Although butterfly wing patterns are highly complex, it is believed that they are produced by simple rules that determine the fate of immature scale cells fixed in a two-dimensional plane. Among the colour-pattern elements that constitute the overall wing pattern, eyespots are conspicuous symmetric elements. Partly for this reason, characterisation of eyespots via physical damage and transplantation methods has been intensively performed, with the focus on the forewing eyespots of the nymphalid butterflies Junonia coenia
] and Bicyclus anynana
]. Two other nymphalid butterflies, Junonia orithya
and Ypthima argus
, were employed in a similar study [6
The experimental results obtained in these studies have been explained by the concentration gradient model for positional information, the theoretical basis of which was proposed by Wolpert [7
]. Such explanations necessarily exclude alternative models, such as the cascade model, which addresses serial inductive signals, and the wave model, in which signals have an autonomous wave-like character [8
]. The main reason for this exclusion is the relatively long period of focus dependence in eyespot formation; this phenomenon was revealed when it was observed that focal damage in the early pupal stage resulted in smaller eyespots [1
]. Following this line of discussion, the putative morphogenic molecules Wingless and TGF-β have been shown to be expressed in at least some eyespots [10
]. These molecules are believed to be secreted from prospective eyespot foci and to determine eyespot rings [10
], although there is currently no functional evidence that the expression of these molecules affects butterfly colour-pattern determination.
Although rarely discussed in the literature, there are several experimental findings and natural colour-pattern variations that have not been explained by the conventional gradient model [14
]. Therefore, based on colour-pattern analyses of various nymphalid butterflies, the induction model was proposed as a more realistic alternative [14
]. In this model, autonomous wave-like signals for dark rings are released from the focus. They are self-enhancing at a short range and induce inhibitory signals at a long range during their expansion and after their settlement, as originally proposed by Gierer and Meinhardt [16
]. These dark-ring and inhibitory signals may be mutually stabilised and then translated into colour-pattern expression. These processes were simulated computationally using reaction-diffusion equations [15
]. The induction model was also shown to be consistent with the results of experimental disruption of eyespots [19
Nevertheless, there is one important point that has not yet been sufficiently explained by the induction model: how a released wave-like signal "finds" a proper position in which to settle. Because the released wave can progress indefinitely unless it is equipped with a settlement mechanism, this point is directly related to how to organise colour-pattern elements on a wing surface and how to rearrange the nymphalid groundplan in a species-specific fashion.
It is certainly highly likely that the butterfly eyespot system is constructed based on the reaction-diffusion system, and it appears that this settlement problem could be solved by further exploration of the reaction-diffusion model, which could suggest possible molecular interactions. In hydra, the Wnt signal responsible for head development is explained by a reaction-diffusion model [20
]. The shell colour patterns of molluscs are also simulated by this type of model [24
]. In both cases, moving signals can be slowed down and become stable under specific conditions. However, these conditions are unlikely to be directly applicable to the butterfly system, considering that in comparison to the hydra system, the butterfly system is much larger, and that in contrast to molluscan systems, it is stable and predictable in a given species, with small individual variations.
It is important to recognise that in any modelling study, the object to be modelled must be understood. That is, structural features of actual butterfly eyespots must be studied first and then the related reaction-diffusion simulations can be explored. Therefore, before approaching to a final solution for the settlement problem, in the present study, I turned my attention to actual colour patterns and did not consider the hypothetical physicochemical nature of signalling molecules included in the gradient and reaction-diffusion models. I do not intend to reject the previously proposed reaction-diffusion mechanism, but I found that more descriptive mechanics are helpful to understand butterfly wings, at least at this point, because of the lack of studies that describe butterfly wing colour patterns in detail. The present study is an attempt to faithfully describe and simulate the observed behaviour of natural and experimentally induced colour patterns using simple equations, without stringently focusing on their hypothetical physicochemical or molecular bases.
Throughout this paper, I focus on the eyespots and parafocal elements (PFEs) of the peacock pansy butterfly, Junonia almana
(Figure ). The eyespots of this nymphalid butterfly have several notable features that are not found in the eyespots of other butterflies. Specifically, unlike some satyrine butterflies, including Bicyclus
, that exhibit "typical" symmetric eyespots, J. almana
shows remarkable intra-individual variation in eyespot size and morphology [26
]. This variation makes morphological comparison between eyespots both possible and fruitful. Based on a colour-pattern analysis of J. almana
, I present a simple descriptive mathematical model to explain the possible behaviour of morphogenic signals in light of the induction model that is consistent with both observational and experimental results.
Wing colour patterns of Junonia almana. Each eyespot (ES) is referred to as indicated for convenience. Terms for eyespot substructures and peripheral elements are indicated on the right side.