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Summary
The hexagonal cells constructed by honey bees and social wasps are an instance of adaptive structure; hexagons decrease materials use, whereas maximizing cupboard space and structural stability. Hexagon constructing advanced independently within the bees and wasps, however in some species of each teams, the hexagonal cells are dimension dimorphic—small employee cells and enormous reproductive cells—which forces the builders to hitch in a different way sized hexagons collectively. This inherent tiling drawback creates a novel alternative to analyze how comparable architectural challenges are solved throughout unbiased evolutionary origins. We investigated how 5 honey bee and 5 wasp species solved this drawback by extracting per-cell metrics from 22,745 cells. Right here, we present that each one species used the identical constructing methods: intermediate-sized cells and pairs of non-hexagonal cells, which improve in frequency with rising dimension dimorphism. We then derive a easy geometric mannequin that explains and predicts the noticed pairing of non-hexagonal cells and their fee of incidence. Our outcomes present that regardless of totally different constructing supplies, comb configurations, and 179 million years of unbiased evolution, honey bees and social wasps have converged on the identical options for a similar architectural issues, thereby revealing elementary constructing properties and evolutionary convergence in development habits.
Quotation: Smith ML, Loope KJ, Chuttong B, Dobelmann J, Makinson JC, Saga T, et al. (2023) Honey bees and social wasps attain convergent architectural options to nest-building issues. PLoS Biol 21(7):
e3002211.
https://doi.org/10.1371/journal.pbio.3002211
Educational Editor: Abderrahman Khila, Centre Nationwide de la Recherche Scientifique, FRANCE
Acquired: December 13, 2022; Accepted: June 21, 2023; Revealed: July 27, 2023
Copyright: © 2023 Smith et al. That is an open entry article distributed beneath the phrases of the Artistic Commons Attribution License, which allows unrestricted use, distribution, and copy in any medium, offered the unique creator and supply are credited.
Knowledge Availability: All photographs and knowledge used on this research will be discovered at Auburn College’s open entry on-line repository, AUrora: http://dx.doi.org/10.35099/aurora-605.
Funding: This work was supported by the Nationwide Science Basis (grant quantity 2216835 to MLS, 2042411 to KHP, 1846340 to NN), the Deutsche Forschungsgemeinschaft (DFG, German Analysis Basis) beneath Germany’s Excellence Technique – EXC 2117 – 422037984 (MLS), the Packard Fellowship for Science and Engineering (KHP) and GETTYLABS (KHP). The funders had no function in research design, knowledge assortment and evaluation, choice to publish, or preparation of the manuscript.
Competing pursuits: The authors have declared that no competing pursuits exist.
Social insect nests are outstanding feats of development; their construction is crucial to colony success, however the scale is past the scope of any particular person builder [1–3]. Nest structure varies throughout species, reflecting elements of their pure historical past that result in constructing challenges (e.g., constructing nests inside an outlined cavity, uncovered to the weather, or together with nest options that present protection [4–6]). Distinctive constructing situations make it tough to straight examine nest structure throughout species, however there’s a shared problem for all species that construct hexagonal cells that exhibit dimension dimorphism [4,5,7]. Staff in these colonies usually construct small hexagonal cells for rearing staff, and enormous hexagonal cells for rearing reproductives, which creates an inherent tiling drawback: tips on how to be part of non-uniform hexagons inside a single lattice. Evaluating how totally different species be part of non-uniform hexagons right into a single array gives a uncommon alternative to disclose how comparable architectural challenges are solved throughout unbiased evolutionary origins.
Honey bees and social wasps symbolize unbiased origins of each eusociality and comb constructing [8]. In each teams, staff construct hexagonal cells, although honey bees (Apis) construct a double-sided array out of wax, whereas wasps usually construct a single-sided array out of paper [4,6,9,10]. Constructing supplies, nonetheless, will not be free—honey bees should devour honey to supply wax, and wasps should spend time accumulating wooden and water to supply pulp [4,10–12]. Consequently, staff are anticipated to be frugal of their use of constructing materials, and common hexagonal cells are in reality optimum within the absence of boundary situations [13]. We’d additionally anticipate them to have advanced near-optimal behaviors when introduced with the problem of transitioning between cell sizes. Whereas non-hexagonal cells exist within the hexagonal arrays of each honey bees and social wasps [7,14–17], these irregular shapes are normally ignored in earlier analysis as a result of measuring them is notoriously tedious. Nevertheless, it’s precisely in these conditions, the place development of cells requires a non-trivial trade-off between materials and performance, that we could assess their expertise as builders.
Right here, we examine hexagonal lattices constructed by 5 species of honey bees and 5 species of social wasps, representing at the very least 4 transitions between hexagonal cell dimension monomorphism and dimorphism in lineages separated by 179 million years [8] (Fig 1). Staff construct reproductive cells when colonies attain a crucial colony dimension, time of yr, or colony situation, however the 2 cell sizes are constructed throughout the identical hexagonal lattice [18–23]. For every species, we collected nest photographs that included each employee and reproductive cells (male cells for Apis, queen cells for Vespula). Specialised hexagonal reproductive cells will not be constructed haphazardly all through the comb, however as an alternative seem clustered, with a transparent transition between employee and reproductive cells. After figuring out worker-to-reproductive transitions, we used a customized software program [7] to determine and extract per-cell metrics. Every picture and its corresponding metrics had been verified by hand, to verify the accuracy of each hexagonal and non-hexagonal cells. Utilizing this dataset, which incorporates 22,745 cells extracted from 115 photographs, we investigated how every species solves the issue of tiling non-uniform hexagons right into a single lattice.
Fig 1. Measurement dimorphism within the hexagonal cells of honey bees and social wasps.
(A) Phylogeny of the species investigated, with measurements of dimension dimorphism, and examples of comb photographs at proper. Blue + signifies an unbiased origin of dimension dimorphism. Caste dimorphism advanced early within the eusocial wasps; thus, we assume a lack of caste dimorphism (pink −) within the M. mesoamerica lineage. In Apis, 2 origins of dimorphism are parsimonious. (B) Histograms of cell space normalized by median employee cell space. Word that in species with a bimodal distribution (A. mellifera onward), the bigger cells are used for rearing reproductives (black star at 0; coloured triangle marks the median cell space for reproductive cells). Underlying knowledge will be discovered at http://dx.doi.org/10.35099/aurora-605.
Hexagonal dimension dimorphism throughout species
We first measured dimension dimorphism in every species by evaluating the relative dimension of employee cells and reproductive cells (Fig 1). Dimorphism scores, i.e., extra space of reproductive cells relative to employee cells (D = [Reproductive cell area] / [Worker cell area] − 1) ranged from D = 0.0 (Metapolybia mesoamerica, no dimension dimorphism, based mostly on lack of morphological queen caste [24]) to D = 1.70 (Apis andreniformis, highest dimension dimorphism; reproductive cells are 2.70 instances bigger than the median employee cell; Fig 1). For two species, Apis cerana and Apis dorsata, dimension dimorphism is actually nonexistent (dimorphism scores of D = 0.01 and D = 0.05, respectively). Reproductive cells in these species had been solely identifiable by the capping on the cell (perforated in A. cerana; domed in A. dorsata) or by eradicating the cell capping to disclose the growing brood inside (Apis males have enlarged compound eyes [5]). Given the phylogeny in Fig 1, and that Apis mellifera does have dimension dimorphism (D = 0.34), these outcomes counsel that 2 origins of hexagonal cell dimorphism in Apis is essentially the most parsimonious state of affairs. Extra sophisticated options embody (a) lack of dimension dimorphism within the shared ancestor of A. mellifera, A. cerana, and A. dorsata and a achieve in A. mellifera; or (b) 2 unbiased losses of dimension dimorphism in A. cerana and A. dorsata. In social wasps, reproductive employee dimorphism is ancestral [25], and the monomorphism of M. mesoamerica is due to this fact resulting from a lack of dimorphism. These outcomes allowed us to prepare species in accordance with the magnitude of their cell dimorphism, with the prediction that larger dimension dimorphism leads to a harder tiling drawback than species with low dimension dimorphism.
In species with out dimension dimorphism, reproductives are reared alongside staff and distributed all through the comb [5]. As dimension dimorphism will increase, nonetheless, the architectural benefit of clustering specialised cells will increase, as this reduces wasted house. As soon as reproductives are reared in specialised areas, staff also can alter their rearing situations (e.g., temperature profile and developmental time [26–28]). Subsequently, the constraints imposed by dimension dimorphism not solely alter the general group of the nest but in addition facilitate developmental divergence within the reproductives, which presumably can result in additional dimorphism.
Totally different species use the identical architectural options
In species with dimension dimorphism, clusters of reproductive cells show a transparent transition from their worker-sized counterparts. To disclose what modifications in constructing habits occurred throughout this transition, we aligned every cell relative to its distance from the transition level (the purpose at which cells change from employee to reproductive; vertical dotted strains in Fig 2). Strikingly, all species exhibited a mix of two development behaviors to kind the transition: Staff constructed intermediate-sized hexagonal cells to adapt to the larger-sized reproductive cells and integrated non-hexagonal cells on the transition level (Fig 2). These non-hexagonal cells included 4-, 5-, 7-, and 8-sided cells, although 5- and 7-sided cells had been the commonest, collectively accounting for over 97.17% of all non-hexagonal cells (whole non-hexagonal cells: 566; 4-sided: 2, 0.35%; 5-sided: 291, 51.41%; 7-sided: 259, 45.76%, 8-sided: 14, 2.47%).
Fig 2. Particular person cells throughout the worker-to-reproductive cell transition.
Species are ordered left to proper in accordance with rising dimension dimorphism (rows 1 and three: normalized cell areas; rows 2 and 4: proportion of non-hexagonal cells). In M. mesoamerica, knowledge factors proven in gray, as employee and reproductive cells are indistinguishable. In A. cerana and A. dorsata, reproductive cells (blue) are constructed intermingled with employee cells (pink). In species with dimension dimorphism (A. mellifera onward), reproductive cells are clumped. Employee cells on the left, reproductive cells on the correct, with the vertical dotted line exhibiting the transition level. Coloration denotes variety of cell partitions. Cell place (x-axis) is plotted as wall lengths for every species. In species with out dimension dimorphism (no vertical dotted line), cells are centered relative to the picture. Underlying knowledge will be discovered at http://dx.doi.org/10.35099/aurora-605.
Non-hexagonal cells had been utterly absent within the 3 species with negligible dimension dimorphism (M. mesoamerica, A. cerana, A. dorsata). Because the magnitude of dimension dimorphism elevated, nonetheless, so too did the proportion of non-hexagonal cells on the transition level (Figs 2 and 3). This relationship held for all species (Fig 3) and was vital even when controlling for non-independence resulting from phylogeny (PGLS: ML lambda = 0.06; F1, 8 = 45.5; p < 0.001; adjusted R2 = 0.83). The sample can also be current when contemplating solely the 7 dimorphic species, indicating a transparent optimistic relationship not solely pushed by the contrasts between the presence and absence of dimorphism (PGLS: ML lambda = 0.0; F1, 5 = 14.8; p = 0.01; adjusted R2 = 0.70). Subsequently, regardless of a number of unbiased evolutions of comb constructing, and dimension dimorphism, their architectural options all fall upon the identical continuum of non-uniform cell development.
Fig 3. Staff construct non-hexagonal cells.
(A) Examples of 5–7 pairs within the species with dimension dimorphism, oriented with the course of comb progress dealing with downwards. Word how the decrease wall of the 5-sided cell has been truncated. (B) As dimension dimorphism will increase, so too does the proportion of non-hexagonal cells on the transition level. Error bars present 95% CI. (C) The commonest non-hexagonal cells are 5- and 7-sided, that are in-built pairs. (D) Place of 5- and 7-sided cells exhibits that 5-sided cells are constructed first, adopted by a 7-sided cell. Underlying knowledge will be discovered at http://dx.doi.org/10.35099/aurora-605.
Constant configurations of non-hexagonal cells
The commonest non-hexagonal cells had been 5- and 7-sided. Curiously, the proportions of those cells had been matched inside every species (Fig 3). Pairs of 5- and 7-sided cells are recognized to exist in A. mellifera and are even present in graphene, an abiotic hexagonal lattice, which suggests a fundamental geometric motive for this pairing [7,15,17,29,30]. Right here, we present that paired 5- and 7-sided cells are an architectural characteristic shared throughout all species with dimension dimorphism.
Staff usually construct worker-sized cells earlier than they transition to reproductive-sized cells, which allowed us to orient every array of cells in accordance with the order wherein every cell was constructed. Viewing cells within the course of comb progress exhibits that when inserting non-hexagonal cells, staff construct a 5-sided cell first, adopted by a 7-sided cell (Fig 3). The 5-sided cell is fashioned by truncating the decrease vertex; changing 2 partitions and a vertex with a single wall. This wall is the primary to kind the 7-sided cell. This truncation–enlargement can also be mirrored in cell areas: 5-sided cells are smaller, and 7-sided cells are bigger, as in comparison with hexagonal cells (Figs 2 and S1). Certainly, 7-sided cells are bigger even than reproductive cells, although the hexagonal lattice stabilizes at reproductive-sized cells (Fig 2).
Modeling predicts scale and kind of architectural answer
We current a mannequin that explains the actual pairing of 5- and 7-sided cells, the anticipated fee of such pairs based mostly on the quantity of dimension dimorphism, and the presence of intermediate-sized cells (Fig 4). Quite than specializing in a mechanistic mannequin of how the animals construct their nest, we begin from 3 observations: (1) patches of employee and reproductive cells are typically common hexagonal and defect-free when removed from the transition space; (2) the two patches of hexagonal cells usually have the identical cell orientation; and (3) the transitions occur alongside a straight line. Given these observations, we then reasoned how non-hexagonal cells would come up if the builders acted optimally when transitioning between patches of employee and reproductive cells. The mathematical argument relies on research of cells in A. mellifera, the place the variety of partitions for every cell relies on the place of optimally spaced cell facilities [7,30]. These cells are effectively represented by the polygons in a Voronoi partition, and within the following, we leverage the truth that on this development the neighborhood graph of cell facilities is a Delaunay triangulation, i.e., the circumscribing circle of every triangle within the graph comprises no different facilities (see Strategies). Fig 4A exhibits an instance of how cell facilities from employee and reproductive cells will be mixed whereas upholding (left grey disk) or violating (proper grey disk) the Delaunay triangulation.
Fig 4. Modeling totally different merge methods.
(A–C) Connection sample for transition between employee and reproductive cells, the place the middle of every extra reproductive cell is offset by Δ, the mismatch between employee and reproductive cells. (A) Lateral displacement of cell facilities rising left to proper. The grey disks present the circumscribing circles for middle triplets that will be related in a daily lattice. Word that the disk on the correct violates the Delaunay situation (disk comprises 4, not 3, cell facilities). (B) This lateral displacement introduces a 5–7 pair. (C) A transition area introduces intermediate-sized hexagonal cells, however no non-hexagonal cells (3 transition layers, in inexperienced). (D) Noticed transition in A. florea exhibiting a sudden transition between employee and reproductive cells, which incorporates 5–7 pairs (unshaded cells). (E) Noticed transition in V. shidai exhibiting a transition that makes use of intermediate-sized hexagonal cells, and no 5–7 pairs. (F) Noticed fee of non-hexagonal cells per employee cell as a perform of the noticed dimension dimorphism (n = 115 photographs). The pink line is the expected fee based mostly on a one-step worker-to-reproductive transition for a mean transition size (technique proven in A and B). The shaded space corresponds to transitions that outperform one of the best one-step transition when it comes to non-hexagonal cells. Species with excessive dimension dimorphism don’t construct transition layers (A. andreniformis, A. florea; D), whereas species with moderate-sized dimorphism produce fewer non-hexagonal cells (“higher than optimum” shaded area), by using intermediate-sized cells throughout the transition area (technique proven in C, E). (G) Regression of predicted non-hexagonal cells based mostly on the next: dimension dimorphism, noticed merge size, and depth of the transition area, versus the noticed non-hexagonal cells, for every picture. Marker dimension corresponds to the variety of knowledge factors that fall on the identical coordinate. Within the mannequin prediction (x-axis), every 5–7 pair produces 2 non-hexagonal cells. The noticed non-hexagonal cells (y-axis) counts all non-hexagonal cells, together with 4-sided, 8-sided, and unpaired cells. Underlying knowledge will be discovered at http://dx.doi.org/10.35099/aurora-605.
When there’s dimension dimorphism, the mismatch of cell sizes on both aspect of the transition makes common connection patterns unattainable (e.g., there are 7 employee cells and 6 reproductive cells alongside the transition line in Fig 4A). We are able to quantify this impact as follows. First, assume cells on both aspect of the transition are aligned as they need to be in a daily connection sample (Fig 4A “0”) after which rely the variety of employee cell neighbors shifting laterally away from this level ([1, 2, … N]). Every pair of employee cell neighbors and one reproductive cell ought to kind one triangle of the triangulation. With every step the mismatch, , between the cell facilities accumulates, and as soon as this mismatch is bigger than half a reproductive cell (Fig 4A “N”), the circumscribing circle would embody each the present and the earlier reproductive cell middle. Thus, to keep up the Delaunay situation, a employee cell primarily loses an edge within the graph and turns into 5-sided, whereas the earlier reproductive cell good points an edge and turns into 7-sided (Fig 4B).
Following this line of reasoning leads to an anticipated fee of 5-7 pairs. We present {that a} common hexagonal connection sample can’t be sustained for greater than employee cell lengths earlier than a 5-7-sided pair should be integrated, with the 5-sided cell on the employee aspect of the transition and the 7-sided cell on the reproductive aspect (see Strategies). Because of this for a given one-step transition, we might anticipate to see
5-7 pairs, the place L is the size of the transition measured in variety of employee cells lengths. When dimension dimorphism is low, staff can keep away from constructing any non-hexagonal cells if the variety of reproductive cells within the comb is small (i.e., the variety of fractional predicted pairs will be rounded down, as a result of the buildup of mismatch, Δ, has not but pressured a 5-7 pairing; Fig 4A and 4B).
The noticed knowledge additionally present that some species use a second, complementary technique to resolve the transition drawback: the incorporation of layers of intermediate-sized hexagonal cells to reduce the incidence of non-hexagonal cells (Fig 4C, 4D and 4E). The geometrical argument additionally works for this technique, principally treating the intermediate layers as a collection of consecutive one-step low-dimorphism transitions.
In high-dimorphism species, e.g., A florea and A. andreniformis, we don’t observe layers of intermediate-sized cells as this technique would require many layers of cells with areas which can be far exterior the distributions of both employee or reproductive cells and due to this fact are presumably unsuitable for rearing brood [31]. We solely observe intermediate-sized cell transitions in species with average dimorphism; these species incorporate 5-7 pairs at a decrease fee as a result of transitional layers primarily lower the dimorphism rating that forces 5-7 pairs. The truth that species with average dimorphism outperform the optimum variety of 5-7 pairs in a one-step transition will be seen in Fig 4F, the place nearly all of knowledge factors for average dimorphism species fall within the “higher than optimum” shaded area. We estimate the variety of transition layers by becoming a gradual 4-parameter dimension transition mannequin for every comb picture. This produces the transition space, which is used to suit optimum transition layers—optimum within the sense that the dimensions distinction between consecutive layers is as huge as attainable with out introducing 5-7 pairs. The “efficient” dimorphism rating, De, now accounts for the remaining dimension distinction that was not accommodated for by these transition layers (Fig 4G, Strategies).
Though the geometric argument makes simplifying assumptions, the noticed knowledge match the mannequin effectively (Fig 4G). It predicts that non-hexagonal cells ought to happen as 5-7 pairs; of the noticed 5- and 7-sided cells, 87.8% are in pairs as predicted (85.2% of all non-hexagonal cells). The mannequin additionally recovers the noticed optimistic relationship between the dimorphism rating and the proportion of non-hexagonal cells. The sudden transitions produced by the high-dimorphism species, A. andreniformis and A. florea (D > 1.50), don’t embody many intermediate-sized hexagonal cells (Figs 1 and 2), and the noticed fee of 5-7-sided pairs matches the one-step prediction (Figs 3 and 4). Species with no/low dimorphism (D < 0.10) don’t produce any non-hexagonal cells, and species with average dimorphism (0.10 ≤ D ≤ 1.50) use layers of intermediate-sized hexagonal cells, as indicated by the cell dimension distribution and gradual transition (Figs 1 and 2), modeled in Fig 4G. Along with being derived from easy geometric assumptions, utilizing our mannequin to foretell the noticed variety of non-hexagonal cells throughout all photographs and species produces a greater correlation (R2 = 0.93) than data-driven fashions that regress on the dimensions dimorphism alone (Figs 4G and S2).
Conclusions
Investigating how architectural challenges are overcome in several organic programs gives a uncommon alternative to straight examine the outcomes of collective constructing. Whereas our work has centered on cell configurations in accomplished combs, a completed product of collective habits, future work might handle the mechanistic and behavioral processes of non-hexagonal cell development. For instance, are 5-7 pairs constructed of their entirety in the course of the transition to reproductive cells, or do staff rearrange partitions to realize this configuration? In Apis, the comb constructing space is obscured by a thick layer of bees, however X-ray tomography can be utilized to visualise comb progress beneath the cluster [32]. Social wasps, nonetheless, don’t cluster as tightly, and so could possibly be a most popular system, particularly in species that exhibit excessive charges of dimension dimorphism, and, due to this fact, 5-7 pairs are extra frequent. It’s attainable that the constructing supplies themselves, wax versus paper, affect the methods wherein builders should adapt their constructing habits to accommodate non-hexagonal configurations inside their latticework.
Right here, we present that regardless of totally different constructing supplies, comb configurations, prices, and evolutionary origins, the honey bees and social wasps all remedy the architectural drawback of dimension dimorphism with the identical scalable options; intermediate-sized cells, and non-hexagonal cells, organized in configurations that may be predicted by assuming near-optimal constructing habits in simplified geometries. In contrast to species-specific guidelines [33], these outcomes counsel that the advanced habits of all species have converged upon the identical answer for a elementary geometrical motive. Fashions of distributed development are likely to concentrate on higher-level buildings in idealized situations [34]. By specializing in constructing responses to biologically related cell configurations, the place “completely” symmetrical hexagons can’t be constructed, we present proof for convergent evolution in architectural outcomes throughout unbiased origins of comb development.
Strategies
Picture assortment
Every nest picture needed to have individually discernable cells, together with cell partitions and vertices. For managed species of Apis, combs needed to be constructed with out embossed wax or plastic basis, as this could alter their constructing habits. For the social wasps, nests needed to be freed from picket sticks or different supplies that will alter cell shapes. Most combs in Vespula nests consist totally of employee or queen cells; for our evaluation, we solely chosen photographs of comb that embody areas of each cell varieties.
Photos of A. andreniformis and A. cerana nests had been taken within the Mae Rim district of Chiang Mai, and Mae Fah Leuang College (Thailand). Photos of A. dorsata nests had been taken within the Mae Rim district of Chiang Mai (Thailand). Photos of A. florea nests had been taken within the Mae Rim district of Chiang Mai, Mae Fah Leuang College, and the Ratchaburi campus of King Mongkut College (Thailand). Photos of A. mellifera nests had been taken on the Liddell Subject Station of Cornell College, in Ithaca, NY (USA). Photos of M. mesoamerica nests had been taken at La Selva Organic Station (Costa Rica; allow quantity 018-2010-SINAC). Photos of V. flavopilosa nests had been taken in Ithaca NY (USA). Photos of V. maculifrons nests had been taken in Georgia and North Carolina (USA). Photos of V. shidai nests had been taken at Gifu and Nagano (Japan). Photos of V. vulgaris nests had been taken on the Nelson Lakes Nationwide Park (New Zealand), and the Katholieke Universiteit Leuven (Belgium). We additional supplemented our dataset by performing internet seek for nest photographs of frequent species (e.g., A. florea) and by contacting particular person researchers.
For every species, we collected the next variety of photographs and cells: A. andreniformis, 9 photographs, 1,875 cells; A. cerana, 12 photographs, 1,030 cells; A. dorsata, 9 photographs, 1,085 cells; A. florea, 33 photographs, 8,046 cells; A. mellifera, 8 photographs, 1,619 cells; M. mesoamerica, 8 photographs, 1,834 cells; V. flavopilosa, 5 photographs, 477 cells; V. maculifrons, 3 photographs, 442 cells; V. shidai, 19 photographs, 4,511 cells; V. vulgaris, 9 photographs, 1,827 cells.
Semi-automated picture evaluation of comb cells
To extract per-cell metrics, we used a customized GUI to determine cell facilities, vertices, and partitions (see particulars in [7]). The unique software program was meant for photographs taken beneath managed lighting situations, however the photographs used on this research had extra heterogeneous situations. Subsequently, every cell in every picture was verified, by hand, to verify that the place of the cell middle, vertices, and partitions precisely matched the cells within the picture. This semi-automated technique allowed us to incorporate cells from photographs that will in any other case not have been possible (e.g., nest photographs that also had the occasional grownup employee atop the comb, so long as the cell was nonetheless seen beneath). For every picture, we recognized the placement of employee cells, reproductive cells, and the transition between the 2. In some cases, the cells throughout the complete transition area couldn’t be recognized (e.g., if a number of consecutive cells had been broken, or a overseas object was embedded within the nest).
Statistical analyses
Statistical analyses had been carried out in Python (ver. 3.8.9), and R (ver. 4.0.2). Scale bars weren’t current in most photographs, so we as an alternative normalized our measurements relative to worker-sized cells (i.e., cell space 0 = median employee cell space; cell space 1.5 = cell is 1.5 instances bigger than the median employee cell). To maintain plots constant throughout species, knowledge are plotted in models of worker-wall lengths (e.g., distance from transition level, in models of worker-wall lengths). This made comparisons throughout species possible, because the models of measurement are scaled to every species, and unbiased of picture high quality.
Phylogeny and comparative strategies
First, we created a phylogenetic tree of our species utilizing divergence dates (in tens of millions of years in the past) from key nodes in [8] (break up between Vespidae and Apidae; break up between Vespinae and Polistinae) and [35] (splits between species of Apis). We couldn’t discover a time-calibrated tree containing our species of Vespula. As a substitute, we used the topology described in [36] with V. flaviceps as a proxy for V. shidai (these are sister species [37]), with node ages set to at least one lower than the variety of descending species [38]. We then linearly scaled the department lengths of the Vespula clade in order that the divergence time between V. shidai and the remaining species was 8 mya, as this occurred previous to the divergence between V. germanica and V. vulgaris [36], estimated at 6 mya [39].
To manage for non-independence of species-level knowledge resulting from shared evolutionary historical past, we used phylogenetic least squares regression (PGLS) to find out if the diploma of cell dimension dimorphism predicted the share of cells which can be non-hexagonal. We used the pgls perform within the bundle caper to concurrently estimate the utmost chance worth for the branch-length scaling parameter lambda, which finds one of the best diploma of phylogenetic scaling that minimizes the phylogenetic correlation of the residuals of the regression, whereas additionally becoming the regression between dimorphism and proportion of non-hexagonal cells [40]. We additionally created a PGLS mannequin with solely the species with substantial dimension dimorphism (excluding M. mesoamerica, A. cerana, and A. dorsata) to see if we might discover proof for an impact of the diploma of dimorphism on the proportion of non-hexagonal cells.
Geometric mannequin for non-uniform hexagonal lattice construction
This materials covers particulars associated to the geometrical argument for the way different-sized hexagons will be joined in a single lattice construction, as associated to comb constructed by honey bees and social wasps.
1. Models.
In species with sexual dimension dimorphism, reproductive cells are bigger than employee cells, by an element α, in order that the common hexagonal sample of a comb is disturbed when the colony transitions to constructing reproductive cells. Right here, α refers back to the linear scale distinction between employee and reproductive cells. Word that some figures report the world improve D, as acceptable within the context of the dialogue (Figs 1 and 2). To transform between the 2: D = α2−1 and . All through the mannequin description, we additional seek advice from L because the size of the transition line between the employee and reproductive cell patches, measured in wall-to-wall employee cell widths.
2. Assumptions and mannequin.
Our aim is to derive a easy relation that may clarify the noticed cell geometries within the transition areas of naturally constructed comb, i.e., what number of non-hexagonal cells to anticipate alongside a transition line between employee and reproductive cells (Fig 4A–4C). This relation is then used to foretell the variety of anticipated non-hexagonal cells for all species, whether or not or not they use layers of intermediate sized cells.
After we assume that the transition occurs immediately, from one employee cell to the subsequent reproductive cell, we will derive the relation because the variety of non-hexagonal cells we might anticipate per employee cell alongside a transition line. Moreover, we might anticipate 5-7-sided cell pairs the place the 5-sided cell is a deformed employee cell and the 7-sided cell is a deformed reproductive cell. This agrees with present and former observations [7,30].
To begin, we assume a patch of recurrently spaced employee cells and a (not but constructed) patch of reproductive cells. We then look at how these 2 patches could possibly be related over a sudden transition, and when the distinction in cell dimension requires non-hexagonal cells. As a substitute of straight analyzing the position of cell partitions, we have a look at the twin illustration of a comb when it comes to its cell facilities (marked with black factors in Fig 4A–4C). To narrate the cell middle illustration to cells, we use the facilities to generate a Voronoi partition to decompose the aircraft into polygonal areas, i.e., polygons which have the properties that each one the factors inside a polygon are closest to the cell middle that’s contained in the partition, and all the perimeters are equidistant from 2 cell facilities. Prior research on irregular nest development discovered that the Voronoi partition generated by cell facilities is in good settlement to the noticed cell partitions [30]. To derive a certain on the variety of vital non-hexagonal cells, we use the truth that the graph generated by connecting facilities whose Voronoi partition polygons share an edge is a Delaunay triangulation, i.e., a graph the place any circumscribing circle of a triangle comprises no different vertices. This equivalence is useful because the connection topology of cell facilities is less complicated to investigate and we will derive bounds based mostly on imposing the Delaunay situation.
Subsequent, we analyze the graph of cell facilities for each patches and assume a partial Delaunay triangulation the place all edges which can be purely between employee cell facilities and reproductive cell facilities are already added. Now we will add the remaining edges and analyze the connection topology as allowed by the Delaunay situation to compute the fraction of non-hexagonal cells alongside the transition line (Fig 4A–4C).
From the attitude of a employee cell edge that faces the reproductive cell patch, we have to select a cell middle from the reproductive cells that completes the triangle. Since we already know 2 factors of the triangle, we all know that the middle of the circumscribing circle must be positioned on the bisecting regular of that edge (and if the reproductive cells had been the identical dimension, the middle could be on that bisecting line). By symmetry across the bisecting line, the reproductive cell middle that’s chosen can’t be additional away than from the bisecting line because the circumscribing circle would in any other case include one other vertex and violate the Delaunay situation.
To compute the variety of hexagonal cells that may be related earlier than non-hexagonal cells have to be launched, assume that we join one cell and that the reproductive cell is completely coincident with the bisecting line (marked by a “0” in Fig 4A and 4B). Subsequently, the subsequent reproductive cell middle is Δ = (α−1) away from the bisecting line, and the subsequent cell is 2(α − 1), and so on. After N employee cell edges, the corresponding reproductive cell middle is N(α−1) from the bisecting line. When α > 1, this may ultimately violate the Delaunay situation and such that the triangulation ought to as an alternative connect with the N−1st reproductive cell middle (grey shaded disk on the correct in Fig 4A). This dislocation from the common connection sample removes one edge from a employee cell and provides one edge to a reproductive cell leading to a 5-7-sided pair of cells (i.e., 2 non-hexagonal cells per dislocation; pink and blue shaded cells in Fig 4B). The identical reasoning applies whether or not we contemplate cells to the correct or the left of the 0th reproductive cell middle, in order that for
, there should be a dislocation leading to a 5-7-sided cell pair. In different phrases, on common, we might anticipate
dislocations per employee cell alongside the transition line, and twice as many non-hexagonal cells.
3. Intermediate-sized cell transitions and blended methods.
As a result of discretization of 5-7-sided cell pairs, it’s attainable to realize transitions alongside a size of with out having to introduce any non-hexagonal cells. For instance, for low-dimorphism species, the place α is shut to at least one, Lmax is giant, and we’d not anticipate to want non-hexagonal cells. Conversely, given a size L, we will compute the most important dimension distinction than will be achieved with out having to introduce dislocations (and related 5-7 pairs). For a selected L, the utmost αL that may be transitioned to with out introducing any dislocations is
, and we will use this expression to look at methods that embody constructing intermediate sized cells.
Let T denote the variety of “transition layers”, i.e., hexagonal cells of intermediate dimension that accomplish the worker-reproductive transition with out introducing non-hexagonal cells. Given an α related to a given dimension dimorphism and an αL related to a selected transition, then a dislocation-free transition could possibly be achieved with T successive steps such that αLT = α. The multiplicative nature of this relation is because of the truth that within the first layer, all of the cells have the dimensions αL and rising to the second layer the most important cells that may be achieved with out introducing a 5-7 pair must be αL * αL, and αL * αL * αL for the third layer, and so on.
We explored a number of methods for estimating the variety of transition layers from the picture knowledge. Provided that common cells have dimension variation, deciding if a selected cell is a daily cell or a transition cell is considerably arbitrary, and we discovered that totally different confidence stage cutoffs produced totally different outcomes. As a substitute, we used a easy transition mannequin with 4 parameters (Aemployee, Arep, Oemployee, Orep) to suit the areas of hexagonal cells utilizing least squares, the place the variables A correspond to the areas of employee and reproductive calls and the variables O correspond to the offsets from a line match to the transition. All cells which can be additional than Oemployee away from the transition line in a single course are assumed to be of Aemployee; all cells which can be additional away than Orep within the different course are assumed to be of dimension Arep. The cells between the two offsets are linearly interpolated between the employee and reproductive cell sizes. The gap between the two offsets on this match is the measured size of the transition and used to estimate Tobs Whereas considerably easy, this mannequin has no extra tuning parameters and solely assumes the general geometry of the merge as an alternative of attempting to categorise particular person cells as being a part of the common comb or transition areas.
When measuring transition layers from picture knowledge, the space between the two offsets seldom precisely suits multiples of αL. In that case, we interpolate the final (largest) layer to estimate the entire variety of transition layers Tobs. For the reason that introduction of non-hexagonal cells will deform the adjoining hexagonal cells, leading to intermediate sizes, we require that the transition area include at the very least 50% hexagonal cells. If it doesn’t, we set Tobs to zero. That is solely a difficulty for high-dimorphism species the place the one-step transition has a excessive fee of 5-7 pairs and the place constructing intermediate sized cells is clearly not a part of their technique (however does happen due to the adjoining non-hexagonal cells). Setting Tobs to zero in these instances prevents these transitions from being handled like a blended technique, described subsequent.
Utilizing Tobs, we will additionally mannequin a blended technique the place some intermediate sized cells are launched in the course of the transition, however the place Tobs is simply too small to finish the transition with out introducing 5-7 pairs. For such “incomplete” transition layers which have each intermediate sized cells and 5-7 pairs, we will compute an efficient αE, i.e., the dimensions distinction left after utilizing transitional layers to slender the distinction, αE αLTobs = α or αE = = . On this state of affairs, we might anticipate a fee of 5-7 pairs of
(Fig 4G).
We observe totally different relative desire of getting non-hexagonal or intermediate-sized cells depending on species. For species the place the employee and reproductive sized cells are extensively separated, reminiscent of A. florea, a pure intermediate cell dimension transition would produce many layers of cells which can be exterior the vary of both employee or reproductive cell sizes, which isn’t noticed (Fig 4D and 4F). For species with intermediate-sized dimorphism, we do observe transition layers and intermediate-sized cells (Fig 4E). Evaluating the standard of the mannequin with and with out transition layers is proven in S2 Fig, together with the transition layer and utilizing αE does a greater job (R2 = 0.93 versus R2 = 0.86) of predicting the speed of non-hexagonal cells for all the dataset.
Comparability to various fashions
In the primary paper, we introduced the best-fit mannequin out of 4. Right here, we elaborate on the choice fashions evaluated (S2 Fig). The primary and second mannequin predicts the speed of non-hexagonal cells whether it is straight proportional to the noticed dimorphism rating, D, and the noticed mismatch between cells Δ = (α−1), respectively. Each had been included just for comparability; they take a easy data-driven strategy and don’t clarify the actual fee or configuration of the non-hexagonal cells. The third mannequin predicts the speed of non-hexagonal cells as a one-step transition, , and the fourth mannequin features a blended technique that may incorporate intermediate sized cells if they’re noticed within the picture,
. We see that the third (one-step transition) mannequin predicts the actual configuration of 5-7-sided cells however predicts the next fee of non-hexagonal cells for average dimorphism species. The fourth mannequin contains noticed transition layers and does a greater job of predicting the incidence fee of non-hexagonal cells in medium-dimorphism species that construct intermediate-sized cells. Together with each medium- and high-dimorphism species, the R2 match rating of 0.93 is best than all the opposite approaches, and the coefficient of proportionality is shut to at least one, indicating that our derived mannequin does a superb job of straight predicting the noticed variety of non-hexagonal cells regardless of the simplifying assumptions. Provided that our mannequin assumes optimum habits, it inherently underestimates the variety of non-hexagonal cells indicated by a coefficient better than 1.
Supporting data
S2 Fig. Evaluating fashions for predicting the incidence fee of non-hexagonal cells.
The x-axis for every plot is the expected variety of cells based mostly on the model-predicted fee and the noticed size of the transition areas. The y-axis is the noticed variety of non-hexagonal cells. The dashed black line is the best-fit linear regression for the prediction and the noticed variety of non-hexagonal cells. For the reason that numbers are all integers, a number of knowledge factors can fall on the identical coordinate, so the world of the marker is proportional to the variety of knowledge factors (colours denote species). The fashions proven in (A, B) are data-driven, whereas the fashions in (C, D) clarify each the speed and specific configuration of non-hexagonal cells. Underlying knowledge will be discovered at http://dx.doi.org/10.35099/aurora-605.
https://doi.org/10.1371/journal.pbio.3002211.s002
(PDF)
Acknowledgments
We thank Tom Seeley for supplying A. mellifera colonies to supply the pure combs that impressed this work. D Anderson, JS Ascher, CP Borowski, A Brockmann, EA George, MAD Goodisman, V Hegde, Z Huang (beetography.com), RS Jacobson, D Leap (Bees Limitless), S Parichehreh, and GR Williams offered supplementary nest photographs. RA Kelleher offered childcare that enabled knowledge processing on the Punta Ratón discipline station. Cissy Ballen, Peter Marting, Walter Tschinkel, and Tom Seeley offered useful suggestions.
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