Beyond diffusion: Patterns

Transcription

1 Beyond diffusion: Patterns L5 dunkel@mit.edu

2 2D diffusion equation

3

4 Plankton ESA cost of Ireland

5 Animals

6 Shells

7 Compare: vibrated granular media

8 Compare: vibrated granular media

9 Slime mold aggregation of a starving slime mold (credit: Florian Siegert)

10 Belousov-Zhabotinsky reaction feature=player_detailpage&v=bh6brt4xjcw Mix of potassium bromate, cerium(iv) sulfate, malonic acid and citric acid in dilute sulfuric acid the ratio of concentration of the cerium(iv) and cerium(iii) ions oscillated This is due to the cerium(iv) ions being reduced by malonic acid to cerium(iii) ions, which are then oxidized back to cerium(iv) ions by bromate(v) ions

11 eferred handedness [40, 48, 49, 50]. For trajectory of a swimming cell can arxiv:exhibit a pr nd Caulobacter [48] have been observed 2d Swift-Hohenberg example,model the bacteria Escherichia coli [40] an broken vita tensor, i = / xi for i = 1, 2, and we use ij denotes the Cartesian components of the Levi-Civ reflection-symmetry a summation convention for equal indices throughout. Sea urchin sperm cells near surface (high concentration) b=0 Riedel et al (2007) Science 24, 14. Monday, (a) February Stationary hexagonal lattice of the pseudo-figure 2. Effect of symmetry breaking

12 arxiv: Symmetry breaking near boundaries

13 Broken reflection-symmetry at surfaces Sea urchin sperm Gibbons (1980) JCB in bulk (dilute) near surface (dilute) similar for bacteria (E. coli): Di Luzio et al (2005) Nature

14 2d Swift-Hohenberg model arxiv: reflection-symmetry t = U ( ) ( 2 ) 2 b =0 a>0 U( ) = a b c 4 4 (t, x) = v a<0

15 arxiv: d Swift-Hohenberg model tropic fourth-order model for a non-conserved scalar or reflection-symmetry r (t, x), given by 2 b(1)= 0, t = U ( ) ( ) 2 he time derivative, and = 2 is the d-dimensional ved from a Landau-potental U ( ) 0 a 2 b 3 c 4 U ( ) = + +, (2) e rhs. of Equation (1) can also be obtained by variational ed energy functional. In the context of active suspensions, x) =fluctuations, v tify local (t, energy local alignment, phase will assume throughout that the system is confined to d of volume (3) Figure 1. Numerical illustration of structural transitions in the order-pa

16 2d Swift-Hohenberg model arxiv: broken reflection-symmetry t = U ( ) ( 2 ) 2 b =0 a>0 U( ) = a b c 4 4 (t, x) = v a<0 b<

17 trajectory of a swimming cell can arxiv:exhibit a pre 2d Swift-Hohenberg example,model the bacteria Escherichia coli [40] an broken tropic fourth-order model for a non-conserved scalar or ij denotes the Cartesian components of the Levi-Civ reflection-symmetry r (t, x), given by a summation convention for equal indices throughout. 2 b(1)=, t = U ( ) ( ) he time derivative, and = 2 is the d-dimensional ved from a Landau-potental U ( ) a 2 b 3 c 4 U ( ) = + +, (2) e rhs. of Equation (1) can also be obtained by variational ed energy functional. In the context of active suspensions, x) =fluctuations, v tify local (t, energy local alignment, phase will assume throughout that the system is confined to d of volume (3) Figure 2. Effect of symmetry breaking.

18 eferred handedness [40, 48, 49, 50]. For trajectory of a swimming cell can arxiv:exhibit a pr nd Caulobacter [48] have been observed 2d Swift-Hohenberg example,model the bacteria Escherichia coli [40] an broken vita tensor, i = / xi for i = 1, 2, and we use ij denotes the Cartesian components of the Levi-Civ reflection-symmetry a summation convention for equal indices throughout. Sea urchin sperm cells near surface (high concentration) b=0 Riedel et al (2007) Science 24, 14. Monday, (a) February Stationary hexagonal lattice of the pseudo-figure 2. Effect of symmetry breaking

19 Turing model A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London. B 327, (1952)

20 Turing model

21 Turing pattern Kondo S, & Miura T (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329 (5999),

22 The matching of zebrafish stripe formation and a Turing model Kondo S, & Miura T (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329 (5999),