Checkerboard color patterns are rare but present in many diverse taxa of Eukaryotes, notably plant and animal species

To perform a survey of species bearing checkerboard patterns, we first had to define them properly. We chose to consider that three characteristics are necessary and sufficient: i) motifs organized in square lattices (i.e., when each unit has 4 closest neighbors), ii) square-shaped motifs and iii) an alternation of two colors or shades (Fig 1B). Perfect checkerboard patterns are never reached by biological patterns; therefore, we chose to work with a tolerance to variation as biological patterns could not match perfect mathematical ones. Consequently, we include species with slightly diverging characteristics, like unperfect squares or partly irregular color alternation (Fig 1A). Since no exhaustive survey was possible, we chose to use keywords to find the species in the literature and biodiversity databases (like ‘checkerboard’, ‘checkered’, ‘check’, ‘square’ and ‘Fritillaria’, the latter being used in species names to pinpoint the presence of this particular pattern, or one close to it), together with surveys also sent to expert researchers in various taxonomic groups (typically working in natural history museums). Unlike previous studies based on a near-exhaustive examination of certain groups, such as the search for PGTCPs (Putative Growth Turing Colour Patterns) in mammals or orchids [14] this study is based on a keyword search and does not allow for the estimation of a rarity metric for checkerboard patterns beyond stating that they are absent from the 6,649 mammal species and virtually absent from the 29,573 orchid species, except in certain genera such as Bulbophyllum or Huntleya, in forms very different from the perfect theoretical checkerboard (see Fig 1A).

Thanks to this, we described several species and genera, summarized in Fig 1, and qualitatively represented in a diagram thanks to the more or less perfectness of their characteristics. Both Angiosperms and animals were found to exhibit checkerboard-like patterns, but we were not able to find any species belonging to Fungi. Regarding this last point, several hypotheses could be put forward—none of which are mutually exclusive—but we believe this may be linked to the close connections that appear to exist between checkerboard patterns and Turing patterns (which we will discuss in relation to both Fritillaria and Sarcophaga in the following sections) and to the fact that no Turing pattern (periodic spots, stripes or mazes and their PGTCPs derivatives) has ever been described in fungi, particularly on the surface of carpophores. Functional reasons linked to the specific characteristics of fungal tissues have been suggested, potentially making it difficult for a Turing system to emerge in these organisms [14]. However, one type of structure—the hymenophore—could be a candidate for the presence of checkerboard patterns in Fungi, as a study shows that some observed patterns (poroid, odontoid, lamellate or labyrinthic) are consistent with Turing patterns [15]. On the contrary, very diverse animal species (vertebrates, insects, echinoderms, cephalopods, gastropods) exhibiting checkerboards were found, but few plant species seem to bear such patterns. The two most perfect checkerboard-like patterns were found in Sarcophaga flies genus, exhibited on their upper abdomen, and Fritillaria genus on their tepals.

We then chose these two groups to explore checkerboard patterns formation.

Color patterns of the 164 recognized species of Fritillaria exhibit a large diversity of hues and shapes, including checkerboard whose morphogenesis is compatible with a Turing-like mechanism constrained by a parallel nervure frame

Before focusing our study on Fritillaria meleagris, we decided to characterize the flower color patterns of the 164 recognized species according to the Kew Botanical Garden POWO website [16]. By harmonizing the descriptive terms for colors and using various sources of photographs, we described fourteen different colors of backgrounds and motifs (brown, purplish-brown, purple, pink, red, reddish-brown, green, yellowish-green, greenish-yellow, yellow-green, yellow, orange, blue and white, see Fig 2A). Certain color combinations of pattern and background were more common than others, and in an intuitive way, strong contrasts in terms of brightness and complementary colors seem to have been favored during evolution (such as dark red-brown patterns on a light yellowish-green background, see Table 1).

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Table 1. Motif and background color combinations in Fritillaria species. Each number corresponds to the number of recognized species exhibiting this combination.

https://doi.org/10.1371/journal.pone.0352968.t001

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Fig 2. Diversity of colors, shapes and tissue location of Fritillaria color patterns.

A. The 14 harmonized hues of Fritillaria color patterns and their reference pictures. B. Four variations of pattern location, from left to right: peduncle, both sides of petal and tepal, both sides of petal only, internal sides of petal and tepal. C. Presence of a checkerboard pattern and motif and background colors mapped on Fritillaria phylogeny. D. The seven pattern categories and their possible combination with checkerboard patterns. Pictures sources: Courtesy of Laurence Hill (http://www.fritillariaicones.com/icones/Icones.html).

https://doi.org/10.1371/journal.pone.0352968.g002

Furthermore, the diversity of these color patterns is not limited to colors alone, but also extends to shapes, and indeed, the checkerboard pattern is not the only color pattern that exists in Fritillaria tepals. We have described seven families of patterns (Fig 2D), which can be grouped into four categories: small periodic patterns (checkerboard, stains, spots), large area patterns (bull’s eye and striped), vascular network coloration (or related to it) and 3D papillae (not color patterns per se, but visible because of their 3D shape and/or the shadows generated by the reliefs).

Interestingly, combinations of several pattern families exist, but some have never been observed, either for adaptive reasons or because they are mechanistically impossible. In particular, the three patterns of small periodic motifs (checkerboard, stained and spotted) are almost never observed simultaneously in an individual, and even if they are, they are never superimposed. This led us to hypothesize that these three families of patterns are produced by the same morphogenetic mechanism, and that the internal or external parameters of the system determine one or the other output.

Finally, the diversity of these color patterns also extends to their location, as shown in Fig 2B. Some species have a visible pattern on both sepals and petals, others only on petals (the reverse has not been observed), some have patterns only on the inner surfaces of tepals, and surprisingly, in at least one species, the pattern is absent from the tepals but present on the peduncle.

The phylogeny of Fritillaria mapped with these color and shape characters reveals a probable presence of checkerboard pattern in the common ancestor of the genus (Fig 2C). We also explored color patterns in the Fritillaria sister-genus Lilium, together with all their sub-family Lilioideae genera. Many periodic patterns are present but no checkerboard pattern was found, suggesting an evolutionary innovation after the separation of Fritillaria and Lilium lineages:

Tribe: Medeoleae

Clintonia: No particular color pattern found

Medeola: No particular color pattern found on flowers but reddish bullseyes on bractea in some species like Medeola virginiana

Tribe: Lilieae

Cardiocrinum: Patterns on the base of the tepals, with spotted and vein patterns.

Erythronium: Patterns on the base of the tepals, spotted patterns. Spotted or network patterns on the leaves (e.g., Erythronium dens-canis)

Fritillaria: (detailed earlier)

Gagea: Patterns on the base of the tepals, vein patterns (e.g., Gagea serotina)

Lilium: Spots patterns only above the veins, in particular above vein intersections

Nomocharis: PGTCPs (Putative-Growth Turing-like Color Patterns), including rosettes (like in Nomocharis aperta).

Notholirion: Patterns on the base of the tepals with a small spot, sometimes shaped like a circumflex accent (e.g., Notholirion thomsonianum)

Tulipa: Bullseye patterns

To extract clues on the morphogenetic processes producing Fritillaria checkerboard-like patterns, we performed observations on fresh and dissected individuals presenting pattern variability on color and shape (Fig 2A-C). Fig 2D (and the corresponding numbered arrows in Fig 2A-C) summarizes the results in nine pattern observations organized in three axes with their respective deductions and hypotheses: i) organ and tissue location, ii) relationships with vein patterning and iii) geometry characteristics.

Concerning the first axe, we found that in F. meleagris the checkerboard pattern (at least its purplish colored part) is borne by the vacuoles of the puzzle-shaped epidermal cells (Fig 3C and 3D), in both upper and lower epidermis (Fig 3C and 3B). Interestingly, upper and lower patterns are superimposed (Fig 3C, right), suggesting a common morphogenesis, either in one or other of these tissues, in both, or even in a third tissue, such as the precursor of the parenchyma which separates the two epidermises. Because the lacunar nature of the spongy parenchyma [17] seems unsuitable for the establishment of a continuous pattern such as the checkerboard pattern of Fritillaria, we therefore hypothesize that it is established quite early during tepal development, before the differentiation of the parenchyma. Since the colors are produced late, this also implies a pre-pattern, coded in a way that remains to be determined.

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Fig 3. Fritillaria checkerboard patterns.

A. Pictures of Fritillaria meleagris from a wild population of Eastern Britanny, France (left) and color diversity from the same population (right). B. Comparative pictures of sepals (first row) and petals (second row) from two individuals (red morph and white morph), with the respective abaxial/outside and adaxial/inside sides. C. Magnifications of Fritillaria thunbergii (two left pictures) and Fritillaria meleagris (four right pictures) color patterns. D. The nine pattern characteristics with their respective deductions and hypotheses. Numbered blue arrows are dispatched on A-C panels to illustrate the observations. E. Summary of Fritillaria meleagris checkerboard pattern morphogenesis hypotheses. F. Modeling of Fritillaria checkerboard pattern by constraining a Turing pattern in a parallel stripes mask. From left to right: system equations, initial and boundary conditions, modeling without constraint, shape of the constraint mask, modeling with the constraint.

https://doi.org/10.1371/journal.pone.0352968.g003

Next, we described a clear link between checkerboard and vein patterns: lateral borders of checkerboard squares exclusively coincide with veins, which are sub-parallel in most of the tepals’ surface (Fig 3C and 3D), a characteristic of many monocots [18]. It remains then to determine whether this correlation is a sign of chance, of a common inheritance of a third structure that remains currently unknown, or, as we hypothesize, of the influence of one on the other, in this case of the vein pattern on the color pattern. While the vein pattern appears to act as (or to produce) a boundary, it must not be entirely permeable, since the alternation of patterns between two consecutive veins leads us to hypothesize that possible signals can cross (Fig 3C). We have to keep in mind veins are located outside the epidermis, and the latter is continuous so whether they act as a physical, biochemical, or any other type of boundary signal, it will have to be taken into account in the hypotheses. In particular, these observations could acknowledge the hypothesis of a pattern establishment taking place in the parenchyma, as its space is restrained where veins are present.

Finally, while the vein pattern seems to play a direct or indirect role in the final geometry of the checkerboard, it does not seem sufficient to explain all of its characteristics, including the anteroposterior borders of the squares, as well as the alternation of the motifs and their anteroposterior periodicity. To make hypotheses on the characteristics of the morphogenetic process(es) at work, we took advantage of the variations in the geometry of the pattern, whether within the same tepal, or between individuals. We found that the motif geometry of i) the colored areas is a continuum from squares to little triangles via bowties shapes and ii) for the unpigmented areas, from squares to continuous background via truncated disks (Fig 3C and 3D). In some individuals, we observed a very interesting feature in the places where the inter-vein distance is the widest (and/or when the pattern periodic wavelength is smaller): a non-colored spot pattern on a colored background is observed instead. Given the periodicity of this pattern, and the round geometry of the motifs, the variability and the complementarity between colored and non-colored regions, it shares all the characteristics of a Turing-like pattern [19]. Indeed, at that time, the latter is the only mechanism demonstrated to produce periodic color dots patterning in Angiosperms [20]. Turing pattern relies classically on reaction and diffusion of morphogens (but can be generalized to other scales and natures [21]), and it seems that veins could act as constraining semi-permeable barriers, which therefore shape the pattern into square-like shapes. This notion of constraint in a Turing system has been demonstrated in zebrafish with the role of a lateral structure transforming a spot pattern into stripes [22], and is believed to occur in the lizard Timon lepidus to form their scale color pattern [23].

Using Visual PDE [24] and a custom-made Turing system based on Schnakenberg equations, we simulated the effect of a parallel striped constraining on Turing pattern formation (Fig 3E). From an initial classical hexagonal lattice, the constraint transforms it into a square lattice, and spots motifs transformed into rectangle-like motifs, leading to an overall checkerboard-like pattern (Fig 3E).

Fig 3. E them sums up the deductions and hypotheses we made from our observations and simulations.

To produce a final checkerboard-like pattern, several mechanisms need to interact: first, the lateral sides of the squares are produced by the straightness and parallelism of the tepal veins. Secondly, the two other sides are more or less straight, depending on the species and the individual, and might depend on the Turing parameters of the system. Finally, the same Turing-like mechanism seems to be involved to produce an alternation of colored and uncolored zones, characteristics of the checkerboard-like patterns.

Sarcophaga checkerboard patterns are relying on a structural coloration generated by planar polarity of hair, compatible with the interference between two stripe patterns

Unlike the genus Fritillaria, the phenotypic diversity of Sarcophaga color patterns is much more limited, both in terms of colors (in this case grey shades, at least for the human eye) and shapes [25]. Furthermore, we observe that many species of other genera of the Sarcophagidae family exhibit patterns that are more or less similar to checkerboards, sometimes without the alternation of dark and light squares, sometimes with spots or triangles instead of squares, or sometimes with only stripes (typically two per segment, aligned from right to left, light in the anterior side and dark in the posterior side). Given this relatively low variability in shapes and colors, we decided against conducting an exhaustive study of the genus and focused instead on a restricted number of species.

Like for Fritillaria checkerboards, we perform an in-depth patterns analysis we then summarized in nine observations grouped in 3 categories (1. Macroscopic observations, 2. Links with setae phenotypes and 3. Correlations between colors and setae phenotype) presented in Fig 3D and in the corresponding arrows (numbered 1–9).

First, the checkerboard is located on the cuticle of the dorsal part of the abdomen (Fig 4A and 4D, arrow 1), which is not the only part with a pattern, as the thorax cuticle has stripes, or even intermediate bands, a category of PGTCPs (Putative-Growth Turing-like Color Patterns). We counted 28 squares, divided into three and a half batches of eight squares, across four segments (Fig 4B and 4D, arrow 2). Some sides of the squares match other structural boundaries: the midline and the boundaries between segments. Then, we describe an unexpected feature of Sarcophaga checkerboards by observing them under a binocular microscope: some squares could change their colors (or more precisely their shade of grey) depending on the incident light direction (Fig 4A and 4D, arrow 3) and the angle of observation. Some squares can change from “light” to “dark” through intermediate shade (6 for each octet of squares), while others remain dark regardless of the angle of incidence (the two located in the posterior and lateral corner). In addition, the midline is marked by a thin ever black stripe, as well as two very thin ever black stripes on either side of the midline and parallel to it, located between the first and second columns of squares. Together, this led us to postulate a “physical” coloration [26], as it depends on the angles of illumination and observation, and a morphogenesis that may rely (at least partially) on existing boundaries, such as the midline, and that use the right-left symmetry of the abdomen.

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Fig 4. Sarcophaga checkerboard patterns.

A. Pictures of Sarcophaga sp. (up) and detail of the abdomen checkerboard pattern from an individual of the MNHN Diptera collection (down). The eight pictures correspond to the same abdomen taken with different incident light angles. A. Correspondence between color and setae hair phenotypes, under a 90 degrees angle of light illumination. Up left picture and fourth row: light microscopy, all the other pictures: SEM. B.Assembly of SEM pictures and superimposition of colors corresponding to the phenotype of the setae of the 2^nd abdomen segment. C. The nine pattern characteristics with their respective deductions and hypotheses. Numbered blue arrows are dispatched on A-C panels to illustrate the observations. D. Summary of Sarcophaga sp. checkerboard pattern morphogenesis hypotheses. Picture source: https://commons.wikimedia.org/wiki/File:Sarcophaga_variagata,_Gop_Hill,_North_Wales,_July_2013_(17168726209).jpg.

https://doi.org/10.1371/journal.pone.0352968.g004

To further explore the origin of this changing pattern composed of grey shades, we took a microscopic look by examining samples of Sarcophaga sp. abdomens under a scanning electron microscope. In particular, we explored the characteristics of the small epidermal hairs covering much of the external surface of the abdomen, called setae, which are then strongly suspected of contributing to the checkerboard pattern.

First, by examining partially damaged museum specimens, we were able to demonstrate that areas where the setae are damaged lose their ability to change their shade and remain constantly dark (Fig 4A and 4D, arrow 4). In contrast, when the large “bristles” (another type of epidermal hair) are damaged, this has no visible effect on the pattern or the coloration. Next, we observed that the setae were organized into large areas characterized by size (large and overlapping, or small and revealing the surface underneath) and preferred direction (also known as planar polarity [27]), and that the boundaries between areas corresponded perfectly to the edges of the checkerboard squares (Fig 4A, 4C and 4D, arrow 5). Finally, we observed a right-left symmetry in these microscopic characteristics, with long setae oriented N-W (~135°), N-E (~225°), W-E (~90°) or E-W (~270°) (Fig 4B and 4D, arrow 6). We can note that the areas closest in terms of direction are located diagonally to each other, which surely plays a role in the alternation of dark and light squares in the checkerboard pattern. There is therefore a perfect correlation between microscopic boundaries based on the setae phenotypes and macroscopic boundaries based on color. To provide clues as to the existence of a causality between micro and macro scales, we explored in detail the microscopic characteristics of setae in relation to the direction of light.

We first observed that the areas that remain dark regardless of the incident light conditions (i.e., the midline, the two parallel lines, and the lateral-posterior squares) always have short setae that do not cover the entire surface, revealing the hairless parts of the epidermis (Fig 4A, 4B and 4D, arrow 7). On the contrary, the squares which change of shade depending on the light are composed of long, lying setae partially overlapping and hiding the hairless part of the epidermis. When incident light rotates through 360 degrees, each zone passes once through a maximum brightness leading to a light square and once through a minimum brightness leading to a dark square, with shades of gray in between (Fig 4A, 4B and 4D, arrow 8). Finally, squares are at maximum brightness when the light points in the same direction, and at minimum brightness when the light is anti-parallel (Fig 4A, 4B and 4D, arrow 9). All these observations lead us to propose the following hypotheses:

The hairless part of the epidermis absorbs light (for example, via melanin pigmentation), while the hairs reflect incident light (without any particular wavelength preference, hence the shades of gray). When light is parallel to the direction of the hairs, it is reflected on their surface and reaches the observer. Conversely, when it is anti-parallel to the hairs, it penetrates inside and beneath the setae network and is absorbed or prevented from escaping, leading to a lack of light to the observer.

This ability to display a change in colour (or rather, shades of grey) depending on the viewing angle appears to bear strong similarities to so-called structural or physical colouration; however, it seem that this is not the case here for several reasons: i) what creates the dark shade could be pigment-based (for example, melanin in the epidermis beneath the setae), and what creates the light shade could be structural (e.g., light scattering on the surface of the setae) so both pigments and structural could be important, but more importantly ii) it is ‘macroscopic’ structures (at least relative to the wavelength of visible light), the setae, that appear to control the switch from dark to light, and not microscopic structures, typically of the same order of magnitude as visible light, as in the case of classic structural colouration. As a result, this is more akin to biological tabula scalata, images with two pictures divided into strips on different sides of a corrugated carrier [28].

Fig 4E sums up the deductions and hypotheses we made from our observations.

First, we simplify the pattern as the repetition of a unit of four independent squares, then doubled using left-right symmetry, then multiplied by 3.5 using the abdomen segmentation, leading finally to 28 squares. The outer boundaries of the 4-square unit all coincide with an existing structural boundary (midline, boundaries between segments and dorso-ventral boundary), so we hypothesize that they are used for the pattern. On the contrary, the two internal perpendicular boundaries do not coincide with any observable structural frontier. However, many species possess lateral stripes on the abdomen [29,30], therefore the mechanism that produces them could be reused. Concerning the anterior-posterior boundaries, we hypothesize that the mechanism that produces the thorax [31] ones could be co-opted here in the abdomen. Finally, the combination of these perpendicular stripes could code for the length and orientation of the setae, which by extension codes for their behavior with respect to the angle of incident light their and therefore for the resulted color (or shade). To propose an initial model of interaction between anteroposterior and lateral bands, we could describe the “long setae” trait as dominant (“+” by genetics analogy) over “short setae” (“-”). This model would explain why, of the four independent squares making up a unit, only one is always black, and it would therefore be located at the intersection of two “black” bands (-/-). Two other squares would be (+/-), and would therefore possess long setae and the ability to change colour, as would the last one, which would be (+/+). A similar line of reasoning could be applied to the preferred orientation angle of the setae, but in the absence of concrete data on the factors influencing planar polarity, we have chosen to leave it at that. A natural next step following this study would be to move to the functional level, using a genetic and cellular approach to decipher the mechanisms of planar polarity at work.

Can checkerboard patterns be evolutionary attractors in relation to the other associated repeated patterns, or are they simply by-products of morphogenetic constraints?

Whilst their geometric characteristics make them easily identifiable by the experimenter alongside other periodic patterns such as spots or stripes, one might ask whether i) checkerboard patterns enable, in some cases, an improvement in the associated biological functions (typically the visibility of the pattern to pollinators in the case of Fritillaria) compared to other patterns, or whether ii) do they offer no particular improvement, are not subject to positive selection compared to other patterns, and does their presence result solely from developmental constraints (such as the presence of a parallel vein pattern in Fritillaria that serves other biological functions)? This study does not claim to provide a definitive answer to these questions but offers arguments that point in one direction or the other. Firstly, in the same way that certain Putative-Growth Turing-like Colour Patterns (PGTCPs [14]), angular patterns such as the squares found in checkerboard patterns are rare in both biotic and abiotic environments, and as a result can make the species displaying them more easily detectable by visual systems compared to surrounding species, in a sort of ‘rarity bonus’. This is in addition to the numerous periodicities found in checkerboard patterns which, as mentioned in the introduction, and like other patterns such as Turing patterns, facilitate their detection by many visual systems, including the human one. By combining several characteristics that may facilitate their detection, we can therefore hypothesise that, at least in certain species, checkerboard patterns may have been selected during evolution over other patterns. Furthermore, in both Fritillaria and Sarcophaga, it appears that several independent processes act in concert to form squares. In Fritillaria, the antero-posterior edges are determined by the vein pattern, whilst the lateral edges are generated by the Turing system. In Sarcophaga, depending on the square in question and the edge of that square, entirely different and seemingly independent processes come into play and act in concert. In this context, it is more difficult to attribute this to chance or the contingency of developmental constraints, and the hypothesis of positive selection for square shapes is strengthened. These questions, at the interface between morphogenesis, evolutionary biology and ecology, will obviously require the design of dedicated functional experiments to resolve them, but we believe they merit further investigation in the future.

Comparing the morphogenesis of Fritillaria and Sarcophaga checkerboards in a standardized flowchart called FORMula

Through this study, it became clear that the morphological similarity of the checkerboards of the two genera studied was not reflected at the level of morphogenetic processes, where evolutionary convergence appears to be at work. We defend the idea that a finite number of morphogenetic processes are responsible for the forms and patterns of the living world, and that these can be defined and classified into universal categories [11,32]. Furthermore, by summarizing the morphogenesis of a form or a pattern into different sequential stages, we can synthesize it using a flowchart, which we propose to call the FORMula (Fig 5A for the general representation and Fig 5B and 5C for Fritillaria and Sarcophaga examples).

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Fig 5. FORMula, a standardized and logical representation of morphogenesis.

Yellow rectangle: shape, pink trapeze and blue bracket-shaped rectangle: morphogenetic processes (respectively generator and modulator). A. General principles of FORMula: visual version (top) and unabridged and abridged text versions (bottom). B. FORMula of Fritillaria meleagris checkerboard pattern (top), equivalent unabridged and abridged textual FORMula for Fritillaria meleagris, and equivalent classic text description (bottom). C. FORMula of Sarcophaga sp. checkerboard pattern. Picture source: https://commons.wikimedia.org/wiki/File:Sarcophaga_variagata,_Gop_Hill,_North_Wales,_July_2013_(17168726209).jpg.

https://doi.org/10.1371/journal.pone.0352968.g005

In this formalism, an elementary morphological event could be visually represented by an arrow: at the start, the initial shape, which is modified by a morphogenetic process (or several acting in concert) represented in the middle along the arrow, and at the end the final shape. Alternatively, in the case of shape creation ex nihilo (e.g., by a Turing-like system), the symbol “∅” could indicate the absence of an initial shape.

We defined two types of morphogenetic processes, generator and modulator (the first one being sufficient to produce a shape contrary to the second one), respectively represented with a pink trapeze and a blue rectangle with brackets. Shapes (which include both forms and patterns, the latter being characterized by two or more internal states possible, typically colors) are represented by a yellow rectangle. If two morphogenetic processes interact to form a shape, the symbol “×” may indicate it, “+” if their effects are additive without interaction. A process may also have a shape, which can itself be produced by other morphogenetic processes (e.g., putative reduced diffusion in Fritillaria possess the shape of the vein sub-parallel network). In this way, two “lineages” of shapes can merge. Furthermore, when two or more mechanisms act at the same time, it can be interesting to separate their respective effects on the final form in the representation. To this end, an “unfolded” version of FORMula could represent this, acting as if they were acting one after the other, rather than all at once. The use of doted lines in the visual representation would be essential as a reminder that the intermediate forms in the unfolded version are the result of thought experiments and not of a real separation of processes during morphogenesis.

For the final shape of interest, here the checkerboard pattern, we detail the three necessary and sufficient characteristics, to be able to precise the respective roles of morphogenetic processes. For example, in Fritillaria, we hypothesize that the reduced diffusion caused by the vein network is not responsible for color alternation, but for both the square lattice and the square shapes of the motifs. Putative Turing-like process participates to the establishment of all of the three characteristics.

All of these visual representations could also be transcribed into a character string with the same amount of essential information (Fig 5A and 5B, bottom), to be processed by an algorithm, for example. In this textual version of the FORMula, by visual analogy, shapes could be transcribed to « [shape name] », generators to « /generator name\ » and modifiers to « {modifier name} ». Descriptive names of processes and illustrations are excluded for clarity reasons in the textual FORMula. Fig 5B also presents an example of conventional text that would contain the same amount of morphogenetic information, for comparison. It seems reasonable to assume that the advantages of clarity (in the visual version of the FORMula) and conciseness (in the textual versions of the FORMula) over a conventional text are all the greater when a significant number of forms and processes, as well as interactions, come into play.

By extension, we hypothesize that the entire morphological development of an individual could then be represented through FORMula, and an appropriate name might be “morphogenetic tree” by analogy with phylogenetic trees and cell lineage trees. Of course, this representation is an intellectual construct of processes that are more complex than the information contained in the FORMula and, moreover, until functional studies provide direct evidence, many areas of the diagram remain at the hypothesis stage. Nevertheless, it should allow knowledge and hypotheses to be synthesized while clearly differentiating between them, and facilitates comparative analyses between species. Finally, like in this study, FORMula allows to pinpoint the links between different shapes to be directly represented. We hypothesize that many biological structures, once formed, serve by their own shape as constraints, bases, boundaries, and/or signals for the morphogenesis of other biological structures. This intertwining of shapes is understudied in morphogenesis, particularly due to the obvious focus on ‘generator’ morphogenetic processes in the first instance, whereas what we have referred to above are more like ‘modulators’ of shapes. To further explore the possibilities and limits of this logical representation and the potential ways of improving it, we therefore must test it on as many examples as possible, extracted from the literature of functional morphogenesis studies.

Scanning Electron Microscopy

A total of three abdomens were sampled from Sarcophaga sp. museum specimens (National Museum of Natural History, Paris, France). Samples were mounted on the aluminum stubs with colloidal graphite, sputter-coated with platinum using an EM ACE600 fine coater (Leica, Wetzlar, Germany), and observed using a SU3500 scanning electron microscope (SEM, Hitachi, Tokyo, Japan).

Light Microscopy and photography

Fritillaria thunbergii: a total of 20 tepals were sampled from 8 specimens of Fritillaria thunbergii, harvested in the botanical gardens of the University of Tokyo (Koishikawa Botanical Gardens, Tokyo, Japan). Tepals gas were extracted by a vacuum pump to make the tissues more transparent and suitable for observation. Abaxial and adaxial sides were mounted in water and observed under light microscopy.

Fritillaria meleagris, herbarium specimens: A total of 8 tepals were sampled from 3 specimens of Fritillaria meleagris, from museum specimens (National Museum of Natural History, Paris, France), after a survey of hundreds of museum specimens. They were rehydrated and observed under light microscopy.

Fritillaria meleagris, fresh specimens: A total of 31 specimens were harvested from a wild population in Eastern Britanny, France. They were chosen to be representative of all the developmental stages of tepal morphogenesis and to present as much color and shape variability as possible. All tepals (8 per flower, one flower per specimen) were mounted in water between two glass sheets and photographed on both abaxial and adaxial sides with scales. 16 sepals and 16 petals were selected to be mounted in distilled water between a slide and a cover slip. They were observed under a binocular microscope. To observe independently abaxial and adaxial epidermis of the same tepal, we separate them with the sharp edge of a glass cover slip.

Biodiversity surveys among Eukaryotes and Fritillaria genus

In order to take as comprehensive an approach as possible, we used multiple sources: first, scientific literature, using keywords such as “checkerboard,” “checkered,” “check,” “square,” and “Fritillaria” with the word “pattern,” then adding the names of clades in Latin or English, such as ‘insecta’ or “insect.” We then consulted several experts from different taxonomic groups, particularly those working at the National Museum of Natural History of Paris, France. Finally, we use secondary sources like online encyclopedias, Wikipedia (https://en.wikipedia.org/) and Wikimedia Commons, and image browsers. For each candidate species, we double checked with scientific sources. In addition, some groups had already been explored for another study focused on other patterns, PGTCPs, but since the author had already begun this study on checkerboards, he systematically noted those he found. Thus, the 6,500 species of mammals were explored, as were the 30,000 species of orchids, and all families of angiosperms and orders of animals were explored using keywords. The fact that we found checkerboard patterns in small numbers but in groups that are highly representative of the macroscopic biodiversity of angiosperms and animals reinforces our belief that we have not overlooked too large a proportion of the clades exhibiting these patterns.

Phylogenetic relationships between species were built with phyloT (V2, https://phylot.biobyte.de/) based on NCBI taxonomy. Phylogenetic trees were plotted and annotated in iTOL (Interactive Tree Of Life, version 6.8.1, https://itol.embl.de/)[33]

Modeling

All the model parameters are available at: https://visualpde.com/sim/?mini=lR1qNaKv

We use the software Visual PDE [24] and a custom-made Turing system based on Schnakenberg equations, and we simulated the effect of a parallel striped constraining on Turing pattern formation thanks to a mask containing 16 black stripes on a white background designed on GIMP (version 3.0.4). Each black pixel corresponds to a reduction of 60% of the diffusion rate compared to the control, whereas white pixels kept 100% of the diffusion rate.