Untangling Complex Systems, page 51
∂ t
Activator
∂ y = k
2
2
4 b − k 3 x y + DY ∇ y
∂ t
Inhibitor
9.4 TURING PATTERNS IN A CHEMICAL LABORATORY
The first example of a Turing structure in a chemical laboratory occurred in 1990, nearly forty years
after Turing’s seminal paper, thanks to De Kepper and his colleagues who were working with the
chlorite-iodide-malonic acid (CIMA) reaction in an open unstirred gel reactor, in Bordeaux. The
mechanism of the CIMA reaction is quite complicated. After a relatively brief initial period, chlo-
rine dioxide and iodine build up within the gel and play the role of reactants whose concentrations
252
Untangling Complex Systems
vary relatively slowly compared with those of ClO_2 and I−. This observation allows us to state that
it is indeed the chlorine-dioxide-iodine-malonic acid (CDIMA) reaction that governs the formation
of Turing patterns. Fortunately, the CDIMA and CIMA reactions may be described by a model that
consists of only three stoichiometric processes and their empirical laws (see equations, from [9.25]
up to [9.30]). Such model holds because there is no significant interaction among the intermediates
of the three main stoichiometric processes and because none of the intermediates builds up to high
concentrations (Lengyel and Epstein 1991).
−
+
CH2(COOH)2 + I2 → ICH(COOH)2 + I + H
[9.25]
k a [
(
) ][ ]
1
CH2 COOH 2 I
v
2
[9.26]
1 =
k b
]
1 + [I2
−
−
1
ClO2 + I → ClO2 + I
[9.27]
2
2
−
v2 = k [
]
2 ClO2
I
[9.28]
ClO−
−
+
−
2 + 4I + 4H
→ Cl + I
2 2 + 2H2O
[9.29]
ClO−
−
2
I2 I
v
−
−
+
[ ]
3 = k
3 a ClO2
I
H
k 3
+ b
[9.30]
α + − 2
I
The malonic acid serves only to generate iodide via reaction [9.25]. The reaction [9.27] between
chlorine dioxide and iodide produces iodine. The reaction [9.29] is the key process: it is autocata-
lytic in iodine and inhibited by iodide (see equation [9.30]). The differential equations [9.26], [9.28]
and [9.30] contain six variables. They are [CH
−
2 (COOH)2 ], [I2 ], [ClO2 ], [I− ], [H+ ], and [ClO2 ]. The
system of three differential equations ([9.26], [9.28] and [9.30]) have been integrated numerically,
and the computational results have reproduced quite well the experimental results and the possibil-
ity of having oscillations (Lengyel et al. 1990). The numerical analysis has also revealed that during
each cycle of oscillation, [CH
−
2 (COOH)2 ], [I2 ], [ClO2 ], and [H+ ] change very little, while [ClO2 ] and
[I−] vary by several orders of magnitude. This evidence suggests that the model can be reduced to a
two-variable system by treating the four slightly varying concentrations as constants. The simplified
model looks like the following set of schematic reactions, wherein X = −
[I ], Y =
−
[ClO2 ] and A = I2.
A k′
1→
X
X
k′
2→
Y
[9.31]
X + Y
k′
3
4
→
P
The new rate laws are:
k
1 a CH2 (COOH )
I
′
2
v
0
0
1 = k′1 =
2
1
k b + I20
′
v2 = k′2 X = k 2 ClO
2
X
[9.32]
0
X Y
X Y
v′ = k′ =
3
3
k b I
2
3 2 0
2
α + X
α + X
The Emergence of Order in Space
253
In the definition of ′
v3 in [9.32], the first term of [9.30] has been neglected, since it is much smaller
than the second term under the conditions of interest. From the mechanism [9.31] and equation [9.32],
it derives that the system of partial differential equations describing the spatiotemporal development
for [ X ] and [ Y] is
∂ X
X Y
∂2 X
= k′1 − k′2
X
4 k 3
+
D
∂
− ′
t
α +
X
2
2
X
∂ r
∂ Y
X Y
∂2 Y
[9.33]
= k′2 X k
3
+ D
− ′
Y
2
2
t
∂
α + X
r
∂
where r is the spatial coordinate. In [9.33], we have four variables (two are independent, i.e., t and r, and two dependent, i.e., [ X] and [ Y]) and six constants. We may simplify further the system of differential equations [9.33] if we nondimensionalize it.4 The final result is
∂ X *
* *
2
*
*
X Y
=
X
γ a − X − 4
∂τ
1+ *2 + ∂
X ∂ρ2
∂
[9.34]
Y *
* *
*
=
*
bX Y
γ
Y
2
bX −
d
*2 +
∂
2
∂τ
1+
X
∂ρ
In [9.34], X * = (1 α )[ X ], Y* = ( k′ α
2
3 k′2
) Y
, τ = ( DX L ) t, ρ = (1 L) r are the dimension-
less variables and d = ( D
( k 2
( ′ / ′ α =( ′ / ′ α
3
2
)
1
2
)
2 L DX ), a = k
k
, b
k k
are the new four
Y DX ), γ =
′ /
parameters. To observe Turing structures, the homogeneous system of ODEs
dX *
* *
*
X Y
= a − X − 4
2 = f ( X *, Y * ) = 0
dτ
1+
X *
[9.35]
dY *
* *
*
bX Y
= bX −
= g ( X * Y *
,
) = 0
*2
dτ
1+
X
must have a steady-state ( X *
a
*
a
1
2
that is stable to homogeneous perturbation. In
5
( 25)
s = ( ), Ys =
+
other words, the steady-state solution ( * s, *
X Ys ) must have tr( J ) < 0, det( J ) > ,
0 and one species must
play as the activator whereas the other as the inhibitor.
TRY EXERCISE 9.9
4 The first step of the procedure of nondimensionalization requires the definition of dimensionless variables. For the system of differential equations [9.33], the dimensionless variables are X * = X [ ], Y * = Y [ ], τ = t , ρ = rCr. Introducing C t
C Y
C X
these new variables in [9.33], we obtain:
∂ X *
X
*
*
*
2
2
C
X
X Y
1
DX rC ∂
=
X *
k′1
− k′2
− 4 k′3
+
∂τ
t
2
C
tC
tC YC
X *
tC
∂ρ2
α +
XC
∂ Y *
Y
* *
2
2 *
C
*
1
X Y
DY rC ∂
=
Y
k′2
X − k′3
+
∂τ
X
2
2
CtC
XCtC
X *
tC
∂ρ
α +
XC
The next step is to choose the nondimensionalizing constants. If L is the extension of the spatial coordinate, r = ( 1 ). Introducing the definition of r , we derive that t
DX
= (
. If we fix X = ( 1 , Y
k
=
′
( 3 )
Y
= ( ),
α )
2 )
C
L
C
C
L
C
C
k′2α , and d
D
DX
γ = ( k′ 2
, a = k′ /
α , b = k′
α , the differential equations can be written into dimensionless form [9.34].
3 k
/ ′
( 2 )
1
k′
( 2 )
2 L / DX )
254
Untangling Complex Systems
Since (∂ g ∂ Y *) < 0 and (∂ f ∂ Y *) < 0, the species Y * =
−
ClO2 is the inhibitor of itself and X *. The
species X * = I − plays as the self-activator (or auto-catalyst) if a > 5 5 3. The steady state becomes a saddle point in the presence of the diffusion, if the determinant of the new Jacobian J ′ (containing
the extra terms of diffusion in its diagonal elements) is negative, i.e., if
∂ f
2
∂ f
K
* −
*
∂ X s
∂ Y
det ( J′) = det
s
<
[9.36]
0
∂ g
∂ g
K 2 d
*
* −
∂ X s
∂ Y s
The determinant of J ′ is a negative real number if the relations [9.37] and [9.38] are true.
∂ g
∂ f
0
[9.37]
* + d
* >
∂ Y
∂ X
s
s
∂ g
f
f ∂ g f ∂ g
2
0
[9.38]
* +
∂
d
* >
∂
d
*
* −
∂
>
∂ Y
∂ X
∂ X
∂ Y
∂ *
*
X
s ∂
Y
s
s
s
s
s
If we use plausible values of the kinetic constants for the CDIMA reaction (Lengyel and Epstein
1991), we find that the ratio d = ( D
)
Y DX must be at least 10 for the relation [9.38] to hold. If you
want to verify this result, try to solve exercise 9.10.
In aqueous solution, substantially all small molecules and ions have diffusion coefficients that
lie within a factor of two of 2 10 5
2
1
× −
−
cm s . Therefore, it seems impossible that the diffusion coef-
ficient of ClO−2 can be ten times greater than that of I −. What did it make possible to get around this
problem? Here, serendipity entered. De Kepper and colleagues (Castets et al. 1990) were working
with the CIMA reaction in an unstirred continuous flow gel reactor. In such a reactor, the two broad
faces of a cylindrical slab of gel, 2 or 3 mm thick and made of either polyacrylamide or agar (just to
make a pair of examples), are in contact with two solutions of different compositions: one contain-
ing iodine and the other chlorine dioxide and malonic acid. The reactants diffuse into the gel, and
they encounter in a region near the middle of the gel, where the pattern, less than 1 mm thick, can
emerge. The continuous flow of fresh reactants maintains the system as open, and the gel prevents
convective motion. De Kepper and colleagues employed starch as an indicator to increase the color
contrast between the oxidized and reduced states of the reaction. They introduced starch into the
acrylamide monomer solution before polymerization to the gel because the bulky and heavy starch
molecules cannot diffuse into the gel from the reservoirs. The starch ( St) forms a blue complex with
tri-iodide ( I −3):
St + I + I − → ( StI −
2
3 )
[9.39]
The complex ( StI −3) is practically immobile. We may imagine the starch molecules being dispersed
randomly throughout the gel. When an iodide encounters a “trap” made of starch and iodine, it
remains blocked until the complex breaks up. The net result is that the effective diffusion rate of
iodide decreases noticeably. If the concentration of starch is high enough, it provides the way of
slowing down the diffusion rate of the activator iodide with respect to the inhibitor chlorite, which,
on the other hand, maintains the typical diffusion rate it has in aqueous solution. It was the fortu-
itous choice of starch as the indicator that produced the first experimental Turing structure
(for a quantitative treatment of the starch’s effect read the paper by Lengyel and Epstein (1992)).
The patterns, observed for the CIMA and CDIMA reactions performed in a gel reactor, appear to
occur in a single plane parallel to the faces of the gel at which the reactor is fed. The patterns are
The Emergence of Order in Space
255
essentially bi-dimensional. Why? The formation of Turing patterns requires that concentrations of
the species involved in the reaction lie within ranges that allow the four inequalities (I) tr( J ) < 0, (II) det( J) > 0, (III) [9.37] and (IV) [9.38] to be satisfied. In a gel reactor, the values of the concentrations of all the species are position-dependent, ranging from their input feed values on one side of
the gel where they enter to essentially zero on the other side. Evidently, the conditions for generating
Turing structures can be satisfied only within a thin slice of the gel, if at all.
Recently, Epstein and his collaborators (Bánsági et al. 2011) at Brandeis University have
obtained the first examples of three-dimensional Turing structures in a chemical laboratory. Such
wonderful three-dimensional Turing patterns are based on the Belousov-Zhabotinsky (BZ) reac-
tion carried out in an oil-water mixture in the presence of the amphiphilic surfactant sodium
bis(2-ethylhexyl) sulfosuccinate, known as Aerosol OT (AOT). In an AOT-water-oil system when
the ratio R
w/o = ([water]/[oil]) is low, a reverse microemulsion forms, in which droplets of water,
surrounded by a monolayer of surfactant molecules, are dispersed in oil. Such droplets have
