Seminars in Cell & Developmental Biology 20 (2009) 82–89

Contents lists available at ScienceDirect

Seminars in Cell & Developmental Biology
journal homepage: www.elsevier.com/locate/semcdb

Review

Theoretical analysis of mechanisms that generate the pigmentation patternof animals
Shigeru Kondo ∗ , Hideaki Shirota
Division of Biological Science, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japana r t i c l e

i n f oArticle history:
Available online 19 October 2008
Keywords:
Pigment pattern
Pigment cell
Reaction–diffusion
Theoretical analysis
TuringSimulation
Zebraﬁsha b s t r a c t
Mechanisms of animal skin pigment pattern formation have long been of interest to developmental and mathematical biologists. Although there has been a well-studied theoretical hypothesis—thereaction–diffusion system—that is able to reproduce the variety of skin patterns, a lack of molecular evidence has kept it just a hypothesis. In this review, we summarize the results of theoretical studies to date
for researchers not familiar with their mathematical underpinnings, and we discuss future approaches
that will more fully integrate mathematical models and experimental analyses.
© 2008 Elsevier Ltd. All rights reserved.

Contents
1.
2.

3.

4.

5.
6.

7.

8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Common characteristics of animal skin patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.
Diversity among the closely related species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.
Autonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.
Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A mathematical model that satisﬁes characteristics of animal pigment patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.
The RD system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.
Necessary conditions for stationary pattern formation in the RD system: “local enhancement and long-range inhibition” . . . . . . . . . . . . . . .
Ability of the RD system to form the variety of observed patterns in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.
Basic three patterns: spots, stripes, and polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.
Complex 2D patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1.
Leopard spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2.
Directionality of the stripes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3.
More complex patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Autonomy of the RD pattern and the real skin pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robustness of the RD pattern and the real skin pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.
Rearrangement of the pattern induced by the growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.
Rearrangement of the pattern induced by artiﬁcial perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RD model as the working hypothesis for the molecular genetic study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.
Other mathematical models those are able to form the similar patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.
Molecular genetic studies on the pigmentation of zebraﬁsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.
Consistency of the mathematical model with the experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author. Tel.: +81 52 789 2503.
E-mail address: skondo@bio.nagoya-u.ac.jp (S. Kondo).
1084-9521/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.semcdb.2008.10.008

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1. Introduction
Many animals have fascinating color patterns on their skin,which play important roles in many kinds of behavior such as
shoaling, mate choice, camouﬂage, and antagonistic displays. How
these skin patterns form is a long-held but still unsolved question. There have been two principle avenues towards solving the
problem. Classic genetic studies have long been done for withpigmented domestic mammals, birds and ﬁshes. Many strains ofdifferent pigmentation patterns have been isolated and relationships between mutants analyzed mainly by the breeding [1–3].
However, as the species used in such analysis were not standard
model animal of molecular genetic studies, it had been difﬁcult
to identify the underlying molecular mechanism in such animals.
There is also a long history of theoretical study on this subject [4–6].
With many characteristic pattern shapes, mathematical methods
have shed light on the logic of underlying mechanisms: althoughmathematical studies cannot identify the involved genes or signaling pathways that function in the pattern-forming event, they can
predict fundamental behaviors of skin patterns and can suggest thenecessary conditions for their formation. Emergence of zebraﬁsh
as a model organism for developmental genetics has made it possible to combine the two kinds of studies because both “molecular
genetic” and “mathematical” approaches are possible with thisspecies [7–10]. In this review, we summarize theoretical studiesof the reaction–diffusion (RD) system, a representative mathematical model that explains the patterning phenomena, and we show
how this model can explain the speciﬁc characteristics of animalskin patterns.
2. Common characteristics of animal skin patterns
Although there is large diversity in skin pattern, there are somecommon characteristics that can suggest underlying mechanisms.
2.1. Diversity among the closely related species
Most skin pigment patterns can be roughly classiﬁed into threebasic patterns: spots, stripes, and polygons (i.e., reverse spots). As
many animals have the skin of continuous coloration, “uniform pattern” can be counted as another skin pattern. It is interesting to notethat some species in the same family or genus often have different
patterns. Catﬁsh species shown in Fig. 1 belong to the same family,
Loricariidae [11] and their skin patterns differ despite morphological similarities in other respects. As they are phylogenetically very
close, molecular mechanisms at work are likely to be very similar;
i.e., an essentially similar pattern-forming system can form very
different patterns, such as spots, stripes, and polygons. There aremany other examples showing this pattern diversity in the familyof ﬁsh (Tetraodontidae, Siganidae, Labridae, and Serranidae [12])and frogs (Dendrobatidae [13]), suggesting that ﬂexibility of theunderlying mechanism is a common feature among the pigmented
animals.
2.2. Autonomy
The second characteristic of animal skin patterns is that they
appear independent of the internal tissues or the body structures.
For example, the ﬁshes shown in Fig. 1 have uniform skin patterns
across their entire bodies. In spite of the structural difference of theinternal tissues, both the patterns and the length of the repetitive
units are almost identical. This fact is very interesting when considering the nature of the underlying patterning mechanisms, because
the distance of the repetitive units is much larger than the size of asingle cell. How do cells know how wide stripes should be? In the83

case of striped pattern of segmentation genes in Drosophila, positional information comes from morphogen gradients that function
as a prepattern. However, in the case of skin pattern, it has not beenpossible to ﬁnd such a prepattern, suggesting that many skin patterns are formed autonomously without reference to an external
prepattern.
2.3. Robustness
The third interesting characteristic of animal skin pattern is its
robustness against perturbation. In some tropical ﬁsh, when skinpatterns are disturbed by injury or transplantation, or when growthof the ﬁsh enlarges the spacing of the patterns, rearrangement of thepattern is induced, and this restores the original spacing [14–16].
The third interesting characteristic of animal skin pattern is its
robustness against perturbation. In some tropical ﬁsh, when skinpatterns are disturbed by injury or transplantation, or when growthof the ﬁsh enlarges the spacing of the patterns, rearrangement of
the pattern is induced, and this restores the original spacing.
3. A mathematical model that satisﬁes characteristics ofanimal pigment patterns
From the characteristics of the skin pattern explained earlier
and extensive empirical analyses [16–18], it is clear that formationof skin pigment patterns cannot be explained by the function of single molecules or a simple prepattern-based mechanism. Although
there is no known molecular mechanism that is capable of explaining all the characteristics of pigment patterns, a well-establishedmathematical model, called the RD system, can explain these characteristics very well [4].
3.1. The RD system
In 1952, Turing presented a ground-breaking idea that a combination of reaction and diffusion of the putative chemical substances
can generate spatial patterns in his work entitled “The chemicalbasis of morphogenesis” [4]. He studied the behavior of a complex
system in which two substances interact with each other and diffuse at different diffusion rates, which is known as the RD system.
Turing proved mathematically that such a system can form stable periodic patterns in the ﬁeld even from a uniform state. In the
1970s [4,5,19,20], theoretical studies of the RD system with computer simulation showed that this model can produce most of the2D patterns found in the skin of animals.
3.2. Necessary conditions for stationary pattern formation in the
RD system: “local enhancement and long-range inhibition”
The RD system is described by partial differential equations,which mathematically deal with the synthesis, decay and diffusion of the hypothetical substances. Although many different
types of equations can generate the stable periodic pattern, theyshare a common property, “local enhancement and long-range
inhibition,” which is thought to be the necessary condition for pattern formation [21,22]. The simplest type of these equations, theactivator–inhibitor model, is shown in Fig. 2a and is represented asfollows [6]:∂uu2
= a − bu ++ Du u∂tv(1 + u2 )∂v
= −v + u2 + Dv v∂t
Here u and v are the concentrations of activator and inhibitor.
a, b, Du , and Dv represent the parameters. In the model, the acti-

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Fig. 1. Pigment pattern diversity in catﬁsh, family Loricariidae. The most frequently seen skin pigment patterns are spots, stripes and polygons. This diversity of the patternis often seen in the species belonging to an identical family or genus. Pictures are species in the family Loricariidae.

vator is supposed to activate the synthesis of both activator andinhibitor, whereas the inhibitor inhibits the function of activator,
and the diffusion rate of inhibitor is set much larger than that of
the activator. The dynamic behavior of the system is represented
schematically in Fig. 2b–d. Graphs in Fig. 2b–d show the concentration of activator and inhibitor in the 1D ﬁeld. Fig. 2b shows how
the periodic peaks emerge from a random pattern. In the central
region of Fig. 2a, the concentration of the activator is set relatively
higher than in other regions as an initial condition. By the selfenhancing nature of the activator, the concentration of activatorincreases at the center, followed by the increase of inhibitor at theregion (Fig. 2c). As the diffusion rate of inhibitor is much larger thanthat of the activator, substantial amounts of inhibitor move toward
the lateral regions, and depress the activator function, decreasing
the activator concentration there. Eventually, this system ampliﬁes the minor concentration difference and stabilizes the borderof the concentration peaks and valleys (Fig. 2d). This system also
has the ability to regulate the width of the peak or valley regions.
When a wide region has a high concentration of activator, diffusion
cannot decrease the inhibitor concentration at the center. Then thehigh inhibitor stops activator synthesis, and decreases the activator
concentration to divide the wide peak. At the valley of the activator
concentration, activator production is suppressed by the inhibitor
diffusing from the neighboring region. However, if the low activator region is too wide, inhibitor cannot reach to the middle of
the region; then the inhibition of the activator synthesis is lost at
this region and the activator synthesis starts to insert a new peak.
These effects occur continuously at all places in the ﬁeld, and giverise to evenly distributed wave-like patterns, the spacing of which
is determined by parameter values of the equations.
(This explanation is only for giving a rough image of the RD
mechanism. For more “mathematical” explanations, please refer[5,23,24])
4. Ability of the RD system to form the variety of observed
patterns in nature
4.1. Basic three patterns: spots, stripes, and polygons
Shapes of the RD patterns are sensitive to parameter values ofthe differential equations, and using this property, it is possible
to generate various patterns with almost identical equations. The
relationship between the parameter values and the 2D patterns has
been studied extensively using computer simulations. Among theparameters in the equation, Meinhardt pointed out that the valuethat determines the upper limit of activator synthesis is particularly
important to pattern outcome [24]. Fig. 3 shows the relationshipbetween the shape of 1D wave and the 2D patterns. When theupper limit of activator synthesis is high, the wave peaks become
high and pointed. On the contrary, when the upper limit is low,
the wave peaks become bold, and the valleys become pointed. The
former parameter value makes the spot pattern, whereas the latterparameter value makes polygons. When this parameter is set at anintermediate value, the system forms a stripe pattern. Therefore,
it is possible to generate the three basic periodic patterns—spots,
stripes, and polygons, just by changing a single parameter in the RD

Fig. 2. Schematic explanation of the reaction–diffusion mechanism. (a) Interaction network of a simple RD model called “activator–inhibitor model”. (b) A process of periodic
pattern formation in the 1D ﬁeld originated from a random pattern. (c) A process of division of a broad peak. (d) A process of new peak insertion. The dynamic properties shownin this ﬁgure are common among most of RD models that satisﬁes the necessary conditions for the stable pattern formation, “local activation and long-range inhibition”.
Solid lines and dotted lines represent the concentration of activator and inhibitor, respectively.

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equation. It is notable that most of the pigmentation patterns seenin animal skin belong roughly to one of these three patterns.
4.2. Complex 2D patterns
Although the pure RD systems can generate only the basic patterns described earlier, it is possible to generate more realistic andcomplex patterns by adding some additional conditions.

Fig. 3. Difference in the shape of the 1D waves and 2D patterns depending of aparameter value in the equation. When the upper limit of the activator synthesis ishigh, the wave peaks becomes pointed and spot pattern forms in the 2D ﬁeld. Whenthe parameter is low, the wave peaks become rounded and polygon (reverse spots)
pattern forms. Stripes form when the parameter value is intermediate. Simulations
are performed as described in [10].

4.2.1. Leopard spots
One of the complex but relatively common animal patterns is
the so-called leopard spots. To generate the leopard pattern, one
ﬁrst needs to make a spot pattern, and then the ﬁeld should beenlarged, which corresponds to the growth of the animal. Enlargement (growth) of the ﬁeld (skin) enlarges each spot. But as the RD
system has a tendency to keep the size of the spots, they divide
into 2–4 daughter spots of original size to form the characteristic
leopard spots, which is very similar to the skin pattern of a leopard(Fig. 4b and d). With the parameter value that favors stripes insteadof spots, enlargement of the ﬁeld generates rings, which is similar
to the patterns seen in the skin of jaguars (Fig. 4a and c). Such spots
and rings also can be made by changing the diffusion rate after thesimple spot pattern is formed [25].

Fig. 4. Dividing spots and ring pattern in the animals and simulations (a) skin pattern of a jaguar; (b) skin pattern of a leopard; (c and d) simulated pattern of RD mechanismin the growing ﬁeld. Simulations are performed as described in [10].

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Fig. 5. Directionality of the stripes near the boundary stripes tend to run along or vertical to the ﬁeld boundary made by the eyes of ﬁsh and the simulation. (a) Arthoron
mappa, (b) Acanthrus lineatus, and (c and d) simulated pattern besides the circle boundary. Simulations are performed as described in [10] with the following modiﬁcationof speciﬁc boundary conditions. In the simulation shown in Fig. 3c, the activator concentration in the center part of the ﬁeld (black circle) is ﬁxed at low level. This boundary
condition induces the quick formation of stripes along the boundary. As the newly formed stripes induce additional stripes in the neighboring region, concentric ring pattern
was generated. In the simulation shown in Fig. 3d, it was supposed that the pattern formation occurred by the two steps. In the ﬁrst step, the reaction and diffusion occurred
only around the edge of the center circle boundary, which formed the periodic around the edge. Then the reaction and diffusion were made active in all other regions. The
periodic pattern around the center circle acted as the patterning cores and formed the radial stripes. Finer simulations of the ﬁsh pattern around the ﬁsh eyes are describedby Sanderson et al. [29]. Fish pictures are courtesy of Tokai University Press.

4.2.2. Directionality of the stripes
Most stripe patterns seen in animals have speciﬁc directionality.
As the RD system lacks directional information, additional conditions are required to orient the stripes. Shoji et al. [26,27] haveshown that adding even very small directionality to the diffusion
can orient stripes, and suggested that the scales of the ﬁsh may
be the origin of this directionality. Boundary conditions are alsoable to specify stripe directionality near the edge of the patterningﬁeld [28]. Stripes of the RD system tend to run parallel or verticalto a ﬁxed boundary, which is a common characteristic of nonlinear
waves. In the ﬁsh skin, the eyes create such an edge, and as shown in
Fig. 5, the stripes become concentric or radial to the round boundary[29].
4.2.3. More complex patterns
The additional conditions described here can be used in combination to generate more complex patterns. For example, a gradualchanging of the parameter value which determines the basal production of activator causes the resulting pattern to change fromspots to polygons, which are seen in Plectorhinchus lineatus (Fig. 6aand c), and a gradual enlargement of the ﬁelds in a speciﬁc direction is capable of making a complex pattern like that of whale sharks(Fig. 6b and d). Extensive studies with RD models thus suggest thatan RD system can generate nearly all animal skin patterns having
spatial periodicity [30,31].

5. Autonomy of the RD pattern and the real skin pattern
One of the remarkable properties of the RD mechanism is its
independence from the prepattern. As explained earlier, the system ampliﬁes the minor difference of concentration and regulates
the width of the peaks and valleys to get the periodic pattern of
uniform spacing. The shape and the spacing are determined onlyby the equation and the parameter values. Pigment patterns onreal skin also can appear largely independent of internal structure, suggesting the autonomy of the pattern-forming process.
Experimental results with zebraﬁsh have shown more directly
the autonomous property of the underlying mechanism. Usinga temperature sensitive allele of the panther mutant, Parichy et
al. eliminated two classes of pigment cells (blacks melanophoreand yellow xanthophores) in the adult zebraﬁsh, and observed
the regeneration of the pattern in the body and on the tail ﬁn.
Their result on the body is consistent with a patterning mechanism that is autonomous to the pigment cells or the presence of
persisting cues [9]. On the ﬁn, however, the directionality of the
stripes was lost, showing the autonomous nature of the patternforming process (Fig. 7a). Using a laser to kill pigment cells ina speciﬁc area of body trunk, Yamaguchi et al. got almost an
identical result (Fig. 7b). When all the pigment cells in a broad
region were killed, the regenerated stripes sometimes lost theirdirectionality and the resulting patterns were very similar to that

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Fig. 6. Complex 2D patterns seen in the ﬁsh (a) Plectorhinchus lineatus. (b) Whale shark. (c) A parameter value corresponding to the basal synthesis of the parameter in theactivator inhibitor model is gradually changed in the 2D ﬁeld. (d) Stripe pattern is formed in the growing 2D ﬁeld. Details of the simulation and the used equations were
described in [42]. Fish pictures are courtesy of Tokai University Press.

generated by an RD mechanism from a random initial condition[15].

skin pattern, one can further test whether an RD mechanism really
underlies pattern formation or not in vivo.

6. Robustness of the RD pattern and the real skin pattern

6.1. Rearrangement of the pattern induced by the growth

Another important characteristic of the stationary pattern madeby an RD mechanism is its robustness against the perturbation. As
far as the patterning mechanism functions, the resulting pattern is
not ﬁxed, but is able to autonomously rearrange to recover original
spacing in a manner that is very speciﬁc to the RD mechanism.
By comparing the simulation and the rearrangement of the real

When an animal with a stripe pattern grows, the spacing of
the stripes becomes wider. Therefore, growth of the body continuously changes the spacing of the pattern. If the pattern is madeby an RD mechanism, some rearrangement must occur to keep
the original spacing. According to RD simulations, the manner ofrearrangement differs depending on the types of stripes. If stripes

Fig. 7. Results of pattern regeneration experiments. (a) Pigment cells in the ﬁn were eliminated by temperature-sensitive mutation [9]. (b) Pigment cells (both melanophores
and xanthophores) were killed by laser [15]. (c) Dorsal movement of a melanophore stripe induced by melanophore elimination in the dorsal region [15]. (d) Division of
stripes induced by the ﬁsh growth [14].

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exhibit branch points, their division occurs by the horizontal movement of the branch points; if the stripes contain no branch points,
new stripes emerge between the old stripes. Kondo et al. observedin vivo exactly these changes predicted by simulation in tropicalangelﬁsh, Pomacanthus imperators, and Pomacanthus semicirculatus(Fig. 7d) [10,14].
6.2. Rearrangement of the pattern induced by artiﬁcial
perturbation
Yamaguchi et al. used laser ablation of the pigment cells to artiﬁcially perturb the pattern, and observed a strange but predictedchange to the striped pattern of zebraﬁsh [15]. They ablated themelanophores in the dorsal two stripes, and continuously eliminated new melanophores developing in the area. This operation
made a broad region lacking melanophores, and induced a dynamic
dorsal movement of the remaining ventral melanophore stripe to
ﬁll the space. The rearrangement started by bending and dorsally
moving of the more ventral band, and ended by forming a bellshaped pattern to ﬁll the vacant space (Fig. 7c). It is interesting
to note that the region of xanthophores reappeared at the ventral
side of moving black band, suggesting that the black stripe did not
simply get wider but moved by keeping the width. This dynamic
movement of the pattern also possesses the typical property of an
RD mechanism, and is reproducible by the simulation.
7. RD model as the working hypothesis for the molecular
genetic study
7.1. Other mathematical models those are able to form the similar
patterns
In the RD mechanism, signals of the reactions are spatiallytransferred by diffusion. By applying different values for diffusion
constants to the substances, it is possible to satisfy the necessary conditions of Turing pattern formation, “local activation andlong-range inhibition”. It is notable, however, that diffusion is notessential to achieve this necessary condition. It is also possible
to satisfy the necessary condition with neuronal networks [32],mechanical pressure [33], or cell migration [34], and mathematicalmodels using these factors have been developed. As there models do not require “diffusion” per se, these are not RD models in
the formal sense. However, as long as these models satisfy theidentical necessary condition, the mathematical equations become
similar, and their pattern-forming abilities are almost identical to
RD models. In order to ﬁnd out to which mathematical model thebiological systems resemble most, identiﬁcation of the signalingmechanisms is required. If future experiments reveal that patterns
result only from local interactions, different theoretical models willneed to be constructed. In this regard, Caicedo-Carvajal and Shinbrot presented an interesting mathematical model in which thepigment patterns are made by only the local effect among two
kinds of particles, which represent the pigment cells in the ﬁshskin [35]. It is supposed that the cell–cell binding afﬁnities between
the homophilic and heterophilic contact are different. They showedthat their model can reproduce most of zebraﬁsh pigment patterns from the random conditions. However, the pattern-forming
ability of their model has a serious limitation because it does
not contain the long-range interaction. The stripe pattern madeby the simulation is only two-cell width because the mechanism cannot specify the distance wider than three cells. It seems
to be impossible to reproduce the rearrangement of the stripes
observed in the tropical ﬁsh (Kondo S., personal communication). From the experimental observation so far made, involvementof the long-range effect should be essential for many animal
patterns.
7.2. Molecular genetic studies on the pigmentation of zebraﬁsh
As the zebraﬁsh is a model animal on which the advanced molecular technology is available, ﬁnding the speciﬁc dynamics of an RD
pattern opens the possibility of identifying the molecular level network of the hypothetical mechanism. Recent progress has made it
possible to investigate the molecular bases of pigment pattern formation in zebraﬁsh. The pattern of skin pigmentation in zebraﬁshis determined by the distribution of three types of pigment cellin the hypodermis: melanophores, the main component of dark
stripes; xanthophores, the main component of light stripes; and iridophores [9,17,36,37]. To understand how the stripe pattern forms,
several mutant genes responsible for abnormal skin pigmentation
have been isolated [38–45] and the molecular functions of thesegenes have been investigated [9,46,47,48,18,49,50].
These mutant genes can be classiﬁed into two categories: thosewith a defect in the development of pigment cells; and thosewith normal development of pigment cells, but a disrupted pattern in adult ﬁsh. Molecular genetic studies of the mutants in theﬁrst category have shown that nacre (mitf), sparse (kit), shady (ltk),
picasso (erbb3) and rose (ednrb1) are required for the developmentof melanophores, and that panther (fms) is required for the development of xanthophores [38,40,39,45,51,41]. Interestingly, wheneither type of pigment cell misdevelops, other pigment cells failto localize to their normal positions [40,9]. This fact suggests thatinteraction between pigment cells plays a critical role in the generation of skin pigmentation pattern, a notion that has been conﬁrmed
by cell transplantation studies [16,47]. Mutants of jaguar (obelix)[47] and leopard [8] belong to the second category. In these mutants,
development and distribution of the pigment cells is normal inembryos and young ﬁsh. In adult ﬁsh, although the developmentof pigment cells is normal, their spatial arrangement (pattern) ischanged, suggesting that these genes are speciﬁcally required forpattern formation. Positional cloning of these latter two mutants
identiﬁed the affected genes as encoding a potassium channel,
Kir7.1 [44], and a gap junction, connexin41.8 [43], respectively. How
these molecules are involved in the pigment pattern formation isstill unknown.
7.3. Consistency of the mathematical model with theexperimental results
Many different types of hypothetical network that are capable ofgenerating the RD pattern have been presented to date. In most ofthem, the putative network is composed of the interactions among
the diffusible chemicals. Meanwhile, molecular genetic studies
have suggested that two types of pigment cells, melanophores andxanthophores, play a major role in the patterning mechanism. Thisdifference is not a serious one because it is also possible to compose the equivalent network with the interacting two types of cells(melanophores and xanthophores). For example, if the pattern islarge relative to the cell size, the density of the cells can be usedas the concentration of the chemicals in the original RD equation.
Therefore, there is no theoretical difﬁculty in applying the RD modelto the experimental data of zebraﬁsh.
8. Future directions
As shown earlier in this article, the RD system is capable of generating most of the animal pigmentation patterns, and explaining
many of their characteristics. However, as there are many different
models with equivalent ability to form the patterns, it is impossible

S. Kondo, H. Shirota / Seminars in Cell & Developmental Biology 20 (2009) 82–89

to identify the real mechanism only by the mathematical analysis. On the other hand, the dynamic characteristics of the skinpattern suggest that the usual techniques of the molecular biology are not enough to clarify the hidden mechanism. Therefore,
combinatorial use of both strategies should be required to clarify
the hidden mechanism. The mathematical studies have shown thatthe necessary conditions for the skin pattern formation are “local
enhancement and long-range inhibition [24].” Therefore, according to the theory, two kinds of interactions, which differ in theeffective distance and function oppositely, must exist among thepigment cells. Finding such interaction in the ﬁsh skin, and theidentiﬁcation of the molecules involved in the interaction will solve
the long-asked question, “how does the animal pigment pattern
form?”
Acknowledgement
We thank very much for Dr. Parichy DM for his help and suggestion to complete this review.
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