Fig 1.
Ventral eyespot patterns of the butterfly Ypthima arugus (Nymphalidae, Satyrinae) at rest (left), and the extended adult specimen (right).
The right-hand side photo: courtesy of Mr.Toru Tokiwa.
Fig 2.
Development of eyespot focus points in the wing disc of Junonia coenia (Nymphalidae, Nymphalinae).
Numbers 1~7 in the photos show a time course of the Notch expression pattern during the focus point development. The expression pattern by antibody staining were visualized on a fluorescent light microscope and digitally photographed. Black arrows in photo numbers 1, 2, and 5 indicate pre-veins, which finally evolve to become veins of the adult butterfly wing. White arrows in photo 6 show two peaks of N-related chemicals along the centerline of each wing cell, the right-hand one of which evolves into a focus point afterwards (in photo number 7) while no focus point remains on the left-hand wing cell. Photos: courtesy of Prof. Fred Nijhout of Duke University. For more details on the adult forewing of J.coenia butterfly, see Fig 7 in Section 3.3.
Fig 3.
A wing disc in the larval stage (up left) and its venation system (down left) of Papilio polyxenes (Papilionidae).
Both photos: courtesy of Prof. Fred Nijhout of Duke University. (Right) Hypothetical wing disc and its venation system with rectangular approximation to wing cells in which eyespot focus point selection occurs.
Table 1.
Parameter values used for all the simulations of Eq (3.1).
Fig 4.
Proximal boundary conditions may govern eyespot focus point determination.
The figure shows snapshots of the activator concentration corresponding to the solution of Eq (3.1). The boundary conditions on the proximal boundary (top) of the rectangular cell for the activator are of the form kpa1ss where kp = 0, 1 and 2 (reading from left to right in each row) and a1ss 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 3.3). 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.
Fig 5.
(Top row): Examples of proximal boundary condition: concave (left) and convex (right) profiles. (Bottom row): Numerical simulations on the influence of proximal boundary profile on eyespot focus point determination.
The figure shows snapshots of the activator concentration corresponding to the solution of Eq (3.1) on wing cells with proximal boundary conditions. The wing cells are taken to be rectangular of length 3 and width 2. In each subfigure, the left hand plot corresponds to the concave proximal boundary condition and the right hand plot the convex proximal boundary condition (c.f., Fig 5 (Top row)). We observe the formation of a spot in the concave case whilst the midline peak completely recedes leaving behind no spot in the convex case.
Fig 6.
Steady state values of the activator concentration in simulations of Eq (3.1) on a domain of increasing width in the proximal-distal direction (top to bottom) and with curved proximal (top) and wing margin (bottom) boundaries.
The left hand figure corresponds to constant boundary conditions equal to zero on the proximal boundary curve. The right hand figure corresponds to proximal boundary conditions equal to twice the activator steady state. The observed behavior is analogous to the rectangular domain case.
Fig 7.
Development of focus points in the wing disc during eyespot determination.
(a) Time series of Notch expression patterns in Junonia coenia wing discs for the final instar eyespot determination. The Notch expression patterns were obtained by anti-N mouse monoclonal antibody and were visualized on a fluorescent light microscope [18]. (Upper row) The five panels show stained wing discs. (Bottom row) The five panels show the wing cells extracted from the respective figures in the upper panels. Regarding the orientation of bottom panels, the upper side corresponds to the proximal boundary and the bottom side corresponds to the distal boundary of the wing cell, respectively. Insets in the panels detail gene expression in the wing cells marked by white arrows. (b) The corresponding adult forewing of J.coenia. (c) Simulation results of Fig 7 (a) by use of Eq (3.1). The initial data and boundary conditions are perturbed by uniformly distributed noise which leaves the qualitative features of the results unchanged. In Fig 7 (a), we could see a migration of the focal point into the distal direction from the 3rd stage (middle) to the 4th stage (next to the middle). Both photos (a) and (b): courtesy of Dr. Robert Reed of Cornell University.
Fig 8.
Incomplete vein development leaves two focus points with an eyespot covering two focus points.
(a) Normal (left) and abnormal (right) eyespot patterns on the hind wing of the butterfly Ypthima arugus. (b) Sketch of the abnormal venation system and an arrow to show two distinct focus points. (c) Simulations of the abnormal case of incomplete vein development shown in (a) (right) by use of Eq (3.1). This incomplete vein development leads to two focus points forming close to both the incompletely developed vein’s end point. The eventual pattern observed on the butterfly wing is that of a single eyespot generated by two focus points that are in close proximity. The corresponding normal pattern is of two distinct eyespots with orally separated foci. Photos (a) and the sketch (b): courtesy of Mr.Toru Tokiwa.
Fig 9.
One eyespot splits into two eyespots through the addition of a vein.
(a) Normal (left) and abnormal (right) eyespot patterns on the hind wing of the butterfly Ypthima arugus. (b) Sketch of the abnormal venation system and the corresponding patterning, where arrows show an additional vein. (c) Simulations of the abnormal case of an additional vein development in the middle of adjacent two veins shown in (a) (right) by use of Eq (3.1). Photos (a) and the sketch (b): courtesy of Mr.Toru Tokiwa.
Fig 10.
Focus points on the ventral hind wing of Bycyclus anynana and numerical simulation results by the 2-stage model.
(a) Steady state values of activator concentration (u1) of the 1D RDS (4.1) with γ(x) = 0.01. (b) Ventral hindwing of B. anynana. (c) Steady state values of the activator concentration (a1) for the seven independent bulk RDSs (3.1) with proximal boundary conditions given by (1/3)u1 where u1 is the steady state activator concentration shown above. (d) The hind wing imaginal disc of B. anynana with focus points labelled. (Left hand column) Simulation results of the 2-stage model for focus point formation with a small constant value of the reaction rate γ appearing in the 1D RDS (4.1) (γ(x) = 0.01). The model generates a focus point in every wing cell. (Right hand column) The adult ventral hind wing of B. anynana with seven eyes-pots (top) and the fifth-instar hind wing imaginal disc displaying a pre-pattern with seven foci (bottom), which correspond to eyespots positions on the adult ventral hind wing. Experimental figures: from Brakefield et al. [2] with permission by the publisher.
Fig 11.
Simulation results of the 2-stage model for focus point formation with a large constant value of the reaction rate γ appearing in the 1D RDS (4.1) (γ(x) = 1).
The model generates no focus points. (a) Steady state values of activator concentration (u1) of the 1D RDS (4.1) with γ(x) = 1. (b) Steady state values of the activator concentration (a1) for the seven independent bulk RDSs (3.1) with proximal boundary conditions given by (1/3)u1 where u1 is the steady state activator concentration shown above.
Fig 12.
Focus points on the dorsal hind wing of Precis coenia and numerical simulation results by the 2-stage model.
(a) Steady state values of activator concentration (u1) of the 1D RDS (4.1) with a constant value of the function γ(x). (b) Dosal hindwing of P. coenia. (c) Steady state values of the activator concentration (a1) for the seven independent bulk RDSs (3.1) with proximal boundary conditions given by (1/3)u1 where u1 is the steady state activator concentration shown above. (d) The hind wing imaginal disc of P.coenia with focus points labelled. Two white arrows point two Dll stained focus points. (Left hand column) Simulation results of the 2-stage model for focus point formation with aconstant value of the reaction rate γ appearing in Eq (4.1) (γ(x) = 5.4). The model generates the formation of foci in wing cells 2 and 5 and no foci in the other wing cells similar to the experimental observations. (Right hand column) The adult P. coenia dorsal hindwing with two eyespots (top) and the fifth-instar hindwing imaginal disc displaying a pre-pattern with two foci (bottom), which correspond to eyespots positions on the adult dorsal hindwing. Experimental figures: from Brakefield et al. [2] with permission by the publisher.