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Discovery of I-WP minimal-surface-based photonic crystal in the scale of a longhorn beetle

Yuka Kobayashi

Yuka Kobayashi

Department of Physics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda 278-8510, Japan

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Ryosuke Ohnuki

Ryosuke Ohnuki

Department of Physics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda 278-8510, Japan

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Shinya Yoshioka

Shinya Yoshioka

Department of Physics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda 278-8510, Japan

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Published:https://doi.org/10.1098/rsif.2021.0505

    Abstract

    The structural colours of certain insects are produced by three-dimensional periodic cuticle networks. The topology of the cuticle network is known to be based on the mathematically well-defined triply periodic minimal surface. In this paper, we report the discovery of an I-WP minimal-surface-based photonic crystal on the scale of a longhorn beetle. In contrast to gyroid or diamond surfaces, which are found in butterfly and weevil scales, respectively, the I-WP surface is an unbalanced minimal surface, wherein two subspaces separated by the surface are different in terms of shape and volume fraction. Furthermore, adjacent photonic crystal domains were observed to share a particular crystal plane as their domain boundary, indicating that they were developed as twin crystals. These structural features pose certain new questions regarding the development of biological photonic crystals. We also performed an optical analysis of the structural colour of the longhorn beetle and successfully explained the wavelength of reflection by the photonic bandgap of the I-WP photonic crystal.

    1. Introduction

    Biological membranes can take various shapes [1]. The most striking example is the smooth endoplasmic reticulum (SER) in a eukaryotic cell, which can be shaped as interconnected discs, cylinders and complicated organized structures [2,3]. Biological membranes are thought to control their curvature via membrane proteins and their interactions with lipids to transform their shape [3,4]. However, detailed shape-forming mechanisms and their relationship with biological functions remain to be understood. In addition, these questions are also interesting from the perspectives of mathematics, physics and material sciences. Biological membranes can assume the shape of mathematically well-defined triply periodic minimal surfaces (TPMS) [5], functioning as a template for cuticle secretion, which later becomes a photonic crystal exhibiting brilliant structural colour [6,7].

    It is known that certain insect species use submicrometre structures to produce brilliant colours through optical interference [811]. While the periodic multilayer in the elytrum of the jewel beetle is one of the most common colour-causing microstructures, certain species use a three-dimensional (3D) complicated cuticle network, which can be considered as an example of a natural photonic crystal. Cuticle networks are based on different TPMSs depending on the insect species, for example, the butterfly wing scale has a cuticle network corresponding to the gyroid minimal surface [7,12,13], while many weevil species have been reported to possess photonic crystals based on the diamond minimal surface [1417]. It has been hypothesized that in the butterfly wing scale, the SER and plasma membranes are folded to form double gyroid surfaces, and later or concurrently, the nascent cuticle is secreted into the extracellular space, which eventually dries up to form the photonic crystal structure [6,7,18]. Although there remain many unanswered questions regarding the detailed mechanisms, scientific studies have already begun to synthesize similar photonic structures using a self-organizing method and to produce optical materials using soft matter such as lipids, block copolymers and liquid crystals [19,20].

    Sranathane et al. have explored photonic structures in a wide variety of arthropod species using small-angle X-ray scattering (SAXS) experiments [21]. They reported that the morphology of the photonic structures found in these species is unexpectedly wide, covering most of the phases found in the lyotropic lipid–water system. In this paper, we report the discovery of another photonic structure found in the scale of a structurally coloured longhorn beetle Sternotomis callais (figure 1a).

    Figure 1.

    Figure 1. Longhorn beetle Sternotomis callais. (a) Photograph. (b) Optical micrograph of the scale. Scale bar: (a) 0.5 cm and (b) 20 μm.

    Longhorn beetles have been reported to possess various photonic structures. For example, a ball-and-stick structure in the face-centred cubic (FCC) geometry has been reported in Prosopocera lactator [22], while an amorphous packing of small particles has been observed in other species [23,24]. Further, a disordered bicontinuous network has also been reported on the scale of Sphingnotus mirabilis [25]. Interested in these varieties, we started research on the structural colour of S. callais. We carefully characterized the microstructure inside the scale using focused ion beam (FIB) milling and scanning electron microscopy (SEM). The 3D reconstruction revealed that the photonic crystal structure in the scale is based on the I-WP-type minimal surface, which was mathematically found by Schoen in 1970 [26]. The I-WP type minimal surface is characterized by its topology that is different from that of gyroid and diamond surfaces. In this paper, the importance and implications of this type of photonic crystal are discussed, along with an explanation of the coloration mechanisms using the photonic band diagram.

    2. Material and methods

    2.1. Sectioning and SEM observation of the scale

    We employed an ion-beam cross-section polisher (JEOL IB-19530CP) to prepare the cross-section of the sample. This apparatus uses a several-millimetre-wide Ar+ ion beam for milling. The acceleration voltage was 3.5 kV. The milling process required several hours, but the sections obtained were found to be quite smooth. The exposed section of the scale was observed using SEM (JEOL JSM-6500F).

    2.2. FIB milling and three-dimensional reconstruction

    We employed an FIB-SEM system (Helios NanoLab TM 600i) for three-dimensional structural investigations. Sequential FIB milling and SEM imaging (slice-and-view observation) were performed after the initial section was prepared using the above cross-section polisher. Following our previous study [17], the FIB milling process was performed under a beam current of 33 pA and an accelerating voltage of 30 kV. The slice thickness was set to 17 nm. After the first SEM image was obtained, the slice-and-view process was repeated 44 times to obtain a total sliced thickness of 0.75 μm. The 45 SEM images obtained were corrected for the view angle as they were taken from an oblique direction: the angle between the electron beam and focused ion beam was 52°. The drift during the milling processes was also corrected, which can be noticed as the SEM images were taken for a larger area than the milling area. These corrected images were binarized and finally used for three-dimensional rendering, which was performed using commercial software (Wolfram Research MATHEMATICA v. 12.1). This software was also used for the theoretical modelling of the I-WP-type photonic crystal.

    2.3. Microspectrophotometry

    Microspectrophotometry was used to determine the reflectance spectrum of a small region within a scale. The experimental system consisted of an optical microscope (Olympus BX51) and a fibre optic spectrometer (Ocean Optics USB2000). The microscope was equipped with a xenon lamp and an objective lens with 50 × magnification (Olympus SLMPlan N, NA 0.35). The fibre diameter was 200 μm, enabling the examination of a 4 μm diameter region on the scale. We placed a 200-μm-diameter pinhole in the plane of the aperture stop of the microscope such that the illumination became nearly collimated at the sample position [27]. The reflectance spectrum was determined as the ratio of the observed spectrum to that of a diffuse reflection standard (Labsphere Spectralon).

    2.4. Three-dimensional modelling of the I-WP photonic crystal

    We used the following mathematical function to model the I-WP minimal surface-based photonic crystal according to a study [28]:

    0.0652+0.5739(cosXcosY+cosYcosZ+cosZcosX)0.1712(cos2X+cos2Y+cos2Z)0.1314(cos2XcosYcosZ+cosXcos2YcosZ+cosXcosYcos2Z)+0.0184(cos2Xcos2Y+cos2Ycos2Z+cos2Zcos2X)t,2.1
    where X = (2π/a)x, Y = (2π/a)y and Z = (2π/a)z, x, y, z are Cartesian coordinates, and a is the lattice constant of the cubic lattice. This is a modified formula from that reported in [29] to render the mean curvature closer to zero.

    2.5. Photonic band calculation

    The photonic band diagram of the I-WP-type photonic crystal was calculated using a self-developed code [17] based on the plane-wave expansion method [30]. The expansion was made using electromagnetic waves corresponding to 1055 reciprocal lattice points around the origin in the reciprocal space. We confirmed that this number was sufficient to obtain the converged results of the calculation. The refractive index of the cuticle was assumed to be 1.555, which was experimentally estimated using the Becke line test.

    3. Results

    3.1. Structural determination

    First, we observed the scales on the elytrum using an optical microscope. Figure 1b shows that each scale exhibits a brilliant green colour, and strong reflection is particularly observed around the basal part of the scale. However, the scale morphology is rather different from those of butterflies and weevils; in these species, scales are usually thin and flat. The SEM micrographs show that the longhorn beetle has a more three-dimensional shape (figure 2a,b); it is similar to an elongated particle with a thick basal part as illustrated in figure 2c,d. In addition, there exist ribs on both sides of the scale that run obliquely with respect to the longer axis of the scale.

    Figure 2.

    Figure 2. Scanning electron micrograph. Scales were observed from the (a) top and (b) lateral directions. (c,d) Schematic illustration. (c) Top view and (d) side view with typical dimensions Scale bar: (a) 20 μm and (b) 20 μm.

    Thereafter, we prepared a cross-section of the scale to investigate its internal structure. We employed an experimental apparatus called an ion-beam cross-section polisher. The obtained cross-section was found to be smooth (electronic supplementary material, figure S1); thus, we could reliably compare the surface morphology with theoretical models. Figure 3 shows the internal structure of the scale that was sectioned approximately parallel to the scale base (see also electronic supplementary material, figure S1(a) for a low-magnification image). The SEM image clearly shows that there is a photonic crystal-like periodic cuticle network. Focusing on the periodically arranged oval holes shown in figure 3b, it is evident that the orientation of the cuticle network is different depending on the position; their boundaries are shown by white dotted lines. Thus, the photonic crystal is separated into small crystal domains, as such multidomain photonic crystals have been commonly found in the scale of butterfly and weevil species. However, the direction of the oval holes appear to be set at a certain angle with respect to the domain boundary, similar to the oval hole patterns in two adjacent domains that are related by a mirror operation. Figure 3b shows that the directions along the oval holes are at 33.2° and 38.3° with respect to the domain boundary. Similar angles have been observed for, not all, but many domain boundaries, and some of them at different scales are shown in the electronic supplementary material, figure S2. These observations indicate that the photonic crystals developed as twin crystals, as is discussed later.

    Figure 3.

    Figure 3. Cross section of the scale. (a) Illustration showing the height of the sectioning plane for the observations shown in (b,d,f). (b,d,f) SEM images and (c,e,g) model structures assuming the I-WP minimal-surface-based photonic crystal. (c,e,g) correspond to the (110), (111) and (100) planes of the photonic crystal, respectively. Scale bar, (b) 1 μm (d) 300 nm (f) 300 nm.

    Figure 3a schematically shows the height of the sectioning plane; figure 3b shows an example of the case when the sectioning plane is near the basal part of the scale, and the arrangement of oval-shaped holes is observed. Such a pattern is observed in many scales when they are sectioned at the basal part (electronic supplementary material, figure S2). This implies that a particular crystal direction is preferably oriented, and this preferred direction possibly explains the observation that the basal part of the scale is particularly reflective (figure 1b). Figure 3d,f shows the different cuticle patterns that can be observed when the scale is sectioned at a higher part; the height of the sectioning plane can be inferred based on whether the contour of the scale has an undulation due to the ribs (see also electronic supplementary material, figure S3 for low magnification images), and also from the fact that the sectioning plane is slightly tilted with respect to the elytrum surface such that the sectioning height gradually changes from scale to scale (electronic supplementary material, figure S1a). The cuticle patterns of figure 3d,f appear to possess the threefold and fourfold symmetries, respectively. These observations suggest that the photonic crystal network consists of a cubic lattice, which is the only case among the seven crystal systems that has both the three- and fourfold axes.

    We compared the observed cuticle patterns with various structural models for photonic crystals based on TPMSs. It is known that a TPMS can be approximately expressed by an equation F(x, y, z) = 0, where F(x, y, z) is a Fourier series including space coordinates x, y and z, which are often truncated by several leading terms [29]. We can obtain a mathematical formula that approximately expresses the photonic crystal based on a TPMS by substituting the above equation into an inequality (a spatial position satisfying the inequality is filled with cuticle [12]) and by replacing the zero on the right-hand side with a parameter t, which is called the level set parameter, to adjust the volume fraction of the cuticle. Schnering and Nesper reported a Fourier series approximating 19 minimal surfaces, including primitive, gyroid and diamond types [29]. Using these formulae, we simulated the surface structures of the (100), (110) and (111) planes of 19 cubic photonic crystals and compared them with the SEM images. The best matching structure was determined to be the photonic crystal based on the I-WP-type minimal surfaces, as shown in figure 3c,e,g. Considering the comparison with the SEM images, the lattice constant a was determined to be 315 nm, which is the average of the analyses for 10 SEM images.

    To investigate the three-dimensional structure, we performed successive FIB milling and SEM observations (slice and view method). Several SEM images obtained during this process are shown in the electronic supplementary material, figure S4. By analysing the 45 images obtained, we reconstructed the three-dimensional network of the cuticle, as shown in figure 4 and found that it can be modelled appropriately by the I-WP photonic crystal. In fact, it is possible to cut out from the reconstructed three-dimensional structure of a unit cell of the cubic lattice, with a network exactly matching that of the I-WP-type photonic crystal, as shown in figure 4c,d. The unit cell possesses body-centre symmetry, and the cuticle network connects the body centre position to the eight surrounding corners (see electronic supplementary material, movie S1 for three-dimensional comparison). In addition, we also confirmed that many similar unit cells can be obtained by simply translating the cutting-out position with translation vectors of the cubic lattice (electronic supplementary material, figure S5). From these results, we conclude that the photonic crystal of this longhorn beetle is based on the I-WP minimal surface.

    Figure 4.

    Figure 4. Three-dimensional cuticle network of the scale. (a) Structure reconstructed by successive FIB milling and SEM observations. (b) Theoretical model for (a) based on the I-WP-type photonic crystal. (c,d) Cubic unit cell cut out from (a,b), respectively. Note that to observe the topology more clearly, the cuticle network is made thinner using the erosion command in Mathematica. Correspondingly, the parameter t is assumed to be 0.4 in (b,d), which leads to a lower cuticle volume fraction than the experimental result.

    The volume fraction of the cuticle was determined to be 0.44 from the analysis of the three-dimensional reconstruction. This value corresponds to the parameter value t = 0.19 in equation (2.1); the relationship between the volume fraction and the value of t parameter is shown in the electronic supplementary material, figure S6.

    3.2. Optical analysis

    Optical reflection was characterized by measuring the reflectance spectrum using a microspectrophotometer. Because the basal part of the scale was observed to be glittering, such positions were examined for several scales, as shown in the inset of figure 5. The obtained spectra show a strong reflection band in the wavelength range of 520–600 nm. Subsequently, to theoretically understand the reflection wavelength, we calculated the photonic band diagram assuming the structure of the I-WP surface-based photonic crystal. The refractive index of the cuticle was assumed to be 1.555, which was determined using the Becke line test, and the parameter t = 0.19 was used in equation (2.1). The calculated photonic-band diagram shown in figure 6 confirms that the photonic band gap exists at the N point ([110] direction); furthermore, the estimated wavelength range corresponding to this gap, which is shown as a green shade in figure 5, agrees well with the reflection band. The photonic bandgap frequency generally differs depending on the direction. Thus, we conclude that the orientation preference of the (110) surface of the I-WP photonic crystal caused the green colour of the longhorn beetle.

    Figure 5.

    Figure 5. Reflectance spectra obtained for 10 scales using a microspectrophotometer. The green shade shows that the wavelength range corresponds to the frequency of the photonic band gap, as shown in figure 5. The inset shows the examined positions of the scale, which are chosen to be the basal part of the scales as the basal part looks brightly reflective. Scale bar, 20 μm.

    Figure 6.

    Figure 6. Photonic band diagram for the I-WP minimal surface based photonic crystal. A set of horizontal dotted lines in the Γ–N direction indicates the position of the photonic band gap. The parameter t is assumed to be 0.19, and the refractive index of the cuticle is 1.555.

    4. Discussion

    We investigated in detail the structure of the scale of the longhorn beetle Sternotomis callais and found three interesting features. First, the cuticle network can be modelled as a photonic crystal based on the I-WP minimal surface. Second, when the basal part of the scale was sectioned, the (110) planes were observed to preferably face upward. Third, the crystal orientations in the adjacent crystal domains are not random; they are often observed to possess a certain relationship with each other. Here, we discuss the importance and implications of these findings, combined with the previous knowledge of biological photonic crystals.

    The I-WP photonic structure has a body-centred cubic (BCC) lattice with a space group of Im3¯m symmetry. This structural assignment is different from the previously reported Pm3¯m symmetry using the SAXS experiment [21]. However, it is generally difficult to distinguish these two space groups from the SAXS patterns because the difference in the scattering pattern only appears in the presence or absence of the peaks with higher Millar index planes, which can often be broadened because of structural irregularities and buried in the background. Thus, our results might not contradict with the previous study.

    The I-WP minimal surface is different from the three major minimal surfaces, primitive, gyroid, and diamond surfaces, which are abbreviated as P, G and D surfaces, respectively, in terms of the volume fraction of the two subspaces. In the three major surfaces, two subspaces separated by the minimal surface have equal volume fractions (=0.5), and they are symmetrically related to each other. For the P and D surfaces, one of the two subspaces can be translated to overlap with the other. In the case of the G surface, one subspace is related to the other by the inversion operation, and thus, the two subspaces have opposite chirality. However, the I-WP surface separates the space into two subspaces with different volume fractions, and the two subspaces are completely different from each other; thus, they are not related to symmetrical operation. In fact, the name I-WP is derived from the two different networks of I and WP; the letter I derived from expressing the body centre symmetry and WP derived from a ‘wrapped package’ from its similarity in the pattern on the faces of the cube (see electronic supplementary material, figure S7 for the difference between I and WP subspaces) [31]. In this longhorn beetle, it was observed that the I-subspace was filled with a cuticle and the WP-subspace corresponded to air.

    The asymmetry between the two subspaces poses a question regarding the developmental scenario as follows. It has been considered that the pair of the SER and plasma membranes in the butterfly wing scale are folded to form the double gyroid surface, and the nascent cuticle is secreted into the extracellular space, which later becomes the dried cuticle network [7]. In contrast, in the case of the longhorn beetles, based on the high volume fractions of cuticle, it has been hypothesized that the plasma membrane only invaginates to form the template of the cuticle secretion [21]. The volume fraction determined in this study is 0.44, which is higher than that of butterfly wing scales and is consistent with the previous study. Thus, the new question is to determine why the specific subspace (I-subspace) can always be the extracellular space when the plasma membrane invaginates. The expression of membrane proteins may control the membrane curvature, but the curvature alone does not specify one of the two possible subspaces to become the extracellular space. In the case of the butterfly wing scale, both chiralities of the gyroid structure have been observed in different crystal domains on a single scale, although the left-handed gyroid is more abundant than the other [18,32]. This implies that both the subspaces of the gyroid structure can be the extracellular space. However, in this longhorn beetle, there exist certain unknown factors that cause the extracellular space to be the I-subspace and the intracellular space to be WP-subspace.

    The second finding of the present study is the preference for the (110) crystal plane in the basal part of the scale. This feature directly strengthens the green reflection of the scale. Such an orientation preference has been observed in the butterfly wing scale [33,34] and a weevil species [17], although the orientation-selection mechanisms during development are unknown. In addition, as the third finding, two adjacent crystal domains were observed to possess a particular orientation with the domain boundary, as shown in figure 3b. The direction along the oval holes was found to be approximately 35° with respect to the domain boundary; thus the angle between the two lines, which are along the oval holes in the two domains, becomes approximately 70°. By theoretically analysing the cuticle pattern in the (110) plane, it was found that adjacent domains share the (11¯2) planes as their boundary, as shown in figure 7 ([11¯2] direction is perpendicular to both the [110] and [1¯11] directions). Thus, this orientation relation indicates that the two neighbouring crystals developed as twin crystals. The geometry of the observed twin crystal is an example that can be explained by the theory of the coincident site lattice [35], wherein the coincident overlap of the lattice point is considered when the parent crystal lattice is rotated. In fact, for a cubic crystal, the rotation along the axis of the [110] direction by 70.5° is a case named Σ3 wherein the density of the coincident sites becomes high, indicating a lower energy of the domain boundary.

    Figure 7.

    Figure 7. Geometry of twin crystal. (a) The (110) plane is shown in a cubic unit cell. (b) Two vectors in the (110) plane are shown, and the angle between them is 35.3°. (c) Twin crystal. The crystal orientation of the upper part is rotated by 2 × 35.3° from that of the lower crystal. The rotation axis is along the [110] direction (normal to the paper).

    However, such twin crystals have not been observed on the butterfly wing scale. Wilts et al. reported a detailed study of the butterfly Thecla opisena [18], showing that each gyroid crystal domain is developed as a single crystal that grows from a nucleus and the orientations are not related in different domains. In the case of this longhorn beetle, it is hypothesized that nucleation does not occur at many points, but lesser nucleation results in a larger crystal domain. Furthermore, during the growth process, the direction of growth is bent to form a twin crystal (growth twin). However, we cannot exclude the possibility that a developed single crystal becomes a twin crystal owing to the external force (deformation twin).

    The cuticle network of the experimentally determined unit cells matches that of the I-WP photonic crystal (figure 4 and electronic supplementary material, figure S5). However, as the surfaces of the unit cells do not look perfectly smooth, they are not exactly the I-WP surface. In addition, the photonic crystal structure has irregularities such as bond length distribution and deformation of the lattice. Research into these irregular aspects will be important for better understanding of the developmental process.

    5. Conclusion

    This study revealed the structural characteristics of photonic crystals in the longhorn beetle Sternotomis callais. We showed that the physical origin of the structural colour is the photonic bandgap in the Γ–N direction of the I-WP minimal surface-based photonic crystal. However, the unbalanced nature of the I-WP minimal surface provides us with new challenging questions regarding the development of biological photonic crystals. Further, twin crystals were observed, indicating that the development of the multidomain crystal was different from that of the butterfly wing scale.

    Data accessibility

    All other data needed to evaluate the conclusions in the paper are present in the paper and/or the electronic supplementary materials.

    Authors' contributions

    All authors planned the study. Y.K. carried out the experiments. Y.K. and R.O. carried out the data analysis. S.Y. wrote the manuscript and all the authors reviewed it.

    Competing interests

    We declare we have no competing interests.

    Funding

    This study was supported by the Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientific Research no. 18H01191).

    Acknowledgements

    We thank Dr Tawara for the assistance with sequential FIB milling and SEM imaging.

    Footnotes

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.5680556.

    Published by the Royal Society. All rights reserved.

    References