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Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Journal of Applied Nonlinear Dynamics
https://lhscientificpublishing.com/Journals/JAND-Default.aspx
Reaction-diffusion Dynamics and Biological Pattern Formation
Kishore Dutta†
Department of Physics, Handique Girls’ College, Guwahati, India
Submission Info
Communicated by A.C.J. Luo
Received 19 September 2016
Accepted 20 September 2017
Available online 1 January 2018
Keywords
Spatio-temporal pattern
Spontaneous symmetry breaking
Reaction-diffusion system
Diffusion-driven instability
Abstract
The spontaneous formation of a wide variety of natural patterns with
different shapes and symmetries in many physical and biological sys-
tems is one of the deep mysteries in science. This article describes
the physical principles underlying the formation of various intriguing
spatio-temporal patterns in Nature with special emphasis on some
biological structures. We discuss how the spontaneous symmetry
breaking due to diffusion driven instability in the reaction dynamic-
s lead to the emergence of such complicated natural patterns. The
mechanism of the formation of various animal coat patterns is ex-
plained via the Turing-type reaction-diffusion models.
©2017 L&H Scientific Publishing, LLC. All rights reserved.
1 Introduction
One of the most spectacular features of many biological organism is their capability to generate various
intriguing and visually fascinating complex spatio-temporal patterns. The exotic beauty and bewil-
dering complexity of animal coat patterns, namely, zebra stripes [1, 2], patterns on snake skin [2, 3],
coloration of butterfly wings [4–6], coat patterns of the giraffe [7], leopard [8], stripe patterns on fish
skin [9–11] have attracted every curious mind from ancient times. Many such natural color patterns
are treasured objects in certain communities of the world and they are sometimes used for the iden-
tification of species. The problem of pattern formation remains as a multidisciplinary challenge of
great technological and scientific interest, and has been studied in recent years by physicists, chemists,
biologists, ecologists, mathematicians, computer scientists, and neuroscientists as well.
Understanding of spatial pattern formation is an enduring endeavour in basic sciences as it involves
complex interactions of many physical and chemical processes [12, 13]. A number of questions arise
in curious minds: How nature fabric various patterns? What mathematical rules does nature follow
while designing such beautiful patterns? Can they be created by simple physical laws? What roles
do initial conditions, boundaries, thermal noise, and dissipation play in pattern formation? How do
animals acquire their various skin patterns?
An essential prerequisite for the formation of any kind of order in a system is the symmetry breaking.
The simplest way to break the symmetry of a uniform two-dimensional system is to apply a periodic
†Corresponding author.
Email address: kdkishore77@gmail.com
ISSN 2164 −6457,eISSN 2164 −6473/$-see front materials ©2017 L&H Scientific Publishing,LLC.All rights reserved.
DOI : 10.5890/JAND.2017.12.009
548 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Fig. 1 Zebra stripes – a striking biological pattern.
Fig. 2 Emergence of stripe pattern due to the symmetry breaking under an imposed periodic disturbance to
the initial homogeneous state of the system.
(wave-like) disturbance in just one direction (Fig. 2). This symmetry breaking in a minimal way creates
stripes or rolls. The broken symmetry in this case travels only along the direction perpendicular to
the stripes. This type of parallel stripes are often visible in sand ripples. After breaking the symmetry
periodically in one dimension, the imposition of a second periodic variation breaks the symmetry in
both directions. If these two periodic variations are imposed at right angles to each other, square
cells will emerge, and if the two variations are at 600angles, triangles or hexagonal cells will emerge,
as displayed in Fig. 3. Physicists have succeeded in identifying some universal properties of complex
patterns and relating them to broken symmetries. Numerous non-living self-organized structures such
as the emergence of magnetization in ferromagnet below the Curie temperature, formation of high sand
dunes or of sharply contoured rivers, formation of snowflakes, glass cracks, mud cracks, etc. can best
be described from simple symmetry breaking arguments [14].
Patterns are one kind of spatial structures that emerge due to the organized motion of large numbers
of small constituents. Such self-organization is possible only for nonlinear dissipative systems that are
far way from thermodynamic equilibrium. Self-organized systems obey evolution equations that are
nonlinear in the macroscopic configuration variables. The appearance of order and structure in such
systems highly relies on two fundamental physical processes competing with each other, namely, a
driving (or, energy-adding) and a dissipating process. When the driving and the dissipation are of the
same order of magnitude, and when the key behaviour of the system is not a linear function of the
driving action, order may appear. To understand the appearance of order in self-organized system,
one has to observe the evolution of an observable A(x,t) describing the amplitude of the pattern (e.g.,
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 549
(a) (b)
Fig. 3 Breaking of symmetry in both directions creates (a) square (b) triangular or hexagonal cells.
chemical concentration). The evolution equation typically has a general form
∂
A(x,t)
∂
t=
λ
A−
μ
|A|2A+
κ
∇2A+higher orders.(1)
In the right-hand side of Eq. (1), the first term
λ
Ais the driving term where
λ
represents the strength
of driving; the second term is the nonlinear term where
μ
represents the strength of nonlinearity in A
and the third term is a typical diffusive (and hence dissipative) term. In cases where the dissipative
term plays no role (
κ
=0), a temporal oscillation appears as the driving parameter
λ
increases above
zero. This is the case for a stable cycle with non-vanishing amplitude. In cases where dissipation plays a
role, the solution A=0becomes spatially unstable and the above equation describes how an amplitude
for a spatial oscillation appear for positive value of
λ
. The onset of order in both cases is called
bifurcation, because at this critical value of the driving parameter
λ
, the situation with amplitude zero
(the homogeneous state) becomes unstable, and the ordered state becomes stable. Thus the appearance
and disappearance of order depends on the strength of the driving force [15].
2 Few examples of non-living patterns
Fascinations for patterns in Nature is as old as civilization. One such example of geometrical pattern
with astonishing symmetry is the hexagonal honeycomb of honeybees (Fig. 4). Honeybees try to fill
the honeycomb without leaving gap. We know that filling up a plane space with identical equal-
sided and equal-angled cells, without leaving any gap, is possible only with triangles, squares, or
hexagons [14, 16]. It is a clever natural choice of honeybees that they construct hexagonal cells to filled
up their honeycombs with less effort because the total length of the cell walls for hexagonal cells filling
a given area is less than that of square or triangular cells enclosing the same area. Thus less material
is needed to make hexagonal cells.
The formation of ripples in sand dunes [Fig. 5] is another example of pattern-forming process.
Approximately periodic ripples, found in wind-swept sand dunes, are the consequence of their driven-
dissipative nature. The driving comes from the wind blowing over the sand. When a very fast moving
wind lifts sand grains into the air, the grains acquire translational and rotational energy from the wind.
While falling back to earth, their energy get dissipated into heat by friction as they collide with other
sand grains [17]. Thus ripples in sand are self-organized patterns formed by wind-blown sand transport.
Such self-organized ripples are also visualized in laboratory experiments and computer simulations by
550 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Fig. 4 Hexagonal pattern in honeycomb.
(a) (b)
Fig. 5 Pattern in sand dunes.
tracking the motion of thousands of mathematical grains colliding according to some predefined specific
rules [18]. In such experiments the ripples began to develop on a flat surface and migrates across the
surface in the direction of the wind. The sand grains, in fact, possesses another astonishing feature —
when a mixture of sand and sugar is poured onto a heap, it forms regular layered structures that in
cross-section look like zebra stripes [19].
Another natural pattern whose beautiful symmetric complexity has captivated scientists for cen-
turies is the snowflakes [Fig. 6]. It is an example of crystal growth in a supercooled or supersaturated
environment and, thus, a far-from equilibrium process. The non-equilibrium structure of snowflake
emerges in the presence of surrounding air supersaturated with water vapor. Each flake grows by
adsorbing water molecules onto its surface from the surrounding air. There is a competition between
the diffusion of water molecules towards the flake and the microscopic dynamics of crystal growth at
the solid-vapor interface. The diffusion kinetics tend to drive the system towards irregular shapes with
maximal interfacial area, while the microscopic dynamics gives rise to surface tension, surface kinetics
and growth anisotropy. Competition between these two microscopic phenomena imposes characteristic
length scales and overall symmetries on the resultant patterns. The rate at which the flake grows and
its overall shape are determined in a complex way by how rapidly water molecules in the surrounding
air can diffuse to the crystalline flake, and by how rapidly the heat released by adsorption can be dissi-
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 551
Fig. 6 Snowflake– a non-living pattern.
pated by diffusion within the air. The overall hexagonal symmetry of snowflake reflects the underlying
crystalline structure of water molecules while the intricate structure of the similar but not identical
branches is a consequence of the instability of the flakes, propagating at an approximately constant
speed into the surrounding supersaturated water vapor [20, 21].
Common in all these non-living pattern formation is that a small deviation from a homogeneous
distribution has a strong feedback that enhances the deviation further. However, formation of such
highly ordered patterns from almost homogeneous initial conditions is not unique to non-living systems.
Diverse living biological systems, as pointed out at the very beginning, exhibit various intriguing spatio-
temporal structures [1–7]. There is no doubt that chemical and mechanical processes are involved,
but elucidating just how they conspire to bring about morphogenetic movements has proved elusive.
Needless to say, the collective motion of animal groups ranging from fish schools to bird flocks to
ants also sometimes display interesting moving patterns. We had already illustrated this fascinating
problem as the dynamics of flocking in Ref. [22]. In this article, we shall specifically concentrate only
on physical processes that lead to the emergence of inherent natural patterns in biological systems,
particularly the animal skin patterns. Mathematical modeling plays an essential role in elucidating the
underlying physical processes responsible for the emergence of various beautiful biological patterns. In
the following sections, an overview of the paradigm models for pattern formation is provided.
3 Turing’s reaction-diffusion model
The concept of pattern formation in mathematical biology arose when the famous naturalist D’Arcy
Thompson [16] tackled the challenge of how to account for the shape and form of organisms. He discov-
ered that new patterns in morphogenesis could be understood as self-organizing systems. Thompson’s
work inspired continued theoretical work, and in 1952, the modeling of pattern formation took an unex-
pected turn when Alan Turing (the founder of modern computer science) demonstrated using a simple
mathematical model that under certain conditions two interacting chemicals, stable in the absence of
diffusion, could be driven unstable by diffusion [23]. If one of the interacting substances diffuses much
faster than the other, then starting from near-uniform initial distributions, this diffusion driven insta-
bility can generate stable concentration patterns within a region in the parameter space (called the
Turing space). This remarkable result was contrary to our intuitive idea, since diffusion generally leads
to a stable equilibrium by smoothing out concentration differences (e.g., a droplet of ink dispersing into
water). Turing considered the reaction of two diffusive chemicals having concentrations uand vwith
552 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
their corresponding diffusion rates Duand Dv, governed by the coupled partial differential equations
∂
u(x,t)
∂
t=f(u,v)+Du∇2u(x,t),
∂
v(x,t)
∂
t=g(u,v)+Dv∇2v(x,t),(2)
where f(u,v)andg(u,v) are the nonlinear reaction kinetics which can be derived from chemical reaction
formula by using the law of mass action and other physical conditions [2].
The homogeneous steady-state values of uand vare solutions of the reaction terms f(u,v)=0
and g(u,v)=0. As the reaction starts, diffusion plays a crucial role. It drives the chemical system
to instability as soon as the feed rate of a chemical exceeds certain threshold value. This diffusion
driven instability is responsible for spontaneous symmetry breaking leading to a final state which is
a highly organized pattern with respect to the chemical concentration. The condition for instability
can be obtained from a standard linear stability analysis [24] on the governing dynamical equations as
follows. Assuming small perturbations
δ
u,
δ
vabout the equilibrium stable state (u0,v0)asu=u0+
δ
u,
and v=v0+
δ
vand making Taylors expansion of the nonlinear functions f(u,v)andg(u,v)aboutthe
stationary state, one can write
∂δ
u
∂
t=(
∂
f
∂
u)(u0,v0)
δ
u+(
∂
f
∂
v)(u0,v0)
δ
v+Du∇2
δ
u,
∂δ
v
∂
t=(
∂
g
∂
u)(u0,v0)
δ
u+(
∂
g
∂
v)(u0,v0)
δ
v+Dv∇2
δ
v,(3)
to the leading order of the perturbation. In order to seek the solution, the perturbations can be assumed
to be harmonic in space, the spatial variation of which can be expressed as eiknx, while the temporal
variation, which allow this perturbation to grow, diminish, or oscillate, can be represented by e
ω
nt.
Thus one can write
δ
u=
δ
u0eiknxe
ω
nt,
δ
v=
δ
v0eiknxe
ω
nt,
which leads to an eigenvalue equation of the form
A
δ
X=
ω
n
δ
X,(4)
with
A=fu−Duk2
nfv
gugv−Dvk2
n,
δ
X=
δ
u0
δ
v0.
Here fu,fv,gu,andgvare used as the partial derivatives with respect to the subscripted variable
evaluated at the stationary state (u0,v0). The perturbation amplitudes
δ
u0and
δ
v0can be nonzero if
and only if det(A−
ω
nI)=0. This characteristic polynomial gives the eigen values as
ω
2
n+(Du+Dv)k2
n−fu−gv
ω
n+DuDvk4
n−k2
n(Dvfu+Dugv)+ fugv−fvgu=0.(5)
A fluctuation associated with the eigenvalue
ω
ngrows if Re(
ω
n)>0. Thus at the onset of instability,
Re(
ω
n)=0. In that case, the term independent of
ω
nin Eq. (5) reduces to zero. This determines the
critical wavenumber kcas
k2
c=Dvfu+Dugv±(Dvfu+Dugv)2−4DuDv(fugv−fvgu)
2DuDv
,(6)
which, in turn, determines the critical length Lc(n)=n
π
/kc. Thus, the fluctuations associated with the
frequency wnare amplified in the system of length Lif L>Lc(n). This is a necessary condition for the
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 553
development of an inhomogeneous state by growth of fluctuations associated with a wavenumber kn.A
growing organism develops a nonuniform pattern only if its size becomes greater than Lc(1). Further,
the most unstable wavenumber, kmax, can be obtained from the condition
∂
∂
kRe[
ω
(k)] = 0,at k=kmax
giving
kmax =[Dvfu+Dugv
2DuDv
]1/2=[fugv−fvgu
DuDv
]1/4.(7)
From the expressions (6) and (7) with imposed periodic boundary conditions, Turing finally arrived
at the following conditions for diffusion-driven instability (also known as Turing instability)
fu+gv<0,(8)
fugv−fvgu>0,(9)
Dugv+Dvfu>0,(10)
(Dvfu+Dugv)2>4DuDv(fugv−fvgu).(11)
These inequalities bound a regime in the parameter space known as Turing space. This instability
criteria of the spatially homogeneous state determine the pattern forming processes in nonequilibrium
systems.
Turing, for the first time, intuitively offered a visionary idea about the mechanism that can generate
order in a non-equilibrium system. It influenced many biologists and mathematicians who, in subse-
quent times, established the fact that Turing systems provide a powerful and generic mechanism for
imitating many natural pattern formations. In order to explain morphogenesis, i.e., the development
of shape or form in plants and animals, Turing labeled the concentration of chemicals as morphogens
which react and diffuse within the cell clusters. The existence of morphogens have already been estab-
lished in experiments by developmental biologists [25]. There are strong experimental evidences that
the pigment patterns on the skin of the angelfish [9–11] and the hair-follicle patterns in mice [26] could
be modeled by a Turing mechanism. All the subsequent theoretical (mathematical) developments are
based on Turing revolutionary idea that the mutual interaction of elements under certain conditions
results in spontaneous pattern formation. Since the specific characteristics of Turing pattern are more
evident in two dimensions than in one, experimental demonstrations have focused on pattern formation
in the skin. Following Turing’s conjecture, a number of Turing-type mathematical models have been
proposed to account for spatial pattern formation in mammals, fish, bacterial colonies, phylotaxis, and
many others [2]. In order to give the reader a unique flavor of this enthralling area, below we shall
discuss how Turing-type system could explain the formation of morphogenetic pattern.
4 Turing-type reaction-diffusion models
4.1 Activator-Inhibitor Model
The study of biological pattern formation has gained immense popularity since the early seventies, when
Alfred Gierer and Hans Meinhardt [27, 28] offered a biologically justified formulation of Turing model
and studied its properties via computer simulations. For the two morphogens involved in the Turing
model [Eq. (2)], they introduced two essential physical mechanisms, namely, short-range activation and
long-range inhibition (Fig. 7), which are now widely accepted as a general patterning principle.
554 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Fig. 7 Schematic representation of an activation-inhibition mechanism. The activator is an autocatalytic, i.e, it
catalyzes its own production. It also activated the production of its rapidly spreading antagonist, the inhibitor.
Pattern formation in this system is possible if the activator diffuses much more slowly than the inhibitor.
In order to have an intuitive understanding of these key physical features, let us consider the
concentration u(x,t)ofaself-enhancing or auto-catalytic chemical substance (morphogen) whose small
increment over its homogeneous steady-state concentration induces a further increase of u.Onceu
begins to increase at a given position, its positive feedback would lead to an overall activation. In
order to prevent the spread of the self-enhancement of u, it has to be complemented by the action of
a fast diffusing antagonist (morphogen) with concentration v(x,t). Two types of antagonistic reactions
are conceivable: either an inhibitory substance is produced by the activator that, in turn, slows down
the activator production, or a substrate is consumed during autocatalysis. Its depletion slows down
the self-enhancing reaction. Thus, the model consists of a pair of reacting chemicals – an activator
with concentration u(x,t)andaninhibitor with concentration v(x,t), and hence it is commonly known
as the activator-inhibitor model. The activator molecule promotes its own production (autocatalysis)
and activates the production of the inhibitor. The antagonistic inhibitor diffuses rapidly into the
surroundings. As a result, there is a local increase in the activation and a long-range antagonistic effect
that restricts the self-enhancing reaction and keeps it localized. It leads to a local high concentration of
activator surrounded by a ring of high inhibitor concentration — a spatial pattern. Roughly speaking,
the system needs to satisfy the condition of local self-enhancement and long-range inhibition in order
to form a stationary pattern.
An essential feature of this theory is the conceptual difference between the morphogen concentration
and the concentration of their sources. It is assumed that the densities of morphogen sources are due to
the distribution of certain cell types releasing morphogens. The cell differentiation is a slower process
and, consequently, the densities of sources of activators and inhibitors changes at a slower rate than
the production or release of effective activators and inhibitors from established sources which might
proceed even in the absence of cell differentiation. For an even source distribution, a slight local peak of
activator concentration u(x,t) would lead to further increase of u. If the activator source density forms
a gradient, and if activator is evenly distributed initially and inhibition extends over the entire area,
the region of high source density will enhance the production of activator and thus fire the gradient. In
this way a shallow gradient of source density can determine the polarity of the pattern of morphogens.
Gierer and Meinhardt [27] suggested the forms of reaction kinetics f(u,v)andg(u,v) appearing in
Eq. (2) as
f(u,v)=
ρ
u(u2
1+
κ
au2)1
v−
μ
uu+
σ
u,
g(u,v)=
ρ
vu2−
μ
vv+
σ
v,(12)
where the coefficients
μ
uand
μ
vare the removable rates,
σ
uand
σ
vthe production terms,
ρ
uand
ρ
v
the cross reaction coefficients, and
κ
ais the saturation constant. The autocatalytic production term
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 555
(a) (b)
Fig. 8 The coat pattern in (a) leopard (Panthera pardus) and (b) cheetah (Acinonyx jubatus).
is nonlinear (u2) since it must overcome disappearance by linear decay (−
μ
uu). The autocatalysis is
slowed down by the action of the inhibitor (1/v). The term
κ
aleads to a saturation of the self-enhancing
reaction at higher activator concentration and thus to an upper limit of activator production. A small
activator-independent activator production term
σ
uis inserted so that the activator never sinks to zero
and enables the reformation of an activator maximum after removal of an established maximum. It is
important for the initiation of the autocatalytic reaction at low activator concentration. Although the
constant
κ
ahas a strong impact on the actual shape, it is not necessary for pattern formation. Further,
the self production rate of v,namely
σ
v, is unimportant for pattern formation. Thus, one can set ka=0
and
σ
v=0in Eq. (12) and arrive at a simplified form of the governing dynamical equations of motion
in terms of dimensionless variables given by
∂
u
∂
t=D∇2u+u2
v−u+
σ
,
∂
v
∂
t=∇2v+
μ
u2−v,(13)
where Dis the ratio of two diffusion rates Duand Dv,
μ
the ratio of
μ
uand
μ
v,and
σ
is the production
rate of u. The long-range inhibition of uby vrequires that the antagonist vdiffuse faster than the
self-enhanced substance u.Thus,DvDuand hence D1. In addition, the inhibitor must also adapt
rapidly to any change of the activator. This is the case if
μ
u>
μ
v. Thus, in order to obtain a stable
pattern, the activator and inhibitor system given by Eq. (13), must satisfy the conditions
D1,and
μ
>1.
It is here important to note that although Turing’s theory was not originally based on self-activation
and lateral inhibition, the equation with which Turing exemplified his theory can also satisfy the
condition of autocatalysis and lateral inhibition. From Eq. (7), one can see that
k2
max =1
2(fu
Du
+gv
Dv
).
Now, considering fu>0and gv<0[following Eq. (8)] and defining l2
1=Du/fu,l2
2=Dv/|gv|,wecan
write
k2
max =1
2(1
l2
1
−1
l2
2
)
which is positive for l1<l2or equivalently Dv/Du>|gv|/fu. In this case, the system becomes stationary
in time (
ω
n=0) and periodic in space (kn=0). For fu>0and gv<0,uis referred as the activator and
vas inhibitor and the condition l1<l2implies short-range activation and long-range inhibition.
556 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Fig. 9 Giraffe coat patterns.
4.2 Numerical simulation of activator-inhibitor models
In order to give a thrill of pattern formation by activator-inhibitor model, here we provide a simple
procedure that can be simulated even in Microsoft Excel. For simplicity, we can assume that the
molecules are distributed in a one-dimensional rod-like embryonic tissue with unit horizontal length.
This rod-like structure is divided into small pieces or cells each of horizontal length dx 1and vertical
length dy 1. Time is discretized in steps of dt. The spatial distribution of activator and inhibitor
in the cells is assumed to be homogeneous. There are two factors that affect the concentration of
activator and inhibitor in these small pieces: (i) the interaction of activator and inhibitor within each
element and (ii) the transfer of activator and inhibitor between each element and its two nearest
neighbours. Accordingly, the concentration of activator molecule in the nth tissue element from the left
end boundary at time m×dt can be describes by p(n,m) and similarly, the concentration of inhibitor
by q(n,m). Thus, the concentration of activator in the tissue element to the right will be p(n+1,m)so
that the amount of molecule transferred from element nto n+1is proportional to the concentration
gradient [p(n+1,m)−p(n,m)]/dx and transverse length element dy [Fig. 10]. If dprepresents the
diffusion coefficient of the activator, the amount of activator that is transferred from element nto n+1
can be expressed as
dp×p(n+1,m)−p(n,m)
dx ×dy×dt,
and similarly from element n−1to nas
dp×p(n−1,m)−p(n,m)
dx ×dy×dt.
Taking these two quantities together and then dividing it by the area (dx ×dy) of the tissue element,
the change in activator concentration in element nbetween time m×dt and (m+1)×dt is obtained as
dp×p(n+1,m)+p(n−1,m)−2×p(n,m)
dx2×dt.
Similarly, the change of inhibitor concentration is
dq×q(n+1,m)+q(n−1,m)−2×q(n,m)
dx2×dt,
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 557
p(n,m) p(n+1,m)
dx
dy
p(n-1,m)
Fig. 10 Transfer of activator (black dots) or inhibitor (pink dots) between neighbouring tissue pieces.
where dqis the diffusion coefficient of the inhibitor. Upon considering both reaction and diffusion, the
changes in concentrations of activator and inhibitor during the time interval (m×dt,(m+1)×dt)can
be written as
p(n,m+1)−p(n,m)= f(p(n,m),q(n,m)) + dp(p(n+1,m)+p(n−1,m)−2×p(n,m))/dx2×dt,
q(n,m+1)−q(n,m)=g(p(n,m),q(n,m)) + dq(q(n+1,m)+q(n−1,m)−2×q(n,m))/dx2×dt,
where the functions f(p,q)andg(p,q) correspond to the nonlinear reaction kinetics appearing in the
general reaction-diffusion equation [Eq. (2)]. From the above two equations the concentrations of
activator and inhibitor at any arbitrary time (m+1)×dt can be obtained provided their initial spatial
distribution are known.
The above equations hold for 1<n<Nwhere Nis the total number of cells in the tissue. At the
two extreme ends, p(1,m)andp(N,m) have only one neighbour and hence they can not be treated as
above. The problem now depends on the imposed boundary condition. In case of periodic boundary
condition, the left-most tissue and the right-most tissue are assumed to be connected so that the value
of p(1,m)andp(N,m) are equal for all m. In case of zero-flux boundary condition, the boundary is
assumed to be impermeable, i.e., material transfer is not allowed across it. In this case the changes of
p(1,m)andp(N,m) are defined as:
p(1,m+1)=p(1,m)+ f(p(1,m),q(1,m)) + dp(p(2,m)−p(1,m))/dx2×dt,
p(N,m+1)=p(1,m)+ f(p(N,m),q(N,m)) + dp(p(N−1,m)−p(N,m))/dx2×dt.
In case of fixed boundary condition, specific values for p(1,m)andp(N,m) are assumed for each iteration
of the simulation.
With this background, let us see how the numerical simulation can be performed for a reaction-
diffusion system given by
∂
u
∂
t=0.8u−v−u3+0.0004
∂
2u
∂
x2,
∂
v
∂
t=1.6u−1.8v+0.02
∂
2v
∂
x2,(14)
on spatial domain size [0,1] with zero-flux boundary conditions. In Eq. (14), the model parameters
satisfy the conditions for pattern formation, namely, DuDvand l1<l2. Let the domain be discretized
with dx =0.05 and dt =0.1so that we have 1/0.05 =20 cells. The initial distribution of activator and
inhibitor in these 20 cells are defined by assigning 20 random numbers for each of the activator and
inhibitor. These numbers can be recorded in Excel spreadsheet where each column represents the
concentration of activator or inhibitor in a specific tissue cell. Let’s enter these initial distributions
558 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
in the spreadsheet columns B1-U2. For dx =0.05 and dt =0.1, the discrete version of the governing
equation is
p(n,m+1)=p(n,m)+0.8p(n,m)−q(n,m)−p(n,m)3+0.0004(p(n+1,m)+p(n−1,m)
−2×p(n,m))/0.052×0.01,
q(n,m+1)=q(n,m)+1.6p(n,m)−1.8q(n,m)+0.02(q(n+1,m)+q(n−1,m)
−2×q(n,m))/0.052×0.01.
This numerical simulation can be performed using Excel spreadsheet. Let’s use the above two equations
to calculate the concentration for the initial distribution given in column C only so that p(n,m)=C1,
p(n−1,m)=B1,p(n+1,m)=D1,q(n,m)=C2,q(n−1,m)=B2,q(n+1,m)=D2. Substituting them in
the above two equation, we obtain the equations in columns C4 and C5 as
C4:=C1+(0.8∗C1)−C2−C1∗C1∗C1+0.0004 ∗(D1+B1−2∗C1)/0.05/0.05)∗0.01
C5:=C2+(1.6∗C1−1.8∗C2)+0.02 ∗(D2+B2−2∗C2)/0.05/0.05)∗0.01
These give the concentration of activator and inhibitor in the tissue at column C. The same equations
can be applied to evaluate the concentrations in other tissue cells. The zero-flux boundary condition
canbeexpressedinB4andB5as
B4:=B1+(0.8∗B1−B2−B2∗B1∗B1+0.0004 ∗(C1−B1)/0.5/0.5)∗0.01
B5: =B2+(1.6∗B1−1.8∗B2+0.02 ∗(C2−B2)/0.5/0.5)∗0.01
andinU4andU5as
U4:=U1+(0.8∗U1−B2−U1∗U1∗U1+0.0004 ∗(T1−U1)/0.5/0.5)∗0.01
U5: =U2+(1.6∗U1−1.8∗U2+0.02 ∗(T2−U2)/0.5/0.5)∗0.01
By evaluating these cells, we can obtain the complete spatial distribution of activator and inhibitor at
time dt. To obtain the distribution of activator and inhibitor at a later time t, we have to carry out
the process repeatedly (if t=ndt , we must iterate ntimes). Fortunately Excel automatically converts
the equations to appropriate forms by simply copying and pasting the whole row to the rows below.
By doing this 100 −200 times repeatedly, one can observe that the concentrations gradually form a
periodic structure, as shown schematically in Fig. 11.
Thus we see that the numerical simulation consists of many steps (iterations) in which the com-
puted concentration change within a short time interval is added to existing concentrations, and the
resulting new concentration is used to calculate the next concentration change, and so on. However
this computation is rather time consuming and, therefore, researchers usually use powerful software
like Mathematica. The Mathematica programs for reaction-diffusion system are provided by some
research group on request, see for example the personal web-page of Hans Meinhardt (Max-Planck-
Institut f¨
ur Entwicklungsbiologie, T¨
ubingen, Germany). For those who are familiar with the computer
programming in FORTRAN, Chapter 17 of Ref. [29] will be very useful for simulations.
4.3 Applicability of activator-inhibitor models
There are some common characteristics of animal skin pattern: (i) they can be classified into three basic
patterns, namely, spots, stripes and polygons (ii) they appear independent of the internal tissues or the
body structure. For example, many fishes have uniform skin patterns across their entire bodies despite
the structural difference of the internal tissues (iii) their robustness against external perturbation. In
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 559
Fig. 11 A schematic view of the concentration pattern obtained via numerical simulation of activator-inhibitor
model. The thick line represents the activator concentration and the thin line for the inhibitor concentration.
some tropical fish, when skin patterns are disturbed by injury or transplantation, or when growth of
the fish enlarges the spacing of the patterns, there is some rearrangement mechanism that restores the
original spacing [9,30]. These features indicate the nature of underlying patterning mechanism.
A wide variety of remarkable coat patterns of mammals such as the leopard, the cheetah (Fig.
8), the jaguar, the zebra (Fig. 1) and the giraffe (Fig. 9), correspond to regions of differently colored
hair. Hair color is the characteristic trait of specialized pigment cells called melanocytes,whichare
found in the innermost layer of the epidermis. The melanocytes generate a pigment called melanin.
In mammals there are essentially only two kinds of melanin – eumelanin, which makes hair colour
black or brown, and phaeomelanin, which makes hairs yellow or reddish orange. It is believed that
the production of melanin depends on the presence of chemical activators and inhibitors. In reaction-
diffusion models it is assumed that one of the morphogens is an activator that causes the melanocytes
to produce one kind melanin, say black, and other is an inhibitor that results in the pigment cells
producing no melanin. Suppose the reactions are such that the activator increases its concentration
locally and simultaneously generates the inhibitor. If the inhibitor diffuses faster than the activator, it
creates an island of high activator concentration within a region of high inhibitor concentration. Detail
mathematical analysis and numerical simulations of activator-inhibitor model indicate its capability of
reproducing an enormous range of biological as well as chemical patterns. This success continues to
inspire and motivate biologists, physicists, mathematicians, and computer scientists as well [12].
By computer simulations of Turing-type reaction-diffusion model, Murray [7] was able to generate
various mammalian coat patterns such as spots of the leopard (Panthera pardus), striped tails of the
cheetah (Acinonyx jubatus), the jaguar (Panthem onca), and the genet (Genetta genetta). He showed
that such patterns depend strongly on the geometry and scale of the domain where the chemical reaction
takes place. Very few spots are formed in the simulation when the inhibitor is allowed to diffuse quickly
across the domain. In addition, the simulated pattern is too small to see when the domain is very large.
From these simulation results, Murray explained why very small and very large animals tend not to be
patterned, while medium sized animals have many spots or stripes. Murray’s calculations also showed
that the markings on the tail of an animal must always change from spots where the tail is thick near
the body to stripes where the tail is thin at the tip. Thus, it explains why a spotty animal can have a
560 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
Fig. 12 Pattern in butterfly wings.
stripy tail, but a stripy animal can never have a spotty tail.
The spectacular color patterns in butterfly wings (Fig. 12) have also been studied via Turing-type
models [5, 6]. In general, there exist two different kind of pattern in butterfly wings– color pattern and
the spatial arrangement of scale cells. Color patterns are formed by the colors of the regularly arranged
scale cells. The color of scale cells is mainly due to the presence of chemical pigments (particularly those
known as pterins), but can sometimes result from diffraction of light in the physically fine structured
scale. It is known that the formation of the color pattern is independent of the arrangement pattern
of scale cells. The time-scales on which these two patterns are generated are different from each other.
The arrangement pattern of scale cells occurs in the early stage of pupation, while color patterns
appear in the last stage of pupation after completion of cell arrangement [5]. Numerical simulations of
Gierer-Meinhardt model on a geometrically accurate wing domain [6] showed that the wing coloration
is due to a simple underlying stripe-like pattern of some pigment inducing morphogen. The simulated
patterns, obtained by choosing different types of boundary conditions, are found to be consistent with
many of those observed on the butterfly.
An interesting fact about the activator-inhibitor models is that they are not only able to explain
the formation of various biological patterns observed in Nature, but also explain some non-biological
patterns such as in sand dunes. In Section 2, we have already discussed about the structure of sand
dunes. Hans Meinhardt suggested that such dune formation is also akin to an activator-inhibitor
system, in which short-range activation competes with long-range inhibition. Dunes are formed by
deposition of wind-blown sand. As a dune gets bigger, it enhances its own prospects for growth by
capturing more sand from the air and providing more wind shelter for the grains on the leeward side.
But in doing so, the dune removes the sand from the wind and so suppresses the formation of other
dunes in the vicinity. The balance between these two processes establishes a constant mean distance
between dunes which depends on wind speed, sand grain size (and thus their mobility in the wind) and
so forth.
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 561
5 Growth dependent pattern
A distinctive feature of most of the animal coat patterns is the growth dependent change of the pattern.
For example, at the very young stage, the leopards and jaguars have spot (fleck) patterns. As they
grow in size, these spots gradually transform into rosettes (small spots organized into patterns of six or
fewer spots) in leopards and blotches (irregular shaped areas of dark on a usually lighter background) in
jaguar at their adult stage [8]. Numerous observations of fish skin markings [9,10, 31] also demonstrated
the alternation of patterns during growth. Experiments on zebrafish skin [30] suggest that the stripe
formation takes place due to an autonomous mechanism that satisfies local self-enhancement and long-
range inhibition, and is independent of prepattern. The pattern remains dynamic in growing fish and
with increasing size, new stripes emerge with the stripe spacing remains roughly constant. A more
recent investigation by Venkataraman et al. [32] on Amago skin pigmentation patterns suggested that
a reaction-diffusion system on an evolving surface is a viable model for describing the emergence of
stripe patterns on fish skin.
To capture the features of such growing patterns on animal skins, a number of two-stage models
describing two-stages of patterning have been proposed [8,11,33]. One such coupled equation of motions
involving the quadratic and cubic interactions between two morphogens uand vis given by [33]
∂
u
∂
t=D
δ
∇2u+
α
u+v−r2uv −
α
r3uv2,(15)
∂
v
∂
t=
δ
∇2v+
β
v+
γ
u+r2uv +
α
r3uv2,(16)
where D=Du/Dvis the ratio of the diffusion coefficient of the activator uto that of the inhibitor v,
and
δ
is a scaling variable giving the spatio-temporal scale.
For numerical simulation of the above equations, the concentrations of the morphogens are assumed
to be of the form
u(x,t)∼u0e
ω
nteikn·x,v(x,t)∼v0e
ω
nteikn·x,
where the temporal growth rate
ω
ncan be obtained as a function of possible modes knfrom the
eigenvalue equation. Standard linear stability analysis on these dynamical equations about the steady
state (0,0) yields conditions on the parameters D,
δ
,
α
,
β
,
γ
,r2,andr3for the diffusion driven
instability. The initial values of uand vat each point are assigned randomly between 0 and 1. Only
discrete number of modes are allowed due to zero-flux boundary condition. For the values of the
parameters D=0.45,
δ
=6,
α
=0.899,
β
=−0.91,r2=2,r3=3.5, the simulation generates a spotted
pattern [8]. This pattern is used as an initial distribution for the morphogens concentration uand vin
the second stage of the model. This way the temporal evolution of the coat markings (from spots to
rosettes and from spots to blotches) in leopard and jaguar at different stages of growth are generated [8]
by tuning parameters of the model in steps. In these simulations, the density of spots is found to be
enhanced with increasing the value of r2(the coefficient of the quadratic term), indicating that the
quadratic terms in Eqs. (15) and (16) favor spots. The effect of growth can be seen in this two-stage
model by tuning the parameter
δ
as it is inversely related to the domain scale.
Unlike mammal skin patterns, which simply enlarge proportionally during their body growth [Note
that, in the above model simulation, the number of rosettes or blotches on the adult leopard and jaguar
is roughly the same with the number of spots on younger animals], pattern alternations during growth
in several phylogenetically distant fish and reptiles have been observed. As the fish grow in size, division
and insertion of new spots or stripes occurred to maintain the spot or stripe spacing. For example,
as the size of the marine angelfish Pomocanthus doubled, new stripes along the skin develops between
the old ones [9]. In order to take into account the growth of fish pigmentation pattern in the marine
562 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
angelfish, Kondo and Asai [9] proposed a linear Turing-type model with saturation, given by
∂
u
∂
t=x(u,v)−du+Du∇2u,
∂
v
∂
t=y(u,v)−gv +Dv∇2v,(17)
with
x(u,v)=au −bv +c,y(u,v)=eu −f,0x,yP.(18)
Here Pis the upper limit of the synthesis of uand v,and fis the threshold of the inhibitor production.
The upper and lower limit of x(u,v)andy(u,v) are set to avoid unrealistic situations. In this simulation,
the value of the parameters are chosen as Du=0.007,Dv=0.1,d=0.03,g=0.06,a=b=0.08,c=0.05,
e=0.1,and f=0.15. The upper and lower limits for the synthesis rate of activator x(u,v) and inhibitor
y(u,v)aresetas0<x<0.18 and 0<y<0.5. A comparison of stripe rearrangement and numerical
simulations of the model showed close agreement. Using a similar model, Asai et al. [10] numerically
investigated how the mutations in the zebrafish change the pattern from stripe to spots. In this
simulation, with a gradual increase in the value of the parameter c[appearing in Eq. (18)] from 0.04
to 0.44, the striped pattern gradually changes to meandered lines, fused spots, isolated spots, and then
small spots. This led to suggest that stripe patterns on fish skin depend on some spatial regulatory
mechanism that can be usefully described by the reaction- diffusion model. For more details of the
method and results of numerical simulation, readers are referred to Ref. [10].
The above mentioned numerical and theoretical works strongly indicate the capability of Turing-
type reaction-diffusion models in predicting the dynamic process of color pattern rearrangement in some
animals. The color patterns of such animals are dynamically and autonomously rearranged to resolve
pattern inconsistency due to body growth, indicating the flexible property of underlying mechanisms.
It is, indeed, really interesting and surprising how such a simple model generates a rich variety of
patterns just by tuning only few model parameters, boundary conditions, and spatial domains. The
striking similarity between the actual and the simulated pattern strongly suggests the viability of
reaction-diffusion systems.
6 Conclusion
The emergence of various spatio-temporal natural patterns in physical or biological systems from the co-
ordination of events occurring across molecular, cellular, and tissue scales, is an exciting topic in physics
and many other disciplines such as chemistry, ecology, animal biology, mathematics, and neuroscience.
From the above mathematical and physical descriptions, we have seen that the complex nonlinear and
feedback interactions of various physical and chemical processes, under certain conditions, result in a
wide variety of biological patterns. The minimum requirements for such nonequilibrium patterns are
the presence of nonlinearities and of spatial coupling. It is, therefore, not possible to understand pat-
tern formation in any depth without having a solid background in the mathematical underpinnings of
nonlinear equations. Mathematical modeling serves as an important task in developmental biology due
to its ability to suggest and test mechanisms by which complex biological patterns can arise. Although
the development of a novel approach to study this complex process remains an intellectual challenge,
various in-depth investigations as described in this article, demonstrated that the reaction-diffusion
mechanism, embedded in the short-range activator and long-range inhibitor, fabric the spatio-temporal
patterns. Thus, Turing-type reaction-diffusion model serves as a feasible framework for understanding
the dynamics of many biological pattern formation.
The fascinating structural patterns of many biological species play important role in their behavior
such as shoaling, mate choice, concealing, and antagonistic display. The ability to physiological and
Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564 563
morphological change in pattern (or color) is essential for survival in many species. Investigating the
basic mechanism for such physiological and morphological changes in pattern is not only of academic
interest but also have practical implications in clinical science. A number of skin diseases are attributed
to defective pigmentation. Vitiligo is one such acquired skin disease due to destruction of melanocytes
and characterized by patches of unpigmented skin (often surrounded by a heavily pigmented border).
Some other examples are keratonosis (an abnormal condition of the epidermis), melanosis (an abnormal
deposits of melanin), leukoderma (appearance of spots or bands in unpigmented skin), xanthosis (an
abnormal yellow discoloration on the skin), livedo (an abnormal patchy bluish discolorations on the
skin), and albinism ( the congenital absence of pigmentation in the eyes, skin, and hair). Understanding
the mechanisms by which pigment cells migrate and proliferate to pattern the skin may lead to a
more successful course of treatment. We expect that the enormous advances in biotechnology and
computational power might led to unravel the deeper insights to the underlying mechanisms. We
conclude with a hope that this introductory article will provide a glimpse to the beginners who would
like to endeavour exciting and challenging research work on this enthralling area of science.
References
[1] Bard, J.B.L. (1981), A model for generating aspects of zebra and other mammalian coat patterns, Journal
of Theoret ical Biology,93, 363-385.
[2] Murray, J.D. (2003), Mathematical Biology, Vol. II, Springer-Verlag, Berlin.
[3] Murray, J.D. and Myerscough, M.R. (1991), Pigmentation pattern formation on snakes, Journal of Theoretical
Biology,149, 339-360.
[4] Murray, J.D. (1981), On pattern formation mechanism for lepidopteran wing patterns and mammalian coat
markings, Philos.trans.R.Soc.London,Ser.B,295, 473-496.
[5] Nijhout, H.F. (1990), A comprehensive model for color pattern formation in butterflies, Proc. R. Soc. London
B,239, 81-113.
[6] Sekimura T., Madzvamuse A., Wathen A. J., and Maini P. K. (2000), A model for color pattern formation
in the butterfly wing of Papilio dardanus, Proc.R.Soc.Lond.B,267, 851-859.
[7] Murray, J.D. (1981), A pre-pattern formation mechanism for animal coat markings, Journal of Theoretical
Biology,88, 161-199.
[8] Liu, R.T., Liaw, S.S., and Maini, P.K. (2006), Two-stage Turing model for generating pigment patterns on
the leopard and the jaguar, Phys. Rev. E,74, 011914.
[9] Kondo, S. and Asai, R. (1995), A Reaction-diffusion wave on the marine angelfish Pomacanthus, Nature,
376, 765-768.
[10] Asai, R., et al. (1999), Zebrafish Leopard gene as a component of the putative reaction-diffusion system,
Mechanisms of Development,89, 87-92.
[11] Barrio, R.A., et al. (2009), Modeling the skin pattern of fishes, Phys. Rev. E,79, 031908.
[12] Walgraef, D. (1997), Spatio-Temporal Pattern Formation, Springer, New York.
[13] Harrison, L.G. (2011), The Shaping of Life, Cambridge University Press, New York.
[14] Cross, M. and Greenside H. (2009), Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge
University Press, Cambridge.
[15] Walgraef, D. (1996), Spatiotemporal Pattern Formation, With Examples in Physics, Chemistry and Materials
Science, Springer.
[16] Thompson, W. D’Arcy (1961), On Growth and Form, Cambridge University Press, Cambridge (U. K.).
[17] Bagnold, R.A. (1941), The Physics of Blown Sand and Desert Dunes, Chapman and Hall, London.
[18] Forrest, S.B. and Haff, P.K. (1992), Mechanics of wind ripple stratigraphy, Science,255, 1240.
[19] Andreotti, B. (2004),The song of dunes as a wave-particle mode locking, Physical Review Letter 92, 238001.
[20] Ball, P. (1999),The Self-Made Tapestry: Pattern Formation in Nature, Oxford University Press, Oxford.
[21] Jacob, E.B. and Levine, H. (2001), The artistry of nature, Nature,409, 985-986.
[22] Dutta, K. (2010), How birds fly together: the dynamics of flocking, Resonance,15, 1097-1110.
[23] Turing, A.M. (1952), On the chemical basis of morphogenesis, Philosophical Transactions of the Royal Society
of London, Series B, Biological Sciences, 237, 37-72.
[24] Nicolis, G. (1995), Introduction to nonlinear science, Cambridge University Press, Cambridge.
[25] Koch, A.J. and Meinhardt H. (1994), Biological pattern formation: from basic mechanisms to complex
structures, Rev. Modern Phys.,66, 1481-1507.
564 Kishore Dutta / Journal of Applied Nonlinear Dynamics 6(4) (2017) 547–564
[26] Sick, S., et al. (2006), WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism,
Science, 314, 1447-1450.
[27] Gierer, A. and Meinhardt, H. (1972), A theory of biological pattern formation, Kybernetik,12, 30-39.
[28] Meinhardt, H. (2003), The Algorithmic Beauty of Sea Shells, Third Edition, Springer, Berlin.
[29] Meinhardt, H. (1982), Models of Biological Pattern Formation, Academic Press, London.
[30] Yamaguchi, M., Yoshimoto, E., and Kondo, S. (2007), Pattern regulation in the stripe of zebrafish suggests
an underlying dynamic and autonomous mechanism, PNAS,104, 4790-4793.
[31] Shoji, H. et al. (2003), Origin of directionality in the fish stripe pattern, Developmental Dynamics,226,
627-633.
[32] Venkataraman, C., et al. (2011), Modeling parr-mark pattern formation during the early development of
Amago trout, Phys. Rev. E.,84, 041923.
[33] Barrio, R.A., Varea, C., Arag´on, J.L., and Maini, P.K. (1999), A two-dimensional numerical study of spatial
pattern formation in interacting Turing systems, Bulletin of Mathematical Biology,61, 483-505.
