Methods for analyzing the furrows in horn primordia
In our previous work, we reconstructed the three-dimensional mesh of horn primordia from serial cross-sectional images of actual horn primordia samples. When pressure was applied to the 3D mesh from the inside, the folds were unfolded and their shape transformed into the pupal horn shape. Here, we analyzed the relationship between the folding patterns and transformation using the 3D mesh model and computer simulation in the ways described below.
First, the three-dimensional mesh was used to determine the positional correspondence between the pupal horn and the horn primordium. Since the surface of biological samples of horn primordia has a very complex nested structure with macro folds and micro furrows, it is very difficult to identify the correspondence of a given position in the folded primordium after deployment. However, using the 3D mesh model facilitates this task. After marking some regions of the expanded 3D mesh, the physical simulation was rewound to show exactly which positions in the folded primordium corresponded to those regions. (Fig.1, red broken lines). This analysis is referred to as “correspondence analysis” in the following sections.
The second operation was the visualization of the furrow patterns on the 3D mesh model. Micro furrows are located on the surface of complex three-dimensional macro shapes. Therefore, it is not possible to observe all furrow patterns even if the light is shed from one direction to create a shadow. It is also necessary to know where and in what direction the furrows run in the structure both before and after deployment. To solve these problems, we used smoothing algorithms, which were originally used to reduce noise from the data and create a smooth mesh surface. By comparing the position of the surface before and after smoothing, we were able to color the ridges and valleys of the furrows (Fig.1, blue dotted line). In the following sections, we refer to this operation as “furrow visualization”. By using furrow visualization together with the correspondence analysis, it is also possible to colorize the unfolded primordium depending on whether the positions are ridges or valleys when folded.
Third, we used the mesh and applied the smoothing algorithm to investigate how the furrows at each position contribute to transformation. The smoothing algorithm can make the furrows at a specific position shallower. Thus, the contribution of specific furrows to transformation can be understood by comparing the original mesh with the operated mesh after unfolding (Fig.1 yellow solid line). In the following sections, we refer to this operation as “furrow removal analysis.”
The details of each algorithm are provided in the supplementary material.
Overview of morphological changes
First, we compared the external shapes of larvae, peeled larvae (larvae after removal of the outermost exoskeleton), and pupae (Fig.2a, b). The proximodistal length of the pupal horn was about six times longer than that of a horn primordium, while the lateral width and the dorsoventral length were almost unchanged (Fig.2b). In addition to length change, a four-branched structure not present in the primordia appeared in the pupal horn (Fig.2a). Next, we performed a forced expansion treatment to lessen the effect of transformation in parts other than horn primordia (Fig.2c, c’). In the treatment, we pushed the larval abdomen to expand horn primordia after removing the larval head capsules17. This treatment, in which the mandibles were used as a reference, revealed that the angle of the pupal horn was dorsally declined compared to that of the primordia (details on the definition of the orientation are given in the Supplementary Figure 2).
Thus, the transformation includes three types of changes: shape, length, and angle.
Division of the primordia into three regions
Next, we divided the horn primordia into three regions likely to contribute to each of the morphological changes described above (Fig.2e). The distal region contained four-branched tips. This region was named “cap”. The middle region following the cap was cylindrical with a nearly constant radius. This part was named “stalk”. The proximal region was named “base”. This region connected the horn to the body. The top and bottom edges of the cylindrical part were set as the boundaries (see Supplementary Information 1 for details of the algorithm).
Each corresponding region in the folded primordia was retrospectively identified in the “correspondence analysis” (Fig.2e and Supplementary Movie 1), which allowed us to observe how each region had been transformed separately. The cap, which creates a four-branched structure after unfolding, was shaped like a mushroom umbrella with large overhangs on the left and right sides in the primordia (Supplementary Movie 2). The stalk was very flat and complexly folded in the primordia. This indicated that the elongation along the long axis mainly depends on this region (Supplementary Movie 3). The folded shape of the base region was rather flat and its distal boundary ran almost parallel to its proximal boundary. However, after unfolding, the two boundaries became almost perpendicular to each other (Supplementary Movie 4). From this, it could be inferred that the angular change during transformation depends mainly on this region.
In the following sections, we examined in detail how the deployment of the folding contributed to the transformation of each region.
Structure of the base
The base connects a horn with the body. As mentioned above, during transformation from horn primordia to pupal horns, the overall orientation changes to the dorsal side, to which the unfolding of the furrows in this region should contribute. First, we visualized the furrow pattern in this region using the “furrow visualization” method (Fig.3a and b). Apparently, the ventral side had more furrows than the dorsal side (the simplest example is shown in Fig.3b), which could be the cause of dorsal bending. Next, we performed the “furrow removal analysis” (Fig.3c). When all the furrows in the base were made shallower, the orientation of the unfolded horn changed to the ventral side (Fig.3c), suggesting the contribution of the furrows in this region to dorsal bending.
Structure of the cap
As mentioned above, the cap is a region that generates four branches in the distal region. Figure 4A shows a superimposed drawing of the cap before and after the expansion, with the angle adjusted to the same orientation (seen from the ventral side). We then subdivided the cap into four subregions according to the characteristics of the expanded structure (Fig.4b and c). The lateral tips were colored in red and the medial tips were colored in white. The subregion between the medial tips was named “upper surface” (colored in dark pink) and the underside of the overhangs was named “bottom surface” (colored in light pink). Next, the area in the folded primordia corresponding to each subregion was identified in the “correspondence analysis” and the functions of the furrow were examined in the “furrow removal analysis.”
In the medial tips, the furrow showed a concentric semicircular pattern (Fig.4d). When the furrow pattern was removed, the medial tips became much smaller than the original ones (Fig.4d’). To understand the function of the furrow pattern, we compared the concentric semicircle furrow pattern with a perfectly concentric circle pattern (Fig.4h-k). When the simplified models were expanded, perfect circles formed a cone perpendicular to the plane (Fig.4i), while concentric semicircular furrows had an oblique cone overhang (Fig.4k). The reason that the angle of the medial tip was tilted outward is that it was an outwardly vacant concentric semicircle. In contrast to the medial tip, the furrows in the lateral tips had no distinct directionality (Fig.4e). The shape of the unfolded horn changed little even after removal of the furrows (Fig.4e’). This means that the morphology of the lateral tips is determined by the macroscopic shape of the horn primordia rather than the furrow pattern. On the other hand, the medial tips were not present in the folded primordia and were formed by the furrow pattern. Therefore, the mechanisms for making the two types of tips were different.
The upper surface subregion had some bow-shaped furrows facing each other (Fig.4f). When the furrows were removed, the angle of the central groove became smaller than the original one (Fig.4f’). In the bottom surface subregion, there were many parallel linear furrows running in the dorsoventral direction (Fig.4g). When the furrow pattern was removed in this area, the angle of the central groove became bigger than the original one (Fig.4g’). This transformation is explained as follows: when the parallel furrows in the bottom surface unfold, the lower surface of the overhang elongates. On the other hand, the top surface does not elongate because the furrow pattern consists of concentric semicircles. The difference in the elongation between the top and lower surface raises the branches.
Thus, by using the 3D mesh data, we were able to connect the morphological changes to the furrow patterns in each part of the cap region (Fig.4l).
Structure of the stalk
As described above, the stalk was the region that mainly contributed to the elongation of the horn. We first compared the stalk region before and after unfolding (Fig.5a, b; after adjusting the orientation). The proximodistal length of the unfolded stalk was about six times larger than that of the folded stalk (Fig.5a). Seen from the top side, the folded stalk had an elongated shape that was flattened in the left-right axis (Fig.5b). However, the dorsoventral length of the folded stalk was almost the same as that of the unfolded stalk (Fig.5b).
Since the shape of the unfolded stalk was a cylinder, the furrows in this area did not create a specific three-dimensional shape (because the Gaussian curvature of a cylinder is zero) but contributed to the elongation of the cylinder along the proximodistal axis. Therefore, the direction of the folding in most of the stalk region was parallel to the cylinder’s circumference (Fig.5e). However, on the ventral and dorsal sides of the stalk (especially on the ventral side), two more complicated patterns were observed. One consisted of wavy folds (Fig.5d) and the other of zigzag folds (ridges/valleys alternately coming from the left and right side, Fig.5f). The zigzag fold, also known as Yoshimura fold19, is a famous folding pattern that arises inevitably when a cylinder-like plane is compressed in the axial direction (Fig.5h). Since the stalk is surrounded by the cap and the base, the growth of a cell sheet in the axial direction is thought to result in relative longitudinal compression, causing the zigzag folds. A wavy fold also raised autonomously when a sheet with folds was bent (Fig.5g). Since the lateral tips of the cap occupy both the left and right spaces beside the stalk, the growing cap is thought to exert bending forces to the ventral side of the stalk.