Chemical systems:
• Relatively easy to identify species• Mechanisms tend to be much simpler • Seemed easier to find/design systems supporting Turing patterns• Reality: first Turing patterns reported in 1990 – Clorite-Iodide-Malonic acid
reaction (V. Castets, E. Dulos, J. Boissonade, P. De Kepper, 1990)
Examples from Biology:
• Disposition of feather buds in chick (H. S. Jung, 1998)• Hair follicles in mice (S. Sick, S. Reinker, J. Timmer, T. Schlake, 2006)• Skin pattern regeneration in zebra fish (M. Yamaguchi, E. Yoshimoto, S. Kondo,
2007)
Turing patterns in the BZ reactionOregonator model
X activator
Y inhibitor
Z oxidized form of catalyst
Oregonator model in dimensionless form
Turing patterns in the BZ reaction1D Oregonator reaction-diffusion system
Dx=Dy=Dz=1
Turing patterns in the BZ reaction1D Oregonator reaction-diffusion system – Homogeneous perturbation
Dx=0.01, Dy=Dz=1
Turing patterns in the BZ reaction1D Oregonator reaction-diffusion system – Inhomogeneous perturbation
Dx=0.01, Dy=Dz=1
It is in the model but how can we “slow down” the activator or “speed up” the inhibitor?
Turing patterns – BZ-AOT system
AOT – Aerosol OT - sodium bis(2-ethylhexyl)
sulfosuccinate
Aqueous BZ chemicals
Oil (Octane)
Water-in-oil microemulsionCommunication between droplets• collision (fusion and fission) ~ 10-3 s time scale (exchange of polar species)• nonpolar species in oil ~ 10-4 - 10-5 s time scale
Role of Br2
• produced in the reaction• quickly diffuses in the oil phase• its reaction with malonic acid gives bromide (Y)(Thorough review – V. K. Vanag and I. R. Epstein, 2008)
Long range inhibition
Rh
= 5
-20
nm
Turing patterns – BZ-AOT system
2D
3D
Experiments (oil: cyclooctane)
Reconstruction(inverse Radon transform)
Reconstructed patterns
Numerical results in an Oregonator-based model (T. Bansagi, V. K. Vanag, I. R. Epstein, 2011)