Toshio Sekimura

A model for colour pattern formation in the butterfly wing of Papilio dardanus

The butterfly Papilio dardanus is well known for the spectacular phenotypic polymorphism in the female of the species. We show that numerical simulations of a reaction diffusion model on a geometrically accurate wing domain produce spatial patterns that are consistent with many of those observed on the butterfly. Our results suggest that the wing coloration is due to a simple underlying stripe–like pattern of some pigment–inducing morphogen. We focus on the effect of key factors such as parameter values for mode selection, threshold values which determine colour, wing shape and boundary conditions. The generality of our approach should allow us to investigate other butterfly species. The relationship between these key factors and gene activities is discussed in the context of recent biological advances.

The morphogenesis of liposomes viewed from the aspect of bending energy

It is known that liposomes transform their shapes sequentially through one of several transformation pathways. Using the mechanical principle of the least bending energy of membranes, we investigate the stability and shape transformation of liposomes with geometrical symmetry. We have done this by computer simulations and theoretical analyses, in which three-dimensional liposome shapes have been generated by the modified Cassini equation. We show first that there are energetically stable liposome shapes having intrinsic geometrical symmetry. We find that by reducing the volume, the stable shape can change from a circular biconcave shape as in red blood cells, to elliptical, triangular, square, and other polygonal shapes. It is also found that the preceding two results hold true irrespective of the overall surface area of liposome.

The effect of population density on shoot morphology of herbs in relation to light capture by leaves

Plants change their shapes, depending on their environment, for example, plant height increases with increasing population density. We examined the density-dependent plasticity in shoot morphology of herbs by analysing a mathematical model which identifies a number of key factors that influence shoot morphology, namely (i) solar radiation captured by leaves; (ii) shading from neighbouring plants; and (iii) utilisation efficiency of resource by leaves, stems and veins. An optimisation theory was used to obtain optimal shoot morphology in relation to maximal light capture by leaves, under trade-offs of resource partition among organs. We first evaluated the solar radiation flux per unit leaf area per day for different shoot forms. Our model predicts that the optimal internodal length of the stem that brings about the maximal light capture by leaves increases with plant population density, and this is consistent with experimental data. Moreover, our simple model can also be extended to explain the morphological plasticity in other herbs (i.e. stemless plants) that are different from our model plants with a stem. These findings illustrate how optimisation theory can be used for the analysis of plasticity in shoot morphology of plants in response to environmental changes, as well as the analysis of diversity in morphology.

The moving grid finite element method applied to biological problems

This paper presents a novel numerical technique, the moving grid finite element method, to solve generalised Turing [20] reaction-diffusion type models on continuously deforming growing domains. Applications to the development of bivalve ligaments and pigmentation colour patterns in the wing of the butterfly Papilio dardanus will be considered, by way of examples.

A Model for Selection of Eyespots on Butterfly Wings.

Unsolved Problem The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not. Key Idea and Model We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature. Result We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.

Diversity and Evolution of Butterfly Wing Patterns

The border ocelli and adjacent parafocal elements are among the most diverse and finely detailed features of butterfly wing patterns. The border ocelli can be circular, elliptical, and heart-shaped or can develop as dots, arcs, or short lines. Parafocal elements are typically shaped like smooth arcs but are also often “V,” “W,” and “M” shaped. The fusion of a border ocellus with its adjacent parafocal element is a common response to temperature shock and treatment with chemicals such as heparin and tungstate ions. Here I develop a new mathematical model for the formation of border ocelli and parafocal elements. The models are a reactiondiffusion model based on the well-established gradient-threshold mechanisms in embryonic development. The model uses a simple biochemical reaction sequence that is initiated at the wing veins and from there spreads across the field in the manner of a grass-fire. Unlike Turing-style models, this model is insensitive to the size of the field. Like real developmental systems, the model does not have a steady state, but the pattern is “read out” at a point in development, in response to an independent developmental signal such as a pulse of ecdysone secretion, which is known to regulate color pattern in butterflies. The grass-fire model reproduces the sequence of Distal-less expression that determines the position of eyespot foci and also shows how a border ocellus and its neighboring parafocal element can arise from such a single focus. The grass-fire model shows that the apparent fusion of ocellus and parafocal element is probably due to a premature termination of the normal process that separates the two and supports the hypothesis that the parafocal element is the distal band of the border symmetry system.

A model for population dynamics of the mimetic butterfly Papilio polytes in the Sakishima Islands, Japan

We present a mathematical model for population dynamics of the mimetic swallowtail butterfly Papilio polytes in the Sakishima Islands, Japan. The model includes four major variables, that is, population densities of three kinds of butterflies (two female forms f. cyrus, f. polytes and the unpalatable butterfly Pachliopta aristolochiae) and their predator. It is well-known that the non-mimic f. cyrus resembles and attracts the male most, and the mimic f. polytes mimics the model butterfly P. aristolochiae. Based on experimental evidence, we assume that two forms f. cyrus and f. polytes interact under intraspecific competition for resources including the male, and the growth rate of f. cyrus is higher than that of f. polytes. We further assume that both the benefit of mimicry for the mimic f. polytes and the cost for the model are dependent on their relative frequencies, i.e. the motality of the mimic by predation decreases with increase in frequency of the model, while the motality of the model increases as the frequency of the mimic increases. Taking the density-dependent effect through carrying capacity into account, we set up a model system consisting of three ordinary differential equations (ODEs), analyze it mathematically and provide computer simulations that confirm the analytical results. Our results reproduce field records on population dynamics of P. polytes in the Miyako-jima Island. They also explain the positive dependence of the relative abundance (RA) of the mimic on the advantage index (AI) of the mimicry in the Sakishima Islands defined in Section 2.

Pigmentation Pattern Formation in Butterfly Wings:Global Patterns on Fore- and Hindwing

Pigmentation patterns in butterfly wings are one of the most spectacular and vivid examples of pattern formation in biology. In this chapter, we devote our attention to the mechanisms for generating global patterns. We focus on the relationship between pattern forming mechanisms for the fore- and hindwing patterns. Through mathematical modeling and computational analysis of Papilio dardanus and polytes, our results indicate that the patterns formed on the forewing need not correlate to those of the hindwing in the sense that the formation mechanism is the same for both patterns. The independence of pattern formation mechanisms means that the coordination of united patterns of fore- and hindwings is accidental. This is remarkable, because from Oudemans’s principle [10], patterns appearing on the exposed surface of fore- and hindwing at the natural resting position are often integrated to form a composite and united adaptive pattern with their surrounding environment.

Morphogenesis of liposomes and bending energy of lipid bilayer

Based on the principle of the least bending energy of membranes, we study theoretically the morphogenesis of liposomes. We show first that there are bending-energetically stable shapes having geometrical symmetry. We find that by reducing the volume, the stable shape can change from a circular biconcave shape as in red blood cells to elliptical, triangular, square, and higher polygonal shapes. It is also found that the preceding two results hold true irrespective of the liposome size.

Spatial Variation in Boundary Conditions Can Govern Selection and Location of Eyespots in Butterfly Wings

Despite being the subject of widespread study, many aspects of the development of eyespot patterns in butterfly wings remain poorly understood. In this work, we examine, through numerical simulations, a mathematical model for eyespot focus point formation in which a reaction-diffusion system is assumed to play the role of the patterning mechanism. In the model, changes in the boundary conditions at the veins at the proximal boundary alone are capable of determining whether or not an eyespot focus forms in a given wing cell and the eventual position of focus points within the wing cell. Furthermore, an auxiliary surface reaction-diffusion system posed along the entire proximal boundary of the wing cells is proposed as the mechanism that generates the necessary changes in the proximal boundary profiles. In order to illustrate the robustness of the model, we perform simulations on a curved wing geometry that is somewhat closer to a biological realistic domain than the rectangular wing cells previously considered, and we also illustrate the ability of the model to reproduce experimental results on artificial selection of eyespots.