Beyond diffusion:
Pattern formation
18.354 - L07
dunkel@mit.edu
2D diffusion equation
dunkel@math.mit.edu
dunkel@math.mit.edu
Natural & artificial patterns
Paddock (2001) Biotechniques
Chan & Crosby (2006) Advanced Materials
Plankton
ESA cost of Ireland
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Animals
dunkel@math.mit.edu
Shells
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e to the static angle of repose and subseque
Compare: vibrated granular media
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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 li
longitudinal vortices in rapid chute flows !Forterre and
Pouliquen, 2001", see Fig. 8, long modulation waves
container. One of the
dunkel@math.mit.edu
for this class of syste
Slime mold
aggregation of a starving slime mold (credit: Florian Siegert)
dunkel@math.mit.edu
Physarum developmental cycle
Slime mold (physarum plasmodium)
Tero et al (2010) Science
Tero et al (2010) Science
Slime mold
aggregation of a starving slime mold (credit: Florian Siegert)
dunkel@math.mit.edu
Movie credit: Stephen Morris
Belousov-Zhabotinsky reaction
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 oscillates.!
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.
dunkel@math.mit.edu
Yin et al (2014) Sci Rep
Breid & Crosby (2013)Soft Matter
Surface buckling
d
2
Figure 1 | SEM images of PDMS(1051) microspheres before (a) and after (b–e) the PI processing. Zoome
,2.25 mm; d) ,4.75 mm; e) ,15 mm. Frame f shows the lR , R relationship for different EPDMS (n:1) .
e
f
onstrate that these surface textures belong to typical wrinkle
morphologies (Fig. 1c–e). Furthermore, when the radius R of
PDMS(1051) microspheres increases from ,2.25 mm to ,4.75 mm,
the corresponding wrinkles change from the dimple (or buckyballlike) (Fig. 1c) to labyrinth patterns (Fig. 1d). A careful examination
further
shows that under the same conditions, no surface wrinkling is
1cm
induced on a small radius of PDMS(1051) microsphere (circled one in
Fig. 1b), owing to the requirement of a larger critical wrinkling stress
28,29
according
to
Eqs.2,3
Stoop, Lagrange,Terwagne, Reis &. Dunkel, 2014
The influence of the Young’s modulus of PDMS(n:1) microspheres
It is known that whe
the surface oxidation r
the formation of an oxi
PDMS substrate applied
system composed of th
microsphere is generate
the thickness hR of the
different radii of PDM
dimple patterns are form
with a larger hR/R (i.e., V
FIG. 1: Macroscopic and microscopic wrinkling morphologies of sti↵ thin films on spherically curved soft
a
Film
Substrate
b
h
R
pi
u
pe
c
Air
channel
R
FIG. 2: Notation and experimental system.
a,
Schematic of a curved thin film adhering to a soft spherical substrate of outer radius R. b, The film (thickness h) is
driven towards a wrinkling instability by the compressive film
stress , leading to wrinkling pattern with wavelength and
radial displacement u. c, The experimental system consists
of two merged hemispherical caps. An air channel allows to
Reis lab @ MIT
Pattern selec
nonlinear proce
linear stability
Eq. (1) yield a
ary states that
hexagonal phas
intermediate co
the transition fr
stood through a
in Eq. (1) break
lutions under th
1 are controlle
curvature-induc
of . Furthermo
SB terms cause
nal patterns in t
to expect a hex
labyrinths at sm
To obtain a
boundaries, w
standard SH
lished results
Theory vs. experiment
2
b
c
g
100%
h
60%
i
50%
R/h=75
R/h=200
d
e
f
Experiment
R/h=50
umin/h
u/h
umax/h
Increasing excess lm stress
e
Theory
a
Increasing e ective radius R/h
FIG. 1: Macroscopic and microscopic wrinkling morphologies of sti↵ thin films on spherically curved soft
substrates. a-c, Theoretical predictions based on numerical steady-state solutions of Eq. (1). Color red (blue) signals inward
(outward) wrinkles. Simulation parameters: (a) 0 = 0.029, a = 0.00162, c = 0.0025, (b) 0 = 0.04, a = 1.26 · 10 6 ,
Stoop et al., Nature Materials, 2015
c = 0.002, (c) 0 = 0.02, a = 1.49 · 10 4 , c = 0.0025 (see Table I). d-f, Experimentally observed patterns confirm the
transition from hexagonal (d) to labyrinth-like wrinkles (f) via a bistable region (e) when the radius-to-thickness ratio R/h (see
Fig. 2) is increased. Scale bars: 10 mm. Parameters: Ef = 2100 kPa, R = 20 mm, ⌫ = 0.5 and (d) Es = 230 kPa, h = 0.630 mm;
Higher-order PDE models
&
symmetry-breaking
Mathematical description
1. higher-than-second-order PDEs
2. many coupled second-order PDEs
dunkel@math.mit.edu
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
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
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or
⇤(t, x), given by
0
⇤
2
2
⇤,
⇤t ⇥ =
U (⇥) +
0⇥
2
⇥
2 (⇥
) ⇥(1)
2 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
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional.! In the context of active suspensions,
tify localreflection-symmetry
energy fluctuations, local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
a>0
a<0
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or
⇤(t, x), given by
0
⇤
2
2
⇤,
⇤t ⇥ =
U (⇥) +
0⇥
2
⇥
2 (⇥
) ⇥(1)
2 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
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional.! In the context of active suspensions,
tify localreflection-symmetry
energy fluctuations, local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or
⇤(t, x), given by
0
⇤
2
2
⇤,
⇤t ⇥ =
U (⇥) +
0⇥
2
⇥
2 (⇥
) ⇥(1)
2 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
(2)
a>0
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
tify local energybroken
fluctuations,
local alignment, phase
reflection-symmetry
will assume
throughout that the
system is confined to
d
of volume
a<0
(3)
−1
0
1
trajectory of a swimming cell can exhibit a pre
2d Swift-Hohenberg
example,model
the bacteria Escherichia coli [40] an
tropic fourth-order model for a non-conserved scalar or
⌃ ij denotes the Cartesian components of the Levi-Civ
⇤(t, x), given by
0
⇤
2
2
⇤,
⇤t ⇥ =
a summation convention for equal indices throughout.
U (⇥) +
0⇥
2
⇥
2 (⇥
) ⇥(1)
2 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
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
tify local energybroken
fluctuations,
local alignment, phase
reflection-symmetry
will assume
throughout that the
system is confined to
d
of volume
(3)
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)
Riedel et al (2007) Science
b=0
Mathematical description
1. higher-than-second-order PDEs
2. many coupled second-order PDEs
dunkel@math.mit.edu
?
dunkel@math.mit.edu
NATURE|Vol 448|26 July 2007
Maternal input
From: Reinitz
a
Bicoid
affects
level of
b
orations of their
photo-bleached
ce within a few
ear Bicoid is in
oplasmic pool.
del of diffusionsed it to predict
nt for Bicoid in
he recovery of
ached region of
this prediction.
ght to be exceparly patterns of
cause its distri-
Protein concentration
Hunchback
c
xbcd
xhb
Anterior–posterior position
NEWS 1&| VIEWS
Figure
The role of Bicoid in dictating levels
NATURE|Vol 448|26 July 2007
Intensity profiles
Kni (488nm)
Kr (546nm)
Gt (594nm)
Hb (633nm)
Intensity (a.u.)
A
4000
B
2000
0
4
I (a.u.)
10
C
2
10
0
10
−2
10
1
Dubois et al (2013) Mol Sys Biol
D
Gregor Lab, Princeton
dunkel@math.mit.edu
http://www.iephb.nw.ru/hoxpro/ftz.html
The Chemical Basis of Morphogenesis
A. M. Turing
Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol.
237, No. 641. (Aug. 14, 1952), pp. 37-72.
Stable URL:
http://links.jstor.org/sici?sici=0080-4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-I
Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences is currently published by The Royal
Society.
THE CHEMICAL BASIS OF MOKPHOGENESIS
BY A. M. TURING, F.R.S. University qf Manchester
(Received
9 November
195Conditions
1-Revised of Use,
15 March
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It is suggested that a system of chemical substances, called morphogens, reacting together and
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diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Such aany
system,
may Publisher
originallycontact
be quite
homogeneous,
may later
Please contact the publisher regarding
furtheralthough
use of thisit work.
information
may be obtained
at develop a pattern
or structure due to an instability of the homogeneous equilibrium, which is triggered off by
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random disturbances. Such reaction-diffusion systems are considered in some detail in the case
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screen or printed
of an
isolated ring
cells,the
a same
mathematically
convenient,
though
unusual system.
page of such transmission.
The investigation is chiefly concerned with the onset of instability. It is faund that there are six
essentially different forms which this may take. In the most interesting form stationary waves
appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns
on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered.
Such a system
appears
account afor
gastrulation.
Another journals.
reactionFor
system in two
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waves in two dimensions could account for the phenomena of phyllotaxis.
The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote
may determine the anatomical structure of the resulting organism. The theory does not make any
new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account
for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be
experts in all of these subjects, a number of elementary facts are explained, which can be found in
text-books, but whose omission would make the paper difficult reading.
I n this section a mathematical model of the growing embryo will be described. This model
will be a simplification and an idealization, and consequently a falsification. I t is to be
hoped that the features retained for discussion are those of greatest importance in the
present state of knowledge.
The model takes two slightly different forms. In one of them the cell theory is recognized
1912-1954
Google: 8721 citations
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The Chemical Basis of Morphogenesis
A. M. Turing
Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol.
237, No. 641. (Aug. 14, 1952), pp. 37-72.
Stable URL:
http://links.jstor.org/sici?sici=0080-4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-I
Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences is currently published by The Royal
Society.
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained
prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in
the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/journals/rsl.html.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For
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Web of Science >4800 citations
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Turing model
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Turing model
A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London. B 327, 37–72 (1952)
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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
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
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