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Mathematical modeling of the eyespots in butterfly wings

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Abstract

Butterfly wing color patterns are a representative model system for studying biological pattern formation, due to their two-dimensional simple structural and high inter- and intra-specific variabilities. Moreover, butterfly color patterns have demonstrated roles in mate choice, thermoregulation, and predator avoidance via disruptive coloration, attack deflection, aposematism, mimicry, and masquerade. Because of the importance of color patterns to many aspects of butterfly biology and their apparent tractability for study, color patterns have been the subjects of many attempts to model their development. Early attempts focused on generalized mechanisms of pattern formation such as reaction-diffusion, diffusion gradient, lateral inhibition, and threshold responses, without reference to any specific gene products. As candidate genes with expression patterns that resembled incipient color patterns were identified, genetic regulatory networks were proposed for color pattern formation based on gene functions inferred from other insects with wings, such as Drosophila. Particularly detailed networks incorporating the gene products, Distal-less (Dll), Engrailed (En), Hedgehog (Hh), Cubitus interruptus (Ci), Transforming growth factor-β (TGF-β), and Wingless (Wg), have been proposed for butterfly border ocelli (eyespots) which helps the investigation of the formation of these patterns. Thus, in this work, we develop a mathematical model including the gene products En, Hh, Ci, TGF-β, and Wg to mimic and investigate the eyespot formation in butterflies. Our simulations show that the level of En has peaks in the inner and outer rings and the level of Ci has peaks in the inner and middle rings. The interactions among these peaks activate cells to produce white, black, and yellow pigments in the inner, middle, and outer rings, respectively, which captures the eyespot pattern of wild type Bicyclus anynana butterflies. Additionally, our simulations suggest that lack of En generates a single black spot and lack of Hh or Ci generates a single white spot, and a deficiency of TGF-β or Wg will cause the loss of the outer yellow ring. These deficient patterns are similar to those observed in the eyespots of Vanessa atalanta, Vanessa altissima, and Chlosyne nycteis. Thus, our model also provides a hypothesis to explain the mechanism of generating the deficient patterns in these species.

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This chapter discusses that within the conceptual framework of positional information, a new and a simple way of looking at pattern formation may be obtained. In pressing the possibility of universality, the chapter deliberately takes an extreme stand, but at least it serves to counterbalance the special-substance inductive view of pattern formation. Also, in order to show its possible relevance to pattern formation and even cell movement, procrustean view of the data is taken. One of the virtues of the positional information mechanism of pattern formation is that, with the same system for positional information one can generate an enormous number of different patterns, by changing the cell's rules for interpretation. Since interpretation will be gene determined, there is little difficulty in seeing how this can be achieved. In fact, the concept of positional information makes excellent use of a central feature of development, that all the cells carry the same genetic information.
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Two different kinds of patterns have been studied in lepidopteran wings -- "color pattern" and "the spacing pattern of scale cells". These patterns exist on two different length scales. In the early stages of adult development, precursors of scale cells differentiate throughout each epithelial monolayer and migrate into rows that are roughly parallel to the body axis and regularly spaced about 50 µm. We develop a mathematical model for the formation of these parallel rows of scale cells in the developing adult wings of lepidoptera. We show that the inclusion of biologically realistic adhesive properties of cells, as specified by their positions, is sufficient to generate in a robust manner a series of scale rows along the length of the wing in the correct orientation. We next look briefly at the biology of color pattern formation, and we review some mathematical models for this phenomenon, which, in contrast to the spatial arragement of scale cells, involves interactions among cells that operate over longer distances.
Article
1) The North American butterflies, Limenitis arthemis and L. astyanax undergo phenotypic intergradation in a relatively narrow zone of approximately two degrees of latitude across the northeastern United States and southern Ontario. 2) Three adult color-pattern characters (white bands, dorsal blue-green iridescence, and dorsal red-orange spotting) were analyzed quantitatively in seven populations from Quebec to Virginia. Phenotypic stability occurs in the south for all three characters, but in the north for only the white bands. All three characters exhibit extreme variance in the intergradation zone, and the iridescence and spotting continue to be highly variable in the north. The mean values for all three characters shift from broad white bands, extensive red-orange spotting, and extreme reduction of iridescence in the north, to a southern population which is uniformly dark, lacking both the white bands and red-orange spotting, but being highly iridescent. 3) Hardy-Weinberg analyses of population samples from areas within the intergradation zone show that breeding is at random. Laboratory crosses show no evidence of heterogametic inviability, and studies of the male genitalia indicate complete overlapping in all characters, there being only a clinal increase in size from northern to southern populations. The two butterflies therefore are conspecific, and their relationship to each other is considered one of primary intergradation. 4) The color-pattern of L. astyanax and the dark female form of another North American butterfly, Papilio glaucus, closely resemble Battus philenor, and it is widely held that both are Batesian mimics of this highly unpalatable troidine butterfly. Both mimics are extensively sympatric with the model. In the zone where L. astyanax intergrades to L. arthemis, the black form of P. glaucus gives way to a monomorphic yellow population. In this same area, B. philenor becomes increasingly rare, and is absent to the north where, however, L. arthemis and the non-mimetic form of P. glaucus remain common. 5) It is proposed that selection favoring mimicry in the south gives way to selection favoring disruptive coloration to the north as the model becomes increasingly rare to absent. The intergradation zone of the two Limenitis butterflies thus appears to represent an area of phenotypic reversal from one adaptive peak to another. 6) It is likely that an arthemis-like ancestor spread eastward and southward, and evolved the astyanax coloration when once inside the range of B. philenor, the latter being of neotropical origin. 7) Selection is probably favoring the evolution of clear-cut dominance for the white band character, but not for the iridescence and red-orange spotting, since variation per se in the latter two probably enhances the disruptive color-pattern of the northern form. It is therefore unlikely that the three characters will ever come under the control of a super-gene.
Article
Visual signaling in animals can serve many uses, including predator deterrence and mate attraction. In many cases signals used to advertise unprofitability to predators are also used for intraspecific communication. Although aposematism and mate choice are significant forces driving the evolution of many animal phenotypes, the interplay between relevant visual signals remains little explored. Here we address this question in the aposematic passion-vine butterfly Heliconius erato by using color- and pattern-manipulated models to test the contributions of different visual features to both mate choice and warning coloration. We found that the relative effectiveness of a model at escaping predation was correlated with its effectiveness at inducing mating behavior, and in both cases wing color was more predictive of presumptive fitness benefits than wing pattern. Overall, however, a combination of the natural (local) color and pattern was most successful for both predator deterrence and mate attraction. By exploring the relative contributions of color versus pattern composition in predation and mate preference studies, we have shown how both natural and sexual selection may work in parallel to drive the evolution of specific animal color patterns.This article is protected by copyright. All rights reserved.
Article
The angled sunbeam butterfly, Curetis acuta (Lycaenidae), is a distinctly sexually dimorphic lycaenid butterfly from Asia. The dorsal wings of female and male butterflies have a similar pattern, with a large white area in the female and an orange area in the male, framed within brown–black margins. The ventral wings of both sexes are silvery white, which is caused by stacks of overlapping, non‐pigmented, and specular‐reflecting scales. With oblique illumination, the reflected light of the ventral wings is strongly polarized. We show that the silvery reflection facilitates camouflage in a shaded, foliaceous environment. The ecological function of the silvery reflection is presumably two‐fold: for intraspecific signalling in flight, and for reducing predation risk at rest and during hibernation. © 2013 The Linnean Society of London, Biological Journal of the Linnean Society, 2013, 109, 279–289.
Article
SYNOPSIS. This paper describes a case study of adaptation, constraint, and evolutionary innovation in pierid butterflies. I develop a framework for discussing these issues that focuses on the questions: What is the form of the adaptive landscape relating fitness to phenotypic characters? How do such landscapes differ for evolutionarily related groups? I examine the evolution of wing pigment patterns and thermoregulatory behavior for butterflies in two subfamilies in the family Pieridae, with three principal results. First, I show that thermoregulation can be an important component of fitness in pierids, and that wing color and thermoregulatory behavior are important phenotypic characters determining thermoregulatory performance and the adaptive landscape. Second, I show how limits on possible variation in wing color and behavior constrain evolution within one subfamily of pierids, and how these constraints are set by the physical and biochemical mechanisms of adaptation. Third, I show how evolutionary innovation may have resulted from the addition of a new, behavioral dimension to the landscape, and how this addition has altered the functional interrelations among various elements of the wing color pattern. I suggest that comparative analyses of the form and determinants of the adaptive landscape may be useful in identifying evolutionary innovations, and complement theoretical analyses of evolutionary dynamics on such fitness surfaces.
Article
A formal model is presented that demonstrates how the color patterns of the wings of butterflies and moths can be analysed in terms of current concepts of pattern formation. A review of pertinent literature on this little-known developmental system is provided. Crucial to the understanding of color pattern formation in lepidopteran wings is the realization that the wing pattern is a mosaic. The pattern in each wing cell (i.e., the area bordered by wing veins) is determined independently from the pattern in other cells. Within a cell, pigments are deposited in a definite relation to a central focus. A focus always lies on the cell midline and is often visible as a small pigmented dot. In the simplest condition the color pattern is laid down around a focus as a system of perfect concentric circles (eyespots). More often these circles are considerably distorted and have an axis of bilateral symmetry that parallels the wing veins. The nature and extent of these distortions from circularity account, to a large degree, for the species-specific character of the wing pattern. The determination of such patterns is most readily explained if it is assumed that a focus represents the reference point with respect to which positional information (for pigment deposition) is specified. Circular patterns are thus obtained if all points equidistant from a focus undergo identical differentiation. Deviations from circularity of the pattern around a focus indicate that either the specification or interpretation of positional information is not the same in all areas around a focus. This phenomenon is most conveniently described as an “interpretation landscape”: a gradient system whose value at any point in a morphogenetic field is a measure of how positional information is to be interpreted at that location. Examples are provided of how species-specific wing patterns can be generated by modest alterations in the shape of the interpretation landscape. As a rule the field of a focus is limited by the margins of the wing cell in which it is centered except in cases where no focus is present in an adjacent cell. Under those circumstances one focus can determine the pattern in several adjoining wing cells.
Article
The “deflection hypothesis” asserts that conspicuous marginal patches on insect wings function to deflect predator attacks toward such patches and away from more vital body parts. As a result of selection from predator attacks, these marks are predicted to increase the probability of escape by tearing relatively easily. To test if a conspicuous marginal patch is weak relative to a homologous wing area without such a patch, hindwing tear weight was compared among three Pierella species (Satyrinae) differing in the presence of a conspicuous patch in the hindwing tornus. The species with a conspicuous white hindwing patch (P. astyoche) had significantly lower tear weights than the two species lacking the patch (P. lamia and P. lena). Forewing length did not explain variation in wing‐tear weight, but wing‐tear weight was positively related to insect age in a manner consistent with the deflection hypothesis. Older individuals of P. lamia and P. lena had higher tear weight, whereas this relationship was absent in P. astyoche. These results represent the first direct evidence that deflection marks on butterfly wings are relatively weak and should have an increased tendency to tear when handled by a predator. RESUMEN La “hipótesis de desvio” propone que las conspicuas manchas marginales en alas de insectos funcionan para desviar los ataques de depredadores hacia tales manchas y lejos de partes del cuerpo más vitales. Como resultado de selección por ataques de depredadores, se predice que estas manchas deberian rompese fácilmente para aumentar la probabilidad de escape. Para probar si un mancha marginal conspicua es débil comparada con un área homóloga de alas sin tales manchas, se comparó el peso de rotura del ala posterior entre tres especies de Pierella (Satyrinae) que difieren en la presencia de un mancha visible en el tornus del ala posterior. Las alas posteriores de la especie con una mancha blanca visible ( P astyoche ) se rasgaron con pesos significativamente más bajos comparadas con las alas posteriores de las dos especie sin mancha ( P. lamia y P. lena ). La longitud del ala anterior no explicó la variación en el peso de rotura, pero el peso de rotura estuvo relacionado positivamente con la edad del insecto de una manera consistente con la hipotesis de desvio. Los individuos más viejos de P. lamia y P. lena tuvieron los pesos de rotura más altos, mientras que esta relación no se encontró en P. astyoche . Estos resultados representan la primera evidencia directa de que las manchas de desvio en alas de mariposas son relativamente débiles y deben tener una tendencia aumentada a romperse cuando son manipuladas por un depredador.
Article
Ischemic cardiovascular disease represents one of the largest epidemics currently facing the aging population. Current literature has illustrated the efficacy of autologous, stem cell therapies as novel strategies for treating these disorders. The CD34+ hematopoetic stem cell has shown significant promise in addressing myocardial ischemia by promoting angiogenesis that helps preserve the functionality of ischemic myocardium. Unfortunately, both viability and angiogenic quality of autologous CD34+ cells decline with advanced age and diminished cardiovascular health. To offset age- and health-related angiogenic declines in CD34+ cells, we explored whether the therapeutic efficacy of human CD34+ cells could be enhanced by augmenting their secretion of the known angiogenic factor, sonic hedgehog (Shh). When injected into the border zone of mice after acute myocardial infarction, Shh-modified CD34+ cells (CD34(Shh)) protected against ventricular dilation and cardiac functional declines associated with acute myocardial infarction. Treatment with CD34(Shh) also reduced infarct size and increased border zone capillary density compared with unmodified CD34 cells or cells transfected with the empty vector. CD34(Shh) primarily store and secrete Shh protein in exosomes and this storage process appears to be cell-type specific. In vitro analysis of exosomes derived from CD34(Shh) revealed that (1) exosomes transfer Shh protein to other cell types, and (2) exosomal transfer of functional Shh elicits induction of the canonical Shh signaling pathway in recipient cells. Exosome-mediated delivery of Shh to ischemic myocardium represents a major mechanism explaining the observed preservation of cardiac function in mice treated with CD34(Shh) cells.
Article
The determination of color patterns of butterfly wing eyespots has been explained by the morphogen concentration gradient model. The induction model has been proposed recently as a more realistic alternative, in which the eyespot-specifying signal does not depend entirely on focal activity. However, this model requires further elaboration and supporting evidence to be validated. Here, I examined various color patterns of nymphalid butterflies to propose the mechanics of the induction model. Based on cases in which an eyespot light ring is identical to the background in color, I propose that eyespots are fundamentally composed of dark rings and non-dark "background" spaces between them. In the induction model, the dark-ring-inducing signal that is released from a prospective eyespot focus (the primary organizing center) as a slow-moving wave effects both selfenhancement and peripheral induction of the dark-ring-inhibitory signal at the secondary organizing centers, resulting in an eyespot that has alternate dark and light rings. Moreover, there are cases in which an unseen "imaginary light ring" surrounds an eyespot proper and in which PFEs are integrated into the eyespot. It appears that PFEs constitute a periodic continuum of eyespot dark rings; thus, a background space between the eyespot and a PFE is mechanistically equivalent to eyespot light rings. The eyespot dark-ring-inducing signals and PFE-inducing signal are likely to be identical in quality, but released at different times from the same organizing center. Computer simulations based on the reaction-diffusion system support the feasibility of the induction model.
Article
Butterfly wing color patterns consist of many color-pattern elements such as eyespots. It is believed that eyespot patterns are determined by a concentration gradient of a single morphogen species released by diffusion from the prospective eyespot focus in conjunction with multiple thresholds in signal-receiving cells. As alternatives to this single-morphogen model, more flexible multiple-morphogen model and induction model can be proposed. However, the relevance of these conceptual models to actual eyespots has not been examined systematically. Here, representative eyespots from nymphalid butterflies were analyzed morphologically to determine if they are consistent with these models. Measurement of ring widths of serial eyespots from a single wing surface showed that the proportion of each ring in an eyespot is quite different among homologous rings of serial eyespots of different sizes. In asymmetric eyespots, each ring is distorted to varying degrees. In extreme cases, only a portion of rings is expressed remotely from the focus. Similarly, there are many eyespots where only certain rings are deleted, added, or expanded. In an unusual case, the central area of an eyespot is composed of multiple "miniature eyespots," but the overall macroscopic eyespot structure is maintained. These results indicate that each eyespot ring has independence and flexibility to a certain degree, which is less consistent with the single-morphogen model. Considering a "periodic eyespot", which has repeats of a set of rings, damage-induced eyespots in mutants, and a scale-size distribution pattern in an eyespot, the induction model is the least incompatible with the actual eyespot diversity.
Article
We have studied the development of the eyespot colour pattern on the adult dorsal forewing of the nymphalid butterflies, Bicyclus safitza and B. anynana, by cauterising the presumptive eyespot centres (the foci) on the pupal wing. The effects on pattern depended on age at cautery. Early focal cautery (at 1-12 hours after pupation) usually reduced or eliminated the eyespot, while cautery at a non-focal site usually had no effect. These results resemble those of a previous study on another species but, in addition, we find that a later cautery (at 12-24 hours) had the converse effect of generating pattern, so that focal cautery enlarged the anterior eyespot (but usually not the large posterior eyespot) and nonfocal cautery induced a new ectopic eyespot. The effects of cautery on patterning are more extensive by an order of magnitude than the cell death which is caused, so implicating a long-range mechanism, such as a morphogen gradient, in eyespot development. The focus clearly acts to establish the normal eyespot pattern, but a simple source/diffusion model is not supported by the response to late cautery. We suggest two alternative forms of gradient model in which late damage can mimic and augment the action of a focus. In the Source/Threshold model, the focus is a morphogen source, and cautery can remove the focus but also transiently lowers the response threshold in surrounding cells. In the Sink model, the focus generates the gradient by removing morphogen, and cautery can eliminate the focus but it also causes a transient destruction or leakage of morphogen. These models can explain most features of the results of cautery.
Article
The problem of pattern is considered in terms of how genetic information can be translated in a reliable manner to give specific and different spatial patterns of cellular differentiation. Pattern formation thus differs from molecular differentiation which is mainly concerned with the control of synthesis of specific macromolecules within cells rather than the spatial arrangement of the cells. It is suggested that there may be a universal mechanism whereby the translation of genetic information into spatial patterns of differentiation is achieved. The basis of this is a mechanism whereby the cells in a developing system may have their position specified with respect to one or more points in the system. This specification of position is positional information. Cells which have their positional information specified with respect to the same set of points constitute a field. Positional information largely determines with respect to the cells' genome and developmental history the nature of its molecular differentiation. The specification of positional information in general precedes and is independent of molecular differentiation. The concept of positional information implies a co-ordinate system and polarity is defined as the direction in which positional information is specified or measured. Rules for the specification of positional information and polarity are discussed. Pattern regulation, which is the ability of the system to form the pattern even when parts are removed, or added, and to show size invariance as in the French Flag problem, is largely dependent on the ability of the cells to change their positional information and interpret this change. These concepts are applied in some detail to early sea urchin development, hydroid regeneration, pattern formation in the insect epidermis, and the development of the chick limb. It is concluded that these concepts provide a unifying framework within which a wide variety of patterns formed from fields may be discussed, and give new meaning to classical concepts such as induction, dominance and field. The concepts direct attention towards finding mechanisms whereby position and polarity are specified, and the nature of reference points and boundaries. More specifically, it is suggested that the mechanism is required to specify the position of about 50 cells in a line, relatively reliably, in about 10 hours. The size of embryonic fields is, surprisingly, usually less than 50 cells in any direction.
Article
The patterns on wings of Lepidoptera can be generated with a few pattern elements, but no mechanism has been suggested for producing them. I consider two of the basic patterns, namely, central symmetry and dependent patterns. A biochemically plausible model mechanism is proposed for generating major aspects of these patterns, based on a diffusing morphogen that activates a gene or colour-specific enzyme in a threshold manner to generate a stable heterogeneous spatial pattern. The model is applied to the determination stream hypothesis of Kühn & von Engelhardt (Wilhelm Roux Arch. Entw Mech. Org. 130, 660 (1933)), and results from the model compared with their microcautery experiments on the pupal wing of Ephestia kühniella. In the case of dependent patterns, results are compared with patterns on specific Papilionidae. For the same mechanism and a fixed set of parameters I demonstrate the important roles of geometry and scale on the spatial patterns obtained. The results and evidence presented here suggest the existence of diffusion fields of the order of several millimetres, which are very much larger than most embryonic fields. The existence of zones of polarizing activity is also indicated. Colour patterns on animals are considered to be genetically determined, but the mechanism is not known. I have previously suggested that a single mechanism that can exhibit an infinite variety of patterns is a candidate for that mechanism, and proposed that a reaction-diffusion system that can be diffusively driven unstable could be responsible for the laying down of the spacing patterns that generates the pre-pattern for animal coat markings. For illustrative purposes I consider, a practical reaction mechanism, which exhibits substrate inhibition, and show that the geometry and scale of the domain (part of the epidermis) play a crucial role in the structural patterns that result. Patterns are obtained for a selection of geometries, and general features are related to the coat colour distribution in the spotted Felidae, giraffe, zebra and other animals. The patterns depend on the initial conditions, but for a given geometry and scale are qualitatively similar, a positive feature of the model and a necessary model attribute in view of the pattern individuality on animals of the same species.
Article
The formation of the wing pigmentation patterns of three species of butterflies has been modelled using a mechanism based on a tripod of assumptions. First, that there may be morphogen sources in the foci of eyespots and morphogen sinks at some parts of the wing margin, all other cells being passive. Second, that the morphogen has a finite half life and diffuses simply and freely away from the sources throughout a wing of hexagonally packed cells. Third, that the overt pattern derives from cells interpreting the local morphogen concentration with respect to thresholds which determine scale colours. The final pattern thus follows lines of constant morphogen concentration and may, depending on the distribution of sources, comprise rings, curves or bands. With such a model, we have been able to compute stable patterns having the essential topology of the compound spots of Tenaris domitilla, the large rings of Diaethria marchalii and the pattern of eyespots, rings and asymmetric bands of Ragadia minoa. Quantitative analysis of the pattern-forming process shows that, with a biologically realistic diffusion constant (approximately 5.10(-7) cm2 sec-1) and a morphogen half life less than 6h, the patterns form within approximately 12h over a wing of approximately 1000 cells in length. The limitations of the model are that the exact morphology of the eyespots and bands do not match precisely those of the original wings, that there are edge distortions and that optimal patterns may be critically dependent on the exact positions of sources and sinks. An explanation for part of the discrepancy is that we have assumed an adult wing shape and foci coordinates in modelling a process that took place earlier in development. Nevertheless, the limitations of the model argue against a mechanism based on a single morphogen operating in vivo. However, as the model can generate many features of butterfly wing patterns, it may be considered as a degenerate case of that mechanism.
Article
Color patterns on lepidopteran wings are believed to be organized around a series of hypothetical foci. Foci presumably serve as sources of positional information, directing synthesis of appropriate pigments in their vicinity. Species-specific color patterns arise by variations in the number of foci on the wing and variations in the rules by which positional information is interpreted. The present paper provides experimental evidence for the existence of a focus that determines the large eyespot on the forewing of the Buckeye butterfly,Precis coenia. Cautery of a small group of about 300 cells at the center of the presumptive eyespot, early during wing development, can completely inhibit development of the eyespot. The same group of cells can be transplanted to another region of the wing and induce ring-shaped pigment patterns in the host tissue around the graft. These findings show that a focus is a physiological entity. Color pattern induction is an intrinsic property of a small group of cells. Presumably these cells produce a factor that affects the production of enzymes involved in pigment synthesis.
Article
A widely used mechanism for pattern formation is based on positional information: cells acquire positional identities as in a coordinate system and then interpret this information according to their genetic constitution and developmental history. In Drosophila maternal factors establish the axes and set up a maternal system of positional information on which further patterning is built. There is a cascade of gene activity which leads both to the development of periodic structures, the segments, and to their acquiring a unique identity. This involves the binding of transcription factors to regulatory regions of genes to produce sharp thresholds. Many of the genes involved in these processes, particularly the Hox complex, are also involved in specifying the body axis and limbs of vertebrates. There are striking similarities in the mechanisms for specifying and recording positional identity in Drosophila and vertebrates.