The avian egg exhibits general allometric invariances in mechanical design

The avian egg exhibits extraordinary diversity in size, shape and color, and has a key role in avian adaptive radiations. Despite extensive work, our understanding of the underlying principles that guide the “design” of the egg as a load-bearing structure remains incomplete, especially over broad taxonomic scales. Here we define a dimensionless number C, a function of egg weight, stiffness and dimensions, to quantify how stiff an egg is with respect to its weight after removing geometry-induced rigidity. We analyze eggs of 463 bird species in 36 orders across five orders of magnitude in body mass, and find that C number is nearly invariant for most species, including tiny hummingbirds and giant elephant birds. This invariance or “design guideline” dictates that evolutionary changes in shell thickness and Young’s modulus, both contributing to shell stiffness, are constrained by changes in egg weight. Our analysis illuminates unique reproductive strategies of brood parasites, kiwis, and megapodes, and quantifies the loss of safety margin for contact incubation due to artificial selection and environmental toxins. Our approach provides a mechanistic framework for a better understanding of the mechanical design of the avian egg, and may provide clues to the evolutionary origin of contact incubation of amniote eggs.

Since the stiffness is a function of shell thickness, Young's modulus, egg size and aspect ratio, this poses a fundamental challenge of quantifying eggshell stiffness in a way that allows meaningful intraspecific and interspecific comparisons. For example, the stiffness of an elephant bird egg is much larger than that of a hummingbird egg, but is it large enough to withstand the massive incubating bird? Here we ask whether there exist general "guidelines" that dictate the mechanical design of avian eggs. We propose a dimensionless number ≡ C where K is the stiffness (N m −1 ) along the long axis, A and B the breadth and length of the egg, respectively, W the egg weight (N), and A 2 /B a size/shape factor (see Table 1 for terminology and definitions). This approach has the important property of quantifying diverse eggs using a single metric with clear physical meaning-how stiff an eggshell is with respect to its egg size after removing the GIR 14 . Thus, eggs with dramatically different shapes, sizes and material properties can be compared (Fig. 1a).
Here we study 463 bird species in 35 extant orders and one extinct order (Supplementary Dataset 1). We use two methods to determine shell stiffness K: (i) In Method 1, we experimentally compress freshly laid egg samples, and (ii) in Method 2, we create finite element models of eggs using published egg images and data, and run numerical simulations, resembling the compression tests. Once K is determined, the C number is determined by definition.

Results
C number based on freshly laid eggs (Method 1). We experimentally compressed over 400 freshly laid intact egg samples from 40 bird species (11 orders, 16 families; Supplementary Dataset 1) using quasi-static compression tests (Fig. 1b, Supplementary Fig. S2), and obtained their stiffness K, defined as the initial slope of the load-displacement curve (Fig. 1d). We used this method for two reasons. First, this is a standard measurement used by the poultry industry so the measured stiffness is very repeatable, accurate, and with high resolution. There is no black box nor hidden parameters that might affect the results, once the apparatus is properly calibrated (Supplementary Information). Second, this measurement gives us the local stiffness of the shell near the contact area (approximately a few mm for a chicken egg) between the shell and plates. This stiffness indicates how much a shell may deform at the contacting regions due to an external force, be it the weight of incubating bird, the impact from other eggs in the nest, or the loading of a rigid plate in the experimental apparatus. The external force applied by the standard compression test mimics the selection pressures experienced by wild birds since their eggs constantly experience external forces.
To obtain the Young's modulus, E, for each eggshell, we performed FEM simulations (Fig. 1c) using the same loading and boundary conditions as those used in the compression tests. The Young's modulus at which the simulated load-displacement curve coincides with the experimental value is regarded as the Young's modulus of that particular eggshell (Fig. 1d). Note that the determination of C number does not require the Young's modulus, where K, W, A, and B are all measured directly. The Young's modulus is largely invariant, with an average value of E ≈ 32 ± 5 GPa, for the 40 species studied here ( Fig. 1h; Table 2; Supplementary Dataset 1). This is an important result. Since most studies in the literature focused on domestic fowls 13 , the Young's moduli of wild birds were previously unexplored. Figure 2a shows that the C number remains nearly constant C = 15,200 ± 4,400 and C ∝ M −0.062 across four orders of magnitude of body mass M from 12 to 100,000 g. This invariance is remarkable, considering that an ostrich's egg (~1500 g) is nearly 830-fold heavier than a Scaly-breasted munia's egg (~1.8 g). We map the experimental C numbers onto the phylogenetic tree, and observe that the C numbers are distributed between 10,000 and 20,000 ( Supplementary Fig. S3). Substitution of basic scalings of K, W, A, B into C number yields We further quantify the safety margin of the eggs, subject to the force from the incubating bird, by defining a factor of safety F.S. (see Table 1 and more discussion below). The safety margin for contact incubation is categorized into three regions: safe (F.S. > 2), marginal (1 < F.S. < 2), and unsafe (F.S. < 1). All species in Fig. 2b have a F.S. greater than 2, except the artificially-bred turkey, which is included to serve as an interesting contrast to the wild turkey (see more discussion below).

C number based on published egg images and data (Method 2).
We develop a new computational method (Methods and Supplementary Information) that enables us to study any egg provided that the egg profile, length, mass, shell thickness, and E are available. This approach greatly expands the number of species that one can study without the need to acquire freshly laid eggs, and proves useful to study the eggs of inaccessible and even extinct species. Here we extend our study to include 430 species (36 orders, 104 families; Supplementary Dataset 1) using published egg images 11 and data 16 . We assumed a constant E = 30 GPa for all simulations for simplicity. The stiffness K was obtained by performing compression simulations, resembling the experimental setup. The simulated C = 15,200 ± 5,300 and its scaling relation C ∝ M −0.075 (Fig. 3a) are consistent with those obtained experimentally. Several species are observed to have very small C numbers, which will be discussed as special cases below, but their F.S. are all in the safe region (Fig. 3b).

Evolution of body mass and C number.
Body mass, or equivalently body size, is one of the most fundamental attributes of organisms, and influences many aspects of life 17 . As such, body mass is key to our understanding of egg evolution. Having quantified the egg stiffness using C number, we investigated the evolution of avian body mass and C number by assembling a recent comprehensive tree 18 onto which we mapped body mass and C number of 416 species (35 orders) in our sample for which DNA sequencing data exist (Fig. 4). An investigation of the tree highlights three key observations. First, body mass varies significantly across the tree, with small and large body masses occurring in parallel across different lineages. For example, extremely small body mass (<20 g) evolved independently in hummingbirds and in perching birds, whereas large body mass evolved in ratites and in some raptors. Second, the trend of body mass evolution is not singular. The body mass may monotonically increase such as ostriches, or decrease such as hummingbirds. It may also decrease and then increase such as cinereous vultures (Aegypius monachus), or increase and then decrease such as kiwis. Third, the body mass of the common ancestor of living birds, emerged around 113.2 million years ago 18 , is ~300 g (Fig. 4). Although fossil record of modern birds (Neornithes) from the Cretaceous Period is limited, the recently discovered fossils of Mesozoic birds, such as Archaeornithura meemannae 19 (Ornithuromorpha; ~77 g) and Chongmingia zhengi 20 (Ornithothoraces; ~290 g), corroborate our analysis and the view that small body size is common among birds throughout the Cretaceous Period 21 .
Despite the complicated evolutionary pattern of body mass, the C number, by contrast, remains largely invariant, suggesting that the "mechanical design" of avian eggs was well preserved throughout the process. We note that some clades, such as shorebirds (Charadriiformes), possess relatively small C numbers, which is due to their large eggs relative to body mass (≈15%). However, their F.S. are in the safe region (Fig. 3b). and shell thickness (80−1,730 μm). The moduli of some species, however, are considerably larger or smaller than the average, e.g., the ostriches (48 GPa, n = 7) and African penguins (23 GPa, n = 4), which may be related to differences in the ultrastructure and composition of the shells (Supplementary Fig. S9; Supplementary Information). Error bars are the intraspecific maximum and minimum values. Bird images and silhouettes not to scale. See Supplementary Dataset 2 for details on the source of bird images. The egg images were taken by the authors.
Our analyses may shed light on the interesting evolution of kiwis and elephant birds. The elephant bird and kiwi are sister taxa 22 ; they diverged ~50 million years ago and shared a common ancestor, which was probably flighted and capable of over-water dispersal. The body size of their ancestor was likely between those of the elephant bird and kiwi. We suggest that the gigantism of the elephant bird 22 was accompanied with the gigantism of its egg to maintain a typical W/M ≈ 2%, whereas the kiwi egg did not shrink as much as the body mass did over evolutionary time, but instead the shell thickness was reduced to maintain a proper F.S.

Mechanics of elastic shells.
Why is C number nearly constant across a wide range of body mass? Consider an ellipsoidal shell loaded by the force W at its poles (local curvature κ = 2B/A 2 ). The shell is locally spherical and the indentation response is locally identical to the indentation of a spherical shell of radius κ = r 1/ (ref. 14 ). It follows that  9 , thickness includes dried membranes. e Ar et al. 9 , based on the data in Handbuch der Oologie 16 ; thickness was estimated from the eggshell mass and surface area.  indicates that δ subjected to W falls within a certain range (~65 × 10 −6 ). We substitute the vertical stiffness of an ellipsoidal elastic shell 14 into C and obtain ≡ = , where ν is the Poisson ratio. The factor κ in C eliminates the GIR, and allows for decoupling of contributions of shape/size and thickness/materials to the overall shell stiffness. Substituting for the measured allometric relations of t, W, E, and M from Fig. 1h and Supplementary Fig. S1, we arrive at The invariance of C is largely achieved by scaling t and W with respect to M, since E is found to be generally invariant.
Fracture, buckling, and factor of safety. Similar to engineering structures, the safety margin of the shell should be sufficiently high to account for environmental perturbations and incidental contact between eggs other than the static loading of incubating birds, but not too high to allow successful hatching and save of materials. Is there a lower limit for the thickness, and accordingly stiffness and C number, for a given body mass? We consider two failure modes: (i) fracture, and (ii) buckling. Figure 5a shows the experimental fracture force scales as F f ∝ M 0.591 and simulated buckling force scales as F b ∝ M 0.679 . The buckling force is generally larger than the fracture force, indicating that the egg generally fractures before buckling, and buckling force may be used as an upper bound for the load an egg can withstand. Hence we use buckling as a criterion to estimate a lower limit for the shell thickness (critical thickness t cr ), and in turn the critical stiffness K cr and critical dimensionless number for a given body mass ( Supplementary Fig. S4).
We define a factor of safety F.S. ≡ C/C cr to quantify the load-bearing capacity of an egg beyond the expected load from the incubating bird. Factor of safety is often used in engineering with the intent to provide a safeguard to failure. The term usually refers to the ratio of the load that would cause failure of a structure to the load that is imposed upon it in service. The term may also be used to represent the ratio of the failure to service value of speed, deflection, temperature variation, or other stress-producing quantities.
The results shown in Figs 2b and 3b indicate that the F.S. decreases slightly with increasing body mass. This trend may be explained by the fact that smaller eggs (e.g. Scaly-breasted munia, F f = 1.6 N) are more likely to be damaged by environmental disturbances or predator attacks than the larger eggs (e.g. ostrich, F f = 383 N), and hence require extra safety margin.
Several other mechanical quantities are also found to exhibit allometric invariances irrespective of size and origin, and are summarized in Table 2.
Case Studies. Several cases of special biological implications are elaborated in the following and are summarized in Table 3.
Artificial selection. Artificial selection can produce breeds, all developed from the same wild species, that differ widely in appearance in just a few generations 23 . For example, the body mass of an artificially-bred turkey (M ≈ 20,000) is much larger than that of a wild turkey (M ≈ 4,000 g), which provides a unique contrast to test C number and F.S. We experimentally tested freshly laid eggs of both wild turkeys and artificially-bred turkeys, and found that, despite the great difference in M, their eggs are very similar in all aspects, including C ≈ 13,000 and E ≈ 33 GPa (Table 3), indicating that the shell mechanical characteristics remain unchanged through the artificial selection. The eggshell can no longer withstand the much larger body mass of the artificially-bred turkeys. The F.S. < 1 suggests that such a breed cannot survive without human assistance as the eggs must be artificially incubated. Brood parasitism. The common cuckoo Cuculus canorus is a generalist obligate parasite, laying their eggs in nests of several passerine species, and has evolved numerous host-specific races, each having eggs with color and pattern mimicking the eggs of their respective host species 25,26 . Our results show that despite the fact that the cuckoo is four-fold heavier than one of its hosts, the white wagtail (Motacilla alba), their eggs are almost identical in appearance with cuckoo's shell being slightly thicker. We obtain C = 19,400 and 17,353 for the cuckoo and wagtail, respectively. Interestingly, the F.S. of the cuckoo egg incubated by a wagtail (F.S. = 9.9) is significantly higher than that of the wagtail egg (F.S. = 3.7), supporting that parasitic cuckoo eggs are indeed stronger than the host' eggs to withstand puncture rejection 27 . Loss of safety margin due environmental toxins. The C number may be used to quantify the detrimental impact of environmental toxins on eggshell stiffness and safety margin due to shell thinning 28 . Reproductive failure of brown pelicans (Pelecanus occidentalis) occurred in 1969 and a 35% decrease in shell thickness was associated with 71 p.p.m. of DDE in the egg 29 . Our simulation shows that such shell-thinning results in 53% drop in C, and accordingly, a significant drop of F.S. from 3.8 (safe) to 1.8 (marginal). The eggs were no longer strong enough for contact incubation.
Exceptions. The C numbers of megapodes, kiwis, and wandering albatrosses are substantially smaller than the norm (Fig. 4a) mainly because their shells are thin (hence K is small) with respect to their egg weight W.
, a smaller K results in a smaller C number. These three species are exceptions of the constancy of C number. However, it is interesting to note that their F.S. are in the normal range (>2), strong enough for contact incubation, because their eggs are large relative to body mass in comparison to the ratios for other birds (Supplementary Dataset 1). One question is: The megapode does not practice contact incubation; why is its F.S. still in the normal range (F.S. = 3.2) and not smaller to save more materials? The answer remains unknown, but we hypothesize that this may be related to megapodes' large clutch size (8−12 eggs). The large clutch size increases the possibility of incidental contact between eggs, so that the shell must be sufficiently thick to avoid damage. We note that all three species have the longest incubation period among birds (77 days, 74-84 days, and 60 days, for wondering albatrosses, the kiwis, and megapodes, respectively). The precise causality of the ecological adaptation of long incubation and small C number is unknown, and requires further studies. Nevertheless, the C number offers a single quantitative measure to identify species with special reproductive strategies.

Discussion
The avian egg provides an excellent model system in which to explore the structural evolution of load-bearing biological structures. We define the C number, a measure of how stiff an egg is with respect to its weight after removing the shape effect, and find that it is generally invariant, suggesting that bird eggs are effectively working under a particular "design" irrespective of size or origin. This invariance is the result of two largely invariant mechanical quantities: E and t 2 /W. Our conclusion is supported by the large dataset complied by Ar et al. 9 -they analyzed 3,434 species (over one-third of all living bird species) and showed that t 2 /W is an allometric invariant ( Table 2). From a biological perspective, as the body mass of a bird evolves to fit into a particular ecological niche, its egg weight and shell thickness evolve accordingly to maintain a constant t 2 /W and a proper W/M ratio, which ensures safe contact incubation and saves precious shell materials. Similar allometric invariances are ubiquitous in nature. Examples include the urination time of large animals 30 , the Young's modulus to density ratio in green woods 31 , and the height of a jump 32 . The existence of such invariances is mainly driven by the predominate force on earth -the gravity.
Considering the many requirements that the egg must fulfill, it is not surprising that the diversity of eggs is based on some elaboration of an "optimal design"-an optimal compromise between different selection pressures. Given the need to redesign the entire system whenever body mass changes, either through ontogeny or speciation, certain deviations from the general trend, and even exceptions, will occur. However, as body mass varies across many orders of magnitude, these design guidelines are followed with remarkable precision. Our findings presented here can be tested and provide a rich arena for future study on other amniote eggs 33 , structure-function relationship of eggshells 34 , avian reproductive strategies, and the evolution of amniote eggs and contact incubation 35 .  After the compression test, three small fragments of each eggshell (one near the pointed pole, one near the blunt pole, the other near the equator) were prepared, and the shell thickness was measured per fragment, with and without shell membrane, using a digital tube micrometer (Mitutoyo 395-271, 0−25 mm range, 1 μm resolution with spherical/spherical anvils). We assume that the membrane has negligible contribution to the stiffness of whole egg, and does not include it when using the FEM simulations to obtain the Young's modulus E of the shell.
All methods were carried out in accordance with relevant guidelines and regulations. All experimental protocols were approved by the Institutional Animal Care and Use Committee (IACUC) at National Taiwan University.

Quasi-Static Compression Tests (Method 1).
We vertically compressed the freshly laid egg along its long axis, with its pointed end facing up, using an electromechanical universal test system (MTS Criterion Model 42) at room temperature. The egg was placed between two smooth steel plates. The bottom plate was fixed; the top plate was connected to a 250-N load cell. The loading was conducted at a constant compression speed of 1 μm s −1 until the egg fractured ( Fig. 1 and Supplementary Fig. S2). The compressive load F, resulted from the compression by a prescribed displacement δ, was recorded using the load cell. Representative load-displacement curves are shown in Supplementary Fig. S2. The load-displacement curve was, in general, very linear until the egg fractured, at which a sudden drop in the load was observed and was often accompanied by a cracking sound. Examination of the tested sample shows that the fracture almost always occurred at the pointed pole, where concentric or radial cracks were formed (Fig. 1f) due to the high tensile stress generated on the inner surface of the shell during the test (Supplementary Fig. S7). The load at the first fracture was denoted as fracture force F f . We define the experimental stiffness K as the initial slope of the load-displacement curve. Although we compressed the egg sample until it fractured to measure the fracture force, the shell stiffness and C number can be measured using a small force without breaking the shell for nondestructive diagnosis.

Numerical compression simulations (Method 2).
In Method 2, we analyzed egg stiffness of 430 species, covering 36 orders and 104 families, by numerical FEM simulations using published egg images 11 and data 16 . To construct the model, we need the two-dimensional (2D) egg profile, egg length and mass, shell thickness, and Young's modulus of the shell. The Young's modulus of most avian species is absent in the literature; only a few common species have been studied 13 . In the present study, we found that the Young's modulus is largely invariant, with an average value of E ≈ 32 ± 5 GPa, for the 40 species that we have tested (Table 2). For all simulations in Method 2, we assume a constant E = 30 GPa for simplicity. This approach greatly expands the number of species that one can study without the need to acquire freshly laid eggs, and proves useful to study the eggs of inaccessible and even extinct species. The stiffness K was obtained by compression simulations of the egg model with loading and boundary conditions resembling the experimental setup (Fig. 1c). To validate this approach, we randomly selected seven species and compared the egg characteristics with experimental results (Supplementary Table S1). Most parameters are, in general, in good agreement for the two methods. In particular, the deviation of the C number for 6 of the 7 species is within 25%. The only species whose C shows marked deviation is the ostrich, which is due to the difference in the Young's modulus (30 and 48 GPa for Method 2 and Method 1, respectively). If E = 48 GPa is used, Method 2 predicts C = 19,700, which is very close to the average experimental value of 18,800 (<4.8%). This agreement is remarkable, considering the dramatically different nature of these two methods and substantial variation between species. This confirms that Method 2 is adequate for studying the mechanical design of the eggs of inaccessible and extinct species.
Finite Element Method (FEM) Simulation. We used FEM for four purposes: (i) estimating the Young's modulus, E, of a particular eggshell by fitting its experimental load-displacement curve (Fig. 1d); (ii) performing numerical compression simulations (Method 2); (iii) predicting the buckling force, F b , and factor of safety, F.S. (Fig. 5a,b, and Supplementary Fig. S8); and (iv) calculating the stress distribution and fracture strength, σ f (Fig. 5c, and Supplementary Fig. S7). The FEM simulations were cross-validated by experiments and shell theory (Tables S1 and S2, Fig. S6).
Failure due to Buckling. Although fracture force is a more direct measure of the eggshell strength and can be obtained from compression tests, it is more difficult to predict (micro-crack initiation, crack propagation, macroscopic catastrophic rupture), whereas existing shell theory 36 and FEM simulations allow for more accurate predictions of the onset of buckling and buckling force, F b . The prediction of minimum stiffness, defined as the critical stiffness K cr , is based on the following scenario: When an eggshell is subjected to a force equal to the body weight, the maximum thickness that induces the eggshell to buckle is the critical thickness t cr ( Supplementary  Fig. S8b) that determines K cr (Supplementary Fig. S4a). In this sense, we applied a point force M at the pole and determined t cr by adjusting the shell thickness ( Supplementary Fig. S8). We then created a new eggshell model with t = t cr , and conducted the compression simulations, resembling the experimental condition, to obtain K cr . The critical dimensionless number C cr and factor of safety F.S. are readily obtained for this eggshell.
ScienTific REPORTS | 7: 14205 | DOI:10.1038/s41598-017-14552-0 Data Availability. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Ethics.
All methods were carried out in accordance with relevant guidelines and regulations. All experimental protocols were approved by the Institutional Animal Care and Use Committee (IACUC) at National Taiwan University.