@C1
@ 2 C1
= R1 (C1 , C2 ) + D1
@t
@x2
@C2
@ 2 C2
= R2 (C1 , C2 ) + D2
@t
@x2
where
C 1 , C2
R 1 , R2
D 1 , D2
are the concentrations of two chemicals
are the reaction terms
are the diffusion coefficients
Pattern formation
At least two reacting chemicals
Diffusion can destabilize the uniform state
Instability can cause the growth of a structure of
a particular wavelength
The diffusion coefficients of the two reactants
differ significantly
Diffusion can destabilize the uniform state
that is, in the absence of diffusion the uniform state is
stable
@C1
@ 2 C1
= R1 (C1 , C2 ) + D1
@t
@x2
2
@C2
@ C2
= R2 (C1 , C2 ) + D2
@t
@x2
let (C1 , C2 ) be the uniform steady state
@C1
=0
@t
@ 2 C1
=0
2
@x
@C2
=0
@t
@ 2 C2
=0
2
@x
Stability of the linearized system
@C10
= a11 C10 + a12 C20
@t
0
@C2
0
0
= a21 C1 + a22 C2
@t
where,
0
C1 (t, x)
0
C2 (t, x)
= C1 (t, x)
= C2 (t, x)
C1
C2
are perturbations around the fixed point (C1 , C2 )
a11
a21
@R1
@R2
=
, a12 =
@C1
@C1
@R2
@R2
=
, a22 =
@C1
@C2
Eigenvalues determine the linear stability
a11
det
a21
a12
a22
stability implies
= a11 + a22 < 0
= a11 a22
a12 a21 > 0
=0
2
2
Stability of the linearized system with diffusion
terms
0
@C1
0
a11 C1
0
a12 C2
0
@ C1
D2
@x2
2
@t
2 0
@C10
@
C2
0
0
= a21 C1 + a22 C2 + D2
@t
@x2
The above equation can be written as
@C10
@t0
@C1
@t
a11
a21
0
a12
C1
D1
+
0
a22
C2
0
0
D2
@ C10
2
@x
@ 2 C20
@x2
2
Any function can expanded as a Fourier series
f (x) =
A(q) exp(iqx)
The value of A(q) determines which frequencies/
modes contribute to the function f (x)
Assume a solution of the form
C1
C20
X
q
1
exp(
2
Fourier Series
q t) exp(iqx)
A(q)
The eigenvalue q determines which frequencies/
modes contribute to the solution.
Substitute
C1
C20
in
@C10
@t0
@C1
@t
X
q
1
exp(
2
a11
a21
0
a12
C1
D1
+
0
a22
C2
0
For each q we get
1
2
= a11 1 + a12 2
D1 q 1
= a21 1 + a22 2
D2 q 2 2
collecting terms
1 (
q t) exp(iqx)
a11 + D1 q ) + 2 ( a12 ) = 0
1 ( a21 ) + 2 (
a22 + D2 q ) = 0
0
D2
@ C10
2
@x
@ 2 C20
@x2
2
if
1 , 2 6= 0
then
1 (
a11 + D1 q 2 ) + 2 ( a12 ) = 0
1 ( a21 ) + 2 (
a22 + D2 q ) = 0
is satisfied only if
det
a11 + D1 q 2
a21
a12
a22 + D2 q 2
i.e.
2
((a11
+ ( a22 + D2 q
D1 q 2 )(a22
a11 + D1 q ) +
D2 q 2 )
a12 a21 ) = 0
=0
trace
2
+ ( a22 + D2 q 2 a11 + D1 q 2 ) +
2
2
((a11 D1 q )(a22 D2 q ) a12 a21 ) = 0
determinant
< 0 implies
a11 + a22
2
(a11
D1 q )(a22
2 2
H(q ) = D1 D2 (q )
Instability implies
2
H(q ) < 0
D1 q
2
D2 q )
always true
D2 q < 0
a12 a21 > 0
2
(D1 a22 + D2 a11 )(q ) + (a11 a22
a12 a21 )
Since the uniform state is stable in the absence of diffusion
= a11 + a22 < 0
= a11 a22 a12 a21 > 0
H(q 2 ) = D1 D2 (q 2 )2
(D1 a22 + D2 a11 )(q 2 ) + (a11 a22
a11 > 0, a22 < 0, a12 > 0, a21 < 0
a11 > 0, a22 < 0, a12 < 0, a21 > 0
a11
a21
a12
a22
is of the form
or
+
+
+
a12 a21 )
To test this hypothesis, we built a twodimensional (2D) finite-element model based
+
+
@C1
@t
@C2
@t
a11
a22
a21
a12
+C1
+C1
C2
C2
text) (25). We determined
reaction parameter of the
effective target for waveleng
S4 and S5) and chose the
on inhibitor production (gu)
(see supplementary text). Our
that digit bifurcations occu
case (red arrowhead Fig. 1
furcations were shown wh
wavelength scaling (Fig. 1E
observed mutant patterns. O
that, in the absence of Gli3
tion of Hoxa13 produces a
wavelength (w), thereby in
ber, but also causes a shallo
w, which explains the observ
Because there is no eviden
v
in
the
form:
shape of the Gli3 -/- mutant. We used an outline of a representative specimen
u
2
and generated a triangular grid with the software Gmesh. The resulting trian=
f
(u,
v)
+
d
u
u
gular grid is shown in Figure S13.
t
v
= g(u, v) + dv 2 v
t
The biological implementation of the reaction kinetics f
For this reason, model
we used the general model developed in [25]
reaction-diusion
linear approximation around the steady state (0, 0). In add
al reaction-diusion
model
madethe
of growth
two reactants
u and
(u3 ) was used
to limit
of the activator.
We obta
form:
Figureu
S13: The triangular limb grid made from
Gli3
-/- v)
shape= f u + f v u3
2 the experimental f
(u,
u
v
= f (u, v) + du u
t
g(u, v) = gu u + gv(1)
v
Wev
mapped the experimental expression pattern
of Hoxa13 into this realistic
2
v)
+
d
domain shape=
by g(u,
using the
Vtk
library
[35].vEventually, we normalized the
v
Any 0reaction-diusion
model of two species can be ap
tpattern between
expression
and 1. This expression pattern was used as an
approximation for Hox genes in the model. In addition, we simulated an Fgf
general
form
by
Taylor
expansion
up
to
the
cubic
term.
Ac
signaling
gradient
by
diusing
a
substance
from
a
region
corresponding
to
the
ementation of the reaction kinetics f and g is unknown.
AER into themodel
mesenchyme.
The patterns ofaHox
expression and Fgf signalinginstability when the follo
produces
diusion-driven
are showed
in Figuremodel
S14.
d the
general
developed in [25] that is obtained by
satisfied:
round the steady state (0, 0). In addition, a cubic term
the growth of the activator.
fu +We
gv obtained
< 0, fu gfv and
fvgguin>the
0
d f + d3 g > 0,
(d f + d g ) 4d(f g f
ally,
ong
no
MBER 2012
VOL 338
SCIENCE
www.sciencemag.org
