Transcript Bansagi
BZ and the Turing Instability Tamas Bansagi BZ Boot camp @ Brandeis What are these two patterns? Tropical fish Turing patterns in a 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). Consider: du f (u , v) Du 2u dt dv g (u, v) Dv 2 v dt Kinetics+ u activator Nonlinearity v inhibitor Di diffusion coefficients f , g kinetic rate equations Diffusive Transport 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) Alan Turing’s theory Morphogenesis (development of pattern and form) Chemical pre-patterning through diffusion driven instability. Positional information template Cell differentiation, migration, shape change Formation and development of embryo Early stage Turing patterns in experiment Living 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 experiment Living 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 reaction Oregonator model X Y Z Oregonator model in dimensionless form activator inhibitor oxidized form of catalyst Turing patterns in the BZ reaction 1D Oregonator reaction-diffusion system Dx=Dy=Dz=1 Turing patterns in the BZ reaction 1D Oregonator reaction-diffusion system – Homogeneous perturbation Dx=0.01, Dy=Dz=1 Turing patterns in the BZ reaction 1D 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 Water-in-oil microemulsion Rh = 5-20 nm Communication 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) Oil (Octane) Aqueous BZ chemicals AOT – Aerosol OT sodium bis(2-ethylhexyl) sulfosuccinate Long range inhibition Turing patterns – BZ-AOT system 2D 3D Experiments (oil: cyclooctane) Reconstructed patterns Reconstruction (inverse Radon transform) Numerical results in an Oregonator-based model (T. Bansagi, V. K. Vanag, I. R. Epstein, 2011)