Hexagonal Patterning of the Insect Compound Eye: Facet Area Variation, Defects, and Disorder

Tending to the Facial Surfaces of a Mathematical Biology Head-Scratcher: Why Does the Head of the Sea Turtle Natator depressus Resemble a Convex Zygomorphic Dodecahedron?

Two convex polyhedra that markedly resemble the head of the flatback sea turtle hatchling are identified. The first example is a zygomorphic tetragonal dodecahedron, while the other, an even better matching structure, is a related tetradecahedron, herein speculated to arise from this particular dodecahedron via known mechanisms gleaned from studies of the behavior of foams. A segmented, biomorphic, convex polyhedral model to address cephalic topology is thus presented stemming from solid geometry, anatomical observations, and a recently computed densest local packing arrangement of fifteen slightly oblate spheroids in which fourteen oblate spheroids surround a central such spheroid. This particular array of oblate spheroids shares salient structural features with the aforementioned dodecahedron. Successful testing of the model has been achieved by converting this array of fifteen oblate spheroids constructed with putty to the cephaloid dodecahedron in a process involving ventral elongation induced by stretching in the anterior direction along the anteroposterior axis (convergent extension). During convergent extension, the two left most anterolateral oblate spheroids that are in direct contact with the ventral spheroid of the array merge into a single lateral facet of the incipient dodecahedron, while the corresponding two right such oblate spheroids do the same. Thus, the fourteen outer oblate spheroids of the array give rise to the twelve facets of the finalized dodecahedron, while the central oblate spheroid remnant assumes an interior dodecahedral position. The hypothetical dodecahedron to tetradecahedron transformation entails the collapse of a tetravalent vertex (which is known to occur in foams as part of a T1 transition) followed by bilateral facet splitting. Remarkably, a model stipulating that convexity is to be retained in connection with this sequence of steps necessitates that the starting dodecahedral template undergoes modification to become a tetradecahedron in possession of precisely the highly ordered feature found at the top of the head of numerous specimens of the flatback sea turtle hatchling, namely, a fused medial pentagon-heptagon pair in the form of a pentagonal frontal scute and heptagonal frontoparietal scute. Such a possible new instance of geometric biomorphy, taken together with the correct anticipation of the cephalic pentagon-heptagon pair, might serve to instill further confidence in renewed efforts to shed light on morphogenesis with foam embryo models.

A nuclear jamming transition in vertebrate organogenesis

Jamming of cell collectives and associated rigidity transitions have been shown to play a key role in tissue dynamics, structure and morphogenesis. Cellular jamming is controlled by cellular density and the mechanics of cell-cell contacts. However, the contribution of subcellular organelles to the physical state of the emergent tissue is unclear. Here we report a nuclear jamming transition in zebrafish retina and brain tissues, where physical interactions between highly packed nuclei restrict cellular movements and control tissue mechanics and architecture. Computational modelling suggests that the nuclear volume fraction and anisotropy of cells control the emerging tissue physical state. Analysis of tissue architecture, mechanics and nuclear movements during eye development show that retina tissues undergo a nuclear jamming transition as they form, with increasing nuclear packing leading to more ordered cellular arrangements, reminiscent of the crystalline cellular packings in the functional adult eye. Our results reveal an important role of the cell nucleus in tissue mechanics and architecture.

From soap bubbles to multicellular organisms: Unraveling the role of cell adhesion and physical constraints in tile pattern formation and tissue morphogenesis

Tile patterns, in which numerous cells are arranged in a regular pattern, are found in a variety of multicellular organisms and play important functional roles. Such regular arrangements of cells are regulated by various cell adhesion molecules. On the other hand, cell shape is also known to be regulated by physical constraints similar to those of soap bubbles. In particular, circumference minimization plays an important role, and cell adhesion negatively affects this process, thereby regulating tissue morphogenesis based on physical properties. Here, we focus on the Drosophila compound eye and the mouse auditory epithelium, and summarize the mechanisms of tile pattern formation by cell adhesion molecules such as cadherins, Irre Cell Recognition Modules (IRMs), and nectins. Phenomena that cannot be explained by physical stability based on cortical tension alone have been reported in the tile pattern formation in the compound eye, suggesting that previously unexplored forces such as cellular concentric expansion force may play an important role. We would like to summarize perspectives for future research on the mechanisms of tissue morphogenesis.

Diffractive lensless imaging with optimized Voronoi-Fresnel phase

Lensless cameras are a class of imaging devices that shrink the physical dimensions to the very close vicinity of the image sensor by replacing conventional compound lenses with integrated flat optics and computational algorithms. Here we report a diffractive lensless camera with spatially-coded Voronoi-Fresnel phase to achieve superior image quality. We propose a design principle of maximizing the acquired information in optics to facilitate the computational reconstruction. By introducing an easy-to-optimize Fourier domain metric, Modulation Transfer Function volume (MTFv), which is related to the Strehl ratio, we devise an optimization framework to guide the optimization of the diffractive optical element. The resulting Voronoi-Fresnel phase features an irregular array of quasi-Centroidal Voronoi cells containing a base first-order Fresnel phase function. We demonstrate and verify the imaging performance for photography applications with a prototype Voronoi-Fresnel lensless camera on a 1.6-megapixel image sensor in various illumination conditions. Results show that the proposed design outperforms existing lensless cameras, and could benefit the development of compact imaging systems that work in extreme physical conditions.

Structural Measures as Guides to Ultrastable States in Overjammed Packings

Jammed, disordered packings of given sets of particles possess a multitude of equilibrium states with different mechanical properties. Identifying and constructing desired states, e.g., of superior stability, is a complex task. Here, we show that in two-dimensional particle packings the energy of all metastable states (inherent structures) is reliably classified by simple scalar measures of local steric packing. These structural measures are insensitive to the particle interaction potential and so robust that they can be used to guide a modified swap algorithm that anneals polydisperse packings toward low-energy metastable states exceptionally fast. The low-energy states are extraordinarily stable against applied shear, so that the approach also efficiently identifies ultrastable packings.

EyeVolve, a modular PYTHON based model for simulating developmental eye type diversification

Vision is among the oldest and arguably most important sensory modalities for animals to interact with their external environment. Although many different eye types exist within the animal kingdom, mounting evidence indicates that the genetic networks required for visual system formation and function are relatively well conserved between species. This raises the question as to how common developmental programs are modified in functionally different eye types. Here, we approached this issue through EyeVolve, an open-source PYTHON-based model that recapitulates eye development based on developmental principles originally identified in Drosophila melanogaster. Proof-of-principle experiments showed that this program’s animated timeline successfully simulates early eye tissue expansion, neurogenesis, and pigment cell formation, sequentially transitioning from a disorganized pool of progenitor cells to a highly organized lattice of photoreceptor clusters wrapped with support cells. Further, tweaking just five parameters (precursor pool size, founder cell distance and placement from edge, photoreceptor subtype number, and cell death decisions) predicted a multitude of visual system layouts, reminiscent of the varied eye types found in larval and adult arthropods. This suggests that there are universal underlying mechanisms that can explain much of the existing arthropod eye diversity. Thus, EyeVolve sheds light on common principles of eye development and provides a new computational system for generating specific testable predictions about how development gives rise to diverse visual systems from a commonly specified neuroepithelial ground plan.

Tiling mechanisms of the Drosophila compound eye through geometrical tessellation

Tiling patterns are observed in many biological structures. The compound eye is an interesting example of tiling and is often constructed by hexagonal arrays of ommatidia, the optical unit of the compound eye. Hexagonal tiling may be common due to mechanical restrictions such as structural robustness, minimal boundary length, and space-filling efficiency. However, some insects exhibit tetragonal facets.^1-4 Some aquatic crustaceans, such as shrimp and lobsters, have evolved with tetragonal facets.^5-8 Mantis shrimp is an insightful example as its compound eye has a tetragonal midband region sandwiched between hexagonal hemispheres.^9,10 This casts doubt on the naive explanation that hexagonal tiles recur in nature because of their mechanical stability. Similarly, tetragonal tiling patterns are also observed in some Drosophila small-eye mutants, whereas the wild-type eyes are hexagonal, suggesting that the ommatidial tiling is not simply explained by such mechanical restrictions. If so, how are the hexagonal and tetragonal patterns controlled during development? Here, we demonstrate that geometrical tessellation determines the ommatidial tiling patterns. In small-eye mutants, the hexagonal pattern is transformed into a tetragonal pattern as the relative positions of neighboring ommatidia are stretched along the dorsal-ventral axis. We propose that the regular distribution of ommatidia and their uniform growth collectively play an essential role in the establishment of tetragonal and hexagonal tiling patterns in compound eyes.

Shaping an optical dome: The size and shape of the insect compound eye

The insect compound eye is the most abundant eye architecture on earth. It comes in a wide variety of shapes and sizes, which are exquisitely adapted to specific ecosystems. Here, we explore the organisational principles and pathways, from molecular to tissular, that underpin the building of this organ and highlight why it is an excellent model system to investigate the relationship between genes and tissue form. The compound eye offers wide fields of view, high sensitivity in motion detection and infinite depth of field. It is made of an array of visual units called ommatidia, which are precisely tiled in 3D to shape the retinal tissue as a dome-like structure. The eye starts off as a 2D epithelium, and it acquires its 3D organisation as ommatidia get into shape. Each ommatidium is made of a complement of retinal cells, including light-detecting photoreceptors and lens-secreting cells. The lens cells generate the typical hexagonal facet lens that lies atop the photoreceptors so that the eye surface consists of a quasi-crystalline array of these hexagonal facet-lenses. This array is curved to various degree, depending on the size and shape of the eye, and on the region of the retina. This curvature sets the resolution and visual field of the eye and is determined by i) the number and size of the facet lens - large ommatidial lenses can be used to generate flat, higher resolution areas, while smaller facets allow for stronger curvature of the eye, and ii) precise control of the inter facet-lens angle, which determines the optical axis of the each ommatidium. In this review we discuss how combinatorial variation in eye primordium shape, ommatidial number, facet lens size and inter facet-lens angle underpins the wide variety of insect eye shapes, and we explore what is known about the mechanisms that might control these parameters.

Predicting the characteristics of defect transitions on curved surfaces

The energetically optimal position of lattice defects on intrinsically curved surfaces is a complex function of shape parameters. For open surfaces, a simple condition predicts the critical size for which a central disclination yields lower energy than a boundary disclination. In practice, this transition is modified by activation energies or more favorable intermediate defect positions. Here it is shown that these transition characteristics (continuous or discontinuous, first or second order) can also be inferred from analytical, general criteria evaluated from the surface shape. A universal scale of activation energy is found, and the criteria are generalized to predict transition order as surface shape symmetry is broken. The results give practical insight into structural transitions to disorder in many cellular materials of technological and biological importance.

Defect patterns on the curved surface of fish retinae suggest a mechanism of cone mosaic formation

The outer epithelial layer of zebrafish retinae contains a crystalline array of cone photoreceptors, called the cone mosaic. As this mosaic grows by mitotic addition of new photoreceptors at the rim of the hemispheric retina, topological defects, called "Y-Junctions", form to maintain approximately constant cell spacing. The generation of topological defects due to growth on a curved surface is a distinct feature of the cone mosaic not seen in other well-studied biological patterns like the R8 photoreceptor array in the Drosophila compound eye. Since defects can provide insight into cell-cell interactions responsible for pattern formation, here we characterize the arrangement of cones in individual Y-Junction cores as well as the spatial distribution of Y-junctions across entire retinae. We find that for individual Y-junctions, the distribution of cones near the core corresponds closely to structures observed in physical crystals. In addition, Y-Junctions are organized into lines, called grain boundaries, from the retinal center to the periphery. In physical crystals, regardless of the initial distribution of defects, defects can coalesce into grain boundaries via the mobility of individual particles. By imaging in live fish, we demonstrate that grain boundaries in the cone mosaic instead appear during initial mosaic formation, without requiring defect motion. Motivated by this observation, we show that a computational model of repulsive cell-cell interactions generates a mosaic with grain boundaries. In contrast to paradigmatic models of fate specification in mostly motionless cell packings, this finding emphasizes the role of cell motion, guided by cell-cell interactions during differentiation, in forming biological crystals. Such a route to the formation of regular patterns may be especially valuable in situations, like growth on a curved surface, where the resulting long-ranged, elastic, effective interactions between defects can help to group them into grain boundaries.

The evolution and development of eye size in flies

The compound eyes of flies exhibit striking variation in size, which has contributed to the adaptation of these animals to different habitats and their evolution of specialist behaviors. These differences in size are caused by differences in the number and/or size of ommatidia, which are specified during the development of the retinal field in the eye imaginal disc. While the genes and developmental mechanisms that regulate the formation of compound eyes are understood in great detail in the fruit fly Drosophila melanogaster, we know very little about the genetic changes and mechanistic alterations that lead to natural variation in ommatidia number and/or size, and thus overall eye size, within and between fly species. Understanding the genetic and developmental bases for this natural variation in eye size not only has great potential to help us understand adaptations in fly vision but also determine how eye size and organ size more generally are regulated. Here we explore the genetic and developmental mechanisms that could underlie natural differences in compound eye size within and among fly species based on our knowledge of eye development in D. melanogaster and the few cases where the causative genes and mechanisms have already been identified. We suggest that the fly eye provides an evolutionary and developmental framework to better understand the regulation and diversification of this crucial sensory organ globally at a systems level as well as the gene regulatory networks and mechanisms acting at the tissue, cellular and molecular levels. This article is categorized under: Establishment of Spatial and Temporal Patterns > Regulation of Size, Proportion, and Timing Invertebrate Organogenesis > Flies Comparative Development and Evolution > Regulation of Organ Diversity.

Characterization of the Genetic Architecture Underlying Eye Size Variation Within Drosophila melanogaster and Drosophila simulans

The compound eyes of insects exhibit striking variation in size, reflecting adaptation to different lifestyles and habitats. However, the genetic and developmental bases of variation in insect eye size is poorly understood, which limits our understanding of how these important morphological differences evolve. To address this, we further explored natural variation in eye size within and between four species of the Drosophila melanogaster species subgroup. We found extensive variation in eye size among these species, and flies with larger eyes generally had a shorter inter-ocular distance and vice versa We then carried out quantitative trait loci (QTL) mapping of intra-specific variation in eye size and inter-ocular distance in both D. melanogaster and D. simulans This revealed that different genomic regions underlie variation in eye size and inter-ocular distance in both species, which we corroborated by introgression mapping in D. simulans This suggests that although there is a trade-off between eye size and inter-ocular distance, variation in these two traits is likely to be caused by different genes and so can be genetically decoupled. Finally, although we detected QTL for intra-specific variation in eye size at similar positions in D. melanogaster and D. simulans, we observed differences in eye fate commitment between strains of these two species. This indicates that different developmental mechanisms and therefore, most likely, different genes contribute to eye size variation in these species. Taken together with the results of previous studies, our findings suggest that the gene regulatory network that specifies eye size has evolved at multiple genetic nodes to give rise to natural variation in this trait within and among species.

Threshold response to stochasticity in morphogenesis

During development of biological organisms, multiple complex structures are formed. In many instances, these structures need to exhibit a high degree of order to be functional, although many of their constituents are intrinsically stochastic. Hence, it has been suggested that biological robustness ultimately must rely on complex gene regulatory networks and clean-up mechanisms. Here we explore developmental processes that have evolved inherent robustness against stochasticity. In the context of the Drosophila eye disc, multiple optical units, ommatidia, develop into crystal-like patterns. During the larva-to-pupa stage of metamorphosis, the centers of the ommatidia are specified initially through the diffusion of morphogens, followed by the specification of R8 cells. Establishing the R8 cell is crucial in setting up the geometric, and functional, relationships of cells within an ommatidium and among neighboring ommatidia. Here we study an PDE mathematical model of these spatio-temporal processes in the presence of parametric stochasticity, defining and applying measures that quantify order within the resulting spatial patterns. We observe a universal sigmoidal response to increasing transcriptional noise. Ordered patterns persist up to a threshold noise level in the model parameters. In accordance with prior qualitative observations, as the noise is further increased past a threshold point of no return, these ordered patterns rapidly become disordered. Such robustness in development allows for the accumulation of genetic variation without any observable changes in phenotype. We argue that the observed sigmoidal dependence introduces robustness allowing for sizable amounts of genetic variation and transcriptional noise to be tolerated in natural populations without resulting in phenotype variation.

A simple landscape of metastable state energies for two-dimensional cellular matter

The mechanical behavior of cellular matter in two dimensions can be inferred from geometric information near its energetic ground state. Here it is shown that the much larger set of all metastable state energies is universally described by a systematic expansion in moments of the joint probability distribution of size (area) and topology (number of neighbors). The approach captures bounds to the entire range of metastable state energies and quantitatively identifies any such state. The resulting energy landscape is invariant across different classes of energy functionals, across simulation techniques, and across system polydispersities. The theory also finds a threshold in tissue adhesion beyond which no metastable states are possible. Mechanical properties of cellular matter in biological and technological applications can thus be identified by visual information only.

Mechanics of epithelial tissue formation

A key process in the life of any multicellular organism is its development from a single egg into a full grown adult. The first step in this process often consists of forming a tissue layer out of randomly placed cells on the surface of the egg. We present a model for generating such a tissue, based on mechanical interactions between the cells, and find that the resulting cellular pattern corresponds to the Voronoi tessellation of the nuclei of the cells. Experimentally, we obtain the same result in both fruit flies and flour beetles, with a distribution of cell shapes that matches that of the model, without any adjustable parameters. Finally, we show that this pattern is broken when the cells grow at different rates.

Universal Features of Metastable State Energies in Cellular Matter

Mechanical equilibrium states of cellular matter are overwhelmingly metastable and separated from each other by topology changes. Using theory and simulations, it is shown that for a wide class of energy functionals in 2D, including those describing tissue cell layers, local energy differences between neighboring metastable states as well as global energy differences between initial states and ground states are governed by simple, universal relations. Knowledge of instantaneous length of an edge undergoing a T1 transition is sufficient to predict local energy changes, while the initial edge length distribution yields a successful prediction for the global energy difference. An analytical understanding of the model parameters is provided.

Geometric constraints alter cell arrangements within curved epithelial tissues

Organ and tissue formation are complex three-dimensional processes involving cell division, growth, migration, and rearrangement, all of which occur within physically constrained regions. However, analyzing such processes in three dimensions in vivo is challenging. Here, we focus on the process of cellularization in the anterior pole of the early Drosophila embryo to explore how cells compete for space under geometric constraints. Using microfluidics combined with fluorescence microscopy, we extract quantitative information on the three-dimensional epithelial cell morphology. We observed a cellular membrane rearrangement in which cells exchange neighbors along the apical-basal axis. Such apical-to-basal neighbor exchanges were observed more frequently in the anterior pole than in the embryo trunk. Furthermore, cells within the anterior pole skewed toward the trunk along their long axis relative to the embryo surface, with maximum skew on the ventral side. We constructed a vertex model for cells in a curved environment. We could reproduce the observed cellular skew in both wild-type embryos and embryos with distorted morphology. Further, such modeling showed that cell rearrangements were more likely in ellipsoidal, compared with cylindrical, geometry. Overall, we demonstrate that geometric constraints can influence three-dimensional cell morphology and packing within epithelial tissues.