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

A Predictive Model for Color Pattern Formation in the Butterfly Wing of Papilio dardanus

Previously, we have proposed a mathematical model based on a modified Turing mechanism to account for pigmentation patterning in the butterfly wing of Papilio dardanus, well-known for the spectacular phenotypic polymorphism in the female of the species (Sekimura, et al., Proc. Roy. Soc. Lond. B 267, 851-859 (2000)). In the present paper, we use our model to predict the outcome of a number of different types of cutting experiments and compare our results with those of a model based on different hypotheses.\ud \ud This paper is dedicated to Professor Masayasu Mimura on his sixtieth birthda

A model for selection of eyespots on butterfly wings

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. 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. 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: An Integrative Approach

This book facilitates an integrative understanding of the development, genetics and evolution of butterfly wing patterns. To develop a deep and realistic understanding of the diversity and evolution of butterfly wing patterns, it is essential and necessary to approach the problem from various kinds of key research fields such as “evo-devo,” “eco-devo,” ”developmental genetics,” “ecology and adaptation,” “food plants,” and “theoretical modeling.” The past decade-and-a-half has seen a veritable revolution in our understanding of the development, genetics and evolution of butterfly wing patterns. In addition, studies of how environmental and climatic factors affect the expression of color patterns has led to increasingly deeper understanding of the pervasiveness and underlying mechanisms of phenotypic plasticity. In recognition of the great progress in research on the biology, an international meeting titled “Integrative Approach to Understanding the Diversity of Butterfly Wing Patterns (IABP-2016)” was held at Chubu University, Japan in August 2016. This book consists of selected contributions from the meeting. Authors include main active researchers of new findings of corresponding genes as well as world leaders in both experimental and theoretical approaches to wing color patterns. The book provides excellent case studies for graduate and undergraduate classes in evolution, genetics/genomics, developmental biology, ecology, biochemistry, and also theoretical biology, opening the door to a new era in the integrative approach to the analysis of biological problems

Pigmentation pattern formation in butterfly wings: Global patterns of fore- and hind-wing.

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Morphogenesis and pattern formation in biological systems: experiments and models

A central goal of current biology is to decode the mechanisms that underlie the processes of morphogenesis and pattern formation. Concerned with the analysis of those phenomena, this book covers a broad range of research fields, including developmental biology, molecular biology, plant morphogenesis, ecology, epidemiology, medicine, paleontology, evolutionary biology, mathematical biology, and computational biology. In Morphogenesis and Pattern Formation in Biological Systems: Experiments and Models, experimental and theoretical aspects of biology are integrated for the construction and investigation of models of complex processes. This collection of articles on the latest advances by leading researchers not only brings together work from a wide spectrum of disciplines, but also provides a stepping-stone to the creation of new areas of discovery

Proximal boundary conditions may govern eyespot focus point determination.

<p>The figure shows snapshots of the activator concentration corresponding to the solution of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e003" target="_blank">3.1</a>). The boundary conditions on the proximal boundary (top) of the rectangular cell for the activator are of the form <i>k</i>_<i>p</i><i>a</i>₁^<i>ss</i> where <i>k</i>_<i>p</i> = 0, 1 and 2 (reading from left to right in each row) and <i>a</i>₁^<i>ss</i> is the (activator) steady state value. The veins (left and right boundaries of each wing cell) have Dirichlet (fixed) boundary conditions for the activator with constant values at twice the steady state. Initially in all the wing cells a vertical stripe of high activator concentration is generated originating from the zero-flux distal boundary (bottom). In the wing cells with lowest activator values at the proximal boundary (left hand), a spot forms and this spot eventually moves towards the center of the cell (see also Section <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#sec013" target="_blank">3.3</a>). In the wing cells with medium activator values at the proximal boundary (middle), we have both the formation of a spot from the receding midline peak and later the insertion of a new spot that originates from the proximal boundary with the steady state consisting of two spots. In the wing cell with highest activator values at the proximal boundary (right hand), the vertical stripe recedes without leaving behind a spot.</p