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Scientific Reports volume 14, Article number: 31594 (2024 ) Cite this article flat bottom flask
Worldwide museums hold collections of eggshells representing material for descriptive studies. However, an obstacle to this is the lack of information about the original contents and weight of the entire egg (W). This study aimed to fill this gap though development of a methodological mechanism for calculating the volume of the egg interior (Vi), its density (Di) and W. To determine Vi, it is sufficient to measure four geometric dimensions of the egg and shell thickness. The Di value depends on the surface area-to-volume ratio (S/V) and can be calculated from an empirical relationship. For its derivation, data on 454 eggs from 447 avian species, 95 families and 13 orders were used. Imputing data on the contents and shell weight (Ws), we proposed a theoretical relationship for calculating W. We found a negative correlation between Di and S/V (which reflects the egg metabolism level) and suggest that a female in most species maintains the duration of egg incubation at a constant level that has practically an unchanged value for the respective species. A mathematical algorithm for calculating the Di value depending on the S/V ratio provides the missing link in calculating W of a whole egg from archived collection material.
The enduring popularity of investigations into the density of bird eggs’ interior (Di)1,2,3 is mostly due to the fact that Di can be an indirect indicator of quality, particularly important for poultry species3. The value of Di may even be used to estimate the sex of the embryo, which would be useful for poultry management technologies, if applied widely4,5. The use of destructive methods for examining eggs of wild bird species is, of course, unacceptable in view of current environmental protection agenda, ethical issues and conservational concerns. The possibility, therefore, of non-invasive assessment of Di for non-domesticated species could allow for a new wave of ornithological applications to study its relationship with offspring survival, sex ratio, ecological impact, evolutionary principles and many other factors.
The earlier works of authors who explored the structure, physical and biological interdependencies of egg contents are especially relevant, particularly a large number of allometric studies describing the relationship between various parameters of bird eggs and their weight (W). Most were derived by a group of scientists led or coordinated by an American Professor Hermann Rahn (1912–1990; see the respective biographical memoir by Pappenheimer6). In spite of those, Di remained without an appropriate description in terms of mathematical dependencies. Rahn and Paganelli2, examining the variability of this parameter, suggested using a constant value of 1.031 g/cm3 for all eggs by demonstrating the adequacy of this value for eggs of both domestic and wild species.
In our previous work3, we demonstrated a theoretical relationship between Di and other egg parameters as follows:
where Di is density of the egg contents (in g/cm3), W is egg weight (in g), Ws is shell weight (in g), V is egg volume (in cm3), and Vs is shell volume (in cm3).
For further calculations, we used the Paganelli et al.1 allometric relationships regarding Ws and Vs, as well as V, with the latter being obtained from the formula for egg density (D) presented by those authors:
where Vs and V are measured in cm3, and Ws and W are measured in g.
Substituting Eq. 2 to 4 into (1), we can derive the following formula:
Substituting the W values from 0.5 to 1000 g (which approximately corresponds to the entire series of eggs existing in nature) into Eq. 5, the resultant calculation of Di allowed us to obtain the graphical dependence presented in Fig. 1.
Graphical visualization of Eq. 5 for density of egg contents (Di) depending on egg weight (W).
The nature of the resultant dependence in Fig. 1 does not fit into the framework of the usual allometric relationship, i.e., power function, which invoked the Rahn and Paganelli2 decision to recommend taking the value Di in the form of a constant in calculations.
In our preliminary work3, we dwelled in some detail on methods for non-destructive assessment of the Di value. The results obtained turned out to be quite promising, but were purely empirical in nature. Experimental studies were carried out on chicken eggs using destructive assessment methods, suggesting a set of parameters that most accurately predict Di. The resultant computation formulae, however, cannot automatically be transferred to eggs of other species. That is, interspecific trends in the variability of this parameter need to be taken into account. This is because they can, firstly, be pivotal for understanding evolutionary processes. Secondly, they can serve as an impetus for other oological studies limited by a lack of data and. Most importantly, however, we cannot derive real world data for Di of wild bird eggs due to the need to use destructive measurement methods to establish them.
In order to develop a universal methodology for calculating the Di value among multiple avian species, museum collections of eggs (see, for an example, Fig. 2), of which millions of pieces have been accumulated in various storage facilities7,8, provide a most useful resource. Most often, a researcher examining such collections can only deal with the shell remaining from a once whole egg, and reliable information about its W is a thing of the distant past. Knowledge of the Di value of a particular egg that can, nonetheless, contribute to the calculation of the W value of these eggs. From a mathematical point of view, such restoration of a disappeared parameter is relatively straightforward.
Example of the shell of a Kākāpō (Strigops habroptilus) egg from the collection of the Auckland Museum, New Zealand (https://commons.wikimedia.org/wiki/File:Strigops_habroptilus_(AM_LB14427-1).jpg, by Fæ; CC-BY-4.0).
Transforming Eq. 1 into the following formula:
The V value of an egg can be quite easy to determine. Our oological group alone has proposed about seven different approaches for calculating this indicator based on simplified, more complex and most accurate dependencies9,10,11,12,13,14,15,16. Ws can be easily obtained by weighing museum exhibits. With respect to Vs, the question is somewhat more complicated since museum rules prohibit immersing exhibits in water to prevent possible damage. Previously, we laid down theoretical approaches to calculating this parameter non-invasively17,18. However, given the variety of bird egg shapes that exist in nature, the calculation of the Vs value requires some improvement (see below).
Using the theoretically derived formula for a mathematical description of the shape of any bird egg19, we defined the following calculated relationship for determining V of such an ovoid16:
where L is egg length, B is its maximum breadth, w is the parameter that shows the distance between two vertical lines conforming to B and the half length of the egg (L/2), and Dp is diameter measured at a point distant from the pointed end of the egg by L/4.
Transforming Eq. 7 into the following form:
Similar to the course of mathematical transformations presented by us in the study by Narushin et al.18, we find the volume of egg contents (Vi), correspondingly reducing the geometric dimensions in Eq. 8 by twice the shell thickness (T), except for the parameter w, the value of which remains unchanged, as was demonstrated in our previous investigation17.
The mathematical transformation of Eq. 9 resulted in the following equation:
Considering that V of a bird’s egg can be represented as the sum of Vi and Vs, analysis of Eqs. 8 and 10 suggests that:
Thus, to determine the Vs value, one needs to measure the geometric parameters of the egg (B, L, w and Dp) and T. Most often, T values in museum collections (e.g., Fig. 2) can be measured directly or using a special micrometer with an extended pin. For small museum eggs or for shells with very small holes, ultrasonic thickness gauges can be used that are common in many industries (e.g., Deis and Allen20), including commercial poultry production21,22,23. As a result, the Di value (Eq. 6) is the only parameter that impedes the recalculation of W and thereby accurate information about what museum ostraca, sherds and shells were like before they became exhibits.
A promising criterion that seems to be very informative in assessing the correlation relationships with the Di value may be the ratio of the egg surface area to its volume (S/V). Based on the results of studies that analyzed the influence of this index on the indicators of various biological mechanisms (e.g., Cohen et al.24, Cragg25, Harris and Theriot26, Lewis27), the S/V value may prove to be very relevant for predicting the parameters of the egg contents.
The goal of the current studies was to develop a method for non-destructive prediction of the density of the contents in various bird species eggs (i.e., Di) depending on their physical and geometric features.
As part of the research, on the one hand, we used the published results of other researchers who have performed such experiments previously (e.g., Rahn and Paganelli2). On the other, it was important for us to develop our own approach to calculating Di using data from oological measurements of other parameters of bird eggs. Although, among these parameters, there is no direct information on Di, the available other characteristics allow us to judge the Di value indirectly.
The most extensive source of oomorphological data is the reference book by Schönwetter28, used by Paganelli et al.1 when deriving their allometric dependencies. It was especially important for this study that Schönwetter28, in addition to oomorphological parameters, also placed many images of bird eggs, whose data was included here in the reference information. This enabled us to measure all the geometric egg parameters required not only to calculate Vs (Eq. 11), but also V (Eq. 7), as well as S according to the formula from Narushin et al.16:
We described the procedure for measuring images of bird eggs in detail in the results of our previous studies29,30. Briefly, the egg image was measured in pixels using an electronic ruler in Microsoft Office Picture Manager. The pixel measurements were then converted to centimeters according to the eggs’ metric data for L and B given in the tables of Schönwetter28. Schematically, the measurements of images of bird eggs are presented in Fig. 3.
Schematic representation of the measured geometric parameters using the example of the Common Sandpiper egg image (Actitis hypoleucos). (Image source: https://commons.wikimedia.org/wiki/File: Actitis_hypoleucos_MWNH_0255.JPG, by Klaus Rassinger and Gerhard Cammerer, 2012; CC-BY-SA-3.031).
Schönwetter28 in his oological reference book presented images of 434 eggs belonging to 433 avian species. Despite the large array of data obtained, the drawback of their analysis was the relatively low W range of eggs with available images, i.e., from 1 to 100 g. For a more complete analysis, we lacked data on birds laying eggs of greater W values. In this respect, we used the numerical values of such eggs from the reference book Schönwetter28, while relying on images of these eggs obtained from other sources, e.g., the digitized collection of images of bird eggs from the Natural History Collections of the Museum Wiesbaden31. As a result, a total of 454 eggs belonging to 447 avian species, 95 families and 13 orders were represented. The entire list of avian species whose eggs were used in the current study can be found as Supplementary Table S1 online.
The second stage of our experimental analysis of Di values was based on data presented in Rahn and Paganelli2. Their results were based on data from some earlier findings by Roca et al.32, in which the authors studied the density, weight and chemical structure of the main components the egg contents, i.e., albumen and yolk. Although these investigations were carried out on eggs of only 14 bird species representing six families and five orders, they provided invaluable material for a more in-depth analysis and derivation of relationships with those egg parameters, the measurement of which can be performed non-invasively even during field studies. Images for computing geometric characteristics in this group of eggs were obtained from the Natural History Collections of the Museum Wiesbaden31 and the Muséum de Toulouse33.
When performing the experimental procedure, we encountered one undermentioned constraint related to the definition of the concept embedded in the value of the parameter Di. On the one hand, the Di value can be determined by the density of the main components of the contents: albumen and yolk. This is the approach undertaken by Roca et al.32 Perhaps it is applicable when examining freshly laid eggs in which the air cell has not yet formed. However, in practice, this is extremely difficult to achieve, if at all possible.
In our opinion, the contents should include the values of W (and/or V) of the egg after subtracting Ws (and/or Vs) of the shell. That is, this calculation should take into account the fact that if the air cell weight is conditionally equal to 0, its volume can be very significant, especially during the process of hatching or even storing eggs (e.g., Narushin et al.34). In this context, when calculating Di, a certain process naturally occurs that depends on the time elapsed between laying and weighing an egg. Despite this error, a larger array of analyzed data will not greatly distort the eligibility and general trend of possible dependencies.
The next assumption in our calculations was the fact that the shell membrane was classified as a component of the shell. This assumption was also adhered to by Schönwetter28 who placed data on T and Ws in his reference materials. Moreover, the shell membrane is present in the composition of the shell in museum collections, which was the fundamental factor in combining these membranes into a single structural component of the egg used in further calculations here.
To process the results, statistical and mathematical algorithms were used that are available in the STATISTICA 5.5 program (StatSoft, Inc./TIBCO, Palo Alto, CA, USA), as well as applications to the Microsoft Excel program. Thereby, the validity of the obtained relationships was assessed by the value of the Pearson correlation coefficient (R) and regression models using the coefficient of determination (R2) with confirmation of their significance at the level of p < 0.05.
As a result of the analysis of experimental data, we found a significant, although relatively lower, correlation between the Di value and the S/V ratio (R = − 0.277, p < 0.05). A negative sign of the correlation coefficient was indicative of an inverse relationship between these values as also demonstrated by the graphical interpretation of this dependence (Fig. 4).
Graphical visualization of the relationship between Di and S/V.
The S/V value is an indirect criterion for reflecting the level of embryonic metabolism that was examined in detail by us in a number of our recent studies16,35,36. Thus, it can be assumed that eggs with denser contents have a lower metabolic rate. This assumption was accepted by us as a working hypothesis requiring at least a logical justification.
According to various authors32,37, the density of the yolk is slightly higher than that of the albumen. Thus, eggs with greater yolk content also appear to have higher Di values. Taking into account the fact that the yolk contains the main nutritional components necessary for the developing embryo37, it is quite logical to assume that embryo’s maturation occurs at a slightly slower pace, which is regulated by a decrease in the metabolic rate (S/V). This fact can be confirmed by the results of our previous studies30, in which we demonstrated an inverse relationship between incubation time and S/V value using a large interspecific sampling of bird eggs. In other words, in species whose eggs have a relatively lower level of metabolism, hatching periods will be expected to be longer.
We asked the question whether the existence of such a relationship between Di and S/V is possible due to the high level of differences in the weight fractions of oomorphological parameters included in the computation equation used to recalculate the Di value (Eq. 1). To answer this question, we decided to isolate from the general sampling, and analyze separately, eggs of the order Passeriformes characterized by relatively similar values of W and weights of egg’s structural components. For example, within the framework of our current research, the W value of Passeriformes eggs was in the range of 0.5 to 11 g. The results of this analysis are presented in Fig. 5.
Graphical visualization of the relationship between Di and S/V in eggs of the order Passeriformes.
Dependence analysis in Fig. 4 also suggests a similar negative relationship between these parameters, although with a slightly lower but significant correlation coefficient (R = − 0.101, p < 0.05). That is, the nature of this relationship remained unchanged in general.
An analysis of the relationship between the Di value and the S/V ratio was also carried out on the data presented by Roca et al.32 The correlation between these indicators was R = − 0.358, although it was statistically non-significant, probably due to a limited sampling of bird eggs (n = 14). A visualization of the resulting relationship is presented in Fig. 6 and is fully consistent with our results for a more representative sampling (Figs. 4 and 5).
Graphical visualization of the relationship between Di and S/V calculated from the data by Roca et al.26.
In order to completely remove questions about the relationship between Di and S/V, we decided to carry out such an analysis on the eggs of birds of the same species. This experiment could only be carried out on poultry eggs. Considering that we previously performed a study on calculating the morphological parameters of goose eggs38, we used the measurements taken as part of that work (Fig. 7).
Graphical visualization of the relationship between Di and S/V in goose eggs.
The resultant relationship was fully concordant with our hypothesis about a decrease in the level of embryonic metabolism in eggs with denser contents. The correlation between the parameters Di and S/V was at the level of − 0.122, but turned out to be insignificant.
Based on the free energy minimization principle (e.g., Karl39), including minimized costs of reproduction, the mother bird very clearly, even at the species level, invests into the interaction between the size of the yolk that she managed to produce and the parameters V and S of the entire egg. Thereby, she maintains the duration of incubation of her offspring at a constant level that has practically an unchanged value for the respective species.
Since one of the fundamental factors that served as an impetus for carrying out these studies was the possibility of restoring museum shell remains into a full-fledged, albeit virtual, analogue of a whole egg, we need to be able to recalculate the value of Di for these purposes. Based on our present research, the most convenient and logical parameter for carrying out such a calculation turned out to be the S/V ratio. Mathematical approximation of data in Fig. 4 allowed us to obtain the following dependence:
where Di is measured in g/cm3, V in cm3, and S in cm2.
Despite of a lower value of R2, its meaning appeared to be significant and thus can be used for the corresponding recalculations of Di.
Within the set of the studied eggs, the values of the S/V ratio varied from 0.4 to 6.4, due to which the resultant calculated Di (Eq. 13) had a range of approximately 5% (i.e., 0.973 to 1.022). Naturally, the final calculation of the initial W value (Eq. 6) would provide an even smaller value of accuracy variation. However, considering that any study assumes the provision of the most accurate result that can be accepted as a final solution, we suggest following the proposed calculation of Di (Eq. 13). In cases where preference is given to simplicity and/or speed of predicting, one can use the average Di value that, within the data we analyzed, was 1.00 g/cm3.
Thus, we suggest that the process of restoring the lost W value of a whole museum egg should, methodically, consist of the following stages:
A shell from a museum collection is weighed (Ws), its T and geometric dimensions (B, L, w and Dp) are measured.
The V value of the egg is calculated using Eq. 7, its S using Eq. 12 and Vs using Eq. 11.
The Di value is computed using Eq. 13.
The W value of the whole egg is calculated using Eq. 6.
As an outcome of the current study and its results obtained therein on non-destructive prediction of Di in various bird species eggs in relation to their physical/geometric features, two undermentioned postulates can be formulated. Firstly, an evolutionary feature of the bird egg is the inverse relationship between its Di and embryonic metabolism expressed by the S/V ratio. Secondly, a mathematical algorithm for calculating the Di value depending on the S/V ratio provides the missing link in calculating the initial W of a whole egg (Eq. 13), the value of which has hitherto not been available for the extensive collection material of shell membranes stored in the world’s museums.
Conceptualization, data curation, formal analysis, methodology, resources, investigation, software, visualization: VGN. Writing – original draft: VGN and MNR. Writing – review & editing: VGN, MNR, NAM, and DKG. Project administration: MNR. Supervision: DKG.
statement The authors have no relevant financial or non-financial interests to disclose.
Ethics statement This study involved only images of the eggs from the electronic sources. No live animals or natural eggs were used. No compliance with ARRIVE or other relevant guidelines were needed.
Supplementary Information The online version contains supplementary material available at https://doi.org/.
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Research Institute for Environment Treatment, Zaporizhya, 69035, Ukraine
School of Biosciences, University of Kent, Canterbury, Kent, CT2 7NZ, UK
Michael N. Romanov & Darren K. Griffin
Animal Genomics and Bioresource Research Unit (AGB Research Unit), Faculty of Science, Kasetsart University, Chatuchak, Bangkok, 10900, Thailand
Michael N. Romanov & Darren K. Griffin
L. K. Ernst Federal Research Center for Animal Husbandry, Dubrovitsy, Podolsk Municipal District, Moscow Oblast, 142132, Russia
Tisch Family Zoological Gardens in Jerusalem, PO Box 9505, 9109401, Jerusalem, Israel
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Conceptualization, data curation, formal analysis, methodology, resources, investigation, software, visualization: VGN. Writing – original draft: VGN and MNR. Writing – review & editing: VGN, MNR, NAM, and DKG. Project administration: MNR. Supervision: DKG.
Correspondence to Michael N. Romanov.
The authors declare no competing interests.
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Narushin, V.G., Romanov, M.N., Avni-Magen, N. et al. Accurate calculation of the content volume, density and original weight of museum curated eggs. Sci Rep 14, 31594 (2024). https://doi.org/10.1038/s41598-024-75397-y
DOI: https://doi.org/10.1038/s41598-024-75397-y
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