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)

Spots and stripes

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 diffusion

Spatially uniform steady state

Dynamics in a linearized problem

Characteristic equation (1)

Characteristic equation (2)

Very small and very large wavenumbers are stable 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 uniform perturbationslarge wavenumbers Perturbations with very small and large wavenumbers decay. What happens at intermediate wavenumbers?

Condition for instability (1)

Condition for instability (2) When is this form negative? Minimum must be negative What is the minimum?

Condition for instability (3)

Condition for instability (4) Spatially uniform problem Diffusion-induced instability

Physical interpretation (1)

Physical interpretation (2) AI

Short-range activation and long-range inhibition Localized activator increases its own production and production of inhibitor Diffusible inhibitor prevents the spread of autoactivation AI A(x) I(x)

Effect of finite size (1)

Two systems of different size, L 1 <L 2 L1L1 L2L2 First nonzero wavenumber 2 nd 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 no. 5928, pp 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 space time (0-100 mins) space space Science 8 May 2009: Vol no. 5928, pp

Turing patterns in experiments 4 mm Science 8 May 2009: Vol no. 5928, pp

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?