Furthermore, the size of the white center in the anterior eyespots was not plastic at all (Monteiro et al. 2015). Subsequent work (Bhardwaj et al. 2017) examining the plasticity of dorsal eyespots similarly concluded that wandering.
Plasticity Across Populations and Species
Alternatively, truncation selection was used for small eyespots at high temperature and large eyespots at low temperature in the following generation (trying to reduce the slope) (Wijngaarden and Brakefield 2001). Both types of experiments indicated that there was little or no genetic variation for the slope of the response norms.
Conclusions
Evolutionary genetics and developmental basis of wing pattern variation in the butterfly bicyclus-anynana. Windig JJ (1994) Response rates and the genetic basis of phenotypic plasticity in the wing pattern of the butterfly Bicyclus anynana.
Introduction
Modelling
Setting
The model we consider for the formation of focal points is based on the one proposed by Nijhout (1990) and consists of a reaction-diffusion system of the activator-inhibitor type (Gierer and Meinhardt1972) located in each wing cell with time-independent Dirichlet boundary conditions ( i.e. a source of chemicals) on the wing veins and Neumann (zero flux) boundary conditions (i.e. no flux of chemicals) at the wing edge.
Mathematical Model
We take the initial data for both activator and inhibitor to be the positive spatial homogeneous steady state of the Gierer-Meinhardt (GM) equation. For this reason, we allow the response rate to be a function of space, which appears to provide sufficient freedom to generate the necessary source profiles from this one-dimensional model that produces any arbitrary eyespot configuration observed on butterfly wings.
Computational Approximation
The resulting model is a two-stage model for focal spot formation in which the first stage corresponds to the solution of the Schnakenberg surface reaction–diffusion system Eq. 6.2) in the steady state and in the second stage the solution of this model is used to determine the proximal boundary profiles for a1 in the model of the reaction-diffusion system of the eye point Eq.
Results
Gradients in Source Strength on the Wing Veins Can Determine Eyespot Location in the Wing Cell
For the wing veins, we consider a gradient in the Dirichlet boundary condition by considering a linear boundary condition of the formulað Þ ¼~x 2a1ssð1s1x2=3Þ, where x2 denotes the distance in the proximal-distal direction from the wing margin and s1>0 is a parameter that determines the size of the gradient. We see that at constant boundary conditions or when the gradient is small, the centerline peak, characteristic of the Nijhout model, does not extend far from the margin. The focal point forms near the center of the wing cell and migrates toward the wing margin, with the steady state corresponding to a single focal point near the margin.
For larger values of the gradient (s), the tip of the centerline extends much further, almost reaching the proximal border, and the resulting focal spot forms close to the proximal border. The focal spot migrates downward only until around the center of the wing cell, and the resulting steady state is a single point of focus around the center of the wing cell.
A Surface Reaction-Diffusion System Model with Piecewise Constant Reaction Rate Generates
The proximal (Γp) and marginal (Γm) boundaries are curves corresponding to the part of the circumference of two concentric circles with radii 9 and 12, respectively. After solving the Schnakenberg system to a steady state, we assume that the Dirichlet boundary condition on the proximal boundary for the reaction-diffusion system is set in each wing cell, forms. The surface reaction-diffusion system was solved on a tracking grid corresponding to the boundary edges of the common grid; the corresponding one-dimensional mesh had 1793 DOF.
If we allow γ to be large on one half of the proximal boundary and small on the other half, then we create boundary profiles from a one-dimensional system that results in a single eye spot on the half of the wing in which γ is large, similar to the ApandaP phenotypes Figure 6.3. The choice of Dirichlet boundary conditions leads to the formation of substrate troughs at the correct locations for potential eyespots, depending on whether they are front or back; as for zero flux or symmetric Dirichlet boundary conditions, we would expect solutions that are symmetric along the middle of the proximal boundary.
Discussion
Butterfly wings are nearly two-dimensional but contain three-dimensional bumps and bulges that correspond to organizing centers for color pattern elements. A new distortion of epithelial cells is induced in the center of a putative parafocal element through an increase in cell size, producing self-similarity between the eye spot and the parafocal element. The self-similar configuration of the boundary symmetry system further suggests the essence of morphogenesis as the DCG cycle: repeated sequential events of epithelial distortions (D), calcium waves (C) and gene expression changes (G).
Future studies should examine these hypotheses and speculations that constitute the induction model in butterfly wings and the generality of the DCG cycle in other organisms. Keywords Butterfly wing • Color pattern rule • Distortion hypothesis • Eyespot • Induction model • Morphogen • Parafocal element • Pattern formation • Ploidy hypothesis • Self-similarity.
Introduction
Butterfly wing discs at the larval and pupal stages are sheets of epithelial cells (more specifically epidermal cells) that may be ready to accept mechanical changes. In this paper I attempt to extract "the essence of morphogenesis" from the color pattern development of the boundary symmetry system. The border symmetry system is one of the symmetry systems in nymphalid color patterns and consists of border ocelli (eye spots) and parafocal elements (PFEs), which will be briefly explained below.
The eyespots on the surface of one wing are homologous but not self-similar; self-similarity is hierarchical repetition, not parallel repetition. In the following sections, I first introduce the concept of self-similarity in biological entities using plants as examples.
Self-Similarity in Plants and Animals
I use plants because they often manifest similar structures that are relatively easy to detect, and many of them have been well analyzed mathematically (Mandelbrot1983; Barnsley et al.1986; . Ball1999,2016). The spiral flower arrangement of cauliflower romanesco (Brassica oleracea var. botrytis) is another well-known example of self-similarity (Fig.7.1a). It appears that this type of self-similarity in a complex biological entity (i.e. a flower in this example) that is not either simple branching or spiral patterns is relatively rare.
These examples in plants, animals, and other organisms demonstrate that organisms have the ability to form structures similar to themselves. Below I discuss the color patterns of butterfly wings from the point of view of self-similarity, but before discussing self-similarity, I will first discuss the symmetry in the color patterns of butterfly wings.
Part I: Color Pattern Rules
- Symmetry in Butterfly Wing Color Patterns
- The Core-Paracore Rule and Self-Similarity Rule
- The Border Symmetry System and Its Self-Similarity
- Eyespot Pattern Rules: The Binary Rule and Inside- Wide Rule
- Eyespot Pattern Rules: The Uncoupling Rule and Midline Rule
In this diagram, dSMB is part of the distal band of the central symmetry system (dBC) and can therefore be omitted from the map. Based on the core-paracore rule and the self-similarity rule, the diversity of the symmetry system can be understood as different modifications of the basic process of element formation (Fig. 7.3). This configuration of the boundary symmetry system appears to be typical for nymphlid butterflies.
This means that morphogenic signals for the outer ring and PFE can travel long distances from the center of the symmetry system. Similarly, a white eyespot (“focus”) behaves independently of the rest of the eyespot (eyespot body) (Iwata and Otaki2016a).
Part II: Formal Models toward the Induction Model
- Four Steps for Color Pattern Formation as a Starting Frame
- Gradient Model for Positional Information
- Transient Models for TS-Type Modifications and Parafocal Elements
- Heterochronic Uncoupling Model for TS-Type Changes
After determining the PFE location at the periphery of the gradient, the gradient disappears quite quickly and does not form an eyespot in an eyespot-free space. However, the results of damage experiments can be well explained by the revised version of the induction model (see below). The threshold change model is the most popular interpretation of TS-type modifications (Otaki 1998, 2008a; Serfas and Carroll 2005) as well as of physically induced modifications (Nijhout1980a,1985; French and Brakefield1992,1995; Brakefield and French1995).
Because TS-type modifications are interpreted as a series of possible color pattern snapshots during development, the modifications are likely consequences of a delay in the signaling step (slow signal propagation) or an acceleration of the reception step (Otaki 2008a ). This model simply states that TS-type modifications are products of snapshots of propagating signals, which is part of the basis of the induction model (Fig.7.7b).
Part III: Induction Model .1 An Overview
- Early and Late Stages
- Settlement Mechanisms
- Mechanisms for Self-Similarity
- Reality Check
However, TS-type modifications cannot be reproduced by simple threshold changes, as not only the relative locations but also the size and colors of the elements change. The late phase is the induction of activating signals (and their self-reinforcement) and inhibitory signals and their stabilizing interactions. The late stage of the induction model uses the concept of “short-term activation and long-term lateral inhibition” (Figure 7.8b), which is a central concept of the reaction-diffusion model (Gierer and Meinhardt 1972; Meinhardt and Gierer Meinhardt 1982).
In this sense, the rate and level of inhibitory signal induction primarily determine the final size of an eyespot. That is, the timeout mechanism cannot explain the heterochronic behavior of the primary and secondary signal dynamics.
Part IV: Ploidy, Calcium Waves, and Physical Distortions
- Scale Size of Elements
- Ploidy Hypothesis
- Calcium Waves
- Physical Distortion Hypothesis
- Damage-Induced Ectopic Elements
- Focal Damage
Our laboratory also obtained similar results using Junonia and other butterflies (Kusaba and Otaki2009; Iwata and Otaki unpublished data; Kazama et al. 2017). For example, blurred boundaries of pattern elements have been reported in thapsigargin-treated individuals (Otaki et al. 2005b; Ohno and Otaki 2015b). The slow contraction of wing tissue during the early pupal stage, revealed by time-lapse movies (Iwata et al. 2014), probably helps to extend the distortion waves.
Fortunately, mechanobiology is a growing interdisciplinary field between biology and physics (Iskratsch et al.2014). Interestingly, the genes expressed are similar in normal development and in the healing process (Monteiro et al.2006).
Part V: Generalization and Essence
Reinforced Version of the Induction Model
Physical deformation of the epithelial sheet occurs due to changes in the size and deformation of the cells. Calcium oscillations induce unknown inhibitory signals in cells located at the periphery of the oscillations. Cell size increases in the future black ring relative to genome size or ploidy level.
Cell size increases at the future black rings depending on the number of genomes in a cell. Where self-enhancement calcium oscillations are highly active, the high rate of cell size increase occurs, resulting in the formation of a secondary organizing center, often seen in PFEs.
Generalization to Other Systems
DCG Cycle for Self-Similarity and Its Implications
Dhungel B, Otaki JM (2009) Local pharmacological effects of tungstate in the color patterning of butterfly wings: a possible relationship between the eyespot and parafocal element. Iwata M, Otaki JM (2016b) Spatial patterns of correlated scale size and scale color in relation to color pattern elements in butterfly wings. Otaki JM (2007) Reverse type color pattern modifications of butterfly wings: a physiological mechanism of wing-wide color pattern determination.
Otaki JM (2008c) Physiological side effects model for diversification of nonfunctional or neutral traits: a possible evolutionary history of Vanessa butterflies (Lepidoptera, Nymphalidae). Otaki JM (2011b) Color pattern analysis of eyespots on butterfly wings: a critical examination of morphogenic gradient patterns.
