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)