Beyond diffusion:
Patterns
18.354 - L5
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2D diffusion equation
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Monday, February 24, 14
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Monday, February 24, 14
Plankton
ESA cost of Ireland
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Animals
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Shells
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e to the static angle of repose and subseque
Compare: vibrated granular media
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Compare: vibrated granular media
644
I. S. Aranson and L. S. Tsimring: Patterns and collective behavi
FIG. 3
terns in
various
the vib
ral, int
shots o
lique lig
longitudinal vortices in rapid chute flows !Forterre and
Pouliquen, 2001", see Fig. 8, long modulation waves
Monday, February 24, 14
container. One of the
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for this class of system
Slime mold
aggregation of a starving slime mold (credit: Florian Siegert)
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Monday, February 24, 14
Belousov-Zhabotinsky reaction
http://www.youtube.com/watch?
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
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Monday, February 24, 14
eferred handedness [40, 48, 49, 50]. For
trajectory of a swimming cell can
arxiv:exhibit
1208.4464a 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
arxiv: 1208.4464
Symmetry breaking
near boundaries
Monday, February 24, 14
Broken reflection-symmetry at surfaces
Sea urchin sperm
near surface (dilute)
in bulk (dilute)
F I GURE 1
Dark- f i e l d mi crographs o f l ive sperm o f Tr i pneus t es suspended in natura l seawa ter conta i n i ng
0.2 mM EDTA and ad j usted to pH 8 .3 ( refer red to as standard seawa ter ) . The mi crograph , wh i ch was
taken a f ew seconds af ter mov i ng this f ield into the l ight beam, shows some sperm in l i ght - i nduced
qu i escence , and some that are sw i mm i ng. Among those sw i mm i ng , mos t show l i t t le asymme t ry as i nd i cated
by the near st ra ightness of the i r pa ths . Exposure : 1 s. X 380 .
similar for bacteria (E. coli):
ght absorpt i on proper t i es o f the g l ass. By us i ng
ser i es of f i l ters (Ze i ss : UG5 , BG3 , BG12) , the
t i ve wave l ength i n l i ght - i nduced s topp i ng has
enMonday,
de t erm i February
ned to l i e 24,
i n the
14 range 400-500 r i m,
Gibbons (1980) JCB
Di Luzio et al (2005) Nature
Howeve r , we canno t be cer ta i n on th i s po i nt because i t is di f f i cul t to make an accura t e es t i ma t e
of the percent age of qu i escent spe rm when th i s is
<1% . Except at the h i ghes t intens i t i es , the spe rm
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or
reflection-symmetry
r ⇤(t, x), given by
2
b(1)= 0
⇤
⇤,
2 2
2
⇤t ⇥ = U (⇥) + 0 ⇥2 ⇥
(⇥
) ⇥
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
3
4
(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)
Monday, February 24, 14
a>0
a<0
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or
reflection-symmetry
r ⇤(t, x), given by
2
b(1)= 0
⇤
⇤,
2 2
2
⇤t ⇥ = U (⇥) + 0 ⇥2 ⇥
(⇥
) ⇥
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
3
4
(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
Monday, February 24, 14
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalarbroken
or
reflection-symmetry
r ⇤(t, x), given by
2
b(1)= 0
⇤
⇤,
2
2 2
0
2
⇤t ⇥ = U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a>0
a 2 b 3 c 4
U (⇤) = ⇤ + ⇤ + ⇤ ,
2
3
4
(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
−1
Monday, February 24, 14
(3)
0
a<0
b<0
1
trajectory of a swimming cell can
arxiv:exhibit
1208.4464a 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)=
⇤
⇤,
2
2 2
0
2
⇤t ⇥ = U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a 2 b 3 c 4
U (⇤) = ⇤ + ⇤ + ⇤ ,
2
3
4
0
(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)
Monday, February 24, 14
Figure 2. Effect of symmetry breaking.
eferred handedness [40, 48, 49, 50]. For
trajectory of a swimming cell can
arxiv:exhibit
1208.4464a 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
Turing model
A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London. B 327, 37–72 (1952)
dunkel@math.mit.edu
Monday, February 24, 14
Turing model
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Turing pattern
Kondo S, & Miura T (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329 (5999), 1616-20
dunkel@math.mit.edu
Monday, February 24, 14
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), 1616-20
dunkel@math.mit.edu
Monday, February 24, 14