Spatial periodicity induced by a chemical wave train
Shrabani Sen, Pushpita Ghosh, Syed Shahed Riaz,
*
and Deb Shankar Ray
†
Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
Received 27 October 2009; published 14 January 2010
An all-chemical analog of clock-wave-front model for somitogenesis is proposed. The spatial periodicity can
be obtained by arresting the homogeneous oscillations in a typical two-component reaction-diffusion system in
the Hopf region by interacting with a chemical wave front. The patterns can be controlled by tuning the wave
speed of the front.
DOI: 10.1103/PhysRevE.81.017101 PACS numbers: 82.40.Ck, 89.75.Kd, 47.54.-r
The theory of morphogenesis 1 proposed by Turing in
the middle of the last century is based on the search for
symmetry breaking of a homogeneous steady state of a dy-
namical system governed by the reaction-diffusion equa-
tions. The underlying instability owes its origin to short-
range activation and long-range diffusion of two interacting
species obeying activator-inhibitor reaction kinetics and the
resulting symmetry-broken state is characterized by various
spatial regions with distinct physical and chemical features
or conspicuous patterns. The theory and its variants have
found extensive applications in several areas of natural sci-
ences, particularly, in developmental biology 2–11, where
the growth of sequential space-periodic structures along the
anterio-posterior axis of the vertebrate embryo has remained
a focus of major attraction. Somitogenesis 3–15 provides a
typical example of how such spatial periodicity forms during
embryogenesis. Several models 8–14 of somitogenesis are
now known. Among them the clock and wave front model
14 has gained a good testing ground for further experiment.
The model assumes the existence of a biochemical oscillator
within the band of paraxial mesodermal cells. The oscillator
of all the neighboring cells oscillates synchronously. The
model further postulates that a wave front of cell change
sweeps posteriorly through the cells slowly, halting the os-
cillations and thereby inducing maturation of somatites
which are essentially blocks of cells of paraxial mesoderm
arranged as spatially periodic structures.
In the present Brief Report we look for a chemical analog
of the clock-wave-front model of somitogenesis. Chlorine
dioxide-iodine-malonic acid CDIMA system 16–21,
which has remained as a classic paradigm for a wide class of
far-from-equilibrium phenomena in spatially extended sys-
tem, offers itself as an excellent candidate for the model.
With CDIMA systems it has been shown that progressive
arrest of homogeneous oscillations can control the symmetry
as well as the wavelength of spatial structures when the re-
action medium is illuminated with a constant or periodic
source of light 22,23. Our endeavor in this Brief Report is
to suggest an all-chemical analog for exploring the possibil-
ity of formation stationary pattern out of homogeneous os-
cillations in the Hopf region by interaction with a progres-
sive chemical wave front or pulse rather than light field. We
exploit the sensitivity of CDIMA system to the concentration
of some external reducing agent to demonstrate how the
wavelength of the produced patterns changes with the veloc-
ity of the traveling wave and frequency of the temporal os-
cillations.
To begin with we consider the following chemical reac-
tions that are involved in the CDIMA model 16,17;
MA +I
2
→ I MA +I
-
+H
+
,
ClO
2
+I
-
→ ClO
2
-
+
1
2
I
2
,
ClO
2
-
+ 4I
-
+ 4H
+
→ Cl
-
+ 2I
2
+ 2H
2
O.
When starch is added from outside in the reaction medium,
we have the additional equilibrium between starch and io-
dide ions;
S +I
3
-
⇌ SI
3
-
.
Ideally the reaction schemes imply a many variable model
for CDIMA system. However in the course of reaction the
concentrations of all the species but I
-
and ClO
2
-
remain
more or less constant. This effectively reduces the model into
a two-variable one with concentrations of I
-
and ClO
2
-
play-
ing the role of activator and inhibitor respectively. With
I
-
u and ClO
2
-
v and treating concentrations of all
other species as constants Lengeyl and Epstein have sug-
gested the governing reaction-diffusion dynamics of these
two variables as follows:
u
= a - u -4
uv
1+ u
2
+
2
u , 1
v
=
b u -
uv
1+ u
2
+ d
2
v
. 2
Here the parameters a and b are proportional to the con-
centration ratios MA / I
2
and ClO
2
/ I
2
, respectively,
and are related to kinetic parameters. d refers to the ratio of
the diffusion coefficients d = D
ClO
2
- / D
I
-. is the concen-
tration of starch which forms a complex with I
3
-
such that
=1+ KS, where K is the equilibrium constant for the com-
plexation reaction and S is the concentration of starch.
The complexation thus separates the time scales for the
*
Also at Dept. of Chemistry, Belur Ramakrishna Mission Vidya-
mandira, Howrah, 711202, India.
†
pcdsr@iacs.res.in
PHYSICAL REVIEW E 81, 017101 2010
1539-3755/2010/811/0171014 ©2010 The American Physical Society 017101-1