数理解析研究所講究録
1191 巻 2001 年 25-28

25Patterns on the Fish Skins Induced by Anisotropy in Diffusion

九大大学院理学研究院

望月敦史 (MOCHIZUKI Atsushi)

Department of Biology, Faculty of Sciences, Kyushu University
amochscb@mbox.nc. kyushu-u. ac.jP

Most of the stripes observed on fish skins are either parallel or perpendicular to theirantero-posterior axis (Kondo&Asai, 1995). Some species have parallel stripes, some haveperpendicular ones, and small number of species has random stripes, where the direction ofthe stripe is not fixed.

For example, very close two species (Genicanthus melanosphilos andGenicanthus watanabei) show very different pattems; G. melanosphilos shows perpendicularand G. watanabei shows parallel stripes to the antero-posterior axis. On the other hand, thedirection of stripes obtained by simple reaction diffusion systems is basically free. The
stripes patterns generated by the reaction-diffusion mechanism in two-dimensional space has
stable periodicity (Turing, 1952), however, the direction of the stripe is not stable; that isvariable depending on the initial distribution. What makes the strong directionality in theactual fish skin?

Figure 1. An example ofparallel stripe to the antero-posterior axis (G. watanabei)
In fact, there is strong possibility that a fish skin has the property to makedirectionality in its structure. In the structure of fish skin, we can see that each scale comesout to the same direction along the antero-posterior axis. The epidermis is wrapping theexist along dorso-ventral
scales and making zigzag form. As the zigzag structure
axis, the structure is different between the antero-posterior axis and the dorso-ventral axis.
This structural difference may make difference in speed of informational transfer between the$\mathrm{d}\mathrm{o}\mathrm{e}\mathrm{S}\mathrm{n}^{\mathrm{t}}\mathrm{t}$two directions.
From the idea, we developed a modified reaction-diffusion model where thesubstances likely to diffuse faster to the special direction rather than simple homogeneous

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diffusion. By using the anisotropic diffusion-reaction model, we can explain the transitionof the pattem in actual fish.
Model

The anisotropic property of the fish epidermis is modeled by incorporating theanisotropy into the diffusion term. The usual diffusion term is derived by assuming that theflux of substance is linear to the gradient of the substance. In this study, it is assumed thatthe diffusion coefficient is not a constant but a function of an angle between the gradient and aspecial direction. The special direction means the direction where the substances diffuse
faster. In other word, we assumed that the flux would be enhanced if the direction ofgradient of the substance is same as the special direction, and it would be reduced if thedirection of gradient is perpendicular to the special direction.
The assumption is expressed mathematically in the following:where($\frac{\partial u}{\partial t}=\nabla\cdot(D_{\mathcal{U}}(\theta_{u})\nabla u)+_{V}(_{u},\mathcal{V})$(1a)$\frac{\partial v}{\partial t}=d\nabla\cdot(D_{v}(\theta)\mathcal{V}\nabla v)+_{l}g(u,\mathcal{V})$(1b)$\theta_{u}$and$\theta_{v}$indicate$\theta_{u}=\tan^{-1}((\partial \mathcal{U}/\partial y)/(\partial u/\partial x))$andanglestheWe call the parameter

gradient

the$\theta_{v}=\tan^{-1}((\partial v/\partial y)/(\partial v/\partial x))$function of anisotropy in diffusion term.
$D_{\sigma}( \theta_{\sigma})=\{1-\delta_{\sigma}\cos 2(\theta\sigma-\varphi)\}\frac{1}{2}$of, where,$\delta_{\sigma}\dagger \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{i}_{\mathrm{S}}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}^{\uparrow 1}$

The form of the$\varphi$), and$D_{\sigma}(\theta_{\sigma})$

of

the

$D_{\sigma}(\theta_{\sigma})$variables

indicates theused in the analysis is:indicates the angle of most diffusive direction.

which indicates the degree of distortion of thediffusion range from a circle. If the anisotropy is , the diffusion range is just a circle(normal diffusion), and if the value is large the distortion of diffusion range is large. We can). On the otherandchoose different values of anisotropy for the two substances (hand, the diffusive direction
is assumed to be common between the two substances$0$$\delta_{\mathcal{U}}$$\delta_{v}$$\varphi$(because it comes from the structure of epidermis).

This method to incorporate theanisotropy into the diffusion is first modeled by Kobayashi (Kobayashi, 1993), where hestudied pattem formation the dendritic crystal growth like snow crystal.
indicate reaction terms. We use the form studied byThe forms $f$ and$g$

Schnakenberg (Schnakenberg, 1972);$f(u,v)=A-u+u^{2}v$(1c)

27(1d)$g(_{\mathcal{U}},v)=B-uv2$

where A and$\mathrm{B}$are positive constants. We also tried some different formulae later.

ResultWe calculated the modeland derive pattems by using computer simulation.
The used Parameters values are ones that generate stripes in simple reaction-diffusion model.
The boundary condition is periodic. The initial distribution is equilibrium value with small
random fluctuation. To remove the effect of the boundary, the periodic boundary condition
was chosen.
$(\mathrm{l}\mathrm{a})-(\mathrm{l}\mathrm{d})$Figure 2 The obtained pattems by computer simulation
The effect ofpositive same anisotropy
We incorporate the same positive value of anisotropy in both substances (i.e.
. Fig.
shows the simulation results, when the most diffusive direction is parallelto the -axis. We cannot find any directionality in this figure. Thiswas(1)$2\mathrm{a}$$\delta_{u}--_{\delta_{\mathcal{V}}})$$\mathrm{x}$$(\varphi--0)$$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}_{b}\mathrm{t}$observed even if the value of the anisotropy is very large.
(2)
The effect of anisotropic diffusion
Fig.
shows the simulation results when the anisotropy of is positive and that ofis . In the figure, the most diffusive direction is parallel to the -axis. The$\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{S}\mathrm{s}- \mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{s}\mathrm{i}_{\mathrm{V}\mathrm{e}- \mathrm{S}\mathrm{u}}\mathrm{b}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{C}\mathrm{e}\mathcal{U}$$2\mathrm{b}$

$u$

$0$

$v$$x$$(\varphi--\mathrm{o})$direction of the stripe becomes always parallel to the most diffusive direction of , even if wechanged the diffusive direction . The wave-length of the stripe is not influenced by.
$\mathrm{u}$$\delta_{u}$$\varphi$

The effect of anisotropic diffitsion ofshow the results when the anisotropy of is positive and that of is . TheFig.
most diffusive direction is parallel to the -axis. In this case, the direction of the lines(3)$\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{e}- \mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}- \mathrm{S}\mathrm{u}\mathrm{b}_{\mathrm{S}\mathrm{t}}\mathrm{a}\mathrm{n}\mathrm{C}\mathrm{e}\mathcal{V}$$2\mathrm{c}$

$v$

$x$

$u$

$0$

$(\varphi--\mathrm{o})$is likely to be perpendicular to the diffusive direction of . We can remember that
corresponds to what is called inhibitor. Then we can intuitively understand the result that thedirection of the stripes crosses the most diffusive direction of .
$v$

$v$

$v$

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10.80.6$f\mathrm{G}\triangleright \mathrm{O}.4$

O.2$0$

1$\mathrm{O}.5$$\mathrm{O}$$\delta_{Il}$Figure 3 Summary of directionality of obtained stripes
Fig. 3 summarizes the directionality of stripe obtained pattems. The diffusive
direction is fixed to be parallel to the -axis. Each point indicates the direction of theobserved stripe, horizontal, vertical, or not determined. To identify the direction is done byusing a computer algorithm.
The result depends only on the relative magnitude of anisotropy of and . When, the direction of the stripe is horizontal; the direction is parallel to theis larger than
, the direction of stripe isis larger thanmost diffusive direction.
When
$x$

$u$

$\delta_{u}$$v$$\delta_{v}$$\delta_{v}$$\delta_{u}$perpendicular to the most diffusive direction. Only when the anisotropies of both substances
are almost the same, the direction is not determined.
We tested several different conditions. We changed the value of parameter inreaction term, value of diffusion coefficient , the function of anisotropy in diffusion term andalso the form of the reaction term. The result does not depend on these changes. The same
figure was obtained from all the trials we tested. We can say that this result is very general.
$\mathrm{d}$References
Kobayashi R. (1993) Modeling and numerical simulations of dendritic crystal growth.
$Phy_{SiCa}D63$: 410-423

Kondo S. and Asai R. (1995) A reaction-diffusion wave on the marine angelfish Pomacanthus.
Nature 376: 765-768
Schnakenberg J. (1979) Simple Chemical Reaction systems with Limit cycle behavior. .
$J$theor. Biol. 81: 389
Turing A. M. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. London $B237$ :37-72