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Matthew S. Lehnert, Eric Brown, Margie P. Lehnert, Patrick D. Gerard, Huan Yan, Chanjoong Kim, The Golden Ratio Reveals Geometric Differences in Proboscis Coiling Among Butterflies of Different Feeding Habits, American Entomologist, Volume 61, Issue 1, Spring 2015, Pages 18–26, https://doi.org/10.1093/ae/tmv005
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The Golden Ratio is a number that has fascinated people for centuries, from artists and mathematicians to architects and biologists, partially due to its aesthetic appeal when showcasing symmetry. Also known as phi (cp), the Golden Mean, Golden Section, Golden Number, or Divine Number, the Golden Ratio has been categorized as beauty personified, arguably incorporated into esteemed works such as Katsushika Hokusai's The Great Wave off Kanagawa (Fig. 1A), Phidias's Athena Parthenos (Fig. 1B), Salvador Dali's Sacrament of the Last Supper, Leonardo da Vinci's Vitruvian Man, and Frank Lloyd Wright's spiral staircase in the Guggenheim Museum (Livio 2002, Atlas 2003). The Golden Ratio even starred alongside Donald Duck, pi, and a tree with square roots in the educational film Mathmagic Land (Disney 1959).
As a relationship between two numbers, the Golden Ratio is an incommensurable number, which describes the mathematical condition in which two distinct numbers have no common measure; i.e., they cannot be stated as a fraction (Knorr 1974). Similarly to the famous number pi (π), the Golden Ratio also is an irrational number, 1.61803…, meaning it continues forever without ever repeating (Livio 2002) In other words, two numbers are incommensurable if their ratio is irrational, as is the Golden Ratio. According to Wikipedia (the most widely used unacceptable student paper citation source), the Golden Ratio has been calculated out to one million numbers (Wikipedia 2014).
The true mystery of the Golden Ratio, however, is that it resonates throughout the universe and happens organically. Disc-shaped spiral galaxies (Munteanu 2010, Fig. 3A), quasi-crystals (Shechtman et al. 1984), the disc florets of some flowers, and the shell of the mollusk Nautilus pompiliusL. (Fig. 3B) all exhibit characteristics indicative of the Golden Ratio, though contradictory evidence exists on the validity of the ratio in the nautilus shell (Livio 2002, Falbo 2012). The beauty of the Golden Ratio was elegantly shown in a short video created by filmmaker Cristobal Vila (www.theatlantic.com/video/index/373479/nature-by-numbers-visualized/). Some have gone so far as to state that the perception of human beauty is correlated with the closeness of physical features to the Golden Ratio (Seghers et al. 1964). It is likely that the Golden Ratio has yet to be discovered in many other natural systems. We have observed that the coiling action of the butterfly proboscis, for instance, exhibits a spiral similar in appearance to the Golden Ratio, and thus deserves further exploration.
Butterfly Proboscis Structures and Coiling Mechanism
Of the more than 160,000 described species of butterflies and moths (Lepidoptera), approximately 95% use a coilable, tube-like proboscis to feed on nectar, tree sap, fecal moisture, rotting fruits, and other fluids (Adler 1982, Adler and Foottit 2009, Krenn 2010). The proboscis is a noteworthy component of insect-plant coevolutionary relationships (Kritsky 2001, Alexandersson and Johnson 2002, Pellmyr 2003) and lepidopteran fitness (Kunte 2007, Lehnert et al. 2014). The importance of the proboscis in ecological and evolutionary systems has led to numerous structural descriptions (Eastham and Eassa 1955, Krenn and Kristensen 2004, Krenn and Mühlberger 2002) and experimental studies of adaptations that facilitate fluid uptake (Monaenkova et al. 2012; Lehnert et al. 2013, Tsai et al. 2014).
The proboscis is a structurally and functionally complex conduit that is used for transporting fluids from pools and wetted surfaces to the butterfly's gut (Fig. 4A).
The food canal of the proboscis (where fluids travel) is formed when the two strand-like maxillary galeae fasten together via dorsal and ventral linking structures called legulae (Fig. 4B, C) following emergence from the exu-vium (Krenn 1997, 2010). Although its function is often described as straw-like (Kingsolver and Daniel 1995, Bauder et al. 2013), the proboscis is not a sealed, airtight tube (Monaenkova et al. 2012; Tsai et al. 2014; Kwauk et al., 2014), and can be used to feed when partially split (Lehnert et al. 2014). Lepidopteran proboscises are equipped with mechano- and chemosensilla, and their size and shape relate to feeding habits (Krenn et al. 2001, Molleman et al. 2005, Lehnert et al. 2013). Lepidoptera that generally feed from porous surfaces, such as sap, have enlarged chemosensilla (Fig. 4C), e.g., sensilla styloconica (Petr and Stewart 2004), that aid in fluid uptake through capillarity, hydrophilicity, and overall proboscis ellipticity (Lehnert et al. 2013). Enlarged chemosensilla give the distal region of the proboscis a brush-like appearance (Knopp and Krenn 2003, Molleman et al. 2005), which is lacking in the smooth-tipped proboscises of nectar-feeding butterflies (Krenn et al. 2001).
Proboscis movements such as uncoiling, probing, and coiling are controlled by an antagonistic relationship between hydrostatic pressure of hemolymph in the galeae and the intrinsic musculature that is coupled with the elastic properties of the proboscis cuticle (Krenn 1990, 2000; Wannenmacher and Wasserthal 2003). Lepidoptera are able to modify the angulation of the proboscis by modifying the hydrostatic pressure, via maxillary stipites, and increasing or decreasing muscle tension (Krenn 2010). Sideways movements of the proboscis, for instance, are possible by modifying the hydrostatic pressure in each galea.
The aim of this study is to determine if proboscises among six butterfly species fit the Golden Ratio during the coiling process. We predicted that some species might exemplify the Golden Ratio in their coils better than others and hypothesized that this could relate to phylogeny and feeding habits. Although evidence indicates that proboscis structures can be used to assess the evolutionary relationships among Lepidoptera (Kristensen 1984) and delineate feeding habits (Krenn et al. 2001, Molleman et al. 2005, Lehnert et al. 2013), the coiling configuration of proboscises has not been quantified to determine its association with phylogenetic relationships or feeding habits. We also took other geometrical measurements of proboscises to better assess the overall proboscis configuration.
Geometric Studies of Proboscis Coiling Configurations
In order to study proboscis geometry, we used four species of butterflies in the family Nymphalidae: the red-spotted purple (Limenitis arthemis astyanax F., Limenitidinae), painted lady (Vanessa cardui L., Nymphalinae), little wood satyr (Megisto cymela Cramer, Satyrinae), and monarch (Danaus plexippus L., Danainae). We also used eastern tiger swallowtail (Papilio glaucus L., Papilionidae) and cabbage butterflies (Pieris rapae L., Pieridae). In an outdoor enclosure, painted lady and monarch butterflies were reared on Cirsium sp. and Asclepius sp., respectively, during July-August, 2013, near Alliance, OH. The other butterfly species were captured with an aerial net from Beech Creek Botanical Garden & Nature Preserve (Alliance, OH) during July-August, 2013.
Specimens were placed in front of a high-speed camera (Casio Exilim EX-F1, Casio Inc., Dover, NJ) and filmed at 1200 frames per second. During filming, proboscises were uncoiled horizontally using an insect pin and allowed to recoil. Single frames in which the coils of proboscises appeared to fit the Golden Ratio (visually estimated) were isolated from the videos using Adobe Premier Pro CS6 (Adobe Systems Inc., San Jose, CA). A Cartesian coordinate grid was overlaid on the image and the lateral distances between the two innermost coils were measured in pixels using ImageJ software (http://imagej.nih.gov/ij/). The Golden Ratio fit to proboscis coils was determined by calculating the difference between the A/B values and the (A+B)/A values (Fig. 5); individuals with lower values are closer to fitting the Golden Ratio than those with higher values. We also considered the placement of the proboscis coils relative to the knee bend region (Fig. 4A) as a qualified component of proboscis coiling configuration; therefore, we quantified proboscis angle and slope using the same image that was used for the Golden Ratio measurements (Fig. 5). The total lengths of the proboscises were measured using ImageJ to determine if proboscis length is an important component of proboscis fitment to the Golden Ratio. SAS version 9.2 was used for all analysis and a significance level of 0.05 was used for all hypothesis tests.
Fitment of Proboscis Coils to the Golden Ratio among Species
We used analysis of covariance to evaluate the impact of species on Golden Ratio fit, slope, and angle, adjusting for possible effects of proboscis length. Proboscis length, however, was not related to any of the response variables; therefore, analysis of variance (ANOVA) was used to investigate the impact of species on Golden Ratio, slope, and angle, and a Tukey-HSD posthoc test was used to separate means. The fit of the proboscis coils to the Golden Ratio significantly differed between species (F = 3.03; df = 5, 28; P = 0.026), with the proboscis of L. a. astyanax displaying a spiral closer to the Golden Ratio than D. plexippus (Fig. 6A).
The angle produced between the knee bend region and the proboscis tip, when the proboscis best fits the Golden Ratio, also had significant differences among species (F = 28.74; df = 5, 27; P < 0.0001), with the largest angles in D. plexippus and V. cardui, and the smallest angle in M. cymela (Fig. 6B). The slope of proboscises did not significantly differ between species (F = 1.26; df = 5, 27; P = 0.3076; Fig. 6C); however, there was a general pattern where L. a. astyanax and M. cymela had the largest slope values, P. rapae and P. glaucus were intermediate, and V. cardui and D. plexippus have the smallest slope values.
Relationship of the Golden Ratio to Butterfly Feeding Habits
Proboscis geometry lacked a pattern among species when considering phylogenetic relationships. The two butterfly species with the best Golden Ratio fit, M. cymela and L. a. astyanax, are both nymphalid species, but are not closely related and share a more recent common ancestry with the other nymphalid butterflies, D. plexippus and V. cardui (Pohl et al. 2009), both with the least closest fit to the Golden Ratio (Fig. 7). Intermediate closeness of fit to the Golden Ratio was observed in the distantly related P. rapae (Pieridae) and P. glaucus (Papilionidae) (Pohl et al. 2009). The phylogenetic arrangement associated with angle and slope measurements was similar to that of the Golden Ratio fit; therefore, we further explored patterns in proboscis geometry according to species feeding habits.
The six chosen species represent three feeding habits: sap feeders (L. a. astyanax and M. cymela), nectar feeders (V. cardui and D. plexippus), and nectar feeders with male-puddling habits (P. glaucus and P. rapae), referred to hereafter as puddle feeders. The feeding habits represent a spectrum in exposure of fluids. Nectar feeders, for instance, feed primarily on nectar that is confined in floral tubes, whereas sap feeders acquire nutrients from exposed sap on porous surfaces; puddle feeders feed on both liquids that are confined (nectar from flowers) and exposed fluid on porous surfaces (wetted soil near puddles to acquire sodium, Boggs and Dau 2004). An ANOVA specifically using linear contrasts was used to compare puddle feeders to nectar-feeding species, puddle feeders to sap feeders, and species with sap-feeding habits to nectar feeders.
When data were analyzed by feeding habit, the fit of the proboscis coil was significantly closer to the Golden Ratio in sap-feeding butterflies than in nectar-feeding butterflies (t = -3.62; df = 28; P = 0.0011; Fig. 6A), and puddle feeders were intermediate. A trend of increasing angle occurred from sap feeders to nectar feeders, and the angle was significantly different between the three feeding habits (sap feeders vs. puddle feeders, t =–2.68; df = 28; P = 0.0123; sap feeders vs. nectar feeders, t = -11.43l df = 28; P < 0.0001; puddle feeders vs. nectar feeders, t = -7.45; df = 28; P < 0.0001; Fig. 6B). Analysis of slope values between feeding habits revealed significantly higher values in sap-feeding butterflies compared to nectar feeders (t = 2.24; df = 28; P = 0.0337; Fig. 6C), and puddle feeders had intermediate values.
A linear discriminant analysis exposed a classification system that was relatively accurate for feeding habits (Table 1). Nectar feeders were never incorrectly classified as sap feeders, and sap feeders were never incorrectly classified as nectar feeders. Only 12.5% (n = 1) of nectar feeders were incorrectly classified as puddle feeders, while 42.86% (n = 6) of sap feeders were incorrectly classified as puddle feeders. Species labeled as puddle feeders, however, were incorrectly classified as both sap feeders and nectar feeders. The linear discriminant analysis supports our interpretation that sap feeders and nectar feeders represent distinct entities and puddle feeders are intermediate, given the measurements of proboscis geometry used here. Golden Ratio fit, angle, and proboscis length had significant effects (ANOVA, p = 0.05) on feeding habits. A principal components biplot (Khattree and Naik 2000) was used to visualize the relationship between the acquired measurements, as well as to help identify possible clusters of species. When considering geometric measurements and proboscis length, the biplot reveals that species tend to cluster into groups that represent feeding habits rather than phylogeny (Fig. 8).
. | . | Assigned feeding habit (%) . | ||
---|---|---|---|---|
Feeding habit . | n . | Sap . | Puddle . | Nectar . |
Sap | 14 | 57.14 | 42.86 | 0 |
Puddle | 6 | 33.33 | 50.00 | 16.67 |
Nectar | 8 | 0 | 12.5 | 87.50 |
. | . | Assigned feeding habit (%) . | ||
---|---|---|---|---|
Feeding habit . | n . | Sap . | Puddle . | Nectar . |
Sap | 14 | 57.14 | 42.86 | 0 |
Puddle | 6 | 33.33 | 50.00 | 16.67 |
Nectar | 8 | 0 | 12.5 | 87.50 |
. | . | Assigned feeding habit (%) . | ||
---|---|---|---|---|
Feeding habit . | n . | Sap . | Puddle . | Nectar . |
Sap | 14 | 57.14 | 42.86 | 0 |
Puddle | 6 | 33.33 | 50.00 | 16.67 |
Nectar | 8 | 0 | 12.5 | 87.50 |
. | . | Assigned feeding habit (%) . | ||
---|---|---|---|---|
Feeding habit . | n . | Sap . | Puddle . | Nectar . |
Sap | 14 | 57.14 | 42.86 | 0 |
Puddle | 6 | 33.33 | 50.00 | 16.67 |
Nectar | 8 | 0 | 12.5 | 87.50 |
Golden Ratio Fitment in Proboscises of Sap-Feeding Butterflies
Here, we have revealed that the proboscis coiling action of some butterfly species nearly matches the famous Golden Ratio, and patterns of Golden Ratio fit are best described in relation to feeding habits. Sap-feeding butterflies have proboscis conformations that are significantly closer to fitting the Golden Ratio than species with nectar feeding habits. Angle measurements indicated a pattern similar to the Golden Ratio fit; sap-feeding butterflies grouped separately (smaller angles) from the nectar feeders, and puddle feeders had intermediate measurements. A smaller angle indicates that the proboscis, when it is most near the Golden Ratio, is positioned under the base of the proboscis, closer in proximity to the head, whereas a larger angle designates the proboscis is positioned further from the head, almost underneath the knee bend region. Slope measurements also revealed a similar pattern as angle and Golden Ratio fit when considering feeding habits. There was a general trend where sap feeders and nectar feeders represented the extreme measurement values and puddle feeders were intermediate. This trend fits a pattern in which nectar feeders primarily feed on fluids confined in floral tubes, sap feeders feed on exposed fluids of porous surfaces, and puddle feeders feed from both.
Mechanisms of proboscis coiling are dependent on cuticular elasticity and intrinsic proboscis musculature (Hepburn 1971, Krenn 1990, Wannenmacher and Was-serthal 2003). Extant representatives of ancestral Lep-idoptera, such as the Eriocraniidae (Kristensen 1968, Nielsen and Kristensen 1996) have short proboscises and lack the intrinsic musculature needed for coiling (Krenn and Kristensen 2004). The evolution of intrinsic musculature in the higher Lepidoptera, which is likely derived from basal galeal musculature, can be considered a key innovation that facilitated the evolution of longer proboscises that were needed to reach nectar in floral tubes (Krenn and Kristensen 2004); i.e., longer proboscises require musculature for probing movements and directing the proboscis into floral tubes to retrieve nectar. Distinct units of intrinsic musculature and alterations in their attachment sites in the proboscis are characteristics of the Macrolepidoptera and butterflies (Papilionoidae) (Krenn and Mühlberger 2002). We suggest that muscle attachments and cuticular elasticity differ among species with different feeding habits, thus addressing the patterns revealed here at the proximate level; however, there are no studies of which we are aware that have specifically tested this hypothesis. Previous comprehensive examinations of proboscis musculature did not address the relationship between musculature and feeding habits (Krenn 1990, Krenn and Mühlberger 2002).
The selective pressures that might act on proboscis coiling at the ultimate level (i.e., fitness) need further exploring. Nutrition and viscosity differ among the liquid food sources of butterflies (Boggs 1988, Fernández et al. 2006). The proboscis, however, is uncoiled when feeding on fluids; therefore, we are unsure how coiling patterns might relate to fluid uptake mechanisms once fluids are inside the food canal. Proboscis bending, however, has been determined to facilitate fluid uptake along the proboscis length by changing the sizes of interlegular spaces (Kwauk et al., 2014). The coiling patterns shown here might reflect differences in the extent of the proboscis that is applied to surfaces when feeding, which could differ among butterflies of different feeding habits, and facilitate the entrance of liquid residues into the food canal.
In addition, we suggest that the structural orientation of substrates associated with certain food sources, in relation to the butterfly, might act on variations in muscle attachment sites in the proboscis and cuticular elasticity (Krenn 1990, Krenn and Mühlberger 2002), which could affect the coiling configurations. Nectar, for instance, is often confined in the small spaces of floral tubes surrounded by flower petals, a possible obstruction that needs to be bypassed by the proboscis when feeding. The results from this study indicate that a nectar-feeding butterfly has the proboscis more tightly coiled when placed underneath the knee bend region when compared to sap-feeding species, which has its coils under the base of the proboscis, closer to the thorax. We suggest that this geometry could be an adaptation to avoid flower petals and other floral parts when coiling the proboscis; i.e., first lifting and coiling the proboscis on a vertical axis to the floral tube (avoiding floral parts), then coiling the proboscis on a horizontal axis once the proboscis is clear of obstructions. Evidence suggests that substrate shape and structure, such as those associated with flowers (Nilsson et al. 1985, Willmer 2011), acts on proboscis structure.
Nectar feeders, in general, also probe with the proboscis somewhat outstretched in order to retrieve nectar in floral corollas (Krenn 1990,1998), whereas butterflies that feed nearly exclusively on sap might be able to acquire fluids closer in proximity to the body. Puddle feeders, however, fall between these groups because of their mixed feeding modes (e.g., floral tubes with confined nectar and porous substrates such as wetted soil). Other possibilities associated with differences in proboscis conformation could be attributed to grooming behavior (Hikl and Krenn 2011) or self-cleaning ability via altering microbump spacing arrangements on the galea cuticle (Lehnert et al. 2013). These postulations that address coiling patterns at the ultimate level, however, are speculative and require further exploration.
We can describe the overall coiling conformation of proboscises using the Golden Ratio and other geometric measurements. In addition, we suggest that our measurements can be used to predict the feeding habits of unstudied lepidopteran species. A previous study involving the quantification of proboscis-coiling configurations did not examine phylogenetic or behavioral differences among species (Zhou and Zhang 2013), although previous studies did find differences in musculature among lepidopteran species (Krenn 1990, Krenn and Mühlberger 2002). Our study indicates the need for further experimental investigations of proboscis musculature and cuticular elasticity in relation to feeding habits. Although sap-feeding butterflies better express the Golden Ratio (i.e., Divine Number) in their proboscis coils than nectar-feeding butterflies, we suggest not concluding divine favoritism in sap-feeding butterfly species and that subsequent studies use only testable and falsifiable hypotheses.
Acknowledgements
We would like to thank Jim Nero and Beech Creek Botanical Garden & Nature Preserve (Alliance, OH) for their permission and assistance in obtaining butterfly specimens. We also want to thank Catherine P. Mulvane, Meredith Jenkins, and Aubrey Brothers for their assistance collecting specimens in the field, Carrie E. Schweitzer (Department of Geology, KSU-Stark) for providing us with the nautilus shell for imaging, and Richard H. Lehnert and Catherine P. Mulvane for editorial comments on an early version of the manuscript.